Springer Handbook of Atomic, Molecular, and Optical Physics

Springer Handbooks of Atomic, Molecular, and Optical Physics

Springer Handbooks provide a concise compilation of approved key information on methods of research, general principles, and functional relationships in physics and engineering. The world’s leading experts in the fields of physics and engineering will be assigned by one or several renowned editors to write the chapters comprising each volume. The content is selected by these experts from Springer sources (books, journals, online content) and other systematic and approved recent publications of physical and technical information. The volumes will be designed to be useful as readable desk reference books to give a fast and comprehensive overview and easy retrieval of essential reliable key information, including tables, graphs, and bibliographies. References to extensive sources are provided.

Springer

Handbook of Atomic, Molecular, and Optical Physics Gordon W. F. Drake (Ed.) With CD-ROM, 288 Figures and 111 Tables

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Editor: Dr. Gordon W. F. Drake Department of Physics University of Windsor Windsor, Ontario N9B 3P4 Canada Assistant Editor: Dr. Mark M. Cassar Department of Physics University of Windsor Windsor, Ontario N9B 3P4 Canada

Library of Congress Control Number:

ISBN-10: 0-387-20802-X ISBN-13: 978-0-387-20802-2

2005931256

e-ISBN: 0-387-26308-X Printed on acid free paper

c 2006, Springer Science+Business Media, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+ Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in Germany. The use of designations, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Product liability: The publisher cannot guarantee the accuracy of any information about dosage and application contained in this book. In every individual case the user must check such information by consulting the relevant literature. Production and typesetting: LE-TeX GbR, Leipzig Handbook coordinator: Dr. W. Skolaut, Heidelberg Typography, layout and illustrations: schreiberVIS, Seeheim Cover design: eStudio Calamar Steinen, Barcelona Cover production: design&production GmbH, Heidelberg Printing and binding: Stürtz GmbH, Würzburg SPIN 10948934 100/3141/YL 5 4 3 2 1 0

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Handbook of Atomic, Molecular, and Optical Physics Editor Gordon W. F. Drake Department of Physics, University of Windsor, Windsor, Ontario, Canada [email protected] Assistant Editor Mark M. Cassar Department of Physics, University of Windsor, Windsor, Ontario, Canada [email protected]

Advisory Board William E. Baylis – Atoms Department of Physics, University of Windsor, Windsor, Ontario, Canada [email protected] Robert N. Compton – Scattering, Experiment Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA [email protected] M. Raymond Flannery – Scattering, Theory School of Physics, Georgia Institute of Technology, Atlanta, Georgia, USA [email protected] Brian R. Judd – Mathematical Methods Department of Physics, The Johns Hopkins University, Baltimore, Maryland, USA [email protected] Kate P. Kirby – Molecules, Theory Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts, USA [email protected] Pierre Meystre – Optical Physics Optical Sciences Center, The University of Arizona, Tucson, Arizona, USA [email protected]

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Foreword by Herbert Walther

The Handbook of Atomic, Molecular and Optical (AMO) Physics gives an in-depth survey of the present status of this field of physics. It is an extended version of the first issue to which new and emerging fields have been added. The selection of topics thus traces the recent historic development of AMO physics. The book gives students, scientists, engineers, and other interested people a comprehensive introduction and overview. It combines introductory explanations with descriptions of phenomena, discussions of results achieved, and gives a useful selection of references to allow more detailed studies, making the handbook very suitable as a desktop reference. AMO physics is an important and basic field of physics. It provided the essential impulse leading to the development of modern physics at the beginning of the last century. We have to remember that at that time not every physicist believed in the existence of atoms and molecules. It was due to Albert Einstein, whose work we commemorate this year with the world year of physics, that this view changed. It was Einstein’s microscopic view of molecular motion that led to a way of calculating Avogadro’s number and the size of molecules by studying their motion. This work was the basis of his PhD thesis submitted to the University of Zurich in July 1905 and after publication became Einstein’s most quoted paper. Furthermore, combining kinetic theory and classical thermodynamics led him to the conclusion that the displacement of a microparticle in Brownian motion varies as the square root of time. The experimental demonstration of this law by Jean Perrin three years later finally afforded striking proof that atoms and molecules are a reality. The energy quantum postulated by Einstein in order to explain the photoelectric effect was the basis for the subsequently initiated development of quantum physics, leading to a revolution in physics and many new applications in science and technology. The results of AMO physics initiated the development of quantum mechanics and quantum electro-

dynamics and as a consequence led to a better understanding of the structure of atoms and molecules and their respective interaction with radiation and to the attainment of unprecedented accuracy. AMO physics also influenced the development in other fields of physics, chemistry, astronomy, and biology. It is an astonishing Prof. Dr. Herbert Walther fact that AMO physics constantly went through periods where new phenomena were found, giving rise to an enormous revival of this area. Examples are the maser and laser and their many applications, leading to a better understanding of the basics and the detection of new phenomena, and new possibilities such as laser cooling of atoms, squeezing, and other nonlinear behaviour. Recently, coherent interference effects allowed slow or fast light to be produced. Finally, the achievement of Bose–Einstein condensation in dilute media has opened up a wide range of new phenomena for study. Special quantum phenomena are leading to new applications for transmission of information and for computing. Control of photon emission through specially designed cavities allows controlled and deterministic generation of photons opening the way for a secure information transfer. Further new possibilities are emerging, such as the techniques for producing attosecond laser pulses and laser pulses with known and controlled phase relation between the envelope and carrier wave, allowing synthesis of even shorter pulses in a controlled manner. Furthermore, laser pulses may soon be available that are sufficiently intense to allow polarization of the vacuum field. Another interesting development is the generation of artificial atoms, e.g., quantum dots, opening a field where nanotechnology meets atomic physics. It is thus evident that AMO physics is still going strong and will also provide new and interesting opportunities and results in the future.

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IX

Preface

The year 2005 has been officially declared by the United Nations to be the International Year of Physics to commemorate the three famous papers of Einstein published in 1905. It is a fitting tribute to the impact of his work that the Springer Handbook of Atomic, Molecular, and Optical Physics should be published in coincidence with this event. Virtually all of AMO Physics rests on the foundations established by Einstein in 1905 (including a fourth paper on relativity and his thesis) and his subsequent work. In addition to the theory of relativity, for which he is best known, Einstein ushered in the era of quantum mechanics with his explanation of the photoelectric effect, and he demonstrated the influence of molecular collisions with his explanation of Brownian motion. He also laid the theoretical foundations for all of laser physics with his discovery (in 1917) of the necessity of the process of stimulated emission, and his discussions of the Einstein–Podolsky–Rosen Gedanken experiment (in 1935) led, through Bell’s inequalities, to current work on entangled states and quantum information. The past century has been a Golden Age for physics in every sense of the term. Despite this history of unparalleled progress, the field of AMO Physics continues to advance more rapidly than ever. At the time of publication of an earlier Handbook published by AIP Press in 1996 I wrote “The ever increasing power and versatility of lasers continues to open up new areas for study.” Since then, two Nobel Prizes have been awarded for the cooling and trapping of atoms with lasers (Steven Chu, Claude Cohen-Tannoudji, William D. Phillips in 1997), and for the subsequent achievement of Bose–Einstein condensation in a dilute gas of trapped atoms (Eric A. Cornell, Wolfgang Ketterle, Carl E. Wieman in 2001). Although the topic of cooling and trapping was covered in the AIP Handbook, Bose–Einstein condensation was barely mentioned. Since then, the literature has exploded to nearly 2500 papers on Bose–Einstein condensation alone. Similarly, the topics of quantum information and quantum computing barely existed in 1995, and have since become rapidly growing segments of the physics literature. Entirely new topics such as “fast light” and “slow light” have emerged. Techniques for both

high precision theory and measurement are opening the possibility to detect a cosmological variation of the fundamental constants with time. All of these topics hold the promise of important engineering and technological applications that come with advances in fundamental science. The more established areas of AMO Prof. Gordon W. F. Drake Physics continue to provide the basic data and broad understanding of a great wealth of underlying processes needed for studies of the environment, and for astrophysics and plasma physics. These changes and advances provide more than sufficient justification to prepare a thoroughly revised and updated Atomic, Molecular and Optical Physics Handbook for the Springer Handbook Program. The aim is to present the basic ideas, methods, techniques and results of the field at a level that is accessible to graduate students and other researchers new to the field. References are meant to be a guide to the literature, rather than a comprehensive bibliography. Entirely new chapters have been added on Bose–Einstein condensation, quantum information, variations of the fundamental constants, and cavity ring-down spectroscopy. Other chapters have been substantially expanded to include new topics such as fast light and slow light. The intent is to provide a book that will continue to be a valuable resource and source of inspiration for both students and established researchers. I would like to acknowledge the important role played by the members of the Advisory Board in their continuing support of this project, and I would especially like to acknowledge the talents of Mark Cassar as Assistant Editor. In addition to keeping track of the submissions and corresponding with authors, he read and edited the new material for every chapter to ensure uniformity in style and scientific content, and he composed new material to be added to some of the chapters, as noted in the text.

February 2005

Gordon W. F. Drake

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List of Authors

Nigel G. Adams University of Georgia Department of Chemistry Athens, GA 30602-2556, USA e-mail: [email protected] Miron Ya. Amusia The Hebrew University Racah Institute of Physics Jerusalem, 91904, Israel e-mail: [email protected] Nils Andersen University of Copenhagen Niels Bohr Institute Universitetsparken 5 Copenhagen, DK-2100, Denmark e-mail: [email protected] Nigel R. Badnell University of Strathclyde Department of Physics Glasgow, G40NG, United Kingdom e-mail: [email protected] Thomas Bartsch Georgia Institute of Technology School of Physics 837 State Street Atlanta, GA 30332-0430, USA e-mail: [email protected] Klaus Bartschat Drake University Department of Physics and Astronomy Des Moines, IA 50311, USA e-mail: [email protected] William E. Baylis University of Windsor Department of Physics Windsor, ON N9B 3P4, Canada e-mail: [email protected]

Anand K. Bhatia NASA Goddard Space Flight Center Laboratory for Astronomy & Solar Physics Code 681, UV/Optical Astronomy Branch Greenbelt, MD 20771, USA e-mail: [email protected] Hans Bichsel University of Washington Center for Experimental Nuclear Physics and Astrophysics (CENPA) 1211 22nd Avenue East Seattle, WA 98112-3534, USA e-mail: [email protected] Robert W. Boyd University of Rochester Department of Physics and Astronomy Rochester, NY 14627, USA e-mail: [email protected] John M. Brown University of Oxford Physical and Theoretical Chemistry Laboratory South Parks Road Oxford, OX1 3QZ, England e-mail: [email protected] Henry Buijs ABB Bomem Inc. 585, Charest Boulevard East Suite 300 Québec, PQ G1K 9H4, Canada e-mail: [email protected] Philip Burke The Queen’s University of Belfast Department of Applied Mathematics and Theoretical Physics Belfast, Northern Ireland BT7 1NN, UK e-mail: [email protected]

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List of Authors

Denise Caldwell National Science Foundation Physics Division 4201 Wilson Boulevard Arlington, VA 22230, USA e-mail: [email protected] Mark M. Cassar University of Windsor Department of Physics Windsor, ON N9B 3P4, Canada e-mail: [email protected] Kelly Chance Harvard-Smithsonian Center for Astrophysics 60 Garden Street Cambridge, MA 02138-1516, USA e-mail: [email protected] Raymond Y. Chiao 366 Leconte Hall U.C. Berkeley Berkeley, CA 94720-7300, USA e-mail: [email protected] Lew Cocke Kansas State University Department of Physics Manhattan, KS 66506, USA e-mail: [email protected] James S. Cohen Los Alamos National Laboratory Atomic and Optical Theory Los Alamos, NM 87545, USA e-mail: [email protected] Bernd Crasemann University of Oregon Department of Physics Eugene, OR 97403-1274, USA e-mail: [email protected] David R. Crosley SRI International Molecular Physics Laboratory 333 Ravenswood Ave., PS085 Menlo Park, CA 94025-3493, USA e-mail: [email protected]

Derrick Crothers Queen’s University Belfast Department of Applied Mathematics and Theoretical Physics University Road Belfast, Northern Ireland BT7 1NN, UK e-mail: [email protected] Lorenzo J. Curtis University of Toledo Department of Physics and Astronomy 2801 West Bancroft Street Toledo, OH 43606-3390, USA e-mail: [email protected] Alexander Dalgarno Harvard-Smithsonian Center for Astrophysics 60 Garden Street Cambridge, MA 02138, USA e-mail: [email protected] Abigail J. Dobbyn Max-Planck-Institut für Strömungsforschung Göttingen, 37073, Germany Gordon W. F. Drake University of Windsor Department of Physics 401 Sunset St. Windsor, ON N9B 3P4, Canada e-mail: [email protected] Joseph H. Eberly University of Rochester Department of Physics and Astronomy and Institute of Optics Rochester, NY 14627-0171, USA e-mail: [email protected] Guy T. Emery Bowdoin College Department of Physics 15 Chestnut Rd. Brunswick, ME 04011, USA e-mail: [email protected]

List of Authors

Volker Engel Universität Würzburg Institut für Physikalische Chemie Am Hubland Würzburg, 97074, Germany e-mail: [email protected] Paul Engelking University of Oregon Department of Chemistry and Chemical Physics Institute Eugene, OR 97403-1253, USA e-mail: [email protected] Kenneth M.

Evenson†

James M. Farrar University of Rochester Department of Chemistry 120 Trustee Road Rochester, NY 14627-0216, USA e-mail: [email protected] Gordon Feldman The Johns Hopkins University Department of Physics and Astronomy Baltimore, MD 21218-2686, USA e-mail: [email protected] Paul D. Feldman The Johns Hopkins University Department of Physics and Astronomy 3400 N. Charles Street Baltimore, MD 21218-2686, USA e-mail: [email protected] Charlotte F. Fischer Vanderbilt University Department of Electrical Engineering Computer Science PO BOX 1679, Station B Nashville, TN 37235, USA e-mail: [email protected] Victor Flambaum University of New South Wales Department of Physics Sydney, 2052, Australia e-mail: [email protected]

M. Raymond Flannery Georgia Institute of Technology School of Physics Atlanta, GA 30332-0430, USA e-mail: [email protected] David R. Flower University of Durham Department of Physics South Road Durham, DH1 3LE, United Kingdom e-mail: [email protected] A. Lewis Ford Texas A&M University Department of Physics College Station, TX 77843-4242, USA e-mail: [email protected] Jane L. Fox Wright State University Department of Physics 3640 Colonel Glenn Hwy Dayton, OH 45419, USA e-mail: [email protected] Matthias Freyberger Universität Ulm Abteilung für Quantenphysik Albert Einstein Allee 11 Ulm, 89069, Germany e-mail: [email protected] Thomas Fulton The Johns Hopkins University The Henry A. Rowland Department of Physics and Astronomy Baltimore, MD 21218-2686, USA e-mail: [email protected] Alexander L. Gaeta Cornell University Department of Applied and Engineering Physics Ithaca, NY 14853-3501, USA e-mail: [email protected]

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List of Authors

Alan Gallagher JILA, University of Colorado and National Institute of Standards and Technology Quantum Physics Division Boulder, CO 80309-0440, USA e-mail: [email protected]

Donald C. Griffin Rollins College Department of Physics 1000 Holt Ave. Winter Park, FL 32789, USA e-mail: [email protected]

Thomas F. Gallagher University of Virginia Department of Physics 382 McCormick Road Charlottesville, VA 22904-4714, USA e-mail: [email protected]

William G. Harter University of Arkansas Department of Physics Fayetteville, AR 72701, USA e-mail: [email protected]

Muriel Gargaud Observatoire Aquitain des Sciences de l’Univers 2 Rue de l’Observatoire 33270 Floirac, France e-mail: [email protected] Alan Garscadden Airforce Research Laboratory Area B 1950 Fifth Street Wright Patterson Air Force Base, OH 45433-7251, USA e-mail: [email protected] John Glass British Telecommunications Solution Design Riverside Tower (pp RT03-44) Belfast, Northern Ireland BT1 3BT, UK e-mail: [email protected] S. Pedro Goldman The University of Western Ontario Department of Physics & Astronomy London, ON N6A 3K7, Canada e-mail: [email protected] Ian P. Grant University of Oxford Mathematical Institute 24/29 St. Giles’ Oxford, OX1 3LB, UK e-mail: [email protected]

Carsten Henkel Universität Potsdam Institut für Physik Am Neuen Palais 10 Potsdam, 14469, Germany e-mail: carsten.henkel @quantum.physik.uni-potsdam.de Eric Herbst The Ohio State University Departments of Physics 191 W. Woodruff Ave. Columbus, OH 43210-1106, USA e-mail: [email protected] Robert N. Hill 355 Laurel Avenue Saint Paul, MN 55102-2107, USA e-mail: [email protected] David L. Huestis SRI International Molecular Physics Laboratory Menlo Park, CA 94025, USA e-mail: [email protected] Mitio Inokuti Argonne National Laboratory Physics Division 9700 South Cass Avenue Building 203 Argonne, IL 60439, USA e-mail: [email protected]

List of Authors

Takeshi Ishihara University of Tsukuba Institute of Applied Physics Ibaraki 305 Tsukuba, 305-8577, Japan

Kate P. Kirby Harvard-Smithsonian Center for Astrophysics 60 Garden Street MS-14 Cambridge, MA 02138, USA e-mail: [email protected]

Juha Javanainen University of Connecticut Department of Physics Unit 3046 2152 Hillside Road Storrs, CT 06269-3046, USA e-mail: [email protected]

Sir Peter L. Knight Imperial College London Department of Physics Blackett Laboratory Prince Consort Road London, SW7 2BW, UK e-mail: [email protected]

Erik T. Jensen University of Northern British Columbia Department of Physics 3333 University Way Prince George, BC V2N 4Z9, Canada e-mail: [email protected]

Manfred O. Krause Oak Ridge National Laboratory 125 Baltimore Drive Oak Ridge, TN 37830, USA e-mail: [email protected]

Brian R. Judd The Johns Hopkins University Department of Physics and Astronomy 3400 North Charles Street Baltimore, MD 21218, USA e-mail: [email protected] Alexander A. Kachanov Research and Development Picarro, Inc. 480 Oakmead Parkway Sunnyvale, CA 94085, USA e-mail: [email protected] Isik Kanik California Institute of Technology Jet Propulsion Laboratory Pasadena, CA 91109, USA e-mail: [email protected] Savely G. Karshenboim D.I.Mendeleev Institute for Metrology (VNIIM) Quantum Metrology Department Moskovsky pr. 19 St. Petersburg, 190005, Russia e-mail: [email protected]

Kenneth C. Kulander Lawrence Livermore National Laboratory 7000 East Ave. Livermore, CA 94551, USA e-mail: [email protected] Paul G. Kwiat University of Illinois at Urbana-Champaign Department of Physics 1110 West Green Street Urbana, IL 61801-3080, USA e-mail: [email protected] Yuan T. Lee Academia Sinica Institute of Atomic and Molecular Science PO BOX 23-166 Taipei, 106, Taiwan Stephen Lepp University of Nevada Department of Physics 4505 Maryland Pkwy Las Vegas, NV 89154-4002, USA e-mail: [email protected]

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List of Authors

Maciej Lewenstein ICFO–Institut de Ciéncies Fotóniques C. Jordi Ginora 29 Nexus II Barcelona, 08034, Spain e-mail: [email protected] James D. Louck Los Alamos National Laboratory Retired Laboratory Fellow PO BOX 1663 Los Alamos, NM 87545, USA e-mail: [email protected] Joseph H. Macek University of Tennessee and Oak Ridge National Laboratory Department of Physics and Astronomy 401 Nielsen Physics Bldg. Knoxville, TN 37996-1200, USA e-mail: [email protected] Mary L. Mandich Lucent Technologies Inc. Bell Laboratories 600 Mountain Avenue Murray Hill, NJ 07974, USA e-mail: [email protected] Edmund J. Mansky Oak Ridge National Laboratory Controlled Fusion Atomic Data Center Oak Ridge, TN 37831, USA e-mail: [email protected] Steven T. Manson Georgia State University Department of Physics and Astronomy Atlanta, GA 30303, USA e-mail: [email protected] William C. Martin National Institute of Standards and Technology Atomic Physics Division Gaithersburg, MD 20899-8422, USA e-mail: [email protected]

Jim F. McCann Queen’s University Belfast Dept. of Applied Mathematics and Theoretical Physics Belfast, Northern Ireland BT7 1NN, UK e-mail: [email protected] Ronald McCarroll Université Pierre et Marie Curie Laboratoire de Chimie Physique 11 rue Pierre et Marie Curie 75231 Paris Cedex 05, France e-mail: [email protected] Fiona McCausland Northern Ireland Civil Service Department of Enterprise Trade and Investment Massey Avenue Belfast, Northern Ireland BT4 2JP, UK e-mail: [email protected] William J. McConkey University of Windsor Department of Physics Windsor, ON N9B 3P4, Canada e-mail: [email protected] Robert P. McEachran Australian National University Atomic and Molecular Physics Laboratories Research School of Physical Sciences and Engineering Canberra, ACT 0200, Australia e-mail: [email protected] James H. McGuire Tulane University Department of Physics 6823 St. Charles Ave. New Orleans, LA 70118-5698, USA e-mail: [email protected] Dieter Meschede Rheinische Friedrich-Wilhelms-Universität Bonn Institut für Angewandte Physik Wegelerstraße 8 Bonn, 53115, Germany e-mail: [email protected]

List of Authors

Pierre Meystre University of Arizona Department of Physics 1118 E, 4th Street Tucson, AZ 85721-0081, USA e-mail: [email protected] Peter W. Milonni 104 Sierra Vista Dr. Los Alamos, NM 87544, USA e-mail: [email protected] Peter J. Mohr National Institute of Standards and Technology Atomic Physics Division 100 Bureau Drive, Stop 8420 Gaithersburg, MD 20899-8420, USA e-mail: [email protected] David H. Mordaunt Max-Planck-Institut für Strömungsforschung Göttingen, 37073, Germany John D. Morgan III University of Delaware Department of Physics and Astronomy Newark, DE 19716, USA e-mail: [email protected] Michael S. Murillo Los Alamos National Laboratory Theoretical Division PO BOX 1663 Los Alamos, NM 87545, USA e-mail: [email protected] Evgueni E. Nikitin Technion-Israel Institute of Technology Department of Chemistry Haifa, 32000, Israel e-mail: [email protected] Robert F. O’Connell Louisiana State University Department of Physics and Astronomy Baton Rouge, LA 70803-4001, USA e-mail: [email protected]

Francesca O’Rourke Queen’s University Belfast Department of Applied Mathematics and Theoretical Physics University Road Belfast, BT7 1NN, UK e-mail: [email protected] Ronald E. Olson University of Missouri-Rolla Physics Department Rolla, MO 65409, USA e-mail: [email protected] Barbara A. Paldus Skymoon Ventures 3045 Park Boulevard Palo Alto, CA 94306, USA e-mail: [email protected] Josef Paldus University of Waterloo Department of Applied Mathematics 200 University Avenue West Waterloo, ON N2L 3G1, Canada e-mail: [email protected] Gillian Peach University College London Department of Physics and Astronomy London, WC1 E6BT, UK e-mail: [email protected] Ruth T. Pedlow Queen’s University Belfast Department of Applied Mathematics and Theoretical Physics University Road Belfast, Northern Irland BT7 1NN, UK e-mail: [email protected] David J. Pegg University of Tennessee Department of Physics Nielsen Building Knoxville, TN 37996, USA e-mail: [email protected]

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List of Authors

Ekkehard Peik Physikalisch-Technische Bundesanstalt Bundesallee 100 Braunschweig, 38116, Germany e-mail: [email protected] Ronald Phaneuf University of Nevada Department of Physics MS-220 Reno, NV 89557-0058, USA e-mail: [email protected] Michael S. Pindzola Auburn University Department of Physics Auburn, AL 36849, USA e-mail: [email protected] Eric H. Pinnington University of Alberta Department of Physics Edmonton, AB T6H 0B3, Canada e-mail: [email protected] Richard C. Powell University of Arizona Optical Sciences Center Tuscon, AZ 85721, USA e-mail: [email protected] John F. Reading Texas A&M University Department of Physics College Station, TX 77843, USA e-mail: [email protected] Jonathan R. Sapirstein University of Notre Dame Department of Physics 319 Nieuwland Science Notre Dame, IN 46556, USA e-mail: [email protected]

Stefan Scheel Imperial College London Blackett Laboratory Prince Consort Road London, SW7 2BW, UK e-mail: [email protected] Axel Schenzle Ludwig-Maximilians-Universität Department für Physik Theresienstraße 37 München, 80333, Germany e-mail: [email protected] Reinhard Schinke Max-Planck-Institut für Dynamik & Selbstorganisation Bunsenstr. 10 Göttingen, 37073, Germany e-mail: [email protected] Wolfgang P. Schleich Universität Ulm Abteilung für Quantenphysik Albert Einstein Allee 11 Ulm, 89069, Germany e-mail: [email protected] David R. Schultz Oak Ridge National Laboratory Physics Division Oak Ridge, TN 37831-6373, USA e-mail: [email protected] Michael Schulz University of Missouri-Rolla Physics Department 1870 Miner Circle Rolla, MO 65409, USA e-mail: [email protected] Peter L. Smith Harvard University Harvard-Smithsonian Center for Astrophysics 60 Garden Street Cambridge, MA 02138, USA e-mail: [email protected]

List of Authors

Anthony F. Starace The University of Nebraska Department of Physics and Astronomy 116 Brace Laboratory Lincoln, NE 68588-0111, USA e-mail: [email protected] Glenn Stark Wellesley College Department of Physics 106 Central Street Wellesley, MA 02481, USA e-mail: [email protected] Allan Stauffer Department of Physics and Astronomy York University 4700 Keele Street Toronto, ON M3J 1P3, Canada e-mail: [email protected] Aephraim M. Steinberg University of Toronto Department of Physics Toronto, ON M5S 1A7, Canada e-mail: [email protected] Stig Stenholm Royal Institute of Technology Physics Department Roslagstullsbacken 21 Stockholm, SE-10691, Sweden e-mail: [email protected] Jack C. Straton Portland State University University Studies 117P Cramer Hall Portland, OR 97207, USA Michael R. Strayer Oak Ridge National Laboratory Physics Division Oak Ridge, TN 37831-6373, USA e-mail: [email protected]

Carlos R. Stroud Jr. University of Rochester Institute of Optics Rochester, NY 14627-0186, USA e-mail: [email protected] Arthur G. Suits State University of New York Department of Chemistry Stony Brook, NY 11794, USA e-mail: [email protected] Barry N. Taylor National Institute of Standards and Technology Atom Physics Division 100 Bureau Drive Gaithersburg, MD 20899-8401, USA e-mail: [email protected] Aaron Temkin NASA Goddard Space Flight Center Laboratory for Solar and Space Physics Solar Physics Branch Greenbelt, MD 20771, USA e-mail: [email protected] Sandor Trajmar California Institute of Technology Jet Propulsion Laboratory 3847 Vineyard Drive Redwood City, 94063, USA e-mail: [email protected] Elmar Träbert Ruhr-Universität Bochum Experimentalphysik III/NB3 Bochum, 44780, Germany e-mail: [email protected] Turgay Uzer Georgia Institute of Technology School of Physics 837 State Street Atlanta, GA 30332-0430, USA e-mail: [email protected]

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List of Authors

Karl Vogel Universität Ulm Abteilung für Quantenphysik Albert Einstein Allee 11 Ulm, 89069, Germany e-mail: [email protected]

Martin Wilkens Universität Potsdam Institut für Physik Am Neuen Palais 10 Potsdam, 14469, Germany e-mail: [email protected]

Jon C. Weisheit Washington State University Institute for Shock Physics PO BOX 64 28 14 Pullman, WA 99164, USA e-mail: [email protected]

David R. Yarkony The Johns Hopkins University Department of Chemistry Baltimore, MD 21218, USA e-mail: [email protected]

Wolfgang L. Wiese National Institute of Standards and Technology 100 Bureau Drive Gaithersburg, MD 20899, USA e-mail: [email protected]

Springer Handbook of Atomic, Molecular, and Optical Physics Organization of the Handbook

Part A gathers together the mathematical methods applicable to a wide class of problems in atomic, molecular, and optical physics. The application of angular momentum theory to quantum mechanics is presented. The basic tenet that isolated physical systems are invariant to rotations of the system is thereby implemented into physical theory. The powerful methods of group theory and second quantization show how simplifications arise if the atomic shell is treated as a basic structural unit. The well established symmetry groups of quantum mechanical Hamiltonians are extended to the larger compact and noncompact dynamical groups. Perturbation theory is introduced as a bridge between an exactly solvable problem and a corresponding real one, allowing approximate solutions of various systems of differential equations. The consistent manner in which the density matrix formalism deals with pure and mixed states is developed, showing how the preparation of an initial state as well as the details regarding the observation of the final state can be treated in a systematic way. The basic computational techniques necessary for accurate and efficient numerical calculations essential to all fields of physics are outlined and a summary of relevant software packages is given. The ever present one-electron solutions of the nonrelativistic Schrödinger equation and the relativistic Dirac equation for the Coulomb potential are then summarized.

Part A Mathematical Methods 2 Angular Momentum Theory 3 Group Theory for Atomic Shells 4 Dynamical Groups 5 Perturbation Theory 6 Second Quantization 7 Density Matrices 8 Computational Techniques 9 Hydrogenic Wave Functions

Part B presents the main concepts in the theoretical and experimental knowledge of atomic systems, including atomic structure and radiation. Ionization energies for neutral atoms and transition probabilities of selected neutral atoms are tabulated. The computational methods needed for very high precision approximations for helium are summarized. The physical and geometrical significance of simple multipoles is examined. The basic nonrelativistic and relativistic theory of electrons and atoms in external magnetic fields is given. Various properties of Rydberg atoms in external fields and in collisions are investigated. The sources of hyperfine structure in atomic and molecular spectra are outlined, and the resulting energy splittings and isotope shifts given. Precision oscillator strength and lifetime measurements, which provide stringent experimental tests of fundamental atomic structure calculations, are discussed. Ion beam spectroscopy is introduced, and individual applications of ion beam techniques are detailed A basic description of neutral collisional line shapes is given, along with a discussion of radiation transfer in a confined atomic vapor. Many qualitative features of the Thomas–Fermi model are studied and its later outgrowth into general density functional theory delineated. The Hartree–Fock and multiconfiguration Hartree–Fock theories, along with configuration interaction methods, are discussed in detail, and their application to the calculation of various atomic properties presented. Relativistic methods for the calculation of atomic structure for general many-electron atoms are described. A consistent diagrammatic method for calculating the structure of atoms and the characteristics of different atomic

Part B Atoms 10 Atomic Spectroscopy 11 High Precision Calculations for Helium 12 Atomic Multipoles 13 Atoms in Strong Fields 14 Rydberg Atoms 15 Rydberg Atoms in Strong Static Fields 16 Hyperfine Structure 17 Precision Oscillator Strength and Lifetime Measurements 18 Spectroscopy of Ions Using Fast Beams and Ion Traps 19 Line Shapes and Radiation Transfer 20 Thomas–Fermi and Other Density-Functional Theories 21 Atomic Structure: Multiconfiguration Hartree–Fock Theories 22 Relativistic Atomic Structure 23 Many-Body Theory of Atomic Structure and Processes 24 Photoionization of Atoms

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processes is given. An outline of the theory of atomic photoionization and the dynamics of the photon–atom collision process is presented. Those kinds of electron correlation that are most important in photoionization are emphasized. The process of autoionization is treated as a quasibound state imbedded in the scattering continuum, and a brief description of the main elements of the theory is given. Green’s function techniques are applied to the calculation of higher order corrections to atomic energy levels, and also of transition amplitudes for radiative transitions of atoms. Basic quantum electrodynamic calculations, which are needed to explain small deviations from the solution to the Schrödinger equation in simple systems, are presented. Comparisons of precise measurements and theoretical predictions that provide tests of our knowledge of fundamental physics are made, focussing on several quantitative tests of quantum electrodynamics. Precise measurements of parity nonconserving effects in atoms could lead to possible modifications of the Standard Model, and thus uncover new physics. An approach to this fundamental problem is described. The problem of the possible variation of the fundamental constants with time is discussed in relation to atomic clocks and precision frequency measurements. The most advanced atomic clocks are described, and the current laboratory constraints on these variations are listed.

Part B Atoms 25 Autoionization 26 Green’s Functions of Field Theory 27 Quantum Electrodynamics 28 Tests of Fundamental Physics 29 Parity Nonconserving Effects in Atoms 30 Atomic Clocks and Constraints on Variations of Fundamental Constants

Part C begins with a discussion of molecular structure from a theoretical/computational perspective using the Born–Oppenheimer approximation as the point of departure. The key role that symmetry considerations play in organizing and simplifying our knowledge of molecular dynamics and spectra is described. The theory of radiative transition probabilities, which determine the intensities of spectral lines, for the rotationally-resolved spectra of certain model molecular systems is summarized. The ways in which molecular photodissociation is studied in the gas phase are outlined. The results presented are particularly relevant to the investigation of combustion and atmospheric reactions. Modern experimental techniques allow the detailed motions of the atomic constituents of a molecule to be resolved as a function of time. A brief description of the basic ideas behind these techniques is given, with an emphasis on gas phase molecules in collision-free conditions. The semiclassical and quantal approaches to nonreactive scattering are outlined. Various quantitative approaches toward a description of the rates of gas phase chemical reactions are presented and then evaluated for their reliability and range of application. Ionic reactions in the gas phase are also considered. Clusters, which are important in many atmospheric and industrial processes, are arranged into six general categories, and then the physics and chemistry common to each category is described. The most important spectroscopic techniques used to study the properties of molecules are presented in detail.

Part C Molecules 31 Molecular Structure 32 Molecular Symmetry and Dynamics 33 Radiative Transition Probabilities 34 Molecular Photodissociation 35 Time-Resolved Molecular Dynamics 36 Nonreactive Scattering 37 Gas Phase Reactions 38 Gas Phase Ionic Reactions 39 Clusters 40 Infrared Spectroscopy 41 Laser Spectroscopy in the Submillimeter and Far-Infrared Regions 42 Spectroscopic Techniques: Lasers 43 Spectroscopic Techniques: Cavity-Enhanced Methods 44 Spectroscopic Techniques: Ultraviolet

Part D collects together the topics and approaches used in scattering theory. A handy compendium of equations, formulae, and expressions for the classical, quantal, and semiclassical approaches to elastic scattering is given; reactive systems and model potentials are also considered. The dependence of scattering processes on the angular orientation of the reactants and products is discussed through the analysis of scattering experiments which probe atomic collision theories at a fundamental level.

Part D Scattering Theory 45 Elastic Scattering: Classical, Quantal, and Semiclassical 46 Orientation and Alignment in Atomic and Molecular Collisions 47 Electron–Atom, Electron–Ion, and Electron–Molecule Collisions

XXIII

The detailed quantum mechanical techniques available to perform accurate calculations of scattering cross sections from first principles are presented. The theory of elastic, inelastic, and ionizing collisions of electrons with atoms and atomic ions is covered and then extended to include collisions with molecules. The standard scattering theory for electrons is extended to include positron collisions with atomic and molecular systems. Slow collisions of atoms or molecules within the adiabatic approximation are discussed; important deviations from this model are presented in some detail for the low energy case. The main methods in the theoretical treatment of ion-atom and atom–atom collisions are summarized with a focus on intermediate and high collision velocities. The molecular structure and collision dynamics involved in ion–atom charge exchange reactions is studied. Both the perturbative and variational capture theories of the continuum distorted wave model are presented. The Wannier theory for threshold ionization is then developed. Studies of the energy and angular distribution of electrons ejected by the impact of high-velocity atomic or ionic projectiles on atomic targets are overviewed. A useful collection of formulae, expressions, and specific equations that cover the various approaches to electron-ion and ion-ion recombination processes is given. A basic theoretical formulation of dielectronic recombination is described, and its importance in the interpretation of plasma spectral emission is presented. Many of the equations used to study theoretically the collisional properties of both charged and neutral particles with atoms and molecules in Rydberg states are collected together; the primary approximations considered are the impulse approximation, the binary encounter approximation, and the Born approximation. The Thomas mass-transfer process is considered from both a classical and a quantal perspective. Additional features of this process are also discussed. The theoretical background, region of validity, and applications of the classical trajectory Monte Carlo method are then delineated. One-photon processes are discussed and aspects of line broadening directly related to collisions between an emitting, or absorbing, atom and an electron, a neutral atom or an atomic ion are considered.

Part D Scattering Theory 48 Positron Collisions 49 Adiabatic and Diabatic Collision Processes at Low Energies 50 Ion–Atom and Atom–Atom Collisions 51 Ion–Atom Charge Transfer Reactions at Low Energies 52 Continuum Distorted Wave and Wannier Methods 53 Ionization in High Energy Ion–Atom Collisions 54 Electron–Ion and Ion–Ion Recombination 55 Dielectronic Recombination 56 Rydberg Collisions: Binary Encounter, Born and Impulse Approximations 57 Mass Transfer at High Energies: Thomas Peak 58 Classical Trajectory and Monte Carlo Techniques 59 Collisional Broadening of Spectral Lines

Part E focuses on the experimental aspects of scattering processes. Recent developments in the field of photodetachment are reviewed, with an emphasis on accelerator-based investigations of the photodetachment of atomic negative ions. The theoretical concepts and experimental methods for the scattering of low-energy photons, proceeding primarily through the photoelectric effect, are given. The main photon–atom interaction processes in the intermediate energy range are outlined. The atomic response to inelastic photon scattering is discussed; essential aspects of radiative and radiationless transitions are described in the two-step approximation. Advances such as cold-target recoil-ion momentum spectroscopy are also touched upon. Electron–atom and electron–molecule collision processes, which play a prominent role in a variety of systems, are presented. The discussion is limited to electron collisions with gaseous targets, where single collision conditions prevail, and to low-energy impact processes. The physical principles and experimental methods used to investigate low energy ion–atom collisions are outlined. Inelastic processes which occur in collisions between fast, often highly charged, ions and atoms, are described. A summary of the methods commonly employed in scattering experiments

Part E Scattering Experiment 60 Photodetachment 61 Photon–Atom Interactions: Low Energy 62 Photon–Atom Interactions: Intermediate Energies 63 Electron–Atom and Electron–Molecule Collisions 64 Ion–Atom Scattering Experiments: Low Energy 65 Ion–Atom Collisions – High Energy 66 Reactive Scattering 67 Ion–Molecule Reactions

XXIV

involving neutral molecules at chemical energies is presented. Applications of single-collision scattering methods to the study of reactive collision dynamics of ionic species with neutral partners are discussed. Part F presents a coherent collection of the main topics and issues found in quantum optics. Optical physics, which is concerned with the dynamical interactions of atoms and molecules with electromagnetic fields, is first discussed within the context of semiclassical theories, and then extended to a fully quantized version. The theoretical techniques used to describe absorption and emission spectra using density matrix methods are developed. Applications of the dark state in laser physics is briefly mentioned. The basic concepts common to all lasers, such as gain, threshold, and electromagnetic modes of oscillation are described. Recent developments in laser physics, including single-atom lasers, two-photon lasers, and the generation of attosecond pulses are also introduced. The current status of the development of different types of lasers – including nanocavity, quantum-cascade and free-electron lasers – are summarized. The important operational characteristics, such as frequency range and output power, are given for each of the types of lasers described. Nonlinear processes arising from the modifications of the optical properties of a medium due to the passage of intense light beams are discussed. Additional processes that are enabled by the use of ultrashort or ultra-intense laser pulses are presented. The concept of coherent optical transients in atomic and molecular systems reviewed; homogeneous and inhomogeneous relaxation in the theory are properly distinguished. Multiphoton and strong-field processes are given a theoretical description. A discussion of the generation of sub-femtosecond pulses is also included. General and specific theories for the control of atomic motion by light are presented. Various traps used for the cooling and trapping of charged and neutral particles and their applications are discussed. The fundamental physics of dilute quantum degenerate gases is outlined, especially in connection with Bose–Einstein condensation. de Broglie optics, which concerns the propagation of matter waves, is presented with a concentration on the underlying principles and the illustration of these principles. The fundamentals of the quantized electromagnetic field and applications to the broad area of quantum optics are discussed. A detailed description of the changes in the atom–field interaction that take place when the radiation field is modified by the presence of a cavity is given. The basic concepts needed to understand current research, such as the EPR experiment, Bell’s inequalities, squeezed states of light, the properties of electromagnetic waves in cavities, and other topics depending on the nonlocality of light are reviewed. Applications to cryptography, tunneling times, and gravity wave detectors are included, along with recent work on “fast light” and “slow light.” Correlations and quantum superpositions which can be exploited in quantum information processing and secure communication are delineated. Their link to quantum computing and quantum cryptography is given explicitly.

Part F Quantum Optics 68 Light–Matter Interaction 69 Absorption and Gain Spectra 70 Laser Principles 71 Types of Lasers 72 Nonlinear Optics 73 Coherent Transients 74 Multiphoton and Strong-Field Processes 75 Cooling and Trapping 76 Quantum Degenerate Gases 77 De Broglie Optics 78 Quantized Field Effects 79 Entangled Atoms and Fields: Cavity QED 80 Quantum Optical Tests of the Foundations of Physics 81 Quantum Information

Part G is concerned with the various applications of atomic, molecular, and optical physics. A summary of the processes that take place in photoionized gases, collisionally ionized gases, the diffuse interstellar medium, molecular clouds, circumstellar shells, supernova ejecta, shocked regions, and the early

Part G Applications 82 Applications of Atomic and Molecular Physics to Astrophysics

XXV

Universe are presented. The principal atomic and molecular processes that lead to the observed cometary spectra, as well as the needs for basic atomic and molecular data in the interpretation of these spectra, are focused on. The basic methods used to understand planetary atmospheres are given. The structure of atmospheres and their interaction with solar radiation are detailed, with an emphasis on ionospheres. Atmospheric global change is then studied in terms of the applicable atomic and molecular processes responsible for these changes. A summary of the well-known prescriptions for atomic structure and ionization balance, and a discussion of the modified transition rates for ions in dense plasmas are given. A review of current simulations being used to address a wide array of issues needed to accurately describe atoms in dense plasmas is also presented. The main concepts and processes of the physics and chemistry of the conduction of electricity in ionized gases are described. The physical models and laser diagnostics used to understand combustion systems are presented. Various applications of atomic and molecular physics to phenomena that occur at surfaces are reviewed; particular attention is placed on the application of electron- and photon-atom scattering processes to obtain surface specific structural and spectroscopic information. The effect of finite nuclear size on the electronic energy levels of atoms is also detailed; and conversely, the electronic structure effects in nuclear physics are discussed. A discussion of the concepts needed in the operation of charged particle detectors and in describing radiation effects is introduced. The description is restricted to fast charged particles. The key topics in basic radiation physics are then treated, and illustrative examples are given.

Part G Applications 83 Comets 84 Aeronomy 85 Applications of Atomic and Molecular Physics to Global Change 86 Atoms in Dense Plasmas 87 Conduction of Electricity in Gases 88 Applications to Combustion 89 Surface Physics 90 Interface with Nuclear Physics 91 Charged-Particle–Matter Interactions 92 Radiation Physics

XXVII

Contents

List of Tables.............................................................................................. XLVII List of Abbreviations ................................................................................. LV 1 Units and Constants William E. Baylis, Gordon W. F. Drake ....................................................... 1.1 Electromagnetic Units .................................................................... 1.2 Atomic Units ................................................................................. 1.3 Mathematical Constants ................................................................ References...............................................................................................

1 1 5 5 6

Part A Mathematical Methods 2 Angular Momentum Theory James D. Louck ........................................................................................ 2.1 Orbital Angular Momentum............................................................ 2.2 Abstract Angular Momentum.......................................................... 2.3 Representation Functions .............................................................. 2.4 Group and Lie Algebra Actions ....................................................... 2.5 Differential Operator Realizations of Angular Momentum ................ 2.6 The Symmetric Rotor and Representation Functions ........................ 2.7 Wigner–Clebsch–Gordan and 3-j Coefficients ................................. 2.8 Tensor Operator Algebra................................................................. 2.9 Racah Coefficients ......................................................................... 2.10 The 9–j Coefficients ....................................................................... 2.11 Tensor Spherical Harmonics ........................................................... 2.12 Coupling and Recoupling Theory and 3n–j Coefficients.................... 2.13 Supplement on Combinatorial Foundations .................................... 2.14 Tables ........................................................................................... References...............................................................................................

9 12 16 18 25 28 29 31 37 43 47 52 54 60 69 72

3 Group Theory for Atomic Shells Brian R. Judd .......................................................................................... 3.1 Generators .................................................................................... 3.2 Classification of Lie Algebras .......................................................... 3.3 Irreducible Representations ........................................................... 3.4 Branching Rules ............................................................................ 3.5 Kronecker Products........................................................................ 3.6 Atomic States ................................................................................ 3.7 The Generalized Wigner–Eckart Theorem ........................................ 3.8 Checks .......................................................................................... References...............................................................................................

75 75 76 77 78 79 80 82 83 84

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Contents

4 Dynamical Groups Josef Paldus ............................................................................................ 4.1 Noncompact Dynamical Groups ...................................................... 4.2 Hamiltonian Transformation and Simple Applications ..................... 4.3 Compact Dynamical Groups ............................................................ References...............................................................................................

87 87 90 92 98

5 Perturbation Theory Josef Paldus ............................................................................................ 5.1 Matrix Perturbation Theory (PT) ...................................................... 5.2 Time-Independent Perturbation Theory ......................................... 5.3 Fermionic Many-Body Perturbation Theory (MBPT) .......................... 5.4 Time-Dependent Perturbation Theory ............................................ References...............................................................................................

101 101 103 105 111 113

6 Second Quantization Brian R. Judd .......................................................................................... 6.1 Basic Properties............................................................................. 6.2 Tensors ......................................................................................... 6.3 Quasispin ...................................................................................... 6.4 Complementarity........................................................................... 6.5 Quasiparticles ............................................................................... References...............................................................................................

115 115 116 117 119 120 121

7 Density Matrices Klaus Bartschat ....................................................................................... 7.1 Basic Formulae.............................................................................. 7.2 Spin and Light Polarizations .......................................................... 7.3 Atomic Collisions ........................................................................... 7.4 Irreducible Tensor Operators .......................................................... 7.5 Time Evolution of State Multipoles ................................................. 7.6 Examples ...................................................................................... 7.7 Summary ...................................................................................... References...............................................................................................

123 123 125 126 127 129 130 133 133

8 Computational Techniques David R. Schultz, Michael R. Strayer .......................................................... 8.1 Representation of Functions .......................................................... 8.2 Differential and Integral Equations ................................................ 8.3 Computational Linear Algebra ........................................................ 8.4 Monte Carlo Methods ..................................................................... References...............................................................................................

135 135 141 148 149 151

9 Hydrogenic Wave Functions Robert N. Hill ........................................................................................... 9.1 Schrödinger Equation .................................................................... 9.2 Dirac Equation ..............................................................................

153 153 157

Contents

9.3 The Coulomb Green’s Function ....................................................... 9.4 Special Functions .......................................................................... References...............................................................................................

159 162 170

Part B Atoms 10 Atomic Spectroscopy William C. Martin, Wolfgang L. Wiese ....................................................... 10.1 Frequency, Wavenumber, Wavelength............................................ 10.2 Atomic States, Shells, and Configurations ....................................... 10.3 Hydrogen and Hydrogen-Like Ions ................................................. 10.4 Alkalis and Alkali-Like Spectra ....................................................... 10.5 Helium and Helium-Like Ions; LS Coupling ..................................... 10.6 Hierarchy of Atomic Structure in LS Coupling ................................... 10.7 Allowed Terms or Levels for Equivalent Electrons............................. 10.8 Notations for Different Coupling Schemes ....................................... 10.9 Eigenvector Composition of Levels .................................................. 10.10 Ground Levels and Ionization Energies for the Neutral Atoms .......... 10.11 Zeeman Effect ............................................................................... 10.12 Term Series, Quantum Defects, and Spectral-Line Series .................. 10.13 Sequences .................................................................................... 10.14 Spectral Wavelength Ranges, Dispersion of Air ................................ 10.15 Wavelength (Frequency) Standards ................................................ 10.16 Spectral Lines: Selection Rules, Intensities, Transition Probabilities, f Values, and Line Strengths .......................................................... 10.17 Atomic Lifetimes............................................................................ 10.18 Regularities and Scaling ................................................................ 10.19 Spectral Line Shapes, Widths, and Shifts......................................... 10.20 Spectral Continuum Radiation........................................................ 10.21 Sources of Spectroscopic Data ........................................................ References...............................................................................................

175 176 176 176 177 177 177 178 179 181 182 183 184 185 185 186 186 194 194 195 196 197 197

11 High Precision Calculations for Helium Gordon W. F. Drake .................................................................................. 11.1 The Three-Body Schrödinger Equation............................................ 11.2 Computational Methods ................................................................ 11.3 Variational Eigenvalues ................................................................. 11.4 Total Energies ............................................................................... 11.5 Radiative Transitions ..................................................................... 11.6 Future Perspectives ....................................................................... References...............................................................................................

199 199 200 205 208 215 218 218

12 Atomic Multipoles William E. Baylis ...................................................................................... 12.1 Polarization and Multipoles ........................................................... 12.2 The Density Matrix in Liouville Space ..............................................

221 222 222

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Contents

12.3 Diagonal Representation: State Populations ................................... 12.4 Interaction with Light.................................................................... 12.5 Extensions .................................................................................... References...............................................................................................

224 224 225 226

13 Atoms in Strong Fields S. Pedro Goldman, Mark M. Cassar ........................................................... 13.1 Electron in a Uniform Magnetic Field .............................................. 13.2 Atoms in Uniform Magnetic Fields .................................................. 13.3 Atoms in Very Strong Magnetic Fields ............................................. 13.4 Atoms in Electric Fields .................................................................. 13.5 Recent Developments .................................................................... References...............................................................................................

227 227 228 230 231 233 234

14 Rydberg Atoms Thomas F. Gallagher ................................................................................ 14.1 Wave Functions and Quantum Defect Theory................................... 14.2 Optical Excitation and Radiative Lifetimes ...................................... 14.3 Electric Fields ................................................................................ 14.4 Magnetic Fields ............................................................................. 14.5 Microwave Fields ........................................................................... 14.6 Collisions ...................................................................................... 14.7 Autoionizing Rydberg States .......................................................... References...............................................................................................

235 235 237 238 241 242 243 244 245

15 Rydberg Atoms in Strong Static Fields Thomas Bartsch, Turgay Uzer .................................................................... 15.1 Scaled-Energy Spectroscopy ........................................................... 15.2 Closed-Orbit Theory ....................................................................... 15.3 Classical and Quantum Chaos ......................................................... 15.4 Nuclear-Mass Effects ..................................................................... References...............................................................................................

247 248 248 249 251 251

16 Hyperfine Structure Guy T. Emery ............................................................................................ 16.1 Splittings and Intensities ............................................................... 16.2 Isotope Shifts ................................................................................ 16.3 Hyperfine Structure ....................................................................... References...............................................................................................

253 254 256 258 259

17 Precision Oscillator Strength and Lifetime Measurements Lorenzo J. Curtis ....................................................................................... 17.1 Oscillator Strengths ....................................................................... 17.2 Lifetimes....................................................................................... References...............................................................................................

261 262 264 268

Contents

18 Spectroscopy of Ions Using Fast Beams and Ion Traps Eric H. Pinnington, Elmar Träbert ............................................................. 18.1 Spectroscopy Using Fast Ion Beams ................................................ 18.2 Spectroscopy Using Ion Traps ......................................................... References...............................................................................................

269 269 272 277

19 Line Shapes and Radiation Transfer Alan Gallagher ........................................................................................ 19.1 Collisional Line Shapes .................................................................. 19.2 Radiation Trapping........................................................................ References...............................................................................................

279 279 287 292

20 Thomas–Fermi and Other Density-Functional Theories John D. Morgan III ................................................................................... 20.1 Thomas–Fermi Theoryand Its Extensions ........................................ 20.2 Nonrelativistic Energies of Heavy Atoms ......................................... 20.3 General Density Functional Theory ................................................. 20.4 Recent Developments .................................................................... References...............................................................................................

295 296 300 301 303 304

21 Atomic Structure: Multiconfiguration Hartree–Fock Theories Charlotte F. Fischer .................................................................................. 21.1 Hamiltonians: Schrödinger and Breit–Pauli .................................... 21.2 Wave Functions: LS and LSJ Coupling .............................................. 21.3 Variational Principle ...................................................................... 21.4 Hartree–Fock Theory...................................................................... 21.5 Multiconfiguration Hartree–Fock Theory ......................................... 21.6 Configuration Interaction Methods ................................................. 21.7 Atomic Properties .......................................................................... 21.8 Summary ...................................................................................... References...............................................................................................

307 307 308 309 309 313 316 318 322 322

22 Relativistic Atomic Structure Ian P. Grant ............................................................................................ 22.1 Mathematical Preliminaries ........................................................... 22.2 Dirac’s Equation ............................................................................ 22.3 QED: Relativistic Atomic and Molecular Structure ............................. 22.4 Many-Body Theory For Atoms ........................................................ 22.5 Spherical Symmetry ....................................................................... 22.6 Numerical Approximation of Central Field Dirac Equations............... 22.7 Many-Body Calculations ................................................................ 22.8 Recent Developments .................................................................... References...............................................................................................

325 326 328 329 334 337 344 350 354 355

23 Many-Body Theory of Atomic Structure and Processes Miron Ya. Amusia .................................................................................... 23.1 Diagrammatic Technique ...............................................................

359 360

XXXI

XXXII

Contents

23.2 Calculation of Atomic Properties ..................................................... 23.3 Concluding Remarks ...................................................................... References...............................................................................................

365 375 376

24 Photoionization of Atoms Anthony F. Starace ................................................................................... 24.1 General Considerations .................................................................. 24.2 An Independent Electron Model ..................................................... 24.3 Particle–Hole Interaction Effects .................................................... 24.4 Theoretical Methods for Photoionization ........................................ 24.5 Recent Developments .................................................................... 24.6 Future Directions ........................................................................... References...............................................................................................

379 379 382 384 386 387 388 388

25 Autoionization Aaron Temkin, Anand K. Bhatia ............................................................... 25.1 Introduction ................................................................................. 25.2 The Projection Operator Formalism ................................................. 25.3 Forms of P and Q ........................................................................... 25.4 Width, Shift, and Shape Parameter ................................................ 25.5 Other Calculational Methods .......................................................... 25.6 Related Topics ............................................................................... References...............................................................................................

391 391 392 393 394 396 398 399

26 Green’s Functions of Field Theory Gordon Feldman, Thomas Fulton .............................................................. 26.1 The Two-Point Green’s Function .................................................... 26.2 The Four-Point Green’s Function.................................................... 26.3 Radiative Transitions ..................................................................... 26.4 Radiative Corrections ..................................................................... References...............................................................................................

401 402 405 406 408 411

27 Quantum Electrodynamics Jonathan R. Sapirstein ............................................................................. 27.1 Covariant Perturbation Theory........................................................ 27.2 Renormalization Theory and Gauge Choices .................................... 27.3 Tests of QED in Lepton Scattering .................................................... 27.4 Electron and Muon g Factors.......................................................... 27.5 Recoil Corrections .......................................................................... 27.6 Fine Structure ............................................................................... 27.7 Hyperfine Structure ....................................................................... 27.8 Orthopositronium Decay Rate......................................................... 27.9 Precision Tests of QED in Neutral Helium ......................................... 27.10 QED in Highly Charged One-Electron Ions........................................ 27.11 QED in Highly Charged Many-Electron Ions ..................................... References...............................................................................................

413 413 414 416 416 418 420 421 422 423 424 425 427

Contents

28 Tests of Fundamental Physics Peter J. Mohr, Barry N. Taylor ................................................................... 28.1 Electron g-Factor Anomaly............................................................. 28.2 Electron g-Factor in 12 C5+ and 16 O7+ .............................................. 28.3 Hydrogen and Deuterium Atoms .................................................... References...............................................................................................

429 429 432 437 445

29 Parity Nonconserving Effects in Atoms Jonathan R. Sapirstein ............................................................................. 29.1 The Standard Model ...................................................................... 29.2 PNC in Cesium ............................................................................... 29.3 Many-Body Perturbation Theory .................................................... 29.4 PNC Calculations ............................................................................ 29.5 Recent Developments .................................................................... 29.6 Comparison with Experiment ......................................................... References...............................................................................................

449 450 451 451 452 453 453 454

30 Atomic Clocks and Constraints

on Variations of Fundamental Constants Savely G. Karshenboim, Victor Flambaum, Ekkehard Peik .......................... 30.1 Atomic Clocks and Frequency Standards ......................................... 30.2 Atomic Spectra and their Dependence on the Fundamental Constants ...................................................................................... 30.3 Laboratory Constraints on Time the Variations of the Fundamental Constants ....................................................... 30.4 Summary ...................................................................................... References...............................................................................................

455 456 459 460 462 462

Part C Molecules 31 Molecular Structure David R. Yarkony ..................................................................................... 31.1 Concepts ....................................................................................... 31.2 Characterization of Potential Energy Surfaces.................................. 31.3 Intersurface Interactions: Perturbations ......................................... 31.4 Nuclear Motion ............................................................................. 31.5 Reaction Mechanisms: A Spin-Forbidden Chemical Reaction............ 31.6 Recent Developments .................................................................... References............................................................................................... 32 Molecular Symmetry and Dynamics William G. Harter ..................................................................................... 32.1 Dynamics and Spectra of Molecular Rotors ...................................... 32.2 Rotational Energy Surfaces and Semiclassical Rotational Dynamics...................................................................... 32.3 Symmetry of Molecular Rotors ........................................................

467 468 470 476 480 484 486 486

491 491 494 498

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Contents

32.4 Tetrahedral-Octahedral Rotational Dynamics and Spectra ............... 32.5 High Resolution Rovibrational Structure ......................................... 32.6 Composite Rotors and Multiple RES................................................. References...............................................................................................

499 503 507 512

33 Radiative Transition Probabilities David L. Huestis ....................................................................................... 33.1 Overview....................................................................................... 33.2 Molecular Wave Functions in the Rotating Frame ............................ 33.3 The Energy–Intensity Model ........................................................... 33.4 Selection Rules .............................................................................. 33.5 Absorption Cross Sections and Radiative Lifetimes........................... 33.6 Vibrational Band Strengths ............................................................ 33.7 Rotational Branch Strengths .......................................................... 33.8 Forbidden Transitions .................................................................... 33.9 Recent Developments .................................................................... References...............................................................................................

515 515 516 518 521 524 525 526 530 531 532

34 Molecular Photodissociation Abigail J. Dobbyn, David H. Mordaunt, Reinhard Schinke .......................... 34.1 Observables .................................................................................. 34.2 Experimental Techniques ............................................................... 34.3 Theoretical Techniques .................................................................. 34.4 Concepts in Dissociation ................................................................ 34.5 Recent Developments .................................................................... 34.6 Summary ...................................................................................... References...............................................................................................

535 537 539 540 541 543 544 545

35 Time-Resolved Molecular Dynamics Volker Engel ............................................................................................ 35.1 Pump–Probe Experiments ............................................................. 35.2 Theoretical Description .................................................................. 35.3 Applications .................................................................................. 35.4 Recent Developments .................................................................... References...............................................................................................

547 548 548 550 551 552

36 Nonreactive Scattering David R. Flower ....................................................................................... 36.1 Definitions .................................................................................... 36.2 Semiclassical Method..................................................................... 36.3 Quantal Method ............................................................................ 36.4 Symmetries and Conservation Laws ................................................ 36.5 Coordinate Systems ....................................................................... 36.6 Scattering Equations...................................................................... 36.7 Matrix Elements ............................................................................ References...............................................................................................

555 555 556 556 557 557 558 558 560

Contents

37 Gas Phase Reactions Eric Herbst ............................................................................................... 37.1 Normal Bimolecular Reactions ....................................................... 37.2 Association Reactions .................................................................... 37.3 Concluding Remarks ...................................................................... References...............................................................................................

561 563 570 572 573

38 Gas Phase Ionic Reactions Nigel G. Adams ........................................................................................ 38.1 Overview....................................................................................... 38.2 Reaction Energetics ....................................................................... 38.3 Chemical Kinetics .......................................................................... 38.4 Reaction Processes ........................................................................ 38.5 Electron Attachment ...................................................................... 38.6 Recombination.............................................................................. References...............................................................................................

575 575 576 578 578 582 583 585

39 Clusters Mary L. Mandich ...................................................................................... 39.1 Metal Clusters................................................................................ 39.2 Carbon Clusters ............................................................................. 39.3 Ionic Clusters................................................................................. 39.4 Semiconductor Clusters .................................................................. 39.5 Noble Gas Clusters ......................................................................... 39.6 Molecular Clusters ......................................................................... 39.7 Recent Developments .................................................................... References...............................................................................................

589 590 593 596 597 599 602 603 604

40 Infrared Spectroscopy Henry Buijs .............................................................................................. 40.1 Intensities of Infrared Radiation .................................................... 40.2 Sources for IR Absorption Spectroscopy ........................................... 40.3 Source, Spectrometer, Sample and Detector Relationship ................ 40.4 Simplified Principle of FTIR Spectroscopy ........................................ 40.5 Optical Aspects of FTIR Technology .................................................. 40.6 The Scanning Michelson Interferometer .......................................... 40.7 Recent Developments .................................................................... 40.8 Conclusion .................................................................................... References...............................................................................................

607 607 608 608 608 611 612 613 613 613

41 Laser Spectroscopy in the Submillimeter

and Far-Infrared Regions

Kenneth M. Evenson† , John M. Brown ....................................................... 41.1 Experimental Techniques using Coherent SM-FIR Radiation ............. 41.2 Submillimeter and FIR Astronomy .................................................. 41.3 Upper Atmospheric Studies ............................................................ References...............................................................................................

615 616 620 620 621

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Contents

42 Spectroscopic Techniques: Lasers Paul Engelking ........................................................................................ 42.1 Laser Basics................................................................................... 42.2 Laser Designs ................................................................................ 42.3 Interaction of Laser Light with Matter............................................. 42.4 Recent Developments .................................................................... References...............................................................................................

623 623 625 628 630 631

43 Spectroscopic Techniques: Cavity-Enhanced Methods Barbara A. Paldus, Alexander A. Kachanov ............................................... 43.1 Limitations of Traditional Absorption Spectrometers ....................... 43.2 Cavity Ring-Down Spectroscopy ..................................................... 43.3 Cavity Enhanced Spectroscopy ........................................................ 43.4 Extensions to Solids and Liquids .................................................... References...............................................................................................

633 633 634 636 639 640

44 Spectroscopic Techniques: Ultraviolet Glenn Stark, Peter L. Smith ....................................................................... 44.1 Light Sources................................................................................. 44.2 VUV Lasers ..................................................................................... 44.3 Spectrometers ............................................................................... 44.4 Detectors ...................................................................................... 44.5 Optical Materials ........................................................................... References...............................................................................................

641 642 645 647 648 651 652

Part D Scattering Theory 45 Elastic Scattering: Classical, Quantal, and Semiclassical M. Raymond Flannery .............................................................................. 45.1 Classical Scattering Formulae ......................................................... 45.2 Quantal Scattering Formulae .......................................................... 45.3 Semiclassical Scattering Formulae .................................................. 45.4 Elastic Scattering in Reactive Systems ............................................. 45.5 Results for Model Potentials........................................................... References...............................................................................................

659 659 664 675 683 684 689

46 Orientation and Alignment in Atomic

and Molecular Collisions Nils Andersen .......................................................................................... 46.1 Collisions Involving Unpolarized Beams .......................................... 46.2 Collisions Involving Spin-Polarized Beams ...................................... 46.3 Example ....................................................................................... 46.4 Recent Developments .................................................................... 46.5 Summary ...................................................................................... References...............................................................................................

693 694 699 702 703 703 703

Contents

47 Electron–Atom, Electron–Ion, and Electron–Molecule Collisions Philip Burke ............................................................................................ 47.1 Electron–Atom and Electron–Ion Collisions..................................... 47.2 Electron–Molecule Collisions .......................................................... 47.3 Electron–Atom Collisions in a Laser Field ........................................ References...............................................................................................

705 705 720 723 727

48 Positron Collisions Robert P. McEachran, Allan Stauffer .......................................................... 48.1 Scattering Channels ....................................................................... 48.2 Theoretical Methods ...................................................................... 48.3 Particular Applications................................................................... 48.4 Binding of Positrons to Atoms ........................................................ 48.5 Reviews ........................................................................................ References...............................................................................................

731 731 733 735 737 738 738

49 Adiabatic and Diabatic Collision Processes at Low Energies Evgueni E. Nikitin .................................................................................... 49.1 Basic Definitions ........................................................................... 49.2 Two-State Approximation .............................................................. 49.3 Single-Passage Transition Probabilities: Analytical Models .............. 49.4 Double-Passage Transition Probabilities and Cross Sections ............. 49.5 Multiple-Passage Transition Probabilities ....................................... References...............................................................................................

741 741 743 746 749 751 752

50 Ion–Atom and Atom–Atom Collisions A. Lewis Ford, John F. Reading ................................................................. 50.1 Treatment of Heavy Particle Motion ................................................ 50.2 Independent-Particle Models Versus Many-Electron Treatments .................................................. 50.3 Analytical Approximations Versus Numerical Calculations ................ 50.4 Description of the Ionization Continuum ........................................ References...............................................................................................

755 756 758 759

51 Ion–Atom Charge Transfer Reactions at Low Energies Muriel Gargaud, Ronald McCarroll ............................................................ 51.1 Molecular Structure Calculations..................................................... 51.2 Dynamics of the Collision ............................................................... 51.3 Radial and Rotational Coupling Matrix Elements ............................. 51.4 Total Electron Capture Cross Sections .............................................. 51.5 Landau–Zener Approximation........................................................ 51.6 Differential Cross Sections .............................................................. 51.7 Orientation Effects ......................................................................... 51.8 New Developments........................................................................ References...............................................................................................

761 762 765 766 767 769 769 770 772 772

753 754

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Contents

52 Continuum Distorted Wave and Wannier Methods Derrick Crothers, Fiona McCausland, John Glass, Jim F. McCann, Francesca O’Rourke, Ruth T. Pedlow .......................................................... 52.1 Continuum Distorted Wave Method ................................................ 52.2 Wannier Method ........................................................................... References...............................................................................................

775 775 781 786

53 Ionization in High Energy Ion–Atom Collisions Joseph H. Macek, Steven T. Manson .......................................................... 53.1 Born Approximation ...................................................................... 53.2 Prominent Features ....................................................................... 53.3 Recent Developments .................................................................... References...............................................................................................

789 789 792 796 796

54 Electron–Ion and Ion–Ion Recombination M. Raymond Flannery .............................................................................. 54.1 Recombination Processes............................................................... 54.2 Collisional-Radiative Recombination.............................................. 54.3 Macroscopic Methods .................................................................... 54.4 Dissociative Recombination ........................................................... 54.5 Mutual Neutralization.................................................................... 54.6 One-Way Microscopic Equilibrium Current, Flux, and Pair-Distributions ................................................................... 54.7 Microscopic Methods for Termolecular Ion–Ion Recombination................................................................. 54.8 Radiative Recombination............................................................... 54.9 Useful Quantities ........................................................................... References...............................................................................................

812 817 824 824

55 Dielectronic Recombination Michael S. Pindzola, Donald C. Griffin, Nigel R. Badnell ............................. 55.1 Theoretical Formulation................................................................. 55.2 Comparisons with Experiment........................................................ 55.3 Radiative-Dielectronic Recombination Interference ........................ 55.4 Dielectronic Recombinationin Plasmas ........................................... References...............................................................................................

829 830 831 832 833 833

799 800 801 803 807 810 811

56 Rydberg Collisions: Binary Encounter,

Born and Impulse Approximations Edmund J. Mansky ................................................................................... 56.1 56.2 56.3 56.4 56.5 56.6 56.7

Rydberg Collision Processes ............................................................ General Properties of Rydberg States .............................................. Correspondence Principles ............................................................. Distribution Functions ................................................................... Classical Theory ............................................................................. Working Formulae for Rydberg Collisions ........................................ Impulse Approximation .................................................................

835 836 836 839 840 841 842 845

Contents

56.8 Binary Encounter Approximation ................................................... 56.9 Born Approximation ...................................................................... References...............................................................................................

852 856 860

57 Mass Transfer at High Energies: Thomas Peak James H. McGuire, Jack C. Straton, Takeshi Ishihara .................................. 57.1 The Classical Thomas Process .......................................................... 57.2 Quantum Description ..................................................................... 57.3 Off-Energy-Shell Effects................................................................. 57.4 Dispersion Relations ...................................................................... 57.5 Destructive Interference of Amplitudes ........................................... 57.6 Recent Developments .................................................................... References...............................................................................................

863 863 864 866 866 867 867 868

58 Classical Trajectory and Monte Carlo Techniques Ronald E. Olson ....................................................................................... 58.1 Theoretical Background ................................................................. 58.2 Region of Validity .......................................................................... 58.3 Applications .................................................................................. 58.4 Conclusions ................................................................................... References...............................................................................................

869 869 871 871 874 874

59 Collisional Broadening of Spectral Lines Gillian Peach ........................................................................................... 59.1 Impact Approximation ................................................................... 59.2 Isolated Lines ................................................................................ 59.3 Overlapping Lines .......................................................................... 59.4 Quantum-Mechanical Theory ......................................................... 59.5 One-Perturber Approximation........................................................ 59.6 Unified Theories and Conclusions ................................................... References...............................................................................................

875 875 876 880 882 885 888 888

Part E Scattering Experiments 60 Photodetachment David J. Pegg .......................................................................................... 60.1 Negative Ions ................................................................................ 60.2 Photodetachment ......................................................................... 60.3 Experimental Procedures ............................................................... 60.4 Results.......................................................................................... References...............................................................................................

891 891 892 893 895 898

61 Photon–Atom Interactions: Low Energy Denise Caldwell, Manfred O. Krause .......................................................... 61.1 Theoretical Concepts ...................................................................... 61.2 Experimental Methods...................................................................

901 901 907

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Contents

61.3 Additional Considerations .............................................................. References...............................................................................................

911 912

62 Photon–Atom Interactions: Intermediate Energies Bernd Crasemann .................................................................................... 62.1 Overview....................................................................................... 62.2 Elastic Photon-Atom Scattering ...................................................... 62.3 Inelastic Photon-Atom Interactions................................................ 62.4 Atomic Response to Inelastic Photon-Atom Interactions .................. 62.5 Threshold Phenomena................................................................... References...............................................................................................

915 915 916 918 919 923 925

63 Electron–Atom and Electron–Molecule Collisions Sandor Trajmar, William J. McConkey, Isik Kanik ....................................... 63.1 Basic Concepts .............................................................................. 63.2 Collision Processes ......................................................................... 63.3 Coincidence and Superelastic Measurements .................................. 63.4 Experiments with Polarized Electrons ............................................. 63.5 Electron Collisions with Excited Species .......................................... 63.6 Electron Collisions in Traps ............................................................. 63.7 Future Developments .................................................................... References...............................................................................................

929 929 933 936 938 939 939 940 940

64 Ion–Atom Scattering Experiments: Low Energy Ronald Phaneuf ...................................................................................... 64.1 Low Energy Ion–Atom Collision Processes ....................................... 64.2 Experimental Methods for Total Cross Section Measurements ............................................. 64.3 Methods for State and Angular Selective Measurements .................. References...............................................................................................

945 947 948

65 Ion–Atom Collisions – High Energy Lew Cocke, Michael Schulz ........................................................................ 65.1 Basic One-Electron Processes ......................................................... 65.2 Multi-Electron Processes ................................................................ 65.3 Electron Spectra in Ion–Atom Collisions .......................................... 65.4 Quasi-Free Electron Processes in Ion–Atom Collisions...................... 65.5 Some Exotic Processes ................................................................... References...............................................................................................

951 951 957 959 961 962 963

66 Reactive Scattering Arthur G. Suits, Yuan T. Lee ...................................................................... 66.1 Experimental Methods................................................................... 66.2 Experimental Configurations .......................................................... 66.3 Elastic and Inelastic Scattering ....................................................... 66.4 Reactive Scattering ........................................................................ 66.5 Recent Developments .................................................................... References...............................................................................................

967 967 971 976 978 980 980

943 943

Contents

67 Ion–Molecule Reactions James M. Farrar ....................................................................................... 67.1 Instrumentation............................................................................ 67.2 Kinematic Analysis ........................................................................ 67.3 Scattering Cross Sections ................................................................ 67.4 New Directions: Complexity and Imaging........................................ References...............................................................................................

983 985 985 987 991 992

Part F Quantum Optics 68 Light–Matter Interaction Pierre Meystre .......................................................................................... 68.1 Multipole Expansion ...................................................................... 68.2 Lorentz Atom ................................................................................ 68.3 Two-Level Atoms ........................................................................... 68.4 Relaxation Mechanisms ................................................................. 68.5 Rate Equation Approximation ........................................................ 68.6 Light Scattering ............................................................................. References...............................................................................................

997 997 999 1000 1003 1005 1006 1007

69 Absorption and Gain Spectra Stig Stenholm .......................................................................................... 69.1 Index of Refraction........................................................................ 69.2 Density Matrix Treatment of the Two-Level Atom ............................ 69.3 Line Broadening............................................................................ 69.4 The Rate Equation Limit................................................................. 69.5 Two-Level Doppler-Free Spectroscopy ............................................ 69.6 Three-Level Spectroscopy............................................................... 69.7 Special Effects in Three-Level Systems ............................................ 69.8 Summary of the Literature ............................................................. References...............................................................................................

1009 1009 1010 1011 1013 1015 1016 1018 1020 1020

70 Laser Principles Peter W. Milonni ...................................................................................... 70.1 Gain, Threshold, and Matter–Field Coupling ................................... 70.2 Continuous Wave, Single-Mode Operation ...................................... 70.3 Laser Resonators ........................................................................... 70.4 Photon Statistics ........................................................................... 70.5 Multi-Mode and Pulsed Operation ................................................. 70.6 Instabilities and Chaos .................................................................. 70.7 Recent Developments .................................................................... References...............................................................................................

1023 1023 1025 1028 1030 1031 1033 1033 1034

71 Types of Lasers Richard C. Powell ..................................................................................... 1035 71.1 Gas Lasers ..................................................................................... 1036 71.2 Solid State Lasers........................................................................... 1039

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Contents

71.3 Semiconductor Lasers .................................................................... 71.4 Liquid Lasers ................................................................................. 71.5 Other Types of Lasers ..................................................................... 71.6 Recent Developments .................................................................... References...............................................................................................

1043 1044 1045 1046 1048

72 Nonlinear Optics Alexander L. Gaeta, Robert W. Boyd .......................................................... 72.1 Nonlinear Susceptibility ................................................................. 72.2 Wave Equation in Nonlinear Optics................................................. 72.3 Second-Order Processes ................................................................. 72.4 Third-Order Processes .................................................................... 72.5 Stimulated Light Scattering ............................................................ 72.6 Other Nonlinear Optical Processes .................................................. References...............................................................................................

1051 1051 1054 1056 1057 1059 1061 1062

73 Coherent Transients Joseph H. Eberly, Carlos R. Stroud Jr. ......................................................... 73.1 Optical Bloch Equations ................................................................. 73.2 Numerical Estimates of Parameters ................................................ 73.3 Homogeneous Relaxation .............................................................. 73.4 Inhomogeneous Relaxation ........................................................... 73.5 Resonant Pulse Propagation .......................................................... 73.6 Multi-Level Generalizations ........................................................... 73.7 Disentanglement and “Sudden Death” of Coherent Transients ........ References...............................................................................................

1065 1065 1066 1066 1068 1069 1071 1074 1076

74 Multiphoton and Strong-Field Processes Kenneth C. Kulander, Maciej Lewenstein ................................................... 74.1 Weak Field Multiphoton Processes.................................................. 74.2 Strong-Field Multiphoton Processes ............................................... 74.3 Strong-Field Calculational Techniques ............................................ References...............................................................................................

1077 1078 1080 1086 1088

75 Cooling and Trapping Juha Javanainen ..................................................................................... 75.1 Notation ....................................................................................... 75.2 Control of Atomic Motion by Light .................................................. 75.3 Magnetic Trap for Atoms ................................................................ 75.4 Trapping and Cooling of Charged Particles ...................................... 75.5 Applications of Cooling and Trapping ............................................. References...............................................................................................

1091 1091 1092 1099 1099 1103 1105

76 Quantum Degenerate Gases Juha Javanainen ..................................................................................... 1107 76.1 Elements of Quantum Field Theory ................................................. 1107 76.2 Basic Properties of Degenerate Gases ............................................. 1110

Contents

76.3 Experimental ................................................................................ 76.4 BEC Superfluid ............................................................................... 76.5 Current Active Topics...................................................................... References...............................................................................................

1115 1117 1119 1123

77 De Broglie Optics Carsten Henkel, Martin Wilkens ................................................................ 77.1 Overview....................................................................................... 77.2 Hamiltonian of de Broglie Optics .................................................... 77.3 Principles of de Broglie Optics ........................................................ 77.4 Refraction and Reflection .............................................................. 77.5 Diffraction .................................................................................... 77.6 Interference .................................................................................. 77.7 Coherence of Scalar Matter Waves .................................................. References...............................................................................................

1125 1125 1126 1129 1131 1133 1135 1137 1139

78 Quantized Field Effects Matthias Freyberger, Karl Vogel, Wolfgang P. Schleich, Robert F. O’Connell 78.1 Field Quantization ......................................................................... 78.2 Field States ................................................................................... 78.3 Quantum Coherence Theory ........................................................... 78.4 Photodetection Theory................................................................... 78.5 Quasi-Probability Distributions ...................................................... 78.6 Reservoir Theory ............................................................................ 78.7 Master Equation ............................................................................ 78.8 Solution of the Master Equation ..................................................... 78.9 Quantum Regression Hypothesis .................................................... 78.10 Quantum Noise Operators .............................................................. 78.11 Quantum Monte Carlo Formalism ................................................... 78.12 Spontaneous Emission in Free Space .............................................. 78.13 Resonance Fluorescence ................................................................ 78.14 Recent Developments .................................................................... References...............................................................................................

1141 1142 1142 1146 1147 1148 1151 1152 1154 1156 1157 1159 1159 1160 1162 1163

79 Entangled Atoms and Fields: Cavity QED Dieter Meschede, Axel Schenzle ................................................................. 79.1 Atoms and Fields........................................................................... 79.2 Weak Coupling in Cavity QED .......................................................... 79.3 Strong Coupling in Cavity QED......................................................... 79.4 Strong Coupling in Experiments ..................................................... 79.5 Microscopic Masers and Lasers ....................................................... 79.6 Micromasers.................................................................................. 79.7 Quantum Theory of Measurement .................................................. 79.8 Applications of Cavity QED .............................................................. References...............................................................................................

1167 1167 1169 1173 1174 1175 1178 1180 1181 1182

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Contents

80 Quantum Optical Tests of the Foundations of Physics Aephraim M. Steinberg, Paul G. Kwiat, Raymond Y. Chiao ......................... 80.1 The Photon Hypothesis .................................................................. 80.2 Quantum Properties of Light .......................................................... 80.3 Nonclassical Interference ............................................................... 80.4 Complementarity and Coherence.................................................... 80.5 Measurements in Quantum Mechanics ........................................... 80.6 The EPR Paradox and Bell’s Inequalities ......................................... 80.7 Quantum Information.................................................................... 80.8 The Single-Photon Tunneling Time ................................................. 80.9 Gravity and Quantum Optics .......................................................... References...............................................................................................

1185 1186 1186 1188 1191 1193 1195 1200 1202 1206 1207

81 Quantum Information Peter L. Knight, Stefan Scheel ................................................................... 81.1 Quantifying Information ................................................................ 81.2 Simple Quantum Protocols ............................................................. 81.3 Unitary Transformations ................................................................ 81.4 Quantum Algorithms ..................................................................... 81.5 Error Correction ............................................................................. 81.6 The DiVincenzo Checklist ................................................................ 81.7 Physical Implementations.............................................................. 81.8 Outlook......................................................................................... References...............................................................................................

1215 1216 1218 1221 1222 1223 1224 1225 1227 1228

Part G Applications 82 Applications of Atomic and Molecular Physics to Astrophysics Alexander Dalgarno, Stephen Lepp ........................................................... 82.1 Photoionized Gas .......................................................................... 82.2 Collisionally Ionized Gas ................................................................ 82.3 Diffuse Molecular Clouds ................................................................ 82.4 Dark Molecular Clouds ................................................................... 82.5 Circumstellar Shells and Stellar Atmospheres .................................. 82.6 Supernova Ejecta........................................................................... 82.7 Shocked Gas.................................................................................. 82.8 The Early Universe ......................................................................... 82.9 Recent Developments .................................................................... 82.10 Other Reading ............................................................................... References...............................................................................................

1235 1235 1237 1238 1239 1241 1242 1243 1244 1244 1245 1245

83 Comets Paul D. Feldman ...................................................................................... 83.1 Observations ................................................................................. 83.2 Excitation Mechanisms .................................................................. 83.3 Cometary Models ........................................................................... 83.4 Summary ...................................................................................... References...............................................................................................

1247 1247 1250 1254 1256 1257

Contents

84 Aeronomy Jane L. Fox .............................................................................................. 84.1 Basic Structure of Atmospheres ...................................................... 84.2 Density Distributions of Neutral Species .......................................... 84.3 Interaction of Solar Radiation with the Atmosphere ........................ 84.4 Ionospheres .................................................................................. 84.5 Neutral, Ion and Electron Temperatures ......................................... 84.6 Luminosity .................................................................................... 84.7 Planetary Escape ........................................................................... References...............................................................................................

1259 1259 1264 1265 1271 1281 1284 1287 1290

85 Applications of Atomic and Molecular Physics

to Global Change Kate P. Kirby, Kelly Chance ....................................................................... 85.1 Overview....................................................................................... 85.2 Atmospheric Models and Data Needs .............................................. 85.3 Tropospheric Warming/Upper Atmosphere Cooling .......................... 85.4 Stratospheric Ozone....................................................................... 85.5 Atmospheric Measurements ........................................................... References...............................................................................................

1293 1293 1294 1295 1298 1300 1301

86 Atoms in Dense Plasmas Jon C. Weisheit, Michael S. Murillo ............................................................ 86.1 The Dense Plasma Environment ..................................................... 86.2 Atomic Models and Ionization Balance ........................................... 86.3 Elementary Processes .................................................................... 86.4 Simulations................................................................................... References...............................................................................................

1303 1305 1308 1311 1313 1316

87 Conduction of Electricity in Gases Alan Garscadden ..................................................................................... 87.1 Electron Scattering and Transport Phenomena ................................ 87.2 Glow Discharge Phenomena .......................................................... 87.3 Atomic and Molecular Processes ..................................................... 87.4 Electrical Discharge in Gases: Applications ...................................... 87.5 Conclusions ................................................................................... References...............................................................................................

1319 1320 1327 1328 1330 1333 1333

88 Applications to Combustion David R. Crosley ....................................................................................... 88.1 Combustion Chemistry ................................................................... 88.2 Laser Combustion Diagnostics ........................................................ 88.3 Recent Developments .................................................................... References...............................................................................................

1335 1336 1337 1342 1342

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Contents

89 Surface Physics Erik T. Jensen ........................................................................................... 89.1 Low Energy Electrons and Surface Science....................................... 89.2 Electron–Atom Interactions ........................................................... 89.3 Photon–Atom Interactions ............................................................. 89.4 Atom–Surface Interactions............................................................. 89.5 Recent Developments .................................................................... References...............................................................................................

1343 1343 1344 1346 1351 1352 1353

90 Interface with Nuclear Physics John D. Morgan III, James S. Cohen .......................................................... 90.1 Nuclear Size Effects in Atoms.......................................................... 90.2 Electronic Structure Effects in Nuclear Physics ................................. 90.3 Muon-Catalyzed Fusion ................................................................. References...............................................................................................

1355 1356 1358 1359 1369

91 Charged-Particle–Matter Interactions Hans Bichsel ............................................................................................ 91.1 Experimental Aspects .................................................................... 91.2 Theory of Cross Sections ................................................................. 91.3 Moments of the Cross Section......................................................... 91.4 Energy Loss Straggling ................................................................... 91.5 Multiple Scattering and Nuclear Reactions ...................................... 91.6 Monte Carlo Calculations................................................................ 91.7 Detector Conversion Factors ........................................................... References...............................................................................................

1373 1374 1376 1378 1381 1384 1384 1385 1385

92 Radiation Physics Mitio Inokuti ........................................................................................... 92.1 General Overview .......................................................................... 92.2 Radiation Absorption and its Consequences.................................... 92.3 Electron Transport and Degradation ............................................... 92.4 Connections with Related Fields of Research................................... 92.5 Supplement .................................................................................. References...............................................................................................

1389 1389 1390 1392 1397 1397 1398

Acknowledgements ................................................................................... About the Authors ..................................................................................... Detailed Contents...................................................................................... Subject Index.............................................................................................

1401 1405 1425 1471

XLVII

List of Tables

1

Units and Constants Table 1.1 Table 1.2 Table 1.3 Table 1.4 Table 1.5

Table of physical constants. Uncertainties are given in parentheses ......................................................................... The correlation coefficients of a selected group of constants based on the 2002 CODATA ..................................................... Conversion factors for various physical quantities .................. Physical quantities in atomic units with ~ = e = m e = 4π0 = 1, and α−1 = 137.035 999 11(46)............... Values of e, π, Euler’s constant γ , and the Riemann zeta function ζ(n) ........................................................................

2 3 4 5 6

Part A Mathematical Methods 2

Angular Momentum Theory Table 2.1

Table 2.2 Table 2.3 Table 2.4 3

The solid and spherical harmonics Ylm , and the tensor harmonics Tµk (labeled by k = l and µ = m) for l = 0, 1, 2, 3, and 4 ................................................................................... The 3– j coefficients for all M’s = 0, or J3 = 0, 12 ..................... The 3– j coefficients for J3 = 1, 32 , 2 ........................................ The 6– j coefficients for d = 0, 12 , 1, 32 , 2, with s = a + b + c.......

69 69 70 71

Group Theory for Atomic Shells Table 3.1 Table 3.2 Table 3.3 Table 3.4

Generators of the Lie groups for the atomic l shell ................. Dimensions D of the irreducible representations ([irre]IR’s) of various Lie groups................................................................. Eigenvalues of Casimir’s operator C for groups used in the atomic l shell ....................................................................... The states of the d shell ........................................................

77 78 79 81

Part B Atoms 10 Atomic Spectroscopy Table 10.1 Atomic structural hierarchy in L S coupling and names for the groups of all transitions between structural entities ............... Table 10.2 Allowed J values for l Nj equivalent electrons ( jj) coupling ...... Table 10.3 Ground levels and ionization energies for the neutral atoms . Table 10.4 Selection rules for discrete transitions ................................... Table 10.5 Wavelengths λ, upper energy levels E k , statistical weights gi and gk of lower and upper levels, and transition probabilities Aki for persistent spectral lines of neutral atoms....................

178 178 182 187

187

XLVIII

List of Tables

Table 10.6 Conversion relations between S and Aki for forbidden transitions............................................................................ Table 10.7 Relative strengths for lines of multiplets in L S coupling ......... Table 10.8 Some transitions of the main spectral series of hydrogen ....... Table 10.9 Values of Stark-broadening parameter α1/2 of the Hβ line of hydrogen (4861 Å) for various temperatures and electron densities .............................................................................. 11 High Precision Calculations for Helium Table 11.1 Formulas for the radial integrals c e−αr1 −βr2  I0 (a, b, c; α, β) =
r1a r2br12 rad and log a b c ln r e−αr1 −βr2  I0 (a, b, c; α, β) =
r1 r2 r12 12 rad ........................... Table 11.2 Nonrelativistic eigenvalue coefficients ε0 and ε1 for helium .... Table 11.3 Eigenvalue coefficients ε2 for helium ..................................... Table 11.4 Values of the reduced electron mass ratio µ/M ...................... Table 11.5 Nonrelativistic eigenvalues E = ε0 + (µ/M)ε1 + (µ/M)2 ε2 for helium-like ions................................................................... Table 11.6 Expectation values of various operators for He-like ions for the case M = ∞.................................................................... Table 11.7 Total ionization energies for 4 He, calculated with R M = 3 289 391 006.715 MHz ................................................... Table 11.8 QED corrections to the ionization energy included in Table 11.7 for the S- and P-states of helium ......................................... Table 11.9 Quantum defects for the total energies of helium with the ∆Wn term subtracted (11.54) .................................................. Table 11.10 Formulas for the hydrogenic expectation value
r − j  ≡
nl|r − j |nl ................................................................. Table 11.11 Oscillator strengths for helium............................................... Table 11.12 Singlet–triplet mixing angles for helium................................ 13 Atoms in Strong Fields Table 13.1 Relativistic ground state binding energy −E gs /Z 2 and finite nuclear size correction δE nuc /Z 2 of hydrogenic atoms for various magnetic fields B......................................................
 1 Table 13.2 Relativistic binding energy −E
2S,−1/21 for the 2S1/2 mj = − 2 and −E 2P,−1/2 for the 2P1/2 m j = − 2 excited states of hydrogen in an intense magnetic field B ............................... Table 13.3 Relativistic corrections δE = (E − E NR )/|E R | to the nonrelativistic energies E NR for the ground state and n = 2 excited states of hydrogen in an intense magnetic field B...... Table 13.4 Relativistic dipole polarizabilities for the ground state of hydrogenic atoms ............................................................. 17 Precision Oscillator Strength and Lifetime Measurements Table 17.1 Measured np 2 P J lifetimes.....................................................

192 193 195

196

203 205 207 207 207 208 210 211 212 214 216 217

230

231

231 233

266

List of Tables

21 Atomic Structure: Multiconfiguration Hartree–Fock Theories Table 21.1 The effective quantum number and quantum defect parameters of the 2snd Rydberg series in Be .......................... Table 21.2 Observed and Hartree–Fock ionization potentials for the ground states of neutral atoms, in eV .............................. Table 21.3 Comparison of theoretical and experimental energies for Be 1s 2 2s 2 1 S in hartrees ............................................................. Table 21.4 Specific mass shift parameter and electron density at the nucleus as a function of the active set ................................... Table 21.5 MCHF Hyperfine constants for the 1s 2 2s2p 1 P state of B II .................................................................................. Table 21.6 Convergence of transition data for the 1s 2 2s 2 2p 2 P o → 1s 2 2s2p 2 2 D transition in Boron with increasing active set ............................................................. 22 Relativistic Atomic Structure Table 22.1 Relativistic angular density functions .................................... Table 22.2 Nonrelativistic angular density functions ............................... Table 22.3 Spectroscopic labels and angular quantum numbers .............. Table 22.4 Radial moments
ρs  ............................................................. Table 22.5 j N configurational states in the seniority scheme .................. 25 Autoionization Table 25.1 Test of sum rule (25.15) for the lowest He− (1s2s 2 2 S) autodetachment state ([25.4]) ............................................... Table 25.2 Comparison of methods for calculating the energy of the lowest He− (1s2s 2 2 S) autodetachment state ................ Table 25.3 Energies Es of the He(2s2p 1P 0 ) autoionization states below He+ (n = 2) threshold from the variational calculations of O’Malley and Geltman [25.13] ............................................ Table 25.4 Comparison of high precision calculations with experiment for the resonance parameters of the He(1P 0 ) resonances below the n = 2 threshold ..................................................... Table 25.5 Comparison of resonance parameters obtained from different methods for calculating 1 D e states in H− ......... Table 25.6 Resonance energies EF (Ry) and widths (eV) for 1 P states of He below n = 2 threshold (−1 Ry) of He+ ...........................

312 313 317 319 320

322

339 339 340 342 351

393 393

394

396 397 398

27 Quantum Electrodynamics Table 27.1 Contributions to of C2 in Yennie gauge ..................................

417

28 Tests of Fundamental Physics Table 28.1 Theoretical contributions and total for the g-factor of the electron in hydrogenic carbon 12 based on the 2002 recommended values of the constants...................................

433

XLIX

L

List of Tables

Table 28.2 Theoretical contributions and total for the g-factor of the electron in hydrogenic oxygen 16 based on the 2002 recommended values of the constants................................... Table 28.3 Relevant Bethe logarithms ln k0 (n, l) ...................................... Table 28.4 Values of the function G SE (α) ............................................... Table 28.5 Values of the function G (1) VP (α) ............................................... Table 28.6 Values of N .......................................................................... Table 28.7 Values of bL and B60 ............................................................. Table 28.8 Measured transition frequencies ν in hydrogen......................

433 439 440 441 442 442 445

30 Atomic Clocks and Constraints

on Variations of Fundamental Constants Table 30.1 Table 30.2 Table 30.3

Table 30.4 Table 30.5

Limits on possible time variation of frequencies of different transitions in SI units ............................................................ Magnetic moments and relativistic corrections for atoms involved in microwave standards .......................................... Limits on possible time variation of the frequencies of different transitions and their sensitivity to variations in α due to relativistic corrections ......................................... Model-independent laboratory constraints on the possible time variations of natural constants ...................................... Model-dependent laboratory constraints on possible time variations of fundamental constants .....................................

459 460

460 461 462

Part C Molecules 32 Molecular Table 32.1 Table 32.2 Table 32.3 Table 32.4 Table 32.5

Symmetry and Dynamics

Tunneling energy eigensolutions ........................................... Character table for symmetry group C2 .................................. Character table for symmetry group D2 .................................. Character table for symmetry group O ................................... Eigenvectors and eigenvalues of the tunneling matrix for the (A1 , E, T1 ) cluster with K = 28 .............................................. Table 32.6 Spin − 12 basis states for SiF4 rotating about a C4 symmetry axis ......................................................................................

497 498 498 500 503 506

38 Gas Phase Ionic Reactions Table 38.1 Examples illustrating the range of ionic reactions that can occur in the gas phase ..........................................................

576

42 Spectroscopic Techniques: Lasers Table 42.1 Fixed frequency lasers .......................................................... Table 42.2 Approximate tuning ranges for tunable lasers ........................

627 627

44 Spectroscopic Techniques: Ultraviolet Table 44.1 Representative third-order frequency conversion schemes for generation of tunable coherent VUV light ..............................

647

List of Tables

Part D Scattering Theory 45 Elastic Scattering: Classical, Quantal, and Semiclassical Table 45.1 Model interaction potentials .................................................

685

46 Orientation and Alignment in Atomic

and Molecular Collisions Table 46.1 Summary of cases of increasing complexity, and the orientation and alignment parameters necessary for unpolarized beams .......................................................... Table 46.2 Summary of cases of increasing complexity for spin-polarized beams .................................................................................. 49 Adiabatic and Diabatic Collision Processes at Low Energies Table 49.1 Selection rules for the coupling between diabatic and adiabatic states of a diatomic quasimolecule (w = g, u; σ = +, −) .............................................................. Table 49.2 Selection rules for dynamic coupling between adiabatic states of a system of three atoms ..........................................

698 702

745 746

56 Rydberg Collisions: Binary Encounter,

Born and Impulse Approximations Table 56.1 General n-dependence of characteristic properties of Rydberg states .................................................................. Table 56.2 Coefficients C(n i i → n f f ) in the Born capture cross section formula (56.284) ................................................................... Table 56.3 Functions F(n i i → n f f ; x) in the Born capture cross section formula (56.284) ...................................................................

837 860 860

Part E Scattering Experiments 62 Photon–Atom Interactions: Intermediate Energies Table 62.1 Nomenclature for vacancy states ...........................................

920

66 Reactive Scattering Table 66.1 Collision numbers for coupling between different modes .......

968

Part F Quantum Optics 71 Types of Lasers Table 71.1 Categories of lasers ............................................................... 1035 75 Cooling and Trapping Table 75.1 Laser cooling parameters for the lowest S1/2 –P3/2 transition of hydrogen and most alkalis (the D2 line) ............................ 1092

LI

LII

List of Tables

81 Quantum Information Table 81.1 BB84 protocol for secret key distribution ............................... 1219

Part G Applications 82 Applications of Atomic and Molecular Physics to Astrophysics Table 82.1 Molecules observed in interstellar clouds ............................... 1240 84 Aeronomy Table 84.1 Homopause characteristics of planets and satellites ............... Table 84.2 Molecular weights and fractional composition of dry air in the terrestrial atmosphere ................................................ Table 84.3 Composition of the lower atmospheres of Mars and Venus ............................................................................ Table 84.4 Composition of the lower atmospheres of Jupiter and Saturn ........................................................................... Table 84.5 Composition of the lower atmospheres of Uranus and Neptune ........................................................................ Table 84.6 Composition of the lower atmosphere of Titan ....................... Table 84.7 Composition of the atmosphere of Triton ............................... Table 84.8 Number densities of species at the surface of Mercury ............ Table 84.9 Ionization potentials (IP ) of common atmospheric species ...... Table 84.10 Exobase properties of the planets .........................................

1260 1261 1262 1263 1263 1263 1263 1264 1273 1289

86 Atoms in Dense Plasmas Table 86.1 Some plasma quantities that depend on its ionization balance ................................................................................ 1308 88 Applications to Combustion Table 88.1 Combustion chemistry intermediates detectable by laser-induced fluorescence .............................................. 1339 90 Interface with Nuclear Physics Table 90.1 Resonant (quasiresonant if negative) collision energies res (in meV) calculated using (90.35) ........................................... 1365 Table 90.2 Comparison of sticking values ............................................... 1368 91 Charged-Particle–Matter Interactions Table 91.1 The coefficient τ(β) = M0 β 2 /(NZkR ) for pions with Mπ = 139.567 MeV/c2 , calculated in the FVP approximation .... 1378 Table 91.2 Calculated most probable energy loss ∆mp of pions with Z 1 = ±1 and
kinetic energy T passing througha distance x of argon gas at 760 Torr, 293 K,  = 1.66 g/dm3 ..................... 1381 Table 91.3 Calculated values of Γ (fwhm) of the straggling function F(∆) (see Table 91.2) ..................................................................... 1382

List of Tables

92 Radiation Physics Table 92.1 The mean number N j of initial species produced in molecular hydrogen upon complete degradation of an incident electron at 10 keV, and the energy absorbed E abs ............................... 1391 Table 92.2 Condensed matter effects ...................................................... 1396

LIII

LV

List of Abbreviations

2P/2H

two-particle/two-hole

A AA ACT ADDS ADO AES AI AL ALS AMO ANDC AO AOM AS ASD ASF ATI AU

average atom activated complex theory angular distribution by Doppler spectroscopy average dipole orientation Auger electron spectroscopy adiabatic ionization absorption loss advanced light source atomic, molecular, and optical arbitrarily normalized decay curve atomic orbital acoustooptic modulator active space atomic spectra database atomic state functions above threshold ionization absorbance units

B BEA BEC BF BI BL BO BS BW

binary encounter approximation Bose–Einstein condensate (or condensation) body-fixed Bell’s inequality Bethe log Born–Oppenheimer Bethe–Salpeter Brillouin–Wigner

C CARS CAS CASPT CAUGA CC CCA CCC CCD CCO CDW CEAS CES CES

coherent anti-Stokes Raman scattering complete active space complete active space perturbation theory Clifford algebra unitary group approach coupled cluster coupled cluster approximation convergent close coupling coupled cluster doubles coupled-channels optical continuum distorted wave cavity enhanced absorption spectroscopy cavity enhanced spectroscopy constant energy surface

CETS CFCP CG CH CI CIS CL CM CMA COA CODATA COIL COLTRIMS CP CPA CQC CRDS CSDA CSF CTF CTMC CW CW-CRDS CX CXO

cavity enhanced transmission spectroscopy free–free molecular Franck–Condon Clebsch–Gordan Clauser–Horne configuration interaction constant ionic state constant log center-of-mass cylindrical mirror analyzer classical oscillator approximation Committee on Data for Science and Technology chemical-oxygen-iodine cold-target recoil-ion momentum spectroscopy central potential chirped-pulsed-amplification classical-quantal coupling cavity ring-down spectroscopy continuous slowing down approximation configurational state functions common translation factor classical trajectory Monte Carlo continuous wave continuous-Wave Cavity Ring-Down Spectroscopy charge exchange Chandra X-ray Observatory

D DB DCS DDCS DF DFB DFS DFT DFWM DLR DODS DR DSPB

detailed balance differential cross sections doubly differential cross sections Dirac–Fock distributed feedback decoherence free subspace discrete Fourier transform degenerate four wave mixing dielectronic recombination different orbitals for different spins dielectronic recombination distorted wave strong potential Born approximation

E EA EBIT EBS

excitation-autoionization electron beam ion traps eikonal Born series

LVI

List of Abbreviations

ECP ECS EEDF EOM EPR ESM ESR EUV EW-CRDS EXAFS

effective core potential exterior complex scaling electron energy distribution functions equation of motion Einstein–Podolsky–Rosen elastic scattering model experimental storage ring extreme ultraviolet evanescent-wave CRDS extended X-ray absorption fine structure

first Born approximation full-core plus correlation free-electron lasers fast Fourier transform free induction decay far-infrared frequency modulation first-order theory for oscillator strengths fine-structure Fourier transform Fourier transform infrared spectroscopy Fourier transform mass spectrometry Fourier transform spectroscopy far ultraviolet spectroscopic explorer far ultraviolet Fermi virtual photon full width at half maximum

G GBT GFA GGA GHZ GI GIB GOME GOS GPE GRPAE

HRTOF HUM

IERM IPCC

IPM IPP IR IR IRI IRREP ISO

intermediate coupling inertial confinement fusion integrated cavity output spectroscopy international conference on spectral line shapes intermediate energy R-matrix intergovernmental panel on climate change inverse photoemission spectroscopy independent-processes and isolated-resonance independent particle model impact parameter picture irreducible representations infrared international reference ionosphere irreducable representation infrared space observatory

J JB

Jeffrey–Born

K KS KTA KTP

Kohn–Sham potassium titanyl arsenate potassium titanyl phosphate

L generalized Brillouin’s Theorem Green’s function approach generalized gradient approximation Greenberger, Horne, Zeilinger gauge invariant guided ion beam global ozone monitoring experiment generalized oscillator strength Gross–Pitaevskii equation generalized random phase approximationwith exchange

H HEDP HF HF HG HOM HREELS

IC ICF ICOS ICSLS

IPES IPIR

F FBA FCPC FEL FFT FID FIR FM FOTOS FS FT FTIR FTMS FTS FUSE FUV FVP FWHM

I

high energy-density physics Hartree–Fock equations Hellman–Feynman harmonic generation Hong–Ou–Mandel high resolution electron energy loss spectroscopy H-atom Rydberg time-of-flight Hylleraas–Undheim–MacDonald theorem

LieA LA L-CETS LDA LEED LER LG LHC LHV LIF LIGO LISA LL LM LMR LPT LRL LTE LYP LZ

Lie algebras linear algebraic locked cavity enhanced transmission spectroscopy local density approximation low energy electron diffraction laser electric resonance Lie groups left-hand circular local hidden variable laser-induced-fluorescence laser interferometer gravitational-wave observatory laser interferometer space antenna Landau–Lifshitz Levenberg–Marquardt laser magnetic resonance laser photodetachment threshold Laplace–Runge–Lenz local thermodynamic equilibrium Lee, Yang, and Parr Landau–Zener

List of Abbreviations

M MBE MCP MDAL MBPT MCDHF MCHF MCSCF MEMS MFP MIGO MIM MKSA MM MMCDF MO MOPA MOT MOX MP2 MP3 MPI MQDT MR MR-SDCI MUV

molecular beam epitaxy microchannel plate minimum detectable absorption loss many-body perturbation theory multiconfigurational Dirac–Hartree–Fock multiconfiguration Hartree–Fock multiconfigurational self-consistent field microelectromechanical systems mean free path matter–wave interferometric gravitational-wave observatory metal-insulator-metal meters, kilograms, seconds, and amperes Massey–Mohr multichannel multiconfiguration Dirac–Fock molecular orbital master oscillator power amplifier magneto-optical trap molecular orbital X-radiation second order Møller–Plesset perturbation theory third order Møller–Plesset perturbation theory multiphoton ionization multichannel quantum defect theory multireference multireference singles/doubles configuration interaction middle ultraviolet

NIM NIST NMR NNS NR NRQED

nonadiabatic region noise-equivalent power near-edge X-ray absorption fine structure non-dispersive infrared nonfine-structure noise-immune, cavity-enhanced optical heterodyne molecular spectroscopy normal incidence monochromator National Institute of Standards and Technology nuclear magnetic resonance systems nearest-neighbor energy level spacings nonrelativistic NR quantum electrodynamics

O OAO-2 OB OBE OBK

one-component plasma one-and-a-half centered expansion ozone monitoring instrument optical parametric oscillator

P P-CRDS PADDS PAH PBS PCDW PDM PEC PES PES PH/HP PI PID PIMC PMT PNC PPT PR PSD PSS PT PWBA PZT

pulsed-cavity ringdown spectroscopy angular distribution by Doppler spectroscopy polycyclic aromatic hydrocarbon polarizing beam splitters projectile continuum distorted wave approximation phase diffusion model potential energy curves photoelectron spectroscopy potential energy surface particle–hole/hole–particle photoionization particle identification path-integral Monte Carlo photomultiplier tubes parity nonconservation positive partial transposes polarization radiation postion senitive detectors perturbed stationary state perturbation theory plane wave Born approximation piezo-electric transducer

Q

N NAR NEP NEXAFS NDIR NFS NICE-OHMS

OCP OHCE OMI OPO

orbiting astronomical observatory ordinary Bremsstrahlung optical Bloch equations Oppenheimer–Brinkman–Kramers

QCD QED QIP QKD QMC QND QS QSS

quantum chromodynamics quantum electrodynamics quantum information processing quantum key distribution quantum Monte Carlo quantum nondemolition quasistatic quasi-steady state

R RATIP RDC READI REC REDA REMPI RES RHC RHIC

relativistic atomic transition and ionization properties ring-down cavity resonant excitation auto-double ionization radiative electron capture resonant excitation double autoionization resonance-enhanced multiphoton ionization rotational energy surface right-hand circular relativistic heavy ion collider

LVII

LVIII

List of Abbreviations

RIMS RMI RMPS RNA RPA RPA RPAE RR RRKM RSE RSPT RT RTE RWA

recoil-ion momentum spectroscopy relativistic mass increase R-matrix with pseudostates Raman–Nath approximation random-phase approximation retarding potential analyzer random phase approximation with exchange radiative recombination Rampsberger–Rice–Karplus–Marcus radial Schrödinger equation Rayleigh–Schrödinger perturbation theory Ramsauer–Townsend resonant transfer and excitation rotating wave approximation

S SA-MCSCF

state averaged multiconfiguration self-consistent field SACM statistical adiabatic channel model SBS stimulated Brillouin scattering SCA semiclassical approximation SCF self-consistent field SCIAMACHY scanning imaging absorption spectrometer for atmospheric chartography SD spin-dependent SD single and double SDS singly differential cross section SDTQ single, double, triple, quadruple SE Schrödinger equation SEP stimulated emission pump SEPE simultaneous electron photon excitation SEXAFS surface extended X-ray absorption fine structure SF space-fixed SI spin-independent SIAM Society for Industrial and Applied Mathematics SM submillimeter SM-FIR submillimeter far-infrared SMS specific mass shift SOHO solar and heliospheric observatory SP stationary phase SPA stationary phase approximations SQL standard quantum limit SQUID superconducting quantum interference detector SR synchrotron radiation SRS stimulated Raman scattering SS strong-short

STIRAP STO STP

stimulated Raman adiabatic passage Slater type orbital standard temperature and pressure

T TCDW TDCS TDHF TDS TDSE TEA TF TOF TOP TPA TSR TuFIR

target continuum distorted wave triply differential cross section time-dependent Hartree–Fock thermal desorption spectroscopy time dependent Schrödinger equation transverse-excitation-atmosphericpressure toroidal field time-of-flight time orbiting potential two-photon absorption test storage ring tunable far-infrared

U UGA UHF UPS UV UV-VIS

unitary group approach unrestricted Hartree–Fock ultraviolet photoelectron spectroscopy ultraviolet ultraviolet-visible

V VASP VCSEL VECSEL VES VUV

Vienna ab-initio simulation package vertical-cavity surface-emitting laser vertical external cavity surface-emitting laser vibrational energy surfaces vacuum ultraviolet

W WCG WDM WKB WL WMAP WPMD

Wigner–Clebsch–Gordan warm dense matter Wentzel, Kramers, Brillouin weak-long Wilkinson microwave anisotropy probe wavepacket molecular dynamics

X XPS

X-ray photoelectron spectroscopy

Y YAG

Yttrium Aluminum Garnet

1

The currently accepted values for the physical constants are listed in Table 1.1, based on the 2002 CODATA (Committee on Data for Science and Technology) recommendations [1.1]. The quoted values are based on all data available through 31 December 2002, and replace the earlier 1998 CODATA set. Because the uncertainties are correlated, the correlation matrix, given in Table 1.2, must be used in calculating

1.1

Electromagnetic Units ..........................

1

1.2

Atomic Units .......................................

5

1.3

Mathematical Constants ....................... 1.3.1 Series Summation Formula .........

5 5

References ..................................................

6

uncertainties for any quantities derived from those tabulated [1.1].

1.1 Electromagnetic Units The standard electromagnetic units adopted by most scientific journals and elementary texts belong to the système international (SI) or rationalized MKSA (meters, kilograms, seconds, and amperes) units. However, many authors working with microscopic phenomena prefer Gaussian units, and theoretical physicists often use Heaviside–Lorentz (H–L) units. In this Handbook, SI units are used together with atomic units. The current section is meant as a reference relating these different systems. The relations among different sets of units are not simple conversions since the same symbol in different systems can have different physical dimensions. To clarify the meanings of the units, we summarize basic electromagnetic relations for SI, Gaussian, and H–L systems below. The Coulomb law for the magnitude F of the force acting on each of two static charges q and Q separated by a distance r in a homogeneous medium of permittivity
can be written as F=

1 qQ , 4π
r 2

where in a vacuum,
is   2 −1 , SI    µ0 c
0 = (4π)−1 , Gaussian ,    1, H–L

(1.1)

(1.2)

with the closely related permeability of vacuum given by  −7 2   4π × 10 N/A , SI (1.3) µ0 = 4π , Gaussian .    1, H–L (We deviate here from Jackson [1.2] who takes
0 = µ0 = 1 in Gaussian units and must introduce additional constants to relate the units. The physically important quantities are the relative values
r ≡
/
0 and µr ≡ µ/µ0 , which in traditional Gaussian-unit notation are written without the r subscript.) Note that
0 and µ0 are dimensionless in H–L and Gaussian units, but not in the SI units. Current or electric charge is an independent quantity in the MKSA system but can be expressed in purely mechanical dimensions in the H–L and Gaussian systems. Thus, in Gaussian units, 1 statcoulomb = 1 dyne(1/2) cm, but in SI, even though the ampere is defined in terms of the attractive force between thin parallel wires carrying equal currents, there is no mechanical equivalent for the ampere or the coulomb. To establish such an equivalence, one can supplement the SI units by assigning a dimensionless number to
0 or to µ0 . Gaussian and H–L units arise from two different assignments. The result of assigning a number to
0 is analogous to the relation 1 s = 3˙ × 108 m established between time and distance units if one sets the speed of light c = 1, a convention often used in conjunction with H–L units. (Note that for simplicity, the pure number

Introduction

Units and Con 1. Units and Constants

2

Part

Introduction

Table 1.1 Table of physical constants. Uncertainties are given in parentheses Quantity

Symbol

Value

Units

Speed of light in vacuum Gravitational constant Planck constant

c G h

2.997 924 58 6.6742(10) 6.626 0693(11) 1.054 571 68(18) 1.602 176 53(14) 4.803 204 40(42) 137.035 999 11(46) 2.067 833 72(18) 1.660 538 86(28) 931.494 043(80) 9.109 3826(16) 5.485 799 0945(24) 0.113 428 9264(30) 1.007 276 466 88(13) 1.008 664 915 60(55) 2.013 553 212 70(35) 4.001 506 179 149(56) 1.097 373 156 8525(73) 3.289 841 960 360(22) 13.605 692 3(1 2) 2.179 872 09(37) 0.529 177 2108(18) 27.211 3845(23) 6.579 683 920 721(44) 2.194 746 313 705(15) 3.861 592 678(26) 2.817 940 325(28) 0.665 245 873(13) 9.274 009 49(80) 5.788 381 804(39) − 1.001 159 652 1859(38) − 4.841 970 45(13) 1.521 032 206(15) − 1.041 875 63(25) 0.466 975 4567(50) − 2.002 319 304 3718(75) − 2.002 331 8396(12) 2.675 222 05(23) 6.022 1415(10) 9.648 533 83(83) 1.380 6505(24) 8.617 343(15) 3.166 8153(55) 8.314 472(15)

108 m s−1 10−11 m3 kg−1 s−2 10−34 J s 10−34 J s 10−19 C 10−10 esu

~ = h/2π Elementary charge

e

Inverse fine structure constant [4π
0 ]~c/e2 Magnetic flux quantum h/2e   1 Atomic mass constant 12 m 12 C = 1 u

α−1 Φ0 mu m u c2 me

Electron mass Muon mass Proton mass Neutron mass Deuteron mass α-particle mass Rydberg constant m e cα2 2h

mµ mp mn md mα R∞ R∞ c R∞ hc

Bohr radius α/4πR∞ Hartree energy e2 /[4π
0 ]a0 = 2R∞ hc

Compton wavelength αa0 Classical electron radius α2 a0 Thomson cross section 8πre2 /3 Bohr magneton [c]e~/2m e c

a0 Eh E h /h E h /hc λC = λC /2π re σe µB

Electron magnetic moment Muon magnetic moment Proton magnetic moment Neutron magnetic moment Deuteron magnetic moment Electron g factor −2(1 + ae ) Muon g factor −2(1 + aµ ) Proton gyromagnetic ratio 2µp /~ Avogadro constant Faraday constant NA e Boltzmann constant R/NA

µe /µB µµ /µB µp /µB µn /µB µd /µB ge gµ γp NA F kB

Molar gas constant Molar volume (ideal gas) RT/P T = 273.15 K, P = 101.325 kPa T = 273.15 K, P = 100 kPa

kB /E h R Vm Vm

0.022 413 996(39) 0.022 710 981(40)

10−15 Wb 10−27 kg MeV 10−31 kg 10−4 u u u u u u 107 m−1 1015 Hz eV 10−18 J 10−10 m eV 1015 Hz 107 m−1 10−13 m 10−15 m 10−28 m2 10−24 J T−1 10−5 eV T−1 10−3 10−3 10−3 10−3

108 s−1 T−1 1023 mol−1 104 C mol−1 10−23 J K−1 10−5 eV K−1 10−6 K−1 J mol−1 K−1 m3 mol−1 m3 mol−1

Units and Constants

1.1 Electromagnetic Units

Quantity

Symbol

Value

Units

Stefan–Boltzmann constant π 2 kB4 /(60~3 c2 ) First radiation constant 2πhc2 Second radiation constant hc/kB Wien displacement law constant c2 λmax T = 4.965 114 231...

σ c1 c2 b

5.670 400(40) 3.741 771 38(64) 0.014 387 752(25) 2.897 7685(51)

10−8 W m−2 K−4 10−16 W m2 mK 10−3 m K

Introduction

Table 1.1 Table of physical constants. Uncertainties are given in parentheses, cont.

Table 1.2 The correlation coefficients of a selected group of constants based on the 2002 CODATA [1.1] α h e me NA m e /m p F

α

h

e

me

NA

me /mp

F

− 0.010 0.029 −0.029 0.029 −0.249 0.087

− − 1.000 0.999 −0.999 −0.002 −0.995

− − − 0.998 −0.998 −0.007 −0.993

− − − − −1.000 0.007 −0.998

− − − − − −0.007 0.998

− − − − − − −0.022

− − − − − − −

2.997 924 58, equal numerically to the defined speed of light in vacuum in units of 108 m/s, is represented by 3˙ .) Thus, although within the Gaussian system, where the assignment 4π
0 = 1 is made, it is justified to assert that 1 coulomb equals 3˙ × 109 statcoulombs, this is not true in pure SI, where there is no equivalent mechanical unit for charge. Maxwell’s macroscopic equations can be written as λ∇ · D = ρ , ∂D λc ∇ × H − λ = j, ∂t ∂B =0, c ∇ × E + ∂t ∇·B=0,

(1.4)

with the macroscopic field variables related to the polarizations P and M by λD =
0 E + P =
E λH = µ−1 0 B − M = B/µ

(1.5)

(the last equalities for D and H hold only for homogeneous media) and  1, SI λ= (1.6) 
= µ−1 , Gaussian or H–L , 0

0

 1, SI c = c, Gaussian or H–L .

In Gaussian or H–L units, the fields E, B, D, H, and polarizations (dipole moments per unit volume) P, M all have the same dimensions, whereas in SI units the microscopic fields E and B have dimensions that are generally distinct from each other as well as from P (or D) and M (or H), respectively. In all three unit systems, the dimensionless ratio
/
0 is called the dielectric constant (or relative permittivity) of the medium, and the (dimensionless) fine-structure constant is α=

(1.7)

1 e2 , 4π
0 ~c

3

(1.8)

with a numerical value α−1 = 137.035 999 11 (46). In atomic units (Sect. 1.3), the factor e2 / (4π
0 ), the electron mass m e , and ~, Planck’s constant divided by 2π, are all equal to 1. In Gaussian and H–L systems, these conditions determine a numerical value for all electro-mechanical units. Thus in Gaussian units, the electronic √ charge is e = 1, whereas in H–L atomic units e = 4π. In the SI system, on the other hand, the three conditions e2 / (4π
0 ) = m e = ~ = 1 determine numerical values for mechanical units but not for electromagnetic ones. A complete determination of values requires that
0 also be assigned a value. The choice most consistent with previous work is to take e = 1 = 4π
0 . This choice is made here. Since a volt is a joule/coulomb and a statvolt is an erg/statcoulomb, 1 volt corresponds to (but is not generally equal to, since the physical dimensions may

4

Part

Introduction

Table 1.3 Conversion factors for various physical quantities SI units

Gaussian units

Natural H–L units: ~ = c = 0 = 1

Length

1m

= 102

=1m

Mass

1 kg

= 103 g

Time

1s

=1s

↔ 2.842 788 82(49) × 1042 m−1 ↔ 3˙ × 108 m

Velocity

1 m s−1

= 102 cm s−1

↔ 3˙ −1 × 10−8

Energy

1 J = 1 kg m2 s−2 1 Js

Quantity

Action

J m−1

Force

1N=1

Power

1 W = 1 J s−1

Intensity

1

W m−2

1 C = 1 As

Charge

1V=1

Potential Electric field Magnetic field

1

V m−1

1T=1

J C−1 =1

N C−1

N A−1

m−1

cm

=

107

erg

↔ 3.163 029 14(54) × 1025 m−1

=

107

erg s

↔ 0.948 252 28(16) × 1034

=

105

dyne

↔ 3.163 029 14(54) × 1025 m−2

=

107

erg s−1

↔ 1.055 072 95(18) × 1017 m−2

=

103

erg cm−2

↔ 1.055 072 95(18) × 1017 m−4

↔ 3˙ × 109 statcoul ↔ (3˙ × 102 )−1 statvolt

↔ 1.890 067 14(16) × 1018

↔ (3˙ × 104 )−1

↔ 1.673 500 94(14) × 107 m−2

↔

104

gauss

differ) 107 erg 3˙ × 109 statcoulomb

=

1 statvolt . 3˙ × 102

(1.9)

In Gaussian units, the unit of magnetic field, namely the Gauss (B) or Oersted (H) has the same physical size and dimension as the unit of electric field, namely the statvolt/cm, which in turn corresponds to an SI field of 3˙ × 104 V/m. However, the tesla (1 T = 1 weber/m2 ), the SI unit of magnetic field B (older texts refer to B as the magnetic induction), has the physical dimensions of V s/m2 . To find the correspondence to Gaussian units, one must multiply by the speed of light c: 1 T c = 3˙ × 108 V/m ,

(1.10)

which corresponds to 104 statvolt/cm and hence to 104 gauss. Tables 1.3 and 1.4 related basic mechanical and electromagnetic quantities in the different unit systems. Caution is required both because the same symbol often stands for quantities of different physical dimensions in different systems of units, and because factors of 2 π sometimes enter frequencies, depending on whether the units are cycles/s (Hz) or radians/s. The double-headed arrows (↔) indicate a correspondence between quantities whose dimensions are not necessarily equal. Thus for example, the force on an electron due to a Gaussian electric field of 1 statvolt/cm is the same as due to an SI electric field of 3˙ × 104 V/m. The correspondences between Gaussian and SI electrostatic quantities become equalities if and only if 4π
0 = 1. Thus they are equalities

statvolt cm−1

↔ 1.673 500 94(14) × 107 m−1 ↔ 5.017 029 61(43) × 1015 m−2

within the Gaussian system but not within the less constrained SI scheme. The SI and Gaussian units of magnetic field have different dimensions unless both
0 and c are set equal to dimensionless numbers. Natural H–L units can be considered SI units supplemented by the conditions
0 = c = ~ = 1. They are listed here in units of meters, although eV are also often used: 1 eV = 5.067 731 04(43) × 106 m−1 × ~c. The correspondences may be considered equalities within the natural H–L system but not within SI. Note that the electronic charge √ in the natural H–L system has the magnitude e = 4πα. More electromagnetic conversions can be found in Jackson [1.2]. The data here are based on the 2002 adjustment by Mohr and Taylor [1.1]. A few additional energy conversion factors are 1 eV = 1.602 176 53 (14) × 10−19 J = 2.417 989 40 (21) × 1014 Hz × h = 8065.544 45 (69) cm−1 × hc = 3.674 932 45 (31) × 10−2 E h = 1.160 4505 (20) × 104 K × kB = 96.485 3383 (83) kJ mol−1 The basic unit of temperature, the kelvin, is equivalent to about 0.7 cm−1 , i. e., the value of the Boltzmann constant kB expressed in wavenumber units per kelvin is 0.695 0356(12) cm−1 K−1 . Since K is the internationally accepted symbol for the Kelvin [1.3], this suggests that the use of the letter K as a symbol for 1 cm−1 (1 Kayser) should be discontinued.

Units and Constants

1.3 Mathematical Constants

Atomic and molecular calculations based on the Schrödinger equation are most conveniently done in atomic units (a.u.), and then the final result converted to the correct SI units as listed in Table 1.4. In atomic units, ~ = m e = e = 4π
0 = 1. The atomic units of length, velocity, time, and energy are then length: a0 =

~ 4π
0 ~2 , = αm e c m e e2

velocity: vB = time: τ0 =

e2 4π
0 ~

= αc ,

16π 2
02 ~3 ~ = 2 , m e e4 α m e c2

energy: E h =

e2 = α2 m e c2 , 4π
0 a0

where, from the definition (1.8), the numerical value of c is α−1 = 137.035 999 11(46) a.u. For the lowest 1s state of hydrogen (with infinite nuclear mass), a0 is the Bohr radius, vB is the Bohr velocity, 2πτ0 is the time to complete a Bohr orbit, and E h (the Hartree energy) is twice the ionization energy. To include the effects of a finite nuclear mass M, one must replace the electron mass m e by the reduced electron mass µ = m e M/(M + m e ). Atomic energies are often expressed in units of the Rydberg (Ry). The Rydberg for an atom having nuclear mass M is 1 Ry = R M =

µ R∞ = M(M + m e )−1 R∞ , me (1.11)

Table 1.4 Physical quantities in atomic units with ~ = e = m e = 4π
0 = 1, and α−1 = 137.035 999 11(46) Quantity

Unit

Value

Length Mass Time Velocity Energy Action Force Power Intensity Charge Electric potential Electric field Magnetic flux density

a0 me ~/E h vB ≡ αc Eh

0.529 177 2108(18) × 10−10 m 0.910 938 26(16) × 10−30 kg 2.418 884 326 505(16) × 10−17 s 2.187 691 2633(73) × 106 m s−1 4.359 744 17(75) × 10−18 J 1.054 571 68(18) × 10−34 J s 0.823 872 25(14) × 10−7 N 0.180 237 811(31) W 64.364 091(11) × 1018 W m−2 1.602 176 53(14) × 10−19 C 27.211 3845(23) V

~

E h /a0 E h2 /~ E h2 /~a02 e E h /e E h /ea0 = α~c/ea02 E h /ea0 αc

0.514 220 642(44) × 1012 V m−1 2.350 517 42(20) × 105 T

with R∞ =

m e cα2 = 10 973 731.568 525 (73) m−1 . 2h (1.12)

The Rydberg constant R∞ is thus the limiting value of R M for infinite nuclear mass, and hcR∞ is 12 a.u., which is equivalent to 13.605 6923(12) eV. The energy equivalent of the electron mass, m e c2 , is 0.510 998 918(44) MeV. This energy is a natural unit for relativistic atomic theory. For example, for innershell energies in the heaviest elements, the binding energy of the 1s electron in hydrogenic Lr (Z = 103) is 0.338 42 m e c2 .

1.3 Mathematical Constants A selection of the most important mathematical constants is listed in Table 1.5. More extensive tabulations and formulas can be found in the standard mathematical works [1.4, 5]

convergent series of the form ∞  Ti . S=

(1.13)

i=1

For example, suppose that the series

1.3.1 Series Summation Formula ∞

The Riemann zeta function defined by ζ(n) = i=1 i −n (Table 1.5) is particularly useful in summing slowly

Ti = t2 i −2 + t3 i −3 + · · ·

(1.14)

for the individual terms in S is rapidly convergent for i > N, where N is some suitably large integer.

Introduction

1.2 Atomic Units

5

6

Part

Introduction

Table 1.5 Values of e, π, Euler’s constant γ , and the Riemann zeta function ζ(n)

Then S=

N 

Ti + t2 ζ N (2) + t3 ζ N (3) + · · · ,

(1.15)

i=1  N −n ζ N (n) = ζ(n) − i=1 i

where is the zeta function with the first N terms subtracted. For N sufficiently large, only the first few t j coefficients need be known, and they can be adequately estimated by solving the system of equations TN = t2 N −2 + t3 N −3 + · · · + tk+2 N −k−2 , TN−1 = t2 (N − 1)−2 + t3 (N − 1)−3 + · · · + tk+2 (N − 1)−k−2 , .. . TN−k = t2 (N − k)−2 + t3 (N − k)−3 + · · · + tk+2 (N − k)−k−2 ,

(1.16a)

(1.16b)

(1.16c)

where k + 1 ≤ N is the number of terms retained in (1.14).

Constant

Value

e

2.718 281 828 459 045 235 360 287 471 352 66

π

3.141 592 653 589 793 238 462 643 383 279 50

π 1/2

1.772 453 850 905 516 027 298 167 483 341 14

γ

0.577 215 664 901 532 860 606 512 090 082 40

ζ(2)

1.644 934 066 848 226 436 472 415 166 646 02

ζ(3)

1.202 056 903 159 594 285 399 738 161 511 45

ζ(4)

1.082 323 233 711 138 191 516 003 696 541 16

ζ(5)

1.036 927 755 143 369 926 331 365 486 457 03

ζ(6)

1.017 343 061 984 449 139 714 517 929 790 92

ζ(7)

1.008 349 277 381 922 826 839 797 549 849 80

ζ(8)

1.004 077 356 197 944 339 378 685 238 508 65

ζ(9)

1.002 008 392 826 082 214 417 852 769 232 41

ζ(10)

1.000 994 575 127 818 085 337 145 958 900 31

References 1.1

1.2

P. J. Mohr, B. N. Taylor: Rev. Mod. Phys. 77, 1 (2005); see also www.physicstoday.org/guide/ fundcon.html; all of the values, as well as the correlation coefficients between any two constants, are available online in a searchable database provided by NIST’s fundamental constants data center. The internet address is http://physics.nist.gov/constants J. D. Jackson: Classical Electrodynamics, 3rd edn. (Wiley, New York 1999)

1.3

1.4

1.5

B. N. Taylor (Ed.): The International System of Units (SI), NIST Spec. Publ. 330 (U.S. Government Printing Office, Washington 2001) p. 7 M. Abramowitz, I. A. Stegun: Handbook of Mathematical Functions (Dover, New York 1965) I. S. Gradshteyn, I. M. Ryzhik: Table of Integrals, Series, and Products (Academic, New York 1965)

7

Part A

Mathemati Part A Mathematical Methods

2

Angular Momentum Theory James D. Louck, Los Alamos, USA

3 Group Theory for Atomic Shells Brian R. Judd, Baltimore, USA

6 Second Quantization Brian R. Judd, Baltimore, USA 7

Density Matrices Klaus Bartschat, Des Moines, USA

4 Dynamical Groups Josef Paldus, Waterloo, Canada

8 Computational Techniques David R. Schultz, Oak Ridge, USA Michael R. Strayer, Oak Ridge, USA

5 Perturbation Theory Josef Paldus, Waterloo, Canada

9 Hydrogenic Wave Functions Robert N. Hill, Saint Paul, USA

9

Angular Mome 2. Angular Momentum Theory

2.1

Orbital Angular Momentum .................. 2.1.1 Cartesian Representation ........... 2.1.2 Spherical Polar Coordinate Representation .........................

12 12

2.2

Abstract Angular Momentum ................

16

2.3

Representation Functions..................... 2.3.1 Parametrizations of the Groups SU(2) and SO(3,R) ... 2.3.2 Explicit Forms of Representation Functions ....... 2.3.3 Relations to Special Functions..... 2.3.4 Orthogonality Properties ............ 2.3.5 Recurrence Relations ................. 2.3.6 Symmetry Relations ...................

18

2.4

Group and Lie Algebra Actions .............. 2.4.1 Matrix Group Actions ................. 2.4.2 Lie Algebra Actions .................... 2.4.3 Hilbert Spaces ........................... 2.4.4 Relation to Angular Momentum Theory.....

2.5 2.6 2.7

2.8

15

18 19 21 21 22 23 25 25 26 26 26

2.9

Differential Operator Realizations of Angular Momentum .........................

28

The Symmetric Rotor and Representation Functions ..............

29

Wigner–Clebsch–Gordan and 3-j Coefficients ............................. 2.7.1 Kronecker Product Reduction ...... 2.7.2 Tensor Product Space Construction ............................. 2.7.3 Explicit Forms of WCG-Coefficients ................... 2.7.4 Symmetries of WCG-Coefficients in 3-j Symbol Form ................... 2.7.5 Recurrence Relations ................. 2.7.6 Limiting Properties and Asymptotic Forms ............... 2.7.7 WCG-Coefficients as Discretized Representation Functions...........

31 32 33 33 35 36 36 37

Tensor Operator Algebra....................... 2.8.1 Conceptual Framework .............. 2.8.2 Universal Enveloping Algebra of J ......................................... 2.8.3 Algebra of Irreducible Tensor Operators ... 2.8.4 Wigner–Eckart Theorem ............. 2.8.5 Unit Tensor Operators or Wigner Operators...................

37 37

Racah Coefficients ............................... 2.9.1 Basic Relations Between WCG and Racah Coefficients............... 2.9.2 Orthogonality and Explicit Form .. 2.9.3 The Fundamental Identities Between Racah Coefficients ........ 2.9.4 Schwinger–Bargmann Generating Function and its Combinatorics ................ 2.9.5 Symmetries of 6–j Coefficients .... 2.9.6 Further Properties .....................

43

2.10 The 9–j Coefficients ............................. 2.10.1 Hilbert Space and Tensor Operator Actions ....... 2.10.2 9–j Invariant Operators ............. 2.10.3 Basic Relations Between 9–j Coefficients and 6–j Coefficients .

38 39 39 40

43 43 44

44 45 46 47 47 47 48

Part A 2

Angular momentum theory is presented from the viewpoint of the group SU(1) of unimodular unitary matrices of order two. This is the basic quantum mechanical rotation group for implementing the consequences of rotational symmetry into isolated complex physical systems, and gives the structure of the angular momentum multiplets of such systems. This entails the study of representation functions of SU(2), the Lie algebra of SU(2) and copies thereof, and the associated Wigner– Clebsch–Gordan coefficients, Racah coefficients, and 1n–j coefficients, with an almost boundless set of inter-relations, and presentations of the associated conceptual framework. The relationship to the rotation group in physical 3-space is given in detail. Formulas are often given in a compendium format with brief introductions on their physical and mathematical content. A special effort is made to inter-relate the material to the special functions of mathematics and to the combinatorial foundations of the subject.

10

Part A

Mathematical Methods

Part A 2

2.10.4 Symmetry Relations for 9–j Coefficients and Reductionto 6–j Coefficients 2.10.5 Explicit Algebraic Form of 9–j Coefficients ..................... 2.10.6 Racah Operators ........................ 2.10.7 Schwinger–Wu Generating Function and its Combinatorics ................ 2.11

2.12

Tensor Spherical Harmonics .................. 2.11.1 Spinor Spherical Harmonics as Matrix Functions ................... 2.11.2 Vector Spherical Harmonics as Matrix Functions ................... 2.11.3 Vector Solid Harmonics as Vector Functions ................... Coupling and Recoupling Theory and 3n–j Coefficients ........................... 2.12.1 Composite Angular Momentum Systems ................................... 2.12.2 Binary Coupling Theory: Combinatorics...........................

2.12.3

49 49 49

51 52 53 53 53 54 54 56

Angular momentum theory in its quantum mechanical applications, which is the subject of this section, is the study of the group of 2 × 2 unitary unimodular matrices and its irreducible representations. It is the mathematics of implementing into physical theory the basic tenet that isolated physical systems are invariant to rotations of the system in physical 3-space, denoted R3 , or, equivalently, to the orientation of a Cartesian reference system used to describe the system. That it is the group of 2 × 2 unimodular matrices that is basic in quantum theory in place of the more obvious group of 3 × 3 real, orthogonal matrices representing transformations of the coordinates of the constituent particles of the system, or of the reference frame, is a consequence of the Hilbert space structure of the state space of quantum systems and the impossibility of assigning overall phase factors to such states because measurements depend only on the absolute value of transition amplitudes. The exact relationship between the group SU(2) of 2 × 2 unimodular unitary matrices and the group SO(3, R) of 3 × 3 real, proper, orthogonal matrices is an important one for keeping the quantum theory of angular momentum, with its numerous conventions and widespread applications across all fields of quantum

Implementation of Binary Couplings ................... 2.12.4 Construction of all Transformation Coefficients in Binary Coupling Theory .......... 2.12.5 Unsolved Problems in Recoupling Theory ................. 2.13 Supplement on Combinatorial Foundations........................................ 2.13.1 SU(2) Solid Harmonics................ 2.13.2 Combinatorial Definition of Wigner–Clebsch–Gordan Coefficients .............................. 2.13.3 Magic Square Realization of the Addition of Two Angular Momenta ........... 2.13.4 MacMahon’s and Schwinger’s Master Theorems ....................... 2.13.5 The Pfaffian and Double Pfaffian. 2.13.6 Generating Functions for Coupled Wave Functions and Recoupling Coefficients ....... 2.14 Tables ................................................ References ..................................................

57

58 59 60 60

61

63 64 65

66 69 72

physics, free of ambiguities. These notations and relations are fixed at the outset. Presentation of a point in R3 : x = col (x1 , x2 , x3 ) x = (x1 , x2 , x3 ) 
x1 − ix2 x3 X= x1 + ix2 −x3 T

column matrix , row matrix ,

2 × 2 traceless Hermitian matrix ; Cartan’s representation . A one-to-one correspondence between the set R3 of points in 3-space and the set H2 of 2 × 2 traceless Hermitian matrices is obtained from xi = 12 Tr (σi X), where the σi denote the matrices (Pauli matrices)   


0 1 0 −i 1 0 , σ2 = , σ3 = . σ1 = 1 0 i 0 0 −1 (2.1)

Mappings of R3 onto itself: x → x = Rx , X → X  = UXU † ,

Angular Momentum Theory

where † denotes Hermitian conjugation of a matrix or an operator. Two-to-one homomorphism of SU(2) onto SO(3, R): (2.2)

V = { f | f is a polynomial satisfying ∇ 2 f(x) = 0} .

0


 ξ † ∗ = A (U × U )A , x

(2.3)

where ξ is an indeterminate, A is the unitary matrix given by   1 0 0 1  1   0 1 −i 0  A= √  , 20 1 i 0 1 0

Inner or scalar product:    f, f  = f ∗ (x) f  (x) dS , unit sphere

where f(x) = f(X) for x presented in the Cartan matrix form X. Group actions:   each f ∈ V , (O R f )(x) = f R−1 x ,   (TU f )(X) = f U † XU ,

0 −1

U × U ∗ denotes the matrix direct product, and ∗ denotes complex conjugation. There is a simple unifying theme in almost all the applications. The basic mathematical notions that are implemented over and over again in various contexts are: group action on the underlying coordinates and momenta of the physical system and the corresponding group action in the associated Hilbert space of states; the determination of those subspaces that are mapped irreducibly onto themselves by the group action; the Lie algebra and its actions as derived from the group actions, and conversely; the construction of composite objects from elementary constituents, using the notion of tensor product space and Kronecker products of representations, which are the basic precepts in quantum theory for building complex systems from simpler ones; the reduction of the Kronecker product of irreducible representations into irreducibles with the associated Wigner–Clebsch–Gordan and Racah coefficients determining not only this reduction, but also having a dual role in the construction of the irreducible state spaces themselves; and, finally, the repetition of this process for many-particle systems with the attendant theory of 3n − j coefficients. The universality of this methodology may be attributed to being able, in favorable situations, to separate the particular consequences of physical law (e.g., the Coulomb force) from the implications of symmetry imposed on the system by our underlying conceptions of space and time. Empirical models based on symmetry that attempt to identify

each x ∈ R3 , each f ∈ V , each X ∈ H2 .

Operator properties:

• • • • •

O R is a unitary operator on V; that is, (O R f, O R f  ) = ( f, f  ). TU is a unitary operator on V; that is, (TU f, TU f  ) = ( f, f  ). R → O R is a unitary representation of SO(3, R); that is, O R1 O R2 = O R1 R2 . U → TU is a unitary representation of SU(2); that is, TU1 TU2 = TU1 U2 . O R(U ) = TU = T−U is an operator identity on the space V.

One parameter subgroups: U j (t) = exp(−itσ j /2) , t ∈ R , j = 1, 2, 3 ; R j (t) = R(U j (t)) = exp(−itM j ) , t ∈ R , j = 1, 2, 3 ; where



0  M1 = i  0 0  0  M3 = i  1 0

 0 0  0 −1  , 1 0  −1 0  0 0 . 0 0



 0 0 1   M2 = i  0 0 0  , −1 0 0 (2.4)

Part A 2

 1  Rij = Rij (U ) = Tr σi Uσ j U † , 2   1 0 0 0

   ξ ξ  0 = = 0 x x R(U ) 

the more important ingredients underlying observed physical phenomena are also of great importance. The group actions in complex systems are often modeled after the following examples for the actions of the groups SO(3, R) and SU(2) on functions defined over the 2-sphere S2 ⊂ R3 : Hilbert space:

11

12

Part A

Mathematical Methods

momentum operator:

Infinitesimal generators:

Part A 2.1

L j = i(dTU j (t) /dt)t=0 ,

∂ ∂ (L j f )(x) = −i xk f(x) , − xl ∂xl ∂xk j, k, l cyclic in 1, 2, 3 .

(in units of ~) .

L = −i r × ∇

L j = i(dO R j (t) /dt)t=0 ,

(2.5)

Historically, the algebra of angular momentum came about through the quantum rule of replacing the linear momentum p of a classical point particle, which is located at position r, by p → −i~∇, thus replacing the classical angular momentum r × p about the origin of a chosen Cartesian inertial system by the angular

(2.6)

The quantal angular momentum properties of this simple one-particle system are then to be inferred from the properties of these operators and their actions in the associated Hilbert space. This remains the method of introducing angular momentum theory in most textbooks because of its simplicity and historical roots. It also leads to focusing the developments of the theory on the algebra of operators in contrast to emphasizing the associated group transformations of the Hilbert space, although the two viewpoints are intimately linked, as illustrated above. Both perspectives will be presented here.

2.1 Orbital Angular Momentum The model provided by orbital angular momentum operators is the paradigm for standardizing many of the conventions and relations used in more abstract and general treatments. These basic results for the orbital angular momentum operator L = −i r × ∇ acting in the vector space V are given in this section both in Cartesian coordinates x = col (x1 , x2 , x3 ) and spherical polar coordinates: x = (r sin θ cos φ, r sin θ sin φ, r cos θ) , 0 ≤ r < ∞ , 0 ≤ φ < 2π , 0 ≤ θ ≤ π .

2.1.1 Cartesian Representation Commutation relations: Cartesian form: [L 1 , L 2 ] = iL 3 , [L 3 , L 1 ] = iL 2 .

[L 2 , L 3 ] = iL 1 ,

Cartan form: [L 3 , L + ] = L + , [L 3 , L − ] = −L − , [L + , L − ] = 2L 3 . Squared orbital angular momentum: L2 = L 21 + L 22 + L 23 = L − L + + L 3 (L 3 + 1) = L + L − + L 3 (L 3 − 1) = −r 2 ∇ 2 + (x · ∇)2 + (x · ∇) .

L2 , L 3 form a complete set of commuting Hermitian operators in V with eigenfunctions 1 2 2l + 1 (l + m)!(l − m)! 4π  (−x1 −ix2 )k+m (x1 −ix2 )k x l−m−2k 3 , × 22k+m (k + m)!k!(l − m − 2k)! 

Ylm (x) =

k

where l = 0, 1, 2, . . . , ; m = l, l − 1, . . . , −l. Homogeneous polynomial solutions of Laplace’s equation: Ylm (λx) = λl Ylm (x) , (x · ∇)Ylm (x) = lYlm (x) , ∇ 2 Ylm (x) = 0 . Complex conjugate: ∗ Ylm (x) = (−1)m Yl,−m (x) .

Action of angular momentum operators: 1

L ± Ylm (x) = [(l ∓ m)(l ± m + 1)] 2 Yl,m±1 (x) , L 3 Ylm (x) = mYlm (x) , L2 Ylm (x) = l(l + 1)Ylm . Highest weight eigenfunction: L + Yll (x) = 0 , L 3 Yll (x) = lYll (x) ,

1 1 (2l + 1)! 2 Yll (x) = l (−x1 − ix2 )l . 2 l! 4π

Angular Momentum Theory

Generation from highest weight: Ylm (x) =

(l + m) (2l)!(l − m)!

1 2

L l−m − Yll (x) .

2.1 Orbital Angular Momentum

Product of solid harmonics:  lkl  Yl  ,m+µ (x) Ykµ (x)Ylm (x) = l  ||Yk ||l Cm,µ,m+µ l

 (l  ||Yk ||l) = l




l k l m µ −m − µ





Ylm (x) = r

Ym (x1 , x2 ) 1

× [(2l + 1)(l + m)!(l − m)!/2] 2 × Hl,|m| (x3 /r) ,

(2λ)! λ+ 12 (l + λ)! (λ,λ) C P (z) = (z) , 2λ λ! l−λ 2λl! l−λ 0 ≤ λ ≤ l = 0, 1, 2, . . . ,

Hlλ (z) =

where the Ym (x1 , x2 ) are homogeneous polynomial solutions of degree |m| of Laplace’s equation in 2-space, R2 :  (−x − ix )m /√2π , m ≥ 0 , 1 2 Ym (x1 , x2 ) = (x − ix )−m /√2π , m ≤ 0 . 1 2 (Section 2.1.2 for the definition of Gegenbauer and Jacobi polynomials.) Orthogonal group action: (O R Ylm )(x) = Ylm (R−1 x) =

 m

(TU Ylm )(X) = Ylm (U † XU )  = Dlm  m (U )Ylm  (X) , m

where the functions Dlm  m (U ) are defined in Sects. 2.2 and 2.3. Orthogonality on the unit sphere:

unit sphere

l1 l2 l where Cm 1 m 2 m and 
(−1)l1 −l2 +m l1 l2 l l1 l2 l = √ Cm 1 m 2 m m 1 m 2 −m 2l + 1

denote Wigner–Clebsch–Gordan coefficients and 3– j coefficients, respectively (Sect. 2.7). Vector addition theorem for solid harmonics: Ylm (z + z  ) =  kµ

Dml  m (R)Ylm  (x) ,

where the functions Dml  m (R) = Dlm  m (U(R)) are defined in Sect. 2.3 for various parametrizations of R. Unitary group action:



× (−1)l +m+µ Yl  ,m+µ (x) ,

1  (2l + 1)(2k + 1) 2 lkl  l  ||Yk ||l = r l+k−l C000 , 4π(2l  + 1)

12   l+k−l  (2l + 1)(2k + 1)(2l + 1) (l ||Yk ||l) = r 4π 
 l k l × (−1)l , 0 0 0

Yl∗ m  (x)Ylm (x) dS = δl  l δm  m .

j−k,k,l

Cm−µ,µ,m

4π(2l + 1)! (2l − 2k + 1)!(2k + 1)!

1 2

l−k,k,l Cm−µ,µ,m

× Yl−k,m−µ (z)Ykµ (z  ) , z, z  ∈ C 3 , 

 
 1 2 l +m l −m 2l = . k+µ k−µ 2k

Rotational invariant in two vectors:

 1  2l − 2k k l (−1) Il (x, y) = l 2 k l k × (x · y)l−2k (x · x)k ( y · y)k (1/2)

= (x · x)l/2 ( y · y)l/2 Cl (ˆx · yˆ ) 4π  = (−1)m Ylm (x)Yl,−m ( y) , 2l + 1 m (1/2)

(z) where Cl (Sect. 2.1.2) and xˆ = x/|x| ,

is

a

yˆ = y/|y| ,

Gegenbauer

polynomial

cos θ = xˆ · yˆ .

Part A 2.1

Relation to Gegenbauer and Jacobi polynomials: l−|m|

13

14

Part A

Mathematical Methods

Part A 2.1

Legendre polynomials: 4π  Pl (cos θ) = (−1)m Yl,−m ( yˆ )Ylm (ˆx) , 2l + 1 m

1 2 4π Yl0 (x) 2l + 1 

1  2l − 2k l k x3l−2k (x · x)k = l (−1) 2 l k k = r l Pl (x3 /r) . Rayleigh plane wave expansion: eik·x = 4π jl (kr) =

l ∞  

  ∗ ˆ k Ylm (ˆx) , il jl (kr)Ylm

l=0 m=−l

π 2kr

12

kl

Jl+1/2 (kr) .

l=0

Il (x, y)/r

For R = x − y , 1/R =

 l

(−1)l (y · ∇)l = l! 1

r = (x · x) 2 ,

Pl (cos θ)

sl r l+1

,

The bracket symbols in these relations are 6–j and 9–j coefficients (Sects. 2.9, 2.10). Cartan’s vectors of zero length:    α = − z 21 + z 22 , −i z 21 + z 22 , 2z 1 z 2 ,

1 . r 1

s = ( y · y) 2 , s ≤ r, cos θ = xˆ · yˆ .

Rotational invariants in three vectors:   I(l1 l2 l3 ) x1 , x2 , x3 =

I(l) (x)I(k) (x) 
 3   lα kα jα jα (−1) (2 jα + 1) = 0 0 0 ( j ) α=1    l l l   1 2 3 × (xα · xα )(lα +kα − jα )/2 k1 k2 k3 I( j ) (x) ,     j1 j2 j3   where l = (l1 , l2 , l3 ), etc., x = x1 , x2 , x3 . Coplanar vectors:   I(l) x1 , x2 , αx1 + βx2 1  (2l3 + 1)! 2 = (2l3 − 2k)!(2k)! × αl3 −k β k (−1)l1 +l3 +k (2l + 1) 

 k l2 l l3 − k l3 k l3 − k l1 l × 0 0 0 0 0 0 l2 l l1  1 1 (l1 +l3 −l−k)/2  2 2 (l2 +k−l )/2 × x ·x x ·x  1 2 × Il x , x .

Relations in potential theory: (−1)l (2l)! Ylm (x) 1 = , Ylm (∇) r 2l l! r 2l+1 ∞  1/R = Il (x, y)/r 2l+1 , 2l+1

Product law:

(4π)3/2 1

[(2l1 + 1)(2l2 + 1)(2l3 + 1)] 2
  l1 l2 l3 × m1 m2 m3 m1m2m3  1     × Yl1 m 1 x Yl2 m 2 x2 Yl3 m 3 x3 , 
l1 l2 l3 is a 3–j coefficient (Sect. 2.7). where m1 m2 m3   I(l1 l2 l3 ) x1 , x2 , 0   1 = δl1 l2 δl3 0 (−1)l1 Il1 x1 , x2 2l1 + 1 2 .

α · α = α12 + α22 + α32 = 0 , z = (z 1 , z 2 ) ∈ C 2 . Solutions of Laplace’s equation using vectors of zero length: ∇ 2 (α · x)l = 0 ,

l = 0, 1, . . . .

Solid harmonics for vectors of zero length:

1 (2l)! 2l + 1 2 Plm (z 1 , z 2 ) , (−1)m Yl,−m (α) = l! 4π zl+m zl−m 2 Plm (z 1 , z 2 ) = √ 1 . (l + m)!(l − m)! Orbital angular momentum operators for vectors of zero length: J = −i(α × ∇α ) , ∂ ∂ J+ = z 1 , J− = z 2 , ∂z 2 ∂z 1

1 ∂ ∂ . J3 = z1 − z2 2 ∂z 1 ∂z 2

Angular Momentum Theory

Rotational invariant for vectors of zero length: (α · x)l =

4π 2l + 1

1 2

2l l!



Plm (z)Ylm (x) .

m






m1m2m3

j1

j2

j3

m1 m2 m3



      P j1 m 1 z 1 P j2 m 2 z 2 P j3 m 3 z 3

= [( j1 + j2 + j3 + 1)!]−1/2  12  j1 + j2 − j3  31  j3 + j1 − j2  23  j2 + j3 − j1 z 12 z 12 z 12 × , 1 [( j1 + j2 − j3 )!( j3 + j1 − j2 )!( j2 + j3 − j1 )!] 2 ij z 12

=

j j z i1 z 2 − z 1 z i2

.

This relation is invariant under the transformation   z → Uz = (Uz)1 , (Uz)2 = (u 11 z 1 + u 12 z 2 , u 21 z 1 + u 22 z 2 ) , where U ∈ SU(2). Transformation properties of vectors of zero length: α → Rα,

α = col(α1 , α2 , α3 ) ,

where z → Uz and R is given in terms of U in the beginning of this chapter. Simultaneous eigenvectors of L2 and J 2 : L2 (α · x)l = l(l + 1)(α · x)l ,

l = 0, 1, . . . ,

J (α · x) = l(l + 1)(α · x) ,

l = 0, 1, . . . .

2

l

l

2.1.2 Spherical Polar Coordinate Representation The results given in Sect. 2.1.1 may be presented in any system of coordinates well-defined in terms of Cartesian coordinates. The principal relations for spherical polar coordinates are given in this section, where a vector in R3 is now given in the form x = r xˆ = r(sin θ cos φ, sin θ sin φ, cos θ) , 0 ≤ θ ≤ π,

0 ≤ φ < 2π .

15

Orbital angular momentum operators: ∂ ∂ + i sin φ , ∂φ ∂θ ∂ ∂ − i cos φ , L 2 = i sin φ cot θ ∂φ ∂θ ∂ L 3 = −i , ∂φ

∂ ∂ L ± = e±iφ ± + i cot θ , ∂θ ∂φ

∂ 1 ∂2 1 ∂ sin θ − 2 L2 = − . sin θ ∂θ ∂θ sin θ ∂φ2 L 1 = i cos φ cot θ

Laplacian: ∂ , ∂r 

∂ 2 1 ∂ 2 2 r ∇ = 2 +r − L . ∂r ∂r r  Spherical  harmonics solid harmonics on the unit sphere S2 :

1 2 2l + 1 Ylm (θ, φ) = (−1)m (l + m)!(l − m)! eimφ 4π  (−1)k (sin θ)2k+m (cos θ)l−2k−m × . 22k+m (k + m)!k!(l − 2k − m)! x·∇ = r

k

Orthogonality on the unit sphere: 2π

π dφ

0

dθ sin θ Yl∗ m  (θ, φ) Ylm (θ, φ) = δl  l δm  m .

0

Relation to Legendre, Jacobi, and Gegenbauer polynomials:

1 (2l + 1)(l − m)! 2 Ylm (θ, φ) = (−1)m 4π(l + m)! m × Pl (cos θ) eimφ ,

(l + m)! sin θ m (m,m) m Pl (cos θ) = Pl−m (cos θ) . l! 2 Jacobi polynomials:  

 n +α n +β (α,β) Pn (x) = n −s s s

n−s

x −1 x +1 s × , 2 2 n = 0, 1, . . . ,

Part A 2.1

Spinorial invariant under z i → Uz i (i = 1, 2, 3):

2.1 Orbital Angular Momentum

16

Part A

Mathematical Methods

Part A 2.2

where α, β are arbitrary parameters and   z(z − 1) · · · (z − k + 1)/k! 
    for k = 1, 2, . . . z =  k  1 for k = 0     0 for k = −1, −2, . . . Relations between Jacobi polynomials for n + α, n + β, n + α + β nonnegative integers:

(n + α)!(n + β)! x + 1 −β (α,−β) α,β Pn (x) = Pn+β (x) , n!(n + α + β)! 2

−α (n + α)!(n + β)! x − 1 (−α,β) Pnα,β (x) = Pn+α (x) , n!(n + α + β)! 2



x − 1 −α x + 1 −β (−α,−β) Pn+α+β (x) . 2 2 Nonstandard form (α arbitrary):    (−1)s (α + s+1)n−s 1−x 2 s x n−2s , Pn(α,α) (x) = 22s s!(n − 2s)! s Pnα,β (x) =



(z)k = z(z + 1) · · · (z + k − 1) , (z)0 = 1 .

k = 1, 2, . . . ;

Gegenbauer polynomials (α > −1/2):  1  α− ,α− 12 (2α)n Pn 2 (x) Cn(α) (x) = (α + 1/2)n  (−1)s (α)n−s (2x)n−2s = . s!(n − 2s)! s

2.2 Abstract Angular Momentum Abstract angular momentum theory addresses the problem of constructing all finite Hermitian matrices, up to equivalence, that satisfy the same commutation relations [J1 , J2 ] = iJ3 ,

[J2 , J3 ] = iJ1 ,

[J3 , J1 ] = iJ2 (2.7)

as some set of Hermitian operators J1 , J2 , J3 appropriately defined in some Hilbert space; that is, of constructing all finite Hermitian matrices Mi such that under the correspondence Ji → Mi (i = 1, 2, 3) the commutation relations are still obeyed. If M1 , M2 , M3 is such a set of Hermitian matrices, then AM1 A−1 , AM2 A−1 , AM3 A−1 , is another such set, where A is an arbitrary unitary matrix. This defines what is meant by equivalence. The commutation relations (2.7) may also be formulated as: [J3 , J± ] = ±J± ,

[J+ , J− ] = 2J3 ,

J± = J1 ± iJ2 ,

†

J+ = J− .

(2.8)

The squared angular momentum J 2 = J12 + J22 + J32 = J− J+ + J3 (J3 + 1) = J+ J− + J3 (J3 − 1)

(2.9)

commutes with each Ji , and J3 is, by convention, taken with J 2 as a pair of commuting Hermitian operators to be diagonalized. Examples of matrices satisfying relations (2.7) are provided by Ji → σi /2 [the 2 × 2 Hermitian Pauli matrices defined in (2.1)] and Ji → Mi [the 3 × 3 matrices

defined in (2.4)], these latter matrices being equivalent to those obtained from the matrices of the orbital angular momentum operators for l = 1. One could determine all Hermitian matrices solving (2.7) and (2.8) by using only matrix theory, but it is customary in quantum mechanics to formulate the problem using Hilbert space concepts appropriate to that theory. Thus, one takes the viewpoint that the Ji are linear Hermitian operators with an action defined in a separable Hilbert space H such that Ji : H → H. One then seeks to decompose the Hilbert space into a direct sum of subspaces that are irreducible with respect to this action; that is, subspaces that cannot be further decomposed as a direct sum of subspaces that all the Ji leave invariant (map vectors in the space into vectors in the space). In this section, the solution of this fundamental problem for angular momentum theory is given. These results set the notation and phase conventions for all of angular momentum theory, in all of its varied realizations, and the relations are therefore sometimes referred to as standard. The method most often used to solve the posed problem is called the method of highest weights. The solution of this problem is among the most important in quantum theory because of its generality and applicability to a wide range of problems. The space H can be written as a direct sum  H= ⊕ n jHj , j=0, 12 ,1,...

each H j ⊥ H j  ,

j = j  ,

(2.10)

Angular Momentum Theory

{| jm | m = − j, − j + 1, . . . , j} . jm  | jm = δm  ,m .

(2.11) (2.12)

Operator in H corresponding to a rotation by angle ψ about direction nˆ in R3 : TU(ψ,n) ˆ · J) , ˆ = exp(−iψ n nˆ · nˆ = n 21 + n 22 + n 23 = 1 , nˆ · J = n 1 J1 + n 2 J2 + n 3 J3 , U(ψ, n) ˆ = exp(−iψ nˆ · σ/2)   = σ0 cos 12 ψ − i(nˆ · σ) sin 12 ψ
     =

  (−in 1 − n 2 ) sin 12 ψ     , cos 12 ψ + in 3 sin 12 ψ

− in 3 sin 12 ψ   (−in 1 + n 2 ) sin 12 ψ

cos

1 2ψ

0 ≤ ψ ≤ 2π,

(2.15)

where σ0 denotes the 2 ×2 unit matrix. Action of TU(ψ,n) ˆ on H j :  j Dm  m (U )| jm  , TU | jm =

(2.16)

m

j

in which U = U(ψ, n) ˆ and Dm  m (U ) denotes a homogeneous polynomial of degree 2 j defined on the elements u ij = Uij (ψ, n) ˆ in row i and column j of the matrix U(ψ, n) ˆ given by (2.15). The explicit form of this polynomial is j

Dm  m (U )

Simultaneous eigenvectors:

1

J 2 | jm = j( j + 1)| jm ,

J3 | jm = m| jm . (2.13)

Action of angular momentum operators: J+ | jm = [( j − m)( j + m + 1)]1/2 | jm + 1 , (2.14)

Defining properties of highest weight vector: J+ | jj = 0 ,



( j + m)! (2 j )!( j − m)!

1/2

j−m

J−

| jj .

Necessary property of lowest weight vector: J− | j, − j = 0 ,

(2.17)

α11 α21

α12 α22

j + m j − m .

j +m j −m

J3 | jj = j| jj .

Generation of general vector from highest weight: | jm =

= [( j + m)!( j − m)!( j + m  )!( j − m  )!] 2  (u 11 )α11 (u 12 )α12 (u 21 )α21 (u 22 )α22 . × α11 !α12 !α21 !α22 ! α

The notation α symbolizes a 2 × 2 array of nonnegative integers with certain constraints:

J− | jm = [( j + m)( j − m + 1)]1/2 | jm − 1 .

J3 | j, − j = − j | j, − j .

17

In this array the αij are nonnegative integers subject to the row and column constraints (sums) indicated by the (nonnegative) integers j ± m, j ± m  . Explicitly, α11 + α12 = j + m  ,

α21 + α22 = j − m  ,

α11 + α21 = j + m ,

α12 + α22 = j − m .

The summation is over all such arrays. Any one of the αij may serve as a single summation index if one wishes to

Part A 2.2

in which H j denotes a vector space of dimension 2 j + 1 that is invariant and irreducible under the action of the set of operators Ji , i = 1, 2, 3, and where the direct sum is over all half integers j = 0, 12 , 1, . . . . There may be multiple occurrences, n j in number, of the same space H j for given j, or no such space, n j = 0, in the direct sum. Abstractly, in so far as angular momentum properties are concerned, each repeated space H j is identical. Such spaces may, however, be distinguished by their properties with respect to other physical observables, but not by the angular action of momentum operators themselves. The result, (2.10), applies to any physical system, no matter how complex, in which rotational symmetry, hence SU(2) symmetry, is present, even in situations of higher symmetry where SU(2) is a subgroup. Indeed, the resolution of the terms in (2.10) for various physical systems constitutes “spectroscopy” in the broadest sense. The characterization of the space H j with respect to angular momentum properties is given by the following results, where basis vectors are denoted in the Dirac braket notation. Orthonormal basis:

2.2 Abstract Angular Momentum

18

Part A

Mathematical Methods

eliminate the redundancy inherent in the square-array notation. The form (2.17) is very useful for obtaining symmetry relations for these polynomials (Sect. 2.3.6). Unitary property on H: TU Ψ |TU Ψ = Ψ |Ψ ,

denotes the element in row j − m  + 1 and column j − m + 1. Then, dimension of D j (U ) = 2 j + 1 and D j (U )D j (U  ) = D j (UU  ) , U ∈ SU(2) , U  ∈ SU(2) , (D j (U ))† = (D j (U ))−1 = D j (U † ) .

each Ψ ∈ H .

Part A 2.3

Kronecker (direct) product representation: Irreducible unitary matrix representation of SU(2): j

(D j (U )) j−m  +1, j−m+1 = Dm  m (U ) , m  = j, j − 1, . . . , − j ;

m = j, j − 1, . . . , − j , (2.18)

D j1 (U ) × D j2 (U ) is a (2 j1 + 1)(2 j2 + 1) dimensional reducible representation of SU(2). One can also effect the reduction of this representation into irreducible ones by abstract methods. The results are given in Sect. 2.7.

2.3 Representation Functions The R ∈ SO(3, R) corresponding to this U in the twoto-one homomorphism given by (2.2) is:

2.3.1 Parametrizations of the Groups SU(2) and SO(3,R) The irreducible representations of the quantal rotation group, SU(2), are among the most important quantities in all of angular momentum theory: These are the unitary matrices of dimension 2 j + 1, denoted by D j (U ), where this notation is used to signify that the j elements of this matrix, denoted Dm  m (U ), are functions of the elements u ij of the 2 × 2 unitary unimodular matrix U ∈ SU(2). It has become standard to enumerate the rows and columns of these matrices in the order j, j − 1, . . . , − j as read from top to bottom down the rows and from left to right across the columns [see also (2.18)]. These matrices may be presented in a variety of parametrizations, all of which are useful. In order to make comparisons between the group SO(3, R) and the group SU(2), it is most useful to parametrize these groups so that they are related according to the two-to-one homomorphism given by (2.2). The general parametrization of the group SU(2) is given in terms of the Euler–Rodrigues parameters corresponding to points belonging to the surface of the unit sphere S3 in R4 , α02 + α12 + α22 + α32 = 1 .

(2.19)

Each U ∈ SU(2) can be written in the form:
U(α0 , α) =

α0 − iα3 −iα1 − α2 −iα1 + α2 α0 + iα3

= α0 σ0 − iα · σ .



(2.20)

R(α0 , α) =

  2 2 2 2 2α1 α3 + 2α0 α2  α0 + α1 − α2 − α3 2α1 α2 − 2α0 α3    2α1 α2 + 2α0 α3 α02 + α22 − α32 − α12 2α2 α3 − 2α0 α1 .   2α1 α3 − 2α0 α2 2α2 α3 + 2α0 α1 α02 + α32 − α12 − α22 (2.21)

The procedure of parametrization is implemented uniformly by first parametrizing the points on the unit sphere S3 so as to cover the points in S3 exactly once, thus obtaining a parametrization of each U ∈ SU(2). Equation (2.21) is then used to obtain the corresponding parametrization of each R ∈ SO(3, R), where one notes that R(−α0 , −α) = R(α0 , α). Because of this twoto-one correspondence ±U → R, the domain of the parameters that cover the unit sphere S3 exactly once will cover the group SO(3, R) exactly twice. This is taken into account uniformly by redefining the domain for SO(3, R) so as to cover only the upper hemisphere (α0 ≥ 0) of S3 . In the active viewpoint (reference frame fixed with points being transformed into new points), an arbitrary vector x = col(x1 , x2 ,x3 ) ∈ R3 is transformed to the new vector x = col x1 , x2 , x3 by the rule x = Rx, or, equivalently, in terms of the Cartan matrix: X  = UXU † . In the passive viewpoint, the basic inertial reference system, which is taken to be a righthanded triad of unit vectors (ˆe1 , eˆ 2 ,eˆ 3 ), is transformed  by R to a new right-handed triad fˆ1 , fˆ2 , fˆ3 by the

Angular Momentum Theory

2.3 Representation Functions

19

Euler angle parametrization:

rule fˆj =



Rij eˆ i ,

U(αβγ) = e−iασ3 /2 e−iβσ2 /2 e−iγσ3 /2   

i = 1, 2, 3 ,

i

x1 eˆ 1 + x2 eˆ 2 + x3 eˆ 3 = x1 fˆ1 + x2 fˆ2 + x3 fˆ3 , so that x = R T x. Rotation about direction nˆ ∈ S2 by positive angle ψ (right-hand rule):

1 1 (α0 , α) = cos ψ, nˆ sin ψ , 0 ≤ ψ ≤ 2π , 2 2

1 U(ψ, n) = exp −i ψ n · σ = ˆ ˆ 2
 cos 12 ψ − in 3 sin 12 ψ (−in 1 − n 2 ) sin 12 ψ , (−in 1 + n 2 ) sin 12 ψ cos 12 ψ + in 3 sin 12 ψ R(ψ, n) ˆ = exp(−iψ nˆ · M) ,

0≤ψ ≤π

= I3 − i sin ψ(nˆ · M) − (nˆ · M) (1 − cos ψ)   R11 R12 R13   =  R21 R22 R23  , 2

R31 R32 R33   R11 = n 21 + 1 − n 21 cos ψ , R21 = n 1 n 2 (1 − cos ψ) + n 3 sin ψ , R31 = n 1 n 3 (1 − cos ψ) − n 2 sin ψ , R12 = n 1 n 2 (1 − cos ψ) − n 3 sin ψ ,   R22 = n 22 + 1 − n 22 cos ψ , R32 = n 2 n 3 (1 − cos ψ) + n 1 sin ψ , R13 = n 1 n 3 (1 − cos ψ) + n 2 sin ψ , R23 = n 2 n 3 (1 − cos ψ) − n 1 sin ψ ,   R33 = n 23 + 1 − n 23 cos ψ . The unit vector nˆ ∈ S2 can be further parametrized in terms of the usual spherical polar coordinates: nˆ = (sin θ cos φ, sin θ sin φ, cos θ) , 0 ≤ θ ≤ π , 0 ≤ φ < 2π .

0 ≤ α < 2π ,

0≤β≤π

or 2π ≤ β ≤ 3π ,

Part A 2.3

so that eˆ i · fˆj = Rij . In this viewpoint, the coordinates of one and the same point P undergo a redescription under the change of frame. If the coordinates of P are (ˆe1 ,eˆ 2 , eˆ 3 ) and    (x1 , x2, x3 ) relative to the frame x1 , x2 , x3 relative to the frame fˆ1 , fˆ2 , fˆ3 , then



e−iα/2 cos 12 β e−iγ/2 − e−iα/2 sin 12 β eiγ/2 ,   = eiα/2 sin 12 β e−iγ/2 eiα/2 cos 12 β eiγ/2

0 ≤ γ < 2π , U(α, β + 2π, γ) = −U(αβγ) ; R(αβγ) = e−iαM3 e−iβM2 e−iγM3    cos α − sin α 0 cos β 0 sin β    =  sin α cos α 0   0 1 0  0 0 1 − sin β 0 cos β   cos γ − sin γ 0   ×  sin γ cos γ 0  0 0 1 



 − sin α cos γ  − sin α sin γ   =  sin α cos β cos γ − sin α cos β sin γ sin α sin β   + cos α sin γ + cos α cos γ 

       

cos α cos β cos γ − cos α cos β sin γ cos α sin β

− sin β cos γ

0 ≤ α < 2π ,

sin β sin γ

0≤β≤π ,

cos β

0 ≤ γ < 2π .

This matrix corresponds to the sequence of frame rotations given by rotate by γ about eˆ 3 = (0, 0, 1) , rotate by β about eˆ 2 = (0, 1, 0) , rotate by α about eˆ 3 = (0, 0, 1) . Equivalently, it corresponds to the sequence of frame rotations given by rotate by α about nˆ 1 = (0, 0, 1) , rotate by β about nˆ 2 = (− sin α, cos α, 0) , rotate by γ about nˆ 3 = (cos α sin β, sin α sin β, cos β) . This latter sequence of rotations is  depicted in Fig. 2.1 in obtaining the frame fˆ1 , fˆ2 , fˆ3 from (ˆe1 , eˆ 2 , eˆ 3 ). The four complex numbers (a, b, c, d) = (α0 + iα3 , iα1 − α2 , iα1 + α2 , α0 − iα3 )

20

Part A

Mathematical Methods

eˆ 3 = nˆ 1 nˆ 3 = fˆ3 β fˆ2

Part A 2.3

γ

eˆ 1

The (ψ, n) ˆ parameters:

α α

β

Quaternionic multiplication rule for points on the sphere S3 :      α0 , α (α0 , α) = α0 , α , α0 = α0 α0 − α · α , α = α0 α + α0 α + α × α ;     D j α0 , α D j (α0 , α) = D j α0 , α . 1 1 α0 = cos ψ , α = nˆ sin ψ . 2 2 Euler angle parametrization:

nˆ 2

eˆ 2 γ



Dm  m (αβγ) = e−im α dm  m (β) e−imγ , j

fˆ1

dm  m (β) = jm  | e−iβJ2 | jm j

Fig. 2.1 Euler angles. The three Euler angles (αβγ ) are defined by a sequence of three rotations. Reprinted with the permission of Cambridge University Press, after [2.1]

are called the Cayley–Klein parameters, whereas the four real numbers (α0 , α) defining a point on the surface of the unit sphere in four-space, S3 , are known as the Euler–Rodrigues parameters. The three ratios αi /α0 form the homogeneous or symmetric Euler parameters.

2.3.2 Explicit Forms of Representation Functions The general form of the representation functions is given in its most basic and symmetric form in (2.17). This form applies to every parametrization, it being necessary only to introduce the explicit parametrizations of U ∈ SU(2) or R ∈ SO(3, R) given in Sect. 2.3.1 to obtain the explicit results given in this section. A choice is also made for the single independent summation parameter in the α-array. The notation for functions is abused by writing D j (ω) = D j (U(ω)) , ω = set of parameters of U ∈ SU(2) . ! " Euler–Rodrigues representation (α0 , α) ∈ S3 :

1

= [( j + m  )!( j − m  )!( j + m)!( j − m)!] 2  2 j+m−m  −2s  1  (−1)m −m+s cos 2 β × ( j + m − s)!s!(m  − m + s)! s  m  −m+2s sin 12 β × . ( j − m  − s)!

1

Dm  m (Z) = [( j + m  )!( j − m  )!( j + m)!( j − m)!] 2 j

×

2  

(z ij )αij /(αij )! ,

(2.24)

i, j=1

D j (Z  )D j (Z) = D j (Z  Z) .

1

= [( j + m )!( j − m  )!( j + m)!( j − m)!] 2  (α0 − iα3 ) j+m−s (−iα1 − α2 )m  −m+s × ( j + m − s)!(m  − m + s)! s 

(−iα1 + α2 )s (α0 + iα3 ) j−m −s . s!( j − m  − s)!

(2.23)

Explicit matrices:   1 1 β − sin β cos 1  2 2  d 2 (β) =  1 , 1 sin β cos β 2 2  1 + cos β − sin β 1 − cos β  √   2 2 2  sin β  − sin β   cos β √ d 1 (β) =  √ .   2 2  1 − cos β sin β 1 + cos β  √ 2 2 2 Formal polynomial form (z ij are indeterminates):

α

j Dm  m (α0 , α) 

×

j

(2.22)

Boson operator form: j j Put ai = z ij (i, j = 1, 2) in (2.24). Let a¯i denote the Hermitian conjugate boson so that # $ # $ # $ j j j alk , ai = 0 , a¯lk , a¯i = 0 , a¯lk , ai = δk j δli .

Angular Momentum Theory

Then the boson polynomials are orthogonal in the boson inner product: j

¯    0 | Dµ µ ( A)D m  m (A) | 0 = (2 j)!δ j j δµ m δµm . j

Inner (scalar) product:  (Ψ, Φ) = dΩ Ψ ∗ (x)Φ(x) ,

 j j dm  m (β) = (−1)m −m d−m  ,−m (β)  j j = (−1)m −m dmm  (β) = dmm  (−β) .



1 (l − m)! 2 m Pl (cos β) (l + m)!

1 (l + m)! 2 −m Pl (cos β) . = (l − m)!

0≤θ ≤π ,

0 ≤ φ < 2π ,

0≤χ ≤π ,

dω = dφ sin θ , dθ = invariant surface measure for S2 ; 2π π dφ dθ sin θ 0

1 2

1

l dm0 (β) = (−1)m [(l + m)!(l − m)!] 2   (2m)!  sin β m (m+1/2)   Cl−m × (cos β) , m! 2 m≥0.

Solutions of Laplace’s equation in R4 (Sect. 2.5): ∇42 Dm  m (x0 , x) = 0 ,

(x0 , x) ∈ R4 ,

3  ∂2 = . ∂xµ2

=

2π 2 δ jj  δm  µ δmµ . 2j +1

Coordinates (ψ, n) ˆ for S3 :

ψ ψ (α0 , α) = cos , nˆ sin , 2 2 0 ≤ ψ ≤ 2π ,

nˆ · nˆ = 1 ,

dΩ = dS(n) ˆ sin2

ψ dψ , 2 2

dS(n) ˆ = dω for nˆ = (sin θ cos φ, sin θ sin φ, cos θ) , 

2π dS(n) ˆ

µ=0

Replace the Euler–Rodrigues parameters (α0 , α) in (2.22) by an arbitrary point (x0 , x) ∈ R4 .

j

0

1

Gegenbauer polynomials:

j∗

dχ sin2 χ Dm  m (α0 , α)Dµ µ (α0 , α)

×

l eimα dm0 (β)

2l + 1 2 l∗ Dm0 (αβγ) , 4π ∗ Ylm (βα) = (−1)m Yl,−m (βα) .

∇42

(cos χ, cos φ sin θ sin χ, sin φ sin θ sin χ, cos θ sin χ) ,

π

=

j

(α0 , α) =

0

Spherical harmonics:



Spherical polar coordinate for S3 :

dΩ = dω sin χ dχ ,

Dlm0 (β) = (−1)m

2l + 1 4π

S3

2

Legendre polynomials:



dΩ = 2π 2 .

dψ 2



ψ 2 j∗ j sin Dm  m (ψ, n)D ˆ µ µ (ψ, n) ˆ 2

0

=

2π 2 2j +1

δ jj  δm  µ δmµ ,

Part A 2.3



1

1 m−m ( j + m)!( j − m)! 2 j sin β dm  m (β) = ( j + m  )!( j − m  )! 2

m  +m  ,m+m  ) 1 × cos β P (m−m (cos β) , j−m 2

Ylm (βα) =

dΩ = invariant surface measure for S3 ,





21

2.3.4 Orthogonality Properties

2.3.3 Relations to Special Functions Jacobi polynomials (see Sect. 2.1.2):

2.3 Representation Functions

22

Part A

Mathematical Methods

Part A 2.3

Euler angles for S3 (SU(2)): 1 β 1 β (α0 , α) = cos cos (γ + α), sin sin (γ − α), 2 2 2 2

1 β 1 β sin cos (γ − α), cos sin (γ + α) , 2 2 2 2 1 dΩ = dα dγ sin β dβ , (2.25) 8 2π 2π π 1 j∗ j dα dγ dβ sin βDm  m (αβγ)Dµ µ (αβγ) 8 0

+

1 8

0

2π

0

2π dα

0

3π

0

j

j∗

dβ sin βDm  m (αβγ)Dµ µ (αβγ)

dγ 2π

2π 2 δ jj  δm  µ δmµ . = 2j +1

(2.26)

Euler angles for hemisphere of S3 (SO(3, R); j  and j both integral): 2π

2π dα

0

π

0

j∗

j

dβ sin βDm  m (αβγ)Dµ µ (αβγ)

dγ 0

8π 2 δ jj  δm  µ δmµ . = 2j +1

(2.27)

Formal polynomials (2.24):  j j Dm  m , Dµ µ = (2 j )!δ jj  δm  µ δmµ , with inner product ∂  ∗ P  (Z)| Z=0 , (P, P ) = P ∂Z ∂  is the complex conjugate polynomial P ∗ where P ∗ ∂Z of P in which each z ij is replaced by ∂z∂ij . Boson polynomials: & % ' j & j Dm  m &Dµ µ = (2 j )!δ jj  δm  µ δmµ ,   with inner product P|P  = 0|P ∗ A¯ P  (A)|0 .

2.3.5 Recurrence Relations Many useful relations between the representation functions may be derived as special cases of general relations between these functions and the WCG-coefficients given in Sect. 2.7.1. The simplest of these are obtained from the Kronecker reduction 1

D j × D 2 = D j+1/2 ⊕ D j−1/2 .

Such relations are usually presented in terms of the Euler angle realization of U, leading to the following relations j between the functions dm  ,m (β): 1 1 j+1/2 β dm  −1/2,m−1/2 (β) ( j − m + 1) 2 cos 2 1 1 j+1/2 β dm  −1/2,m+1/2 (β) + ( j + m + 1) 2 sin 2 1

j

1

j

= ( j − m  + 1) 2 dm  m (β) , 1 1 j+1/2 β dm  +1/2,m−1/2 (β) − ( j − m + 1) 2 sin 2 1 1 j+1/2 2 β dm  +1/2,m+1/2 (β) + ( j + m + 1) cos 2 = ( j + m  + 1) 2 dm  m (β) , 1 1 j−1/2 β dm  −1/2,m−1/2 (β) ( j + m) 2 cos 2 1 1 j−1/2 2 β dm  −1/2,m+1/2 (β) − ( j − m) sin 2 1

j

1

j

= ( j + m  ) 2 dm  m (β) , 1 1 j−1/2 2 β dm  +1/2,m−1/2 (β) ( j + m) sin 2 1 1 j−1/2 β dm  +1/2,m+1/2 (β) + ( j − m) 2 cos 2 = ( j − m  ) 2 dm  m (β) . Two useful relations implied by the above are: 1

j

[( j − m)( j + m + 1)] 2 sin β dm  ,m+1 (β) 1

j

+ [( j + m)( j − m + 1)] 2 sin β dm  ,m−1 (β) = 2(m cos β − m  )dm  m (β) , j

1

j

[( j + m)( j − m + 1)] 2 dm  ,m−1 (β) "1 j ! + ( j + m  )( j − m  + 1) 2 dm  −1,m (β) 1 j  β dm  m (β) . = (m − m ) cot 2 By considering D j × D1 = D j+1 ⊕ D j ⊕ D j−1 , one can also readily derive the matrix elements of the direction cosines specifying the orientation of the body  fixed frame fˆ1 , fˆ2 , fˆ3 of a symmetric rotor relative to

Angular Momentum Theory

the inertial frame (ˆe1 , eˆ 2 , eˆ 3 ): 1  2 j +1 2 j λµ,ν Ψm,m  = 2 j + 1  j

j1 j 

j1 j 

j

× Cmµm+µ Cm  νm  +ν Ψm+µ,m  +ν ,

2.3.6 Symmetry Relations Symmetry relations for the representation functions j Dm  m (Z) defined by (2.24) are associated with the action of a finite group G on the set M(2, 2) of complex 2 × 2 matrices: g : M(2, 2) → M(2, 2), g ∈ G. Equivalently, if Z ∈ M(2, 2) is parametrized by a set Ω of parameters ω ∈ Ω (parameter space), then g may be taken to act directly in the parameter space g : Ω → Ω. The action, denoted
, of a group G = {e, g, g , . . . } (e = identity) on a set X = {x, x  , . . . } must satisfy the rules g : X → X e
x = x, all x ∈ X , g
(g
x) = (g g)
x, all g , g ∈ G, all x ∈ X . (2.28)

Using · to denote the action of G on M(2, 2) and
to denote the action of G on Ω, one has the relation: (g · Z)(ω) = Z(g−1
ω) . Only finite subgroups G of the unitary group U(2) (group of 2 × 2 unitary matrices) are considered here: G ⊂ U(2). Generally, when G acts on M(2, 2), it effects a( unitary linear transformation of the set of functions j ) Dm  m ( j fixed) defined over Z ∈ M(2, 2). For certain groups G, for some elements of G, a single function ( or j j ) Dµ µ ∈ Dm  m occurs in the transformation, so that  j j g · Dm  m (Z) = Dm  m (g−1
Z) j

= gm  m Dµ µ (Z) , 

(2.29)

(µ µ) ∈ {(λ m  , λm), (λm, λ m  )|λ = ±1, λ = ±1} ,

23

where gm  m is a complex number of unit modulus. Rej lation (2.29) is called a symmetry relation of Dm  m with respect to g. Usually not all elements in G correspond to symmetry relations. In a symmetry relation, the action of the group is effectively transferred to the discrete quantum labels themselves: g : m  → µ = m  (g) , m → µ = m(g) .

(2.30)

In terms of a parametrization Ω of M(2, 2), relation (2.29) is written  j j gDm  m (ω) = Dm  m (g−1
ω) j

= gm  m Dµ µ (ω) .

(2.31)

In practice, action symbols such as · and
are often dropped in favor of juxtaposition, when the context is clear. Moreover the set of complex matrices M(2, 2) may be replaced by U(2) or SU(2) whenever the action conditions (2.28) are satisfied. Relations (2.29–2.31) are illustrated below by examples. There are several finite subgroups of interest with various group-subgroup relations between them: 1. Pauli group: P = {σµ , −σµ , iσµ , −iσµ |µ = 0, 1, 2, 3} , |P| = 16 . Each element of this group is an element of U(2). The action of the group P may therefore be defined on the group U(2) by left and right actions as discussed in Sect. 2.4.1. 2. Symmetric groupS4 : S4 = { p| p is a permutation of the four & & Euler– Rodrigues parameters (α0 , α1 , α2 , σ3 )}, & S4 &= 24. Points in S3 are mapped to distinct points in S3 ; hence, one can take Z ∈ SU(2), and define the group action directly from U(α0 , α) in (2.20). It is simpler, however, to define the action of the group directly on the representation functions (2.22). Not all elements of this group define a symmetry in the sense defined by (2.29) (see below). 3. Abelian group T : T = {(t0 , t1 , t2 , t3 )| each tµ = ±1} , |T | = 16 . Group multiplication is component-wise multiplication and the identity is (1,1,1,1). The action of an element of T is defined directly on the

Part A 2.3

where the wave functions are those defined for integral j by (2.37), for half-integral j by (2.36), and  1 ∗ , µ, ν = −1, 0, +1 ; λµ,ν = eˆ µ · fˆν∗ = Dµ,ν √ eˆ +1 = −(ˆe1 + iˆe2 )/ 2 , eˆ 0 = eˆ 3 , √ eˆ −1 = (ˆe1 − iˆe2 )/ 2 ,  √  fˆ+ 1 = − fˆ1 + i fˆ2 / 2 , fˆ0 = fˆ3 ,  √  fˆ− 1 = fˆ1 − i fˆ2 / 2 .

2.3 Representation Functions

24

Part A

Mathematical Methods

Euler–Rodrigues parameters by component-wise multiplication, thus mapping points in S3 to points in S3 ; hence, one can take Z ∈ SU(2). This group is isomorphic to the direct product group S2 × S2 × S2 × S2 , S2 = symmetric group on two distinct objects. 4. Group G:

Part A 2.3

G = R, C, T , K ,

|G| = 32 ,

where R, C, T , K denote the operations of row interchange, column interchange, transposition, and conjugation (see below) of an arbitrary matrix.
 a b Z= c d The notation designates that the enclosed elements generate G. It is impossible to give here all the interrelationships among the groups defined in (1)–(4). Instead, some relaj tions are listed as obtained directly from either Dm  m (Z) j  defined by (2.24) or Dm m (α0 , α) defined by (2.22). The actions of the groups T and G defined in (3) and (4) are fully given. Abelian group T of order 16: Generators: T = t0 , t1 , t2 , t3 ,

t0 = (−1, 1, 1, 1) ,

t1 = (1, −1, 1, 1) ,

t2 = (1, 1, −1, 1) ,

Subgroup H: H = R, C, T = {1, R, C, T , RC = CR, T R = CT ,T C = RT, RCT } with relations in H given by R2 = C 2 = T 2 = 1 , T RC = T CR = RCT = CRT , RT C = CT R = T . Adjoining the idempotent element K to H gives the group G of order 32: G = {H, HK, HKR, HKRK} .

t3 = (1, 1, 1, −1) . Group action:

Symmetry relations:

t · a = (t0 α0 , t1 α1 , t2 α2 , t3 α3 ) , each t = (t0 , t1 , t2 , t3 ) ∈ T , each a = (α0 , α1 , α2 , α3 ) ∈ S3 , (tF)(a) = F(t · a) , j t0 Dm  m j t1 Dm  m j t2 Dm  m j t3 Dm  m

Generator actions:  

row c d a b , =F (RF ) interchange a b c d  

column b a a b , =F (C F ) interchange d c c d  

a c a b , transposition =F (T F ) b d c d  

d −c a b , conjugation =F (K F ) −b a c d

m  −m

j = (−1) D−m−m   j = (−1)m −m Dmm  , j = Dmm  , j = D−m−m  .

Group G of order 32: Generators: G = R, C, T , K ,

j

j

j

j

j

j

RDm  m = D−m  m , C Dm  m = Dm  −m , T Dm  m = Dmm  , 

K Dm  m = (−1)m −m D−m  −m . j

,

j

(2.32) j

These function relations are valid for Dm  m defined over the arbitrary matrix Z defined by (2.24). They are also true for Z = U ∈ SU(2), but now the operations R and C change the sign of the determinant of the matrix Z so that the transformed matrix no longer belongs to SU(2). It does, however, belong to U(2), the group of all 2 × 2 unitary matrices. The special irreducible representation functions of U(2) defined by (2.24), j

Dm  m (U ) ,

U ∈ U(2) ,

Angular Momentum Theory



= (−1)m −m D−m  −m (U ) . j

(1, 2)(α0 , α1 , α2 , α3 ) = (α0 , α2 , α1 , α3 ) . Symmetry relations: 

j



j

(1, 2)Dm  m = (−i)m −m Dmm  . j

(2.33)

0 ≤ χ ≤ 2π ,

where U(α0 , α) ∈ SU(2) is the Euler–Rodrigues parametrization, the actions of R, C, T , and K correspond to the following transformations in parameter space: R : χ → χ + π, (α0 , α1 , α2 , α3 ) → (−α1 , α0 , −α3 , α2 ) , C : χ → χ + π, (α0 , α1 , α2 , α3 ) → (−α1 , α0 , α3 , −α2 ) , T : χ → χ, (α0 , α1 , α2 , α3 ) → (α0 , α1 , −α2 , α3 ) , C : χ → χ, (α0 , α1 , α2 , α3 ) → (α0 , −α1 , α2 , −α3 ) . χ

The new angle = χ + π is to be identified with the corresponding point on the unit circle so that these mappings are always in the parameter space, which is the sphere S3 together with the unit circle for χ. Observe that the following identities hold for functions over SU(2); hence, over U(2): C = Tt1 Tt3 ,

(0, 3)(α0 , α1 , α2 , α3 ) = (α3 , α1 , α2 , α0 ) ,

j

Relations (2.32) and (2.33) are valid in an arbitrary parametrization of U ∈ U(2). In terms of the parametrization U(χ, α0 , α) = eiχ/2 U(α0 , α) ,

Abelian subgroup of S4 : Generators: K = (0, 3), (1, 2) , where (0, 3) and (1, 2) denote transpositions in S4 , |K | = 4. Group action in parameter space:

(0, 3)Dm  m = (−i)m +m D−m−m  ,

j∗

= (det U )2 j Dm  m (U )

T = Tt2 .

Diagonal subgroup Σ of the direct product group P × P (P = Pauli group): Group elements: Σ = {(σ, σ)|σ ∈ P} ,

|Σ| = 16 .

Group action: (σ, σ) : U → σUσ T ,

each σ ∈ P ,

[(σ, σ)F] (U ) = F(σ Uσ) . T

Example: σ = iσ2 : (σ, σ) : (α0 , α1 , α2 , α3 ) → (α0 , −α1 , α2 , −α3 ) , [(σ, σ)F] (α0 , α1 , α2 , α3 ) = F(α0 , −α1 , α2 , −α3 ) , (σ, σ) = t1 t2 on functions over U(2) . The relations presented above barely touch on the interrelations among the finite groups introduced in (1)–(4). Symmetry relations (2.32) and (2.33), howj ever, give the symmetries of the dm  m (β) given in Sect. 2.3.3 in the Euler angle parametrization. In general, it is quite tedious to present the above symmetries in terms of Euler angles, with χ adjoined when necessary, because the Euler angles are not uniquely determined by the points of S3 .

2.4 Group and Lie Algebra Actions The concept of a group acting on a set is fundamental to applications of group theory to physical problems. Because of the unity that this notion brings to angular momentum theory, it is well worth a brief review in a setting in which a matrix group acts on the set of complex matrices. Thus, let G ⊆ G L(n, C)

25

and H ⊆ G L(m, C) denote arbitrary subgroups, respectively, of the general linear groups of n × n and m × m nonsingular complex matrices, and let M(n, m) denote the set of n × m complex matrices. A Z ∈ M(n, m)   matrix has row and column entries z iα , i = 1, 2, . . . , n; α = 1, 2, . . . , m.

Part A 2.4

possess each of the 32 symmetries corresponding to the operations in the group G. [There exist other irreducible representations of U(2), involving det U.] The operation K is closely related to complex conjugation, since for each U ∈ U(2), U = (u ij ), one can write
 u 22 −u 21 ∗ −1 , U = (det U ) −u 12 u 11  j j K Dm  m (U ) = (det U )2 j Dm  m (U ∗ )

2.4 Group and Lie Algebra Actions

26

Part A

Mathematical Methods

Basis set:

2.4.1 Matrix Group Actions Left and right translations of Z ∈ M(n, m) : L g Z = gZ ,

each g ∈ G ,

each Z ∈ M(n, m) ,

Rh Z = Zh ,

each h ∈ H ,

each Z ∈ M(n, m) .

T

Part A 2.4

(T denotes matrix transposition.) Left and right translations commute: 

Z = L g (Rh Z) = Rh (L g Z) , Z ∈ M(n, m) .

each g ∈ G, h ∈ H ,

Equivalent form as a transformation on z

∈ C nm :

z  = (g × h)z ,

DX = DY =

n  i, j=1 m 

xij Dij ,

X = (xij ) ,

yαβ Dαβ ,

α,β=1 m  ∂ Dij = z iα α ∂z j α=1 n  ∂ Dαβ = z iα β ∂z i=1 i

Y = (yαβ ) ,

, .

Commutation rules: [Dij , Dkl ] = δ jk Dil − δil Dk j ,

where × denotes the direct product of g and h; the column matrix z (resp., z  ) is obtained from the columns of Z (resp., Z  ), z α , α = 1, 2, . . . , m, of the n × m matrix Z as successive entries in a single column vector z ∈ C nm . Left and right translations in function space: (Lg f )(Z) = f(gT Z) , each g ∈ G , (Rh f )(Z) = f(Zh) , each h ∈ H ,   where f(Z) = f z iα , and the commuting property holds for all well-defined functions f : Lg (Rh f ) = Rh (Lg f ) .

[Dαβ , Dγ ] = δβγ Dα − δα Dγβ , [Dij , Dαβ ] = 0 , where i, j, k,l =1, 2, . . . , n and α,β,γ,  = 1, 2, . . . , m. The operator sets {Dij } and {Dαβ } are realizations of the Weyl generators of G L(n, C) and G L(m, C), respectively.

2.4.3 Hilbert Spaces Space of polynomials with inner product: (P, P  ) = P ∗ (∂/∂Z)P  (Z)| Z=0 .

2.4.2 Lie Algebra Actions Lie algebra of left and right translations: d  T (D X f )(Z) = i f e−itX Z |t=0 , dt d  Y (D f )(Z) = i f Z e−itY |t=0 ; dt  D X = Tr Z T X∂/∂Z , each X ∈ L(G) ,   DY = Tr Y T Z T ∂/∂Z , each Y ∈ L(H) , L(G) = Lie algebra of G , L(H) = Lie algebra of H .

Bargmann space of entire functions with inner product:  F, F  = F ∗ (Z)F  (Z) dµ(Z) ,     dµ(Z) = π −nm exp − z iα∗ z iα  dxiα dyiα , i,α

i,α

z iα = xiα + iyiα , i = 1, 2, . . . , n ; α = 1, 2, . . . , m . Numerical equality of inner products: (P, P  ) = P, P  .

2.4.4 Relation to Angular Momentum Theory

Linear derivations: DαX+βX  = αD X + βD X  , [D X , D X  ] = D[X,X  ] ,

α, β ∈ C ,

DY obeys these same rules. Commuting property of left and right derivations: " ! D X , DY = 0 , X ∈ L(G) , Y ∈ L(H ) .

Spinorial Realization of Sects. 2.4.2 and 2.4.3:

G = SU(2) , H = (1) , Z ∈ M(2, 1) , z = col(z 1 , z 2 ) , X = set of 2 × 2 traceless, Hermitian matrices, (RU f )(z) = f(U T z) , Dσi /2 = (z T σi ∂/∂z)/2 ,

Angular Momentum Theory

J± = Dσ1 /2 ± iDσ2 /2 , J3 = Dσ3 /2 , J+ = z 1 ∂/∂z 2 , J− = z 2 ∂/∂z 1 , J3 = (1/2)(z 1 ∂/∂z 1 − z 2 ∂/∂z 2 ) , (P, P  ) = P ∗ (∂/∂z 1 , ∂/∂z 2 )P(z 1 , z 2 )|z1 =z2 =0 . Orthonormal basis:

Standard action:

Mutual commutativity of Lie algebras: [Mi , K j ] = 0 ,

(P, P  ) = P ∗ (Z)P  (∂/∂Z)| Z=0 , Orthogonal basis (2.24): j

j = 0, 1/2, 1, 3/2, . . . ,

m = j, j − 1, . . . , − j ; 1 2

J± P jm (z) = [( j ∓ m)( j ± m + 1)] P j,m±1 (z) . Group transformation:  j (RU P jm )(z) = Dm  m (U )P jm  (z) ,

m  = j, j − 1, . . . , − j ;  j j Dmm  , Dµµ = (2 j )!δ jj  δmµ δm  µ . Equality of Casimir operators:

m

where the representation functions are given by (2.17). The 2-Spinorial Realization of Sects. 2.4.2 and 2.4.3:

G = H = SU(2) , ! " Z = z1 z2 ,

i, j = 1, 2, 3 .

Inner product:

Dmm  (Z) ,

J 2 P jm (z) = j( j + 1)P jm (z) , J3 P jm (z) = m P jm (z) ,

M2 = K 2 = M12 + M22 + M32 = K 12 + K 22 + K 32 . Standard actions: j

j

j

j

M3 Dmm  (Z) = m Dmm  (Z) , K 3 Dmm  (Z) = m  Dmm  (Z) , j

X = Y = set of 2 × 2 traceless, Hermitian matrices , (RU f )(Z) = f(U T Z) , U, V ∈ SU(2) ,

(LV f )(Z) = f(ZV ) ,

j

1

j

1

K ± Dmm  (Z) = [( j ∓ m  )( j ± m  + 1)] 2 j

j

× Dm,m  ±1 (Z) .

Dσi /2 = Tr(σi Z T ∂/∂Z)/2 .

Special values:

M± = Dσ1 /2 ± iDσ2 /2 , K± = D M+ =

2 

± iD

σ2 /2

z α1 ∂/∂z α2 ,

M3 = Dσ3 /2 ,

,

K3 = D M− =

α=1

2 

σ3 /2

α=1 2

α=1

K+ =

z i1 ∂/∂z i2 ,

i=1

K− =

2 

z i2 ∂/∂z i1 ,

i=1

 1  1 z i ∂/∂z i1 − z i2 ∂/∂z i2 . K3 = 2 2

i=1

,

z α2 ∂/∂z α1 ,

 1  α z 1 ∂/∂z α1 − z α2 ∂/∂z α2 , M3 = 2 2 

j

M± Dmm  (Z) = [( j ∓ m)( j ± m +1)] 2 Dm±1,m  (Z) ,

Dσi /2 = Tr(Z T σi ∂/∂Z)/2 ,

σ1 /2

j

M2 Dmm  (Z) = K 2 Dmm  (Z) = j( j + 1)Dmm  (Z) ,

Z ∈ M(2, 2) ,   z α = col z α1 z α2 ,

27

Part A 2.4

1

j+m j−m

P jm (z 1 , z 2 ) = z 1 z 2 /[( j + m)!( j − m)!] 2 , j = 0, 1/2, 1, 3/2, . . . ; m = j, j − 1, . . . , − j .

2.4 Group and Lie Algebra Actions


 1 0 = I2 j+1 = unit matrix , D 0 1 
z1 0 j = δ jm  P jm (z 1 , z 2 ) , Dmm  z2 0
 0 z1 j Dmm  = δ jm P jm  (z 1 , z 2 ) , 0 z2
 z1 0 j j+m j−m Dmm  = δmm  z 1 z 2 , 0 z2  2 j j D jj (Z) = z 11 . j

28

Part A

Mathematical Methods

Symmetry relation: $T #   D j (Z) = D j Z T .

Generating functions: (xT Z y)2 j /(2 j )! =

Part A 2.5

Generation from highest weight:

12 ( j + m  )! ( j + m)! j × Dmm  (Z) = (2 j )!( j − m)! (2 j )!( j − m  )! j−m

× M−

j−m 

K−

j

D jj (Z) .

exp(tx Z y) = T

 j



j

P jm (x)Dmm  (Z)P jm  ( y) ,

mm   2j

t

mm 

j

P jm (x)Dmm  (Z)P jm  ( y) ,

x = col(x1 x2 ) , y = col(y1 y2 ) ,   Z = z iα , i, α = 1, 2 ; all indeterminates .

2.5 Differential Operator Realizations of Angular Momentum Differential operators realizing the standard commutation relations (2.7) and (2.8) can be obtained from the 2-spinorial realizations given in Sect. 2.4.4 by specializing the matrix Z to the appropriate unitary unimodular matrix U ∈ SU(2) and using the chain rule of elementary calculus. Similarly, one obtains the exj plicit functions Dmm  simply by substituting for Z the parametrized U in (2.24). This procedure is used in this section to obtain all the realizations listed. The notations M = (M1 , M2 , M3 ) and K = (K 1 , K 2 , K 3 ) and the associated M± and K ± refer to the differential operators given by the 2-spinorial realization now transformed to the parameters in question. Euler angles with Z = U(αβγ) (Sect. 2.3.1): M3 = i∂/∂α , K 3 = i∂/∂γ ,  1  iα e M+ − e−iα M− 2  ∂ 1  −iγ e K − − eiγ K + = , = 2 ∂β  1  iα e M+ + e−iα M− 2 = − (cot β) M3 + (sin β)−1 K 3 ,  1  −iγ e K − + eiγ K + = (cot β) K 3 − (sin β)−1 M3 , 2 M+ = e−ia [∂/∂β − (cot β) M3 + (sin β)−1 K 3 ] , M− = eiα [−∂/∂β − (cot β) M3 + (sin β)−1 K 3 ] , K + = e−iγ [−∂/∂β + (cot β) K 3 − (sin β)−1 M3 ] , K − = eiγ [∂/∂β + (cot β) K 3 − (sin β)−1 M3 ] . Euler angles with Z = U ∗ (αβγ) [replace i by −i in the above relations]: M3 = −i∂/∂α , K 3 = −i∂/∂γ , (2.34) # $ M± = e±iα ±∂/∂β − (cot β)M3 + (sin β)−1 K 3 , # $ K ± = e±iγ ∓∂/∂β + (cot β)K 3 − (sin β)−1 M3 .

Since D j (U ∗ ) = (D j (U ))∗ , which is denoted these operators have the standard action j∗ on the complex conjugate functions Dmm  (U ). Quaternionic variables. (x0 , x) ∈ R4 :      x0 ,x (x0 , x) = x0 x0 − x · x, x0 x + x0 x + x × x ; 

x0 − ix3 −ix1 − x2 z 11 z 12 = ; Z= z 21 z 22 −ix1 + x2 x0 + ix3 
∂ ∂/∂z 11 ∂/∂z 12 = ∂Z ∂/∂z 21 ∂/∂z 22 
1 ∂/∂x0 + i∂/∂x3 i∂/∂x1 − ∂/∂x2 ; = 2 i∂/∂x1 + ∂/∂x2 ∂/∂x0 − i∂/∂x3   1  1  Mi = Tr Z T σi ∂/∂Z , K i = Tr σi Z T ∂/∂Z . 2 2 D j∗ (U ),

(The form of ∂/∂Z is determined by the requirement (∂/∂z ij )zlk = δil δ jk ; for example, 12 (∂/∂x0 + i∂/∂x3 )(x0 − ix3 ) = 1). Define the six orbital angular momentum operators in R4 by L jk = −i(x j ∂/∂xk − xk ∂/∂x j ) ,

j < k = 0, 1, 2, 3 ,

which may be written as the orbital angular momentum L in R3 together with the three operators A given by L = −ix × ∇ , L 1 = L 23 , L 2 = L 31 , L 3 = L 12 , A = (A1 , A2 , A3 ) = (L 01 , L 02 , L 03 ) . Then, we have the following relations: K 1 = (L 1 − A1 )/2 , K 2 = (L 2 − A2 )/2 , K 3 = (L 3 − A3 )/2 ; M1 = −(L 1 + A1 )/2 , M2 = (L 2 + A2 )/2 , M3 = −(L 3 + A3 )/2 .

Angular Momentum Theory

Commutation rules: [M j , K k ] = 0 , j, k = 1, 2, 3 , M × M = iM , K × K = iK , L × L = iL , A × A = iL , [L j , Ak ] = ie jkl Al ,

µ=0

j

j

K 0 Dmm  (x0 , x) = jDmm  (x0 , x) , j

∇42 Dmm  (x0 , x) = 0 ;

(M1 , −M2 , M3 ) =
    Ri1 K i , Ri2 K i , Ri3 K i , i



Rij =

i

i

 x02 − x · x δij − 2eijk x0 xk + 2xi x j x02 + x · x

,

each (x0 , x) ∈ R4 . The relation Rij = Rij (x0 , x) is a mapping of all points of four-space R4 (except the origin) onto the group of proper, orthogonal matrices; for x02 + x · x = 1, it is just the Euler–Rodrigues parametrization, (2.21). The operators R = (−M1 , M2 , −M3 ) and K = (K 1 , K 2 , K 3 ) have the standard action on (−1) j+m j D−m,m  (x0 , x), so that the orbital angular momentum in 3 R is given by the addition L = R+ K . Thus, one finds:  mm 

jjL

j

Cmm  M (−1) j+m D−mm  (x0 , x)

= A2 j,L R2 j−L YLM (x)C2(L+1) j−L (x 0 /R) ,

1 4π(2 j − L)! 2 L 2j A2 j,L = (2i) (−1) L! . (2 j + L + 1)!

2.6 The Symmetric Rotor and Representation Functions The rigid rotor is an important physical object and its quantum description enters into many physical theories. This description is an application of angular momentum theory with subtleties that need to be made explicit. It is customary to describe the classical  rotor interms of a right-handed triad of unit vectors fˆ1 , fˆ2 , fˆ3 fixed in the rotor and constituting a principal axes system located at the center of mass. The instantaneous orientation of this body-fixed frame relative to a right-handed triad of unit vector (ˆe1 , eˆ 2 , eˆ 3 ) specifying an inertial frame, also located at the center of mass, is then given, say, in terms of Euler angles (one could use for this purpose any parametrization of a proper, orthogonal matrix). For Euler angles, the relationship is  fˆj = Rij (αβγ)ˆei . (2.35) i

The Hamiltonian for the rigid rotor is then of the form H = AP12 + BP22 + CP32 ,

29

where A, B, and C are physical constants related to the reciprocals of the principal moments of inertia, and the angular momenta P j ( j = 1, 2, 3) are the components of the total angular momentum J referred to the body-fixed frame:  Rij (αβγ)Ji , P j = fˆj · J = i

J = eˆ 1 J1 + eˆ 2 J2 + eˆ 3 J3 . For the symmetric rotor (taking A = B), the Hamiltonian can be written in the form H = aP 2 + bP32 . It is in the interpretation of this Hamiltonian for quantum mechanics that the subtleties already enter, since the nature of angular momentum components referred to a moving reference system must be treated correctly. Relation (2.35) shows that the body-fixed axes cannot commute with the components of the total angular momentum J referred to the frame (ˆe1 , eˆ 2 , eˆ 3 ). A position

Part A 2.6

where e jkl = 1 for j, k, l an even permutation of 1, 2, 3; e jkl = −1 for an odd permutation of 1, 2, 3; e jkl = 0, otherwise. The M = (M1 , M2 , M3 ) and K = (K 1 , K 2 , K 3 ) operators have the standard action given in Sect. 2.2 on the j functions Dmm  (x0 , x) defined by (2.22) (Replace α0 by x0 and α by x). Additional relations: 1 K 2 = M2 = − R2 ∇42 + K 02 + K 0 , 4 R2 = x02 + x · x ,  ∂2 ∂2 ∇42 = 2 + ∇ 2 = , ∂xµ2 ∂x0 µ

3 1 1 ∂ T ∂ = xµ ; K 0 = Tr z 2 ∂Z 2 ∂xµ

2.6 The Symmetric Rotor and Representation Functions

30

Part A

Mathematical Methods

Part A 2.6

vector x and the orbital angular momentum L, with components both referred to an inertial frame, satisfy the commutation relations [L j , xk ] = ie jkl xl , and for a rigid body thought of as a collection of point particles rotating together, the same conditions are to be enforced. Relative to the body-fixed frame, the vector x is expressed as   xk eˆ k = ah fˆh , each ah = constant , k

h

xk =



ah Rkh (αβγ) .

h

The direction cosines Rkh = Rkh (αβγ) = eˆ k · fˆh and the physical total angular momentum components referred to an inertial frame must satisfy ! " J j , Rkh = ie jkl Rlh , each h = 1, 2, 3 , in complete analogy to [L j , xk ] = ie jkl xl . The description of the angular momentum associated with a symmetric rigid rotor and the angular momentum states is summarized as follows [compare (2.35)]: Physical total angular momentum J with components referred to (ˆe1 , eˆ 2 , eˆ 3 ): ∂ cos α ∂ ∂ , J1 = i cos α cot β + i sin α − i ∂α ∂β sin β ∂γ ∂ sin α ∂ ∂ , J2 = i sin α cot β − i cos α − i ∂α ∂β sin β ∂γ ∂ J3 = −i . ∂α Physical angular momentum J with components re ferred to fˆ1 , fˆ2 , fˆ3 : ∂ cos γ ∂ ∂ − i sin γ +i , P1 = −i cos γ cot β ∂γ ∂β sin β ∂α ∂ sin γ ∂ ∂ − i cos γ −i , P2 = i sin γ cot β ∂γ ∂β sin β ∂α ∂ P3 = −i . ∂γ Standard commutation of the Ji : [Ji , J j ] = iJk ,

i, j, k cyclic .

Ji can stand to either side:  Rij (αβγ)Ji , Pj = i

Ji =



Ji Rij (αβγ) .

i

The famous Van Vleck factor of −i in the commutation of the Pi : [Pi , P j ] = −iPk ,

i, j, k cyclic .

Mutual commutativity of the J j and Pi : [Pi , J j ] = 0 ,

i, j = 1, 2, 3 .

Same invariant (squared) total angular momentum: P12 + P22 + P32 = J12 + J22 + J32 = J 2

2 ∂ ∂2 ∂2 = − csc2 β + − 2 cos β ∂α∂γ ∂α2 ∂γ 2 2 ∂ ∂ . − 2 − cot β ∂β ∂β Standard actions: j∗

j∗

J 2 Dmm  (αβγ) = j( j + 1)Dmm  (αβγ) , j∗

j∗

J3 Dmm  (αβγ) = m Dmm  (αβγ) , P3 Dmm  (αβγ) = m  Dmm  (αβγ) ; j∗

j∗

1

j∗

J± Dmm  (αβγ) = [( j ∓ m)( j ± m + 1)] 2 j∗

× Dm±1,m  (αβγ) , 1

(P1 − iP2 )Dmm  (αβγ) = [( j − m  )( j + m  + 1)] 2 j∗

j∗

× Dm,m  +1 (αβγ) , 1

(P1 + iP2 )Dmm  (αβγ) = [( j + m  )( j − m  + 1)] 2 j∗

j∗

× Dm,m  −1 (αβγ) . Normalized wave functions: Integral or half-integral j (SU(2) solid body): + , * & & j 2 j + 1 j∗ D  (αβγ) , (2.36) = αβγ && 2π 2 mm mm  with inner product  F|F  = dΩ αβγ |F ∗ αβγ |F  , where dΩ is defined by (2.25) and the integration extends over all α, β, γ given by (2.26). Integral j (collection of “rigid” point particles): , 2 j + 1 j∗ j Ψmm  (αβγ) = D  (αβγ) , (2.37) 8π 2 mm with inner product (Ψ, Ψ  ) =

2π π 2π dα dβ sin β dγ F ∗ (αβγ)F  (αβγ) . 0

0

0

The concept of a solid (impenetrable) body is conceptually distinct from that of a collection of point particles moving collectively together in translation and rotation.

Angular Momentum Theory

2.7 Wigner–Clebsch–Gordan and 3-j Coefficients

31

2.7 Wigner–Clebsch–Gordan and 3-j Coefficients

[ j1 ] × [ j2 ] =

j 1 + j2

⊕[ j]

j=| j1 − j2 |

= [| j1 − j2 |] ⊕ [| j1 − j2 | + 1] ⊕ · · · ⊕ [ j1 + j2 ] . (2.38) 1 2 , 1, . . . }

Given two angular momenta j1 ∈ {0, and j2 ∈ {0, 12 , 1, . . . }, the Clebsch–Gordan (CG) series also expresses the rule of addition of two angular

momenta: j = j1 + j2 , j1 + j2 − 1, . . . , | j1 − j2 | . The integers  j1 j2 j   1 ,  j1 j2 j =   0,

defined by for j1 , j2 , j satisfying the CG-series rule ,

(2.39)

otherwise

are useful in many relations between angular momentum quantities. The notation ( j1 j2 j ) is used to symbolize the CG-series relation between three angular momentum quantum numbers. The representation function and Lie algebra interpretations of the CG-series (2.38) are, respectively:  C(D j1 × D j2 )C T = ⊕ j1 j2 j D j ,  C

j

(j ) (j ) Ji 1 × Ji 2

C = T



(j)

⊕ j1 j2 j Ji

,

j

i = 1, 2, 3 . (j) Ji

The notation with elements

denotes the (2 j + 1) × (2 j + 1) matrix

(j)

Jm  ,m = jm  | Ji | jm ,

m  , m = j, j − 1, . . . , − j . The elements of the real, orthogonal matrix C of dimension (2 j1 + 1)(2 j2 + 1) that effects these reductions are the WCG-coefficients: j j j

(C) jm;m 1 m 2 = Cm11 m2 2 m . The pairs, ( jm) and (m 1 m 2 ), index rows and columns, respectively, of the matrix C: ( jm) : j = j1 + j2 , . . . , | j1 − j2 | , m = j, . . . , − j ; (m 1 m 2 ) : m 1 = j1 , . . . , − j1 ; m 2 = j2 , . . . , − j2 . Sum rule on projection quantum numbers: j j j

Cm11 m2 2 m = 0 ,

for m 1 + m 2 = m .

(2.40)

Clebsch–Gordan series rule on angular momentum quantum numbers: j j j

Cm11 m2 2 m = 0 , for  j1 j2 j = 0 .

(2.41)

Part A 2.7

Wigner–Clebsch–Gordan (WCG) coefficients (also called vector coupling coefficients) enter the theory of angular momentum in several ways: (1) as the coefficients in the real, orthogonal matrix that reduces the Kronecker product of two irreducible representations of the quantal rotation group into a direct sum of irreducibles; (2) as the coupling coefficients for constructing basis states of sharp angular momentum in the tensor product space from basis states of sharp angular momentum spanning the two constituent spaces; (3) as purely combinatoric objects in the expansion of a power of a 3 × 3 determinant; and (4) as coupling coefficients in the algebra of tensor operators. These perspectives are intimately connected, but have a different focus: the first considers the group itself to be primary and views the Lie algebra as the secondary or derived concept; the second considers the Lie algebra and the construction of the vector spaces that carry irreducible representations as primary, and views the representations carried by these spaces as derived quantities; the third is a mathematical construction, at first seeming almost empty of angular momentum concepts, yet the most revealing in showing the symmetry and other properties of the WCG-coefficients; and the fourth is the natural extension of (2) to operators, recognizing that the set of mappings of a vector space into itself is also a vector space. The subject of tensor operator algebra is considered in the next section because of its special importance for physical applications. This section summarizes formulas relating to the first three viewpoints, giving also the explicit mathematical expression of the coefficients in their several forms. Either viewpoint, (1) or (2), may be taken as an interpretation of the Clebsch–Gordan series, which expresses abstractly the reduction of a Kronecker product of matrices (denoted ×) into a direct sum (denoted ⊕) of matrices:

32

Part A

Mathematical Methods

Part A 2.7

In presenting formulas that express relations relating to the conceptual framework described above, it is best to use a notation for a WCG-coefficient giving it as an element of an orthogonal matrix. For the expression of symmetry relations, the 3– j coefficient or 3– j symbol notation is most convenient. The following notations are used here: WCG-coefficient notation:  j j j Cm11 m2 2 m | j1 m 1 ⊗ | j2 m 2 , |( j1 j2 ) jm = m 1 ,m 2

| j1 m 1 ; j2 m 2 = | j1 m 1 ⊗ | j2 m 2 ,

The integers  j1 j2 j ( j3 = j ) are sometimes included in the orthogonality relations (2.43) and (2.45) to incorporate the extended definition (2.41) of the WCGcoefficients.

2.7.1 Kronecker Product Reduction Product form: j

j

Dm1 m (U )Dm2 m (U ) 1 1 2 2  j j j j j2 j 1 2 Cm  ,m  ,m  +m  Cm11 ,m = 2 ,m 1 +m 2

1

1

2

2

1

j

2

1

2

1

2

= δ jj  δmm  .

(2.43)

j j j

j j j

j

j

Cm1 m2  m  Cm11 m2 2 m Dm1 m (U )Dm2 m (U ) 1

2

1

1

2

2

m 1 +m 2 =m  m 1 +m 2 =m

j j j

j

= δ j  j Dm  m (U ) .

j j j 1

jm

j j j

Integral relation:

2



j j j

2 Cm11 m−m C12 = δm 1 m 1 , 1 ,m m  ,m−m  ,m 1

j

j

j

1

j j j

j j j

1 2 Cm−m C12 = δm 2 m 2 . 2 ,m 2 ,m m−m  ,m  ,m 2

(2.44)

2

Orthogonality of 3– j coefficients (symbols):
j1 j2 j3 m1 m2 m3




j1 j2 j3 m 1 m 2 m 3

1

2

2

2π 2 j1 j2 j j j j C    Cm11 m2 2 m , 2 j + 1 m1m2m

=

in any parametrization of U ∈ SU(2) that covers S3 exactly once. Gaunt’s integral:



(2.45) = δ j3 j3 δm 3 m 3 /(2 j3 + 1) ,

  j j j j1 j2 j3 (2 j3 + 1) 1 2 3  m m m m m m 1 2 3 3 1 2 j m 3

= δm 1 m 1 δm 2 m 2 .

j

dΩDm1 m (U )Dm2 m (U )Dm∗ m 1 (U )

1

j

3

j

1

2 = Cm1 ,m  ,m  +m  Dm  +m  ,m (U ) .



Cm11 m2 2 m Cm1 m2  m = δm 1 m 1 δm 2 m 2 ,

m1m2

j

Doubly coupled (reduction) form:

Orthogonality of columns (three forms):



(U ) .

j j j

j j j

(2.42)

m1m2



1 +m 2

Cm11 m2 2 m Dm1 m (U )Dm2 m (U )

m 1 +m 2 =m

Orthogonality of WCG-coefficients: Orthogonality of rows:



2



j j j



2

Singly coupled form:

= (−1) j1 − j2 +m (2 j + 1)−1/2 Cm11 m2 2 m .

j j j j j Cm11 m2 2 m Cm11 m2 2 m 

1

j

The 3– j coefficient notation: 
j1 j2 j m 1 m 2 −m

j

2

× Dm  +m  ,m

( j1 j2 ) j  m  |( j1 j2 ) jm = δ j  j δm  m .



1

j

j j j

Cm11 m2 2 m = j1 m 1 ; j2 m 2 |( j1 j2 ) jm ,

(2.46)

2π

π dα

0

=



∗ sin β dβYlm (βα)Yl1 m 1 (βα)Yl2 m 2 (βα)

0

(2l1 + 1)(2l2 + 1) 4π(2l + 1)

1/2

l1 l2 l l1 l2 l C000 Cm 1 m 2 m  .

Angular Momentum Theory

π sin β dβPl (cos β)Pl1 (cos β)Pl2 (cos β)

2l + 1

2

2

each V ∈ SU(2) ;  j T(U,U ) |( j1 j2 ) jm = Dm  m (U )|( j1 j2 ) jm  ,

.

m



(2l +1)(l1 + l2 − l)!(l1 − l2 + l)!(−l1 + l2 + l)! (l1 + l2 + l + 1)! (−1) L−1 L! , × (L − l1 )!(L − l2 )!(L − l)! 1 L = (l1 + l2 + l) , 2 for l1 + l2 + l even

l1 l2 l C000 =0,

1

each U ∈ SU(2) ,

2

l1 l2 l C000

=

1

m 1 m 2

1

m = j, j − 1, . . . , − j; each U ∈ SU(2) .

2

for l1 + l2 + l odd .

Representation of direct product group SU(2) × SU(2): T(U,V ) T(U  ,V  ) = T(UU  ,VV  ) . Representation of SU(2) as diagonal subgroup of SU(2) × SU(2): T(U,U ) T(U  ,U  ) = T(UU  ,UU  ) , TU = T(U,U ) .

2.7.2 Tensor Product Space Construction Orthonormal basis of H j1 : & ) ( | j1 m 1 & m 1 = j1 , j1 − 1, . . . , − j1 . Orthonormal basis of H j2 : & ( ) | j2 m 2 & m 2 = j2 , j2 − 1, . . . , − j2 . Uncoupled basis of H j1 ⊗ H j2 : & ( | j1 m 1 ⊗ | j2 m 2 &m 1 = j1 , j1 − 1, . . . , − j1 ; ) m 2 = j2 , j2 − 1, . . . , − j2 . Coupled basis of H j1 ⊗ H j2 : & ( |( j1 j2 ) jm & j = j1 + j2 , j1 + j2 − 1, . . . , | j1 − j2 | ; ) m = j, j − 1, . . . , − j ,  j j j Cm11 m2 2 m | j1 m 1 ⊗ | j2 m 2 . |( j1 j2 ) jm = m1m2

Unitary transformations of spaces:  j & Dm1 m (U )& j1 m 1 , TU | j1 m 1 = m 1

1

1

m 1 = j1 , j1 − 1, . . . , − j1 , each U ∈ SU(2) ;  j & TV | j2 m 2 = Dm  m (V )& j2 m 2 , m 2

2

2.7.3 Explicit Forms of WCG-Coefficients Wigner’s form: j j j

Cm11 m2 2 m 1

= δ(m 1 + m 2 , m)(2 j + 1) 2

1 ( j + j1 − j2 )!( j − j1 + j2 )!( j1 + j2 − j )! 2 × ( j + j1 + j2 + 1)!

1 2 ( j + m)!( j − m)! × ( j1 + m 1 )!( j1 − m 1 )!( j2 + m 2 )!( j2 − m 2 )! ×

 (−1) j2 +m 2 +s ( j2 + j + m 1 − s)!( j1 − m 1 + s)! . s!( j − j1 + j2 − s)!( j + m − s)!( j1 − j2 − m + s)! s

Racah’s form: j j j

Cm11 m2 2 m = δ(m 1 + m 2 , m) (2 j + 1)( j1 + j2 − j )! ×

2

m 2 = j2 , j2 − 1, . . . , − j2 ,

× each V ∈ SU(2) ;

2

( j1 + j2 + j + 1)!( j + j1 − j2 )!( j + j2 − j1 )!

1 ( j1 − m 1 )!( j2 − m 2 )!( j − m)!( j + m)! 2 ( j1 + m 1 )!( j2 + m 2 )!  (−1) j1 −m 1 +t ( j1 + m 1 + t)!( j + j2 − m 1 − t)!

×

1

t

t!( j − m − t)!( j1 − m 1 − t)!( j2 − j + m 1 + t)!

.

Part A 2.7

=

 l1 l2 l 2 C000

33

T(U,V ) | j1 m 1 ⊗ | j2 m 2 = TU | j1 m 1 ⊗ TV | j2 m 2  j & - & j = Dm1 m (U )Dm2 m (V )& j1 m 1 ⊗ & j2 m 2 ,

Integral over three Legendre functions:

0

2.7 Wigner–Clebsch–Gordan and 3-j Coefficients

34

Part A

Mathematical Methods

α31 = k231 + k321 ,

Van der Waerden’s form: j j j Cm11 m2 2 m

Part A 2.7

= δ(m 1 + m 2 , m)

1 (2 j + 1)( j1 + j2 − j )!( j + j1 − j2 )!( j + j2 − j1 )! 2 × ( j1 + j2 + j + 1)!

! "1 × ( j1 + m 1 )!( j1 − m 1 )!( j2 + m 2 )!( j2 − m 2 )! 2 ! "1 × ( j + m)!( j − m)! 2  ! × (−1)k k!( j1 + j2 − j − k)!( j1 − m 1 − k)! k

× ( j2 + m 2 − k)!( j − j2 + m 1 + k)! "−1 × ( j − j1 − m 2 + k)! Regge’s formula and its combinatoric structure: (det A)k =



A(α)

α

3 

(aij )αij ,

A = (aij ) ,

i, j=1

(2.47)

where the summation is over all nonnegative integers αij that satisfy the row and column sum constraints (2.17) given by α11 α12 α13 α = α21 α22 α23 α31 α32 α33 k

k

α33 = k123 + k213 ,  φ(K) = kπ = k132 + k213 + k321 . π∈A3

The general multinomial coefficient is the integer defined by
  k = k!/k1 !k2 ! · · · ks ! , k = ki . k1 , k2 , . . . , ks i Relation (2.47) generalizes in the obvious way to an n × n determinant, using the symmetric group Sn and (j) its Sn−1 subgroups Sn−1 , where j denotes that this is the permutation group on the integers 1, 2, . . . , n with j deleted [2.2]. Regge’s formula for the 3– j coefficient is: 
j1 j2 j3 m1 m2 m3  1 2 3    (αij )! = δ(m 1 + m 2 + m 3 , 0) i, j=1

A(α)

, 1 k![(k + 1)!] 2 k = j1 + j2 + j3 , ×

α21 = j1 − m 1 , α22 = j2 − m 2 , α23 = j3 − m 3 , (2.48)

The coefficients A(α) are constrained sums over multinomial coefficients:  A(α) = (−1)φ(K) 
k , × k123 , k132 , k231 , k213 , k312 , k321

α31 = j2 + j3 − j1 , α32 = j3 + j1 − j2 , α33 = j1 + j2 − j3 . Equation (2.49) shows that WCG-coefficients and 3– j coefficients are sums over integers, except for a multiplicative normalization factor. Schwinger’s generating function: exp(t det A) =

 tk  k

where the summation is carried out over all nonnegative integers ki1 i2 i3 such that α11 = k123 + k132 ,

α12 = k231 + k213 ,

α13 = k312 + k321 , α21 = k312 + k213 , α23 = k231 + k132 ,

α22 = k123 + k321 ,

(2.49)

α11 = j1 + m 1 , α12 = j2 + m 2 , α13 = j3 + m 3 ,

k k k

k

α32 = k312 + k132 ,

k!

A(α)

α

 (aij )αij . i, j

The general definition of the p Fq hypergeometric function depending on p numerator parameters, q denominator parameters, and a single variable z is:
 ∞  (a1 )n · · · (a p )n z n a1 . . . , a p , = p Fq ;z (b1 )n · · · (bq )n n! b1 . . . , bq n=0

(a)n = a(a + 1) · · · (a + n − 1) ,

(a)0 = 1 .

Angular Momentum Theory

Such a series is terminating if at least one of the numerator parameters is a negative integer (and all other factors are well-defined). Both WCG-coefficients and Racah 6– j coefficients relate to special series of this type, evaluated at z = 1. For WCG-coefficients, we have for α + β = γ :

= [(2c + 1)(a + α)!(a − α)!(b + β)!(b − β)!(c + γ)! 1

× (c − γ)!] 2 (−1)a+b+γ +δ1 ∆(abc) 
1 − δ1 , 2 − δ1 , 3 − δ1 3 F2 ;1 δ2 − δ1 + 1, δ3 − δ1 + 1 × , (δ2 − δ1 )!(δ3 − δ1 )!(δ1 − 1 )!(δ1 − 2 )!(δ1 − 3 )! δ1 = min(a + α + b + β, b − β + c + γ, a + α + c + γ), (δ1 , δ2 , δ3 ) = any permutation of (a + α + b + β, b − β +c + γ, a + α + c + γ), after δ1 is fixed, (1 , 2 , 3 ) = any permutation of (a + α, b + α + γ, c + γ). A somewhat better form can be found in [2.2]. The quantity ∆(abc) =

(a+b−c)!(a−b+c)!(−a+b+c)! (a+b+c+1)!

1 2

.

is called a triangle coefficient. All 72 Regge symmetries are consequences of known properties of the 3 F2 hypergeometric series.

2.7.4 Symmetries of WCG-Coefficients in 3-j Symbol Form There are 72 known symmetries (up to sign changes) of the 3– j coefficient. There are at least four ways of verifying these symmetries: (1) directly from the van der Waerden form of the coefficients; (2) directly from Regge’s generating function; (3) from the known symmetries of the 3 F2 hypergeometric series; and (4) directly from the symmetries of j the representation functions Dmm  (U ). The set of 72 symmetries is succinctly expressed in terms of the coefficient A(α) defined in Sect. 2.8.3 with αij entries given by (2.48) and (2.49) in which m 1 + m 2 + m 3 = 0:  j2 + m 2 j3 + m 3 j1 + m 1   A  j1 − m 1 j2 − m 2 j3 − m 3  . j2 + j3 − j1 j3 + j1 − j2 j1 + j2 − j3 

This coefficient has determinantal symmetry; that is, it is invariant under even permutations of its rows or columns and under transposition, and is multiplied by the factor (−1) j1 + j2 + j3 under odd permutations of its rows or columns. These 72 determinantal operations may be generated from the three operations C12 , C13 , T consisting of interchange of columns 1 and 2, interchange of columns 1 and 3, and transposition, since the first two operations generate the symmetric group S3 of permutations of columns, and the symmetric group S3 of permutations of rows is then given by TS3 T . The transposition T itself generates a group {e, T } isomorphic to the symmetric group S2 . Thus, the 72 element determinantal group is the direct product group S3 × S3 × {e, T }. The three relations between 3– j coefficients corresponding to the generators C12 , C13 , T are

 j1 j2 j3 j2 j1 j3 j1 + j2 + j3 = (−1) , m1 m2 m3 m2 m1 m3

 j1 j2 j3 j3 j2 j1 j1 + j2 + j3 = (−1) , m1 m2 m3 m3 m2 m1 
j1 j2 j3 m1 m2 m3 
j1 + j2 +m 1 +m 2 j1 + j2 −m 1 −m 2 j 3 2 2 . = j1 − j2 +m j1 − j2 −m 1 +m 2 1 −m 2 j2 − j1 2 2 All 72 relations among 3– j coefficients can be obtained from these 3. The 12 “classical” symmetries of the 3– j symbol 
a b c α β γ are expressed by: 1. even permutations of the columns leave the coefficient invariant; 2. odd permutations of the columns change the sign by the factor (−1)a+b+c ; 3. simultaneous sign reversal of the projection quantum numbers changes the sign by (−1)a+b+c . The 72 corresponding symmetries of the WCGcoefficients (up to sign changes and dimensional factors) are best obtained from those of the 3 j-coefficients by using (2.42).

35

Part A 2.7

abc Cαβγ

2.7 Wigner–Clebsch–Gordan and 3-j Coefficients

36

Part A

Mathematical Methods




2.7.5 Recurrence Relations Three-term:


1

[(J + 1)(J − 2 j1 )] 2

j1

j2

j3



Part A 2.7

m1 m2 m3
 j1 j2 − 12 j3 − 12 1 = [( j2 + m 2 )( j3 − m 3 )] 2 m 1 m 2 − 12 m 3 + 12
 j1 j2 − 12 j3 − 12 1 2 − [( j2 − m 2 )( j3 + m 3 )] ; m 1 m 2 + 12 m 3 − 12
 j1 j2 j3 1 2 [(J − 2 j2 )(J + 1 − 2 j3 )] m1 m2 m3 1

+ [( j2 + m 2 + 1)( j3 + m 3 )] 2
 j1 j2 − 12 j3 + 12 × m 1 m 2 − 12 m 3 + 12

j2

j3



j2 − m 3 − j2 m 3

1 2 2 j2 ( j3 − m 3 + 1) =− ( j1 − j3 + j2 )(J − 2 j2 + 1) 
j2 − 12 j3 + 12 j1 , × j2 − m 3 − j2 + 12 m 3 − 12 j3 = j1 + j2 − 1, j1 + j2 − 2, . . . , j1 − j2 for j1 ≥ j2 . Four-term: 1

[(J+1)(J−2 j1 )(J−2 j2 )(J−2 j3+1)] 2 
j1 j2 j3 = [( j2 − m 2 )( j2 + m 2 + 1) × m1 m2 m3 1

× ( j3 + m 3 )( j3 + m 3 − 1)] 2 
j2 j3 − 1 j1 × m1 m2 + 1 m3 − 1

1

+ [( j2 − m 2 + 1)( j3 − m 3 )] 2
 j1 j2 − 12 j3 + 12 × =0; m 1 m 2 + 12 m 3 − 12
 j1 j2 j3 1 2 ( j2 + m 2 ) m1 m m3

− 2m 2 [( j3+m 3 )( j3−m 3 )]

1 2




 j1 j2 j3−1 m1 m2 m3 1

− [( j2+m 2 )( j2−m 2+1)( j3−m 3 )( j3−m 3−1)] 2 
j1 j2 j3 − 1 . × m1 m2 − 1 m3 + 1

1

= [( j3 − j1 + j2 )(J + 1)( j3 − m 3 )] 2
 j1 j2 − 12 j3 − 12 × m 1 m 2 − 12 m 3 − 12

Five-term: bd f

1




j1

− [( j1 − j3 + j2 )(J − 2 j2 + 1)( j3 + m 3 + 1)] 2
 j1 j2 − 12 j3 + 12 × ; m 1 m 2 − 12 m 3 + 12  j1 j2 j3

j2 − m 3 − j2 m 3

1 2 2 j2 ( j3 + m 3 ) =− ( j3 − j1 + j2 )(J + 1)
 j2 − 12 j3 − 12 j1 × , j2 − m 3 − j2 + 12 m 3 − 12

Cβ,δ,β+δ

1 (b+d− f )(b+ f −d+1)(d−δ)( f +β+δ+1) 2 = (2d)(2 f +1)(2d)(2 f +2) bd−1/2 f +1/2

× Cβ,δ+1/2,β+δ+1/2

1 (b+d− f )(b+ f −d+1)(d+δ)( f −β−δ+1) 2 − (2d)(2 f +1)(2d)(2 f +2) bd−1/2 f +1/2

× Cβ,δ−1/2,β+δ−1/2

1 (d+ f −b)(b+d+ f +1)(d+δ)( f +β+δ) 2 + (2d)(2 f +1)(2d)(2 f ) bd−1/2 f −1/2

j3 = j1 + j2 , j1 + j2 − 1, . . . , j1 − j2 + 1

× Cβ,δ−1/2,β+δ−1/2

1 (d+ f −b)(b+d+ f +1)(d−δ)( f −β−δ) 2 + (2d)(2 f +1)(2d)(2 f )

for j1 ≥ j2 ;

× Cβ,δ+1/2,β+δ+1/2 .

bd−1/2 f −1/2

Angular Momentum Theory

This relation may be used to prove the limit relation (2.50) from the similar recurrence relation (2.84c) for the Racah coefficients.

1

lim (−1)a+b+2 j−τ [(2c + 1)(2 j − 2σ + 1)] 2  abc = Cρστ ,

(2.50)

where the brace symbol is a 6– j coefficient (Sect. 2.9). 
cos 12 β sin 12 β jk j+∆ ∆−µ k Cm,µ,m+µ ≈ (−1) Dµ∆ − sin 12 β cos 12 β k = dµ∆ (β) , for large j ; 2 2 1 j +m j −m 1 , sin β = , cos β = 2 2j 2 2j jk j

Cm0m ≈ Pk (cos β) , for large j ; 1 (−1)k [(2 j + 1)(2J + 1)] 2 W( j, k, J + m, J; j, J ) ∼ Pk (cos β), first for large J, then large j (Sect. 2.9).

= δm 1 +m 2 ,m (−1) j1 −m 1

1 (2 j + 1)( j1 + j2 − j )! 2 × ( j1 + j2 + j + 1)!
√  √ j1 + m 1 j2 + m 2 j × Dm, j1 − j2 , √ √ − j1 − m 1 j2 − m 2 symbolic powers (2.51)

where in evaluating this result one first substitutes 3 u 11 = j1 + m 1 , 3 u 12 = j2 + m 2 , 3 u 21 = − j1 − m 1 , 3 u 22 = j2 − m 2 into the form (2.17), followed by the replacement of ordinary powers by generalized powers: √ (± k)s → (±1)s



k! (k − s)!

1 2

.

2.8 Tensor Operator Algebra 2.8.1 Conceptual Framework A tensor operator can be characterized in terms of its algebraic properties with respect to the angular momentum J or in terms of its transformation properties under unitary transformations generated by J. Both viewpoints are essential. A tensor operator T with respect to the group SU(2) is a set of linear operators T = {T1 , T2 , . . . , Tn } , where each operator in the set acts in the space H defined by (2.10) and maps this space into itself Ti : H → H, i = 1, 2, . . . , n, and where this set of operators has the following properties with respect to the angular momentum J, which acts in the same space H in the standard way:

1. Commutation relations with respect to the angular momentum J: [Ji , T j ] =

n 

tk(i)j Tk ,

k=1

where the tk(i)j are scalars (invariants) with respect to J. 2. Unitary transformation with respect to SU(2) rotations: ˆ ˆ Ti eiψ n·J = e−iψ n·J

n 

D ji (U )T j ,

j=1

U = U(ψ, n) ˆ , where the matrix D(U ) is an n × n unitary matrix representation of SU(2). Reduction of this representation into its irreducible constituents gives the notion of an ir-

Part A 2.8

aba+ρ

lim Ca−α,β,a−α+β = δρβ ,

j −τ a j −σ × b j c

2.7.7 WCG-Coefficients as Discretized Representation Functions j j j

a→∞



37

Cm11 m2 2 m

2.7.6 Limiting Properties and Asymptotic Forms

j→∞

2.8 Tensor Operator Algebra

38

Part A

Mathematical Methods

reducible tensor operator T J of rank J. An irreducible tensor operator T J of rank J is a set of 2J + 1 operators & ) ( T J = TMJ & M = J, J − 1, . . . , −J with the following properties with respect to SU(2):

Part A 2.8

1. Commutation relations with respect to the angular momentum J: # $ 1 J J+ , TMJ = [(J − M )(J + M + 1)] 2 TM+1 , $ # 1 J J− , TMJ = [(J + M )(J − M + 1)] 2 TM−1 , $ # J3 , TMJ = MTMJ , #

$$ # Ji , Ji , TMJ = J(J + 1)TMJ .

(2.52)

i

2. Generation from highest “weight”:

1 2 ! " (J + M )! J J− , T JJ (J−M ) , TM = (2J)!(J − M )! where [A, B](k) = [A, [A, B](k−1) ], k = 1, 2, . . . , with [A, B](0) = B, denotes the k-fold commutator of A with B. 3. Unitary transformation with respect to SU(2) rotations: ˆ ˆ e−iψ n·J T J eiψ n·J =  M J D M  M (U )TMJ  , U = U(ψ, n) ˆ .

(2.53)

M

Angular momentum operators act in Hilbert spaces by acting linearly on the vectors in such spaces. The concept of a tensor operator generalizes this by replacing the irreducible space H J by the irreducible tensor T J , and angular momentum operator action on H J by commutator action on T J , as symbolized, respectively, by J : { states } → { states }, { commutator action of J } : { tensor operators } → {tensor operators } . Just as exponentiation of the standard generator action (2.13) and (2.14) gives relation (2.16), so does the exponentiation of the commutator action (2.52) give relation (2.53), when one uses the Baker–Campell– Hausdorff identity: tA

−t A

e Be

=

 tk k

k!

[A, B](k) .

Thus, the linear vector space of states is replaced by the linear vector space of operators. Abstractly, relations (2.13) and (2.52) are identical: only the rule of action and the object of that action has changed. An example of an irreducible tensor of rank 1 is the angular momentum J itself, which has the special property J : H j → H j . Thus, relations (2.52) and (2.53) are realized as: √ T11 = J+1 = −(J1 + iJ2 )/ 2 , T01 = J0 = J3 ,

√ 1 T−1 = J−1 = (J1 − iJ2 )/ 2 ; # $ 1 1 , J+ , Tµ1 = [(1 − µ)(2 + µ)] 2 Tµ+1 $ # 1 1 , J− , Tµ1 = [(1 + µ)(2 − µ)] 2 Tµ−1 $ # J3 , Tµ1 = µTµ1 , µ = 1, 0, −1 ; ˆ ˆ J eiψ n·J = J cos ψ + n( e−iψ n·J ˆ nˆ · J)(1 − cos ψ) − (nˆ × J) sin ψ ,  1 ˆ ˆ e−iψ n·J Tµ1 eiψ n·J = Dνµ (ψ, n)T ˆ ν1 . ν

2.8.2 Universal Enveloping Algebra of J The universal enveloping algebra A(J) of J is the set of all complex polynomial operators in the components Ji of J, or equivalently in (J+ , J3 , J− ). The irreducible tensor operators spanning this algebra are the analogues of the solid harmonics Ylm (x) and are characterized by the following properties: Basis set: Tkk = ak J+k , ak arbitrary constant ,

# $ (k + µ)! J− , J+k Tµk = ak , (k−µ) (2k)!(k − µ)! µ = k, k − 1, . . . , −k ; k = 0, 1, 2, . . . . Standard action with respect to J: # $ 1 k J± , Tµk = [(k ∓ µ)(k ± µ + 1)] 2 Tµ±1 , ! " J3 , Tµk = µTµk , 3  $ #  Ji , Ji , Tµk = k(k + 1)Tµk . i=1

Unitary transformation: ˆ ˆ e−iψ n·J Tµk eiψ n·J =

 ν

k Dνµ (ψ, n)T ˆ νk .

Angular Momentum Theory

2.8.3 Algebra of Irreducible Tensor Operators

1. Multiplication of an irreducible tensor operator of rank k by a complex number or an invariant with respect to angular momentum J gives an irreducible tensor operator of the same rank. 2. Addition of two irreducible tensor operator of the same rank gives an irreducible tensor of that rank. 3. Ordinary multiplication (juxtaposition) of three irreducible tensor operators is associative, but the multiplication of two is noncommutative, in general. 4. Two irreducible tensor operators Sk1 and T k2 of different or the same ranks acting in the same space may be multiplied to obtain new irreducible tensor operators of ranks given by the angular momentum addition rule (Clebsch–Gordan series):  "k ! k Cµk11kµ22kµ Sµk11 Tµk22 , (2.54) S 1 × T k2 µ = µ1 ,µ2

µ = k, k − 1, . . . , −k ; rank = k ∈ {k1 + k2 , k1 + k2 − 1, . . . , |k1 − k2 |} . The following symbol denotes the irreducible tensor operator with the µ-components (2.54): "k ! k S 1 × T k2 . 5. Two irreducible tensor operators Sk1 and T k2 of different or the same ranks acting in different Hilbert spaces, say H and K, may first be multiplied by the tensor product rule so as to act in the tensor product space H ⊗ K, that is, Sµk11 ⊗ Tµk22 : H ⊗ K → H ⊗ K ,

39

and then coupled to obtain new irreducible tensor operators, acting in the same tensor product space H ⊗ K: $k #  Cµk11kµ22kµ Sµk11 ⊗ Tµk22 , Sk1 ⊗ T k2 = µ

µ,µ2

µ = k, k − 1, . . . , −k .

(2.55)

The following symbol denotes the tensor operator with the µ-components (2.55): $k # Sk1 ⊗ T k2 , k ∈ {k1 + k2 , k1 + k2 − 1, . . . , |k1 − k2 |} . 6. The conjugate tensor operator to T J , denoted J† by T J † , is the set of operators with components TM defined by 4   & J † & - 4 & J &   -∗ j m &TM & jm = jm &TM & j m . These components satisfy the following relations: # $ 1 J† J† J± , TM = −[(J ± M )(J ∓ M + 1)] 2 TM∓1 # $ J† J† J3 , TM = −MTM , # $$ #  J† J† Ji , Ji , TM = J(J + 1)TM ; i J†

ˆ ˆ e−iψ n·J TM eiψ n·J =

IJ =





J†

J∗ DM ˆ M ;  M (ψ, n)T


M  invariant operator to J J† TM TM = , SU(2) rotations

M −iψ n·J J iψ n·J ˆ ˆ

e



I e

= IJ .

An important invariant operator is  I k1 k2 k = Cµk11kµ22kµ Tµk11 Tµk22 Tµk† . µ1 µ2 µ

7. Other definitions of conjugation: T J → (−1) J−M T−M , TM

J TMJ → (−1) J+M T−M .

2.8.4 Wigner–Eckart Theorem The Wigner–Eckart theorem establishes the form of the matrix elements of an arbitrary irreducible tensor operator: 4   & J & - 4  5 J 5 - jJ j  j m &TM & jm = j 5T 5 j Cm Mm 
   5 J 5   j J j j+J+m = j 5T 5 j (−1) . m M −m 

Part A 2.8

Irreducible tensor operators possess, as linear operators acting in the same space, properties 1., 2., and 3. below, and an additional multiplication property 4., which constructs new irreducible tensor operators out of two given ones and is called coupling of irreducible tensor operators. Property 4. extends also to tensor operators acting in the tensor product space associated with kinematically independent systems. It is important that associativity extends to the product (2.54), as well as to the product (2.55). Commutativity in these products is generally invalid. The coupling properties given in 4. and 5. are analogous to the coupling of basis state vectors. The operation of Hermitian conjugation of operators, which is the analogue of complex conjugation of states, is also important, and has the properties presented under 5.

2.8 Tensor Operator Algebra

40

Part A

Mathematical Methods

Reduced matrix elements with respect to WCGcoefficients: 4  5 J 5 -  jJ j  4   & J & & & jµ , j 5T 5 j = C  j µ T µMµ

µM

M

Part A 2.8

each µ = j  , j  − 1, . . . , − j  (the reduced matrix element is independent of µ ). Reduced matrix elements with respect to 3– j coefficients: 3 4 5 5  5 J 5  j 5T 5 j = (−1)2J 2 j  + 1 j  5T J 5 j . Examples of irreducible tensor operators include: 1. The solid harmonics with respect to the orbital angular momentum L: Yk (x) = {Ykµ (x) : µ = k, . . . , −k} , 4 5 5 & lkl  &l , m + µ , Ykµ |lm = l  5Yk 5l Cm,µ,m+µ l

where x | lm = Ylm (x) ,

1 4 5 k 5 (2l + 1)(2k + 1) 2 lkl   C000 , l 5Y 5l = r l+k−l 4π(2l  + 1) Ykµ (x)Ylm (x) 4 5 5  l  5Yk 5l C lkl Yl  ,m+µ (x) , = m,µ,m+µ

l

! k "k Y 1 (x) ⊗ Yk2 (x) µ  = Cµk11kµ22kµ Yk1 µ1 (x)Yk2 µ2 (x) , µ1 µ2

5 ! k "k 4 5 Y 1 (x) ⊗ Yk2 (x) µ = k5Yk1 5k2 Ykµ (x) . 2. The polynomial operator T k in the components of J (Sect. 2.8.2): 4  & k & 4 5 5 - jk j j m &Tµ & jm = δ j  j j 5T k 5 j Cmµm  , 4 5 k5 j 5T 5 j

1 2 (2 j + k + 1)!k!k! k = ak (−1) . (2 j + 1)(2 j − k)!(2k)! 3. Polynomials in the components of an arbitrary vector operator V, which has the defining relations: ! " Ji , V j = ieijk Vk , " ! 1 J± , Vµ = [(1 ∓ µ)(2 ± µ)] 2 Vµ±1 , ! " J3 , Vµ = µVµ , √ V+1 = −(V1 + iV2 )/ 2, V0 = V3 , √ V−1 = (V1 − iV2 )/ 2 .

This construction parallels exactly that given in Sect. 2.8.2 upon replacing J by V. The explicit form of the resulting polynomials may be quite different since no assumptions are made concerning commutation relations between the components Vi of V. The solid harmonics in the gradient operator ∇ constitute an irreducible tensor operator with respect to the orbital angular momentum L.

2.8.5 Unit Tensor Operators or Wigner Operators A unit tensor operator is an irreducible tensor operator Tˆ J,∆ , indexed not only by the angular momentum quantum number J, but also by an additional label ∆, which specifies that this irreducible tensor operator has reduced matrix elements given by 4  5 J,∆ 5 j 5Tˆ 5 j = δ j  , j+∆ . This condition is to be true for all j = 0, 1/2, 1, . . . . There is a unit tensor operator defined for each ∆ = J, J − 1, . . . , −J . The special symbol * + J +∆ 2J 0 • denotes a unit tensor operator, replacing the boldface symbol Tˆ J,∆ , while the symbol * + J +∆ 2J 0 , M = J, J − 1, . . . , −J J+M denotes the components. In the same way that abstract angular momentum J and state vectors {| jm } extract the intrinsic structure of all realizations of angular momentum theory, as given in Sect. 2.2, so does the notion of a unit tensor operator extract the intrinsic structure of the concept of irreducible tensor operator by disregarding the physical content of the theory, which is carried in the structure of the reduced matrix elements. Physical theory is regained from the fact that the unit tensor operators are the basis for arbitrary tensor operators, which is the structural content of the Wigner–Eckart theorem. The concept of a unit tensor operator was introduced by Racah, but it was Biedenharn who recognized the full significance of this concept not only for SU(2), but for all the unitary groups. All of the content of physical tensor operator theory can be regained from the properties of unit tensor operators or Wigner operators as summarized below:

Angular Momentum Theory

Notation (double Gel’fand patterns): * + J +∆ M, ∆ = J, J − 1, . . . , −J 2J 0 , 2J = 0, 1, 2, . . . . J+M

(2.56)

for all j = 0, 12 , . . . ; m = j, j − 1, . . . , − j. Conjugation: *

J +∆

2J

J+M

0 | jm

j−∆J j

= Cm−M,M,m | j − ∆, m − M . (2.57)

Orthogonality: 

* 2J

M



J+M

* 2J

∆

J + ∆

J +∆ J + M

+* 0

J +∆

2J

J+M

+† * 0

*

2J

+† = δ∆ ∆ I∆J ,

0

J +∆ J+M

(2.58)

+

0 = δ M M ,

+*

+†

 J +∆ J  + ∆ jm| 2J  0 2J 0 J  + M J+M m 2j +1 δ J  J δ M  M δ∆ ∆ . = 2J + 1

(2.59)

TMJ | jm  * + 4 J +∆ 5 J5 = j + ∆5T 5 j 2J 0  | jm . J+M ∆ (2.62)

Characteristic null space: The characteristic null space of the Wigner operator defined by (2.56) is the set of irreducible subspaces H j ⊂ H given be

Coupling law: * +* +  b+σ a+ρ abc Cαβγ 2b 0 2a 0 b+β a+α αβ * + c+ρ+σ abc = Wρ,σ,ρ+σ 2c 0 , c+γ

(2.63)

abc is an invariant operator (commutes with J) where Wρστ and is called a Racah invariant. Its relationship to Racah coefficients and 6– j coefficients is given in Sect. 2.9. Product law: * +* + b+σ a+ρ 2b 0 2a 0 b+β a+α * +  c+ρ+σ abc abc Wρ,σ,ρ+σ Cα,β,α+β 2c = 0 . c+α+β c

| jm (2.60)

J is defined by its action on an The invariant operator I ∆ arbitrary vector ψ j ∈ H j :

I∆J ψ j =  j−∆,J, j ψ j . Tensor operator property: * + J +∆ −iψ n·J iψ n·J ˆ e 2J 0 e ˆ J+M + *  J +∆ J = D M  M (ψ, n) ˆ 2J 0 . J + M M

Basis property (Wigner–Eckart theorem):

{H j : 2 j = 0, 1, . . . , J − ∆ − 1} .

+†

(2.64)

Racah invariant:  abc abc Wρστ = Cαβγ *

αβγ

× 2b

b+σ b+β

+* 0

2a

a+ρ a+α

+* 0

2c

c+τ c+γ

+† 0

.

(2.65)

(2.61)

41

abc for a Racah invariant is designed The notation Wρστ to “match” that of the WCG-coefficient on the left, the latter being associated with the lower group theoretical labels, for example, 
2a 0 →| aα , a+α

Part A 2.8

Definition (shift action): + * J +∆ jJ j+∆ 2J 0 | jm = Cm,M,m+M | j + ∆, m + M J+M

2.8 Tensor Operator Algebra

42

Part A

Mathematical Methods

the state vector having a group transformation law under the action of SU(2), and the former with the shift labels of a unit tensor operator,
 α+ρ , 2α 0

Part A 2.8

and having no associated group transformation law. The invariant operator defined by (2.65) has real eigenvalues, hence, is a Hermitian operator, abc† abc = Wρστ , Wρστ

(2.66)

which is diagonal on an arbitrary state vector in H j (Sect. 2.9). The Racah invariant operator does not commute with a unit tensor operator, and it makes a difference whether it is written to the left or to the right of such a unit tensor operator. The convention here writes it to the left. abc and Relation (2.65) is taken as the definition of Wρστ the following properties all follow from this expression: Domain of definition: abc Wρστ : a, b, c ∈ {0, 1/2, 1, 3/2, . . . } ;

ρ = a, a − 1, . . . , −a σ = b, b − 1, . . . , −b τ = c, c − 1, . . . , −c ; abc Wρστ = 0, if ρ + σ = τ; if abc = 0 .

Orthogonality relations:  abc abd Wρστ Wρστ  = δcd δττ  abc Iτc ,

(2.67)

ρσ



abc abc ab Wρστ Wρ σ  τ = δρρ δσσ  Iρσ ,

(2.68)

cτ

where the I invariant operators in these expressions have the following eigenvalues on an arbitrary vector ψj ∈ Hj: Iτc ψ j =  j−τ,c, j ψ j , ab Iρσ ψj

=  j−σ−ρ,a, j−σ  j−σ,b, j ψ j .

The orthogonality relations for Racah invariants parallel exactly those of WCG-coefficients. Using the orthogonality relations (2.67) for Racah invariants, the following two relations now follow from (2.63) and (2.64), respectively:

WCG and Racah operator coupling:  abd abc Wρ,σ,ρ+σ Cα,β,α+β ρσ αβ

*

× 2b

b+σ b+β *

+* 0

= δcd abc Iτd 2c

2a c+τ

a+ρ a+α +

+ 0

(2.69) 0 . c+γ Racah operator coupling of shift patterns: * +* +  b+σ a+ρ abc Wρστ 2b 0 2a 0 b+β a+α ρσ * + c+τ abc = Cα,β,α+β 2c (2.70) 0 . c+α+β Relations (2.56–2.70) capture the full content of irreducible tensor operator algebra through the concept of unit tensor operators that have only 0 or 1 for their reduced matrix elements. Using the Wigner– Eckart theorem (2.62), the relations between general tensor operators can be reconstructed. Unit tensor operators were invented to exhibit in the most elementary way possible the abstract and intrinsic structure of the irreducible tensor operator algebra, stripping away the details of particular physical applications, thus giving the theory universal application. It accomplishes the same goal for tensor operator theory that the abstract multiplet theory in Sect. 2.2 accomplishes for representation theory. Physical theory is regained through the concept of reduced matrix element. The coupling rule (2.54) is now transformed to a rule empty of WCG-coefficient content and becomes a rule for coupling of reduced matrix elements using the invariant Racah operators: 4     5! k "k 5 α j 5 S 1 × T k2 5(α) j  kkk = (−1)k1 +k2 −k W j 2−1j, j  − j  , j  − j ( j  ) (α ) j 

-4   5 5 4  5 5 × α j  5 Sk1 5(a ) j  α j  5T k2 5(α) j . (2.71)

This coupling rule is invariant to all SU(2) rotations, and reveals the true role of the Racah coefficients and reduced matrix elements in physical theory as invariant objects under SU(2) rotations. It now becomes imperative to understand Racah coefficients as objects free of their original definition in terms of WCGcoefficients.

Angular Momentum Theory

2.9 Racah Coefficients

43

2.9 Racah Coefficients

abc ( j) = 0 Wρστ

if τ = ρ + σ ,

or abc = 0 ,

abc ( j ) = [(2c + 1)(2 j − 2σ Wρστ

+ 1)]1/2 × W( j − τ, a, j, b; j − σ, c) ,

[(2e + 1)(2 f + 1)]1/2 W(abcd; ef ) bdf

= We−a,c−e,c−a (c) , W(abcd; ef ) = 0 unless the triples of nonnegative integers and half-integers (abe), (cde), (acf ), (bdf ) satisfy the triangle conditions.

2.9.1 Basic Relations Between WCG and Racah Coefficients  βδ

= [(2e + 1)(2 f + 1)]1/2 W(abcd; ef  [(2e + 1)(2 f + 1)]1/2 W(abcd; ef ) a fc

× Cβ,δ,β+δ Cα,β+δ,α+β+δ edc abe = Cα+β,δ,α+β+δ Cα,β,α+β ,

δcc [(2e + 1)(2 f + 1)]1/2 W(abcd; ef )  bdf abe = Cβ,δ,β+δ Cγedc −δ,δ,γ C γ −β−δ,β,γ −δ 

× Cγabc −β−δ,β+δ,γ ,

 [(2e + 1)(2 f + 1)]1/2 W(abcd; ef ) βδe

bdf 

edc abe Cα,β,α+β × Cβδγ Cα+β,δ,α+γ a fc

= δ f f  Cα,γ,α+γ ,  [(2e + 1)(2 f + 1)]1/2 W(abcd; ef ) e

edc abe × Cα+β,δ,α+β+δ Cα,β,α+β bdf

afc

= Cβ,δ,β+δ Cα,β+δ,α+β+δ .

= δ f f  ac f bdf ,

(2.72)

 (2e + 1)(2 f + 1)W(abcd; ef )W(abcd; e f ) f

= δee abe cde .

(2.73)

Definition of 6– j coefficients:   a b e = (−1)a+b+c+d W(abcd; ef ) . d c f

(2.74)

Orthogonality of 6– j coefficients:     a b e a b e (2e + 1)(2 f + 1) d c f d c f e = δ f f  acf bdf ,  [(2e + 1)(2 f + 1)]

(2.75)



 a b e d c f

a b e d c f



(2.76)

Explicit form of Racah coefficients: a fc )Cα,γ,α+γ

βδ

e

= δee abe cde .

bdf

bdf

Orthogonality relations for Racah coefficients:  (2e + 1)(2 f + 1)W(abcd; ef )W(abcd; ef  )

f

edc abe Cβδγ Cα+β,δ,α+γ Cα,β,α+β

f

2.9.2 Orthogonality and Explicit Form

,

W(abcd; ef ) = ∆(abe)∆(cde)∆(acf )∆(bdf )  (−1)a+b+c+d+k (k + 1)! × (k − a − b − e)!(k − c − d − e)! k

1 (k − a − c − f )!(k − b − d − f )! 1 × (a + b + c + d − k)! 1 , × (a + d + e + f − k)!(b + c + e + f − k)!

×

(2.77)

where ∆(abc) denotes the triangle coefficient, defined for every triple a, b, c of integers and half-odd integers satisfying the triangle conditions by: ∆(abc)

1 (a + b − c)!(a − b + c)!(−a + b + c)! 2 = . (a + b + c + 1)! (2.78)

Part A 2.9

Relation (2.65) is taken, initially, as the definition of the Racah coefficient with appropriate adjustments of notations to conform to Racah’s W-notation and to Wigner’s 6– j notation. Corresponding to each of (2.63–2.65), (2.69, 2.70), there is a corresponding numerical relationship between WCG-coefficients and Racah coefficients. Despite the present day popularity of expressing all such relations in terms of the 3– j and 6– j notation, this temptation is resisted here for this particular set of relations because of their fundamental origins. The relation between the Racah invariant notation and Racah’s original W-notation is abc abc Wρστ | jm = Wρστ ( j ) | jm ,

44

Part A

Mathematical Methods

2.9.3 The Fundamental Identities Between Racah Coefficients

Triangle sum rule:

Part A 2.9

Each of the three relations given in this section is between Racah coefficients alone. Each expresses a fundamental mathematical property. The Biedenharn– Elliott identity is a consequence of the associativity rule for the open product of three irreducible tensor operators; the Racah sum rule is a consequence of the commutativity of a mapping diagram associated with the coupling of three angular momenta; and the triangle coupling rule is a consequence of the associativity of the open product of three symplection polynomials [2.1]. As such, these three relations between Racah coefficients, together with the orthogonality relations, are the building blocks on which is constructed a theory of these coefficients that stands on its own, independent of the WCG-coefficient origins. Indeed, the latter is recovered through the limit relation (2.50). Biedenharn–Elliott identity: W(a ab b; c e)W(a ed  d; b c)  (2 f + 1)W(abcd; ef )W(c bd  d; b f ) =



a a c b b e



 = (−1)φ (2 f + 1) f

 ×

a a c f d c









a b e d c f

c b b d d f



,

(2.79b)

Racah sum rule:  (−1)b+d− f (2 f + 1)W(abcd; ef )W(adcb; gf ) f

= (−1)e+g−a−c W(bacd; eg) ,  (−1)e+g+ f (2 f + 1) 

 =

b a e d c g

(−1)

[∆(acf )∆(bdf )]    abe −1 . = (2 f + 1) [∆(abe)∆(cde)] dcf e (2.81b)

2.9.4 Schwinger–Bargmann Generating Function and its Combinatorics 

Triangles associated with the 6– j symbol ( j1 j2 j3 ) ,

( j3 j4 j5 ) ,

( j1 j5 j6 ) ,

.

(2.80a)





 a b e d c f



j1 j2 j3 j4 j5 j6

:

( j2 j4 j6 ) .

Tetrahedron associated with the points: The points define the vertices of a general tetrahedron with lines joining each pair of points that share a common subscript, and the lines are labeled by the product of the common coordinates (Fig. 2.2). Monomial term: Define the triangle monomial associated with a triangle ( ja jb jc ) and its associated point (z a , z b , z c ) in R3 by j + jc − ja jc + ja − jb ja + jb − jc zb zc

(z a , z b , z c )( ja jb jc ) = z ab

φ = f − e + a + a + b + b + c − c + d  − d .

f

(2.81a) −1

a+b+c+d

( j1 j2 j3 ) → (x1 , x2 , x3 ) , ( j3 j4 j5 ) → (y3 , x4 , x5 ) , ( j1 j5 j6 ) → (y1 , y5 , x6 ) , ( j2 j4 j6 ) → (y2 , y4 , y6 ) . (2.79a)

a e b d d c

e

Points in R3 associated with the triangles:

f

× W(a ad  f ; c c) ,

[∆(acf )∆(bdf )]−1  = (2 f + 1) [∆(abe)∆(cde)]−1 W(abcd; ef ) ,

a d g b c f (2.80b)

.

(2.82)

Cubic graph (tetrahedral T4 ) functions: Interchange the symbols x and y in the coordinates of the vertices of the tetrahedron and define the following polynomials on the vertices and edges of the tetrahedron with this modified labeling. Vertex function: multiply together the coordinates of each vertex and sum over all such vertices to obtain V3 = y1 y2 y3 + x3 y4 y5 + x1 x5 y6 + x2 x4 x6 ; Edge function: multiply together the coordinates of a given edge and the opposite edge and sum over all such pairs to obtain E 4 = x1 y1 x4 y4 + x2 y2 x5 y5 + x3 y3 x6 y6 .

Angular Momentum Theory

x5 y5

x6 y6

Since the factor T(∆) is an integer in the expansion (2.83a), this result shows that the 6– j coefficient is an integer, up to the multiplicative triangle coefficient factors.

(y2 y4 y6) x2 y2

2.9.5 Symmetries of 6– j Coefficients

x4 y4 x3 y3

(x1 x2 x3)

(y3 x4 x5)

Fig. 2.2 Labeled cubic graph (tetrahedron) associated with

6– j coefficients

Generating function: (1 + V3 + E 4 )−2 =



T (∆)Z ∆ ,

∆ ( j1 j2 j3 )

Z ∆ = (x1 , x2 , x3 )

(2.83a)

(y3 , x4 , x5 )( j3 j4 j5 )

( j1 j5 j6 )

× (y1 , y5 , x6 ) (y2 , y4 , y6 )( j2 j4 j6 ) ,   ( j1 j2 j3 )   ( j j j ) ∆= 3 4 5  ; ( j1 j5 j6 )

(2.83b)

( j2 j4 j6 )  (−1)k (k + 1) T(∆) = k

k , × k1 , k2 , k3 , k4 , k5 , k6 , k7

ki = k − ti , i = 1, 2, 3, 4 , k j = e j−4 − k , j = 5, 6, 7 ; ti = triangle sum = vertex sum, e j = opposite edge sum, in pairs, t1 = j1 + j2 + j3 , t2 = j3 + j4 + j5 , t3 = j1 + j5 + j6 , t4 = j2 + j4 + j6 , e1 = ( j2 + j5 ) + ( j3 + j6 ) , e2 = ( j1 + j4 ) + ( j3 + j6 ) , e3 = ( j1 + j4 ) + ( j2 + j5 ) .

(2.83c)

There are 144 symmetry relations among the Racah 6– j coefficients. The 24 classical ones, given already by Racah, and corresponding to the tetrahedral point group Td of rotations-inversions (isomorphic to the symmetric group S4 ) mapping the regular tetrahedron onto itself, are realized in the 6– j symbol   a b e d c f as permutations of its columns and the exchange of any pair of letters in the top row with the corresponding pair in the bottom row. Regge discovered the 6-fold increase in symmetry by noting that each term in the summation in (2.77) is invariant not only to the classical 24 symmetries, but also under certain linear transformations of the quantum labels. These symmetries are also implicit in Schwinger’s generating function. The full set, including the original 24 substitutions, of linear transformations of the letters a, b, c, d, e, f thus yields a group of linear transformation isomorphic to S4 × S3 . The column permutations and row-pair interchanges described above applied to each of the six symbols in the equalities below yield the set of 144 relationships:   b + c+ e − f  a b e  a 2 = d c f   d b + c+ f − e 2 a +d + e − f  b  2 = a +d + f − e   c 2

b + e + f − c   2 c+ e + f − b   2 a + e + f − d   2 d + e + f − a  2

45

Part A 2.9

The summation in (2.83b) is over the infinite set of all tetrahedra; that is, over the infinite set of arrays ∆ having nonnegative integral entries. The 6– j coefficients is then given by   j1 j2 j3 j4 j5 j6 T(∆) . = ∆( j1 j2 j3 )∆( j1 j5 j6 )∆( j2 j4 j6 )∆( j3 j4 j5 )

(y1 y5 x6)

x1 y1

2.9 Racah Coefficients

46

Part A

Mathematical Methods

Part A 2.9

a + b + d − c   2 =   a + c+ d − b 2 a + b + d − c   2 =   a + c+ d − b 2 a +d + e − f   2 =   a +d + f − e 2



a + b + c− d  e  2 b + c+ d − a  f 2

×

b + c+ e − f a + e + f − d    2 2 b + c+ f − e d + e + f − a   2 2 a + b + c− d b + e + f − c   2 2 . b + c+ d − a c+ e + f − b   2 2

2.9.6 Further Properties

 a b e 1 1 d − 2 c + 2 f + 12

+ [(b + d − f )

× (b + f − d + 1)(c + d − e)(c + d + e + 1) 1

× (a + c − f )(a + f − c + 1)] 2   a b e − [(d + f − b) × d − 12 c − 12 f + 12 × (b + d + f + 1)(c + d − e)(c + d + e + 1) 1

× (c + f − a)(a + c + f + 1)] 2   a b e × + [(d + f − b) d − 12 c − 12 f − 12 × (b + d + f + 1)(d + e − c)(c + e − d + 1)

Recurrence relations: Three-term:

1

[(a + b + e + 1)(b + e − a)



× (c + d + e + 1)(d + e − c)]

1/2

× (a + f − c)(a + c − f + 1)] 2   a b e × . d − 12 c + 12 f − 12

 a b e d c f

Relation to hypergeometric series:   abe = (−1)a+b+c+d W(abcd; ef ) dcf

= −2e[(b + d + f + 1)(b + d − f )]1/2   a b − 12 e − 12 × d − 12 c f + [(a + b − e + 1)(a + e − b)(c + d − e + 1)   a b e−1 1/2 × (c + e − d)] , (2.84a) d c f [(a + c + f + 1)(c + e − d)

= ∆(abe)∆(cde)∆(acf )∆(bdf ) ×





× (d + e − c + 1)(b + d − f + 1)]1/2

(2.84c)

(−1)β1 (β1 + 1)! (β2 − β1 )!(β3 − β1 )!
4 F3

a b e d c f

×

= [(a + c − f )(a + e − b)

α1 −β1 , α2 −β1 ,

α3 −β1 ,



α4 −β1

−β1 −1, β2 −β1 +1, β3 −β1 +1,

;1

(β1 − α1 )!(β1 − α2 )!(β1 − α3 )!(β1 − α4 )!

,

β1 = min(a+b+c+d, a+d +e+ f, b+c+e+ f ) ,

× (b + f + d + 2)(b + e − a + 1)]1/2   a + 12 b + 12 e . × d + 12 c − 12 f

The parameters β2 and β3 are identified in either way with the pair remaining in the 3-tuple

+ [(c + f − a)(c + e − d)(b − a − c + d + 1)]1/2   a b e × (2.84b) . d + 12 c − 12 f − 12

after deleting β1 . The (α1 , α2 , α3 , α4 ) may be identified with any permutation of the 4-tuple

Five-term: (2c + 1)(2d)(2 f + 1)

(a + b + e, c + d + e, a + c + f, b + d + f ) .





(a + b + c + d, a + d + e + f, b + c + e + f )

a b e d c f

= [(b + d − f )(b + f − d + 1)(d + e − c) 1

× (c + e − d + 1)(c + f − a + 1)(a + c + f + 2)] 2

The 4 F3 series is Saalschützian:  1+ (numerator parameters)  = (denominator parameters) .

Angular Momentum Theory

2.10 The 9–j Coefficients

47

2.10 The 9–j Coefficients 2.10.1 Hilbert Space and Tensor Operator Actions

2.10.2 9– j Invariant Operators

m1m2

(2.85)

The tensor product operator T a (1) ⊗ T b (2) acts in the tensor product space H(1) ⊗ H(2) according to the rule:  T a (1) ⊗ T b (2) (| j1 m 1 ⊗ | j2 m 2 ) = T a (1)| j1 m 1 ⊗ T b (2)| j2 m 2 , so that  T a (1) ⊗ T b (2) | ( j1 j2 ) jm  j j j = Cm11 m2 2 m T a (1)| j1 m 1 ⊗ T b (2)| j2 m 2 . m1m2

(2.86a)

The angular momentum quantities called 9– j coefficients arise when the coupled tensor operators T (ab)c with components γ defined by  abc a T (ab)cγ = Cαβγ Tα (1) ⊗ Tβb (2) , αβ

γ = c, c − 1, . . . , −c , are considered. The quantity T (ab)c

(2.86b)

is an irreducible tensor operator of rank c with respect to the total angular momentum J for all a, b that yield c under the rule of addition of angular momentum.

The entire angular momentum content of relation (2.86b) is captured by taking the irreducible tensor operators T a (1) and T b (2) to be unit tensor operators acting in the respective spaces H(1) and H(2): (ab)c T(ρσ)γ

=

 αβ

* abc Cαβγ

2a

α+ρ a+α

*

+ 0

⊗ 2b 1

b+σ b+β

+ .

0 2

(2.87)

The placement of the unit tensor operators shows in which space they act, so that the additional identification by indices 1 and 2 could be eliminated. For each given c ∈ {0, 1/2, 1, 3/2, 2, . . . } and all a, b such that the triangle relation (abc) is satisfied, and, for each such pair a, b, all ρ, σ with ρ ∈ {a, a − 1, . . . , −a}, σ ∈ {b, b − 1, . . . , −b}, an irreducible tensor operator of rank c with respect to the total angular momentum J with components γ is defined by (2.87). By the Wigner–Eckart theorem, it must be possible to write * + * +  b+σ α+ρ abc Cαβγ 2a 0 0 ⊗ 2b b + β a+α αβ 1 2  * +  abc c+τ = (2.88) 2c 0 . ρστ c+γ τ where: (i) The unit tensor operator on the right-hand side is a irreducible tensor operator with respect to J; that is, has the action on the coupled states given by * 2c

c+τ c+γ

+ 0 |( j1 j2 ) jm

j c j+τ

= Cm,γ,m+γ |( j1 j2 ) j + τ, m + γ ; #

(2.89)

$ abc denotes an invariant ρστ operator with respect to the total angular momentum J. Using the orthogonality of unit tensor operators, we can also write relation (2.88) in the and (ii) the symbol

Part A 2.10

Let T a (1) and T b (2) denote irreducible tensor operators of ranks a and b with respect to kinematically independent angular momentum operators J(1) and J(2) that act, respectively, in separable Hilbert spaces H(1) and H(2). Let H(1) and H(2) be reduced, respectively, into a direct sum of spaces H j1 (1) and H j2 (2). The angular momentum J(1) (has the &standard action on the or-) thonormal basis | j1 m 1 & m 1 = j1 , j1 − 1, . . . , − j1 of H j1 (1), and J(2) & standard action on the or-) ( has the thonormal basis | j2 m 2 & m 2 = j2 , j2 − 1, . . . , − j2 of H j2 (2). The irreducible tensor operators T a (1) and T b (2) also have the standard actions in their respective Hilbert spaces H(1) and H(2), as given by the Wigner– Eckart theorem. The total angular momentum J has the standard action on the coupled orthonormal basis of the tensor product space H j1 ⊗ H j2 :  j j j Cm11 m2 2 m | j1 m 1 ⊗ | j2 m 2 . |( j1 j2 ) jm =

48

Part A

Mathematical Methods

form: 

 abc ρστ

=

*

Part A 2.10

⊗ 2b

 αβγ

* abc Cαβγ

b+σ b+β

2a

+* 0

2c 2

a+ρ a+α c+τ c+γ

2.10.3 Basic Relations Between 9– j Coefficients and 6– j Coefficients

+ 0

Orthogonality of 9– j coefficients:  (2c + 1)(2 f + 1)(2h + 1)(2i + 1)

1

+† 0

.

(2.90)

This form is taken as the definition of the 9– j invariant operator. Its eigenvalues in the coupled basis define the 9– j coefficient:   abc |( j1 j2 ) jm ρστ & & & & & abc & = ( j1 + ρ, j2 + σ) j + τ| & & |( j j ) j & ρστ & 1 2 × |( j1 j2 ) jm 1

= [(2 j + 1)(2c + 1)(2 j1 + 2ρ + 1)(2 j2 + 2σ + 1)] 2    j2 j   j1  |( j1 j2 ) jm . (2.91) × a b c     j1 + ρ j2 + σ j + τ The 9– j invariant operators play exactly the same role in the tensor product space of two irreducible angular momentum spaces as do the Racah invariants in one such irreducible angular momentum space. The full content of the coupling law (2.86b) for physical irreducible tensor operators is regained in the coupling law for reduced matrix elements: 5 4       5 a α1 α2 j1 j2 j 5[T (1) × T b (2)]c 5(α1 α2 j1 j2 ) j   j1 j2 j 4  5 5   =  a b c  α1 j1 5T a (1)5(α1 ) j1 j1 j2 j  5 4  5 × α2 j2 5T b (2)5(α2 ) j2 ;   j1 j2 j   a b c j1 j2 j  !   "1 = 2 j1 + 1 2 j2 + 1 (2 j + 1)(2c + 1) 2      j1 j2 j  . × a b c       j1 j2 j

(2.92a)

(2.92b)

hi

     a b c   a b c   × d e f d e f  = δcc δ f f  ,        h i j h i j

where this relation is to be applied only to triples (abc), (def ), (cf j), (abc ), (def  ), (c f  j) for which the triangle conditions hold. 9– j coefficients in terms of 3– j coefficients:     j11 j12 j13   −1  (2 j33 + 1) δ j33 j33 j21 j22 j23     j31 j32 j33

  j11 j12 j13 j21 j22 j23 = m 11 m 12 m 13 m 21 m 22 m 23 all m ij except m 33




×
×




j31 j32 j33 m 31 m 32 m 33 j12 j22 j32 m 12 m 22 m 32




 j11 j21 j31 m 11 m 21 m 31  j13 j23 j33 m 13 m 23 m 33

9– j coefficients in terms of 6– j coefficients:     j11 j12 j13    (−1)2k (2k + 1) j21 j22 j23 =     k j31 j32 j33    j12 j22 j32 j11 j21 j31 × j32 j33 k j21 k j23   j j j × 13 23 33 . k j11 j12

 . (2.93)

(2.94)

Basic defining relation for 9– j coefficient from (2.88):  
   j12 j22 j32   j31 j32 j33 φ (−1) j11 j21 j31     m 31 m 32 m 33 j13 j23 j33
  j11 j21 j31 = m 11 m 21 m 31 all m m (1i)

(2i)

Angular Momentum Theory


×
× 




j11 j12 j13 m 11 m 12 m 13

 j13 j23 j33 m 13 m 23 m 33



j21 j22 j23 m 21 m 22 m 23

jkl .

, (2.95)

kl

Additional relations:  (−1)2b+l+h− f (2k + 1)(2l + 1) kl

       a b c    a e k  a b c  , × e d f d b l = d e f            g h i g h i k l i

  a  (2c + 1) d   c g  d e = (−1)2 j b j

  b c  a e f   f h i  f g h h j a

 b c i j  i , d

 (2k + 1)(2l + 1)(2m + 1) klm

   k  a b c  × d e f a       k l m d   a d k b × a d  k  b     k a b c      = d e f a        k l m d

l b e e e l b e

 m  c   f   l c f m c f  m  l  m   . c   f

2.10.4 Symmetry Relations for 9– j Coefficients and Reduction 6– j Coefficients The 9– j coefficient     j11 j12 j13   j21 j22 j23     j31 j32 j33 is invariant under even permutation of its rows, even permutation of its columns, and under the interchange of

rows and columns (matrix transposition). It is multiplied by the factor (−1)φ (2.95) under odd permutations of its rows or columns. These 72 symmetries are all consequences of the 72 symmetries of the 3– j coefficient in relation (2.93). Reduction to 6– j coefficients:         e 0 e   0 e e   a b e c d e = f d b = c f a             d f b f c a f f 0         a f c    f b d    f f 0 = d c e = 0 e e = e 0 e             b f d f a c b a e         c e d   e d c    b a e = f f 0 = e b a = a e b             f 0 f 0 f f d c e  1  (−1)b+c+e+ f 2 abe = . [(2e + 1)(2 f + 1)] dcf

2.10.5 Explicit Algebraic Form of 9– j Coefficients     a b c  c+ f − j (dah)(bei)( jhi) d e f = (1)   (def )(bac)( jcf )   h i j ×

 (−1)x+y+z xyz

x!y!z!

×

(2 f − x)!(2a − z)! (2i + 1 + y)!(a + d + h + 1 − z)!

×

(d + e − f + x)!(c + j − f + x)! (e + f − d − x)!(c + f − j − x)!

×

(e + i − b + y)!(h + i − j + y)! (b + e − i − y)!(h + j − i − y)!

×

(b + c − a + z)! (a + d − h − z)!(a + c − b − z)!

×

(a + d + j − i − y − z)! , (d + i − b − f + x + y)!(b + j − a − f + x + z)!

(abc)

1 (a − b + c)!(a + b − c)!(a + b + c + 1)! 2 = . (b + c − a)!

49

Part A 2.10

φ=


j12 j22 j32 m 12 m 22 m 32

2.10 The 9–j Coefficients

50

Part A

Mathematical Methods

is:

2.10.6 Racah Operators

Part A 2.10

A Racah operator is denoted    a+ρ    ρ, σ = a, a − 1, . . . , −a, 2a 0   2a = 0, 1, 2, . . . ,   a+σ and is a special case of the operator defined by (2.87):    a+ρ    2a 0 |( j1 j2 ) jm     a+σ

1 (2a + 1)(2 j2 + 1) 2 (aa)0 T(ρσ)0 |( j1 j2 ) jm . (2.96) = (2 j2 + 2σ + 1) Thus, a Racah operator is an invariant operator with respect to the total angular momentum J. Alternative definitions are:    a+ρ    2a 0     a+σ * + * +†  a−σ a+ρ a+σ = (−1) 2a 0 ⊗ 2a 0 , a+α a+α a    a+ρ  &  2a 0 &( j1 j2 ) jm     a+σ 1

= [(2 j1 + 2ρ + 1)(2 j2 + 1)] 2

× W( j, j1 , j2 + σ, a; j2 , j1 + ρ) × |( j1 + ρ, j2 + σ) jm with conjugate     

a+ρ 2a a+σ

†   & &( j1 j2 ) jm 0   1

= [(2 j1 + 1)(2 j2 − 2σ + 1)] 2 × W( j, j1 − ρ, j2 , a; j2 − σ, j1 ) × |( j1 − ρ, j2 − σ) jm . Racah operators satisfy orthogonality relations similar in form to Wigner operators. The open product rule

  

   

b+σ 2b

0

  

a+ρ 2a

0

  a+α    c+ρ+σ     abc abc = W ρ,σ,ρ+σ W α,β,α+β 2c 0 .     c c+α+β

 

b+β

  

(2.97) abc W ρστ

abc W α,β,α+β

In this result and denote Racah invariants with respect to the angular momenta J(1) and J(2), respectively, so that abc

abc W ρστ |( j1 j2 ) jm = Wρστ ( j1 )|( j1 j2 ) jm , abc abc |( j1 j2 ) jm = Wαβγ ( j2 )|( j1 j2 ) jm . W αβγ

The matrix elements of relation (2.97) lead to the Biedenharn–Elliott identity. There are five versions of this relationship in complete analogy to relations (2.63– 2.65) and (2.69–2.70) for Wigner operators. Racah operators are a basis for all invariant operators acting in the tensor product space spanned by the coupled basis vectors (2.85) and are the natural way of formulating interactions in that space. Their algebra is a fascinating study, initiated already in a different guise in the work of Schwinger [2.3]. Little use has been made of this concept in physical applications. Additional relations between Racah coefficients or 6– j coefficients may be derived from the various versions of the rule (2.97) or directly from relation (2.79b) by using the orthogonality relations (2.75). Two of these are:  (−1)a+b+e (2e + 1) e

 a a b e × b d c g   c b = (−1)φ1 d d 

 a a c  b e d    b a a g g d

 e b d c  c , c

φ1 = g + a + b + c + c + d  + d ;    (−1)a−c +e−e (2e + 1)(2e + 1)(2 f + 1) e,e

   a a c a b e a e e × d c g b e e d d c   a a c = δ fg (−1)φ2 , g d c 

c b e d d f



φ2 = g + a − b + c + d  + d .

Angular Momentum Theory

The W-coefficient form of these relations is obtained by deleting all phase factors and making the substitution (2.74), ignoring the phase factor. There are no phase factors in the corresponding W-coefficient relations.

     j1 j2 j3  Triangles associated with the 9– j coefficient j4 j5 j6 :     j7 j8 j9

vertices to obtain V4 = y1 y2 x6 x9 + y1 y3 x5 x8 + y2 y3 x4 x7 + y4 y5 x3 x9 + y4 y6 x2 x8 + y5 y6 x1 x7 + y7 y8 x3 x6 + y7 y9 x2 x5 + y8 y9 x1 x4 . Edge function: 

 x1 y1 x2 y2 x3 y3   E 6 = det  x4 y4 x5 y5 x6 y6  . x7 y7 x8 y8 x9 y9

Generating function [2.4–6]: ( j1 j2 j3 ) , ( j2 j5 j8 ) ,

( j4 j5 j6 ) , ( j3 j6 j9 ) .

( j7 j8 j9 ) ,

( j1 j4 j7 ) ,

Points in R3 associated with the triangles:

(1 − V4 + E 6 )−2 = 

∆

Z =



C(∆)Z ∆ ,

∆

(z a , z b , z c )( ja jb jc )

[see (2.82, )] ,

all vertices

( j1 j2 j3 ) → (x1 , x2 , x3 ), ( j4 j5 j6 ) → (x4 , x5 , x6 ) , ( j7 j8 j9 ) → (x7 , x8 , x9 ), ( j1 j4 j7 ) → (y1 , y4 , y7 ) , ( j2 j5 j8 ) → (y2 , y5 , y8 ), ( j3 j6 j9 ) → (y3 , y6 , y9 ) . Cubic graph C 6 in R3 associated with the points: The points define the vertices of a cubic graph C 6 on six points with lines joining each pair of points that share a common subscript, and the lines are labeled by the products xi yi , where i is the common subscript (Fig. 2.3). Cubic graph C 6 functions: Interchange the symbols x and y in the coordinates of the vertices of the cubic graph C 6 , and define the following polynomials on the vertices and edges of the C 6 with this modified labeling: Vertex function: multiply together the coordinates of each pair of adjacent vertices, divide out the coordinates with a common subscript, and sum over all pairs of

Fig. 2.3 Labeled cubic graph associated with the 9– j coef-

ficient

51

Part A 2.10

2.10.7 Schwinger–Wu Generating Function and its Combinatorics

2.10 The 9–j Coefficients

  ( j1 j2 j3 )   ( j4 j5 j6 )   ( j7 j8 j9 )  ∆= ( j j j ) ,  1 4 7    ( j2 j5 j8 ) ( j3 j6 j9 )  C(∆) = (−1)k10 +k11 +k12 (k + 1) k




a

k , k1 , . . . , k9 , k10 , . . . , k15 8 where summation
is over all 3 × 3 square arrays of nonnegative integers k j ( j = 1, 2, . . . , 9) with fixed row and column sums given by   k1 k2 k3 k − t1    k4 k5 k6  k − t2 k7 k8 k9 k − t3 k − t4 k − t5 k − t6 8 and for each such array the summation a is over all nonnegative integers a such that the following quantities are nonnegative integers:

×

k10 = −a + k1 − k + j2 + j3 + j4 + j7 , k11 = −a + k6 − k + j3 + j4 + j5 + j9 , k12 = −a + k8 − k + j2 + j5 + j7 + j9 , k13 = a + k5 − k1 − j3 + j6 − j7 + j8 , k14 = a + k2 − k6 + j1 − j4 + j8 − j9 , k15 = a .

52

Part A

Mathematical Methods

Note that 15 

ki = −2k +

i=10

9 

ji .

i=1

The ti are the following triangle sums:

Part A 2.11

t1 = t3 = t4 = t6 =

j1 + j7 + j1 + j3 +

j2 + j8 + j4 + j6 +

j3 , t2 = j4 + j5 + j6 , j9 , j7 , t5 = j2 + j5 + j8 , j9 .

The 9– j coefficient is given by      j1 j2 j3 j4 j5 j6 = ∆( j1 j2 j3 )∆( j4 j5 j6 )∆( j7 j8 j9 )     j7 j8 j9 × ∆( j1 j4 j7 )∆( j2 j5 j8 )∆( j3 j6 j9 )C(∆) . The coefficient C(∆) is an integer associated with each cubic graph C6 that counts the number of occurrences of the monomial term Z ∆ in the expansion of (1 − V4 + E 6 )−2 .

2.11 Tensor Spherical Harmonics Tensor spherical or tensor solid harmonics are special cases of the coupling of two irreducible tensor operators in the tensor product space given in Sect. 2.7.2. They are defined by  ls j Y(ls) jm = Cm−ν,ν,m Yl,m−ν ⊗ ξν

where , denotes the inner product in the space Hl ⊗ Hs , ( , ) the inner product in Hl , and ( , ) the inner product in Hs . Operator actions: J 2 Y(ls) jm = j( j + 1)Y(ls) jm , J3 Y(ls) jm = mY(ls) jm ,

ν

and belong to the tensor product space Hl ⊗ Hs , where the orthonormal bases of the spaces Hl and Hs are: ( ) Ylµ : µ = l, l − 1, . . . , −l , {ξν : ν = s, s − 1, . . . , −s} . The orbital angular momentum L has the standard action on the solid harmonics, and a second set of kinematically independent angular momentum operators S has the standard action on the basis set of Hs . The total angular momentum is: J = L ⊗ 1 + 1 ⊗ S , The set of vectors { Y(ls) jm : m = j, j − 1, . . . , − j; (ls j ) obey the triangle conditions } has the following following properties: Orthogonality: ' %    Y(l s) j m , Y(ls) jm  l s j  ls j Cm  −ν ,ν ,m  Cm−ν,ν,m (Yl  ,m  −ν , Yl,m−ν ) = νν

× (ξν , ξν ) = δ j  j δl  l δm  m ,

(L ⊗ 1 )Y(ls) jm = l(l + 1)Y(ls) jm , 2

(1 ⊗ S2 )Y(ls) jm = s(s + 1)Y(ls) jm , J 2 = L2 ⊗ 1 + 1 ⊗ S2 + 2



L i ⊗ Si ,

i 1

J± Y(ls) jm = [( j ∓ m)( j ± m + 1)] 2 Y(ls) j,m±1 . Transformation property under unitary rotations:  j  Dm  m (ψ, n)Y exp(−iψ nˆ · J)Y(ls) jm = ˆ (ls) jm . m

Special realization: The eigenvectors ξν are often replaced by column matrices: ξν = col(0 · · · 010 · · · 0) , 1 in position s − ν + 1, ν = s, s − 1, . . . , −s . The operators S = (S1 , S2 , S3 ) are correspondingly replaced by their standard (2s + 1) × (2s + 1) matrix representations Si(s) . The tensor product of operators becomes a (2s + 1) × (2s + 1) matrix containing both operators and numerical matrix elements, e.g., Ji = L i I2s+1 + Si(s) , in which L i is a differential operator multiplying the unit matrix, that is, L i is repeated 2s + 1 times along the diagonal.

Angular Momentum Theory

9 ( j−m+1)( j−m+2)

2.11.1 Spinor Spherical Harmonics as Matrix Functions

2( j+1)(2 j+3)

Y 

Y

9 j− 12 , 12

j+ 12 , 12

jm

jm

 = 9

j+m 2j

Y j− 1 ,m− 1

j−m 2j

Y j− 1 ,m+ 1

2

2

  ,

     9    ( j−m+1)( j+m+1) = − Y j+1,m  . ( j+1)(2 j+3)     9  ( j+m+2)( j+m+1) Y j+1,m+1 2( j+1)(2 j+3)

∇ 2 Y(l1) jm = 0 ,

2

2



Eigenvalue properties:

 9  − j−m+1 Y 1 1 j+ ,m− 2 j+2 2 2  = 9 . j+m+1 Y 1 1 j+ ,m+ 2 j+2 2

Y( j+1,1) jm

Y j+1,m−1

2

J 2 Y(l1) jm = j( j + 1)Y(l1) jm , J3 Y(l1) jm = mY(l1) jm , L2 Y(l1) jm = l(l + 1)Y(l1) jm , S2 Y(l1) jm = 2Y(l1) jm .

2.11.2 Vector Spherical Harmonics as Matrix Functions

2.11.3 Vector Solid Harmonics as Vector Functions

Choose ξ+1 = col(1, 0, 0), ξ0 = col(0, 1, 0), ξ−1 = col(0, 0, 1), and S the 3 × 3 angular momentum matrices given by

Vector spherical and solid harmonics can also be defined and their properties presented in terms of the ordinary solid harmonics, using the vectors x, ∇, and L, and the operations of divergence and curl: Defining equations:

 0  S+ = 0 0  1  S3 = 0 0

 √ 2 0 √  0 2 , 0 0  0 0  0 0. 0 −1



 0 0 0 √  S− =  2 0 0 , √ 0 2 0

9 ( j+m−1)( j+m)

Y( j−1,1) jm

Y( j1) jm

1

Y(l+1,1)lm = − [(l + 1)(2l + 1)]− 2 [(l + 1)x + ix × L]Ylm , Y(l1)lm = [l(l + 1)]−1/2 LYlm , 1

r 2 Y(l−1,1)lm = − [l(2l + 1)]− 2 × (−lx + ix × L)Ylm .

The vector spherical harmonics are the following, where j ∈ {0, 1, 2, . . . }: 

Y j−1,m−1 2 j(2 j−1)      9    ( j−m)( j+m) = , Y j−1,m j(2 j−1)     9  ( j−m−1)( j−m) Y j−1,m+1 2 j(2 j−1)  9  j−m+1) − ( j+m)( Y j,m−1 2 j( j+1)         m √ Y = , j,m j( j+1)     9  ( j−m)( j+m+1) Y j,m+1 2 j( j+1)

Eigenvalue properties: J 2 Y(l1) jm = j( j + 1)Y(l1) jm , L2 Y(l1) jm = l(l + 1)Y(l1) jm , S2 Y(l1) jm = 2Y(l1) jm , J3 Y(l1) jm = mY(l1) jm , ∇ 2 Y(l1) jm = 0 , 2iL × Y(l1) jm = [ j( j + 1) − l(l + 1) − 2]Y(l1) jm . Orthogonality:      dSxˆ Y(l 1) j m ∗ (x) · Y(l1) jm (x) = δl  l δ j  j δm  m r l +l , where the integration is over the unit sphere in R3 . Complex conjugation: Y(l1) jm∗ = (−1)l+1− j (−1)m Y(l1) j,−m .

53

Part A 2.11

Choose ξ+1/2 = col(1, 0), ξ−1/2 = col(0, 1), and S = σ/2. The spinor spherical harmonics or Pauli central field spinors are the following, where j ∈ {1/2, 3/2, . . . } : 

2.11 Tensor Spherical Harmonics

54

Part A

Mathematical Methods

Part A 2.12

Vector and gradient formulas:

1 l + 1 2 (l+1,1)lm xYlm = − Y 2l + 1

1 2 l + r 2 Y(l−1,1)lm , 2l + 1 [(l + 1)∇ + i∇ × L](FYlm )

1 1 dF = −[(l + 1)(2l + 1)] 2 Y(l+1,1)lm , r dr [−l∇ + i∇ × L](FYlm )   1 dF + (2l + 1)F Y(l−1,1)lm , = −[l(2l + 1)] 2 r dr

1

l + 1 2 1 dF Y(l+1,1)lm ∇(FYlm ) = − 2l + 1 r dr

1   2 dF l r + (2l + 1)F Y(l−1,1)lm , + 2l + 1 dr

1

l + 1 2 1 dF Y(l+1,1)lm i∇ × L(FYlm ) = −l 2l + 1 r dr

1 2 l − (l + 1) 2l + 1   dF + (2l + 1)F Y(l−1,1)lm . × r dr Curl equations:

1 2 l )=− i∇ × (FY 2l + 1   dF + (2l + 3)F Y(l1)lm , × r dr

1 2 l (l1)lm i∇ × (FY )=− 2l + 1

1 1 dF l +1 2 Y(l+1,1)lm − × r dr 2l + 1   dF + (2l + 1)F Y(l−1,1)lm , × r dr (l+1,1)lm



1

  l + 1 2 1 dF Y(l1)lm . i∇× FY(l−1,1)lm = − 2l + 1 r dr Divergence equations: ∇ · (FY(l+1,1)lm ) =

1   dF l +1 2 r + (2l + 3)F Ylm , − 2l + 1 dr ∇ · (FY(l1)lm ) = 0 ,

1

2 1 dF l (l−1,1)lm Ylm . ∇ · (FY )= 2l + 1 r dr Parity property: Y(l+δ,1)lm (−x) = (−1)l+δ Y(l+δ,1)lm (x) . Scalar product: 





Y(l 1) j m · Y(l1) jm =

12    l+l  −l  (2 j + 1)(2 j + 1)(2l + 1)(2l + 1) r 4π(2l  + 1)  l





 

jj  l 

ll l × (−1)l+ j +l C000 Cm,m  ,m+m      l j 1 Yl  ,m+m  . × j l l 

Cross product:

 √   l+l  −l     r Y(l 1) j m × Y(l1) jm = − i 2 l  j 

1 (2 j + 1)(2 j  + 1)(3)(2l + 1)(2l  + 1) 2 4π    l 1 j         ll  l  jj j × C000 Cm,m  ,m+m  l  1 j  Y(l 1) j ,m+m .      l 1 j 

×

Conversion to spherical harmonic form: Y(l+δ,1)lm (x) = r l+δ Y(l+δ,1)lm (ˆx) , with appropriate modification of F to account for the factor r l+δ .

2.12 Coupling and Recoupling Theory and 3n–j Coefficients 2.12.1 Composite Angular Momentum Systems

vector spaces H j with orthonormal basis

An “elementary” angular momentum system is one whose state space can be written as a direct sum of

on which the angular momentum J has the standard action, and which under unitary transformation by

{| jm |m = j, j − 1, . . . , − j}

Angular Momentum Theory

exp(−iψ nˆ · J) undergoes the standard unitary transformation. A composite angular momentum system is one whose state space is a direct sum: of the tensor product spaces H j1 j2 ··· jn of dimension nα=1 (2 jα + 1) with orthonormal basis in the tensor product space of the elementary systems given by (2.98)

each m α = jα , jα − 1, . . . , − jα . The following properties then hold for the composite system: Independent rotations of the elementary parts: ; " ! "< ! exp −iψ1 nˆ 1 · J(1) ⊗ · · · ⊗ exp −iψn nˆ n · J(n) × | j1 m 1 ⊗ · · · ⊗ jn m n " ! = exp −iψ1 nˆ 1 · J(1) | j1 m 1 ⊗ · · · " ! ⊗ exp −iψn nˆ n · J(n) | jn m n $ # D j1 (U1 ) × · · · × D jn (Un )  = & & × & j1 m 1 ⊗ · · · ⊗ & jn m n , #

(2.99)

$ D j1 (U1 ) × · · · × D jn (Un ) j

= Dm1 m (U1 ) · · · Dmn m n (Un ) , 1

1

n

Uα = U(ψα , nˆ α ) ∈ SU(2) ,

α = 1, 2, . . . , n .

Multiple Kronecker (direct) product group SU(2) × · · · × SU(2): Group elements: (U1 , . . . , Un ) ,

J = J(1) + J(2) + · · · + J(n) , in which the k-th term in the sum is to be interpreted as the tensor product operator: I1⊗· · ·⊗J(k)⊗· · ·⊗In , Iα = unit operator in H jα . The basic problem for composite systems: The basic problem is to reduce the n-fold direct product representation (2.101) of SU(2) into a direct sum of irreducible representations, or equivalently, to find all subspaces H j ⊂ H j1 j2 ··· jn , j ∈ {0, 1/2, 1, . . . }, with orthonormal bases sets {| jm |m = j, j − 1, . . . , − j} on which the total angular momentum J has the standard action. Form of the solution:  j j ··· jn j | ( j1 j2 · · · jn )(k) jm = Cm11 m2 2 ···m n m (k) all m α mα = m

× | j1 m 1 ⊗ | j2 m 2 ⊗ · · · ⊗ | jn m n ,

(2.102)

J 2 (α) = J12 (α) + J22 (α) + J32 (α), J 2 (α) | ( j1 j2 · · · jn )(k) jm = jα ( jα + 1) | ( j1 j2 · · · jn )(k) jm , α = 1, 2, . . . , n .

(2.103)

Total angular momentum properties imposed:

each Uα ∈ SU(2) .

Multiplication rule:      U1 , . . . , Un (U1 , . . . , Un ) = U1 U1 , . . . , Un Un . Irreducible representations: D j1 (U1 ) × · · · × D jn (Un ) .

8

m = j, j − 1, . . . , − j; index set (k) unspecified. Diagonal operators:

m 1 ···m n ;m 1 ···m n

j

Total angular momentum of the composite system:

m 1 ···m n ;m 1 ···m n

m 1 ···m n

J 2 | ( j1 j2 · · · jn )(k) jm = j( j + 1) | ( j1 j2 · · · jn )(k) jm , J3 | ( j1 j2 · · · jn )(k) jm = m | ( j1 j2 · · · jn )(k) jm ,

(2.100)

J± | ( j1 j2 · · · jn )(k) jm 1

= [( j ∓ m)( j ± m + 1)] 2

Rotation of the composite system as a unit: Common rotation:

× | ( j1 j2 · · · jn )(k) jm ± 1 .

U1 = U2 = · · · = Un = U ∈ SU(2) . Diagonal subgroup SU(2) ⊂ SU(2) × · · · × SU(2) : (U, U, . . . , U ),

each U ∈ SU(2) .

D j1 (U ) × · · · × D jn (U ) .

(2.104)

Properties of the index set (k): Reduction of Kronecker product (2.101):  D j1 × D j2 × · · · × D jn = ⊕n j D j , j

  (2 jα + 1) = n j (2 j + 1) .

Reducible representation of SU(2): (2.101)

55

α

j

(2.105)

Part A 2.12

| j1 m 1 ⊗ | j2 m 2 ⊗ · · · ⊗ | jn m n ,

2.12 Coupling and Recoupling Theory and 3n–j Coefficients

56

Part A

Mathematical Methods

Part A 2.12

For fixed j1 , j2 , . . . , jn , and j, the index set (k) must enumerate exactly n j perpendicular spaces H j . Incompleteness of set of operators: There are 2n commuting Hermitian operators diagonal on the basis (2.98): & ) ( 2 (2.106a) J (α), J3 (α) & α = 1, 2, . . . , n . There are n + 2 commuting Hermitian operators diagonal on the basis (2.102): & ) ( 2 (2.106b) J , J3 ; J 2 (α) & α = 1, 2, . . . , n . There are n − 2 additional commuting Hermitian operators, or other rules, required to complete set (2.106b) and determine the indexing set (k). Basic content of coupling and recoupling theory: Coupling theory is the study of completing the operator set (2.106b), or the specification of other rules, that uniquely determine the irreducible representation spaces H j occurring in the Kronecker product reduction (2.105). Recouping theory is the study of the inter-relations between different methods of effecting this reduction; it is a study of relations between the different ways of spanning the multiplicity space H j ⊕ H j ⊕ · · · ⊕ H j (n j terms) .

2.12.2 Binary Coupling Theory: Combinatorics Binary coupling of angular momenta refers to the selecting any pair of angular momentum operators from the set of individual system angular momenta

n = 2 : J1 + J2 ; n = 3 : (J1 + J2 ) + J3 , J1 + (J2 + J3 ) ; n = 4 : (J1 + J2 ) + (J3 + J4 ) , [(J1 + J2 ) + J3 ] + J4 , [J1 + (J2 + J3 )] + J4 , J1 + [(J2 + J3 ) + J4 ] , J1 + [J2 + (J3 + J4 )] . It is customary to use the ordered sequence j1 j2 · · · jn

(2.108)

of angular momentum quantum numbers in place of the angular momentum operators in (2.107). Thus, the five placement of parentheses for n = 4 becomes: ( j1 j2 )( j3 j4 ) , [( j1 j2 ) j3 ] j4 , [ j1 ( j2 j3 )] j4 , j1 [( j2 j3 ) j4 ] , j1 [ j2 ( j3 j4 )] . (It is also customary to omit the last parentheses pair, which encloses the whole sequence.) A sequence (2.108) into which pairwise insertions of parentheses has been completed is called a binary bracketing of the sequence, and denoted by ( j1 j2 · · · jn ) B . This symbol may also be called a coupling symbol. The total number of coupling symbols, that is, the total number of elements an in the set {( j1 j2 · · · jn ) B |B is a binary bracketing}

{J(1), J(2), . . . , J(n)} , and carrying out the “addition of angular momenta” for that pair by coupling the corresponding states in the tensor product space by the standard use of SU(2) WCGcoefficients; this is followed by addition of a new pair, which may be a pair distinct from the first pair, or the addition of one new angular momentum to the sum of the first pair, etc. If the order 1, 2, . . . , n of the angular momenta is kept fixed in J1 + J2 + · · · + Jn ,

The procedure is clear from the following cases for n = 2, 3, and 4:

(2.107)

one is led to the problem of parentheses. (To avoid misleading parentheses, the notation Jα = J(α) is used in this section.) This is the problem of introducing pairs of parentheses into expression (2.107) that specify the coupling procedure that is to be implemented.

is given by the Catalan numbers: 
1 2n − 2 , n = 2, 3, . . . . an = n n −1 Effect of permuting the angular momenta: Since the position of an individual vector space in the tensor product H j1 ⊗ · · · ⊗ H jn is kept fixed, the meaning of a permutation of the jα in the sequence (2.108) corresponding to a given binary bracketing is to permute the positions of the terms in the summation for the total angular momentum, e.g., ( j1 j2 ) j3 → ( j3 j1 ) j2 corresponds to (J1 ⊗ I2 ⊗ I3 + I1 ⊗ J2 ⊗ I3 ) + I1 ⊗ I2 ⊗ J3 = (I1 ⊗ I2 ⊗ J3 + J1 ⊗ I2 ⊗ I3 ) + I1 ⊗ J2 ⊗ I3 .

Angular Momentum Theory

(2.109)

is cn = n!an = (n)n−1 = n(n + 1) · · · (2n − 2). Caution: One should not assign numbers to the symbols jα , since these symbols serve as noncommuting, nonassociative distinct objects in a counting process. Binary subproducts: A binary subproduct in the coupling symbol ( jα1 jα2 · · · jαn ) B is the subset of symbols between a given parentheses pair, say, {xy}. The symbols x and y may themselves contain binary subproducts. Commutation of a binary subproduct is the operation {xy} → {yx}. For example, the coupling symbol {[( j1 j2 ) j3 ] j4 } contains three binary subproducts, {xy}, [xy], (xy). Equivalence relation: Two coupling symbols are defined to be equivalent ( jα1 jα2 · · · jαn ) B ∼ ( jα1 jα2 · · · jαn ) B if one can be obtained from the other by commutation of the symbols in the binary subproducts. Such commutations change the overall phase of the state vector (2.102) corresponding to a particular coupling symbol, and such states are counted as being the same (equivalent). Number of inequivalent coupling schemes: The equivalence relation under commutation of binary subproducts partitions the set (2.109) into equivalence classes, each containing 2n−1 elements. There are dn = cn /2n−1 = (2n − 3)!! inequivalent coupling schemes in binary coupling theory. Thus, for n = 4, there are 5!! = 5 × 3 × 1 = 15 inequivalent binary coupling schemes. Type of a coupling symbol: The type of the coupling symbol ( jα1 jα2 · · · jαn ) B is defined to be the symbol obtained by setting all the jα equal to a common symbol, say, x. Thus, the type of the coupling symbol {[( j1 j2 ) j3 ] j4 } is {[(x 2 )x]x}. The Wedderburn–Etherington number bn gives the number of coupling symbols of distinct types, counting two symbols as equivalent if they are related by commutation of binary subproducts. A closed form of these numbers is not known, although generating functions exist. The first few numbers are: n 1 2 3 4 5 6 7 8 9 10 bn 1 1 1 2 3 6 11 23 46 98

57

There are 15 nontrivial coupling schemes for 4 angular momenta, and they are classified into 2 types, allowing commutation of binary subproducts: !  " Type x 2 x x [( j1 j2 ) j3 ] j4 , [( j2 j3 ) j1 ] j4 , [( j3 j1 ) j2 ] j4 [( j1 j2 ) j4 ] j3 , [( j2 j4 ) j1 ] j3 , [( j4 j1 ) j2 ] j3 [( j1 j3 ) j4 ] j2 , [( j3 j4 ) j1 ] j2 , [( j4 j1 ) j3 ] j2 [( j2 j3 ) j4 ] j1 , [( j3 j4 ) j2 ] j1 , [( j4 j2 ) j3 ] j1    Type x 2 x 2 ( j1 j2 )( j3 j4 ), ( j1 j3 )( j2 j4 ), ( j2 j3 )( j1 j4 )

2.12.3 Implementation of Binary Couplings Each binary coupling scheme specifies uniquely a set of intermediate angular momentum operators. For example, the intermediate angular momenta associated with the coupling symbol [( j1 j2 ) j3 ] j4 are J(1) + J(2) = J(12) , J(12) + J(3) = J(123) , J(123) + J(4) = J , where J is the total angular momentum. Each coupling symbol ( jα1 jα2 · · · jαn ) B , defines exactly n − 2 intermediate angular momentum operators K (λ), λ = 1, 2, . . . , n − 2. The squares of these operators completes the set of operators (2.106b) for each coupling symbol; that is, the states vectors satisfying (2.103–2.104) and the following equations are unique, up to an overall choice of phase factor: & K 2 (λ) & ( jα1 jα2 · · · jαn ) B (k1 k2 · · · kn−2 ) jm = kλ (kλ + 1)|( jα1 jα2 · · · jαn ) B (k1 k2 · · · kn−2 ) jm , (2.110) λ = 1, 2, . . . , n − 2, n > 2 . The intermediate angular momentum operators K 2 (λ) depend, of course, on the choice of binary couplings implicit in the symbol ( jα1 jα2 · · · jαn ) B . The vectors have the following properties: Orthonormal basis of H j1 (1) ⊗ · · · ⊗ H jn (n) :  δkλ kλ , ( jα ) B (k ) jm|( jα ) B (k) jm = λ

( jα ) = ( jα1 , jα2 , . . . , jαn ) , (k) = (k1 , k2 , . . . , kn−2 ) ,    (k ) = k1 , k2 , . . . , kn−2 . The range of each kλ is uniquely determined by the Clebsch–Gordan series and the binary couplings in the

Part A 2.12

Total number of binary bracketing schemes including permutations: The number of symbols in the set &   & B is a binary bracketing and   &   & ( jα1 jα2 · · · jαn ) B & α1 α2 · · · αn is a permutation &     & of 1, 2, . . . , n

2.12 Coupling and Recoupling Theory and 3n–j Coefficients

58

Part A

Mathematical Methods

coupling symbol. Together these ranges enumerate exactly the multiplicity n j of H j occurring in the reduction of the multiple Kronecker product. Uniqueness of state vectors:

Part A 2.12

The fundamental theorem of binary coupling theory states for inequivalent coupling schemes is: Each recoupling coefficient is expressible as a sum over products of Racah coefficients, the only other quantities occurring in the summation being phase and dimension B | ( jα1 jα2 · · · jαn ) (k1 k2 · · · kn−2 ) jm factors. 
B  In every instance, the summation over projection quan j jα1 · · · jαn = C (k1 , . . . , kn−2 ) tum numbers in the right-hand side of (2.111) is m 8 m α1 · · · m αn re-expressible as a sum over Racah coefficients. m α =m

× | j1 m 1 ⊗ · · · ⊗ | jn m n .  In the C-coefficient, the jα are paired in the binary mα bracketing. Each such C-coefficient is a summation over a unique product of n − 1 SU(2) WCG-coefficients. Equivalent basis vectors: | ( jα1 jα2 · · · jαn ) B (k1 k2 · · · kn−2 ) jm = ± | ( jα1 jα2 · · · jαn ) B (k1 k2 · · · kn−2 ) jm , if and only if ( jα1 jα2 · · · jαn ) B ∼ ( jα1 jα2 · · · jαn ) B . Inequivalent basis vector are orthonormal in all quantum numbers labeling the state vector. Recoupling coefficients: A recoupling coefficient is a transformation coefficient % ' &  & ( jβ ) B & (l) jm & ( jα ) B (k) jm relating any two orthonormal bases of the space H j1 ⊗ · · · ⊗ H jn , say, the one defined by (2.103, 2.104), and (2.110) for a prescribed coupling scheme corresponding to a bracketing B, and a second one, again defined by these relations but for a different coupling scheme corresponding to a bracketing B  . For example, for n = 3, there are 3 inequivalent coupling symbols  and 3 = 3 recoupling coefficients; for n = 4, there 2  are 15 inequivalent coupling symbols and 15 = 105 2 recoupling coefficients. Each coefficient is, of course, expressible as a sum over products of 2(n − 1) WCGcoefficients, obtained simply by taking the inner product: % ' &  ( jβ ) B (l) jm & ( jα ) B (k) jm 
 B   j j · · · j β1 βn C (l) = m 8 m β1 · · · m βn m α =m 
B  j jα1 · · · jαn ×C  (k) . m m α1 · · · m αn (2.111)

2.12.4 Construction of all Transformation Coefficients in Binary Coupling Theory Augmented notation: The coupling symbol ( jα1 jα2 · · · jαn ) B contains all information as to how n angular momenta are to be coupled, but is not specific in how the intermediate angular momentum quantum numbers (k1 k2 · · · kn−2 ), are to be matched with the binary couplings implicit in the coupling symbol. For explicit calculations, it is necessary to remedy this deficiency in notation. This may be done by attaching the n − 2 intermediate angular momentum quantum numbers and the total angular momentum j as subscripts to the n − 1 parentheses pairs in the coupling symbol. For example, for ( j1 j2 j3 j4 j5 ) B = {[( j1 j2 )( j3 j4 )] j5 }, this results in the replacement {[( j1 j2 )( j3 j4 )] j5 }(k1 k2 k3 ) → {[( j1 j2 )k1 ( j3 j4 )k2 ]k3 j5 } j . The basic coupling symbol structure is regained simply by ignoring all inferior letters. Basic rules for commutation and association: Let x, y, z denote arbitrary disjoint contiguous subcoupling symbols {[(x)(y)](z)} contained in the coupling symbol ( jα1 jα2 · · · jαn ) B . Let a, b, c denote the intermediate angular momenta associated with addition of the angular momenta represented in x, y, z, respectively, d the angular momentum representing the sum of a and b, and k the sum of d and c. Symbolically, this subcoupling may be presented as   J(x) = J(a) , J(y) = J(b) ,  J(z) = J(c) , J = · · · {[J(a) + J(b)] + J(c)} · · · ; J(a) + J(b) = J(d); J(d) + J(c) = J(k) with augmented coupling symbol ( jα1 jα2 · · · jαn ) B = · · · {[(x)a (y)b ]d (z)c }k · · · .

Angular Momentum Theory

There are only two basic operations in constructing the recoupling coefficient between any two coupling schemes: commutation of symbols: (x)a (y)b → (y)b (x)a

59

the associative of symbol effects a Racah coefficient transformation: & 4 {[(ab)e c] f d}g & [(ac)h (bd)k ]g = (−1)e+h−a− f [(2 f + 1)(2k + 1)]1/2 W(hbgd; fk) × [(2e + 1)(2h + 1)]1/2 W(bafc; eh) .

| · · · [(x)a (y)b ]d · · ·

9– j coefficient as recoupling coefficient: φ

R

→ (−1)a+b−d | · · · [(y)b (x)a ]d · · · = | · · · [(x)a (y)b ]d · · · . Association of symbols:

φ

R

→ [(ab)c]d → (ab)(cd) , & 4 [(ab)e (cd) f ]g & [(ac)h (bd)k ]g 1

[(x)a (y)b ](z)c → (x)a [(y)b (z)c ] with the transformation of state vector given by | · · · {[(x)a (y)b ]d (z)c }k · · ·  → [(2d + 1)(2e + 1)]1/2 W(abkc; de) e

× | · · · {(x)a [(y)b (z)c ]e }k · · · = | · · · {[(x)a (y)b ]d (z)c }k · · · . The basic result for the calculation of all recoupling coefficients is: Each pair of coupling schemes for n angular momenta can be brought into coincidence by a series of commutations and associations performed on either of the set of coupling symbols defining the coupling scheme. In principle, this result gives a method for the construction of all recoupling transformation coefficients and sets the stage for the formulation of still deeper questions arising in recoupling theory, as summarized in Sect. 2.12.5. The following examples illustrate the content of the preceding abstract constructions. Examples: WCG-coefficient form: & 4 {[(ab)e c] f d}g & [(ac)h (bd)k ]g  ec f f dg abe = Cα,β,α+β Cα+β,γ,α+β+γ Cα+β+γ,δ,m α+β+γ +δ=m

hkg

ach bdk × Cα,γ,α+γ Cβ,δ,β+δ Cα+γ,β+δ,m .

6– j coefficient as recoupling coefficient: φ

R

(ac)(bd) → [(ac)b]d → [b(ac)]d → [(ba)c]d

R

(ac)(bd) → [(ac)b]d → [b(ac)]d → [(ba)c]d φ

→ [(ab)c]d , where φ denotes that the communication of symbols effects a phase factor transformation, and R denotes that

= [(2e + 1)(2 f + 1)(2h + 1)(2k + 1)] 2      e fg bdk ach 2l × (−1) (2l + 1) dlc ghl lbe l    abe    1 = [(2e + 1)(2 f + 1)(2h + 1)(2k + 1)] 2 cd f .     hkg

2.12.5 Unsolved Problems in Recoupling Theory 1. Define a route between two coupling symbols for n angular momenta to be any sequence of transpositions and associations that carries one symbol into the other. Each such route then gives rise to a unique expression for the corresponding recoupling coefficient in terms of 6– j coefficients. In general, there are several routes between the same pair of coupling symbols, leading therefore to identities between 6– j coefficients. How many nontrivial routes are there between two given coupling symbols, leading to nontrivial relations between 6– j coefficients (trivial means related by a phase factor)? 2. Only 6– j coefficients arise in all possible couplings of three angular momenta; only 6– j and 9– j coefficients arise in all possible couplings of four angular momenta; in addition to 6– j and 9– j coefficients, two new “classes” of coefficients, called 12– j coefficients of the first and second kind, arise in the coupling of five angular momenta; in addition to 6– j, 9– j, and the two classes of 12– j coefficients, five new classes of 15– j coefficients arise in the coupling of six angular momenta, · · · . What are the classes of 3n— j coefficients? The nonconstructive answer is that a summation over 6– j coefficients arising in the coupling of n angular momenta is of a new class if it cannot be expressed in terms

Part A 2.12

with the transformation of state vector given by

R

2.12 Coupling and Recoupling Theory and 3n–j Coefficients

60

Part A

Mathematical Methods

Part A 2.13

of previously defined coefficients occurring in the recoupling of n − 1 or fewer angular momenta. 3. Toward answering the question of classes of 3n– j coefficients, one is lead into the classification problem of planar cubic graphs. It is known that every 3n– j coefficient corresponds to a planar cubic graph, but the converse is not true. For small n, the relation between the coupling of n angular momenta, the number of new classes of 3(n − 1)– j coefficients, and the number of nonisomorphic planar cubic graphs on 2(n − 1) points is: n 3 4 5 6 7 8 9

Classes of 3(n − 1) − j coefficients 1 1 2 5 18 84 576

Cubic graphs on 2(n − 1) points 1 2 5 19 87 ? ?

The geometrical object for n = 3 is a planar graph isomorphic to the tetrahedron in 3-space. The classification of all nonisomorphic cubic graphs on 2(n − 1) points is an unsolved problem in mathematics, as is the classification of classes for 3(n − 1)– j coefficients.

b

a

a

b

c c

Fig. 2.4 The fundamental triangle [(ab)c] can be realized by lines or points

4. There are (at least) two methods of realizing the basic triangles of angular momentum theory in terms of graphs. The fundamental structural element [(ab)c] is represented either in terms of its points or in terms of its lines (Fig. 2.4): The right representation leads to the interpretation of recoupling coefficients as functions defined on pairs of labeled binary trees [2.1]; the left to the diagrams of the Jucys school [2.7, 8]. Either method leads to the relationship of recoupling coefficients to cubic graphs. 5. The approach of classifying 3n − j coefficients through the use of unit tensor operator couplings, Racah operators, 9– j invariant operators, and general invariant operators is undeveloped.

2.13 Supplement on Combinatorial Foundations The quantum theory of angular momentum can be worked out using the abstract postulates of the properties of angular momenta operators and the abstract Hilbert space in which they act. The underlying mathematical apparatus is the Lie algebra of the group SU(2) and multiple copies thereof. An alternative approach is to use special Hilbert spaces that realize all the properties of the abstract postulates and perform calculations within that framework. The framework must be sufficiently rich in structure so as to apply to a manifold of physical situations. This approach has been used often in our treatment; it is an approach that is particularly useful for revealing the combinatorial foundations of quantum angular momentum theory. We illustrate this concretely in this supplementary section. The basic objects are the polynomials defined by (2.24), which we now call SU(2) solid harmonics, where we change the notation slightly by interchanging the role of m and m  .

2.13.1 SU(2) Solid Harmonics The SU(2) solid harmonics are defined to be the homogeneous polynomials of degree 2 j in four commuting indeterminates given by j

Dm m  (Z) =

 ZA 3 , α!β! A!

(2.112)

(α:A:β)

in which the indeterminates Z and the nonnegative exponents A are encoded in the matrix arrays

 z 11 z 12 a11 a12 Z= , A= , z 21 z 22 a21 a22 XA =

2 

a

z ijij ,

i, j=1

α! = α1 !α2 ! ,

A! =

2 

(aij )!,

i, j=1

β! = β1 !β2 ! .

Angular Momentum Theory

The symbol (α : A : β) in (2.112) denotes that the matrix array A of nonnegative integer entries has row and column sums of its aij entries given in terms of the quantum numbers j, m, m  by

These SU(2) solid harmonics are among the most important functions in angular momentum theory. Not only do they unify the irreducible representations of SU(2) in any parametrization by the appropriate definition of the indeterminates in terms of generalized coordinates, they also include the popular boson calculus realization of state vectors for quantum mechanical systems, as well as the state vectors for the symmetric rigid rotator. The realization of the inner product is essential. Physical theory demands an inner product that is given in terms of integrations of wave functions over the variables of the theory, as required by the probabilistic interpretation of wave functions. It is the requirement that realizations of angular momentum operators be Hermitian with respect to the inner product for the spaces being used that assures the orthogonality of functions, so that one is able to take results from one realization of the inner product to another with compatibility of relations. Often, in combinatorial arguments, the inner product plays no direct role. The nomenclature SU(2) solid harmonics for the polynomials defined by (2.112) is by analogy with the term SO(3, R) solid harmonics for the polynomials described in Sect. 2.1. The polynomials Ylm (x), x = (x1 , x2 , x3 ) ∈ R3 are homogeneous of degree l. The angular momentum operator L2 is given by L2 = −r 2 ∇ 2 + (x · ∇)2 + (x · ∇) , which is a sum of two commuting operators −r 2 ∇ 2 and (x · ∇)2 + x · ∇, each of which is invariant under orthogonal transformations. The SO(3, R) solid harmonics are homogeneous polynomials of degree l that solve ∇ 2 Ylm (x) = 0, so that L2 Ylm (x) = l(l + 1)Ylm (x). The component angular momentum operators L i then have the standard action on these polynomials, and under real, proper, orthogonal transformations give the irreducible representations of the group SO(3, R). j The polynomials Dm m  (Z), z = (z 11 , z 21 , z 12 , z 22 ) ∈ C 4 are homogeneous of degree 2 j. The angular momentum operator J 2 , with J = (J1 , J2 , J3 ), is given

∂ + J0 (J0 + 1) , J 2 = −(det Z) det ∂Z 1 J0 = z · ∂ , 2

∂ ∂ ∂ ∂ , ∂= , , , ∂z 11 ∂z 21 ∂z 12 ∂z 22

(2.113)

which is a sum of two commuting operators  ∂ −(det Z) det ∂Z and J0 (J0 + 1), each of which is invariant under SU(2) transformations. The SU(2) solid harmonics are homogeneous polynomials of degree 2 j such that det

∂ j D  (Z) = 0 , ∂Z m m j j J 2 Dmm  (Z) = j( j + 1)Dmm  (Z) .

The components of the angular momentum operators J = (J1 , J2 , J3 ) = (M1 , M2 , M3 ) and K = (K 1 , K 2 , K 3 ) then have the standard action on these polynomials as given in Sect. 2.4.4, and under either left or right SU(2) transformations these polynomials give the irreducible representations of the group SU(2).

2.13.2 Combinatorial Definition of Wigner–Clebsch–Gordan Coefficients The SU(2) solid harmonics have a basic role in the interpretation of WCG-coefficients in combinatorial terms. We recall from Sect. 2.7.2 that the basic abstract Hilbert space coupling rule for compounding two kinematically independent angular momenta with components J1 = (J1 (1), J2 (1), J3 (1)) and J2 = (J1 (2), J2 (2), J3 (2)) to a total angular momentum J = (J1 , J2 , J3 ) = J1 + J2 is |( j1 j2 ) j m  j j j C 1 2 | j1 m 1 ⊗ | j2 m 2 . (2.114) = m 1 +m 2 =m m 1 m 2 m This relation in abstract Hilbert space is realized explicitly by spinorial polynomials as follows: ψ( j1

j2 ) jm (Z) =

 m 1 +m 2 =m

C

j1 j2 j m1 m2 m

× ψ j1 m 1 (z 11 , z 21 )ψ j2 m 2 (z 12 , z 22 ) , (2.115)

61

Part A 2.13

a11 + a12 = α1 = j + m, a21 + a22 = α2 = j − m , a11 + a21 = β1 = j + m  , a12 + a22 = β2 = j − m  .

by

2.13 Supplement on Combinatorial Foundations

62

Part A

Mathematical Methods

2 ψ( j1 j2 ) jm (Z) =

2j +1 ( j1 + j2 − j)!( j1 + j2 + j + 1)!

× (det Z) j1 + j2 − j Dm, j1 − j2 (Z) , j

(2.116)

Part A 2.13

x j+m y j−m

ψ jm (x, y) = √ . ( j + m)!( j − m)!

(2.117)

Explicit knowledge of the WCG-coefficients is not needed to prove these relationships. The angular momentum operators ∂ , J+ (1) = z 11 ∂z 21 1 ∂ z 11 J3 (1) = 2 ∂z 11 ∂ J+ (2) = z 12 , ∂z 22 1 ∂ J3 (2) = z 12 2 ∂z 12

∂ J− (1) = z 21 , ∂z

11 ∂ ; − z 21 ∂z 21 ∂ J− (2) = z 22 , ∂z 12

∂ − z 22 ∂z 22

are Hermitian in the polynomial inner product defined in Sect. 2.4.3, and have the standard action on the polynomials ψ j1 m 1 (z 11 , z 21 ) and ψ j2 m 2 (z 12 , z 22 ), respectively, which are normalized in the inner product ( , ). The components of total angular momentum operator J = M = J1 + J2 have the standard action on the polynomials ψ( j1 j2 ) j m (Z), since they have the standard j action on the factor Dm j1 − j2 (Z), as given in Sect. 2.4.4, ! " ∂ and [J, det X] = J, det ∂Z = 0. Thus, we have J 2 ψ( j1 j2 ) jm (Z) = j( j + 1)ψ( j1 j2 ) jm (Z), 3 J± ψ( j1 j2 ) jm (Z) = ( j ∓ m)( j ± m + 1) × ψ( j1 j2 ) jm±1 (Z) . We also note that the two commuting parts of J 2 are diagonal on these functions: J0 (J0 + 1)ψ( j1 j2 ) jm (Z) = ( j1 + j2 )( j1 + j2 + 1)ψ( j1 j2 ) jm (Z) ,

∂ ψ( j1 j2 ) jm (Z) (det Z) det ∂X = ( j1 + j2 − j)( j1 + j2 + j1 + 1)ψ( j1 j2 ) jm (Z) . It is necessary only to verify these properties for the j 2j √ highest weight function D j j (Z) = z 11 / (2 j)!, for which they are seen to hold.

The angular momentum operators K = (K 1 , K 2 , K 3 ) defined in Sect. 2.4.4 with components that commute with those of J = (M1 , M2 , M3 ) and having K 2 = J 2 also have a well-defined action on the functions ψ( j1 j2 ) j m (Z). The action of K + , K − , and K 3 on the quantum numbers   ( j1 , j2 ) is to  effect the shifts to  j1 + 12 , j2 − 12 , j1 − 12 , j2 + 12 , and ( j1 , j2 ), respectively. These actions of Hermitian angular momentum operators satisfying the standard commutation relations K × K = iK are quite unusual in that they depend only on the angular momentum quantum numbers j1 , j2 , j themselves, which satisfy the triangle rule, and give further interesting properties of the modified SU(2) solid harmonics ψ( j1 j2 ) j m (Z). We note these properties in full: K 2 ψ( j1 j2 ) jm (Z) = j( j + 1)ψ( j1 j2 ) jm (Z) , K 3 ψ( j1 j2 ) jm (Z) = ( j1 − j2 )ψ( j1 j2 ) jm (Z) , K + ψ( j1 j2 ) jm (Z) = 3 ( j − j1 + j2 )( j + j1 − j2 + 1)ψ j + 1 j − 1  jm (Z) , 1

K − ψ( j1 j2 ) jm (Z) = 3 ( j + j1 − j2 )( j − j1 + j2 + 1)ψ j

2 2

2

1 1 1 − 2 j2 + 2

 (Z) . jm

These relations play no direct role in our continuing considerations of (2.116) and the determination of the WCG-coefficients, and we do not interpret them further. The explicit WCG-coefficients are obtained by expanding the 2 × 2 determinant in (2.116), multiplying this expansion into the D-polynomial, and changing the order of the summation. These operations are most succinctly expressed in terms of the umbral calculus of Roman and Rota [2.6], using his evaluation operation. The evaluation at y of a divided power x k /k! of a single indeterminate x to a nonnegative integral power k is defined by (y)k y(y − 1) · · · (y − k + 1) xk y = = , eval y = k! k! k! k where (y)k is the falling factorial. This definition is extended to products by n n n  x ki  x ki  yi . = = eval yi eval(y1 ,y2 ,... ,yn ) ki ! ki ! ki i=1

i=1

i=1

It is also extended by linearity to sums of such divided powers, multiplied by arbitrary numbers.

Angular Momentum Theory

The application of these rules to our problem involving four indeterminates gives (det X)n  X A n!

(α:A:α )

A!

k1 +k2 =n

(2.119)

In this result, we do not identify the labels with angular momentum quantum numbers. Relation (2.119) is a purely combinatorial, algebraic identity for arbitrary indeterminates and arbitrary row and column sum constraints  array A as specified by α = (α1 , α2 ) and  on the α = α1 , α2 . There are no square roots involved. We now apply relations (2.118–2.119) to the case at hand: n = j1 + j2 − j, α = ( j + m, j − m), α = ( j + j1 − j2 , j − j1 + j2 ), β = ( j1 + j2 + m, j1 + j2 − m), β  = (2 j1 , 2 j2 ). This gives the following result for the WCG-coefficients: j j j

Cm11 m2 2 m 2 ( j1 + j2 − j )!( j1 − j2 + j )!(− j1 + j2 + j )! = ( j1 + j2 + j + 1)! 2 (2 j + 1)( j + m)!( j − m)! × ( j1 + m 1 )!( j1 − m 1 )!( j2 + m 2 )!( j2 − m 2 )! × eval A

(det X) j1 + j2 − j , ( j1 + j2 − j )!

63

of the divided power (det X) j1 + j2 − j ( j1 + j2 − j )! of a determinant, which is an integer. The abstract umbral calculus of Rota thus finds its way, at a basic level, into angular momentum theory. Relation (2.120) is but a rewriting in terms of evaluations of the well-known Van der Waerden form of the WCG-coefficients.

2.13.3 Magic Square Realization of the Addition of Two Angular Momenta The origin of (2.114), giving the states of total angular momentum by compounding two angular momenta, is usually attributed to properties of the direct sum of two copies of the Lie algebra of the unitary unimodular group SU(2), and to the use of differential operators to realize the Lie algebras and state vectors, as done above. It is an interesting combinatorial result that this structure for adding angular momentum is fully encoded within the properties of magic squares of order 3, and no operators whatsoever are needed, only the condition of being a magic square. We have already noted in Sect. 2.7.4 that Regge observed that the restrictions on the domains of the quantum numbers j1 , m 1 , j2 , m 2 , j, m are encoded in terms of a magic square A with line-sum J = j1 + j2 + j : 

 j1 + m 1 j2 + m 2 j −m   A =  j1 − m 1 j2 − m 2 j +m  . j2 − j1 + j j1 − j2 + j j1 + j2 − j

(2.120)

(2.122)

(det X) j1 + j2 − j eval A ( j1 + j2 − j )!



 j2 + m 2 j1 + m 1 (−1)k2 k1 !k2 ! = k1 k2 k1 +k2 = j1 + j2 − j



j2 − m 2 j1 − m 1 . × (2.121) k2 k1 In summary, we have the following: Up to multiplicative square-root factors, the WCGcoefficient is the evaluation at the point
 j1 + m 1 j2 + m 2 B= j1 − m 1 j2 − m 2

The angular momentum quantum numbers are given in terms of the elements of A = (aij )1≤i, j≤3 by the invertible relations 1 1 j1 = (a11 + a21 ) , j2 = (a12 + a22 ) , 2 2 1 j = (a13 + a23 ) , 2 1 1 m 1 = (a11 − a21 ) , m 2 = (a12 − a22 ) , 2 2 1 m = (a23 − a13 ) . 2 It follows from these definitions and the fact that A is a magic square of line-sum J, that the sum rule m 1 + m 2 = m and the triangle condition are fulfilled.

Part A 2.13

(det X)n X B = , eval B (2.118) n! B! (β:B:β  )   β = α1 + n, α2 + n , β  = (α1 + n, α2 + n), (det X)n eval B n!  b11 b12 b21 b22 k2 . (−1) k1 !k2 ! = k1 k2 k2 k1 

2.13 Supplement on Combinatorial Foundations

64

Part A

Mathematical Methods

Part A 2.13

We use the symbol j1 , j2 , j to denote any triple j1 , j2 , j of angular momentum quantum numbers that satisfy the triangle conditions, where we note that, if a given triple satisfies the triangle conditions, then all permutations of the triple also satisfy the triangle conditions. The number of magic squares for fixed line-sum J is obtained as follows: Define ∆ J = {all triangles j1 , j2 , j | j1 + j2 + j = J} and M( j1 , j2 , j) = {(m 1 , m 2 )| − j1 ≤ m 1 ≤ j1 ; − j2 ≤ m 2 ≤ j2 ; − j ≤ m 1 + m 2 ≤ j}. Then we have the following identity, which gives the number of angular momentum magic squares with line-sum J :



 & & & M( j1 , j2 , j )& = J + 5 − J + 2 . 5 5 j1 , j2 , j ∈∆ J

(2.123)

It is nontrivial to effect the summation on the left-hand side of this relation to obtain the right-hand side, but this expression is known from the theory of magic squares Stanley [2.9, 10]. Not only can the addition of two angular momenta in quantum theory, with its triangle rule for three angular momentum quantum numbers and its sum rule on the corresponding projection quantum numbers, be codified in magic squares of order 3 and arbitrary line-sum, but also the content of the abstract state vector of (2.114) itself can be so expressed: & &1 & (a11 + a21 ) , 1 (a12 + a22 ); 1 (a13 + a23 ) , &2 2 2 = 1 (a23 − a13 ) = 2 & =  &1 1 C(A)&& (a11 + a21 ), (a11 − a21 ) 2 2    a11 a12 

a21 a22

& = &1 1 ⊗ && (a12 + a22 ), (a12 − a22 ) , 2 2

where the summation is over all subsets 
a11 a12 a21 a22 of the magic square of order 3 such that row 3 and column 3 are held fixed. The coefficients C(A) themselves are the WCG-coefficients, which may be regarded as a function whose domain of definition is the set of all magic squares of order 3. The triangle rule j1 , j2 , j and the sum rule on (m 1 , m 2 , m) are implied by the structure of

magic squares of order 3. These rich combinatorial footings of angular momentum theory are completed by the observation that the Clebsch–Gordan coefficients themselves are obtained by the Schwinger–Regge generating function given in Sect. 2.7.3 (see [2.2] for the relation to 3 F2 hypergeometric functions).

2.13.4 MacMahon’s and Schwinger’s Master Theorems Generating functions codify the content of many mathematical entities in a unifying, comprehensive way. These functions are very popular in combinatorics, and Schwinger used them extensively in his treatment of angular momentum theory. In this subsection, we present a natural generalization of the SU(2) solid harmonics to a class of polynomials that are homogeneous in n 2 indeterminates. While these polynomials are of interest in their own right, it is their fundamental role in the addition of n kinematically independent angular momenta that motivates their introduction here. They bring an unexpected unity and coherence to angular momentum coupling and recoupling theory [2.11]. We list in compendium format some of the principal results: Special U(n) solid harmonics:  ZA 3 k Dα,β , (Z) = α!β! (2.124) A! A∈M(α,β)

A = (aij )1≤i, j≤n : matrix of order n in nonnegative integers; A! =

n  i, j=1

aij ! , Z A =

n 

a

z ijij ;

i, j=1

where we employ the notations: α = (α1 , α2 , . . . , αn ): sequence (composition) of nonnegative integers having the sum k, denoted α  k; x α = x1α1 x2α2 · · · xnαn , α! = α1 ! α2 ! · · · αn ! ; β = (β1 , β2 , . . . , βn ): second composition β  k ; M(α, β), set of all matrices A such that the entries in row i sums to αi and those in column j to β j . The significance of the row-sum vector α is that αi is the degree of the polynomial in the variables (z i1 , z i2 , . . . , z in ) in row i of Z, with a similar interpretation for β in terms of columns. k (Z) polynomials: Matrix of the Dα,β The number of compositions  of the  integer k into n nonnegative parts is given by n+k−1 . The compositions in k this set may be linearly ordered by the lexicographical rule α > β, if the first nonzero part of α − β is

Angular Momentum Theory

Dk (XY ) = Dk (X)Dk (Y ) .

(2.125)

Orthogonality in the inner product ( , ) defined in Sect. 2.4.3: % ' k Dα,β , Dαk  ,β = δα,α δβ,β k! . Value on Z = diag(z 1 , z 2 , . . . , z n ) : ! " k diag(z 1 , z 2 , . . . , z n ) = δα,β z α , Dα,β Dk (In ) = In+k−1 .

(2.126)

k

Transposition property: ! "T Dk (Z T ) = Dk (Z) . Special irreducible unitary representations of U(n) : Dk (U )Dk (V ) = Dk (UV ) ,

all U, V ∈ U(n) .

Schwinger’s Master Theorem: For any two matrices X and Y of order n, the following identities hold: & e(∂x :X:∂ y ) e(x:Y :y) & x=y=0

=

∞   k=0 α,βk

=

k k Dα,β (X)Dβ,α (Y )

1 , det(I − XY )

(x : Z : y) = xZ yT =

(2.127) n 

z ij xi y j .

i, j=1

MacMahon’s Master Theorem: Let X be the diagonal matrix X = diag(x1 , x2 , . . . , xn ) and Y a matrix of order n. Then the coefficient of x α in the expansion of 1 equals the coefficient of x α in the product det(I −XY )8 α y , yi = nj=1 yij x j , that is, 1 = det(I − XY )

∞   k=0 αk

Basic Master Theorem: Let Z be a matrix of order n. Then ∞

  1 k = tk Dα,α (Z) . det(I − tZ)

.

(2.128)

(2.129)

αk

k=0

Schwinger’s relation (2.127) follows from the basic relation (2.129) by setting Z = XY and using the multiplication property (2.125); MacMahon’s relation then follows from Schwinger’s result by setting X = diag(x1 , x2 , . . . , xn ) and using property (2.126). Of course, MacMahon’s Master Theorem preceded Schwinger’s result by many years (see MacMahon [2.13]).The unification into the single form by using k (Z) polynomials was pointed out properties of the Dα,β in [2.14]. More surprisingly, relation (2.129) was already discovered for the general linear group in 1897 by Molien [2.15]; its properties are developed extensively in Michel and Zhilinski [2.16] in the context of group theory. For many purposes, it is better in combinatorics to avoid all square roots by using the polynomials L α,β (Z) =

 A∈M(α,β)

ZA A!

k (Z) defined in (2.124). in place of the Dα,β

2.13.5 The Pfaffian and Double Pfaffian Schwinger observed that the calculation √ of 3n − j coefficients involves taking the square root (I − AB), where A and B are skew symmetric (antisymmetric) matrices of order n, but the procedure is rather obscure. The appropriate concepts for taking the square root is that of a Pfaffian and a double Pfaffian, denoted, respectively, by Pf(A) and Pf(A, B). The definitions require the concept of a matching of the set of integers {1, 2, . . . , n}. A matching of {1, 2, . . . , n} is an unordered set of disjoint subsets {i, j} containing two elements. For example, the matchings of 1, 2, 3 are {1, 2}, {1, 3}, and {2, 3}. We then have the following constructs: Pfaffian and double Pfaffian of skew symmetric matrices A = (aij ) and B = (bij ) of order n : Pf(A) =

k Dα,α (Y ) x α

65



ε(i 1 i 2 · · · i n )

{i 1 ,i 2 },{i 3 ,i 4 },... ,{i n−1 ,i n }

× ai1 ,i2 ai3 ,i4 · · · ain−1 ,in ,

(2.130)

Part A 2.13

k (Z) is then the entry positive. The polynomial Dα,β in row α and column β in the matrix Dk (Z) of din+k−1 k , where, following the mension dim D (Z) = k convention for SU(2), the rows are labelled from top to bottom by the greatest to the least sequence, and the columns are labelled in the same manner as read from left to right. There is a combinatorial proof by Chen and Louck [2.12] that these polynomials satisfy the following multiplication rule for arbitrary matrices X and Y :

2.13 Supplement on Combinatorial Foundations

66

Part A

Mathematical Methods

Pf(A, B) = 1 +





k≥1

{i 1 , i 2 }, {i 3 , i 4 }, . . . , {i 2k−1 , i 2k } { j1 , j2 }, { j3 , j4 }, . . . , { j2k−1 , j2k }

ε(i 1 i 2 · i 2k )ε( j1 j2 · · · j2k )

Part A 2.13

× ai1 ,i2 ai3 ,i4 · · · ai2k−1 ,i2k × b j1 , j2 b j3 , j4 · · · b j2k−1 , j2k

(2.131)

where {i 1 , i 2 }, {i 3 , i 4 }, . . . , {i n−1 , i n } is a matching of {1, 2, . . . , n},and ε(i 1 i 2 · · · i n ) is the sign of the permutation (number of inversions). Similarly, in the double Pfaffian, the 2-subsets are matchings of a subset of {1, 2, . . . , n} of even length. Relations of skew symmetric matrices A, B to Pfaffians: 3 √ det A = Pf(A) ; det(I − AB) = Pf(A, B) . (2.132)

2.13.6 Generating Functions for Coupled Wave Functions and Recoupling Coefficients This section is a reformulation, nontrivial extension, and interpretation of results found in Schwinger [2.3]. We first refine the notation used in Sect. 2.12.3. Set of triangles in the coupling scheme: Each coupling scheme, as determined by the bracketing B, has associated with it a unique ordered set of n − 1 triangles TB ( j, k, j ) = { ai , bi , ki |i = 1, 2, . . . , n − 1} , j = ( j1 , j2 , . . . , jn ) , k = (k1 , k2 , . . . , kn−2 ) , kn−1 = j . The third part ki of ai , bi , ki can always be chosen, without loss of generality, as an intermediate angular momentum (kn−1 = j ), and the triangles in the set can be ordered by ai , bi , ki < ai+1 , bi+1 , ki+1 . The remaining pair of angular momentum labels in the triangle ai , bi , ki then fall, in general, into four classes: ai , bi , ki in which ai can be either a jr or a ks , and bi can be either a jr  or ks . The distribution of the j  s and k s among the ai and bi is uniquely determined by the bracketing B that defines the coupling scheme. Clebsch–Gordan coefficients for a given coupling scheme: B
 j k j = Cαaii bβiikqii , (2.133) m q m a ,b ,k ∈T ( j ,k, j ) i

i

i

B

in which the projection quantum numbers αi and βi are m  s and q  s that match the ai and bi . In the given coupling scheme determined by the bracketing B, only ( ji , m i ), i = 1, 2 . . . , n; (ki , qi ), i = 1, 2, . . . , n − 2, and ( j, m) appear in the Clebsch– Gordan coefficients. In fact, if one explicitly implements the sum rule on the projection quantum numbers, it is always possible to express the qi as sums over the m i and m. Coupled angular momentum function for n angular momenta:

Ψ(Bj k) j m (x, y) =

 m


j k j m q m

B

n 

ψ ji m i (xi , yi ) .

i=1

(2.134)



 j1 j2 · · · jn j = , m m1 m2 · · · mn

 k k1 k2 · · · kn−2 = . (2.135) q q1 q2 · · · qn−2 
x1 x2 . . . xn+1 . Z = (z 1 z 2 . . . z n+1 ) = y1 y2 . . . yn+1 (2.136)

Only the first n columns of Z enter into (2.136), but the last column occurs below. The skew symmetric matrix of a coupling scheme: The set of triangles TB ( j, k, j ) = { ai , bi , ki |i = 1, 2, . . . , n − 1}, which is uniquely defined by the bracketing B, can be mapped to a unique skew symmetric matrix of order n + 1. This mapping is one of the most important results for obtaining generating functions for the coupled wave functions (2.134) and the recoupling coefficients given below. The skew symmetric matrix depends on the bracketing B and the detailed manner in which the j  s and k s are distributed among the triangles in TB ( j, k, j ). The rule for constructing the skew symmetric matrix is quite intricate. First, we define a 3 × (n − 1) matrix T of indeterminates by   t11 t12 · · · t1,n−1   T = t21 t22 · · · t2,n−1  . t31 t32 · · · t3,n−1

(2.137)

Second, we associate with each ai , bi , ki ∈ TB ( j, k, j ), a triple of indeterminates (u i , vi , wi ) as

Angular Momentum Theory

a1 , b1 , k1 → (u 1 , v1 , w1 ), with w1 = t21 u 1 + t11 v1 , a2 , b2 , k2 → (u 2 , v2 , w2 ), with w2 = t22 u 2 + t12 v2 , .. .. . .

The indeterminates u i and vi are identified as a column z i = (xi , yi ) of the 2 × (n + 1) matrix Z defined by (2.136), or as one of the w s occurring higher in the display (2.138). The distribution rule is in one-to-one correspondence with the distribution of j  s and k s in the corresponding triangle. Thus, we have = zr , = zr , = wr , = wr ,

if ai if ai if ai if ai

= jr ; = jr ; = kr ; = kr ;

vi vi vi vi

= zs , = ws , = zs , = ws ,

if bi if bi if bi if bi

= js ; = ks ; = js ; = ks .

The explicit identification of all j  s and k s is uniquely determined by the bracketing B. Once this identification has been made, the elements aij , i < j of the skew symmetric matrix A of order n + 1 are uniquely determined in terms of the elements of T by equating coefficients of det(z i , z j ) = xi y j − x j yi on the two sides of the form  aij det(z i , z j ) 1≤i< j≤n+1 n−1 

t3i det(u i , vi ) + det(wn−1 , z n+1 ) , (2.139)

i=1

where (t1i , t2i , t3i ) is the i-th column of the 3 × (n − 1) matrix T of indeterminates. This relation can be inferred from results given by Schwinger. Since the elements of A are determined as monomials in the elements of T, we sometimes denote A by A(T ). It is useful to illustrate the rule for determining A for n = 2, 3, 4: n = 2: Triangle: j1 , j2 , k1 : w1 = t21 z 1 + t11 z 2 a12 det(z 1 , z 2 ) + a13 det(z 1 , z 3 ) + a23 det(z 2 , z 3 ) = t31 det(u 1 , v1 ) + det(w1 , z 3 ) = t31 det(z 1 , z 2 ) + t21 det(z 1 , z 3 ) + t11 det(z 2 , z 3 ) ; a12 = t31 , a13 = t21 , a23 = t11 .

w1 = t21 u 1 + t11 v1 , u 1 = z 1 , v1 = z 2 ; w2 = t22 u 2 + t12 v2 , u 2 = w1 , v2 = z 3 .  aij det(z i , z j ) 1≤i< j≤4

= t31 det(u 1 , v1 ) + t32 det(u 2 , v2 ) + det(w2 , z 4 ) ; a12 = t31 , a13 = t21 t32 , a14 = t21 t22 a23 = t11 t32 , a24 = t11 t22 a34 = t12 n = 4: Ordered triangles: j3 , j1 , k1 , j4 , j2 , k2 , k1 , k2 , k3 : w1 = t21 u 1 + t11 v1 , u 1 = z 3 , v1 = z 1 , w2 = t22 u 2 + t12 v2 , u 2 = z 4 , v2 = z 2 , w3 = t23 u 3 + t13 v3 , u 3 = w1 , v3 = w2 ; w3 = t11 t23 z 1 + t12 t13 z 2 + t21 t23 z 3 + t22 t13 z 4 ,  aij det(z i , z j ) = t31 det(u 1 , v1 ) + t32 1≤i< j≤5

× det(u 2 , v2 ) + t33 det(u 3 , v3 ) + det(w3 , z 5 ) ; a12 = t11 t12 t33 , a13 = −t31 , a14 = t11 t22 t33 , a23 = −t12 t21 t33 , a24 = −t32 , a34 = t21 t22 t33 , a15 = t11 t23 a25 = t12 t13 a35 = t21 t23 a45 = t22 t13 Triangle monomials: Let a, b, c be a triangle of quantum numbers (a, b, c), let (x, y, z) be three indeterminates, and let B denote a binary coupling scheme with the set of triangles TB ( j, k, j ) : Elementary triangle monomial: Φ a,b,c (x, y, z) = {abc}−1 x b+c−a ya+c−b z a+b−c , (2.140)

{abc}

1 (2c + 1)(b + c − a)(a + c − b)!(a + b − c)! 2 = . (a + b + c + 1)! Triangle monomial associated with a given coupling scheme B :  Φ ai ,bi ,ki (t1i , t2i , t3i ) . Φ Bj,k, j (T ) = ai ,bi ,ki ∈TB ( j,k, j )

(2.141)

Part A 2.13

an−1 , bn−1 , kn−1 → (u n−1 , vn−1 , wn−1 ), (2.138) with wn−1 = t2,n−1 u n−2 + t1,n−1 vn−1 .

=

67

n = 3: Ordered triangles: j1 , j2 , k1 , k1 , j3 , k2 :

given by

ui ui ui ui

2.13 Supplement on Combinatorial Foundations

68

Part A

Mathematical Methods

Using the definitions introduced above, we can now give the generating functions for the coupled wave functions and the recoupling coefficients for each coupling scheme as determined by the bracketing B. Generating function for coupled wave functions: 8

ex A(T )y = e 1≤i< j≤n+1 ai, j det(zi ,z j )   = Φ Bj,k, j (T ) (−1) j−m ψ j,−m (xn+1 , yn+1 ) T

Part A 2.13

m

jk

× Ψ(Bjk) jm (x, y) , x = (x1 , x2 , . . . , xn+1 ) ,

(2.142)

y = (y1 , y2 , . . . , yn+1 ) .

Relation to U(n + 1) solid harmonics: ex A(T )y = T

∞   yβ xα k (A(T )) √ . √ Dα,β β! α! k=0 α,βk

(2.143)

Relation of U(n + 1) solid harmonics to triangle monomials: B
 j k j k j−m Dα,β [A(T )] = (−1) Φ Bj,k, j (T ) m q m k (2.144)

αi = ji + m i , βi = ji − m i i = 1, 2, . . . , n ; αn+1 = j − m , βn+1 = j + m ;   αi = βi = j1 + j2 + · · · + jn + j . i

i

The relation between the skew symmetric matrix A(T ) of order n + 1 and the elements of the 3 × (n − 1) matrix T is that described in relations (2.138). Generating function for all recoupling coefficients: 1 (2.145) [Pf(A(T ), A (T  ))]2   = Φ Bj,k, j (T )Φ Bj  ,k , j (T  ) j, k, j|| j  , k , j , j,k,k , j

where j, k, j || j  , k , j denotes the recoupling coefficient that effects the transformation between the coupling schemes corresponding to the bracketing B and the bracketing B  , and where the sequence j  is a permutation of j in accordance with the bracketing B  . We also note that 1 1 =√ Pf(A, A ) det(I − A A ) 1  2   k k Dα,β (A)Dβ,α (A ) , (2.146) = 1 + k≥1 α,βk

for arbitrary skew symmetric matrices of order n. Relation (2.145) generates all recoupling coefficients, the trivial ones (those differing by signs) and all the complicated ones, that is, those corresponding to 3n − j coefficients. It will also be observed that the expansion of the reciprocal of the double Pfaffian effects an infinite sum in which no radicals occur, which in turn implies that the every recoupling coefficient has the form j, k, j || j  , k , j  = {ai , bi , ki } ai ,bi ,ki ∈TB ( j,k, j)

×



(   ) ai , bi , ki

ai ,bi ,ki ∈TB  ( j  ,k , j)  

× I( j, k, j || j , k , j) , where I( j, k, j || j  , k , j) is an integer: Each recoupling coefficient is an integer multiplied by square-root factors that depend on the triangles associated with the coupling scheme. Such features can be very useful in the development of algorithms for the calculation of 3n − j coefficients, including WCG-coefficients [2.17]. Relation (2.145) should be useful for the classification of 3n − j coefficients.

Angular Momentum Theory

2.14 Tables

69

2.14 Tables Excerpts and Fig. 2.1 are reprinted from Biedenharn and Louck [2.1] with permission of Cambridge University Press. Tables 2.2–2.4 have been adapted

from Edmonds [2.18] by permission of Princeton University Press. Thanks are given for this cooperation.

l

m

1

±1 0

2

±2 ±1 0

3

±3 ±2 ±1 0

4

±4 ±3 ±2 ±1 0

√ 4πYlm (r) 9 ∓ 23 (x ± iy) √ 3z 9 1 15 2 2 9 2 (x ± iy) 15 ∓ 2 (x ± iy)z √ 1 2 2 2 5 (3z − r ) √ 1 3 ∓9 4 35 (x ± iy) 1 105 2 2 2 (x ± iy) z ∓ 14 21(x ± iy)(5z 2 − r 2 ) √ 1 2 2 2 7 (5z − 3r )z √ 3 4 16 70 (x ± iy) √ 3 ∓ 4 35 (x ± iy)3 z √ 3 2 2 2 8 10 (x ± iy) (7z − r ) √ 3 2 ∓ 4 5 (x ± iy)(7z − 3r 2 )z 15 8

 4  7z − 6z 2 r 2 + 35 r 4

√ 4π Ylm (θ,ϕ) 9 ∓ 32 e±iϕ sin θ √ 3 cos θ 9 1 15 ±2iϕ sin2 θ 292 e 15 ±iϕ ∓ 2 e sin θ cos θ √ 1 2 2 5 (3 cos θ − 1) √ 1 ±3iϕ sin3 θ ∓9 4 35 e 1 105 ±2iϕ sin2 θ cos θ 2 2 e √ ∓ 14 21 e±iϕ sin θ(5 cos2 θ − 1) √ 1 2 2 7 (5 cos θ − 3) cos θ √ 3 ±4iϕ sin4 θ 16 70 e √ 3 ∓ 4 35 e±3iϕ sin3 θ cos θ √ 3 ±2iϕ sin2 θ(7 cos2 θ − 1) 8 10 e √ ±iϕ 3 ∓ 4 5 e sin θ(7 cos2 θ − 3) cos θ 15 8

  7 cos4 θ − 6 cos2 θ + 35

Tml √ ∓ 2J± 2J3 √ 2 6J± √ ∓ 6J± (2J3 ± 1)   2 3J32 − J 2 √ ∓2 5J±3 √ 2 30J±2 (J3 ± 1) √   ∓2 3J± 5J32 − J 2 ± 5J3 + 2  2  4 5J3 − 3J 2 + 1 J3 √ 70J±4 √ ∓2 35J±3 (2J3 ± 3) √   2 10J±2 7J32 − J 2 ± 14J3 + 9 √  ∓ 5J± 28J33 − 12J 2 J3 ± 42J32  −6J 2 + 38J3 ± 12 70J34 − 60J 2 J32 + 6(J 2 )2 + 50J32 −12J 2

Table 2.2 The 3– j coefficients for all M’s = 0, or J3 = 0, 

1 2



1 1 2  J1 J2 J3  = (−1) 2 J (J1 + J2 − J3 )!(J1 + J3 − J2 )!(J2 + J3 − J1 )! (J + J + J + 1)! 1 2 3 0 0 0 ( 1 J)! ×1  1 2  1  , J even 2 J − J1 ! 2 J − J2 ! 2 J − J3 !    J1 J2 J3  = 0 , J odd, where J = J1 + J2 + J3 0 0 0   1 J J 0   = (−1) J−M (2J + 1)1/2 M −M 0   1/2
1 1 1 J − M + 12 J 2  = (−1) J−M− 2 J + 2 (2J + 2)(2J + 1) M −M − 12 12

Part A 2.14

Table 2.1 The solid and spherical harmonics Ylm , and the tensor harmonics Tµk (labeled by k = l and µ = m) for l = 0, 1, 2, 3, and 4

70

Part A

Mathematical Methods

Table 2.3 The 3– j coefficients for J3 = 1, 32 , 2 

J + 1 M 

J −M − 1

Part A 2.14

J + 1 J M −M  J J

M −M − 1  J J M −M



3 J J + 2 M −M − 32  3 J J + 2



−M − 12

M

1 J J + 2 M −M − 32  1 J J + 2



−M − 12

M

J J + 2 M −M − 2  J J + 2 M  

−M − 1

J + 2 J M −M

J J + 1 M −M − 2  J J + 1 M 

−M − 1

J + 1 J M −M  J J 

M −M − 2

J J M −M − 1  J J M −M



1 2 (J − M )(J − M + 1) 1 = (−1) J−M−1 (2J + 3)(2J + 2)(2J + 1) 1 

1 2(J + M + 1)(J − M + 1) 2 1 = (−1) J−M−1 (2J + 3)(2J + 2)(2J + 1) 0 

1 2(J − M )(J + M + 1) 2 1 = (−1) J−M (2J + 2)(2J + 1)(2J ) 1  M 1 = (−1) J−M 1 0 [(2J + 1)(J + 1)J] 2 
     12 3 1 J − M − 12 J − M + 12 J − M + 32 2  = (−1) J−M+ 2 3 (2J + 4)(2J + 3)(2J + 2)(2J + 1) 2 
     12 3 1 3 J − M + 12 J − M + 32 J + M + 32 2  = (−1) J−M+ 2 1 (2J + 4)(2J + 3)(2J + 2)(2J + 1) 2 
     12 3 1 3 J − M − 12 J − M + 12 J + M + 32 2  = (−1) J−M− 2 3 (2J + 3)(2J + 2)(2J + 1)2J 2  1


2 3 1 J − M + 12 3 2  = (−1) J−M− 2 J + 3M + 1 (2J + 3)(2J + 2)(2J + 1)2J 2 2 

1 (J − M − 1)(J − M )(J − M + 1)(J − M + 2) 2 2 = (−1) J−M (2J + 5)(2J + 4)(2J + 3)(2J + 2)(2J + 1) 2 

1 (J + M + 2)(J − M + 2)(J − M + 1)(J − M ) 2 2 = (−1) J−M (2J + 5)(2J + 4)(2J + 3)(2J + 2)(2J + 1) 1 

1 6(J + M + 2)(J + M + 1)(J − M + 2)(J − M + 1) 2 2 = (−1) J−M (2J + 5)(2J + 4)(2J + 3)(2J + 2)(2J + 1) 0 

1 (J − M − 1)(J − M )(J − M + 1)(J + M + 2) 2 2 = 2(−1) J−M+1 (2J + 4)(2J + 3)(2J + 2)(2J + 1)2J 2 

1 2 (J − M + 1)(J − M ) 2 = (−1) J−M+1 2(J + 2M + 2) (2J + 4)(2J + 3)(2J + 2)(2J + 1)2J 1 

1 2 6(J + M + 1)(J − M + 1) 2 = (−1) J−M+1 2M (2J + 4)(2J + 3)(2J + 2)(2J + 1)2J 0 

1 6(J − M − 1)(J − M )(J + M + 1)(J + M + 2) 2 2 = (−1) J−M (2J + 3)(2J + 2)(2J + 1)(2J )(2J − 1) 2 

1 2 6(J + M + 1)(J − M ) 2 = (−1) J−M (1 + 2M ) (2J + 3)(2J + 2)(2J + 1)(2J )(2J − 1) 1  ! " 2 3M 2 − J(J + 1) 2 = (−1) J−M 1 0 [(2J + 3)(2J + 2)(2J + 1)(2J )(2J − 1)] 2

Angular Momentum Theory

2.14 Tables

71

Table 2.4 The 6– j coefficients for d = 0, 12 , 1, 32 , 2, with s = a + b + c

Part A 2.14

  a b c  −1 = (−1)s [(2b + 1)(2c + 1)] 2 δb f δce 0 e f   

1 a b 2 (s − 2b)(s − 2c + 1) c  = (−1)s  1 c− 1 b+ 1  (2b + 1)(2b + 2)2c(2c + 1) 2 2  2 

1 a b 2 (s + 1)(s − 2a) c  = (−1)s  1 c− 1 b− 1  2b(2b + 1)2c(2c + 1) 2 2 2 

1 a b 2 s(s + 1)(s − 2a − 1)(s − 2a) c  = (−1)s  1 c−1 b−1  (2b − 1)2b(2b + 1)(2c − 1)2c(2c + 1)  

1 a b c 2(s + 1)(s − 2a)(s − 2b)(s − 2c + 1) 2 = (−1)s  1 c−1 b  2b(2b + 1)(2b + 2)(2c − 1)2c(2c + 1)  

1 a b (s − 2b − 1)(s − 2b)(s − 2c + 1)(s − 2c + 2) 2 c  = (−1)s  1 c−1 b+1  (2b + 1)(2b + 2)(2b + 3)(2c − 1)2c(2c + 1)   a b c 2 [b(b + 1) + c(c + 1) − a(a + 1)] = (−1)s+1 , 1 1 c b [2b(2b + 1)(2b + 2)2c(2c + 1)(2c + 2)] 2  

1 a b 2 (s − 1)s(s + 1)(s − 2a − 2)(s − 2a − 1)(s − 2a) c  = (−1)s  3 c− 3 b− 3  (2b − 2)(2b − 1)2b(2b + 1)(2c − 2)(2c − 1)2c(2c + 1) 2 2  2 

1 a b 2 3s(s + 1)(s − 2a − 1)(s − 2a)(s − 2b)(s − 2b + 1) c  = (−1)s  3 c− 3 b− 1  (2b − 1)2b(2b + 1)(2b + 2)(2c − 2)(2c − 1)2c(2c + 1) 2 2  2 

1 a b 3(s + 1)(s − 2a)(s − 2b − 1)(s − 2b)(s − 2c + 1)(s − 2c + 2) 2 c  = (−1)s  3 c− 3 b+ 1  2b(2b + 1)(2b + 2)(2b + 3)(2c − 2)(2c − 1)2c(2c + 1) 2 2  2 

1 a b (s − 2b − 2)(s − 2b − 1)(s − 2b)(s − 2c + 1)(s − 2c + 2)(s − 2c + 3) 2 c  = (−1)s  3 c− 3 b+ 3  (2b + 1)(2b + 2)(2b + 3)(2b + 4)(2c − 2)(2c − 1)2c(2c + 1) 2 2  2  1 a b [2(s − 2b)(s − 2c) − (s + 2)(s − 2a − 1)] [(s + 1)(s − 2a)] 2 c  = (−1)s 1  3 c− 1 b− 1  [(2b − 1)2b(2b + 1)(2b + 2)(2c − 1)2c(2c + 1)(2c + 2)] 2 2 2  2  1 a b [(s − 2b − 1)(s − 2c) − 2(s + 2)(s − 2a)] [(s − 2b)(s − 2c + 1)] 2 c  = (−1)s , 1  3 c− 1 b+ 1  [2b(2b + 1)(2b + 2)(2b + 3)2c(2c + 1)(2c + 2)(2c + 3)] 2 2 2 2  

1 a b 2 (s − 2)(s − 1)s(s + 1) c  = (−1)s  2 c−2 b−2  (2b − 3)(2b − 2)(2b − 1)2b(2b + 1)

1 (s − 2a − 3)(s − 2a − 2)(s − 2a − 1)(s − 2a) 2 × (2c − 3)(2c − 2)(2c − 1)2c(2c + 1)  

1 a b  2 (s − 1)s(s + 1) c = (−1)s 2  2 c−2 b−1  (2b − 2)(2b − 1)2b(2b + 1)(2b + 2)

1 (s − 2a − 2)(s − 2a − 1)(s − 2a)(s − 2b)(s − 2c + 1) 2 × (2c − 3)(2c − 2)(2c − 1)2c(2c + 1)

72

Part A

Mathematical Methods

Table 2.4 The 6– j coefficients for d = 0, 12 , 1, 32 , 2, with s = a + b + c, cont.  a

 c  2 c−2 b 

Part A 2

 a

b

b

c

 

 2 c−2 b+1   a

b

c

 

 2 c−2 b+2   a

b

c

 

 2 c−1 b−1   a

 c  2 c−1 b   a

b

b

c

 

 2 c−1 b+1 

1 2 6s(s + 1)(s − 2a − 1)(s − 2a) (2b − 1)2b(2b + 1)(2b + 2)(2b + 3)

1 2 (s − 2b)(s − 2c + 1)(s − 2c + 2) × (2c − 3)(2c − 2)(2c − 1)2c(2c + 1)

1 (s + 1)(s − 2a)(s − 2b − 2)(s − 2b − 1)(s − 2b) 2 (−1)s 2 2b(2b + 1)(2b + 2)(2b + 3)(2b + 4)

1 (s − 2c + 1)(s − 2c + 2)(s − 2c + 3) 2 × (2c − 3)(2c − 2)(2c − 1)2c(2c + 1)

(s − 2b − 3)(s − 2b − 2)(s − 2b − 1)(s − 2b) 1/2 (−1)s (2b + 1)(2b + 2)(2b + 3)(2b + 4)(2b + 5)

1 (s − 2c + 1)(s − 2c + 2)(s − 2c + 3)(s − 2c + 4) 2 × (2c − 3)(2c − 2)(2c − 1)2c(2c + 1)

=

=

=

=

(−1)s

(−1)s

1

[(2b − 2)(2b − 1)2b(2b + 1)(2b + 2)] 2

1 2 s(s + 1)(s − 2a − 1)(s − 2a) × (2c − 2)(2c − 1)2c(2c + 1)(2c + 2) ! " (a − b + 1)(a − b) − c2 + 1 (−1)s 2 1 [(2b − 1)2b(2b + 1)(2b + 2)(2b + 3)] 2

1 6(s + 1)(s − 2a)(s − 2b)(s − 2c + 1) 2 × (2c − 2)(2c − 1)2c(2c + 1)(2c + 2)

=

=

(−1)s ×

  a b c 2 c b

4 [(a + b)(a − b + 1) − (c − 1)(c − b + 1)]

=

(−1)s

4 [(a + b + 2)(a − b − 1) − (c − 1)(b + c + 2)] 1

[2b(2b + 1)(2b + 2)(2b + 3)(2b + 4)] 2 (s − 2b − 1)(s − 2b)(s − 2c + 1)(s − 2c + 2) (2c − 2)(2c − 1)2c(2c + 1)(2c + 2)

1 2

,

2 [3X(X − 1) − 4b(b + 1)c(c + 1)] 1

[(2b − 1)2b(2b + 1)(2b + 2)(2b + 3)] 2

1 2 1 × , (2c − 1)2c(2c + 1)(2c + 2)(2c + 3)

where X = b(b + 1) + c(c + 1) − a(a + 1)

References 2.1

2.2 2.3

L. C. Biedenharn, J. D. Louck: Encyclopedia of Mathematics and Its Applications, Vol. 8 & 9, ed. by G.-C. Rota (Addison-Wesley, Reading 1981) presently by (Cambridge Univ. Press, Cambridge) J. D. Louck: J. Math. and Math. Sci. 22, 745 (1999) J. Schwinger: On Angular Momentum. U. S. Atomic Energy Commission Report NYO-3071, 1952 (unpublished). In: Quantum Theory of Angular Momentum, ed. by L. C. Biedenharn, H. van Dam (Academic, New York 1965) pp. 229–279

2.4 2.5 2.6 2.7

B. R. Judd, G. M. S. Lister: J. Phys. A 20, 3159 (1987) B. R. Judd: Symmetries in Science, ed. by B. Gruber, R. S. Millman (Plenum, New York 1980) pp. 151–160 S. Roman, G.-C. Rota: Adv. in Math. 27, 95 (1978) A. P. Jucys, I. B. Levinson, V. V. Vanagas: The Theory of Angular Momentum (Israel Program for Scientific Translation, Jerusalem 1962) (Mathematicheskii apparat teorii momenta kolichestva dvizheniya) Translated from the Russian by A. Sen, A. R. Sen (1962)

Angular Momentum Theory

2.8 2.9 2.10 2.11

2.14 2.15

2.16 2.17 2.18 2.19

2.20

2.21 2.22 2.23

2.24

2.25

2.26

2.27 2.28

2.29

2.30 2.31

2.32

2.33 2.34 2.35 2.36

2.37 2.38

2.39

2.40 2.41 2.42

2.43

2.44 2.45 2.46

2.47

2.48

2.49 2.50

I. M. Gel’fand, Z. Ya. Shapiro: Am. Math. Soc. Transl. 2, 207 (1956) A. Erdelyi, W. Magnus, F. Oberhettinger, G. F. Tricomi: Higher Transcendental Functions, Vol. 1 (McGraw-Hill, New York 1953) E. P. Wigner: Application of Group Theory to the Special Functions of Mathematical Physics. Lecture notes, 1955 (unpublished) H. C. Brinkman: Applications of Spinor Invariants in Atomic Physics (North-Holland, Amsterdam 1956) M. E. Rose: Elementary Theory of Angular Momentum (Wiley, New York 1957) U. Fano, G. Racah: Irreducible Tensorial Sets (Academic, New York 1959) E. P. Wigner: Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra (Academic, New York 1959) Translation from the 1931 German edition by J. J. Griffin J. C. Slater: Quantum Theory of Atomic Structure, Vol. 2 (McGraw-Hill, New York 1960) V. Heine: Group Theory and Quantum Mechanics; An Introduction to Its Present Usage (Pergamon, New York 1960) W. T. Sharp: Thesis, Princeton University (1960) (issued as Report AECL-1098, Atomic Energy of Canada, Chalk River, Ontario (1960)) D. M. Brink, G. R. Satchler: Angular Momentum (Oxford Univ. Press, London 1962) M. Hamermesh: Group Theory and Its Applications to Physical Problems (Addison-Wesley, Reading 1962) G. W. Mackey: The Mathematical Foundations of Quantum Mechanics (Benjamin, New York 1963) A. de-Shalit, I. Talmi: Nuclear Shell Theory (Pure and Applied Physics Series), Vol. 14 (Academic, New York 1963) R. P. Feynman: Feynman Lectures on Physics (Addison-Wesley, Reading 1963) Chap. 34 R. Judd: Operator Techniques in Atomic Spectroscopy (McGraw-Hill, New York 1963) A. S. Davydov: Quantum Mechanics (Pergamon, London, Addison-Wesley, Reading 1965) Translation from the Russian of Kvantovaya Mekhanika (Moscow, 1963), with revisions and additions by D. ter Haar I. M. Gel’fand, R. A. Minlos, Z. Ya. Shapiro: Representations of the Rotation and Lorentz Groups and Their Applications (Macmillan, New York 1963) Translated from the Russian by G. Cummins and T. Boddington R. Hagedorn: Selected Topics on Scattering Theory: Part IV, Angular Momentum, Lectures given at the Max-Planck-Institut für Physik, Munich (1963) M. A. Naimark: Linear Representations of the Lorentz Group (Pergamon, New York 1964) L. C. Biedenharn, H. van Dam: Quantum Theory of Angular Momentum (Academic, New York 1965)

73

Part A 2

2.12 2.13

A. P. Jucys, A. A. Bandzaitis: Angular Momentum Theory in Quantum Physics (Moksias, Vilnius 1977) R. P. Stanley: Enumerative Combinatorics, Vol. 1 (Cambridge Univ. Press, Cambridge 1997) A. Clebsch: Theorie der binären algebraischen Formen (Teubner, Leipzig 1872) J. D. Louck, W. Y. C. Chen, H. W. Galbraith: Symmetry, Structural Properties of Condensed Matter, ed. by T. Lulek, B. Lulek, A. Wal (World Scientific, Singapore 1999) pp. 112–137 W. Y. C. Chen, J. D. Louck: Adv. Math. 140, 207 (1998) P. A. MacMahon: Combinatory Analysis (Cambridge Univ. Press, Cambridge 1960) (Chelsia Publishing Co., New York, 1960) (Originally published in two volumes by Cambridge Univ. Press, Cambridge, 1915, 1916) J. D. Louck: Adv. Appl. Math. 17, 143 (1996) T. Molien: Über die Invarianten der linearen Substitutionsgruppen, Sitzungsber. Konig. Preuss. Akad. Wiss. 52, 1152 (1897) L. Michel, B. I. Zhilinskii: Physics Reports 341, 11 (2001) L. Wei: Comput. Phys. Commun. 120, 222 (1999) A. R. Edmonds: Angular Momentum in Quantum Mechanics (Princeton Univ. Press, Princeton 1957) E. Cartan: Thesis (Paris, Nony 1894) [Ouevres Complète, Part 1, pp. 137–287 (Gauthier-Villars, Paris 1952)] H. Weyl: Gruppentheorie und Quantenmechanik (Hirzel, Leipzig 1928) Translated by H. P. Robertson as The Theory of Groups and Quantum Mechanics (Methuen, London 1931) P. A. M. Dirac: The Principles of Quantum Mechanics, 4th edn. (Oxford Univ. Press, London 1958) M. Born, P. Jordan: Elementare Quantenmechanik (Springer, Berlin, Heidelberg 1930) H. B. G. Casimir: Thesis, University of Leyden (Wolters, Groningen 1931) [Koninkl. Ned. Akad. Wetenschap, Proc. 34, 844 (1931)] B. L. van der Waerden: Die gruppentheoretische Methode in der Quantenmechanik (Springer, Berlin, Heidelberg 1932) W. Pauli: Handbuch der Physik, Vol. 24, ed. by H. Geiger, K. Scheel (Springer, Berlin, Heidelberg 1933) Chap. 1, pp. 83–272. Later published in Encyclopedia of Physics, Vol. 5, Part 1, ed. by S. Flügge (Springer, Berlin, Heidelberg 1958), pp. 45, 46 H. Weyl: The Structure and Representations of Continuous Groups, Lectures at the Institute for Advanced Study. Princeton, 1934-1935 (unpublished). Notes by R. Brauer E. U. Condon, G. H. Shortley: The Theory of Atomic Spectra (Cambridge Univ. Press, London 1935) H. A. Kramers: Quantum Mechanics (North-Holland, Amsterdam 1957) Translation by D. ter Haar of Kramer’s monograph published in the Hand- und Jahrbuch der chemischen Physik (1937) G. Szegö: Orthogonal Polynomials (Edwards, Ann Arbor 1948)

References

74

Part A

Mathematical Methods

2.51 2.52 2.53

Part A 2

2.54

2.55 2.56

2.57 2.58

2.59 2.60 2.61

2.62 2.63 2.64 2.65 2.66 2.67 2.68 2.69 2.70 2.71 2.72 2.73

B. L. van der Waerden: Sources of Quantum Mechanics (North-Holland, Amsterdam 1967) B. R. Judd: Second Quantization and Atomic Spectroscopy (Johns Hopkins Press, Baltimore 1967) N. Vilenkin: Special Functions and the Theory of Group Representations, Vol. 22, Translated from the Russian Am. Math. Soc. Transl. (Amer. Math. Soc., Providence, 1968) J. D. Talman: Special Functions: A Group Theoretic Approach (Benjamin, New York 1968) Based on E. P. Wigner’s lectures (see [2.3]) B. G. Wybourne: Symmetry Principles and Atomic Spectroscopy (Wiley-Interscience, New York 1970) E. A. El Baz, B. Castel: Graphical Methods of Spin Algebras in Atomic, Nuclear, and Particle Physics (Dekker, New York 1972) R. Gilmore: Lie Groups, Lie Algebras, and Some of Their Applications (Wiley, New York 1974) D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskiˇı: Quantum Theory of Angular Momentum (Nauka, Leningrad 1975) (in Russian) R. D. Cowan: The Theory of Atomic Structure and Spectra (Univ. Calif. Press, Berkeley 1981) R. N. Zare: Angular Momentum (Wiley-Interscience, New York 1988) G. E. Andrews, R. A. Askey, R. Roy: Special Functions. In: Encyclopidia of Mathematics and Its Applications, Vol. 71, ed. by G.-C. Rota (Cambridge Univ. Press, Cambridge 1999) ¨ P. Gordan: Uber das Formensystem binärer Formen (Teubner, Leipzig 1875) W. Heisenberg: Z. Phys. 33, 879 (1925) M. Born, P. Jordan: Z. Phys. 34, 858 (1925) P. A. M. Dirac: Proc. Soc. A 109, 642 (1925) M. Born, W. Heisenberg, P. Jordan: Z. Phys. 35, 557 (1926) W. Pauli: Z. Phys. 36, 336 (1926) E. P. Wigner: Z. Phys. 43, 624 (1927) E. P. Wigner: Z. Phys. 45, 601 (1927) C. Eckart: Rev. Mod. Phys. 2, 305 (1930) E. P. Wigner: Göttinger Nachr., Math.-Phys. 546, (1932) J. H. Van Vleck: Phys. Rev. 47, 487 (1935) E. P. Wigner: On the Matrices which Reduce the Kronecker Products of Representations of S. R. Groups, 1940 (unpublished). In: Quantum Theory of Angu-

2.74 2.75 2.76 2.77 2.78 2.79 2.80

2.81 2.82 2.83 2.84 2.85 2.86 2.87

2.88 2.89

2.90 2.91 2.92 2.93

2.94 2.95

lar Momentum, ed. by L. C. Biedenharn, H. van Dam (Academic, New York 1965) pp. 87–133 G. Racah: Phys. Rev. 62, 438 (1942) G. Racah: Phys. Rev. 63, 367 (1943) I. M. Gel’fand, M. L. Tseitlin: Dokl. Akad. Nauk SSSR 71, 825 (1950) J. H. Van Vleck: Rev. Mod. Phys. 23, 213 (1951) H. A. Jahn: Proc. R. Soc. A 205, 192 (1951) L. C. Biedenharn, J. M. Blatt, M. E. Rose: Rev. Mod. Phys. 24, 249 (1952) L. C. Biedenharn: Notes on Multipole Fields, Lecture notes at Yale University, New Haven 1952 (unpublished) J. P. Elliott: Proc. R. Soc. A 218, 345 (1953) L. C. Biedenharn: J. Math. Phys. 31, 287 (1953) H. A. Jahn, J. Hope: Phys. Rev. 93, 318 (1954) T. Regge: Nuovo Cimento 10, 544 (1958) T. Regge: Nuovo Cimento 11, 116 (1959) V. Bargmann: Commun. Pure Appl. Math. 14, 187 (1961) A. Giovannini, D. A. Smith: Spectroscopic and Group Theoretic Methods in Physics (Racah Memorial Volume), ed. by F. Block, S. G. Cohen, A. de-Shalit, S. Sambursky, I. Talmi (Wiley–Interscience, New York 1968) pp. 89–97 V. Bargmann: Rev. Mod. Phys. 34, 829 (1962) L. Michel: Lecture Notes in Physics: Group Representations in Mathematics and Physics, Battelle Recontres, ed. by V. Bargmann (Springer, Berlin, Heidelberg 1970) pp. 36–143 A. C. T. Wu: J. Math. Phys. 13, 84 (1972) Ya. A. Smorodinski˘ı, L. A. Shelepin: Sov. Phys. Usp. 15, 1 (1972) Ya. A. Smorodinski˘ı, L. A. Shelepin: Usp. Fiz. Nauk 106, 3 (1972) L. A. Shelepin: Invariant algebraic methods and symmetric analysis of cooperative phenomena. In: Group-Theoretical Methods in Physics, ed. by D. V. Skobel’tsyn (Fourth Internat. Collog., Nijmegen 1975) pp. 1–109 A special research report translated from the Russian by Consultants Bureau, New York, London. Ya. A. Smorodinskiˇı: Sov. Phys. JETP 48, 403 (1978) I. M. Gel’fand, M. I. Graev: Dokl. Math. 33, 336 (2000) Tranl. from Dokl. Akad. Nauk 372, 151 (2000)

75

3. Group Theory for Atomic Shells

Group Theory

3.1

Generators .......................................... 3.1.1 Group Elements ........................ 3.1.2 Conditions on the Structure Constants ................................. 3.1.3 Cartan–Weyl Form ..................... 3.1.4 Atomic Operators as Generators ..

3.2

Classification of Lie Algebras................. 3.2.1 Introduction ............................. 3.2.2 The Semisimple Lie Algebras .......

76 76 76

3.3

Irreducible Representations.................. 3.3.1 Labels ...................................... 3.3.2 Dimensions .............................. 3.3.3 Casimir’s Operator .....................

77 77 77 77

3.4

Branching Rules .................................. 3.4.1 Introduction ............................. 3.4.2 U(n) ⊃ SU(n)............................. 3.4.3 Canonical Reductions................. 3.4.4 Other Reductions ......................

78 78 78 79 79

3.5

Kronecker Products .............................. 3.5.1 Outer Products of Tableaux ......... 3.5.2 Other Outer Products ................. 3.5.3 Plethysms ................................

79 79 80 80

3.6

Atomic States ...................................... 3.6.1 Shell Structure .......................... 3.6.2 Automorphisms of SO(8) ............. 3.6.3 Hydrogen and Hydrogen-Like Atoms ......................................

80 80 81

The Generalized Wigner–Eckart Theorem 3.7.1 Operators ................................. 3.7.2 The Theorem............................. 3.7.3 Calculation of the Isoscalar Factors..................................... 3.7.4 Generalizations of Angular Momentum Theory ....................

82 82 82

3.7

75 75 76 76 76

3.8

81

82 83

Checks ................................................

83

References ..................................................

84

3.1 Generators 3.1.1 Group Elements An element Sa of a Lie group G corresponding to an infinitesimal transformation can be written in the form Sa = 1 + δaσ X σ ,

(3.1)

where the δaσ are the infinitesimal parameters and the X σ are the generators [3.1]. Summation over

the repeated Greek index is implied. Transformations corresponding to finite parameters can be found by exponentiation:


 Sa → exp a1 X 1 exp a2 X 2 · · · exp ar X r . (3.2) The generators necessarily form a Lie algebra, that is, they close under commutation:   X ρ , X σ = cτρσ X τ . (3.3)

Part A 3

The basic elements of the theory of Lie groups and their irreducible representations (IRs) are described. The IRs are used to label the states of an atomic shell and also the components of operators of physical interest. Applications of the generalized Wigner-Eckart theorem lead to relations between matrix elements appearing in different electronic configurations. This is particularly useful in the f shell, where transformations among the seven orbital states of an f electron can be described by the unitary group U(7) and its sequential subgroups SO(7), G2 , and SO(3) with respective IRs [λ], W, U, and L. Extensions to groups that involve electron spin S (like Sp(14)) are described, as are groups that do not conserve electron number. The most useful of the latter is the quasispin group whose generators Q connect states of identical W, U, L and seniority v in the f shell. The symmetries of products of objects (states or operators) that themselves possess symmetries are described by the technique of plethysms.

76

Part A

Mathematical Methods

In terms of the structure constants cτρσ , the metric tensor is defined as µ

gρσ = cρλ cλσµ .

(3.4)

of one set commutes with all the members of the other, that is, if cτρσ = 0 ,

(ρ ≤ p, σ > p) ,

(3.10)

3.1.2 Conditions on the Structure Constants

then the two sets form the generators of two invariant subgroups, H and K. The group G is the direct product of H and K and is written as H × K.

For an Abelian group, all the generators commute with one another:

3.1.3 Cartan–Weyl Form

cτρσ = 0 .

(3.5)

Part A 3.2

The operators X σ , (σ = 1, 2, . . . , p < r) form the generators of a subgroup if [3.2] cτρσ = 0 ,

(ρ, σ ≤ p, τ > p) .

(3.6)

The subgroup is invariant if the stronger condition cτρσ = 0 ,

(ρ ≤ p, τ > p)

(3.7)

(3.8)

All simple groups are semisimple. For semisimple groups, the inverse tensor gµν can be formed, thus permitting suffixes to be raised. The quadratic operator C = gρσ X ρ X σ

[Hi , H j ] = 0 , [Hi , E α ] = αi E α , [E α , E −α ] = α Hi , [E α , E β ] = Nαβ E α+β . i

is satisfied. A group is simple if it contains no invariant subgroup (besides the unit element). A group is semisimple if it contains no Abelian invariant subgroup (besides the unit element). A necessary and sufficient condition that a group be semisimple is that det | gρτ | = 0 .

By taking suitable linear combinations Hi and E α of the generators X σ , the basic commutation relations (3.3) can be thrown into the so-called Cartan–Weyl form [3.1]

(3.9)

commutes with all generators of the group and is called Casimir’s operator [3.1]. If the generators of a group G can be broken up into two sets such that each member

(3.11) (3.12) (3.13) (3.14)

The Roman symbols i, j, . . . run over an l-dimensional space (the weight space of rank l) in which the numbers αi can be visualized as the components of the vectors (called roots). The E α are shift operators, the displacements being specified by the components of α. The operator E α+β in (3.17) is to be interpreted as 0 if α + β is not a root. The coefficient Nαβ depends on the choice of normalization.

3.1.4 Atomic Operators as Generators †

The pairs aξ aη of creation and annihilation operators for either bosons or fermions, as defined in Sect. 6.1.1 close under commutation and form a Lie algebra. The coupled forms W (κk) , defined in Sect. 6.2.2, are often used to play the role of the generators for electrons in an atomic shell.

3.2 Classification of Lie Algebras 3.2.1 Introduction

3.2.2 The Semisimple Lie Algebras

The semisimple Lie algebras have been classified by Cartan [3.3]. They consist of four main classes Al , Bl , Cl , Dl , and five exceptions G 2 , F4 , E 6 , E 7 , E 8 . Each algebra is characterized by an array of roots in the l-dimensional weight space; they are conveniently specified by a set of mutually orthogonal unit vectors ei . The total number of generators (those of type E α plus the l generators of type Hi ) gives the order of the algebra.

Al . The roots are conveniently represented by the vectors ei − e j (i, j = 1, 2, . . . , l + 1). They are all perpendicular to Σek and do not extend beyond the l-dimensional weight space. The order of the algebra is l(l + 2). The group for which this algebra can serve as a basis is the special unitary group SU(l + 1). Bl . The roots are ± ei and ± ei ± e j (i, j = 1, 2, . . . , l; i = j). The order of the algebra is l(2l + 1). A cor-

Group Theory for Atomic Shells

3.3 Irreducible Representations

77

Table 3.1 Generators of the Lie groups for the atomic l shell. The subscripts i and j run over all 4l + 2 states of a single

electron Group

Generators

SO(8l + 5)a SO(8l + 4)a

ai a j , ai a j , ai a j , ai , a j † † † ai a j , ai a j , ai a j

U(4l + 2)b SU(4l + 2)b

W (κk) (κ = 0, 1; k = 0, 1, . . . , 2l) W (κk) (As above, with κ = k = 0 excluded)

Sp(4l + 2)c U(2l + 1)d

W (κk) (As above, with κ + k odd) W (0k) (k = 0, 1, . . . , 2l)

SU(2l + 1)d SO(2l + 1)d

W (0k) (k = 1, 2, . . . , 2l) W (0k) (k = 1, 3, 5, . . . , 2l − 1)

Gd2 SO L (3)e SO S (3)e U A (2l + 1) × U B (2l + 1)f SOλ (2l + 1) × SOµ (2l + 1) × SOν (2l + 1) × SOξ (2l + 1)g Uλ (2l ) × Uµ (2l ) × Uν (2l ) × Uξ (2l )h

W (01) , W (05) (for l = 3) W (01) (or L) W (10) (or S) (0k) (1k) (0k) (1k) W0q + W0q , W0q − W0q (k = 0, 1, . . . , 2l) † (k) (θ θ) (k odd, θ ≡ λ, µ, ν, ξ) † qθ qθ (all components, θ ≡ λ, µ, ν, ξ)

[3.4, 5]

b

[3.1, 6]

c

[3.6, 7]

d

[3.6]

e

[3.8]

f

[3.9]

g

†

†

[3.10] and (6.69)–(6.72)

h

[3.11]

responding group is the special orthogonal (or rotation) group in 2l + 1 dimensions, SO(2l + 1).

E 6 , E 7 , E 8 . The roots are given by Racah [3.1]. The respective orders are 78, 133, and 248.

Cl . The roots are ± 2ei and ± ei ± e j (i, j = 1, 2, . . . , l; i = j). The order of the algebra is l (2l + 1). A corresponding group is the symplectic group in 2l dimensions, Sp(2l). A rotation of the roots yields C2 = B2 .

F4 . The roots consist of the roots of B4 together with the 16 vectors 12 ( ± e1 ± e2 ± e3 ± e4 ). The order of the algebra is 52.

Dl . The roots are ± ei ± e j (i, j = 1, 2, . . . , l ; i = j). The order of the algebra is l(2l − 1). A corresponding group is the special orthogonal (or rotation) group SO(2l). A rotation of the roots yields D3 = A3 . Also, D2 = A1 × A1 .

G 2 . The roots consist of the roots of A2 together with the six vectors ± (2ei − e j − ek ) (i = j = k = 1, 2, 3). The order of the algebra is 14. Examples of Lie groups used in atomic shell theory, together with their generators, are given in Table 3.1.

3.3 Irreducible Representations 3.3.1 Labels If n atomic states of a collection transform among themselves under an arbitrary action of the generators of a group G, then the states form a representation of G. The representation is irreducible if n  linear combinations of the states cannot be found that also exhibit that property, where n  < n. The commuting generators Hi of G can be simultaneously diagonalized within the n states: their eigenvalues (m 1 , m 2 , . . . , m l ) for an eigenstate ψ specify the weight of the eigenstate. weight
 The  above is said to be higher than m 1 , m 2 , . . . , m l if the first non-vanishing term in the sequence m 1 − m 1 , m 2 − m 2 , . . . is positive. An irre-

ducible representation (IR) of a semisimple group is uniquely specified (to within an equivalence) by its highest weight [3.1], which can therefore be used as a defining label.

3.3.2 Dimensions The dimensions of the IRs of various groups are expressed in terms of the highest weights and set out in Table 3.2. General algebraic expressions have been given by Wybourne [3.12, pp. 137]. Numerical tabulations have been made by Butler in the appendix to another book by Wybourne [3.13], and also by McKay and Patera [3.14]. The latter defines the IRs by speci-

Part A 3.3

a

† †

78

Part A

Mathematical Methods

fying the coordinates of the weights with respect to the simple roots of Dynkin [3.15].

3.3.3 Casimir’s Operator The eigenvalues of Casimir’s operator C, defined in (3.9), can be expressed in terms of the highest weights of an IR [3.1]. A complete algebraic listing for all the semisimple Lie groups has been

given by Wybourne [3.12, p. 140]. Sometimes Casimir’s operator is given in terms of the spherical tensors
1 W (κk) , or of their special cases V (k) = 2 2 W (0k) for which the single-electron reduced matrix element satisfies 1

(nl v(k) nl) = (2k + 1) 2 .

(3.15)

The eigenvalues of several operators of that form are given in Table 3.3.

Table 3.2 Dimensions D of the irreducible representations (IR’s) of various Lie groups

Part A 3.4

Group

IR

D

SO(2) SO(3) SO(4) = SO A (3) × SO B (3) SO(5) SO(6)

M DJ D J × DK (w1 w2 ) (w1 w2 w3 )

SO(7)

(w1 w2 w3 )

G2

(u 1 u 2 )

SU(3) or U(3) SU(4) or U(4) Sp(4) Sp(6)

[λ1 λ2 λ3 ] [λ1 λ2 λ3 λ4 ]  σ1 σ2   σ1 σ2 σ3 

1 2J + 1 (2J + 1)(2K + 1) (w1 + w2 + 2)(w1 − w2 + 1)(2w1 + 3)(2w2 + 1)/6 (w1 − w2 + 1)(w1 − w3 + 2)(w2 − w3 + 1) ×(w1 + w2 + 3)(w1 + w3 + 2)(w2 + w3 + 1)/12 (w1 + w2 + 4)(w1 + w3 + 3)(w2 + w3 + 2) ×(w1 − w2 + 1)(w1 − w3 + 2)(w2 − w3 + 1) ×(2w1 + 5)(2w2 + 3)(2w3 + 1)/720 (u 1 + u 2 + 3)(u 1 + 2)(2u 1 + u 2 + 5)(u 1 + 2u 2 + 4) ×(u 1 − u 2 + 1)(u 2 + 1)/120 (λ1 − λ2 + 1)(λ1 − λ3 + 2)(λ2 − λ3 + 1)/2 As for (w1 w2 w3 ) of SO(6)a As for (w1 w2 ) of SO(5)b (σ1 − σ2 + 1)(σ1 − σ3 + 2)(σ1 + σ2 + 5) ×(σ1 + σ3 + 4)(σ2 + σ3 + 3)(σ2 − σ3 + 1) ×(σ1 + 3)(σ2 + 2)(σ3 + 1)/720

a b

Subject to the conditions w1 = (λ1 + λ2 − λ3 − λ4 )/2, w2 = (λ1 − λ2 + λ3 − λ4 )/2, w3 = (λ1 − λ2 − λ3 + λ4 )/2 Subject to the conditions w1 = (σ1 + σ2 )/2, w2 = (σ1 − σ2 )/2

3.4 Branching Rules 3.4.1 Introduction If a group H shares some of its generators with a group G, the first can be considered a subgroup of the second. That is, G ⊃ H. Many of the groups in Table 3.1 can be put in extended group–subgroup sequences. The IRs of a subgroup that together span an IR of the group constitute a branching rule.

3.4.2 U(n) ⊃ SU(n) The group U(n) differs from SU(n) in
that the former contains among its generators a scalar such as W (00) that, by itself, forms an invariant subgroup. Thus U(n)

is not semisimple. The scalar in question commutes with all the generators of the group and so is of type Hi . Its presence enlarges the dimension, l, of the weight space by 1, an extension that can be accommodated by the unit vectors ei of Al given in Sect. 3.2.2. The reduction U(n) ⊃ SU(n) leads to the branching rule [λ1 λ2 · · · λn ] → [λ1 − a, λ2 − a, · · · , λn − a] , (3.16)

where, in the IR of SU(n) on the right, a = (λ1 + λ2 + · · · + λn )/n .

(3.17)

Group Theory for Atomic Shells

3.5 Kronecker Products

79

Table 3.3 Eigenvalues of Casimir’s operator C for groups used in the atomic l shell Group

IR

SU(2l + 1)

[λ]a

SO(2l + 1)

Wb

G2

(u 1 u 2 )

Operator
(k) 2  k>0 V
(k) 2  kodd V 
2
2  1 V (1) + V (5) 4

a

Appropriate for terms of l N with total spin S [3.7], p. 125

b

Defined by the l weights (w1 w2 · · · wl )

3.4.3 Canonical Reductions A group–subgroup sequence of the type U(n) ⊃ U(n − 1) ⊃ U(n − 2) ⊃ · · · ⊃ U(1)

(3.18)

is called canonical [3.17].  The branching rules for  those IRs λ1 λ2 · · · λn−1 of U(n − 1) contained in [λ1 λ2 · · · λn ] of U(n) have been given by Weyl [3.18]

3N + 2Nl − 12 N 2 − 2S(S + 1) − N 2 /(2l + 1) 1 l i=1 wi (wi + 1 + 2l − 2i) 2
2  u 1 + u 22 + u 1 u 2 + 5u 1 + 4u 2 /12

in terms of the “betweenness” conditions λ1 ≥ λ1 ≥ λ2 ≥ λ2 · · · ≥ λn−1 ≥ λn .

(3.19)

The possibility of using the scheme of (3.18) in the theory of complex atomic spectra has been explored by Harter and Patterson [3.19–21], and by Drake and Schlesinger [3.22, 23] (see also Sect. 4.3.1).

3.4.4 Other Reductions The algebraic formulae for U(n) ⊃ SO(n) and U(n) ⊃ Sp(n) have been given by Littlewood [3.24] and in a rather more accessible form by Wybourne [3.13]. Special cases have been tabulated by Butler (in Tables C-1 through C-15 in [3.13]). Another set of tables, in which Dynkin’s labeling scheme is used, has been given by McKay and Patera [3.14]. Descriptions of how to apply the mechanics of the mathematics to the Young tableaux that describe the IRs of U(n) can be found in the articles of Jahn [3.25] [with particular reference to SO(5)] and Flowers [3.26] [for Sp(2 j + 1)]. For the atomic l shell, the reductions SO(2l + 1) ⊃ SO(3) and (for f electrons) SO(7) ⊃ G2 and G2 ⊃ SO(3) are important. The sources cited in the previous Section are useful here. It is important to recognize that the embedding of one group in another can often be performed in inequivalent ways, depending on which generators are discarded in the reduction process. Thus the use of SO(5) ⊃ SO(3) in the atomic d shell involves a different SO(3) group from that derived from the canonical sequence SO(5) ⊃ SO(4) ⊃ SO(3).

3.5 Kronecker Products 3.5.1 Outer Products of Tableaux Consider the tableau [λ1 λ2 · · · λn ], where the total number of cells is N. A preliminary definition is required.

If among the first r terms of any permutation of the N factors of the product, x1 λ1 x2 λ2 · · · xn λn , the number of times x1 occurs is ≥ the number of times x2 occurs ≥ the number of times x3 occurs, etc. for all values of r,

Part A 3.5

To avoid fractional weights, the IRs of SU(n) are frequently replaced by those of U(n) for which the λi are integers. The weights λ1 , λ2 , . . . can be interpreted as the number of cells in successive rows of a Young Tableau. When the n states of a single particle are taken as a basis for the IR [10 . . . 0] of U(n), thus corresponding to a tableau comprising a single cell, the tableaux comprising N cells can be interpreted in two ways, namely, (1) as an IR of U(n) for a system of N particles, and (2) as an IR of S N , the finite group of permutations on N objects. A given tableau corresponds to as many permutations as there are ways of entering the numbers 1, 2, . . . , N in the cells such that the numbers increase going from left to right along the rows, and from top to bottom down the columns. A tableau possessing cells numbered in this way is called standard; it defines a permutation corresponding to a symmetrization with respect to the numbers in the rows, followed by an antisymmetrization with respect to the numbers in the columns [3.16].

Eigenvalue

80

Part A

Mathematical Methods

this permutation is called a lattice permutation. The prescription of Littlewood [3.24] for finding the tableaux appearing in the Kronecker product of [λ1 λ2 · · · λn ] with [µ1 µ2 · · · µm ] is as follows. The acceptable tableaux are those that can be built by adding to the tableau [λ1 λ2 · · · λn ], µ1 cells containing the same symbol α, then µ2 cells containing the same symbol β, etc., subject to two conditions:

Part A 3.6

1. After the addition of each set of cells labeled by a common symbol we must have a permissible tableau with no two identical symbols in the same column; 2. If the total set of added symbols is read from right to left in the consecutive rows of the final tableau, we obtain a lattice permutation of αµ1 β µ2 γ µ3 · · · . Examples of this procedure have been given ([3.24, p. 96], [3.7, p. 136], [3.13, p. 24]). An extensive tabulation involving tableaux with N < 8 has been calculated by Butler and given by Wybourne [3.13, Table B-1].

3.5.2 Other Outer Products The rules for constructing the Kronecker products for U(n) follow by interpreting the Young tableaux of the previous section as IRs of U(n). The known branching rules for reductions to subgroups enable the Kronecker products for the subgroups to be found. Many examples for SO(n), Sp(n), and G2 can be found in the book by Wybourne ([3.13, Tables D-1 through D-15, and E-4].

3.5.3 Plethysms Sometimes a particle can be thought of as being composite [as when the six orbital states s + d of a single electron are taken to span the IR [200] of SU(3)]. When the n  component states of a particle form a basis for

  an IR λ of U(n) other than [10 . . . 0], the process of finding which IRs of U(n) occur for N-particle states whose permutation symmetries are determined by a given Young tableau [λ] with N cells  is called a plethysm [3.24, p. 289] and written as λ ⊗ [λ]. The special techniques for doing this have been described by Wybourne [3.13]. An elementary method, which is often adequate in many cases, runs as follows:  N 1. Expand λ by repeated use of B-1  Table  from [3.13]. The resulting tableaux λ are independent of n. 2. Choose a small value  of n, and strike out all tableaux from the set λ that possess more than n rows [since they are unacceptable as IRs  of  U(n)]. 3. Interpret the remaining tableaux λ as IRs of U(n) and find their dimensions from Tables A-2 through A-17
of [3.13]. N Check that the sum of the dimensions is dim λ . 4. Interpret the various tableaux [λ]
possessing   the same number N of cells as IRs of U dim λ , and find their dimensions from [3.13]. 5. Match the dimensions of parts (3) and (4), remembering that each tableau [λ] occurs as often as the number of its standard forms. Thisdetermines the  possible ways of assigning the IRs λ of U(n) to each [λ]. 6. Proceed to higher n to remove ambiguities and to include the tableaux struck out in step 2. This procedure can be extended to calculate the plethysms for other groups. Examples of the type W ⊗ [λ] and U ⊗ [λ], where W and U are IRs of SO(7) and G2 , have been given for [λ] ≡ [2] and [11] corresponding to the separation of W 2 and U 2 into their symmetric and antisymmetric parts [3.27]. The technique of plethysm is also useful for mixed atomic configurations (Sect. 3.6.1).

3.6 Atomic States 3.6.1 Shell Structure The 24l+2
states of the l shell span the elementary spinor IR 12 21 · · · 12 of
SO(8l + 5), which decomposes into the two IRs 12 21 · · · ± 12 of SO(8l + 4), corresponding to an even and an odd number N of elec  trons [3.4]. The states of l N span the IR 1 N 04l+2−N of U(4l + 2), corresponding to the antisymmetric Young tableau comprising a single column of N cells. The separation of spin and orbit through the subgroup

˜ U(2) × U(2l + 1) yields the tableau products [λ] × [λ], ˜ is the tableau obtained by reflecting [λ] where [λ] in a diagonal line [3.1]. The IRs of the subgroup U(2) × SO(2l + 1) are denoted by S and W [3.6]. An alternative way of reaching this subgroup from U(4l+ 2) involves the intermediary Sp(4l + 2), whose IRs 1v 02l+1−v possess as a basis the states with seniority v [3.7]. A subgroup of SO(2l + 1) is the SO(3) group whose IRs specify L, the total orbital angular momentum.

Group Theory for Atomic Shells

Table 3.4 The states of the d shell dN

MQ

2S+1 [λ]

v

W

L

0

d

− 52

1 [0]

0

(00)

S

d1

−2

2 [1]

1

(10)

D

2

− 32

1 [2]

0 2 2

(00) (20) (11)

S DG PF

1 3 3

(10) (21) (11)

D PDFGH PF

0 2 4 2 4 4

(00) (20) (22) (11) (21) (10)

S DG SDFGI PF PDFGH D

1 3 5 3 5 5

(10) (21) (22) (11) (20) (00)

D PDFGH SDFGI PF DG S

d

3 [11] 3

−1

4

− 12

d

2 [21] 4 [111]

d

1 [22]

3 [211] 5 [1111] 5

d

0

2 [221]

4 [2111] 6 [11111]

by Wybourne [3.30] for the f shell. As Racah [3.6] showed, the group G 2 can be used to help distinguish repeated terms, but a few duplications remain. They are distinguished by Nielson and Koster [3.31] in their tables of spectroscopic coefficients by the letters A and B. The scope for applications of group theory becomes enlarged when the states of a single electron embrace more than one l value. Extensions of the standard model have been made by Feneuille [3.32] with particular reference to the configurations (d + s) N , for which quasiparticles have also been considered [3.33]. The group SU(3) has been used for (d + s) N p M [3.34]. The mixed configurations (s + f )4 have found a use in the quark model of the atomic f shell [3.29]. A brief description of this model has been given by Fano and Rao [3.35].

3.6.2 Automorphisms of SO(8) The quark structure s + f derives  the SO(3) struc
from ture of the elementary spinor 12 21 12 of SO(7). Its eight components span the IR (1000) of SO(8), a group that admits automorphisms [3.36]. This property is exhibited by the existence of the three distinct subgroups SO(7) (Racah’s group), SO(7) , and SO(7) , all of which possess the same G 2 and SO(3) as subgoups. A reversal of the relative phase of the s and f quarks takes SO(7) into SO(7) and vice versa [3.37]. The generators of SO(7) are the sums of the corresponding generators of SO(7) and SO(7) . The phase reversal between the s and f quarks, when interpreted in terms of electronic states, explains the unexpected simplifications found by Racah [3.6] in his equation (87) [3.38], which goes beyond what the Wigner–Eckart theorem for G2 would predict. Similarly, explanations can be found for some (but not all) proportionalities between blocks of matrix elements of components of the spin–other-orbit interaction for f electrons [3.39]. Hansen and Ven have given some examples of still unexplained proportionalities [3.40]. The group SO(7) has proved useful in analyses of the effective three-electron operators used to represent weak configuration interaction in the f shell [3.37].

3.6.3 Hydrogen and Hydrogen-Like Atoms The nonrelativistic hydrogen atom possesses an SO(4) symmetry associated with the invariance of the Runge–Lenz vector, which indicates the direction of the major axis of the classical elliptic orbit [3.5]. The

81

Part A 3.6

Alternatives to this classic sequence are provided by the last three groups listed in Table 3.1, together with their respective subgroups. For U A (2l + 1) × U B (2l + 1), the shell is factored by considering spin-up and spindown electrons as distinct (and statistically independent) particles [3.28]. A further factorization by means of the quasiparticles, θ, leads to four independent spaces. The 2
l states in each space span the elementary spinor 1l 2 of SOθ (2l + 1), which can be regarded as a fictitious particle (or quark), qθ [3.29]. The standard classification of the states of the d-shell is given in Table 3.4. The component M Q of the quasispin Q (defined in (6.33–6.35)) is listed, as well as the seniority, v = 2l + 1 − 2Q , the IRs W of SO(5), and the value of L (as a spectroscopic symbol). Only states in the first half of the shell appear; the classification for the second half is the same as the first except that the signs of M Q are reversed. A general rule for arbitrary l is exemplified by noting that every W [the IR of SO(2l + 1)] occurs with two spins (S1 and S2 ) and two quasispins (Q 1 and Q 2 ) such that S1 = Q 2 and S2 = Q 1 . No duplicated spectroscopic terms appear in Table 3.4. The generators of SO(5) do not commute with the inter-electronic Coulomb interaction; thus the separations effected by SO(5) merely define (to within a phase) a basis. The analog of Table 3.4 has been given

3.6 Atomic States

82

Part A

Mathematical Methods

Part A 3.7

quantum-mechanical form of this vector can be written in dimensionless units as (3.20) a = [(l × p) − ( p × l) + 2Zr/ra0 ] /2 p0 , where a0 = ~2 /me2 is the Bohr radius, Ze is the nuclear charge, p0 is related to the principal quantum number n by p0 = Z/na0 , and where the momentum p and angular momentum l of the electron in its orbit are measured in units of ~. The analysis is best carried out in momentum space [3.41]. The four coordinates to which SO(4) refers can be taken from
(9.43–9.46)  or directly as k p
x , k p y , k pz , and k p0 1 − p2 / p20 /2, where k = 2 p0 / p2 + p20 . The generators of SO(4) are provided by the 6 components of the two mutually commuting vectors (l + a)/2 and (l − a)/2, each of which behaves as an angular momentum vector. The

equivalence SO(4) = SO(3) × SO(3) corresponds to the isomorphism D2 = A1 × A1 of Sect. 3.2.2. Hydrogenic eigenfunctions belonging to various energies can be selected to form bases for a number of groups. The inclusion of all the levels up to a given n yields the IR (n − 1, 0) of SO(5). Levels of a given l and all n form an infinite basis for an IR of the noncompact group SO(2, 1) [3.42]. All the bound hydrogenic states span an IR of SO(4, 2), as do the states in the continuum [3.43]. Subgroups of SO(4, 2) and their generators have been listed by Wybourne [3.12] in his Table 21.2. To the extent that the central potential of a complex atom resembles the r −1 dependence for a bare nucleus, the group SO(4) can be used to label the states [3.44].

3.7 The Generalized Wigner–Eckart Theorem 3.7.1 Operators All atomic operators involving only the electrons can be built from their creation and annihilation operators. The appropriate group labels for an atomic operator acting on N electrons, each with n relevant component states, reduces to working out the various parts of the Kronecker products [10 . . . 0] N × [0 . . . 0 − 1] N of U(n). Subgroups of U(n) can further define these parts, which may be limited by Hermiticity constraints. The group labels for the Coulomb interaction for f electrons were first given by Racah [3.6]. Interactions involving electron spin were classified later [3.45–47]. Operators that represent the effects of configuration interaction on the d and f shells have also been studied [3.27, 48–52].

3.7.2 The Theorem Let the ket, operator T , and bra of a matrix element be labeled by an IR (Ra , Rc , Rb ) of a group G, each with a component (i a , i c , i b ). Suppose the supplementary labels γk are also required to complete the definitions. The generalized Wigner–Eckart theorem is γa Ra i a |T(γc Rc i c )|γb Rb i b  =

Aβ (βRa i a |Rb i b , Rc i c ) ,

(3.21)

β

where β distinguishes the IRs Ra should they appear more than once in the reduction of the Kronecker product Rb × Rc . The reduced matrix element Aβ is independent of the i k [3.6, 8]. The second factor on the right-hand

side of (3.21) is a Clebsch–Gordan (CG) coefficient for the group G. If the specification Ri can be replaced by Rτri, where r denotes an IR of a subgroup H of G, and τ is an additional symbol that may be necessary to make the classification unambiguous, the CG coefficient for G factorizes according to the Racah lemma [3.6] (βRa τa ra i a |Rb τbrb i b , Rc τcrc i c )

= (αra i a |rb i b , rc i c )(βRa τa ra |Rb τbrb + Rc τcrc )α . α

(3.22)

The first factor on the right is a CG coefficient for the group H; the second factor is an isoscalar factor [3.53].

3.7.3 Calculation of the Isoscalar Factors The group H above is often SO(3), whose Clebsch– Gordan coefficients (and their related 3– j symbols) are well-known (Chapt. 2). The principal difficulty in establishing comparable formulae for the isoscalar factors lies in giving algebraic meaning to β. Several methods are available for obtaining numerical results as follows. Extraction from Tabulated Quantities If Rb or Rc correspond to the IRs labeling a single electron, the factorization of the known [3.31] coefficients of fractional parentage (cfp) according to formulae of

Group Theory for Atomic Shells

the type [3.6] 
N d SLv{|d N−1 S L  v =  

N d Sv{|d N−1 S v W  L  + (10)d|WL

3.8 Checks

83

3.7.4 Generalizations of Angular Momentum Theory (3.23)

and W 

Evaluation Using Casimir’s Operator Two commuting copies (b and c) are made of the generators of the group G to form the generators of the direct product Gb × Gc [3.56]. Corresponding generators of Gb and Gc are added to give the generators of Ga . Each quadratic operator (Ta )2 appearing in the expression for Casimir’s operator Ca for Ga (as listed in Table 3.3) is written as (Tb + Tc )2 . On expanding the expressions of this type, the terms (Tb )2 and (Tc )2 yield Casimir’s operators Cb and Cc for Gb and Gc . Their eigenvalues can be written down in terms of the highest weights of the IRs appearing in the isoscalar factor of (3.22). If the cross products of the type (Tb · Tc ) can be evaluated within the uncoupled states |Rb τbrb , Rc τcrc , then our knowledge of the eigenvalues of Ca for the coupled states | βRa τa ra  provides the equations for determining (to within the freedom implied by β) the isoscalar factors relating the uncoupled to the coupled states. The evaluation of the cross products is straightforward when H = SO(3), since the relevant 6– j symbols are readily available [3.57]. Examples of this method can be found in the literature [3.48].

3.8 Checks The existence of numerical checks is useful when using group theory in atomic physics. The CG coefficients, isoscalar factors, and the various generalizations of the n– j symbols are often calculated in ways that conceal the simplicity and structure of the answer. Practitioners are familiar with several empirical rules:

(Wa τa L a |Wb τb L b + Wc τc L c ) = (−1)t (Wa τa L a |Wc τc L c + Wb τb L b ) ,

(3.24)

where t = L a − L b − L c + x, with x dependent on the IRs W only; or, (2) The reciprocity relation of Racah [3.6]: (Wa τa L a |Wb τb L b + Wc τc L c ) = 

1

(−1)t [(2L b + 1) dim Wa /(2L a + 1) dim Wb ] 2 × (Wb τb L b |Wa τa L a + Wc τc L c ) , (3.25) where t  = L a − L b − L c + x  , with x  dependent on the IRs W, but taken to be l by Racah for Wc = (10 . . . 0). Reduced matrix elements in SO(3) can be further reduced by the extraction of isoscalar factors. When Wa occurs once in the decomposition of W × Wb we have  γa Wa τa L a T (WL) γb Wb τb L b =  1 [(2L a + 1)/ dim Wa ] 2 γa Wa |||T (W) |||γb Wb × (Wb τb L b + WL|Wa τa L a ) .

(3.26)

Analogs of the n– j symbols are discussed by Butler [3.58].

√ 1. Numbers √ with different irrationalities, such as 2 and 3, are never added to one another. 2. The denominators of fractions seldom involve high primes. 3. High primes are uncommon, but when they appear, it is usually in diagonal matrix elements rather than off-diagonal ones.

Part A 3.8

yields some isoscalar factors. In this example, W are the IRs of SO(5) defined by the triples NSv and N  S v (with N  = N − 1) as in Table 3.4. This approach can be applied to the f shell to give isoscalar factors for SO(7) and G2 . The many-electron cfp of Donlan [3.54] and the multielectron cfp of Velkov [3.55] further extend the range to IRs Rb and Rc describing many-electron systems. Isoscalar factors found in this way have the advantage that their relative phases as well as the significance of the indices β and τ coincide with current usage.

CG coefficients, n– j symbols, reduced matrix elements, and the entire apparatus of angular momentum theory all have their generalizations to groups other than SO(3). An interchange of two columns of a 3– j symbol has its analog in the interchange of two parts of an isoscalar factor. For IRs W and L of SO(2l + 1) and SO(3), there are two possibilities: (1) The interchange of the two parts separated by the plus sign, namely,

84

Part A

Mathematical Methods

4. A sum of a number of terms frequently factors in what appears to be an unexpected way, and similar sums often exhibit similar factors.

Guided by these rules, one will find that such errors as do arise occur with phases rather than with magnitudes.

References 3.1

3.2 3.3

Part A 3

3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14

3.15 3.16 3.17 3.18 3.19 3.20 3.21

3.22 3.23 3.24 3.25 3.26

G. Racah: Group Theory and Spectroscopy, Springer Tracts in Modern Physics, Vol. 37 (Springer, New York 1965) L. P. Eisenhart: Continuous Groups of Transformations (Dover, New York 1961) E. Cartan: Sur la Structure des Groupes de Transformations Finis et Continus, Thesis (Nony, Paris 1894) B. R. Judd: Group Theory and Its Applications, ed. by E. M. Loebl (Academic, New York 1968) E. U. Condon, H. Odabasi: Atomic Structure (Cambridge Univ. Press, Cambridge 1980) G. Racah: Phys. Rev. 76, 1352 (1949) B. R. Judd: Operator Techniques in Atomic Spectroscopy (Princeton Univ. Press, Princeton 1963) E. P. Wigner: Group Theory (Academic, New York 1959) B. R. Judd: Phys. Rev. 162, 28 (1967) L. Armstrong, B. R. Judd: Proc. R. Soc. London Ser. A 315, 27 and 39 (1970) B. R. Judd, G. M. S. Lister: J. Phys. A 25, 2615 (1992) B. G. Wybourne: Classical Groups for Physicists (Wiley, New York 1974) B. G. Wybourne: Symmetry Principles and Atomic Spectroscopy (Wiley, New York 1970) W. G. McKay, J. Patera: Tables of Dimensions, Indices, and Branching Rules for Representations of Simple Lie Algebras (Dekker, New York 1981) E. B. Dynkin: Am. Math. Soc. Transl. Ser. 2 6, 245 (1965) D. E. Rutherford: Substitutional Analysis (Edinburgh Univ. Press, Edinburgh 1948) M. Moshinsky: Group Theory and the Many-Body Problem (Gordon Breach, New York 1968) H. Weyl: The Theory of Groups and Quantum Mechanics (Dover, New York undated) W. G. Harter: Principles of Symmetry, Dynamics and Spectroscopy 8, 2819 (1973) W. G. Harter: Principles of Symmetry, Dynamics and Spectroscopy (Wiley, New York 1993) W. G. Harter, C. W. Patterson: A Unitary Calculus for Electronic Orbitals, Lect. Notes Phys., Vol. 49 (Springer, Berlin, Heidelberg 1976) G. W. F. Drake, M. Schlesinger: Phys. Rev. A 15, 1990 (1977) R. D. Kent, M. Schlesinger: Phys. Rev. A 50, 186 (1994) D. E. Littlewood: The Theory of Group Characters (Clarendon, Oxford 1950) H. A. Jahn: Proc. R. Soc. London Ser. A 201, 516 (1950) B. H. Flowers: Proc. R. Soc. London Ser. A 212, 248 (1952)

3.27 3.28 3.29 3.30 3.31

3.32 3.33 3.34 3.35 3.36 3.37 3.38 3.39 3.40 3.41 3.42 3.43 3.44 3.45 3.46 3.47 3.48 3.49 3.50 3.51 3.52 3.53 3.54

3.55

B. R. Judd, H. T. Wadzinski: J. Math. Phys. 8, 2125 (1967) C. L. B. Shudeman: J. Franklin Inst. 224, 501 (1937) B. R. Judd, G. M. S. Lister: Phys. Rev. Lett. 67, 1720 (1991) B. G. Wybourne: Spectroscopic Properties of Rare Earths (Wiley, New York 1965) p. 15 C. W. Nielson, G. F. Koster: Spectroscopic Coefficients for the pn , dn , and f n Configurations (MIT Press, Cambridge 1963) S. Feneuille: J. Phys. (Paris) 28, 61, 315, 701, and 497 (1967) S. Feneuille: J. Phys. (Paris) 30, 923 (1969) S. Feneuille, A. Crubellier, T. Haskell: J. Phys. (Paris) 31, 25 (1970) U. Fano, A. R. P. Rao: Symmetry Principles in Quantum Physics (Academic, New York 1996) Sect. 8.3.3 H. Georgi: Lie Algebras in Particle Physics (Benjamin/Cummings, Reading 1982) Chap. XXV B. R. Judd: Phys. Rep. 285, 1 (1997) E. Lo, J. E. Hansen, B. R. Judd: J. Phys. B 33, 819 (2000) B. R. Judd, E. Lo: Phys. Rev. Lett. 85, 948 (2000) J. E. Hansen, E. G. Ven: Mol. Phys. 101, 997 (2003) M. J. Englefield: Group Theory and the Coulomb Problem (Wiley, New York 1972) L. Armstrong: J Phys. (Paris) 31, 17 (1970) C. E. Wulfman: Group Theory and Its Applications, Vol. 2, ed. by E.M. Loebl (Academic, New York 1971) D. R. Herrick: Adv. Chem. Phys. 52, 1 (1982) A. G. McLellan: Proc. Phys. Soc. London 76, 419 (1960) B. R. Judd: Physica 33, 174 (1967) B. R. Judd, H. M. Crosswhite, H. Crosswhite: Phys. Rev. 169, 130 (1968) B. R. Judd: Phys. Rev. 141, 4 (1966) B. R. Judd, M. A. Suskin: J. Opt. Soc. Am. B 1, 261 (1984) B. R. Judd, R. C. Leavitt: J. Phys. B 19, 485 (1986) R. C. Leavitt: J. Phys. A 20, 3171 (1987) R. C. Leavitt: J. Phys. B 21, 2363 (1988) A. R. Edmonds: Proc. R. Soc. London Ser. A 268, 567 (1962) V. L. Donlan: Air Force Material Laboratory Report No. AFML-TR-70-249 (Wright-Patterson Air Force Base, Ohio 1970) D. D. Velkov: Multi-Electron Coefficients of Fractional Parentage for the p, d, and f Shells. Ph.D. Thesis (The Johns Hopkins University, Baltimore 2000) http://www.pha.jhu.edu/groups/cfp/

Group Theory for Atomic Shells

3.56

P. Nutter, C. Nielsen: Fractional parentage coefficients of terms of f n , II. Direct Evaluation of Racah’s Factored Forms by a Group Theoretical Approach, Technical Memorandum T-133 (Raytheon, Waltham 1963) p. 133

3.57

3.58

References

85

M. Rotenberg, R. Bivins, N. Metropolis, J. K. Wooten: The 3-j and 6-j Symbols (MIT Press, Cambridge 1959) P. H. Butler: Phil. Trans. R. Soc. London Ser. A 277, 545 (1975)

Part A 3

87

Dynamical Gro 4. Dynamical Groups

4.1

Noncompact Dynamical Groups ............. 4.1.1 Realizations of so(2,1) ................ 4.1.2 Hydrogenic Realization of so(4,2)

4.2

Hamiltonian Transformation and Simple Applications ........................................ 4.2.1 N-Dimensional Isotropic Harmonic Oscillator ................... 4.2.2 N-Dimensional Hydrogenic Atom 4.2.3 Perturbed Hydrogenic Systems ....

4.3

Compact Dynamical Groups................... 4.3.1 Unitary Group and Its Representations ........................ 4.3.2 Orthogonal Group O(n) and Its Representations ........................ 4.3.3 Clifford Algebras and Spinor Representations ........................ 4.3.4 Bosonic and Fermionic Realizations of U(n) ................... 4.3.5 Vibron Model ............................ 4.3.6 Many-Electron Correlation Problem ................................... 4.3.7 Clifford Algebra Unitary Group Approach ................................. 4.3.8 Spin-Dependent Operators .........

References ..................................................

87 88 88 90 90 91 91 92 92 93 94 94 95 96 97 97 98

initio models of electronic structure) exploit compact LG’s. We follow the convention of designating Lie groups by capital letters and Lie algebras by lower case letters, e.g., the Lie algebra of the rotation group SO(3) is designated as so(3).

4.1 Noncompact Dynamical Groups As an illustration we present basic facts concerning LAs that are useful for centrosymmetric Kepler-type problems, their realizations and typical applications. Recall that a realization of a LA is a homomorphism associating a concrete set of

physically relevant operators with each abstract basis of the given LA. The physical operators we will use are general (intrinsic) position vectors R = (X 1 , X 2 , . . . , X N ) in R N and their corresponding momenta P = (P1 , P2 , . . . , PN ), satisfying the basic

Part A 4

The well known symmetry (invariance, degeneracy) groups or algebras of quantum mechanical Hamiltonians provide quantum numbers (conservation laws, integrals of motion) for state labeling and the associated selection rules. In addition, it is often advantageous to employ much larger groups, referred to as the dynamical groups (noninvariance groups, dynamical algebras, spectrum generating algebras), which may or may not be the invariance groups of the studied system [4.1–7]. In all known cases, they are Lie groups (LGs), or rather corresponding Lie algebras (LAs), and one usually requires that all states of interest of a system be contained in a single irreducible representation (irrep). Likewise, one may require that the Hamiltonian be expressible in terms of the Casimir operators of the corresponding universal enveloping algebra [4.8, 9]. In a weaker sense, one regards any group (or corresponding algebra) as a dynamical group if the Hamiltonian can be expressed in terms of its generators [4.10–12]. In nuclear physics, one sometimes distinguishes exact (baryon number preserving), almost exact (e.g., total isospin), approximate (e.g., SU(3) of the “eightfold way”) and model (e.g., nuclear shell model) dynamical symmetries [4.13]. The dynamical groups of interest in atomic and molecular physics can be conveniently classified by their topological characteristic of compactness. Noncompact LGs (LAs) generally arise in simple problems involving an infinite number of bound states, while those involving a finite number of bound states (e.g., molecular vibrations or ab

88

Part A

Mathematical Methods

commutation relations (~ = 1) 


X j , Pk = iδ jk I . X j , X k = P j , Pk = 0, (4.1)

4.1.1 Realizations of so(2,1) This important LA is a simple noncompact analogue of the well known rotation group LA so(3), (cf. Sects. 2.1 and 3.2). Designating its three generators by T j ( j = 1, 2, 3), its structure constants (Sect. 2.1.1 and Sect. 3.1.1) are defined by [T1 , T2 ] = iγT3 , [T2 , T3 ] = iT1 , [T3 , T1 ] = iT2 ,

Part A 4.1

L2 , L = R∧ P . (4.8) R2 The general form of the desired so(2,1) realization is [4.2, 14–17]    1 T1 = R−ν ν−2 R2 PR2 + ξ ∓ R2ν , 2 T3

1 −1 T2 = 2ν RPR − i(1 − ν−1 )I , (4.9) 2 where ξ is either a c-number (scalar operator) or an operator which commutes with both R and PR , and ν is an arbitrary real number. To interrelate this realization with so(2,1) unirreps D + (k) or D + (−k − 1), we have to establish the connection between the quantum numbers k, q and the parameters ξ and ν. Considering the Casimir operator T 2 in (4.5), we find that in our realization (4.9)  T 2 = ξ + 1 − ν2 /4ν2 , (4.10) P 2 = PR2 +

(4.2)

with γ = −1, while γ = 1 gives so(3). Defining the socalled ladder (raising and lowering) operators T± = T1 ± iT2 , we also have that 
T+ , T− = 2γT3 ,

with unbounded T3 spectra (which may be exploited in scattering problems). For problems involving only central potentials, a useful realization is given in terms of the radial distance R = |R| and the radial momentum   1 i ∂ ∂ R = −i + PR = − R ∂R ∂R R 1 = (R· P − iI) , (4.7) R so that [R, PR ] = iI. Recall that

(4.3)




 T3 , T± = ±T3 .

(4.4)

The Casimir operator then has the form   T 2 = γ T12 + T22 + T32 = γT+ T− + T32 − T3 . (4.5) † With a Hermitian scalar product satisfying T j † 1, 2, 3), so that T± = T∓ , the unitary irreps

= Tj ( j = (unirreps) carried by the simultaneous eigenstates of T 2 and T3 have the form (cf. Sects. 2.1.1 and 2.2) T 2 |kq = k(k + 1)|kq , T3 |kq = q|kq ,  T± |kq = γ(k ∓ q)(k ± q + 1)|k, q ± 1 .

so that k=

(4.6)

D (k) ,

For so(3), (γ = 1), only finite dimensional irreps k = 0, 1, 2, . . . with |q| ≤ k are possible (Sects. 2.2 and 2.3). In contrast, there are no nontrivial finite dimensional unirreps of so(2,1); (for classification, see e.g., [4.2, 14, 15]). The relevant class D + (k) of so(2,1) unirreps for bound state problems has a T3 eigenspectrum bounded from below and is given by q = −k + µ; µ = 0, 1, 2, . . . , and k < 0 or, equivalently, D + (−k − 1) with q = k + 1 + µ; µ = 0, 1, 2, . . . ; k > −1, since k1 = −k − 1 defines an equivalent unirrep and k1 (k1 + 1) = k(k + 1). There exists a similar class of irreps with the T3 spectrum bounded from above and two classes (principal and supplementary)

 

1 −1 ± 4ξ + ν−2 2

(4.11)

and q = q0 + µ , where q0 = k + 1 =

µ = 0, 1, 2, . . .  

1 1 ± 4ξ + ν−2 , 2

(4.12)

k > −1 . (4.13)

4.1.2 Hydrogenic Realization of so(4,2) To obtain suitable hydrogenic realizations of so(4,2) it is best to proceed from so(4) (the dynamical symmetry group for the bound states of the nonrelativistic Kepler problem), and merge it with so(2,1) [4.2, 14, 15].

Dynamical Groups

The so(4) LA can be realized either as a direct sum so(4) = so(3) ⊕ so(3), or by supplementing so(3) with an appropriately scaled quantum mechanical analogue of the Laplace–Runge–Lenz (LRL) vector (cf. Sect. 3.6.2). In the first case, we use two commuting angular momentum vectors M and N (cf. Sect. 2.5),

  M j , Mk = iε jk M , N j , Nk = iε jk N ,
 M j , Nk = 0, ( j, k,  = 1, 2, 3) (4.14) while in the second case we use the components of the total angular momentum vector J and LRL-like vector V with commutation relations 
J j , Jk = iε jk J , 
V j , Vk = iσε jk J , 
(4.15) J j , Vk = iε jk V , ( j, k,  = 1, 2, 3) ,

1 1 (4.16) (J + V) , N = (J − V) , 2 2 so that J = M ⊕ N and V = 2M − J. The two Casimir operators C1 and C2 are M=

C1 = σJ 2 + V 2 = σJ+ J− + V+ V− + V32 + σJ3 (J3 − 2) , C2 = (V · J) = (J · V) 1 = (V+ J− + V− J+ ) + V3 J3 , 2 where again X ± = X 1 ± iX 2 ,

X = J or V .

(4.17)

For so(3,1) and e(3), only infinite dimensional nontrivial irreps are possible, while for so(4), only finite dimensional ones arise. To get unirreps, we require   J and V to be Hermitian. Using J 2 , J3 , C1 , C2 as a complete set of commuting operators for so(4), we label the basis vectors by the four quantum numbers as |γ jm ≡ |( j0 , η) jm, so that J |γ jm = j( j + 1)|γ jm , J3 |γ jm = m|γ jm ,  C1 |γ jm = j02 − η2 − 1 |γ jm , C2 |γ jm = j0 η|γ jm ,

with 2| j0 | being a nonnegative integer and j = | j0 |, | j0 | + 1, . . . , η − 1 ; η = | j0 | + k, k = 1, 2, . . .

2

which commutes with the hydrogenic Hamiltonian H=

1 2 p − Zr −1 . 2

(4.22)

Note that


 ˜ H =0, [L, H] = V,   L · V˜ = V˜ · L = 0 ,  V˜ 2 = 2H L 2 + 1 + Z2 ,

(4.23)

while the components of L and V˜ satisfy the commutation relations 
L j , L k = i jk L  , 
L j , V˜k = i jk V˜ , 
(4.24) V˜ j , V˜k = (−2H )i jk L  . Thus, restricting ourselves to a specific bound state energy level E n , we can replace H by E n and define ( j = 1, 2, 3) ,

(4.25)

obtaining the so(4) commutation relations (4.15) (with J replaced by L). This is Pauli’s hydrogenic realization of so(4) [4.21–23]. [In a similar way we can consider ˜ obcontinuum states E > 0 and define V = (2E)−1/2 V, taining an so(3,1) realization.] The last identity of (4.23) now becomes  V 2 = − L 2 + 1 − Z2 /2E n , (4.26) which immediately implies Bohr’s formula, since V 2 + L 2 = 4M 2 = −1 − Z2 /2E n , so that E n = −Z2 /2(2 j1 + 1)2 = −Z2 /2n 2 ,

(4.19)

(4.20)

(see, e.g., [4.17] for the action of J± , V3 and V± ). To obtain the hydrogenic (or Kepler) realization of so(4), we consider the quantum mechanical analog of the classical LRL vector 1 V˜ = ( p ∧ L − L ∧ p) − Zr −1r 2 1 = r p2 − p(r · p) + r H , 2 (4.21) L=r∧p,

V j = (−2E n )−1/2 V˜ j (4.18)

89

(4.27)

where n = 2 j1 + 1 and j1 is the angular momentum quantum number for M, (4.16). In terms of the ir-

Part A 4.1

with σ = 1. For σ = −1 we obtain so(3,1) (the LA of the homogeneous Lorentz group), which is relevant to the scattering problem of a particle in the Coulomb (or Kepler) potential (see below). For σ = 0 we get e(3) (the LA of the three-dimensional Euclidean group) [4.18– 20]. Note that (4.14) and (4.15) are interrelated by

4.1 Noncompact Dynamical Groups

90

Part A

Mathematical Methods

rep labels (4.20), we have that j0 = 0, η = n, so that |γm = |(0, n)m ≡ |nm,  = 0, 1, . . . , n − 1. Using the stepwise merging of so(4) and so(2,1) [adding first T2 which leads to so(4,1) and subsequently T1 and T3 ], we arrive at the hydrogenic realization of so(4,2) having fifteen generators L, A, B, Γ , T1 , T2 , T3 , namely (cf. [4.2, 14, 15, 17]) L = R∧ P ,  1 A = RP 2 − P(R· P) ∓ R , 2 B

we can write the commutation relations in the following standard form 
L jk , L m =  i g j L km + gkm L j − gk L jm − g jm L k , (4.30)

with the diagonal metric tensor g jk defined by the matrix G = diag[1, 1, 1, 1, −1, −1]. The matrix form (4.29) also implies the subalgebra structure so(4, 2) ⊃ so(4, 1) ⊃ so(4) ⊃ so(3) ,

Γ = RP ,  1 1 T1 = RP 2 ∓ R = RPR2 + L 2 R−1 ∓ R , 2 2 T3 T2 = R· P − iI = RPR .

(4.28)

Part A 4.2

Relabeling these generators by the elements of an antisymmetric 6 × 6 matrix according to the scheme   0 L 3 −L 2 A1 B1 Γ1    0 L 1 A2 B2 Γ2     0 A3 B3 Γ3   L jk ↔  (4.29)  0 T2 T1       0 T3  0

(4.31)

with so(4,1) generated by L, A, B and T2 , and so(4) by L, A. [L, B also generate so(3,1).] The three independent Casimir operators (quadratic, cubic and quartic) are [4.24], (summation over all indices is implied) 1 Q 2 = L jk L jk 2 = L 2 + A2 − B 2 − Γ 2 + T32 − T12 − T22 , 1 Q 3 = εijkmn L ij L k L mn 48 = T1 (B · L) + T2 (Γ · L) + T3 (A · L) + A · (B ∧ Γ ) , Q 4 = L jk L k L m L m j .

(4.32)

For our hydrogenic realization Q 2 = −3, Q 3 = Q 4 = 0. Thus, our hydrogenic realization implies a single unirrep of so(4,2) adapted to the chain (4.31).

4.2 Hamiltonian Transformation and Simple Applications The basic idea is to transform the relevant Schrödinger equation into an eigenvalue problem for one of the operators from the complete set of commuting operators in our realizations, e.g., T3 for so(2,1). Instead of using a rather involved “tilting” transformation ([4.1, p. 20], and [4.2,14,15]), we can rely on a simple scaling transformation [4.16, 25] r = λR ,

p = λ−1 P ,

r = λR ,

pr = λ−1 PR , (4.33)

where

 1/2 N  r = x 2j  ,

(4.34)

with pr defined analogously to PR in (4.7); r and p are the physical operators in terms of which the Hamiltonian of the studied system is expressed. Recall that L 2 has eigenvalues [4.26] ( + N − 2) ,

 = 0, 1, 2, . . .

for

N ≥2, (4.36)

and we can set  = 0 for N = 1 (angular momentum term vanishes in one-dimensional case). The units in which m = e = ~ = 1, c ≈ 137 are used throughout.

4.2.1 N-Dimensional Isotropic Harmonic Oscillator

j=1

and p2 = p2r + r −2



 1 (N − 1)(N − 3) + L 2 , 4

Considering the Hamiltonian (4.35)

H=

1 2 1 2 2 p + ω r , 2 2

(4.37)

Dynamical Groups

with p2 in the form (4.35), transforming the corresponding Schrödinger equation using the scaling transformation (4.33) and multiplying by 14 λ2 , we get for the radial component   1 2 4 2 1 2 1 1 2 −2 P +R ξ + ω λ R − λ E ψR (λR) = 0 , 2 4 R 4 2 (4.38)

with 1 1 (N − 1)(N − 3) + ( + N − 2) . (4.39) 16 4  2 Choosing λ such that ω/2 λ4 = 1, we can rewrite (4.38) using the so(2,1) realization (4.9) with ν = 2 as   1 2 T3 − λ E ψR (λR) = 0 . (4.40) 4 ξ=

E = 4q/λ2 = 2qω ,

(4.41)

with q given by (4.12) and (4.13), i. e., q = q0 + µ , and q0 = k + 1 =

µ = 0, 1, 2, . . .    1 1 1± + N −1 . 2 2

(4.42)

the radial component (after multiplying from the left by λ2 R)   1 2 RPR + R−1 ξ − 2λ2 E R − λZ ψR (λR) = 0 , 2 (4.46)

where now 1 ξ = (N − 1)(N − 3) + ( + N − 2) . (4.47) 4 In this case we must set 2λ2 E = −1 and use realization (4.9) with ν = 1 to obtain (T3 − λZ)ψR (λR) = 0 .

(4.48)

This immediately implies that λZ = q

(4.49)

and 1 (4.50) [1 ± (2 + N − 2)] . 2 Choosing the upper sign [since  ≥ 0 and k > −1], so that q0 =  + 12 (N − 1), and identifying q with the principal quantum number n, we have finally that q0 = k + 1 =

1 Z2 =− 2 . (4.51) 2 2λ 2n The N-dimensional relativistic hydrogenic atom can be treated in the same way, using either the Klein– Gordon or Dirac–Coulomb equations [4.2, 14–17]. E ≡ En = −

(4.43)

1 Now, for N = 1 we set  = 0 so that  1 q0 = 4 and 3 q 0 = 4 , yielding for E = 2qω the values 2 + 2µ ω and 1 + 2µ + 1 ω, µ = 0, 1, 2, . . . . Combining both sets 2 we thus get for N = 1 the well known result   1 E ≡ En = n + ω , n = 0, 1, 2, . . . . (4.44) 2

Similarly, for the general case N ≥ 2 we choose the upper sign in (4.43) [so that k > −1] and get   1 E ≡ E n = n + N ω, n = 0, 1, 2, . . . (4.45) 2 where we identified ( + 2µ) with the principal quantum number n.

4.2.2 N-Dimensional Hydrogenic Atom Applying the scaling transformation (4.33) to the hydrogenic Hamiltonian (4.22) in N-dimensions, we get for

91

4.2.3 Perturbed Hydrogenic Systems The so(4,2) based Lie algebraic formalism can be conveniently exploited to carry out large order perturbation theory (see [4.27–29] and Chapt. 5) for hydrogenic systems described by the Schrödinger equation [H0 + εV(r)] ψ(r) = (E 0 + ∆E)ψ(r) ,

(4.52)

with H0 given by (4.22) and E 0 by E of (4.51). Applying transformation (4.33), using (4.49), (4.51) and multiplying on the left by λ2 R, we get   1 1 RP 2 − q + R + ελ2 RV(λR) − λ2 R∆E Ψ(R) 2 2 =0, (4.53) where we set ψ(λR) ≡ Ψ(R). For the important case of a 3-dimensional hydrogenic atom [N = 3, ξ = ( + 1), q ≡ n] we get using the so(4,2) realization (4.28) [or so(2,1) realization (4.9) with ν = 1] (K + εW − S∆E)Ψ(R) = 0 ,

(4.54)

Part A 4.2

Thus, using the second equation of (4.6) we can interrelate ψR (λR) with |kq and set 14 λ2 E = q, so that

4.2 Hamiltonian Transformation and Simple Applications

92

Part A

Mathematical Methods

with

Z are easily expressed in terms of so(4,2) generators,

K = T3 − n , W = λ2 RV(λR) , S=λ R. 2

Z = B3 − A3 , (4.55)

Part A 4.3

We also have that λ = n/Z and for the ground state case n = q = 1. Although (4.54) has the form of a generalized eigenvalue problem requiring perturbation theory formalism with a nonorthogonal basis (where S represents an overlap), T3 is Hermitian with respect to a (1/R) scalar product, and the required matrix elements can therefore be easily evaluated [4.2, 14, 15, 17, 27–29]. For central field perturbations, V(r) = V(r), the problem reduces to one dimension and since R = T3 − T1 , the so(2,1) hydrogenic realization (ν = 1) can be employed. For problems of a hydrogenic atom in a magnetic field (Zeeman effect) [4.27–30] or a one-electron diatomic ion [4.31], the so(4,2) formalism is required (note, however, that the LoSurdo–Stark effect can also be treated as a one-dimensional problem using parabolic coordinates [4.32]). The main advantage of the LA approach stems from the fact that the spectrum of T3 is discrete, so that no integration over continuum states is required. Moreover, the relevant perturbations are closely packed around the diagonal in this representation, so that infinite sums are replaced by small finite sums. For example, for the LoSurdo–Stark problem when V(r) = F z, where F designates electric field strength in the z-direction, we get (4.54) with ε = F and W = (n/Z)3 RZ, S = (n/Z)2 R. Since both R and

R = T3 − T1 ,

(4.56)

we can easily compute all the required matrix elements [4.2, 14, 15, 17]. Similarly, considering the Zeeman effect with 1 1  V(r) = B L 3 + B 2 r 2 − z 2 , 2 8

(4.57)

where B designates magnetic field strength in the z-direction, we have for the ground state when n = 1,  = m = 0 that ε = 18 B 2 , K = T3 − 1, W = Z−4 R(R2 − Z 2 ) and S = Z−2 R. Again, the matrix elements of W and S are obtained from those of Z and R, (4.56) by matrix multiplication (for tables and programs, see [4.17]). One can treat one-electron diatomic ions [4.2,14,15, 31] and screened Coulomb potentials, including charmonium and harmonium [4.10–12, 17, 33, 34], in a similar way. Note, finally, that we can also formulate the perturbed problem (4.54) in a standard form not involving the “overlap” by defining the scaling factor as λ = (−2E)−1/2 , where E is now the exact energy E = E 0 + ∆E. Equation (4.54) then becomes (T3 + εW − λZ)Ψ(R) = 0 ,

(4.58)

with the eigenvalue λZ. In this case any conventional perturbation formalism applies, but the desired energy has to be found from λZ [4.35].

4.3 Compact Dynamical Groups Unitary groups U(n) and their LAs often play the role of (compact) dynamical groups since

4.3.1 Unitary Group and Its Representations

1. quantum mechanical observables are Hermitian and the LA of U(n) is comprised of Hermitian operators [under the exp(iA) mapping], 2. any compact Lie group is isomorphic to a subgroup of some U(n), 3. “nothing of algebraic import is lost by the unitary restriction” [4.36].

The unitary group U(n) has n 2 generators E ij spanning its LA and satisfying the commutation relations
 E ij , E k = δ jk E i − δi E k j (4.59)

All U(n) irreps have finite dimension and are thus relevant to problems involving a finite number of bound states [4.3–6, 10–12, 36–43].

and the Hermitian property †

E ij = E ji .

(4.60)

They are classified as raising (i < j ), lowering (i > j ) and weight (i = j ) generators according to whether they

Dynamical Groups

raise, lower and preserve the weight, respectively. The weight vector is a vector of the carrier space of an irrep which is a simultaneous eigenvector of all weight generators E ii of U(n) (comprising its Cartan subalgebra), and the vector m = (m 1 , m 2 , . . . , m n ) with integer components, consisting of corresponding eigenvalues, is called a weight. The highest weight mn (in lexical ordering), mn = (m 1n , m 2n , . . . , m nn ) ,

(4.61)

4.3 Compact Dynamical Groups

[m] defined by    m 1n m 2n · · · · · · m nn mn     mn−1   m 1,n−1 · · · · · · m n−1,n−1     ···   ···   [m] =   · · · = ···        m2   m 12 m 22 m1 m 11

93

     ,    

(4.68)

with m 1n ≥ m 2n ≥ · · · ≥ m nn ,

(4.62)

uniquely labels U(n) irreps, Γ (mn ), and may be represented by a Young pattern. Subducing Γ (mr ) of U(r) to U(r − 1), embedded as U(r − 1) ⊕ 1 in U(r), gives [4.41]  Γ (mr ) ↓ U(r − 1) = Γ (mr−1 ) , (4.63)

m ir ≥ m i,r−1 ≥ m i+1,r

Two irreps Γ (mn ) and Γ (m n ) of U(n) yield the same irrep when restricted to SU(n) if m i = m i + h, i = 1, . . . , n. The SU(n) irreps are thus labeled with highest weights with m nn = 0. The dimension of Γ (mn ) of U(n) is given by the Weyl dimension formula [4.36] dim Γ (mn ) =   m in − m jn + j − i 1!2! · · · (n − 1)! . i< j

(4.65)

The U(n) Casimir operators have the form n 

E i1 i2 E i2 i3 · · · E ik−1 ik E ik i1 .

i 1 ,i 2 ,... ,i k =1

(4.66)

The first order Casimir operator is given by the sum of weight generators and equals the sum of the highest weight components. Since U(1) is Abelian, the Gel’fand-Tsetlin [4.42] canonical chain (Sect. 3.4.3) U(n) ⊃ U(n − 1) ⊃ · · · ⊃ U(1)

(4.67)

can be used to label uniquely the basis vectors of the carrier space of Γ (mn ) by triangular Gel’fand tableaux

j=1

(4.69)

while those for other generators are rather involved [4.42, 43]. Note that only elementary (E i,i+1 ) raising generators are required since

(i = 1, . . . , r − 1) . (4.64)

CkU(n) =

j=1

[m ]|E ij |[m] = [m]|E ji |[m ]

(4.70)


E ij = Ei,i+1 , E i+1, j .

(4.71)

and

In special cases required in applications ([4.39, 40] and Sect. 4.3.4) efficient algorithms exist for the computation of explicit representations.

4.3.2 Orthogonal Group O(n) and Its Representations Since O(n) is a proper subgroup of U(n), its representation theory has a similar structure. The suitable generators are Fij = E ij − E ji , F ji = −Fij , Fii = 0 ,

†

Fij = −F ji

(4.72)

and satisfy the commutation relations 
Fij , Fk = δ jk Fi + δi F jk − δik F j − δ j Fik . (4.73)

The canonical chain has the form O(n) ⊃ O(n − 1) ⊃ · · · ⊃ O(2) .

(4.74)

The components of the highest weight mn , mn = (m 1n , m 2n , . . . , m kn ) ,

(4.75)

Part A 4.3

where the sum extends over all U(r − 1) weights mr−1 = (m 1,r−1 , m 2,r−1 , . . . m r−1,r−1 ) satisfying the so-called “betweenness conditions” [4.38]

with entries satisfying betweenness conditions (4.64). Matrix representatives of weight generators are diagonal   i i−1  

[m ]|E ii |[m] = δ[m],[m ]  m ji − m j,i−1 ,

94

Part A

Mathematical Methods

To see the relation with so(m + 1), note that

satisfy the conditions m 1n ≥ m 2n ≥ · · · ≥ m kn ≥ 0 for n = 2k + 1 , (4.76)

and m 1n ≥ m 2n ≥ · · · ≥ |m kn | for n = 2k ,

(4.77)

where m in are simultaneously integers or half-odd integers. The former are referred to as tensor representations (since they arise as tensor products of fundamental irreps), while those with half-odd integer components are called spinor representations. Note that for n = 2k, we have two lowest (mirror-conjugated) spinor representations, namely m(+) = ( 12 , 12 , . . . , 12 ) and m(−) = ( 12 , . . . , 12 , − 12 ). Only tensor representations can be labeled by Young tableaux. Subducing O(n) to O(n − 1), the betweenness conditions (branching rules) have the form

Part A 4.3

m in ≥ m i,n−1 ≥ m i+1,n

(i = 1, . . . , k − 1) (4.78)

together with m k,2k+1 ≥ |m k,2k |

(4.79)

when n = 2k + 1. The m i,n−1 components are integral (half-odd integral) if the m in are integral (half-odd integral). The U(n) ⊃ O(n) [or SU(n) ⊃ SO(n)] subduction rules are more involved [4.44].

4.3.3 Clifford Algebras and Spinor Representations While all reps of U(n) or SL(n) arise as tensor powers of the standard rep, only half of the reps of SO(m) or O(m) arise this way, since SO(m) is not simply connected when m > 2. A double covering of SO(m) leads to spin groups Spin(m). The best way to proceed is, however, to construct the so-called Clifford algebras Cm , whose multiplicative group (consisting of invertible elements) contains a subgroup which provides a double cover of SO(m). The key fact is that C2k is isomorphic with gl(2k ) and C2k+1 with gl(2k ) ⊕ gl(2k ). The reps of Cm thus provide the required spinor reps. A Clifford algebra Cm is an associative algebra generated by Clifford numbers αi satisfying the anticommutation relations {αi , α j } = 2δij

(i, j = 1, . . . , m) .

(4.80)

Since αi2 = 1, dim Cm = 2m and a general element of νm Cm is a product of Clifford numbers α1ν1 α2ν2 · · · αm with νi = 0 or 1.

1 F0k = − iαk , 2 ( j = k)

F jk =

 1 1
α j , αk = α j αk , 4 2 (4.81)

satisfy the commutation relations (4.73). As an example, C2 can be realized by Pauli matrices by setting # # " " 0 1 0 i , α2 = σ2 = . α1 = σ1 = 1 0 −i 0 (4.82)

Clearly, the four matrices 12 , α1 , α2 and α1 α2 are linearly independent (note that σ3 = iσ1 σ2 ), so that C2 is isomorphic to gl(2, C ). Similarly, considering Dirac–Pauli matrices # " −i12 0 = iγ4 , γ0 = 0 −i12 " # 0 iσk (4.83) , (k = 1, 2, 3) γk = −iσk 0 we have that {γi , γ j } = 2δij ,

(i, j = 1, . . . , 4)

(4.84)

so that γi (i = 1, . . . , 4) or (i = 0, . . . , 3) represent Clifford numbers for C4 and 14 , γi , γi γ j (i < j), γi γ j γk (i < j < k) and γ 5 ≡ γ 1 γ 2 γ 3 γ 4 = iγ 0 γ 1 γ 2 γ 3 form an additive basis for gl(4, C ) (the γi themselves are said to form a multiplicative basis). For general construction of Cm Clifford numbers in terms of direct products of Pauli matrices see [4.45, 46].

4.3.4 Bosonic and Fermionic Realizations of U(n) †

Designating by bi (bi ) the boson creation (annihilation) operators rela
(Sect.  6.1.1)
† †satisfying 
the†commutation  tions bi , b j = bi , b j = 0, bi , b j = δij , we obtain a possible U(n) realization by defining its n 2 generators as follows †

G ij = bi b j .

(4.85)

The first order Casimir operator, (4.66) with k = 1, then represents the total number operator Nˆ ≡ C1U(n) =

n  i=1

G ii =

n  i=1

†

bi bi ,

(4.86)

Dynamical Groups

and the physically relevant states, being totally symmetric, carry single row irreps Γ (N 0˙ ) ≡ Γ (N0 · · · 0). Similarly for fermion creation (annihilation) opera† tors X I (X I ) that are associated with some orthonormal spin orbital set {|I}, I = 1, 2, . . . , 2n, and satisfy the † † = X = 0, anticommutation relations X , X , X I J I J  † X I , X J = δ I J , the operators †

eI J = X I X J

(4.87)

again represent the U(2n) generators satisfying (4.59) and (4.60). The first-order$ Casimir then represents the † total number operator Nˆ = I X I X I , while the possible physical states are characterized by totally antisymmetric single column irreps Γ (1 N 0˙ ) ≡ Γ (11 · · · 1 0 · · · 0).


σ, σ † = 1,

ijk

(4.88)

The energy levels (as a function of 0, 1, 2, . . . -body (2) , etc.) are then determined matrix elements h 0 , h ij(1) , h ijk by diagonalizing H in an appropriate space, which is conveniently provided by the carrier space of the totally symmetric irrep Γ (N 0 0 0) ≡ Γ (N 0˙ ) of U(4). The requirement that the resulting states be characterized by angular momentum J and parity P quantum numbers necessitates that the boson operators involved have definite transformation properties under rotations and reflections [4.8]. The boson operators  are thus subdivided into the operators σ † , σ ,  scalar † J = 0, and vector operators πµ , πµ ; µ = 0, ±1 , J = 1 with parity P = (−) J . All commutators vanish except


†  πµ , πµ = δµµ .

(4.89)

Since H preserves the total number of vibrons N = n σ + n π , the second order Hamiltonian (4.88) within the irrep Γ (N 0˙ ) can be expressed in terms of four independent parameters (apart from an overall constant) as

(0) H = e(0) + e(1) π † × π˜ 0 

(0)

(0) (0) † † π + e(2) × π × π × π ˜ ˜ 1 0



(2)

(2) (0) π† × π† × π˜ × π˜ 0 

(0)

(0) (2) † † π ×π + e3 × σ˜ × σ˜ + e(2) 2

(0)

(0) (0) † † + σ ×σ × π˜ × π˜ + · · · , (4.90) 0

= σ, π˜ µ = (−)1−µ π−µ

where σ˜ and square brackets indicate the SU(2) couplings. In special cases the eigenvalue problem for H can be solved analytically, assuming that H can be expressed in terms of Casimir operators of a complete chain of subgroups of U(4) [referred to as dynamical symmetries]. Requiring that the chain contain the physical rotation group O(3), one has two possibilities (I) (II)

U(4) ⊃ O(4) ⊃ O(3) ⊃ O(2) , U(4) ⊃ U(3) ⊃ O(3) ⊃ O(2) .

(4.91)

These imply labels (quantum numbers): N [total vibron number defining a totally symmetric irrep of U(4)], ω = N, N − 2, N − 4, . . . , 1 or 0 [defining a totally symmetric irrep of O(4)] and n π = N, N − 1, . . . , 0 [defining the U(3) irrep], in addition to the O(3) ⊃ O(2) labels J, M; |M| ≤ J. In terms of these labels one finds for the respective Hamiltonians H (I) = F + A C2O(4) + BC2O(3) , H (II) = F + εC1U(3) + αC2U(3) + βC2O(3) ,

(4.92)

where F, A, B, ε, α, β are free parameters and CiU(k) , CiO(k) are relevant Casimir operators, the following expressions [4.8, 50–52] for their eigenvalues E (I) (N, ω, J, M ) = F + Aω(ω + 2) + BJ(J + 1), E (II) (N, n π , J, M ) = F + n π + αn π (n π + 3) (4.93) + βJ(J + 1) .

Part A 4.3

ij

95

for

4.3.5 Vibron Model Similar to the unified description of nuclear collective rovibrational states using the interacting boson model [4.47–49], one can build an analogous model for molecular rotation-vibration spectra [4.8]. For diatomics, an appropriate dynamical group is U(4) [4.8, 50–52] and, generally, for rotation-vibration spectra in r-dimensions one requires U(r + 1). For triatomics, the U(4) generating algebra is generalized to U(4) ⊗ U(4), and for the (k + 1) atomic molecule to U(1) (4) ⊗ · · · ⊗ U(k) (4) [4.8, 50–52]. For the bosonic realization of U(4), we need four † creation (bi , i = 1, . . . , 4) and four annihilation (bi ) operators (Sect. 4.3.4). The Hamiltonian may be generally expressed as a multilinear in terms of boson  † form number preserving products bi b j , so that using (4.85) we can write  (1) 1  (2) H = h (0) + h ij G ij + h ijk G ij G k + · · · . 2

4.3 Compact Dynamical Groups

96

Part A

Mathematical Methods

The limit (I) is appropriate for rigid diatomics and limit (II) for nonrigid ones [4.8, 50–52]. In addition to handling di- and tri-atomic systems, the vibron model was also applied to the overtone spectrum of acetylene [4.53], intramolecular relaxation in benzene and its dimers [4.54, 55], octahedral molecules of the XF6 type (X = S, W, and U) [4.56], and to linear polyatomics [4.57]. Most recently, the experimental (dispersed fluorescence and stimulated emission pumping) vibrational spectra of H2 O and SO2 in their ground states, representing typical local-mode and normalmode molecules, respectively, have been analyzed, including highly excited levels, by relying on the U(2) algebraic effective Hamiltonian approach [4.58–60]. The U(2) algebraic scheme [4.61] also enabled the treatment of Franck–Condon transition intensities [4.62, 63] in rovibronic spectra. The attempts at a similar heuristic phenomenological description of electronic spectra have met so-far with only a limited success [4.64].

Part A 4.3

4.3.6 Many-Electron Correlation Problem In atomic and molecular electronic structure calculations one employs a spin-independent model Hamiltonian H=



h ij

†

X iσ X jσ

1  vij,k 2 i, j,k,

† XI

2 

†

†

X iσ X jτ X τ X kσ ,

(4.94)

σ,τ=1

† X iσ

Eστ =

2  σ=1 n  i=1

eiσ, jσ =

2 

†

X iσ X jσ ,

σ=1

eiσ,iτ =

n 

i, j,k,

(4.96)

We can thus achieve an automatic spin adaptation by exploiting the chain U(2n) ⊃ U(n) ⊗ U(2)

(4.97)

and diagonalize H within the carrier space of twocolumn U(n) irreps Γ (2a 1b 0c ) ≡ Γ (a, b, c) with [4.39, 66] a=

1 N−S, 2

b = 2S ,

1 c = n −a−b = n − N − S , (4.98) 2 considering the states of multiplicity (2S + 1) involving n orbitals and N electrons. The dimension of each spinadapted subproblem equals [4.39, 66] # #" " b+1 n +1 n +1 a b c , dim Γ (2 1 0 ) = n +1 c a m

where ≡ (X I ) designate the creation (annihilation) operators associated with the orthonormal spin orbitals |I ≡ |iσ = |i ⊗ |σ; i = 1, . . . , n; σ = 1, 2
σ = 1, 2 labeling the spin-up and spin-down eigenstates  ˆ j, vij,k = i(1) j(2)|ˆv|k(1)(2) of Sz , and h ij = i|h| are the one- and two-electron integrals in the orbital ba† sis {|i}. As stated in Sect. 4.3.4, e I J ≡ eiσ, jτ = X iσ X jτ may then be regarded as U(2n) generators, and  the appropriate U(2n) irrep for N-electron states is Γ 1 N 0˙ . Similar to the nuclear many-body problem [4.65], one defines mutually commuting partial traces of spin orbital generators e I J , (4.87), E ij =

i, j

(4.99)

σ=1

i, j

+

2 

considered as the generators of the orbital group U(n) and the spin group U(2). The Hamiltonian (4.94) is thus expressible in terms of orbital U(n) generators  1  h ij E ij + vij,k (E ik E j − δ jk E i ) . H= 2

†

X iσ X iτ ,

(4.95)

i=1

which again satisfy the unitary group commutation relations (4.59) and property (4.60), and may thus be

where n designate binomial coefficients. Exploiting simplified irrep labeling by triples of integers (a, b, c), (4.98), at each level of the canonical chain (4.67), one achieves more efficient state labeling by replacing Gel’fand tableaux (4.68) by n × 3 ABC [4.66] or Paldus or Gel’fand–Paldus tableaux [4.40, 67–75] [P] = [ai bi ci ] ,

(4.100)

where ai + bi + ci = i. Another convenient labeling uses the ternary step numbers di , 0 ≤ di ≤ 3 [4.66–68,76,77] di = 1 + 2(ai − ai−1 ) − (ci − ci−1 ) .

(4.101)

An efficient and transparent representation of this basis can be achieved in terms of Shavitt graphs and distinct row tables ([4.67, 68], cf. also [4.10–12, 39, 69]). An efficient evaluation of generator matrix representatives, as well as of their products, is formulated in terms of products of segment values, whose explicit form has been derived in several different ways [4.10–12, 66–69, 73–75, 77, 78]. Since the dimension (4.99) rapidly increases with n and N, various truncated schemes (limited CI) are often employed. The unitary group formalism

Dynamical Groups

that is based either on U(n) or on the universal enveloping algebra of U(n) proved to be of great usefulness in various post-Hartree–Fock approaches to molecular electronic structure [4.79], especially in large-scale CI calculations (in particular in the columbus Program System [4.80]; see also [24–31] in [4.12]) and in the spin-adapted UGA version of the coupled cluster (CC) method [4.81–83] (cf. Chapter 5; for applications, see [4.84–86]), as well as in various other investigations (e.g. quantum dots [4.87], charge migration in fragmentation of peptide ions [4.88, 89]; see also [4.10–12] for other references).

4.3.7 Clifford Algebra Unitary Group Approach

U(2n ) ⊃ Spin(m) ⊃ SO(m) ⊃ U(n) , (m = 2n + 1 or m = 2n)

(4.102)

supplemented, if desired, by the canonical chain (4.67) for U(n). To realize the connection with the fermionic  †Grassmann algebra generated by the creation X I and annihilation (X I ) operators, I = 1, . . . , 2n, note that it is isomorphic with the Clifford algebra C4n when we define [4.12, 25]  † † α I = X I + X I , α I +2n = i X I − X I , (I = 1, . . . , 2n) . (4.103) For practical applications, the most important is the final imbedding U(2n ) ⊃ U(n), (for the role of intermediate groups, see [4.90–92]). All states of an n-orbital model, regardless the electron number N and the total spin S, are contained in a single two-box totally symmetric irrep 20˙  of U(2n ) [4.93, 94]. To simplify the notation, one employs the one-to-one correspondence between the Clifford algebra monomials, labeled by the occupation numbers m i = 0 or m i = 1 (i = 1, . . . , n), and “multiparticle” single-column U(n) states labeled by p ≡ p{m i } = 2n − (m 1 m 2 · · · m n )2 ,

(4.104)

where the occupation number array (m 1 · · · m n ) is interpreted as a binary integer, which we then regard as

97

one-box states | p) of U(2n ). The orbital U(n) generators Λij may then be expressed as simple linear combinations of U(2n ) generators E pq = | p)(q| with coefficients equal to ±1 [4.93, 94]. Generally, any p-column U(n) irrep is contained at least once in the totally symmetric p-box irrep of U(2n ). For many-electron problems, one thus requires a two-box irrep 20˙ . Any state arising in the U(n) irrep Γ(a, b, c) can then be represented as a linear combination of two-box states, labeled by the Weyl tableaux [i| j] ≡ i j . In particular, the
highest  weight state of Γ (a, b, c) is represented by 2c |2b+c . Once this representation is available, it is straightforward to compute explicit representations of U(n) generators, since E pq act trivially on [i| j] [4.94]. Defining unnormalized states (i| j) as  (i| j) = 1 + δij [i| j] , (4.105) we have E pq (i| j ) = δqi ( p| j ) + δq j (i| p) .

(4.106)

The main features of CAUGA may thus be summarized as follows: CAUGA 1. effectively reduces an N-electron problem to a number of two-boson problems; 2. enables an exploitation of an arbitrary coupling scheme (being particularly suited for the valence bond method); 3. can be applied to particle-number nonconserving operators; 4. easily extends to fermions with an arbitrary spin; 5. drastically simplifies evaluation of explicit representations of U(n) generators and of their products; 6. can be exploited in other than shell-model approaches [4.95–101].

4.3.8 Spin-Dependent Operators The spin-adapted U(n)-based UGA is entirely satisfactory in most investigations of molecular electronic structure. However, when exploring the fine structure in high-resolution spectra, the intersystem crossings, phosphorescent lifetimes, molecular predissociation, spin–orbit interactions in transition metals, and like phenomena, the explicitly spin-dependent terms must be included in the Hamiltonian. Since in most cases the total spin S represents a good approximate quantum number, so that the spin-adapted N-electron states render an excellent point of departure, it is necessary to consider the

Part A 4.3

The Clifford algebra unitary group approach (CAUGA) exploits a realization of the spinor algebra of the rotation group SO(2n + 1) in the covering algebra of U(2n ) to obtain explicit representation matrices for the U(n) [or SO(2n + 1) or SO(2n)] generators in the basis adapted to the chain [4.90–94]

4.3 Compact Dynamical Groups

98

Part A

Mathematical Methods

corresponding matrix elements of general spin-orbital U(2n) generators in terms of which the relevant spindependent terms may be expressed. This was first done in the context of the symmetric group and Racah algebra by Drake and Schlesinger [4.78] and later on in terms of the Gel’fand–Paldus tableaux [4.102–106]. In general, the U(2n) generators eiσ, jτ ≡ e I J may be resolved into the spin-shift components e(±) I J that increase (+) or decrease (−) the total spin S by one unit and the zero-spin component e(0) I J that preserves S. The relevant matrix elements can then be expressed in terms of the matrix elements of a single U(n) adjoint ten-

sor operator ∆, which is given by the following second degree polynomial in U(n) generators, ∆ = E(E + N/2 − n − 2),

E = E ij 

(4.107)

and by the well-known matrix elements of U(2) or SU(2) generators in terms of the pure spin states [4.102, 103] (see also [4.107, 108]). The operator (4.107), referred to as the Gould–Paldus operator [4.109], also plays a key role in the determination of reduced density matrices [4.110, 111], and has been recently exploited in the multireference spin-adapted variant of the density functional theory [4.109].

References 4.1

Part A 4

4.2

4.3 4.4 4.5 4.6

4.7 4.8 4.9 4.10

4.11

4.12 4.13 4.14 4.15 4.16 4.17

A. O. Barut: Dynamical Groups and Generalized Symmetries in Quantum Theory, Vols. 1 and 2 (Univ. Canterbury, Christchurch 1971) A. Bohm, Y. Ne’eman, A. O. Barut: Dynamical Groups and Spectrum Generating Algebras (World Scientific, Singapore 1988) R. Gilmore: Lie Groups, Lie Algebras, and Some of Their Applications (Wiley, New York 1974) B. G. Wybourne: Classical Groups for Physicists (Wiley, New York 1974) J.-Q. Chen: Group Representation Theory for Physicists (World Scientific, Singapore 1989) J.-Q. Chen, J. Ping, F. Wang: Group Representation Theory for Physicists, 2nd edn. (World Scientific, Singapore 2002) O. Castaños, A. Frank, R. Lopez-Peña: J. Phys. A: Math. Gen. 23, 5141 (1990) F. Iachello, R. D. Levine: Algebraic Theory of Molecules (Oxford Univ. Press, Oxford 1995) A. Frank, P. Van Isacker: Algebraic Methods in Molecular and Nuclear Structure (Wiley, New York 1994) J. Paldus: Mathematical Frontiers in Computational Chemical Physics, ed. by D. G. Truhlar (Springer, Berlin, Heidelberg 1988) pp. 262–299 I. Shavitt: Mathematical Frontiers in Computational Chemical Physics, ed. by D. G. Truhlar (Springer, Berlin, Heidelberg 1988) pp. 300–349 J. Paldus: Contemporary Mathematics, Vol. 160 (AMS, Providence 1994) pp. 209–236 J. C. Parikh: Group Symmetries in Nuclear Structure (Plenum, New York 1978) B. G. Adams, J. ˇCíˇzek, J. Paldus: Int. J. Quantum Chem. 21, 153 (1982) B. G. Adams, J. ˇCíˇzek, J. Paldus: Adv. Quantum Chem. 19, 1 (1988) J. ˇCíˇzek, J. Paldus: Int. J. Quantum Chem. 12, 875 (1977) B. G. Adams: Algebraic Approach to Simple Quantum Systems (Springer, Berlin, Heidelberg 1994)

4.18 4.19 4.20 4.21 4.22 4.23 4.24 4.25

4.26 4.27 4.28

4.29 4.30

4.31 4.32 4.33 4.34 4.35 4.36 4.37

4.38 4.39

A. Bohm: Nuovo Cimento A 43, 665 (1966) A. Bohm: Quantum Mechanics (Springer, New York 1979) M. A. Naimark: Linear Representations of the Lorentz Group (Pergamon, New York 1964) W. Pauli: Z. Phys. 36, 336 (1926) V. Fock: Z. Phys. 98, 145 (1935) A. O. Barut: Phys. Rev. B 135, 839 (1964) A. O. Barut, G. L. Bornzin: J. Math. Phys. 12, 841 (1971) G. A. Zaicev: Algebraic Problems of Mathematical and Theoretical Physics (Science Publ. House, Moscow 1974) (in Russian) A. Joseph: Rev. Mod. Phys. 39, 829 (1967) M. Bednar: Ann. Phys. (N. Y.) 75, 305 (1973) J. ˇCíˇzek, E. R. Vrscay: Group Theoretical Methods in Physics, ed. by R. Sharp, B. Coleman (Academic, New York 1977) pp. 155–160 J. ˇCíˇzek, E. R. Vrscay: Int. J. Quantum Chem. 21, 27 (1982) J. E. Avron, B. G. Adams, J. ˇCíˇzek, M. Clay, L. Glasser, P. Otto, J. Paldus, E. R. Vrscay: Phys. Rev. Lett. 43, 691 (1979) J. ˇCíˇzek, M. Clay, J. Paldus: Phys. Rev. A 22, 793 (1980) H. J. Silverstone, B. G. Adams, J. ˇCíˇzek, P. Otto: Phys. Rev. Lett. 43, 1498 (1979) E. R. Vrscay: Phys. Rev. A 31, 2054 (1985) E. R. Vrscay: Phys. Rev. A 33, 1433 (1986) H. J. Silverstone, R. K. Moats: Phys. Rev. A 23, 1645 (1981) H. Weyl: Classical Groups (Princeton Univ. Press, Princeton 1939) A. O. Barut, R. Raczka: Theory of Group Representations and Applications (Polish Science Publ., Warszawa 1977) J. D. Louck: Am. J. Phys. 38, 3 (1970) J. Paldus: Theoretical Chemistry: Advances and Perspectives, Vol. 2, ed. by H. Eyring, D. J. Henderson (Academic, New York 1976) pp. 131–290

Dynamical Groups

4.40 4.41 4.42 4.43 4.44 4.45 4.46 4.47 4.48 4.49

4.50 4.51

4.53 4.54 4.55 4.56 4.57 4.58 4.59 4.60 4.61 4.62 4.63 4.64 4.65 4.66 4.67 4.68 4.69

4.70 4.71 4.72

4.73 4.74 4.75 4.76 4.77 4.78 4.79 4.80

4.81 4.82 4.83 4.84 4.85 4.86 4.87 4.88 4.89 4.90 4.91 4.92 4.93 4.94 4.95

4.96

4.97 4.98 4.99 4.100 4.101 4.102 4.103 4.104 4.105 4.106 4.107

M. D. Gould, G. S. Chandler: Int. J. Quantum Chem. 25, 553, 603, 1089 (1984) M. D. Gould, G. S. Chandler: Int. J. Quantum Chem. 26, 441 (1984) M. D. Gould, G. S. Chandler: Int. J. Quantum Chem. 27, 878 (1985), Erratum J. Paldus: Phys. Rev. A 14, 1620 (1976) J. Paldus, M. J. Boyle: Phys. Scr. 21, 295 (1980) G. W. F. Drake, M. Schlesinger: Phys. Rev. A 15, 1990 (1977) M. A. Robb: Theor. Chem. Acc. 103, 317 (2000) R. Shepard, I. Shavitt, R. M. Pitzer, D. C. Comeau, M. Pepper, H. Lischka, P. G. Szalay, R. Ahlrichs, F. B. Brown, J.-G. Zhao: Int. J. Quantum Chem. Symp. 22, 149 (1988) J. Paldus, B. Jeziorski: Theor. Chim. Acta 86, 83 (1993) X. Li, J. Paldus: J. Chem. Phys. 101, 8812 (1994) B. Jeziorski, J. Paldus, P. Jankowski: Int. J. Quantum Chem. 56, 129 (1995) X. Li, J. Paldus: J. Chem. Phys. 104, 9555 (1996) X. Li, J. Paldus: J. Mol. Struct. (theochem) 527, 165 (2000) P. Jankowski, B. Jeziorski: J. Chem. Phys. 111, 1857 (1999) F. Remacle, R. D. Levin: Chem. Phys. Chem. 2, 20 (2001) F. Remacle, R. D. Levin: J. Chem. Phys. 110, 5089 (1999) F. Remacle, R. D. Levin: J. Phys. Chem. A 104, 2341 (2000) R. S. Nikam, C. R. Sarma: J. Math. Phys. 25, 1199 (1984) C. R. Sarma, J. Paldus: J. Math. Phys. 26, 1140 (1985) M. D. Gould, J. Paldus: J. Math. Phys. 28, 2304 (1987) J. Paldus, C. R. Sarma: J. Chem. Phys. 83, 5135 (1985) J. Paldus, M.-J. Gao, J.-Q. Chen: Phys. Rev. A 35, 3197 (1987) J. Paldus: Relativistic and Electron Correlation Effects in Molecules and Solids, NATO ASI Series B, ed. by L. Malli G. (Plenum, New York 1994) pp. 207–282 J. Paldus, X. Li: Symmetries in Science VI: From the Rotation Group to Quantum Algebras, ed. by B. Gruber (Plenum, New York 1993) pp. 573–592 J. Paldus, J. Planelles: Theor. Chim. Acta 89, 13 (1994) J. Planelles, J. Paldus, X. Li: Theor. Chim. Acta 89, 33, 59 (1994) X. Li, J. Paldus: Int. J. Quantum Chem. 41, 117 (1992) X. Li, J. Paldus: J. Chem. Phys. 102, 8059 (1995) P. Piecuch, X. Li, J. Paldus: Chem. Phys. Lett. 230, 377 (1994) M. D. Gould, J. Paldus: J. Chem. Phys. 92, 7394 (1990) M. D. Gould, J. S. Battle: J. Chem. Phys. 99, 5961 (1993) R. D. Kent, M. Schlesinger: Phys. Rev. A 42, 1155 (1990) R. D. Kent, M. Schlesinger: Phys. Rev. A 50, 186 (1994) R. D. Kent, M. Schlesinger, I. Shavitt: Int. J. Quantum Chem. 41, 89 (1992) M. D. Gould, J. S. Battle: J. Chem. Phys. 98, 8843 (1993)

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4.52

F. A. Matsen, R. Pauncz: The Unitary Group in Quantum Chemistry (Elsevier, Amsterdam 1986) H. Weyl: The Theory of Groups and Quantum Mechanics (Dover, New York 1964) I. M. Gel’fand, M. L. Tsetlin: Dokl. Akad. Nauk SSSR 71, 825, 1070 (1950) G. E. Baird, L. C. Biedenharn: J. Math. Phys. 4, 1449 (1963) J. Deneen, C. Quesne: J. Phys. A 16, 2995 (1983) H. Boerner: Representations of Groups, 2nd edn. (North-Holland, Amsterdam 1970) A. Ramakrishnan: L-Matrix Theory or the Grammar of Dirac Matrices (Tata McGraw-Hill, India 1972) A. Arima, F. Iachello: Phys. Rev. Lett. 35, 1069 (1975) F. Iachello, A. Arima: The Interacting Boson Model (Cambridge Univ. Press, Cambridge 1987) F. Iachello, P. Van Isacker: The Interacting Boson– Fermion Model (Cambridge Univ. Press, Cambridge 2004) F. Iachello: Chem. Phys. Lett. 78, 581 (1981) F. Iachello, R. D. Levine: J. Chem. Phys. 77, 3046 (1982) O. S. van Roosmalen, F. Iachello, R. D. Levine, A. E. L. Dieperink: J. Chem. Phys. 79, 2515 (1983) J. Hornos, F. Iachello: J. Chem. Phys. 90, 5284 (1989) F. Iachello, S. Oss: J. Chem. Phys. 99, 7337 (1993) F. Iachello, S. Oss: J. Chem. Phys. 102, 1141 (1995) J.-Q. Chen, F. Iachello, J.-L. Ping: J. Chem. Phys. 104, 815 (1996) T. Sako, D. Aoki, K. Yamanouchi, F. Iachello: J. Chem. Phys. 113, 6063 (2000) T. Sako, K. Yamanouchi, F. Iachello: J. Chem. Phys. 113, 7292 (2000) T. Sako, K. Yamanouchi, F. Iachello: J. Chem. Phys. 114, 9441 (2001) T. Sako, K. Yamanouchi, F. Iachello: J. Chem. Phys. 117, 1641 (2002) F. Iachello, S. Oss: J. Chem. Phys. 104, 6956 (1996) F. Iachello, A. Leviatan, A. Mengoni: J. Chem. Phys. 95, 1449 (1991) T. Müller, P. H. Vaccaro, F. Pèrez-Bernal, F. Iachello: J. Chem. Phys. 111, 5038 (1999) A. Frank, R. Lemus, F. Iachello: J. Chem. Phys. 91, 29 (1989) M. Moshinsky: Group Theory and the Many-Body Problem (Gordon Breach, New York 1968) J. Paldus: J. Chem. Phys. 61, 5321 (1974) I. Shavitt: Int. J. Quantum Chem. Symp. 11, 131 (1977) I. Shavitt: Int. J. Quantum Chem. Symp. 12, 5 (1978) J. Hinze (Ed.): The Unitary Group for the Evaluation of Electronic Energy Matrix Elements, Lect. Notes Chem., Vol. 22 (Springer, Berlin, Heidelberg 1981) R. Pauncz: Spin Eigenfunctions: Construction and Use (Plenum, New York 1979) Chap. 9 S. Wilson: Electron Correlation in Molecules (Clarendon, Oxford 1984) Chap. 5 R. McWeeny: Methods of Molecular Quantum Mechanics, 2 edn. (Academic, New York 1989) Chap. 10

References

100

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Mathematical Methods

4.108 M. D. Gould, J. S. Battle: J. Chem. Phys. 99, 5983 (1993) 4.109 Y. G. Khait, M. R. Hoffmann: J. Chem. Phys. 120, 5005 (2004)

4.110 M. D. Gould, J. Paldus, G. S. Chandler: J. Chem. Phys. 93, 4142 (1990) 4.111 J. Paldus, M. D. Gould: Theor. Chim. Acta 86, 83 (1993)

Part A 4

101

Perturbation T 5. Perturbation Theory

Perturbation theory (PT) represents one of the bridges that takes us from a simpler, exactly solvable (unperturbed) problem to a corresponding real (perturbed) problem by expressing its solutions as a series expansion in a suitably chosen “small” parameter ε in such a way that the problem reduces to the unperturbed problem when ε = 0. It originated in classical mechanics and eventually developed into an important branch of applied mathematics enabling physicists and engineers to obtain approximate solutions of various systems of differential equations [5.1–4]. For the problems of atomic and molecular structure and dynamics, the perturbed problem is usually given by the timeindependent or time-dependent Schrödinger equation [5.5–8].

5.2.2 5.2.3 5.3

5.4 5.1

101 101 102 102 103

Time-Independent Perturbation Theory 103 5.2.1 General Formulation ................. 103

Fermionic Many-Body Perturbation Theory (MBPT)...................................... 5.3.1 Time Independent Wick’s Theorem .................................. 5.3.2 Normal Product Form of PT ......... 5.3.3 Møller–Plesset and Epstein–Nesbet PT .............. 5.3.4 Diagrammatic MBPT ................... 5.3.5 Vacuum and Wave Function Diagrams ................................. 5.3.6 Hartree–Fock Diagrams .............. 5.3.7 Linked and Connected Cluster Theorems ................................. 5.3.8 Coupled Cluster Theory ............... Time-Dependent Perturbation Theory ... 5.4.1 Evolution Operator PT Expansion ............................ 5.4.2 Gell–Mann and Low Formula ...... 5.4.3 Potential Scattering and Quantum Dynamics ............. 5.4.4 Born Series ............................... 5.4.5 Variation of Constants Method ....

105 105 105 106 107 107 108 108 109 111 111 111 111 112 112

References .................................................. 113

5.1 Matrix Perturbation Theory (PT) A prototype of a time-independent PT considers an eigenvalue problem for the Hamiltonian H of the form ∞
εi Vi , (5.1) H = H0 + V , V = i=1

acting in a (finite-dimensional) Hilbert space Vn , assuming that the spectral resolution of the unperturbed operator H0 is known; i. e.,

H0 = ωi Pi , Pi P j = δij P j , Pi = I , i

i

(5.2)

where ωi are distinct eigenvalues of H0 , the Pi form a complete orthonormal set of Hermitian idempotents

and I is the identity operator on Vn . The PT problem for H can then be formulated within the Lie algebra A (see Sect. 3.2) generated by H0 and V [5.9, 10].

5.1.1 Basic Concepts Define the diagonal part X of a general operator X ∈ A by
X = Pi X Pi , (5.3) i

and recall that the adjoint action of adX : A → A, is defined by ad X(Y ) = [X, Y ] ,

(∀ Y ∈ A) ,

X ∈ A, (5.4)

Part A 5

5.2

Matrix Perturbation Theory (PT) ............ 5.1.1 Basic Concepts .......................... 5.1.2 Level-Shift Operators ................. 5.1.3 General Formalism .................... 5.1.4 Nondegenerate Case ..................

Brillouin–Wigner and Rayleigh–Schrödinger PT (RSPT) ... 104 Bracketing Theorem and RSPT ..... 104

102

Part A

Mathematical Methods

where the square bracket denotes the commutator. The key problem of PT is the ‘inversion’ of this operation, i. e., the solution of the equation [5.9–11] adH0 (X) ≡ [H0 , X] = Y .

(5.5)

Assuming that Y  = 0, then X = R(Y ) + A ,

(5.6)

where A ∈ A is arbitrary and
R(Y ) = ∆ij−1 Pi YP j ,

(5.7)

i= j

with ∆ij = ωi − ω j , represents the solution of (5.5) with the vanishing diagonal part R(Y ) = 0.

and Bk designates the Bernoulli numbers [5.13] 1 1 B0 = 1 , B1 = − , B2 = , 2 6 B2k+1 = 0 (k ≥ 1) , 1 1 B4 = , B6 = , etc. (5.15) 30 42

5.1.3 General Formalism Introducing the PT expansion for relevant operators, ∞
X= εi X i , X = E, F, G ; Fi = [H0 , G i ] , i=1

(5.16)

(5.13) leads to the following system of equations E 1 + F1 = V1 ,

5.1.2 Level-Shift Operators

1 E 2 + F2 = V2 + [G 1 , V1 + E 1 ] , 2 1 E 3 + F3 = V3 + [G 1 , V2 + E 2 ] 2 1 + [G 2 , V1 + E 1 ] 2 1 + [G 1 , [G 1 , V1 − E 1 ]] , 12 etc. ,

To solve the PT problem for H, (5.1), we search for a unitary level-shift transformation U [5.9, 10], U † U = UU † = I, UHU † = U(H0 + V )U † = H0 + E ,

(5.8)

where the level-shift operator E satisfies the condition

Part A 5.1

E = E .

(5.9)

To guarantee the unitarity of U, we express it in the form U = eG ,

G † = −G ,

G = 0 .

(5.10)

which can be solved recursively for E i and G i by taking their diagonal part and applying operator R, (5.7), since Ei  = E i , RFi = G i ,

Using the Haussdorff formula A

−A

e Be

∞
= (k!)−1 ( adA)k B ,

(5.11)

G i  = Fi  = 0 , RE i = 0 .

(5.18)

We thus get

k=0

E 1 = V1  ,

and defining the operator F = [H0 , G] ,

1 E 2 = V2  + [RV1 , V1 ] , 2 E 3 = V3  + [RV1 , V2 ] 1 + [RV1 , [RV1 , 2V1 + E 1 ]] , etc. , (5.19) 6

(5.12)

we find from (5.8) that 1 E + F = V + [G, V + E] 2 ∞
+ (k!)−1 Bk ( adG)k (V − E) ,

and G 1 = RV1 , (5.13)

k=2

where we used the identity [5.12] ∞  ∞ 
Bk
1 k k X X =I, k! (k + 1)! k=0

(5.17)

k=0

(5.14)

1 G 2 = RV2 + R[RV1 , V1 + E 1 ] , 2 Since R(X) = RX = 0 , R(XY ) = R(X)Y ,

etc.

(5.20)

R(X)Y  = −X R(Y ) , R(XY) = XR(Y ) , (5.21)

Perturbation Theory

these relationships can be transformed to a more conventional form E 2 = V2  − V1 RV1  , E 3 = V3  − V1 RV2  − V2 RV1  1 + R(V1 )R(V1 )[2V1 + V1 ] 6 1 − R(V1 )[2V1 + V1 ]R(V1 ) 3 1 + [2V1 + V1 ]R(V1 )R(V1 ) , 6

5.2 Time-Independent Perturbation Theory

we get (e1 )ii = (v1 )ii , (e2 )ii = (v2 )ii −


(v1 )ij (v1 ) ji , ∆ ji j


(v1 )ij (v2 ) ji + (v2 )ij (v1 ) ji (e3 )ii = (v3 )ii − ∆ ji j

+

etc.


(v1 )ij (v1 ) jk (v1 )ki j,k

(5.22)

− (v1 )ii

However, in this way certain nonphysical terms arise that exactly cancel when the commutator form is employed (Sect. 5.3.7).

5.1.4 Nondegenerate Case In the nondegenerate case, when Pi = |ii|, with |i representing the eigenvector of H0 associated with the eigenvalue ωi , the level-shift operator is diagonal and its explicit PT expansion (as well as that for the corresponding eigenvectors) is easily obtained from (5.19) and (5.20). Writing xij for the matrix element i|X| j,

∆ ji ∆ki
(v1 )ij (v1 ) ji j

∆2ji

,

etc.,

(5.23)

the prime on the summation symbols indicating that the terms with the vanishing denominator are to be deleted. Note that in contrast to PT expansions which directly expand the level-shift transformation U, U = 1 + εU1 + ε2 U2 + · · · , the above Lie algebraic formulation has the advantage that U stays unitary in every order of PT. This is particularly useful in spectroscopic applications, such as line broadening.

For stationary problems, particularly those arising in atomic and molecular electronic structure studies relying on ab initio model Hamiltonians, the PT of Sect. 5.1 can be given a more explicit form which avoids a priori the nonphysical, size inextensive terms [5.6–8, 14, 15].

5.2.1 General Formulation

(5.26)

the asymmetric energy formula gives ki = κi + Φi |W|Ψi  . Pi = |Φi Φi |, Q i = Pi⊥ = 1 − Pi =

(5.27)




(5.25)

For simplicity, we restrict ourselves to the nondegenerate case (κi = κ j if i = j) and consider only the first order perturbation [see (5.1), εV1 ≡ W, Vi = 0 for i ≥ 2]. Of course, K and K 0 are Hermitian operators acting in a Hilbert space which, in ab initio applications, is finite-dimensional.

|Φ j Φ j | ,

j(=i)

(5.28)

(5.24)

assuming we know those of the unperturbed problem Φi |Φ j  = δij .

Ψi |Φi  = 1 ,

The idempotent Hermitian projectors

We wish to find the eigenvalues and eigenvectors of the full (perturbed) problem

K 0 |Φi  = κi |Φi ,

Using the intermediate normalization for |Ψi ,

commute with K 0 , so that (λ − K 0 )Q i |Ψi  = Q i (λ − ki + W )|Ψi  ,

(5.29)

λ being an arbitrary scalar (note that we write λI simply as λ). Since the resolvent (λ − K 0 )−1 of K 0 is nonsingular on the orthogonal complement of the ith eigenspace, we get Q i |Ψi  = |Ψi  − |Φi  = Ri (λ)(λ − ki + W )|Ψi  , (5.30)

Part A 5.2

5.2 Time-Independent Perturbation Theory

K |Ψi  ≡ (K 0 + W )|Ψi  = ki |Ψi  ,

103

104

Part A

Mathematical Methods

where

where now

Ri ≡ Ri (λ) = (λ − K 0 )−1 Q i


|Φ j Φ j | = Q i (λ − K 0 )−1 = , λ−κj j(=i)

(5.31)

assuming (λ = κ j ). Iterating this relationship, we get prototypes of the desired PT expansion for |Ψi , |Ψi  =

∞


[Ri (λ − ki + W )]n |Φi  ,

(5.32)

n=0

and, from (5.27), for ki , ki = κi +

∞
Φi |W[Ri (λ − ki + W )]n |Φi  . (5.33) n=0

Ri ≡ Ri(RS) =


|Φ j Φ j | . κi − κ j

(5.39)

j(=i)

The main distinction between these two PTs lies in the fact that the BW form has the exact eigenvalues appearing in the denominators, and thus leads to polynomial expressions for ki . Although these are not difficult to solve numerically, since the eigenvalues are separated, the resulting energies are never size extensive and thus unusable for extended systems. They are also unsuitable for finite systems when the particle number changes, as in various dissociation processes. From now on, we thus investigate only the RSPT, which yields a fully size-extensive theory.

5.2.3 Bracketing Theorem and RSPT

5.2.2 Brillouin–Wigner and Rayleigh–Schrödinger PT (RSPT) So far, the parameter λ was arbitrary, as long as λ = κ j ( j = i). The following two choices lead to the two basic types of many-body perturbation theory (MBPT):

Expressions (5.37) and (5.38) are not explicit, since they involve the exact eigenvalues ki on the right-hand side. To achieve an order by order separation, set ki ≡ k =

∞


k( j) ,

|Ψi  ≡ |Ψ  =

j=0

∞


|Ψ ( j)  ,

j=0

Part A 5.2

(5.40)

Brillouin–Wigner (BW) PT Setting λ = ki gives

ki = κi +

∞  n
Φi |W Ri(BW) W |Φi  , n=0

|Ψi  =

∞ 


Ri(BW) W

n

|Φi  ,

(5.34)

(5.35)

X ≡ Φi |X|Φi  .

n=0

where
|Φ j Φ j | Ri(BW) = . ki − κ j

(5.36)

∞  n
Φi |W Ri(RS) (κi − ki + W ) |Φi  , n=0

(5.37) ∞ 
n=0

n Ri(RS) (κi − ki + W ) |Φi  ,

k(0) = κi , k(1) = W ,

Rayleigh–Schrödinger (RS) PT Setting λ = κi gives

|Ψi  =

(5.41)

Substituting the first expansion (5.40) into (5.37) and collecting the terms of the same order in W, we get

j(=i)

ki = κi +

where the superscript ( j) indicates the jth-order in the perturbation W. We only consider the eigenvalue expressions, since the corresponding eigenvectors are easily recovered from them by removing the bra state and the first interaction W [see (5.37) and (5.38)]. We also simplify the mean value notation writing for a general operator X,

(5.38)

k(2) = WRW ,   k(3) = W(RW )2 − W WR2 W ,  k(4) = W(RW )3  

 − W WR(RW )2 + (WR)2 RW   + W2 WR3 W − WRW WR2 W ,

etc. (5.42)

The general expression has the form  k(n) = W(RW )n−1 + R(n) ,

(5.43)

Perturbation Theory

the first term on the right-hand side being referred to as the principal nth-order term, while R(n) designates the so-called renormalization terms that are obtained by the bracketing theorem [5.14, 16] as follows: 1. Insert the bracketings · · ·  around the W, WRW, . . . , WR · · · RW operator strings of the principal term in all possible ways.

5.3 Fermionic Many-Body Perturbation Theory (MBPT)

105

2. Bracketings involving the rightmost and/or the leftmost interaction vanish. 3. The sign of each bracketed term is given by (−1)n B , where n B is the number of bracketings. 4. Bracketings within bracketings are allowed, e.g., 2 WRWRWRWRW = W WR2 W . 5. The total number of bracketings (including the principal term) is (2n − 2)!/[n!(n − 1)!].

5.3 Fermionic Many-Body Perturbation Theory (MBPT) 5.3.1 Time Independent Wick’s Theorem

5.3.2 Normal Product Form of PT

The development of an explicit MBPT formalism is greatly facilitated by the exploitation of the timeindependent version of Wick’s theorem. This version of the theorem expresses an arbitrary product of creation † (aµ ) and annihilation (aµ ) operators (see Chapt. 6)as a normal product (relative to |Φ0 ) and as normal products with all possible contractions of these operators [5.14, 15],

Consider the eigenvalue problem for a general ab initio or semi-empirical electronic Hamiltonian H with oneand two-body components Z and V , namely,

x1 x2 · · · xk = N[x1 x2 · · · xk ] + ΣN[x1 x2 · · · · · · xk ] , (5.44)

where † † aµ aν = aµ aν = 0 , † aµ aν = h(µ)δµν ,

aµ aν† = p(µ)δµν ,

(5.45)

i

and a corresponding unperturbed problem H0 |Φi  = εi |Φi  , H0 = Z + U , Φi |Φ j  = δij ,

h(µ) = 1 , p(µ) = 0 if |µ is occupied in |Φ0  (hole states), h(µ) = 0 , p(µ) = 1 if |µ is unoccupied in |Φ0  (5.46) (particle states) . The N-product with contractions is defined as a product of individual contractions times the N-product of uncontracted operators (defining N[∅] ≡ 1 for an empty set) with the sign given by the parity of the permutation reordering the operators into their final order. Note that the Fermi vacuum mean value of an N-product vanishes unless all operators are contracted. Thus, x1 x2 · · · xk  is given by the sum over all possible fully contracted terms (vacuum terms). Similar rules follow for the expressions of the type (x1 x2 · · · xk )|Φ. Moreover, if some operators on the left-hand side of (5.44) are already in the N-product form, all the terms involving contractions between these operators vanish.

(5.48)

with U representing some approximation to V . In the case that U is also a one-electron operator, U = Σi u(i), the unperturbed problem (5.48) is separable and reduces to a one-electron problem, (z + u)|µ = ωµ |µ ,

and

(5.47)

i< j

(5.49)

which is assumed to be solved. Choosing the orthonormal spin orbitals {|µ} as a basis of the second quantization representation [Chapt. 6, (6.8)], the Nelectron solutions of (5.48) can be represented as † † a† · · · aµ |0 , |Φi  = aµ N 1 µ2

εi =

N


ωµ j ,

(5.50) (5.51)

j=1

the state label i representing the occupied spin orbital set {µ1 , µ2 , . . . , µ N }, while the one- and two-body operators take the form
† X= µ|x|νaµ aν , X = Z, U; x = z, u , µ,ν

1
† † V= µν|v|στaµ aν aτ aσ . 2 µ,ν,σ,τ

(5.52) (5.53)

Part A 5.3

† (xi = aµ or xi = aµi ) i

H|Ψi  = E i |Ψi  ,

z(i) + v(i, j) , H = Z+V =

106

Part A

Mathematical Methods

Considering, for simplicity, a nondegenerate ground † † † state |Φ ≡ |Φ0  = a1 a2 · · · a N |0, referred to as a Fermi vacuum, we define the normal product form of these operators relative to |Φ
† X N ≡ X − X = µ|x|νN aµ aν , (5.54a) µ,ν

(X = Z, U, G;

x = z, u, g)

VN ≡ V − V  − G N

† † 1
= µν|v|στN aµ aν aτ aσ 2 µ,ν,σ,τ † †

1
µν|v|στ A N aµ aν aτ aσ , = 4 µ,ν,σ,τ N


µσ|v|νσ A ,

(5.54b)

(5.55) (5.56)

Part A 5.3

X = Φ|X|Φ, and N[· · · ] designates the normal product relative to |Φ [5.14, 15]. (Recall † † that N[x1 x2 · · · xk ] = ±bµ1 · · · bµi bµi+1 · · · bµk , where † xi = bµi or bµi are the annihilation and creation operators of the particle-hole formalism relative to |Φ, † i. e., bµ = aµ for µ ≤ N and bµ = aµ for µ > N, the sign being determined by the parity of the permutation p : j → µ j .) Defining K = H − H,

K 0 = H0 − H0  = H0 − ε0 , (5.57)

we can return to (5.24) and (5.25), where now ki = E i − H, ε0 =

N


(5.61)

5.3.3 Møller–Plesset and Epstein–Nesbet PT

(5.62)

so that (5.49) represent Hartree–Fock (HF) equations, and ωµ and |µ the canonical HF orbital energies spin orbitals, respectively. Since 

 N and H = µ=1 µ|z|µ + 12 µ|g|µ is the HF energy, k = k0 gives directly the ground state correlation energy. (Note, however, that the N-product form of PT eliminates the first-order contribution k(1) = W in any basis, even when F is not diagonal.) With this choice, W1 = 0, W = VN , and the denominators in (5.39) are given by the differences of HF orbital energies κ0 − κ j =

λ
(ωµi − ωνi ) ≡ ∆{µi }; {ν j } ,

(5.63)

i=1

assuming that |Φ j  is a λ-times excited configuration relative to |Φ obtained through excitations µi → νi , i = 1, . . . , λ. Using the Slater rules (or the second quantization algebra), we can express the second-order contribution in terms of the two-electron integrals and HF orbital energies as

 1
ab|v|rs rs|v|ab − rs|v|ba (2) , k = 2 ωa + ωb − ωr − ωs (5.64)

ωµ ,

(5.58)

and W = K − K 0 = V − U − V − U .

(5.59)

With this choice, W = 0, so that for the reference state |Φ, (5.42) simplify to (we drop the subscript 0 for simplicity) k(1) = 0 ,

k(2) = WRW , k(3) = WRWRW ,   k(4) = W(RW )3 − WRW WR2 W ,

W2 = VN .

a,b,r,s

κi = εi − ε0 ,

µ=1

k(0) = 0,

W1 = G N − U N ,

H0 = Z + G ≡ F ,

σ=1

µν|v|στ A = µν|v|στ − µν|v|τσ ,

W = W1 + W2 ,

Choosing U = G we have

where µ|g|ν =

Note that W is also in the N-product form,

etc. (5.60)

where the summations over a, b (r, s) extend over all occupied (unoccupied) spin orbitals in |Φ. Obtaining the corresponding higher-order corrections becomes more and more laborious and, beginning with the fourth-order, important cancellations arise between the principal and renormalization terms, even when the N-product form is employed. These will be addressed in Sect. 5.3.7. The above outlined PT with H0 given by the HF operator is often referred to as the Møller–Plesset PT [5.17] and, when truncated to the n-th order, is designated by the acronym MPn, n = 2, 3, . . . . In this version, the two-electron integrals enter the denominators only through the HF orbital energies. In an alternative, less often employed variant, referred to as the Epstein– Nesbet PT [5.18, 19], the whole diagonal part of H is

Perturbation Theory

a) µ

b) ν

µ

c) ν

µ

d) σ

µ

a) σ

5.3 Fermionic Many-Body Perturbation Theory (MBPT)

b)

r

c) r

r

s

a ν

τ

ν

τ

Fig. 5.1a–d Diagrammatic representation of one- and two-

b

s

a b

Fig. 5.2a–c The second-order Goldstone (a), (b) and Hugenholtz (c) diagrams

electron operators

used as the unperturbed Hamiltonian, i. e.,
H0 = Φi |H|Φi Pi .

a

b

s

107

(5.65)

i

With this choice, the denominators are given as differences of the diagonal elements of the configuration interaction matrix.

5.3.4 Diagrammatic MBPT

5.3.5 Vacuum and Wave Function Diagrams Applying Wick’s theorem to the strings of operators involved, we represent the individual contractions, (5.45), by joining corresponding oriented lines. To obtain a non-

1. Associate appropriate matrix elements with all vertices and form their product. The outgoing (ingoing) lines on each vertex define the bra (ket) states of a given matrix element, and for the Goldstone diagrams, the oriented lines attached to the same node are associated with the same electron number, (e.g., for the leftmost vertex in diagram (a) of Fig. 5.2 we have ab|ˆv|rs ≡ a(1)b(2)|v|r(1)s(2)). 2. Associate a denominator, (5.63), or its appropriate power, with every neighboring pair of vertices (and, for the wave function diagrams, also with the free lines extending to the left of the leftmost vertex; with each pair of such free lines associate also the corresponding pair of particle creation and hole annihilation operators). 3. Sum over all hole and particle labels. 4. Multiply each diagram contribution by the weight factor given by the reciprocal value of the order of

Part A 5.3

To facilitate the evaluation of higher order terms, and especially to derive the general properties and characteristics of the MBPT, it is useful to employ a diagrammatic representation [5.6–8, 14, 15]. Representing all the operators in (5.42) and (5.43) or (5.60) in the second quantized form, we have to deal with the reference state (i. e., the Fermi vacuum) mean values of the strings of annihilation and creation operators (or with these strings acting on the reference in the case of a wave function). This is efficiently done using Wick’s theorem and its diagrammatic representation via a special form of Feynman diagrams. In this representation we associate with various operators suitable vertices with incident oriented lines representing the creation (outgoing lines) and annihilation (ingoing lines) operators that are involved in their second quantization form. A few typical diagrams representing operators (−U), W1 and V are shown in Fig. 5.1a, Fig. 5.1b and Fig. 5.1c, Fig. 5.1d, respectively. Using the N-product form of PT with HF orbitals (Sect. 5.3.3), we only need the two-electron operator V or VN , which can be represented using either non-antisymmetrized vertices (Fig. 5.1c), leading to the Goldstone diagrams [5.20], or antisymmetrized vertices (Fig. 5.1d), associated with antisymmetrized two-electron integrals (5.56) and yielding the Hugenholtz diagrams [5.21].

vanishing contribution, only contractions preserving the orientation need be considered [cf. (5.45)]. The resulting internal lines have either the left–right orientation (hole lines) or the right–left one (particle lines). Only fully contracted terms, represented by the so-called vacuum diagrams (having only internal lines), can contribute to the energy, while those representing wave function contributions have uncontracted or free lines extending to the left. When the operators involved are in the Nproduct form, no contractions of oriented lines issuing from the same vertex are allowed. The projection-like operators R, (5.39), or their powers, lead to the denominators, (5.63), given by the difference of hole and particle orbital energies associated with, respectively, hole and particle lines passing through the interval separating the corresponding two neighboring vertices. Clearly, there must always be at least one pair of such lines lest the denominator vanish. Thus, for example, the second-order contribution WRW is represented either by the two Goldstone diagrams [5.20] (Fig. 5.2a,b) or by the single Hugenholtz diagram [5.21] (Fig. 5.2c). The rules for the energy (vacuum) diagram evaluation are as follows:

108

Part A

Mathematical Methods

r a

Fig. 5.3 Hugenholtz diagrams for the third-order energy

contribution Fig. 5.5 The second-order one-particle contribution

the group of automorphisms of the diagram (stripped of summation labels) and by the sign (−1)h+ , where h designates the number of internal hole lines and

gives the number of closed loops of oriented lines (for Hugenholtz diagrams, use any of its Goldstone representatives to determine the correct phase). Applying these rules to diagrams (a) and (b) of Fig. 5.2 we clearly recover (5.64) or, using the Hugenholtz diagram of Fig. 5.2, the equivalent expression 1
k(2) = ab|v|rs A rs|v|ab A ∆−1 (a, b; r, s) . 4 a,b,r,s

(5.66)

Part A 5.3

The possible third-order Hugenholtz diagrams are shown in Fig. 5.3 with the central vertex involving particle–particle, hole–hole, and particle–hole interaction [5.14, 15].

5.3.6 Hartree–Fock Diagrams In the general case (non-HF orbitals and/or not normal product form of PT), the one-electron terms, as well as the contractions between operators associated with the same vertex, can occur (the latter are always the hole lines). Representing the W1 and (−U) operators as shown in Fig. 5.1, the one-body perturbation W1 represents in fact the three diagrams as shown in Fig. 5.4. The second-order contribution of this type is then represented by the diagrams in Fig. 5.5, which in fact represents nine diagrams which result when each W1 vertex is replaced by three vertices as shown in Fig. 5.4. Using HF orbitals, all these terms mutually cancel out as seen above. For this reason, the diagrams involving contractions of lines issuing from the same vertex

are referred to as Hartree–Fock diagrams. Note, however, that even when not employing the canonical HF orbitals, it is convenient to introduce W1 vertices of the normal product form PT and replace all nine HF-type diagrams by a single diagram of Fig. 5.5 (clearly, this feature provides even greater efficiency in higher orders of PT).

5.3.7 Linked and Connected Cluster Theorems Using the N-product form of PT, the first nonvanishing renormalization term occurs in the fourth-order [cf. (5.60)]. For a system consisting of N noninteracting species, the energy given by this nonphysical term is proportional to N 2 , and thus violates the size extensivity of the theory. It was first shown by Brueckner [5.22] that in the fourth-order these terms are in fact exactly canceled by the corresponding contributions originating in the principal term. A general proof of this cancellation in an arbitrary order was then given by Goldstone [5.20] using the time-dependent PT formalism (Sect. 5.4). To comprehend this cancellation, consider the fourth-order energy contribution arising from the socalled unlinked diagrams (no such contribution can arise in the second- or the third-order) shown in Fig. 5.6. An unlinked diagram is defined as a diagram containing a disconnected vacuum diagram (for the energy diagrams, the terms unlinked and disconnected are synonymous). The numerators associated with both diagrams being identical, we only consider the denominators. Designating the denominator associated with the

A

a) µ

b) µ

ν =

c) ν

d) ν

µ +

a

ν B

+ a

µ

Fig. 5.4a–d Schematic representation of W1 = G N − U N

Fig. 5.6 The fourth-order unlinked diagrams

Perturbation Theory

top and the bottom part by A and B, respectively, we find for the overall contribution 1 1 1 1 1 1 · · + · · B A+ B B  A A + B B 1 1 1 1 + = = . (5.67) B A (A + B)B AB 2 Thus, the contribution from these terms exactly cancels  that from the renormalization term WRW WR2 W . Generalizing (5.67), we obtain the factorization lemma of Frantz and Mills [5.23], which implies the cancellation of renormalization terms by the unlinked terms originating from the principal term. This result holds for the energy as well as for the wave function contributions in every order of PT, as ascertained by the linked cluster theorem, which states that ∆E = k =

∞


Φ|W|Ψ (n)  =

n=0

∞


W(RW )n L ,

n=0

(5.68)

|Ψ  =

∞


|Ψ (n)  =

n=0

∞
  (RW )n |Φ L ,

(5.69)

n=0

(5.70)

n=1

the subscript C indicating that only contributions from connected diagrams are to be included. Since the general component with r disconnected parts can be shown to be represented by the term (r!)−1 T r |Φ, the general structure of the exact wave function |Ψ  is given by the connected cluster theorem, which states that |Ψ  = eT |Φ .

In other words, the wave operator W which transforms the unperturbed independent particle model wave function |Φ into the exact one according to |Ψ  = W|Φ ,

(5.72)

is given by the exponential of the cluster operator T , W = eT ,

(5.73)

which in turn is given by the connected wave function diagrams. This is in fact the basis of the coupled cluster methods [5.15, 24–28] (Sect. 5.3.8). The contributions to T may be further classified by their excitation rank i, T=

N


Ti ,

(5.74)

i=1

where Ti designates connected diagrams with i pairs of free particle–hole lines, producing i-times excited components of |Ψ  when acting on |Φ.

(5.71)

Summing all HF diagrams (Sect. 5.3.6) is equivalent to solving the HF equations. Depending on the average electron density of the system, it may be essential to sum certain types of PT diagrams to infinite order at the post-HF level. A frequently used approach that is capable of recovering a large part of the electronic correlation energy is based on the connected cluster theorem (Sect. 5.3.7), referred to in this context as the exponential cluster Ansatz for the wave operator. Using this Ansatz, one derives a system of energy-independent nonlinear coupled cluster (CC) equations [5.15, 26–28] determining the cluster amplitudes of T . These CC equations can be regarded as recurrence relations generating the MBPT series [5.15], so that by solving these equations one in fact implicitly generates all the MBPT diagrams and sums them to infinite order. Since the solution of the full CC equations is equivalent to the exact solution of the Schrödinger equation, we must – in all practical applications – introduce a suitable truncation scheme, which implies that only diagrams of certain types are summed. Generally, using the cluster expansion (5.71) in the N-product form of the Schrödinger equation, HN |Ψ  ≡ (H − H)|Ψ  = ∆E|Ψ  , ∆E = E − E 0 ,

(5.75)

Part A 5.3

∞
  (RW )n |Φ C ,

109

5.3.8 Coupled Cluster Theory

where the subscript L indicates that only linked diagrams (or terms) are to be considered. This enables us to obtain general, explicit expressions for the nthorder PT contributions by first constructing all possible linked diagrams involving n vertices and by converting them into the explicit algebraic expressions using the rules of Sect. 5.3.5. Note that linked energy diagrams are always connected, but the linked wave function diagrams are either connected or disconnected, each disconnected component possessing at least one pair of particle–hole free lines extending to the left. To reveal a deeper structure of the result (5.69), define the cluster operator T that generates all connected wave function diagrams, T |Φ =

5.3 Fermionic Many-Body Perturbation Theory (MBPT)

110

Part A

Mathematical Methods

premultiplying with the inverse of the wave operator, and using the Hausdorff formula (5.11) yields ∞
[ad(−T )]n HN = ∆E|Φ . e−T HN eT |Φ = n! n=0

(5.76)

In fact, this expansion terminates, so that using (5.74) and projecting onto |Φ we obtain the energy expression 1 (5.77) ∆E = HN T2  + HN T12 , 2 while the projection onto the manifold of excited states {|Φi } relative to |Φ ≡ |Φ0  gives the system of CC equations 1 Φi |HN + [HN , T ] + [[HN , T ], T ] + · · · |Φ = 0 . 2 (5.78)

Approximating, e.g., T by the most important pair cluster component T ≈ T2 gives the so-called CCD (coupled clusters with doubles) approximation  (2)  1 Φi  HN + [HN , T2 ] + [[HN , T2 ], T2 ]|Φ = 0 , 2 (5.79)

Part A 5.3

the superscript (2) indicating pair excitations relative to |Φ. Equivalently, (5.77) and (5.78) can be written in the form  (5.80) ∆E = HN eT C ,  

  T   Φi HN e (5.81) Φ =0, C

the subscript C again indicating that only connected diagrams are to be considered. The general form of CC equations is

bij t j + cijk t j tk + · · · = 0 , (5.82) ai + j

j≤k

where ai = Φi |HN |Φ0 , bij = Φi |HN |Φ j C , cijk = Φi |HN |Φ j ⊗ Φk C , etc. Writing the diagonal linear term bii in the form (5.83) bii = ∆i + bii , this system can be solved iteratively by rewriting it in the form 


bij t (n) ti(n+1) = ∆i−1 ai + bii ti(n) + j

+




j (n) cijk t (n) j tk + · · ·



.

(5.84)

j≤k

Starting with the zeroth approximation ti(0) = 0, the first iteration is ti(1) = ∆i−1 ai ,

(5.85)

which yields the second-order PT energy when used in (5.77). Clearly, the successive iterations generate higher and higher orders of the PT. At any truncation level, a size extensive result is obtained. The CC methods belong to the most accurate and often used tools in computations of molecular electronic structure and several general-purpose codes are available for this purpose (for reviews see [5.29–32]). The standard approach truncates the cluster operator (5.74) at the singly (S) and doubly (D) excited level (the CCSD method [5.33]) and is often supplemented by a perturbative account of the triply-excited (T) cluster components [the CCSD(T) method] for greater accuracy [5.34]. To avoid the breakdown of the latter method in quasidegenerate situations, one can employ one of the renormalized versions of CCSD(T) [5.35]. The CC ansatz (5.71) has also been exploited in the context of the equation-of-motion (EOM) and the linearresponse formalisms, enabling the computation of the excitation energies and of properties other than the energy (dipole and quadrupole moments, polarizabilities, etc., [5.29–32]. At this stage it is important to recall that the above described MBPT and CC approaches pertain to nondegenerate, lowest-lying closed-shell states of a given symmetry species. Although the CC methods are often used even for open-shell states by relying on the unrestricted HF (UHF) reference [of the different-orbitals-for-different-spins (DODS) type], a proper description of such states requires a multireference (MR) generalization based on the effective Hamiltonian formalism [5.6, 31, 32, 36–38]. Unfortunately, such a generalization is not unambiguous. The two existing formulations, the so-called valence universal [5.6, 37] and state universal [5.38] methods, are computationally demanding and often plagued with the intruder state and other problems [5.15, 36]. For these reasons, no general-purpose codes have yet been developed and very few actual applications have been carried out [5.31, 32] (see, however, the recently formulated SU CC approach for general model spaces [5.39, 40]). Nonetheless, the MR CC formalism proved to be very useful in the formulation of the so-called state selective or state specific approaches (e.g., the reduced MR CCSD method [5.41–46]). Most recently, the CC approach has been used to handle bosonic-type problems of the vibrational structure in molecular spectra and, generally, multimode dynamics [5.47].

Perturbation Theory

5.4 Time-Dependent Perturbation Theory

111

5.4 Time-Dependent Perturbation Theory 5.4.1 Evolution Operator PT Expansion

Iterating we get [5.49, 50] U(t, t0 ) ∞ 
i n = − ~

By introducing the evolution operator U(t, t0 ) |Ψ(t) = U(t, t0 )|Ψ(t0 ) ,

(5.86)

n=0 t

time-dependent Schrödinger equation i~

∂ |Ψ(t) = H|Ψ(t) ∂t

× (5.87)

=

∂ i~ U(t, t0 ) = HU(t, t0 ) . ∂t

n! t

t0

U(t0 , t0 ) = 1 , U(t, t0 ) = U(t, t )U(t , t0 ) , U(t, t0 )−1 = U(t0 , t) = U † (t, t0 ) .

In the interaction picture (subscript I) i  |Ψ(t)I = exp H0 t |Ψ(t) , ~

(5.89)

(5.90)

5.4.2 Gell–Mann and Low Formula For a time-independent perturbation, one introduces the so-called adiabatic switching by writing Hα (t) = H0 + λ e−α|t| V,

(5.91)

(5.92)

∂ |Ψ(t)I = V(t)I |Ψ(t)I , ∂t

∂ U(t, t0 ) = V(t)U(t, t0 ) , ∂t

i ~

(5.94)

t V(t1 )U(t1 , t0 ) dt1 . t0

(5.98)

(t) = λ e−α|t| V

with Uα (t, −∞|λ) obtained with Vα (all in the interaction picture). The desired energy is then given by the Gell–Mann and Low formula [5.51] ∆E = lim i~αλ α→0+

∂ lnΦ0 |Uα (0, −∞|λ)|Φ0  , ∂λ (5.99a)

with the initial condition U(t0 , t0 ) = 1. This differential equation is equivalent to an integral equation U(t, t0 ) = 1 −

(5.97)

so that Hα (t → ±∞) = H0 and Hα (t → 0) = H = H0 + λV . Then

(5.93)

known as Tomonaga-Schwinger equation [5.48]. Analogously, the evolution operator in this picture (we drop the subscript I from now on) satisfies i~

α>0

|Ψ(t)I = Uα (t, −∞|λ)|Φ0  ,

the Schrödinger equation becomes i~

(5.96)

where T [· · · ] designates the time-ordering or chronological operator.

where now H = H0 + V ,

dtn T [V(t1 ) · · · V(tn )] , t0

(5.95)

or ∆E =

∂ 1 lim i~αλ lnΦ0 |Uα (∞, −∞|λ)|Φ0  , 2 α→0+ ∂λ (5.99b)

which result from the asymmetric energy formula (5.27). One can similarly obtain the perturbation expansion for the one- or two-particle Green functions, e.g.,    †

 T aµ (t)aν (t )Uα (∞, −∞|λ) , G µν t, t = lim α→0+ Uα (∞, −∞|λ) (5.100)

Part A 5.4

If the Hamiltonian is time independent then  i  U(t, t0 ) = exp − H(t − t0 ) . ~

t0

dt1 · · ·

×

Clearly,

t0

dtn V(t1 )V(t2 ) · · · V(tn )

∞
(−i/~)n n=0 t

(5.88)

tn−1 dt2 · · ·

dt1 t0

becomes

t1

112

Part A

Mathematical Methods

with the operators in the interaction representation and the expectation values in the noninteracting ground state |Φ0 . Analogous expressions result for G(rt, r t ), etc., when the creation and annihilation operators are replaced by the corresponding field operators.

Again, for causal propagation one chooses the timeretarded or causal Green function or propagator



G (+) 0 (r, r ; t, t ).

5.4.3 Potential Scattering and Quantum Dynamics

Iteration of (5.105) gives the Born sequence

The Schrödinger equation for a free particle of energy E = ~2 k2 /2m, moving in the potential V(r), 

2 ∇ + k2 ψ(k, r) = v(r)ψ(k, r) , 

(5.101) v(r) = 2m/~2 V(r) , has the formal solution ψ(k, r) = Φ(k, r) +



 

 G 0 k, r, r v r ψ k, r dr , (5.102)

where Φ(k, r) is a solution of the homogeneous equation [v(r) ≡ 0] and G 0 (k, r, r ) is a classical Green function

2 



 ∇ + k2 G 0 k, r, r = δ r − r . (5.103)

Part A 5.4

For an in-going plane wave Φ(k, r) ≡ Φki (r) = (2π)3/2 exp(iki · r) with the initial wave vector ki and appropriate asymptotic boundary conditions (outgoing spherical wave with positive phase vel

ocity), when G 0 (k, r, r ) ≡ G (+) 0 (|r − r |) = −(4π|r −

|

−1 ik|r−r r |) e , (5.102) is referred to as the Lippmann– Schwinger equation [5.52]. It can be equivalently transformed into the integral equation for the Green function 









 G (+) r, r = G (+) r, r + G (+) r, r v r 0 0

 (5.104) × G (+) r

, r dr

, representing a special case of the Dyson equation. In the time-dependent case, considering the scattering of a spinless massive particle by a time-dependent potential V(r, t), we get similarly ψ(r, t) =







 Φ(r, t)+ G 0 r, r ; t, t V r , t ψ r , t dr dt ,

5.4.4 Born Series

ψ0 (r, t) = Φ(r, t) ,





 ψ1 (r, t) = Φ(r, t) + G (+) r, r ; t, t

0 



× V r , t Φ r , t dr dt , 



 ψ2 (r, t) = Φ(r, t) + G (+) r, r ; t, t

0 



× V r , t ψ1 r , t dr dt ,

(5.107a)

(5.107b)

and, generally





 ψn (r, t) = Φ(r, t) + G (+) r, r ; t, t

0 



× V r , t ψn−1 r , t dr dt .

(5.108)

Summing individual contributions gives the Born series for ψ(r, t) ≡ ψ (+) (r, t), ψ(r, t) =

∞


χn (r, t) ,

(5.109)

n=0

where χ0 (r, t) = Φ(r, t) , 



 χn (r, t) = Gn r, r ; t, t Φ r , t dr dt , (5.110)

with

 Gn r, r ; t, t =



 G1 r, r

; t, t



 × Gn−1 r

, r ; t

, t dr

dt

, (n > 1)







 r, r ; t, t V r , t . (5.111) G1 r, r ; t, t = G (+) 0 In a similar way we obtain the Born series for the scattering amplitudes or transition matrix elements.

5.4.5 Variation of Constants Method

(5.105)

where the zero-order time-dependent Green function now satisfies the equation  





 ∂ i~ − H0 G 0 r, r ; t, t = δ r − r δ t − t . ∂t (5.106)

An alternative way of formulating the time-dependent PT is the method of variation of the constants [5.53,54]. Start again with the time-dependent Schrödinger equation (5.87) with H = H0 + V , and assume that H0 is time-independent, while V is a time-dependent perturbation. Designating the eigenvalues and eigenstates of

Perturbation Theory

H0 by εi and |Φi , respectively [cf. (5.48)], the general solution of the unperturbed time-dependent Schrödinger equation ∂ i~ |Ψ0  = H0 |Ψ0  (5.112) ∂t has the form  i 
|Ψ0  = c j |Φ j  exp − ε j t , (5.113) ~

j

where the C j (t) are now functions of time. Substituting this Ansatz into the time-dependent Schrödinger equation (5.87) gives
Ck (t)V jk exp[(i/~)∆ jk t] , C˙ j (t) = (i~)−1

C j ≡ C j (t) =

∞


k C (k) j (t)λ ,

(5.118)

k=0

gives the system of first order differential equations
(n)

= (i~)−1 Ck V jk exp (i/~)∆ jk t , C˙ (n+1) j k

n = 0, 1, 2, . . . ,

(5.119)

with the initial condition C˙ (0) implies that j = 0, which (0) C (0) are time independent, so that C = c j , obtainj j ing (5.113) in the zeroth order. The system (5.119) can be integrated to any prescribed order. For example, if the system is initially in a stationary state |Φi , then set  δ for discrete states , ji (0) Cj = δ( j − i) for continuous states , (5.120)

so that t (1) −1 C j(i) (t) = (i~) −∞

k

(5.115)

V ji exp (i/~)∆ ji t dt ,

V jk = Φ j |V |Φk  .

(5.116)

Introducing again the ‘small’ parameter λ by writing the Hamiltonian H in the form H = H0 + λV(t) ,

(5.117)

C (1) j(i) (−∞) = 0.

2 |C (1) j(i) (t)|

assuming Clearly, gives the first order transition probability for the transition from the initial state |Φi  to a particular state |Φ j . These in turn will yield the first order differential cross sections [5.55].

References 5.1 5.2 5.3 5.4

5.5

5.6 5.7

T. Kato: Perturbation Theory for Linear Operators (Springer, Berlin, Heidelberg 1966) H. Baumgärtel: Analytic Perturbation Theory for Matrices and Operators (Akademie, Berlin 1984) E. J. Hinch: Perturbation Methods (Cambridge Univ. Press, Cambridge 1991) V. N. Bogaevski, A. Povzner: Algebraic Methods in Nonlinear Perturbation Theory (Springer, Berlin, Heidelberg 1991) E. M. Corson: Perturbation Methods in the Quantum Mechanics of n-Electron Systems (Blackie & Son, London 1951) I. Lindgren, J. Morrison: Atomic Many-Body Theory (Springer, Berlin, Heidelberg 1982) E. K. U. Gross, E. Runge, O. Heinonen: Many-Particle Theory (Hilger, New York 1991)

5.8

5.9 5.10 5.11 5.12 5.13

5.14 5.15

F. E. Harris, H. J. Monkhorst, D. L. Freeman: Algebraic and Diagrammatic Methods in Many-Fermion Theory (Oxford Univ. Press, Oxford 1992) H. Primas: Helv. Phys. Acta 34, 331 (1961) H. Primas: Rev. Mod. Phys. 35, 710 (1963) M. Rosenblum: Duke Math. J. 23, 263 (1956) G. Arfken: Mathematical Methods for Physicists (Academic, New York 1985) p. 327 Encyclopedic Dictionary of Mathematics, ed. by S. Iyanaga, Y. Kawada (MIT Press, Cambridge 1980) pp. 1494, Appendix B, Table 3 J. Paldus, J. ˇCíˇzek: Adv. Quantum Chem. 9, 105 (1975) J. Paldus: Methods in Computational Molecular Physics, NATO ASI Series B, Vol. 293, ed. by S. Wilson, G. H. F. Diercksen (Plenum, New York 1992) pp. 99–194

Part A 5

(5.121)

where ∆ jk = ε j − εk ,

113

and expanding the ‘coefficients’ C j (t) in powers of λ,

j

with c j representing arbitrary constants, and the sum indicating both the summation over the discrete part and the integration over the continuum part of the spectrum of H0 . In the spirit of the general variation of constants procedure, write the unknown perturbed wave function |Ψ(t), (5.87), in the form  i 
|Ψ(t) = C j (t)|Φ j  exp − ε j t , (5.114) ~

References

114

Part A

Mathematical Methods

5.16 5.17 5.18 5.19 5.20 5.21 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29

5.30

5.31 5.32

5.33 5.34

Part A 5

H. J. Silverstone, T. T. Holloway: J. Chem. Phys. 52, 1472 (1970) C. Møller, M. S. Plesset: Phys. Rev. 46, 618 (1934) P. S. Epstein: Phys. Rev. 28, 695 (1926) R. K. Nesbet: Proc. R. Soc. London A 250, 312 (1955) J. Goldstone: Proc. R. Soc. London A 239, 267 (1957) H. M. Hugenholtz: Physica (Utrecht) 23, 481 (1957) K. A. Brueckner: Phys. Rev. 100, 36 (1955) L. M. Frantz, R. L. Mills: Nucl. Phys. 15, 16 (1960) F. Coester: Nucl. Phys. 7, 421 (1958) F. Coester, H. Kümmel: Nucl. Phys. 17, 477 (1960) J. ˇCíˇzek: J. Chem. Phys. 45, 4256 (1966) J. ˇCíˇzek: Adv. Chem. Phys. 14, 35 (1969) J. Paldus, J. ˇCíˇzek, I. Shavitt: Phys. Rev. A 5, 50 (1972) R. J. Bartlett: Modern Electronic Structure Theory, ed. by D. R. Yarkony (World Scientific, Singapore 1995) pp. 47–108, Part I R. J. Bartlett (Ed.): Recent advances in computational chemistry, Recent Advances in CoupledCluster Methods, Vol. 3 (World Scientific, Singapore 1997) J. Paldus, X. Li: Adv. Chem. Phys. 110, 1 (1999) J. Paldus: Handbook of Molecular Physics and Quantum Chemistry, Vol. 2, Part 3, ed. by S. Wilson (Wiley, Chichester 2003) pp. 272–313 R. J. Bartlett, G. D. Purvis: Int. J. Quantum Chem. 14, 561 (1978) T. J. Lee, G. E. Scuseria: Quantum Mechanical Electronic Structure Calculations with Chemical Accuracy, ed. by S. R. Langhoff (Kluwer, Dordrecht 1995) pp. 47–108

5.35 5.36

5.37 5.38 5.39 5.40 5.41 5.42 5.43 5.44 5.45 5.46 5.47 5.48 5.49 5.50 5.51 5.52 5.53 5.54 5.55

K. Kowalski, P. Piecuch: J. Chem. Phys. 120, 1715 (2004) J. Paldus: Relativistic and Electron Correlation Effects in Molecules and Solids, NATO ASI Series B, Vol. 318, ed. by G. L. Malli (Plenum, New York 1994) pp. 207– 282 I. Lindgren, D. Mukherjee: Phys. Rep. 151, 93 (1987) B. Jeziorski, H. J. Monkhorst: Phys. Rev. A 24, 1686 (1981) X. Li, J. Paldus: J. Chem. Phys. 119, 5320, 5334, 5343 (2003) X. Li, J. Paldus: J. Chem. Phys. 120, 5890 (2004) X. Li, J. Paldus: J. Chem. Phys. 107, 6257 (1997) X. Li, J. Paldus: J. Chem. Phys. 108, 637 (1998) X. Li, J. Paldus: J. Chem. Phys. 110, 2844 (1999) X. Li, J. Paldus: J. Chem. Phys. 113, 9966 (2000) X. Li, J. Paldus: J. Chem. Phys. 118, 2470 (2003) S. Chattopadhyay, D. Pahari, D. Mukherjee, U. S. Mahapatra: J. Chem. Phys. 120, 5968 (2004) O. Christiansen: J. Chem. Phys. 120, 2149 (2004) F. J. Dyson: Phys. Rev. 75, 486 (1949) S. Tomonaga: Prog. Theor. Phys. (Kyoto) 1, 27 (1946) J. Schwinger: Phys. Rev. 74, 1439 (1948) M. Gell-Mann, F. Low: Phys. Rev. 84, 350 (1951) B. A. Lippmann, J. Schwinger: Phys. Rev. 79, 469 (1950) P. A. M. Dirac: Proc. R. Soc. London A 112, 661 (1926) P. A. M. Dirac: Proc. R. Soc. London A 114, 243 (1926) C. J. Joachain: Quantum Collision Theory (Elsevier, New York 1975)

115

Second Quanti 6. Second Quantization

In second quantization, the characteristic properties of eigenfunctions are transferred to operators. This approach has the advantage of treating the atomic shell as the basic unit, as opposed to the electron configuration. The creation and annihilation operators allow one to move from configuration to configuration, exposing an intrinsic shell structure. The introduction of coefficients of fractional parentage (cfp) then allows the calculation of the matrix elements of an operator in one configuration to be expressed in terms of those of the same operator in another configuration; hence the matrix elements of an operator in all configurations may be determined from the knowledge of its matrix elements in but one. This can be viewed as an extension of the usual Wigner-Eckart theorem. The basic concepts of quasispin and quasiparticle are also introduced within this context.

6.1

Basic Properties................................... 115 6.1.1 Definitions ............................... 115 6.1.2 Representation of States ............ 115 6.1.3 Representation of Operators ....... 116

6.2

Tensors ............................................... 6.2.1 Construction ............................. 6.2.2 Coupled Forms .......................... 6.2.3 Coefficients of Fractional Parentage ................................

117

6.3

Quasispin............................................ 6.3.1 Fermions.................................. 6.3.2 Bosons..................................... 6.3.3 Triple Tensors ........................... 6.3.4 Conjugation.............................. 6.3.5 Dependence on Electron Number 6.3.6 The Half-filled Shell ..................

117 117 118 118 118 119 119

6.4

Complementarity ................................. 119 6.4.1 Spin–Quasispin Interchange ....... 119 6.4.2 Matrix Elements ........................ 119

6.5

Quasiparticles ..................................... 120

116 116 116

References .................................................. 121

6.1.1 Definitions †

The creation operator aξ creates the quantum state ξ. The annihilation (or destruction) operator aη annihilates the quantum state η. The vacuum (or reference) state |0
satisfies the equation aη |0
= 0 .

(6.1)

Bosons satisfy the commutation relations
† † aξ , aη = 0 ,

(6.2)

[aξ , aη ] = 0 ,
 aξ , aη† = δ(ξ, η) ,

(6.3) (6.4)

where [A, B] ≡ AB − BA. Fermions satisfy the anticommutation relations
† † (6.5) aξ , aη + = 0 ,

[aξ , aη ]+ = 0 ,
 aξ , aη† + = δ(ξ, η) ,

(6.6) (6.7)

where [A, B]+ ≡ AB + BA.

6.1.2 Representation of States For an electron in an atom, characterized by the quantum number quartet (n  m s m  ), the identification ξ ≡ (n  m s m  ) for fermions is made. For normalized Slater determinants {αβ . . . ν} characterized by the electron states α, β, . . . , ν, the equivalences †

aα† aβ . . . aν† |0
≡ {αβ . . . ν} ,

(6.8)

0|aν . . . aβ aα ≡ {αβ . . . ν}

(6.9)

∗

are valid, where the asterisk denotes the complex conjugate.

Part A 6

6.1 Basic Properties

116

Part A

Mathematical Methods

For a normalized boson state { · · · } in which the label ξ appears Nξ times, the additional factor 1

[Nα !Nβ ! . . . Nν !]− 2

(6.10)

must be included on the left-hand sides of the equivalences (6.8) and (6.9).

6.1.3 Representation of Operators For an operator F, consisting of the sum of operators fi acting on the single electron i,  † aξ ξ| f |η
aη . (6.11) F≡ ξ,η

For an operator G, consisting of the sum of operators gij acting on the pair of electrons i and j, G≡

1  † † aξ aη ξ1 η2 |g12 |ζ1 λ2
aλ aζ . 2

(6.12)

ξ,η,ζ,λ

For an N-particle system |Ψ
,  † aξ aξ |Ψ
= N|Ψ
.

(6.13)

ξ

The representations of single-particle and two-particle operators for bosons are identical to those given above for fermions [6.1].

6.2 Tensors 6.2.1 Construction

6.2.2 Coupled Forms

If the description ξ for a single fermion or boson state includes an angular momentum quantum number t and the corresponding magnetic quantum number m t , then † the 2t + 1 components of a creation operator aσ , where σ ≡ (t, m t ) and −t ≤ m t ≤ t, satisfy the commutation relations of Racah [6.2] for an irreducible spherical tensor of rank t with respect to the total angular momentum T, given by  † (6.14) T= aξ ξ|t|η
aη .

Tensors formed from annihilation and creation operators can be coupled by means of the usual rules of angular momentum theory [6.4]. The double tensor defined for electrons in the  shell by

Part A 6.2

That is, with the phase conventions of Condon and Shortley [6.3],   Tz , aσ† = m t aσ† , (6.15)   1 Tx ± iTy , aσ† = [t(t + 1) − m t (m t ± 1)] 2 aτ† , (6.16)

where τ ≡ (t, m t ± 1). A spherical tensor a constructed from annihilation operators possesses the components a˜σ , which satisfy a˜σ = (−1) p aζ ,

(6.17)

with p = t − m t and ζ ≡ (t, −m t ). The 4 + 2 components of the creation operator for an electron in the atomic  shell form a double tensor of rank 12 with respect to the total spin S, and rank  with respect to the total angular momentum L.

 (κk) W (κk) = − a† a ,

(6.18)

possesses a rank κ with respect to S, and rank k with respect to L. Its reduced matrix element, defined here as in (5.4.1) of Edmonds [6.4], for a single electron in both the spin and orbital spaces, is given by   1 s  || W (κk) || s  = [(2κ + 1)(2k + 1)] 2 .

(6.19)

The connections to tensors whose matrix elements have been tabulated [6.5, 6] are 1

W (0k) = [(2k + 1)/2] 2 U (k) , W

(1k)

1 2

= [2(2k + 1)] V

(k1)

.

For terms with common spin S, say ψ and ψ  ,   ψ||W (0k) ||ψ  =  1 [(2S + 1)(2k + 1)/2] 2 ψ||U (k) ||ψ  .

(6.20) (6.21)

(6.22)

This result is obtained because the ranks assigned to the tensors imply that W (0k) is to be reduced with respect to both the spin S and the orbit L, while U (k) is to be reduced only with respect to L.

Second Quantization

The following relations hold for electrons with azimuthal quantum numbers  [6.7]: 1

S = [(2 + 1)/2] 2 W (10) ,

(6.23) 1 2

L = [2( + 1)(2 + 1)/3] W

(01)

,

(6.24)

  (2) (1) =− si Ci i



( + 1)(2 + 1) 10(2 − 1)(2 + 3)

1 2

W (12)1 , (6.25)

 (si ·
i ) = −[( + 1)(2 + 1)/2]W (11)0 , i

(6.26)

where the tensor C k of Racah [6.2] is related to the spherical harmonics by 1

Cq(k) = [4π/(2k + 1)] 2 Ykq ,

(6.27)

and where the tensors of the type W (κk)K indicate that the spin and orbital ranks are coupled to a resultant K .

6.2.3 Coefficients of Fractional Parentage Let ψ and ψ¯ denote terms of  N and  N−1 character¯ L). ¯ The coefficients of fractional ized by (S, L) and ( S, ¯ of Racah [6.8] allow one to parentage (cfp) (ψ{|ψ) calculate an antisymmetrized function ψ by vectorcoupling ψ¯ to the spin and orbit of the Nth electron: 

 ψ, ¯ 2, SL ψ|}ψ ¯ |ψ
= , (6.28) ¯ L, ¯ and any other where the sum over ψ¯ includes S, quantum numbers necessary to define the spectroscopic

terms of  N−1 . The cfp’s are given by    1 ψ||a† ||ψ¯ = (−1) N [N(2S + 1)(2L + 1)] 2 ψ{|ψ¯ , (6.29)

   1 ¯ ¯ ψ||a||ψ = (−1)g [N(2S + 1)(2L + 1)] 2 ψ|}ψ ,

(6.30)

where g = N + S¯ + L¯ − s − S −  − L. A tabulation for the p, d, and f shells has been given by Nielson and Koster [6.5]. Two-electron cfp are given by   † † (κk)  1 ψ|| a a ||ψ˜ = [N(N − 1)(2S + 1)(2L + 1)] 2   ˜ 2 (κk) , × ψ{|ψ, (6.31) where ψ˜ denotes a term of  N−2 , and the symbols κ and k stand for the S and the L of a term of 2 . A tabulation for the p, d, and f shells has been given by Donlan [6.9]. An extension to all multielectron cfp has been carried out by Velkov [6.10]. If, through successive applications of the twoparticle operators (aa)(00) , a state of  N can be reduced to v , but no further, then v is the seniority number of Racah [6.8]. If the ranks s and  of a† are coupled to S¯ and L¯ of ¯ ψ, the term  † (SL) a ψ¯ (6.32) either vanishes, or is a term of  N characterized by S and L. Such a term is said to possess the godparent ¯ Redmond [6.11] has used the notion of godparents to ψ. generate an explicit formula for the single particle cfp [6.7].

6.3 Quasispin 6.3.1 Fermions For electrons, the components Q ± (≡ Q x ± iQ y ) and Q z of the quasispin Q are defined by [6.7, 12] (00) 1 Q + = [(2 + 1)/2] 2 a† a† , (6.33)

The term quasispin comes from the fact that the components of Q satisfy the commutation relations of an angular momentum vector. The eigenvalues M Q of Q z , for a state of  N , are given by M Q = −(2 + 1 − N )/2 .

(6.36)

Q − = −[(2 + 1)/2] 2 (aa)(00) , (6.34) (00)  † (00)  1
 . Q z = −[(2 + 1)/8] 2 a† a + aa

The shift operators Q + and Q − connect states of the  shell possessing the same value of the seniority v of Racah [6.8]. A string of such connected states defines the extrema of M Q , from which it follows that

(6.35)

(6.37)

1

117

Q = (2 + 1 − v)/2 .

Part A 6.3

ψ¯

6.3 Quasispin

118

Part A

Mathematical Methods

Rudzikas has placed special emphasis on quasispin in his reworking of atomic shell theory, and he has also introduced isospin to embrace electrons differing in their principal quantum numbers n [6.13]. Concise tables of one-electron cfp with their quasispin dependence factored out have been given [6.14], as have the algebraic dependences on ν and S of two-electron cfp [6.15].

Furthermore, the components of X(Kκk) for which M K = 0 are identical to the corresponding components 1 of 2 2 (a† a)(κk) when K + κ + k is odd; and

6.3.2 Bosons

6.3.4 Conjugation †

For real vibrational modes created by aν (ν = 1, 2, . . . , d), the analogs of (6.33–6.35) are 1 † † P+ = − a a , (6.38) 2 ν ν ν 1 P− = aν aν , (6.39) 2 ν  1  † aν aν + aν aν† , Pz = (6.40) 4 ν and P is an angular momentum vector [6.16]. The eigenvalues M P for an n-boson state are given by M P = (2n + d)/4 ,

(6.41)

and can therefore be quarter-integral. Successive application of the operator P+ to a state |n 0
, for which P− |n 0
= 0, generates an infinite ladder of states characterized by P = (2n + d − 4)/4 .

(6.42)

6.3.3 Triple Tensors Part A 6.3

†

The creation and annihilation operators aξ and aξ for a given state ξ can be regarded as the two components of a tensor of rank 12 with respect to quasispin (either Q or P). For electrons, this leads to triple tensors a(qs) (for which q = s = 12 ) satisfying

1

X(Kκk) = −(2 + 1) 2 δ(K, 0)δ(κ, 0)δ(k, 0)

(6.48)

when K + κ + k is even.

Creation and annihilation operators can be interchanged by the operation of the conjugation operator C [6.7, 17]. For electrons in the atomic  shell, (qs)

Caξ

C −1 = (−1)q−m q aη(qs) ,

(6.49)

where ξ ≡ (m q m s m  ) and η ≡ ((−m q )m s m  ). In terms of the tensors a† and a, Ca† C −1 = a ,

CaC −1 = −a† .

(6.50)

Furthermore, (Kκk) CXλ(Kκk) C −1 = (−1) K −M K X µ ,

(6.51)

where λ ≡ (M K Mκ Mk ) and µ ≡ [(−M K )Mκ Mk ], and

C Q M Q = (−1) Q−M Q Q − M Q . (6.52) Thus, from (6.36), the action of C takes N into 4+ 2 − N; that is, C interchanges electrons and holes. When the case κ = k = 0 is excluded, application of (6.51) and (6.52) yields   N ψ || W (κk) ||  N ψ  =  (−1) y  4+2−N ψ||W (κk) || 4+2−N ψ  , (6.53)

where y = κ + k + 12 (v − v) + 1, and where the seniorities v and v are implied by ψ and ψ  . A similar (qs) (qs) (qs) (qs) application to reduced matrix elements of a† and a gives aλ aµ + aµ aλ =       the following relation between cfp: (−1)x+1 δ m q , −m q δ m s , −m s δ m  , −m  ,   (6.43)  N+1 ψ{| N ψ  = (−1)z  4+1−N ψ|} 4+2−N ψ  where λ ≡ (m q m s m  ) , µ ≡ (m q m s m  ) , and x = q + 1

(4 + 2 − N )(2S + 1)(2L  + 1) 2 s +  + m q + m s + m  . In terms of the coupled tensor , × (N + 1)(2S + 1)(2L + 1) (6.44) X(Kκk) = (a(qs) a(qs) )(Kκk) , (6.54)

the angular momenta Q, S, and L are given by 1

Q = −[(2 + 1)/4] 2 X(100) ,

(6.45)

1

S = −[(2 + 1)/4] 2 X(010) , 1 2

L = −[( + 1)(2 + 1)/3] X

(6.46) (001)

.

(6.47)

S + S − s + L + L  −  + 12 (v + v − 1).

where z = The phases y and z stem from the conventions of angular momentum theory, which enter via quasispin. Racah [6.2, 8] did not use this concept, and his phase choices are slightly different from the ones above.

Second Quantization

For a Cartesian component Q u of the quasispin Q, CQ u C

−1

= −Q u .

(6.55)

Thus, C is the analog of the time-reversal operator T , for which TL u T −1 = −L u , TSu T

−1

= −Su .

(6.56) (6.57)

Both C and T are antiunitary; thus, CiC −1 = −i .

6.4 Complementarity

119

seniority and independent of N. These properties were first stated in Eqs. (69) and (70) of [6.8]. Application of these ideas to single-electron cfp yields, for states ψ and ψ¯ with seniorities v and v + 1, respectively,   N ψ{| N−1 ψ¯ =   1 [(N − v)(v + 2)/2N] 2  v+2 ψ  v+1 ψ¯ . (6.60)

(6.58)

6.3.6 The Half-filled Shell

6.3.5 Dependence on Electron Number Application of the Wigner–Eckart theorem to matrix elements whose component parts have well-defined quasispin ranks yields the dependence of the matrix elements on the electron number N [6.18, 19]. For κ + k even and nonzero, the quasispin rank of W (κk) is 1, and   N ψ||W (κk) || N ψ  = (6.59)  (2 + 1 − N ) v  ψ || W (κk) ||  v ψ  . (2 + 1 − v) For κ + k odd, W (κk) is necessarily a quasispin scalar, and the matrix elements are diagonal with respect to the

Selection rules for operators of good quasispin rank K , taken between states of the half-filled shell (for which M Q = 0), can be found by inspecting the 3– j symbol   Q K Q , 0 0 0 which appears when the Wigner-Eckart theorem is applied in quasispin space. This 3– j symbol vanishes unless Q + K + Q  is even. An equivalent result can be obtained for W (κk) by referring to (6.53) and insisting that y be even.

6.4 Complementarity For every γ , Racah [6.20] observed that there are two possible pairs (v1 , S1 ) and (v2 , S2 ) satisfying

6.4.1 Spin–Quasispin Interchange

(qs)

Raξ

R−1 = aη(qs) ,

(6.62)

where λ ≡ (M K Mκ Mk ) and µ ≡ (Mκ M K Mk ). For states of the  shell, R|γQ M Q SM S
= (−1)t |γSM S Q M Q
,

(6.64)

From (6.37) it follows that S1 = Q 2 ,

S2 = Q 1 .

(6.65)

(6.61)

where ξ ≡ (m q m s m  ) and η ≡ (m s m q m  ). For the tensors X(Kκk) defined in (6.44), we get (κKk) , RX λ(Kκk) R−1 = X µ

v1 + 2S2 = v2 + 2S1 = 2 + 1 .

(6.63)

where the quasispin of the ket on the right is S and the spin is Q. The phase factor t depends on S and Q and on phase choices made for the coefficients of fractional parentage. The symbol γ denotes the additional labels necessary to completely define the state in question, including L and M L .

6.4.2 Matrix Elements Application of the complementarity operator R to the component parts of a matrix element leads to the equation

 (6.66) γQ M Q SM S |X λ(Kκk) |γ  Q  M Q S M S = 

(κKk)      (−1) y γSM S Q M Q |X µ |γ S M S Q M Q , where λ and µ have the same significance as in (6.62), and where y, like t of (6.63), depends on the spins and quasispins but not on the associated magnetic quantum numbers. Equation (6.66) leads to a useful special case when M K = Mκ = 0 and the tensors X are converted to

Part A 6.4

The operator R formally interchanges spin and quasispin. The result for the creation and annihilation operators for electrons can be expressed in terms of triple tensors:

120

Part A

Mathematical Methods

those of type W, defined in (6.18). The sum K + κ + k is taken to be odd, with the scalars κ = k = 0 and K = k = 0 excluded. Application of the Wigner-Eckart theorem to the spin and orbital spaces yields  γQ M Q S||W (κk) ||γ  Q  M Q S   = γSM S Q||W (Kk) ||γ  S M S Q    Q K Q −M Q 0 M Q  , (6.67) (−1)z  S κ S −M S 0 M S

where z = y + Q − M Q − S + M S . An equivalent form is   N  γv1 S1 ||W (κk) || N γ  v1 S1   = (6.68)   N γv2 S2 ||W (Kk) || N γ  v2 S2   1 (2 + 1 − v1 ) K 12 (2 + 1 − v1 ) 2   1 1 2 (2 + 1 − N ) 0 2 (N − 2 − 1) , (−1)z  1 (2 + 1 − v2 ) κ 12 (2 + 1 − v2 ) 2   1 1   2 (2 + 1 − N ) 0 2 (N − 2 − 1) where (6.64) is satisfied both for the unprimed and primed quantities.

6.5 Quasiparticles

Part A 6.5

Sets of linear combinations of the creation and annihilation operators for electrons in the  shell can be constructed such that every member of one set anticommutes with a member of a different set. To preserve the tensorial character of these quasiparticle operators with respect to L, it is convenient to define [6.21]   1 † λq† = 2− 2 a 1 + (−1)−q a 1 ,−q , (6.69) 2 2 ,q   1 † µq† = 2− 2 a 1 − (−1)−q a 1 ,−q , (6.70) 2 2 ,q   1 † νq† = 2− 2 a 1 + (−1)−q a− 1 ,−q , (6.71) − 2 ,q 2   1 † ξq† = 2− 2 a 1 − (−1)−q a− 1 ,−q . (6.72) − 2 ,q

2

θ † (≡ λ† , µ† , ν † ,

The four tensors or ξ † ) anticommute with each other; the first two act in the spin-up space, the second two in the spin-down space. The tensors θ, whose components θ˜q are defined as in (6.17) with t =  and m t = q, are related to their adjoints by the equations †

†

λ =λ,

µ = −µ ,

(6.73)

ν† = ν ,

ξ † = −ξ .

(6.74)

Under the action of the complementarity operator R (see (6.61)) [6.22], R λR−1 = λ ,

R µR−1 = µ ,

(6.75)

−1

R ξ R−1 = −ξ .

(6.76)

R νR

=ν,

The tensors λ, µ, and ν, for a given component q, form a vector with respect to S + Q. Every component of ξ is scalar with respect to S + Q [6.23].

The compound quasiparticle operators defined by [6.21] 1
† (6.77) Θq† = 2− 2 θq† , θ0 , where q > 0 and θ ≡ λ, µ, ν, or ξ satisfy the anticommutation relations
† † Θq , Θq  + = 0 , (6.78)
 Θq , Θq  + = 0 , (6.79)
†   Θq , Θq  + = δ(q, q ) , (6.80) †

for q, q  > 0. The Θq with q > 0 can thus be regarded as the creation operators for a fermion quasiparticle with  components. The connection between the creation and annihilation operators for quasiparticles and for quarks (appearing in the last two rows of Table 3.1) is  † (10...0) θ → 2(−1)/2 θ γθ qθ qθ , (6.81) where the γθ are Dirac matrices satisfying γθ γφ + γφ γθ = 2δ(θ, φ) ,

(6.82)

and the θ are phases, to some extent dependent on the definitions (6.69–6.72) [6.24]. The superscript (10 . . . 0) † indicates that q θ and q θ each of which belongs to the elementary spinor ( 12 21 . . . 12 ) of SOθ (2 + 1), are to be coupled to the resultant (10 . . . 0), which matches the group label for θ. In the quark model, the 24+2 states of the atomic  shell are given by †

†

qλ qµ† qν† qξ |0
p p ,

(6.83)

Second Quantization

where p and p are parity labels that distinguish the four reference states |0
corresponding to the evenness and oddness of the number of spin-up and spin-down electrons. The scalar nature of ξ (and hence of q ξ ) with

References

121

respect to S+ Q can be used to derive relations between spin-orbit matrix elements that go beyond those expected from an application of the Wigner–Eckart theorem [6.25].

References 6.1 6.2 6.3 6.4 6.5

6.6

6.7 6.8 6.9

6.10

E. K. U. Gross, E. Runge, O. Heinonen: Many-Particle Theory (Hilger, New York 1991) G. Racah: Phys. Rev. 62, 438 (1942) E. U. Condon, G. H. Shortley: The Theory of Atomic Spectra (Cambridge Univ. Press, New York 1935) A. R. Edmonds: Angular Momentum in Quantum Mechanics (Princeton Univ. Press, Princeton 1957) C. W. Nielson, G. F. Koster: Spectroscopic Coefficients for the pn , dn , and f n Configurations (MIT Press, Cambridge 1963) ˙ Z. Rudzikas, A. P. Jucys: TaR. Karazija, J. Vizbaraite, bles for the Calculation of Matrix Elements of Atomic Operators (Academy Sci. Computing Center, Moscow 1967) B. R. Judd: Second Quantization and Atomic Spectroscopy (Johns Hopkins, Baltimore 1967) G. Racah: Phys. Rev. 63, 367 (1943) V. L. Donlan: Air Force Materials Laboratory Report No. AFML-TR-70-249 (Wright–Patterson Air Force Base, Ohio 1970) D. D. Velkov: Multi-Electron Coefficients of Fractional Parentage for the p, d, and f Shells. Ph.D. Thesis (The Johns Hopkins University, Baltimore 2000) http://www.pha.jhu.edu/groups/cfp/

6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18 6.19 6.20 6.21 6.22 6.23 6.24 6.25

P. J. Redmond: Proc. R. Soc. London A222, 84 (1954) B. H. Flowers, S. Szpikowski: Proc. Phys. Soc. London 84, 673 (1964) Z. Rudzikas: Theoretical Atomic Spectroscopy (Cambridge Univ. Press, New York 1997) G. Gaigalas, Z. Rudzikas, C. Froese Fischer: At. Data Nucl. Data Tables 70, 1 (1998) B. R. Judd, E. Lo, D. Velkov: Mol. Phys. 98, 1151 (2000), Table 4 B. R. Judd: J. Phys. C 14, 375 (1981) J. S. Bell: Nucl. Phys. 12, 117 (1959) H. Watanabe: Prog. Theor. Phys. 32, 106 (1964) R. D. Lawson, M. H. Macfarlane: Nucl. Phys. 66, 80 (1965) G. Racah: Phys. Rev. 76, 1352 (1949), Table I L. Armstrong, B. R. Judd: Proc. R. Soc. London A 315, 27, 39 (1970) B. R. Judd, S. Li: J. Phys. B 22, 2851 (1989) B. R. Judd, G. M. S. Lister, M. A. Suskin: J. Phys. B 19, 1107 (1986) B. R. Judd: Phys. Rep. 285, 1 (1997) B. R. Judd, E. Lo: Phys. Rev. Lett. 85, 948 (2000)

Part A 6

123

Density Matric 7. Density Matrices

The density operator was first introduced by J. von Neumann [7.1] in 1927 and has since been widely used in quantum statistics. Over the past decades, however, the application of density matrices has spread to many other fields of physics. Density matrices have been used to describe, for example, coherence and correlation phenomena, alignment and orientation and their effect on the polarization of emitted radiation, quantum beat spectroscopy, optical pumping, and scattering processes, particularly when spin-polarized projectiles and/or targets are involved. A thorough introduction to the theory of density matrices and their applications with emphasis on atomic physics can be found in the book by Blum [7.2] from which many equations have been extracted for use in this chapter.

7.1

Basic Formulae .................................... 7.1.1 Pure States ............................... 7.1.2 Mixed States ............................. 7.1.3 Expectation Values .................... 7.1.4 The Liouville Equation ............... 7.1.5 Systems in Thermal Equilibrium .. 7.1.6 Relaxation Processes .................

123 123 124 124 124 125 125

Spin and Light Polarizations ................. 125 7.2.1 Spin-Polarized Electrons ............ 125 7.2.2 Light Polarization ...................... 125

7.3

Atomic Collisions ................................. 126 7.3.1 Scattering Amplitudes ................ 126 7.3.2 Reduced Density Matrices ........... 126

7.4

Irreducible Tensor Operators ................. 7.4.1 Definition ................................ 7.4.2 Transformation Properties .......... 7.4.3 Symmetry Properties of State Multipoles ................................ 7.4.4 Orientation and Alignment ......... 7.4.5 Coupled Systems .......................

7.5

Time Evolution of State Multipoles ........ 7.5.1 Perturbation Coefficients............ 7.5.2 Quantum Beats ......................... 7.5.3 Time Integration over Quantum Beats .......................................

127 127 127 128 128 129 129 129 129 130

7.6

Examples ............................................ 130 7.6.1 Generalized STU-parameters ...... 130 7.6.2 Radiation from Excited States: Stokes Parameters ..................... 131

7.7

Summary ............................................ 133

References .................................................. 133

quantum numbers in the final state can be accounted for via the reduced density matrix. Furthermore, expansion of the density matrix in terms of irreducible tensor operators and the corresponding state multipoles allows for the use of advanced angular momentum techniques, as outlined in Chapts. 2, 3 and 12. More details can be found in two recent textbooks [7.3, 4].

7.1 Basic Formulae If |Ψ
is normalized to unity, i.e., if

7.1.1 Pure States Consider a system in a quantum state that is represented by a single wave function |Ψ
. The density operator for this situation is defined as ρ = |Ψ
Ψ | .

(7.1)

Ψ |Ψ
= 1 ,

(7.2)

then ρ2 = ρ .

(7.3)

Part A 7

The main advantage of the density matrix formalism is its ability to deal with pure and mixed states in the same consistent manner. The preparation of the initial state as well as the details regarding the observation of the final state can be treated in a systematic way. In particular, averages over quantum numbers of unpolarized beams in the initial state and incoherent sums over non-observed

7.2

124

Part A

Mathematical Methods

Equation (7.3) is the basic equation for identifying pure quantum mechanical states represented by a density operator. Next, consider the expansion of |Ψ
in terms of a complete orthonormal set of basis functions {|Φn
}, i.e.,
cn |Φn
. (7.4) |Ψ
= n

The density operator then becomes
cn c∗m |Φn
Φm | = ρnm |Φn
Φm | , ρ=

(7.5)

n,m

where the star denotes the complex conjugate quantity. Note that the density matrix elements ρnm = Φn | ρ |Φm
depend on the choice of the basis and that the density matrix is Hermitian, i.e., ∗ = ρnm . ρmn

(7.6)

Finally, if |Ψ
= |Φi
is one of the basis functions, then ρmn = δni δmi ,

(7.7)

where δni is the Kronecker δ. Hence, the density matrix is diagonal in this representation with only one nonvanishing element.

7.1.2 Mixed States The above concepts can be extended to treat statistical ensembles of pure quantum states. In the simplest case, such mixed states can be represented by a diagonal density matrix of the form
wn |Ψn
Ψn | , (7.8) ρ= n

Part A 7.1

where the weight wn is the fraction of systems in the pure quantum state |Ψn
. The standard normalization for the trace of ρ is
Tr{ρ} = wn = 1 . (7.9) n

Since the trace is invariant under unitary transformations of the basis functions, (7.9) also holds if the |Ψn
states themselves are expanded in terms of basis functions as in (7.4). For a pure state and the normalization (7.9), one finds in an arbitrary basis   Tr{ρ} = Tr ρ2 = 1 . (7.10)

7.1.3 Expectation Values The density operator contains the maximum available information about a physical system. Consequently, it can be used to calculate expectation values for any operator A that represents a physical observable. In general, A
= Tr{Aρ}/Tr{ρ} ,

(7.11)

where Tr{ρ} in the denominator of (7.11) ensures the correct result even for a normalization that is different from (7.9). The invariance of the trace operation ensures the same result – independent of the particular choice of the basis representation.

7.1.4 The Liouville Equation Suppose (7.8) is valid for a time t = 0. If the functions |Ψn (r, t)
obey the Schrödinger equation, i.e. ∂ i |Ψn (r, t)
= H(t) |Ψn (r, t)
, (7.12) ∂t the density operator at the time t can be written as ρ(t) = U(t) ρ(0) U † (t) .

(7.13)

In (7.13), U(t) is the time evolution operator which relates the wave functions at times t = 0 and t according to |Ψn (r, t)
= U(t) |Ψn (r, 0)
, and

U † (t)

(7.14)

denotes its adjoint. Note that

U(t) = e−iHt ,

(7.15)

if the Hamiltonian H is time-independent. Differentiation of (7.13) with respect to time and inserting (7.14) into the Schrödinger equation (7.12) yields the equation of motion ∂ i ρ(t) = [H(t), ρ(t)] , (7.16) ∂t where [A, B ] denotes a commutator. The Liouville equation (7.16) can be used to determine the density matrix and to treat transitions from nonequilibrium to equilibrium states in quantum mechanical systems. Especially for approximate solutions in the presence of small time-dependent perturbation terms in an otherwise time-independent Hamiltonian, i.e., for H(t) = H0 + V(t) ,

(7.17)

the interaction picture is preferably used. The Liouville equation then becomes   ∂ (7.18) i ρI (t) = VI (t), ρI (t) , ∂t

Density Matrices

where the subscript I denotes the operator in the interaction picture. In first-order perturbation theory, (7.18) can be integrated to yield t ρI (t) = ρI (0) − i

[VI (τ), ρI (0)] dτ ,

(7.19)

0

and higher-order terms can be obtained through subsequent iterations.

7.1.5 Systems in Thermal Equilibrium According to quantum statistics, the density operator for a system which is in thermal equilibrium with a surrounding reservoir R at a temperature T (canonical ensemble), can be expressed as ρ=

exp(−βH ) , Z

(7.20)

where H is the Hamiltonian, and β = 1/kB T with kB being the Boltzmann constant. The partition sum   Z = Tr exp(−βH) , (7.21) ensures the normalization condition (7.9). Expectation values are calculated according to (7.11), and extensions to other types of ensembles are straightforward.

7.1.6 Relaxation Processes Transitions from nonequilibrium to equilibrium states can also be described within the density matrix formalism. One of the basic problems is to account for irreversibility in the energy (and sometimes particle)

7.2 Spin and Light Polarizations

125

exchange between the system of interest, S, and the reservoir, R. This is usually achieved by assuming that the interaction of the system with the reservoir is negligible and, therefore, the density matrix representation for the reservoir at any time t is the same as the representation for t = 0. Another important assumption that is frequently made is the Markov approximation. In this approximation, one assumes that the system “forgets” all knowledge of the past, so that the density matrix elements at the time t + ∆t depend only on the values of these elements, and their first derivatives, at the time t. When (7.19) is put back into (7.18), the result in the Markov approximation can be rewritten as   ∂ ρSI (t) = − i TrR VI (t), ρSI (0)ρR (0) ∂t t   − dτTrR VI (t),[VI (τ),ρSI (t)ρR (0)] , 0

(7.22)

where TrR denotes the trace with regard to all variables of the reservoir. Note that the integral over dτ contains the system density matrix in the interaction picture, ρSI , at the time t, rather than at all times τ which are integrated over (the Markov approximation), and that the density matrix for the reservoir is taken as ρR (0) at all times. For more details, see Chapter 7 of Blum [7.2] and references therein. Equations such as (7.22) are the basis for the master or rate equation approach used, for example, in quantum optics for the theory of lasers and the coupling of atoms to cavity modes. For more details, see Chapts. 68, 69, 70 and 78.

7.2 Spin and Light Polarizations where N↑ (N↓ ) is the number of electrons with spin up (down) with regard to this axis. An arbitrary polarization state is described by the density matrix   1 1 + Pz Px − iPy ρ= (7.24) , 2 Px + iPy 1 − Pz

7.2.1 Spin-Polarized Electrons

where Px,y,x are the cartesian components of the spin polarization vector. The individual components can be obtained from the density matrix as

The spin polarization of an electron beam with respect to a given quantization axis nˆ is defined as [7.5] N↑ − N↓ Pnˆ = , N↑ + N↓

(7.23)

Pi = Tr{σi ρ} ,

(7.25)

where the σi (i = x, y, z) are the standard Pauli spin matrices.

Part A 7.2

Density matrices are frequently used to describe the polarization state of spin-polarized particle beams as well as light. The latter can either be emitted from excited atomic or molecular ensembles or can be used, for example, for laser pumping purposes.

126

Part A

Mathematical Methods

7.2.2 Light Polarization

where Itot is the total light intensity. Other frequently used names for the various Stokes parameters are

Another important use of the density matrix formalism is the description of light polarization in terms of the socalled Stokes parameters [7.6]. For a given direction of observation, the general polarization state of light can be fully determined by the measurement of one circular and two independent linear polarizations. Using the notation of Born and Wolf [7.7], the density matrix is given by   Itot 1 − P3 P1 − iP2 (7.26) , ρ= 2 P1 + iP2 1 + P3 where P1 and P2 are linear light polarizations while P3 is the circular polarization (see also Sect. 7.6). In (7.26), the density matrix is normalized in such as way that Tr{ρ} = Itot ,

(7.27)

P1 = η3 = M , P2 = η1 = C , P3 = − η2 = S .

(7.28) (7.29) (7.30)

The Stokes parameters of electric dipole radiation can be related directly to the charge distribution of the emitting atomic ensemble. As discussed in detail in Chapt. 46, one finds, for example, L ⊥ = −P3

(7.31)

for the angular momentum transfer perpendicular to the scattering plane in collisional (de-)excitation, and 1 (7.32) γ = arg{P1 + iP2 } 2 for the alignment angle.

7.3 Atomic Collisions 7.3.1 Scattering Amplitudes Transitions from an initial state |J0 M0 ; k0 m 0
to a final state |J1 M1 ; k1 m 1
are described by scattering amplitudes

where ρin is the density operator before the collision. The corresponding matrix elements are given by
k1 ,M  M ρm 0 m 0 ρ M0 M0 (ρout )m  m 1 1 = 1

1

m 0 m 0 M0 M0

× f M1 m 1 ; M0 m 0

× f ∗ M1 m 1 ; M0 m 0 ,

f(M1 m 1 ; M0 m 0 ) = J1 M1 ; k1 m 1 |T |J0 M0 ; k0 m 0
, (7.33)

where T is the transition operator. Furthermore, J0 (J1 ) is the total electronic angular momentum in the initial (final) state of the target and M0 (M1 ) its corresponding z-component, while k0 (k1 ) is the initial (final) momentum of the projectile and m 0 (m 1 ) its spin component.

1

Part A 7.3

7.3.2 Reduced Density Matrices While the scattering amplitudes are the central elements in a theoretical description, some restrictions usually need to be taken into account in a practical experiment. The most important ones are: (i) there is no “pure” initial state, and (ii) not all possible quantum numbers are simultaneously determined in the final state. The solution to this problem can be found by using the density matrix formalism. First, the complete density operator after the collision process is given by [7.2] ρout = T ρin T † ,

(7.34)

(7.35)

where the term ρm 0 m 0 ρ M0 M0 describes the preparation of the initial state (i). Secondly, “reduced” density matrices account for (ii). For example, if only the scattered projectiles are observed, the corresponding elements of the reduced density matrix are obtained by summing over the atomic quantum numbers as follows:
1 M1 . (7.36) (ρout )km1 m = (ρout )km1,M m 1

M1

1

1

The differential cross section for unpolarized projectile and target beams is given by
dσ =C (7.37) (ρout )km11 m 1 , dΩ m 1

where C is a constant that depends on the normalization of the continuum waves in a numerical calculation. On the other hand, if only the atoms are observed (for example, by analyzing the light emitted in optical transitions), the elements 
k1 ,M  M (7.38) (ρout ) M  M = d3 k1 (ρout )m 1 m 11 1 1

1

m1

Density Matrices

determine the integrated Stokes parameters [7.8, 9], i.e., the polarization of the emitted light. They contain information about the angular momentum distribution in the excited target ensemble. Finally, for electron–photon coincidence experiments without spin analysis in the final state, the elements
k1 ,M  M (7.39) (ρout )kM1  M = (ρout )m 1 m 11 1 1

1

m1

simultaneously contain information about the projectiles and the target. This information can be extracted

7.4 Irreducible Tensor Operators

127

by measuring the angle-differential Stokes parameters. In particular, for unpolarized electrons and atoms, the “natural coordinate system”, where the quantization axis coincides with the normal to the scattering plane, allows for a simple physical interpretation of the various parameters [7.10] (see Chapt. 46). The density matrix formalism outlined above is very useful for obtaining a qualitative description of the geometrical and sometimes also of the dynamical symmetries of the collision process [7.11]. Two explicit examples are discussed in Sect. 7.6.

7.4 Irreducible Tensor Operators The general density matrix theory can be formulated in a very elegant fashion by decomposing the density operator in terms of irreducible components whose matrix elements then become the state multipoles. In such a formulation, full advantage can be taken of the most sophisticated techniques developed in angular momentum algebra (see Chapt. 2). Many explicit examples can be found in [7.3, 4].

and the state multipoles or statistical tensors are given by 


†  √ (−1) J −M 2K + 1 T J J KQ = M M



×

K J J M  −M −Q

 J  M  |ρ|JM
. (7.44)

7.4.1 Definition The density operator for an ensemble of particles in quantum states labeled as |JM
where J and M are the total angular momentum and its magnetic component, respectively, can be written as
 J J   ρ= ρM (7.40)  M |J M
JM| , J  JM  M

where 

J J   ρM  M = J M |ρ|JM


(7.41)

|J − J  | ≤ K ≤ J + J  , M − M = Q .

(7.45) (7.46)

Equation (7.44) can be inverted through the orthogonality condition of the 3–j symbols to give J  M  |ρ|JM
=


 √ (−1) J −M 2K + 1 KQ



×

K J J M  −M −Q



(7.47)

J  JK Q

where the irreducible tensor operators are defined in terms of 3–j symbols as


 √ T J J KQ = (−1) J −M 2K + 1 M M



×

J J K M  −M −Q

 |J  M 
JM| , (7.43)



† T J J KQ .

7.4.2 Transformation Properties Suppose a coordinate system (X 2 , Y2 , Z 2 ) is obtained from another coordinate system (X 1 , Y1 , Z 1 ) through a rotation by a set of three Euler angles (γ, β, α) as defined in Edmonds [7.12]. The irreducible tensor operators (7.43) defined in the (X 1 , Y1 , Z 1 ) system are then † related to the operators T(J  J ) K Q
in the (X 2 , Y2 , Z 2 )

Part A 7.4

are the matrix elements. (For simplicity, interactions outside the single manifold of momentum states |JM
are neglected). Alternatively, one may write


†

ρ= T J J KQ T J J KQ , (7.42)

Hence, the selection rules for the 3–j symbols imply that

128

Part A

Mathematical Methods

system by




K T J J KQ = T J  J Kq D(γ, β, α)qQ , (7.48) q

where 

J iM γ J iMα d(β) M D(γ, β, α) M M = e M e

(7.49)

is a rotation matrix (see Chapt. 2). Note that the rank K of the tensor operator is invariant under such rotations. Similarly, 

†
  † K ∗ T J J Kq D(γ, β, α)qQ T J J KQ = q

(7.50)

Furthermore, the transformation property (7.50) of the state multipoles imposes restrictions on nonvanishing state multipoles to describe systems with given symmetry properties. In detail, one finds: 1. For spherically symmetric systems, 

† 

† T J J KQ = T J J KQ rot

(7.57)

for all sets of Euler angles. This implies that only the  †  monopole term T(J )00 can be different from zero. 2. For axially symmetric systems, 

† 

† T J J KQ = T J J KQ

rot

(7.58)

holds for the state multipoles. The irreducible tensor operators fulfill the orthogonality condition





†  Tr T J  J K Q T J  J K  Q  = δ K  K δ Q  Q , (7.51)

for all Euler angles φ that describe a rotation around the z-axis. Since this angle enters via a factor exp(−iQφ) into the general transformation formula (7.50),  it follows

†  that only state multipoles with Q = 0, i.e., T J  J K 0 , can be different from zero in such a situation.

with

3. For planar symmetric systems with fixed J  = J,   ∗ † † T(J ) K Q = (−1) K T(J ) K Q (7.59)







1

δJ J 1 =√ (7.52) 2J + 1 being proportional to the unit operator 1, it follows that all tensor operators have vanishing trace, except for the monopole T(J  J )00 . Reduced tensor operators fulfill the Wigner–Eckart theorem (see Sect. 2.8.4) 

J  M  |T J  J K Q |JM   J K J J  −M  = (−1) −M  Q M    × J TK J , (7.53) T J J

00

where the reduced matrix element is simply given by    1 J TK J = √ . (7.54) 2K + 1

Part A 7.4

7.4.3 Symmetry Properties of State Multipoles The Hermiticity condition for the density matrix implies  

† ∗  † T J  J K Q = (−1) J −J+Q T(JJ  ) K −Q , (7.55)

J

which, for sharp angular momentum = J, yields ∗   † † (7.56) T(J ) K Q = (−1) Q T(J) K −Q .   † Hence, the state multipoles T(J ) K 0 are real numbers.

if the system properties are invariant under reflection in the xz-plane. Hence, state multipoles with even rank K are real numbers, while those with odd rank are purely imaginary in this case. The above results can be applied immediately to the description of atomic collisions where the incident beam axis is the quantization axis (the so-called “collision system”). For example, impact excitation of unpolarized targets by unpolarized projectiles without observation of the scattered projectiles is symmetric both with regard to rotation around the incident beam axis and with regard to reflection in any plane containing this axis.  †   †  Consequently, the state multipoles T(J ) , T(J )20 , 00   † T(J )40 , . . . fully characterize the atomic ensemble of interest. Using (7.50), similar relationships can be derived for state multipoles defined with regard to other coordinate systems, such as the “natural system” where the quantization axis coincides with the normal vector to the scattering plane (see Chapt. 46).

7.4.4 Orientation and Alignment From the above discussion, it is apparent that the description of systems that do not exhibit spherical symmetry requires the knowledge of state multipoles with rank K = 0. Frequently, the multipoles with K = 1 and K = 2 are determined via the angular correlation and the polarization of radiation emitted from

Density Matrices

an ensemble of collisionally excited targets. The state multipoles with K = 1 are proportional to the spherical components of the angular momentum expectation value and, therefore, give rise to a nonvanishing circular light polarization (see also Sect. 7.6). This corresponds to a sense of rotation or an orientation in the ensemble which is therefore called oriented (see Sect. 46.1). On the other hand, nonvanishing multipoles with rank K = 2 describe the alignment of the system. Some authors, however, use the terms “alignment” or “orientation” synonymously for all nonvanishing state multipoles with ranks K = 0, thereby describing any system with anisotropic occupation of magnetic sublevels as “aligned” or “oriented”. For details on alignment and orientation, see Chapt. 46 and [7.3, 4].

129

density operator for two subsystems in basis states |L, M L
and |S, M S
is constructed as [7.2]
  † †  T(L) K Q ⊗ T(S)kq T(L) K Q ⊗ T(S)kq . ρ= K Qkq

(7.60)

If the two systems are uncorrelated, the state multipoles factor as    † †  † † T(L) K Q ⊗ T(S)kq = T(L) K Q T(S)kq ; (7.61) More generally, irreducible representations of coupled operators can be defined in terms of a 9–j symbol as




T J  , J K  Q = Kˆ kˆ Jˆ Jˆ K Q, kq|K  Q  K Qkq

    K k K   × L S J  T(L) K Q ⊗ T(S)kq ,     L S J

7.4.5 Coupled Systems Tensor operators and state multipoles for coupled systems are constructed as direct products (⊗) of the operators for the individual systems. For example, the

7.5 Time Evolution of State Multipoles

(7.62)

√ where xˆ ≡ 2x + 1, and ( j1 m 1 , j2 m 2 | j3 m 3 ) is a standard Clebsch-Gordan coefficient.

7.5 Time Evolution of State Multipoles 7.5.2 Quantum Beats

From the general expansions

An important application of the perturbation coefficients is the coherent excitation of several quantum states which subsequently decay by optical transitions. Such an excitation may be performed, for example, in beamfoil experiments or electron–atom collisions where the energy width of the electron beam is too large to resolve the fine structure (or hyperfine structure) of the target states. Suppose, for instance, that explicitly relativistic effects, such as the spin–orbit interaction between the projectile and the target, can be neglected during a collision process between an incident electron and a target atom. In that case, the orbital angular momentum (L) system of the collisionally excited target states may be oriented, depending on the scattering angle of the projectile. On the other hand, the spin (S) system remains unaffected (unpolarized), provided that both the target and the projectile beams are unpolarized. During the lifetime of the excited target states, however, the spin– orbit interaction within the target produces an exchange of orientation between the L and the S systems, which results in a net loss of orientation in the L system.

ρ(t) =




†

T j  j; t kq T j  j kq

(7.63)

j  jkq

in terms of irreducible components, together with (7.42) for time t = 0 and (7.13) for the time development of the density operator, it follows that 

†
 

† T j  j; t kq = T J J; 0 K Q J  JK Q



Qq × G J  J, j  j; t Kk ,

(7.64)

where the perturbation coefficients are defined as

Qq G J  J, j  j; t Kk 



† = Tr U(t)T J  J K Q U(t)† T( j  j)kq .

(7.65)

Hence, these coefficients relate the state multipoles at time t to those at t = 0.

Part A 7.5

7.5.1 Perturbation Coefficients

130

Part A

Mathematical Methods

This effect can be observed directly through the intensity and the polarization of the light emitted from the excited target ensemble. The perturbation coefficients for the fine structure interaction are found to be [7.2, 13]



exp(−γt)
 G(L; t) K = 2J + 1 2J + 1 2S + 1  J J 2  

L J S cos ω J  − ω J t , × J L K (7.66)

where ω J  − ω J corresponds to the (angular) frequency difference between the various multiplet states with total electronic angular momenta J  and J, respectively. Also, γ is the natural width of the spectral line; for simplicity, the same lifetime has been assumed in (7.66) for all states of the multiplet. Note that the perturbation coefficients are independent of the multipole component Q in this case, and that there is no mixing between different multipole ranks K . Similar results can be derived [7.2, 13] for the hyperfine interaction and also to account for the combined effect of fine and hyperfine structure. The cosine terms represent correlation between the signal from different fine structure states, and they lead to oscillations in the intensity as well as the measured Stokes parameters in a time-resolved experiment. Finally, generalized perturbation coefficients have been derived for the case where both the L and the S systems may be oriented and/or aligned during the collision process [7.14]. This can happen when

spin-polarized projectiles and/or target beams are prepared.

7.5.3 Time Integration over Quantum Beats If the excitation and decay times cannot be resolved in a given experimental setup, the perturbation coefficients need to be integrated over time. As a result, the quantum beats disappear, but a net effect may still be visible through a depolarization of the emitted radiation. For the case of atomic fine structure interaction discussed above, one finds [7.2, 13] ¯ G(L) K =

∞ G(L; t) K dt 0

=



1
 2J + 1 2J + 1 2S + 1  J J 2  γ L J S , × γ 2 + ω2J  J J L K

(7.67)

where ω J  J = ω J  − ω J . Note that the amount of depolarization depends on the relationship between the fine structure splitting and the natural line width. For |ω J  J |  γ (if J  = J), the terms with J  = J dominate and cause the maximum depolarization; for the opposite case |ω J  J |  γ , the sum rule for the 6–j symbols can be applied and no depolarization is observed. Similar depolarizations can be caused through hyperfine structure effects, as well as through external fields. An important example of the latter case is the Hanle effect (see Sect. 17.2.1).

7.6 Examples Part A 7.6

In this section, two examples of the reduced density formalism are discussed explicitly. These are: (i) the change of the spin polarization of initially polarized spin- 12 projectiles after scattering from unpolarized targets, and (ii) the Stokes parameters describing the angular distribution and the polarization of light as detected in projectile-photon coincidence experiments after collisional excitation. The recent book by Andersen and Bartschat [7.4] provides a detailed introduction to these topics, together with a thorough discussion of benchmark studies in the field of electronic and atomic collisions, including extensions to ionization processes, as well as applications in plasma, surface, and nuclear physics.

Even more extensive compilations of such studies can be found in a review series dealing with unpolarized electrons colliding with unpolarized targets [7.10], heavy-particle collisions [7.15], and the special role of projectile and target spins in such collisions [7.16].

7.6.1 Generalized STU-parameters For spin-polarized projectile scattering from unpolarized targets, the generalized STU-parameters [7.11] contain information about the projectile spin polarization after the collision. These parameters can be expressed in terms of the elements (7.36).

Density Matrices

To analyze this problem explicitly, one defines the quantities     m1m0; m1m0 =




1 f M1 m 1 ; M0 m 0 2J0 + 1 M1 M0

∗ (7.68) × f M1 m 1 ; M0 m 0

which contain the maximum information that can be obtained from the scattering process, if only the polarization of the projectiles is prepared before the collision and measured thereafter. Next, the number of independent parameters that can be determined in such an experiment needs to be examined. For spin- 12 particles, there are 2 × 2 × 2 × 2 = 16 possible combinations of {m 1 m 0 ; m 1 m 0 } and, therefore, 16 complex or 32 real parameters (in the most general case of spin-S particles, there would be (2S + 1)4 combinations). However, from the definition (7.68) and the Hermiticity of the reduced density matrix contained therein, it follows that      ∗ m 1 m 0 ; m 1 m 0 = m 1 m 0 ; m 1 m 0 . (7.69) Furthermore, parity conservation of the interaction or the equivalent reflection invariance with regard to the scattering plane yields the additional relationship [7.11]

1 1 f M1 m 1 ; M0 m 0 = (−1) J1 −M1 + 2 −m 1 +J0 −M0 + 2 −m 0

× Π1 Π0 f − M1 − m 1 ;−M0 − m 0 , (7.70)

where Π1 and Π0 are ±1, depending on the parities of the atomic states involved. Hence,       m 1 m 0 ; m 1 m 0 = (−1)m 1 −m 1 +m 0 −m 0   × − m 1 − m 0 ; −m 1 − m 0 . (7.71)

1

0

131

for the scattering of unpolarized projectiles from unpolarized targets and the seven relative parameters   2 1 1 11 − ; , Im (7.73) SA = − σu 2 2 22   11 2 11 ;− , SP = − Im (7.74) σu 22 22     1 1 11 11 1 1 1 − − ; − − ; − , Ty = σu 2 2 22 22 2 2    (7.75) 1 1 11 11 1 1 1 − − ; + − ; − , Tx = σu 2 2 22 22 2 2 (7.76)     11 11 1 11 11 ; − ;− , Tz = (7.77) σu 2 2 2 2 22 22   11 2 11 ;− , Uxz = Re (7.78) σu 22 22   2 1 1 11 − ; Uzx = − Re (7.79) , σu 2 2 22 where Re{x} and Im{x} denote the real and imaginary parts of the complex quantity x, respectively. Note that normalization constants have been omitted in (7.72) to simplify the notation. Therefore, the most general form for the polarization vector after scattering, P  , for an initial polarization vector P = (Px , Py , Pz ) is given by





SP +Ty Py yˆ + Tx Px +Uxz Pz xˆ + Tz Pz −Uzx Px zˆ . 1 + SA Py (7.80)

The physical meaning of the above relation is illustrated in Fig. 7.1. The following geometries are particularly suitable for the experimental determination of the individual parameters; σu and SP can be measured with unpolarized incident projectiles. A transverse polarization compon

ent perpendicular to the scattering plane P = Py yˆ is needed to obtain SA and Ty . Finally, the measurement

of x and Tx , Uzx , Tz , and U requires both transverse P ˆ xz x

longitudinal Pz zˆ projectile polarization components in the scattering plane.

7.6.2 Radiation from Excited States: Stokes Parameters The state multipole description is also widely used for the parametrization of the Stokes parameters that describe the polarization of light emitted in optical decays of excited atomic ensembles. The general case of excitation by spin-polarized projectiles has been treated by

Part A 7.6

Note that (7.70, 71) hold for the collision frame where the quantization axis (ˆz ) is taken as the incident beam axis and the scattering plane is the xz-plane. Similar formulas can be derived for the natural frame (see Sect. 7.3.2) Consequently, eight independent parameters are sufficient to characterize the reduced spin density matrix of the scattered projectiles. These can be chosen as the absolute differential cross section  1
 m1m0; m1m0 (7.72) σu = 2 m ,m

7.6 Examples

132

Part A

Mathematical Methods

y

y x

Py

P

σu (1 + SA Py)

hv

x

P⬘ SP + Ty + Py 1 + SA + Py

Pz Px k0

k1 θ Tz Pz – UzxPz 1 + SA Py

z

Φy

Tx Px + UxzPz 1 + SA Py

Θy

e–, k1

θ

–

e , k0

z

Fig. 7.1 Physical meaning of the generalized STU-

parameters: the polarization function SP gives the polarization of an initially unpolarized projectile beam after the collision while the asymmetry function SA determines a left-right asymmetry in the differential cross section for scattering of a spin-polarized beam. Furthermore, the contraction parameters (Tx , Ty , Tz ) describe the change of an initial polarization component along the three cartesian axes while the parameters Uxz and Uzx determine the rotation of a polarization component in the scattering plane

Part A 7.6

Bartschat and collaborators [7.8]. The basic experimental setup for electron-photon coincidence experiments and the definition of the Stokes parameters are illustrated in Figs. 7.2 and 7.3. For impact excitation of an atomic state with total electronic angular momentum J and an electric dipole transition to a state with J f , the photon intensity in a direction nˆ = (Θγ , Φγ ) is given by  2 (−1) J−J f  † T(J )00 I(Θγ , Φγ ) = C √ 3 2J + 1   1 1 2 − J J Jf 

 † × Re T(J )22 sin2 Θγ cos 2Φγ 

 † − Re T(J )21 sin 2Θγ cos Φγ !  1 † T(J )20 (3 cos2 Θγ − 1) + 6   †

T(J )22 sin2 Θγ sin 2Φγ  "#

 † , + Im T(J )21 sin 2Θγ sin Φγ

− Im

(7.81)

where C=

$2 e2 ω4 $$ J f r J
$ (−1) J−J f 2πc3

(7.82)

Fig. 7.2 Geometry of electron–photon coincidence experi-

ments z Photon detector

nˆ eˆ2 Θy n eˆ1

Φy

y

x

Fig. 7.3 Definition of the Stokes parameters: Photons are

observed in a direction nˆ with polar angles (Θγ , Φγ ) in the collision system. The three unit vectors (n, ˆ eˆ 1 , eˆ 2 ) define the helicity system of the photons, eˆ 1 = (Θγ + 90◦ , Φγ ) lies in the plane spanned by nˆ and zˆ and is perpendicular to nˆ while eˆ 2 = (Θγ , Φγ + 90◦ ) is perpendicular to both nˆ and eˆ 1 . In addition to the circular polarization P3 , the linear polarizations P1 and P2 are defined with respect to axes in the plane spanned by eˆ 1 and eˆ 2 . Counting from the direction of eˆ 1 , the axes are located at (0◦ , 90◦ ) for P1 and at (45◦ , 135◦ ) for P2 , respectively

is a constant containing the frequency ω of the transition as well as the reduced radial dipole matrix element. Similarly, the product of the intensity I and the circular light polarization P3 can be written in terms of

Density Matrices

state multipoles as



I · P3 Θγ , Φγ = − C





1 1 1 J J Jf %   † × Im T(J )11 2sin Θγ sinΦγ 

 † − Re T(J )11 2sin Θγ cosΦγ & √  † + 2 T(J )10 cos Θγ , (7.83)

so that P3 can be calculated as







P3 Θγ , Φγ = I · P3 Θγ , Φγ /I Θγ , Φγ . (7.84)

Note that each state multipole gives rise to a characteristic angular dependence in the formulas for the Stokes parameters, and that perturbation coefficients may need to be applied to deal, for example, with depolarization

References

133

effects due to internal or external fields. General formulas for P1 = η3 and P2 = η1 can be found in [7.8] and, for both the natural and the collision systems, in [7.4]. As pointed out before, some of the state multipoles may vanish, depending on the experimental arrangement. A detailed analysis of the information contained in the state multipoles and the generalized Stokes parameters (which are defined for specific values of the projectile spin polarization) has been given by Andersen and Bartschat [7.4, 17, 18]. They re-analyzed the experiment performed by Sohn and Hanne [7.19] and showed how the density matrix of the excited atomic ensemble can be determined by a measurement of the generalized Stokes parameters. In some cases, this will allow for the extraction of a complete set of scattering amplitudes for the collision process. Such a “perfect scattering experiment” has been called for by Bederson many years ago [7.20] and is now within reach even for fairly complex excitation processes. The most promising cases have been discussed by Andersen and Bartschat [7.4, 17, 21].

7.7 Summary The basic formulas dealing with density matrices in quantum mechanics, with particular emphasis on reduced matrix theory and its applications in atomic physics, have been summarized. More

details are given in the introductory textbooks by Blum [7.2], Balashov et al. [7.3], Andersen and Bartschat [7.4], and the references listed below.

References 7.1 7.2 7.3

7.5 7.6 7.7 7.8 7.9

7.10 7.11 7.12 7.13 7.14 7.15 7.16 7.17 7.18 7.19 7.20 7.21

N. Andersen, J. W. Gallagher, I. V. Hertel: Phys. Rep. 165, 1 (1988) K. Bartschat: Phys. Rep. 180, 1 (1989) A. R. Edmonds: Angular Momentum in Quantum Mechanics (Princeton Univ. Press, Princeton 1957) U. Fano, J. H. Macek: Rev. Mod. Phys. 45, 553 (1973) K. Bartschat, H. J. Andrä, K. Blum: Z. Phys. A 314, 257 (1983) N. Andersen, J. T. Broad, E. E. Campbell, J. W. Gallagher, I. V. Hertel: Phys. Rep. 278, 107 (1997) N. Andersen, K. Bartschat, J. T. Broad, I. V. Hertel: Phys. Rep. 279, 251 (1997) N. Andersen, K. Bartschat: Adv. At. Mol. Phys. 36, 1 (1996) N. Andersen, K. Bartschat: J. Phys. B 27, 3189 (1994); corrigendum: J. Phys. B 29, 1149 (1996) M. Sohn, G. F. Hanne: J. Phys. B 25, 4627 (1992) B. Bederson: Comments At. Mol. Phys. 1, 41,65 (1969) N. Andersen, K. Bartschat: J. Phys. B 30, 5071 (1997)

Part A 7

7.4

J. von Neumann: Göttinger Nachr. 245 (1927) K. Blum: Density Matrix Theory and Applications (Plenum, New York 1981) V. V. Balashov, A. N. Grum–Grzhimailo, N. M. Kabachnik: Polarization and Correlation Phenomena in Atomic Collisions. A Practical Theory Course (Plenum, New York 2000) N. Andersen, K. Bartschat: Polarization, Alignment, and Orientation in Atomic Collisions (Springer, New York 2001) J. Kessler: Polarized Electrons (Springer, New York 1985) W. E. Baylis, J. Bonenfant, J. Derbyshire, J. Huschilt: Am. J. Phys. 61, 534 (1993) M. Born, E. Wolf: Principles of Optics (Pergamon, New York 1970) K. Bartschat, K. Blum, G. F. Hanne, J. Kessler: J. Phys. B 14, 3761 (1981) K. Bartschat, K. Blum: Z. Phys. A 304, 85 (1982)

135

Computationa 8. Computational Techniques

Essential to all fields of physics is the ability to perform numerical computations accurately and efficiently. Whether the specific approach involves perturbation theory, close coupling expansion, solution of classical equations of motion, or fitting and smoothing of data, basic computational techniques such as integration, differentiation, interpolation, matrix and eigenvalue manipulation, Monte Carlo sampling, and solution of differential equations must be among the standard tool kit. This chapter outlines a portion of this tool kit with the aim of giving guidance and organization to a wide array of computational techniques. Having digested the present overview, the reader is then referred to detailed treatments given in many of the large number of texts existing on numerical analysis and computational techniques [8.1–5], and mathematical physics [8.6–10]. We also summarize, especially in the sections on differential equations and computational linear algebra, the role of software

8.1

Representation of Functions................. 8.1.1 Interpolation ............................ 8.1.2 Fitting ..................................... 8.1.3 Fourier Analysis ........................ 8.1.4 Approximating Integrals ............ 8.1.5 Approximating Derivatives .........

135 135 137 139 139 140

8.2

Differential and Integral Equations ....... 8.2.1 Ordinary Differential Equations ... 8.2.2 Differencing Algorithms for Partial Differential Equations . 8.2.3 Variational Methods .................. 8.2.4 Finite Elements ......................... 8.2.5 Integral Equations.....................

141 141 143 144 144 146

8.3

Computational Linear Algebra .............. 148

8.4

Monte Carlo Methods ........................... 8.4.1 Random Numbers ..................... 8.4.2 Distributions of Random Numbers 8.4.3 Monte Carlo Integration .............

149 149 150 151

References .................................................. 151 packages readily available to aid in implementing practical solutions.

8.1 Representation of Functions The ability to represent functions in terms of polynomials or other basic functions is the key to interpolating or fitting data, and to approximating numerically the operations of integration and differentiation. In addition, using methods such as Fourier analysis, knowledge of the properties of functions beyond even their intermediate values, derivatives, and antiderivatives may be determined (e.g., the “spectral” properties).

Given the value of a function f(x) at a set of points x1 , x2 , . . . , xn , the function is often required at some other values between these abscissae. The process known as interpolation seeks to estimate these unknown values by adjusting the parameters of a known

k=0

Part A 8

8.1.1 Interpolation

function to approximate the local or global behavior of f(x). One of the most useful representations of a function for these purposes utilizes the algebraic polynomials, Pn (x) = a0 + a1 x + · · · + an x n , where the coefficients are real constants and the exponents are nonnegative integers. The utility stems from the fact that given any continuous function defined on a closed interval, there exists an algebraic polynomial which is as close to that function as desired (Weierstrass Theorem). One simple application of these polynomials is the power series expansion of the function f(x) about some point, x0 , i. e., ∞
ak (x − x0 )k . (8.1) f(x) =

136

Part A

Mathematical Methods

A familiar example is the Taylor expansion in which the coefficients are given by ak =

f (k) (x0 ) , k!

(8.2)

where f (k) indicates the kth derivative of the function. This form, though quite useful in the derivation of formal techniques, is not very useful for interpolation since it assumes the function and its derivatives are known, and since it is guaranteed to be a good approximation only very near the point x0 about which the expansion has been made. Lagrange Interpolation The polynomial of degree n − 1 which passes through all n points [x1 , f(x1 )], [x2 , f(x2 )], . . . , [xn , f(xn )] is given by

P(x) =

n
k=1

=

n


f(xk )

n  i=1,i=k

x − xi x k − xi

f(xk )L nk (x) ,

(8.3)

(8.4)

k=1

where L nk (x) are the Lagrange interpolating polynomials. Perhaps the most familiar example is that of linear interpolation between the points [x1 , y1 ≡ f(x1 )] and [x2 , y2 ≡ f(x2 )], namely, P(x) =

x − x2 x − x1 y1 + y2 . x1 − x2 x2 − x1

(8.5)

Part A 8.1

In practice, it is difficult to estimate the formal error bound for this method, since it depends on knowledge of the (n + 1)th derivative. Alternatively, one uses iterated interpolation in which successively higher order approximations are tried until appropriate agreement is obtained. Neville’s algorithm defines a recursive procedure to yield an arbitrary order interpolant from polynomials of lower order. This method, and subtle refinements of it, form the basis for most “recommended” polynomial interpolation schemes [8.3]. One important caution to bear in mind is that the more points that are used in constructing the interpolant, and therefore the higher the polynomial order, the greater will be the oscillation in the interpolating function. This highly oscillating polynomial most likely will not correspond more closely to the desired function than polynomials of lower order, and, as a general rule of thumb, fewer than six points should be used.

Cubic Splines By dividing the interval of interest into a number of subintervals and in each using a polynomial of only modest order, one may avoid the oscillatory nature of high-order (many-point) interpolants. This approach utilizes piecewise polynomial functions, the simplest of which is just a linear segment. However, such a straight line approximation has a discontinuous derivative at the data points – a property that one may wish to avoid especially if the derivative of the function is also desired – and which clearly does not provide a smooth interpolant. The solution is therefore to choose the polynomial of lowest order that has enough free parameters (the constants a0 , a1 , . . . ) to satisfy the constraints that the function and its derivative are continuous across the subintervals, as well as specifying the derivative at the endpoints x0 and xn . Piecewise cubic polynomials satisfy these constraints, and have a continuous second derivative as well. Cubic spline interpolation does not, however, guarantee that the derivatives of the interpolant agree with those of the function at the data points, much less globally. The cubic polynomial in each interval has four undertermined coeffitients,

Pi (x) = ai + bi (x − xi ) + ci (x − xi )2 + di (x − xi )3 (8.6)

for i = 0, 1, . . . , n − 1. Applying the constraints, a system of equations is found which may be solved once the endpoint derivatives are specified. If the second derivatives at the endpoints are set to zero, then the result is termed a natural spline and its shape is like that which a long flexible rod would take if forced to pass through all the data points. A clamped spline results if the first derivatives are specified at the endpoints, and is usually a better approximation since it incorporates more information about the function (if one has a reasonable way to determine or approximate these first derivatives). The set of equations in the unknowns, along with the boundary conditions, constitute a tridiagonal system or matrix, and is therefore amenable to solution by algorithms designed for speed and efficiency for such systems (see Sect. 8.3; [8.1–3]). Other alternatives of potentially significant utility are schemes based on the use of rational functions and orthogonal polynomials. Rational Function Interpolation If the function which one seeks to interpolate has one or more poles for real x, then polynomial approximations are not good, and a better method is to use quotients of polynomials, so-called rational functions. This occurs

Computational Techniques

since the inverse powers of the dependent variable will fit the region near the pole better if the order is large enough. In fact, if the function is free of poles on the real axis but its analytic continuation in the complex plane has poles, the polynomial approximation may also be poor. It is this property that slows or prevents the convergence of power series. Numerical algorithms very similar to those used to generate iterated polynomial interpolants exist [8.1,3] and can be useful for functions which are not amenable to polynomial interpolation. Rational function interpolation is related to the method of Padé approximation used to improve convergence of power series, and which is a rational function analog of Taylor expansion. Orthogonal Function Interpolation Interpolation using functions other than the algebraic polynomials can be defined and are often useful. Particularly worthy of mention are schemes based on orthogonal polynomials since they play a central role in numerical quadrature. A set of functions φ1 (x), φ2 (x), . . . , φn (x) defined on the interval [a, b] is said to be orthogonal with respect to a weight function W(x) if the inner product defined by

  φi |φ j =

b φi (x)φ j (x)W(x) dx

(8.7)

a

is zero for i = j and positive for i = j. In this case, for any polynomial P(x) of degree at most n, there exists unique constants αk such that P(x) =

n


αk φk (x) .

(8.8)

k=0

Among the more commonly used orthogonal polynomials are Legendre, Laguerre, and Chebyshev polynomials. Chebyshev Interpolation The significant advantages of employing a representation of a function in terms of Chebyshev polynomials, Tk (x) [8.4, 6] for tabulations, recurrence formulas, orthogonality properties, etc. of these polynomials), i. e., ∞


ak Tk (x) ,

(8.9)

k=0

stems from the fact that (i) the expansion rapidly converges, (ii) the polynomials have a simple form, and (iii) the polynomial approximates very closely the solution

137

of the minimax problem. This latter property refers to the requirement that the expansion minimizes the maximum magnitude of the error of the approximation. In particular, the Chebyshev series expansion can be truncated so that for a given n it yields the most accurate approximation to the function. Thus, Chebyshev polynomial interpolation is essentially as “good” as one can hope to do. Since these polynomials are defined on the interval [−1, 1], if the endpoints of the interval in question are a and b, the change of variable y=

x − 12 (b + a) 1 2 (b − a)

(8.10)

will effect the proper transformation. Press et al. [8.3], for example, give convenient and efficient routines for computing the Chebyshev expansion of a function.

8.1.2 Fitting Fitting of data stands in distinction from interpolation in that the data may have some uncertainty, and therefore, simply determining a polynomial which passes through the points may not yield the best approximation of the underlying function. In fitting, one is concerned with minimizing the deviations of some model function from the data points in an optimal or best fit manner. For example, given a set of data points, even a loworder interpolating polynomial might have significant oscillation, when, in fact, if one accounts for the statistical uncertainties in the data, the best fit may be obtained simply by considering the points to lie on a line. In addition, most of the traditional methods of assigning this quality of best fit to a particular set of parameters of the model function rely on the assumption that the random deviations are described by a Gaussian (normal) distribution. Results of physical measurements, for example the counting of events, is often closer to a Poisson distribution which tends (not necessarily uniformly) to a Gaussian in the limit of a large number of events, or may even contain “outliers” which lie far outside a Gaussian distribution. In these cases, fitting methods might significantly distort the parameters of the model function in trying to force these different distributions to the Gaussian form. Thus, the least squares and chi-square fitting procedures discussed below should be used with this caveat in mind. Other techniques, often termed “robust” [8.3, 11], should be used when the distribution is not Gaussian, or replete with outliers.

Part A 8.1

f(x) =

8.1 Representation of Functions

138

Part A

Mathematical Methods

Least Squares In this common approach to fitting, we wish to determine the m parameters al of some function f(x; a1 , a2 , . . . , am ) depending in this example on one variable, x. In particular, we seek to minimize the sum of the squares of the deviations n


[y(xk ) − f(xk ; a1 , a2 , . . . , am )]2

(8.11)

k=1

by adjusting the parameters, where the y(xk ) are the n data points. In the simplest case, the model function is just a straight line, f(x; a1 , a2 ) = a1 x + a2 . Elementary multivariate calculus implies that a minimum occurs if a1

n


xi2 + a2

k=1

a1

n


xi =

k=1 n
k=1

xi + a2 n =

n


xi yi ,

(8.12)

yi ,

(8.13)

k=1 n
k=1

which are called the normal equations. Solution of these equations is straightforward, and an error estimate of the fit can be found [8.3]. In particular, variances may be computed for each parameter, as well as measures of the correlation between uncertainties and an overall estimate of the “goodness of fit” of the data. Chi-square Fitting If the data points each have associated with them a different standard deviation, σk , the least square principle is modified by minimizing the chi-square, defined as  n 
yk − f(xk ; a1 , a2 , . . . , am ) 2 2 . (8.14) χ ≡ σk k=1

Assuming that the uncertainties in the data points are normally distributed, the chi-square value gives a measure of the goodness of fit. If there are n data points and m adjustable parameters, then the probability that χ 2 should exceed a particular value purely by chance is  n − m χ2 , , Q=Q (8.15) 2 2

Part A 8.1

where Q(a, x) = Γ(a, x)/Γ(a) is the incomplete gamma function. For small values of Q, the deviations of the fit from the data are unlikely to be by chance, and values close to one are indications of better fits. In terms of the chi-square, reasonable fits often have χ 2 ≈ n − m. Other important applications of the chi-square method include simulation and estimating standard de-

viations. For example, if one has some idea of the actual (i. e., non-Gaussian) distribution of uncertainties of the data points, Monte Carlo simulation can be used to generate a set of test data points subject to this presumed distribution, and the fitting procedure performed on the simulated data set. This allows one to test the accuracy or applicability of the model function chosen. In other situations, if the uncertainties of the data points are unknown, one can assume that they are all equal to some value, say σ, fit using the chi-square procedure, and solve for the value of σ. Thus, some measure of the uncertainty from this statistical point of view can be provided. General Least Squares The least squares procedure can be generalized usually by allowing any linear combination of basis functions to determine the model function

f(x; a1 , a2 , . . . , am ) =

m


al ψl (x) .

(8.16)

l=1

The basis functions need not be polynomials. Similarly, the formula for chi-square can be generalized, and normal equations determined through minimization. The equations may be written in compact form by defining a matrix A with elements Ai, j =

ψ j (xi ) , σi

and a column vector B with elements yi . Bi = σi

(8.17)

(8.18)

Then the normal equations are [8.3] m


αk j a j = βk ,

(8.19)

j=1

where [α] = AT A ,

[β] = AT B ,

(8.20)

and a j are the adjustable parameters. These equations may be solved using standard methods of computational linear algebra such as Gauss–Jordan elimination. Difficulties involving sensitivity to round-off errors can be avoided by using carefully developed codes to perform this solution [8.3]. We note that elements of the inverse of the matrix α are related to the variances associated with the free parameters and to the covariances relating them.

Computational Techniques

Statistical Analysis of Data Data generated by an experiment, or perhaps from a Monte Carlo simulation, have uncertainties due to the statistical, or random, character of the processes by which they are acquired. Therefore, one must be able to describe statistically certain features of the data such as their mean, variance and skewness, and the degree to which correlations exist, either between one portion of the data and another, or between the data and some other standard or model distribution. A very readable introduction to this type analysis has been given by Young [8.12], while more comprehensive treatments are also available [8.13].

8.1.3 Fourier Analysis The Fourier transform takes, for example, a function of time, into a function of frequency, or vice versa, namely 1 ϕ(ω) =√ ˜ 2π 1 ϕ(t) = √ 2π

∞ −∞ ∞

ϕ(t) eiωt dt ,

(8.21)

ϕ(ω) e−iωt dω . ˜

(8.22)

−∞

In this case, the time history of the function ϕ(t) may be termed the “signal” and ϕ(ω) the “frequency spectrum”. ˜ Also, if the frequency is related to the energy by E = ~ω, one obtains an “energy spectrum” from a signal, and thus the name spectral methods for techniques based on the Fourier analysis of signals. The Fourier transform also defines the relationship between the spatial and momentum representations of wave functions, i. e., 1 ψ(x) = √ 2π 1 ˜ p) = √ ψ( 2π

∞ −∞ ∞

8.1 Representation of Functions

converted to sums 2N−1
1 ϕ(tk ) eiω j tk , ϕ(ω ˜ j) = √ 2N2π k=0 2N−1
1 ϕ(tk ) = √ ϕ(ω ˜ j ) e−iω j tk , 2N2π j=0

(8.25)

(8.26)

where the functions are “sampled” at 2N points. These equations define the discrete Fourier transform (DFT). Two cautions in using the DFT are as follows. First, if a continuous function of time that is sampled at, for simplicity, uniformly spaced intervals, (i. e., ti+1 = ti + ∆), then there is a critical frequency ωc = π/∆, known as the Nyquist frequency, which limits the fidelity of the DFT of this function in that it is aliased. That is, components outside the frequency range −ωc to ωc are falsely transformed into this range due to the finite sampling. This effect can be remediated by filtering or windowing techniques. If, however, the function is bandwidth limited to frequencies smaller than ωc , then the DFT does not suffer from this effect, and the signal is completely determined by its samples. Second, implementing the DFT directly from the above equations would require approximately N 2 multiplications to perform the Fourier transform of a function sampled at N points. A variety of fast Fourier transform (FFT) algorithms have been developed (e.g., the Danielson–Lanczos and Cooley–Tukey methods) which require only on the order of (N/2) log2 N multiplications. Thus, for even moderately large sets of points, the FFT methods are indeed much faster than the direct implementation of the DFT. Issues involved in sampling, aliasing, and selection of algorithms for the FFT are discussed in great detail, for example, in [8.3, 15, 16].

˜ p) ei px d p , ψ(

(8.23)

8.1.4 Approximating Integrals

ψ(x) e−i px dx .

(8.24)

Polynomial Quadrature Definite integrals may be approximated through a procedure known as numerical quadrature by replacing the integral by an appropriate sum, i. e.,

−∞

b f(x) dx ≈ a

n


ak f(xk ) .

(8.27)

k=0

Most formulas for such approximation are based on the interpolating polynomials described in Sect. 8.1.1, especially the Lagrange polynomials, in which case the

Part A 8.1

Along with the closely related sine, cosine, and Laplace transforms, the Fourier transform is an extraordinarily powerful tool in the representation of functions, spectral analysis, convolution of functions, filtering, and analysis of correlation. Good introductions to these techniques with particular attention to applications in physics can be found in [8.6, 7, 14]. To implement the Fourier transform numerically, the integral tranform pair can be

139

140

Part A

Mathematical Methods

approximates the definite integral of a function multipled by the weight function appropriate to the orthogonal polynomial being used as

coefficients ak are given by b ak =

L nk (xk ) dx .

(8.28)

a

If first or second degree Lagrange polynomials are used with a uniform spacing between the data points, one obtains the trapezoidal and Simpson’s rules, i. e., b f(x) dx ≈

 δ [ f(a) + f(b)] + O δ3 f (2) (ζ) , 2

a

(8.29)

b a



  δ δ f(x) dx ≈ f(a) + 4 f + f(b) 3 2

 + O δ5 f (4) (ζ) ,

(8.30)

respectively, with δ = b − a, and for some ζ in [a, b]. Other commonly used formulas based on low-order polynomials, and generally referred to as Newton–Cotes formulas, are described and discussed in detail in numerical analysis texts [8.1, 2]. Since potentially unwanted rapid oscillations in interpolants may arise, it is generally the case that increasing the order of the quadrature scheme too greatly does not generally improve the accuracy of the approximation. Dividing the interval [a, b] into a number of subintervals and summing the result of application of a low-order formula in each subinterval is usually a much better approach. This procedure, referred to as composite quadrature, may be combined with choosing the data points at a nonuniform spacing, decreasing the spacing where the function varies rapidly, and increasing the spacing for economy where the function is smooth to construct an adaptive quadrature.

Part A 8.1

Gaussian Quadrature If the function whose definite integral is to be approximated can be evaluated explicitly, then the data points (abscissas) can be chosen in a manner in which significantly greater accuracy may be obtained than using Newton–Cotes formulas of equal order. Gaussian quadrature is a procedure in which the error in the approximation is minimized owing to this freedom to choose both data points (abscissas) and coefficients. By utilizing orthogonal polynomials and choosing the abscissas at the roots of the polynomials in the interval under consideration, it can be shown that the coefficients may be optimally chosen by solving a simple set of linear equations. Thus, a Gaussian quadrature scheme

b W(x) f(x) dx ≈

n


ak f(xk ) ,

(8.31)

k=1

a

where the function is to be evaluated at the abscissas given by the roots of the orthogonal polynomial, xk . In this case, the coefficients ak are often referred to as “weights,” but should not be confused with the weight function W(x) (Sect. 8.1.1). Since the Legendre polynomials are orthogonal over the interval [−1, 1] with respect to the weight function W(x) ≡ 1, this equation has a particularly simple form, leading immediately to the Gauss–Legendre quadrature. If f(x) contains as a factor the weight function of another of the orthogonal polynomials, the corresponding Gauss– Laguerre or Gauss–Chebyshev quadrature should be used. The roots and coefficients have been tabulated [8.4] for many common choices of the orthogonal polynomials (e.g., Legendre, Laguerre, Chebyshev) and for various orders. Simple computer subroutines are also available which conveniently compute them [8.3]. Since the various orthogonal polynomials are defined over different intervals, use of the change of variables such as that given in (8.10) may be required. So, for Gauss– Legendre quadrature we make use of the transformation b f(x) dx ≈ a

(b − a) 2

1  (b − a)y + b + a dy . f 2

−1

(8.32)

Other Methods Especially for multidimensional integrals which can not be reduced analytically to seperable or iterated integrals of lower dimension, Monte Carlo integration may provide the only means of finding a good approximation. This method is described in Sect. 8.4.3. Also, a convenient quadrature scheme can easily be devised based on the cubic spline interpolation described in Sect. 8.1.1. since in each subinterval, the definite integral of a cubic polynomial of known coefficients is evident.

8.1.5 Approximating Derivatives Numerical Differentiation The calculation of derivatives from a numerical representaion of a function is generally less stable than the

Computational Techniques

calculation of integrals because differentation tends to enhance fluctuations and worsen the convergence properties of power series. For example, if f(x) is twice continuously differentiable on [a, b], then differentiation of the linear Lagrange interpolation formula (8.5) yields f (1) (x0 ) =

 f(x0 + δ) − f(x0 ) + O δ f (2) (ζ) δ

(centered and forward/backward) are 1 f (1) (x0 ) = f(x0 − 2δ) − 8 f(x0 − δ) 12δ

+ 8 f(x0 + δ) − f(x0 + 2δ)

 + O δ4 f (5) (ζ)

141

(8.34)

1 − 25 f(x0 ) + 48 f(x0 + δ) f (1) (x0 ) = 12δ − 36 f(x0 + 2δ) + 16 f(x0 + 3δ)



− 3 f(x0 + 4δ) + O δ4 f (5) (ζ) . (8.35)

(8.33)

for some x0 and ζ in [a, b], where δ = b − a. In the limit δ → 0, (8.33) coincides with the definition of the derivative. However, in practical calculations with finite precision arithmetic, δ cannot be taken too small because of numerical cancellation in the calculation of f(a + δ) − f(a). In practice, increasing the order of the polynomial used decreases the truncation error, but at the expense of increasing round-off error, the upshot being that threeand five-point approximations are usually most useful. Various three- and five-point formulas are given in standard texts [8.2, 4, 17]. Two common five-point formulas

8.2 Differential and Integral Equations

The second formula is useful for evaluating the derivative at the left or right endpoint of the interval, depending on whether δ is positive or negative, respectively. Derivatives of Interpolated Functions An interpolating function can be directly differentiated to obtain the derivative at any desired point. For example, if f(x) ≈ a0 + a1 x + a2 x 2 , then f (1) (x) = a1 + 2a2 x. However, this approach may fail to give the best approximation to f (1) (x) if the original interpolation was optimized to give the best possible representaion of f(x).

8.2 Differential and Integral Equations The subject of differential and integral equations is immense in both richness and scope. The discussion here focuses on techniques and algorithms, rather than the formal aspect of the theory. Further information can be found elsewhere under the broad catagories of finite element and finite difference methods. The Numerov method, which is particularly useful in integrating the Schrödinger equation, is described in great detail in [8.8].

8.2.1 Ordinary Differential Equations

is a real-valued function y(t) having the following properties: (1) y(t) and its first n derivatives exist, so y(t) and its first n − 1 derivatives must be continuous, and (2) y(t) satisfies the differential equation for all t. A unique

y˙ = f(t, y) ,

y(a) = A .

(8.37)

The methods discussed below can easily be extended to systems of first-order differential equations and to higher-order differential equations. The methods are referred to as discrete variable methods and generate a sequence of approximate values for y(t), y1 , y2 , y3 , . . . at points t1 , t2 , t3 , . . . . For simplicity, the discussion assumes a constant spacing h between t points. We shall first describe a class of methods known as one-step methods [8.19]. They have no memory of the solutions at past times; given yi , there is a recipe for yi+1 that depends only on information at ti . Errors enter into numerical solutions from two sources. The first is discretization error and depends on the method being used. The second is computational error which includes such things as round off error.

Part A 8.2

An ordinary differential equation is an equation involving an unknown function and one or more of its derivatives that depends on only one independent variable [8.18]. The order of a differential equation is the order of the highest derivative appearing in the equation. A solution of a general differential equation of order n,   f t, y, y, (8.36) ˙ . . . , y(n) = 0 ,

solution requires the specification of n conditions on y(t) and its derivatives. The conditions may be specified as n initial conditions at a single t to give an initial value problem, or at the end points of an interval to give a boundary value problem. Consider first solutions to the simple equation

142

Part A

Mathematical Methods

For a solution on the interval [a, b], let the t points be equally spaced; so for some positive integer n and h = (b − a)/n, ti = a + ih, i = 0, 1, . . . , n. If a < b, h is positive and the integration is forward; if a > b, h is negative and the integration is backward. The latter case could occur in solving for the initial point of a solution curve given the terminal point. A general one-step method can then be written in the form yi+1 = yi + h∆(ti , yi ) ,

y0 = y(t0 ) ,

(8.38)

where ∆ is a function that characterizes the method. Different ∆ functions are displayed, giving rise to the Taylor series methods and the Runge–Kutta methods. Taylor Series Algorithm To obtain an approximate solution of order p on [a, b], generate the sequence  yi+1 = yi + h f(ti , yi ) + · · ·  h p−1 + f ( p−1) (ti , yi ) , p! (8.39) ti+1 = ti + h, i = 0, 1, . . . , n − 1

where t0 = a and y0 = A. The Taylor method of order p = 1 is known as Euler’s method: yi+1 = yi + h f(ti , yi ) , ti+1 = ti + h .

(8.40)

Taylor series methods can be quite effective if the total derivatives of f are not too difficult to evaluate. Software packages are available that perform exact differentiation, (ADIFOR, MAPLE, MATHEMATICA, etc.) facilitating the use of this approach.

Part A 8.2

Runge–Kutta Methods Runge–Kutta methods are designed to approximate Taylor series methods [8.20], but have the advantage of not requiring explicit evaluations of the derivatives of f(t, y). The basic idea is to use a linear combination of values of f(t, y) to approximate y(t). This linear combination is matched up as closely as possible with a Taylor series for y(t) to obtain methods of the highest possible order p. Euler’s method is an example using one function evaluation. To obtain an approximate solution of order p = 2, let h = (b − a)/n and generate the sequences  yi+1 = yi + h (1 − γ) f(ti , yi )   h h , yi + f(ti , yi ) , + γ f ti + 2γ 2γ

ti+1 = ti + h ,

i = 0, 1, . . . , n − 1 ,

(8.41)

where γ = 0, t0 = a, y0 = A. Euler’s method is the special case, γ = 0, and has order 1; the improved Euler method has γ = 1/2 and the Euler–Cauchy method has γ = 1. The Adams–Bashforth and Adams–Moulton Formulas These formulas furnish important and widely-used examples of multistep methods [8.21]. On reaching a mesh point ti with approximate solution yi ∼ = y(ti ), there are (usually) available approximate solutions yi+1− j ∼ = y(ti+1− j ) for j = 2, 3, . . . , p. From the differential equation itself, approximations to the derivatives y(t ˙ i+1− j ) can be obtained. An attractive feature of the approach is the form of the underlying polynomial approximation, P(t), to y(t) ˙ because it can be used to approximate y(t) between mesh points

y(t) ∼ = yi +

t P(t) dt .

(8.42)

ti

The lowest-order Adams–Bashforth formula arises from interpolating the single value f i = f(ti , yi ) by P(t). The interpolating polynomial is constant so its integration from ti to ti+1 results in h f(ti , yi ) and the first order Adams–Bashforth formula: yi+1 = yi + h f(ti , yi ) .

(8.43)

This is just the forward Euler formula. For constant step size h, the second-order Adams–Bashforth formula is    3 1 yi+1 = yi + h f(ti , yi ) − f(ti−1 , yi−1 ) . 2 2 (8.44)

The lowest-order Adams–Moulton formula involves interpolating the single value f i+1 = f(xi+1 , yi+1 ) and leads to the backward Euler formula yi+1 = yi + h f (ti+1 , yi+1 ) ,

(8.45)

which defines yi+1 implicitly. From its definition it is clear that it has the same accuracy as the forward Euler method; its advantage is vastly superior stability. The second-order Adams–Moulton method also does not use previously computed solution values; it is called the trapezoidal rule because it generalizes the trapezoidal rule for integrals to differential equations:

h yi+1 = yi + f(ti+1 , yi+1 ) + f(ti , yi ) . (8.46) 2

Computational Techniques

The Adams–Moulton formula of order p is more accurate than the Adams–Bashforth formula of the same order, so that it can use a larger step size; the Adams– Moulton formula is also more stable. A code based on such methods is more complex than a Runge–Kutta code because it must cope with the difficulties of starting the integration and changing the step size. Modern Adams codes attempt to select the most efficient formula at each step, as well as to choose an optimal step size h to achieve a specified accuracy.

8.2.2 Differencing Algorithms for Partial Differential Equations The modern approach to evolve differencing schemes for most physical problems is based on flux conservation methods [8.22]. One begins by writing the balance equations for a single cell, and subsequently applying quadratures and interpolation formulas. Such approaches have been successful for the full spectrum of hyperbolic, elliptic, and parabolic equations. For simplicity, we begin by discussing systems involving only one space variable. As a prototype, consider the parabolic equation ∂ ∂2 u(x, t) = σ 2 u(x, t) , (8.47) ∂t ∂x where c and σ are constants and u(x, t) is the solution. We begin by establishing a grid of points on the xt-plane with step size h in the x direction and step size k in the t-direction. Let spatial grid points be denoted by xn = x0 + nh and time grid points by t j = t0 + jk, where n and j are integers and (x0 , t0 ) is the origin of the space–time grid. The points ξn−1 and ξn are introduced to establish a “control interval”. We begin with a conservation statement

dx r(x, t j+1 ) − r(x, t j )

ξn−1

t j+1 =

dt [q(ξn−1 , t) − q(ξn , t)] .

(8.48)

r(x, t) = cu(x, t) + b

(8.49)

with c and b constants, thus ξn c



dx u(x, t j+1 ) − u(x, t j )

ξn−1

≈ c[u(xn , t j+1 ) − u(xn , t j )]h .

(8.50)

When developing conservation law equations, there are two commonly used strategies for approximating the right-hand-side of (8.48): (i) left end-point quadrature t j+1

dt q(ξn−1 , t) − q(ξn , t) tj



≈ q(ξn−1 , t j ) − q(ξn , t j ) k ,

(8.51)

and (ii) right end-point quadrature t j+1

dt q(ξn−1 , t) − q(ξn , t) tj



≈ q(ξn−1 , t j+1 ) − q(ξn , t j+1 ) k .

(8.52)

Combining (8.48) with the respective approximations yields: from (i) an explicit method

c u(xn , t j+1 ) − u(xn , t j ) h

(8.53) ≈ q(ξn−1 , t j ) − q(ξn , t j ) k , and from (ii) an implicit method

c u(xn , t j+1 ) − u(xn , t j ) h

≈ q(ξn−1 , t j+1 ) − q(ξn , t j+1 ) k .

(8.54)

Using centered finite difference formulas to approximate the fluxes at the control points ξn−1 and ξn yields

tj

q(ξn−1 , t j ) = −σ

u(xn , t j ) − u(xn−1 , t j ) , h

(8.55)

and q(ξn , t j ) = −σ

u(xn+1 , t j ) − u(xn , t j ) h

(8.56)

where σ is a constant. We also obtain similar formulas for the fluxes at time t j+1 .

Part A 8.2

This equation states that the change in the field density on the interval (ξn−1 , ξn ) from time t = t j to time t = t j+1 is given by the flux into this interval at ξn−1 minus the flux out of the interval at ξn from time t j to time t j+1 . This expresses the conservation of material in the case that no sources or sinks are present. We relate the field variable u to the physical variables (the density r

143

and the flux q). We consider the case in which density is assumed to have the form

c

ξn

8.2 Differential and Integral Equations

144

Part A

Mathematical Methods

We have used a lower case u to denote the continuous field variable, u = u(x, t). Note that all of the quadrature and difference formulas involving u are stated as approximate equalities. In each of these approximate equality statements, the amount by which the right side differs from the left side is called the truncation error. If u is a well-behaved function (has enough smooth derivatives), then it can be shown that these truncation errors approach zero as the grid spacings, h and k, approach zero. j If Un denotes the exact solution on the grid, we have from (i) the result     j+1 j j j j c Un − Un h 2 = σk Un−1 + Un+1 − 2Un . (8.57)

This is an explicit method since it provides the solution to the difference equation at time t j+1 , knowing the values at time t j . If we use the numerical approximations (ii) we obtain the result     j+1 j j+1 j+1 j+1 . c Un − Un h 2 = σk Un−1 + Un+1 − 2Un (8.58)

Note that this equation defines the solution at time t j+1 implicitly, since a system of algebraic equations is required to be satisfied.

8.2.3 Variational Methods

(8.59)

Part A 8.2

with x = xi , i = 1, 2, 3 in R, for example, and with u = 0 on the boundary of R. The function f(x) is the source. It is assumed that L is always nonsingular and in addition, for the Ritz method L is Hermitian. The real-valued functions u are in the Hilbert space Ω of the operator L. We construct the functional J[u] defined as  J[u] = dx [u(x)Lu(x) − 2u(x) f(x)] . (8.60) Ω

u n (x) =

n


ci φi (x) .

(8.61)

i=1

We solve for the coefficients ci by minimizing J[u n ] ∂ci J[u n ] = 0 ,

i = 1, . . . , n .

(8.62)

These equations are simply cast into a set of wellbehaved algebraic equations n


Ai, j c j = gi ,

i = 1, . . . , n ,

(8.63)

j=1

  with Ai, j = Ω dxφi (x)Lφ j (x), and gi = Ω dxφi (x) f(x). Under very general conditions, the functions u n converge uniformly to u. The main drawback of the Ritz method is in the assumption of Hermiticity of the operator L. For the Galerkin Method we relax this assumption with no other changes. Thus we obtain an identical set of equations, as above with the exception that the function g is no longer symmetric. The convergence of the sequence of solutions u n to u is no longer guaranteed, unless the operator can be separated into a symmetric part L 0 , L = L 0 + K so that L −1 0 K is bounded.

8.2.4 Finite Elements

Perhaps the most widely used approximation procedures in AMO physics are the variational methods. We shall outline in detail the Rayleigh–Ritz method [8.23]. This method is limited to boundary value problems which can be formulated in terms of the minimization of a functional J[u]. For definiteness we consider the case of a differential operator defined by Lu(x) = f(x)

The variational ansatz considers a subspace of Ω, Ωn , spanned by a class of functions φn (x), and we construct the function u n ≈ u

As discussed in Sect. 8.2.2, in the finite difference method for classical partial differential equations, the solution domain is approximated by a grid of uniformly spaced nodes. At each node, the governing differential equation is approximated by an algebraic expression which references adjacent grid points. A system of equations is obtained by evaluating the previous algebraic approximations for each node in the domain. Finally, the system is solved for each value of the dependent variable at each node. In the finite element method [8.24], the solution domain can be discretized into a number of uniform or nonuniform finite elements that are connected via nodes. The change of the dependent variable with regard to location is approximated within each element by an interpolation function. The interpolation function is defined relative to the values of the variable at the nodes associated with each element. The original boundary value problem is then replaced with an equivalent in-

Computational Techniques

tegral formulation. The interpolation functions are then substituted into the integral equation, integrated, and combined with the results from all other elements in the solution domain. The results of this procedure can be reformulated into a matrix equation of the form n


Ai, j c j = gi ,

i = 1, . . . , n ,

(8.64)

j=1

  with Ai, j = Ω dxφi (x)Lφ j (x), and gi = Ω dxφi (x) f(x) exactly as obtained in Sect. 8.2.3. The only difference arises in the definitions of the support functions φi (x). In general, if these functions are piecewise polynomials on some finite domain, they are called finite elements or splines. Finite elements make it possible to deal in a systematic fashion with regions having curved boundaries of an arbitrary shape. Also, one can systematically estimate the accuracy of the solution in terms of the parameters that label the finite element family, and the solutions are no more difficult to generate than more complex variational methods. In one space dimension, the simplest finite element family begins with the set of step functions defined by  1 xi−1 ≤ x ≤ xi φi (x) = (8.65)  0 otherwise .

functions given by  x − xi−1   xi−1 ≤ x ≤ xi   x  i − xi−1   xi+1 − x φi (x) = xi ≤ x ≤ xi+1   xi+1 − xi      0 otherwise ,

145

(8.66)

and for which the derivative is given by  1   xi−1 ≤ x ≤ xi   x − xi−1  i   d −1 φi (x) = x ≤ x ≤ xi+1  dx  xi+1 − xi i      0 otherwise .

(8.67)

The functions have a maximum value of one at the midpoint of the interval [xi−1 , xi+1 ], with partially overlapping adjacent elements. In fact, the overlaps may be represented by a matrix O with elements ∞ Oij = dx φi (x)φ j (x) . (8.68) −∞

Thus, if i = j xi Oii = xi−1

(x − xi−1 )2 dx + (xi − xi−1 )2

xi+1 dx xi

1 = (xi+1 − xi−1 ) , 3 if i = j − 1 xi+1 Oij =

dx xi

(x − xi )2 (xi+1 − xi )2 (8.69)

(x − xi )(xi+1 − x) (xi+1 − xi )2

1 = (xi+1 − xi ) , 6 if i = j + 1 xi (x − xi−1 )(xi − x) dx Oij = (xi − xi−1 )2

(8.70)

xi−1

1 = (xi − xi−1 ) , (8.71) 6 and Oij = 0 otherwise. The potential energy is represented by the matrix ∞ Vij = dx φi (x)V(x)φ j (x) , (8.72) −∞

Part A 8.2

The use of these simple “hat” functions as a basis provides no advantage over the usual finite difference schemes. However, for certain problems in two or more dimensions, finite element methods have distinct advantages over other methods. Generally, the use of finite elements requires complex, sophisticated computer programs for implementation. The use of higher-order polynomials, commonly called splines, as a basis has been extensively used in atomic and molecular physics. An extensive literature is available [8.25, 26]. We illustrate the use of the finite element method by applying it to the Schrödinger equation. In this case, the linear operator L is H − E where, as usual, E is the energy and the Hamiltonian H is the sum of the kinetic and potential energies, that is, L = H − E = T + V − E and Lu(x) = 0. We define the finite elements through support points, or knots, given by the sequence {x1 , x2 , x3 , . . . } which are not necessarily spaced uniformly. Since the “hat” functions have vanishing derivatives, we employ the next more complex basis, i. e., “tent” functions, which are piecewise linear

8.2 Differential and Integral Equations

146

Part A

Mathematical Methods

which may be well approximated by ∞ Vij ≈ V(xi )

dx φi (x)φ j (x)

(8.73)

−∞

= V(xi )Oij if x j − xi is small. The kinetic energy, T = − 12 d2 / dx 2 , is similarly given by Tij = −

1 2

∞ dxφi (x) −∞

d2 φ j (x) , dx 2

(8.74)

which we compute by integrating by parts since the tent functions have a singular second derivative 1 Tij = 2



∞ dx −∞

 d d φi (x) φ j (x) , dx dx

which in turn is evalutated to yield  xi+1 − xi−1    2(x − x )(x − x )   i i−1 i+1 i     1    Tij = 2(xi − xi+1 )   1      2(xi−1 − xi )      0

(8.75)

i= j i = j −1 i = j +1 otherwise .

Finally, since the Hamiltonian matrix is Hij = Tij + Vij , the solution vector u i (x) may be found by solving the eigenvalue equation (8.77)

8.2.5 Integral Equations Central to much of practical and formal scattering theory is the integral equation and techniques of its solution. For example, in atomic collision theory, the Schrödinger differential equation

Part A 8.2

[E − H0 (r)] ψ(r) = V(r)ψ(r)

(8.78)

where the Hamiltonian H0 ≡ −(~2 /2m)∇ 2 + V0 may be solved by exploiting the solution for a delta function source, i. e., (E − H0 )G(r, r ) = δ(r − r ) .

for which, given a choice of the functions G(r, r ) and χ(r), particular boundary conditions are determined. This integral equation is the Lippmann–Schwinger equation of potential scattering. Further topics on scattering theory are covered in other chapters (see especially Chapts. 47 to 58) and in standard texts such as those by Joachain [8.27], Rodberg and Thaler [8.28], and Goldberger and Watson [8.29]. Owing especially to the wide variety of specialized techniques for solving integral equations, we survey briefly only a few of the most widely applied methods. Integral Transforms Certain classes of integral equations may be solved using integral transforms such as the Fourier or Laplace transforms. These integral transforms typically have the form  f(x) = dx K(x, x )g(x ) , (8.81)

(8.76)

[Hij − EOij ]u i (x) = 0 .

In terms of this Green’s function G(r, r , and any solution χ(r) of the homogeneous equation [i. e. with V(r) = 0], the general solution is  ψ(r) = χ(r) + dr G(r, r )V(r )ψ(r ) (8.80)

(8.79)

where f(x) is the integral transform of g(x ) by the kernel K(x, x ). Such a pair of functions is the solution of the Schrödinger equation (spatial wave function) and its Fourier transform (momentum representation wave function). Arfken [8.6], Morse and Feshbach [8.9], and Courant and Hilbert [8.10] give other examples, as well as being excellent references for the application of integral equations and Green’s functions in mathematical physics. In their analytic form, these transform methods provide a powerful method of solving integral equations for special cases, and, in addition, they may be implemented by performing the transform numerically. Power Series Solution For an equation of the form (in one dimension for simplicity)  ψ(r) = χ(r) + λ dr K(r, r )ψ(r ) , (8.82)

a solution may be found by iteration. That is, as a first approximation, set ψ0 (r) = χ(r) so that  ψ1 (r) = χ(r) + λ dr K(r, r )χ(r ) . (8.83)

Computational Techniques

This may be repeated to form a power series solution, i. e., ψn (r) =

n


λk Ik (r) ,

(8.84)

k=0

I0 (r) = χ(r) , (8.85)  I1 (r) = dr K(r, r )χ(r ) , (8.86)   I2 (r) = dr

dr K(r, r )K(r , r

)χ(r

) , 



(8.87)



dr · · ·

dr

(n)





K(r, r )K(r, r )

· · · K(r (n−1) , r (n) ) . (8.88) If the series converges, then the solution ψ(r) is approached by the expansion. When the Schrödinger equation is cast as an integral equation for scattering in a potential, this iteration scheme leads to the Born series, the first term of which is the incident, unperturbed wave, and the second term is usually referred to simply as the Born approximation.

K(r, r ) =

f k (r)gk (r ) ,

(8.89)

where n is finite, then substitution into the prototype integral equation (8.82) yields  n
ψ(r) = χ(r) + λ f k (r) dr g(r )ψ(r ) . (8.90) k=1

Multiplying by f k (r), integrating over r, and rearranging, yields the set of algebraic equations n


a jk ck ,

(8.91)

k=1

where

bk =

 

(8.95)

The eigenvalues are the roots of the determinantal equation. Substituting these into (1 − λA)c = 0 yields the constants ck which determine the solution of the original equation. This derivation may be found in Arfken [8.6], along with an explicit example. Even if the kernel is not exactly separable, if it is approximately so, then this procedure can yield a result which can be substituted into the original equation as a first step in an iterative solution. Numerical Integration Perhaps the most straightforward method of solving an integral equation is to apply a numerical integration formula such as Gaussian quadrature. An equation of the form  ψ(r) = dr K(r, r )χ(r ) (8.96)

dr gk (r )ψ(r ) ,

(8.92)

dr f k (r)χ(r) ,

(8.93)

n


wk K(r j , rk )χ(rk ) ,

(8.97)

k=1

where wk are quadrature weights, if the kernel is well behaved. However, such an approach is not without pitfalls. In light of the previous subsection, this approach is equivalent to replacing the integral equation by a set of algebraic equations. In this example we have ψj =

n


M jk χk ,

(8.98)

k=1

so that the solution of the equation is found by inverting the matrix M. Since there is no guarantee that this matrix is not ill-conditioned, the numerical procedure may not produce meaningful results. In particular, only certain classes of integral equations and kernels will lead to stable solutions. Having only scratched the surface regarding the very rich field of integral equations, the interested reader is encouraged to explore the references given here.

Part A 8.2

ck =

(8.94)

or, if c and b denote vectors, and A denotes the matrix of constants a jk ,

ψ(r j ) =

k=1

cj = bj +λ

drg j (r) f k (r) ,

can be approximated as

Separable Kernels If the kernel is separable, i. e., n


147

 a jk =

c = (1 − λA)−1 b .

where

In (r) =

8.2 Differential and Integral Equations

148

Part A

Mathematical Methods

8.3 Computational Linear Algebra Previous sections of this chapter have dealt with interpolation, differential equations, and related topics. Generally, discretization methodologies lead to classes of algebraic equations. In recent years enormous progress has been made in developing algorithms for solving linear algebraic equations, and many very good books have been written on this topic [8.30]. Furthermore, a large body of numerical software is freely available via an electronic service called Netlib (www.netlib.org). In addition to the widely adopted numerical linear algebra packages LAPACK, ScaLAPACK, ARPACK, etc., there are dozens of other libraries, technical reports on various parallel computers and software, test data, facilities to automatically translate Fortran programs to C, bibliographies, names and addresses of scientists and mathematicians, and so on. Here we discuss methods for solving systems of equations such as a11 x1 + a12 x2 + · · · + a1n xn = b1 , a21 x1 + a22 x2 + · · · + a2n xn = b2 , .. . am1 x1 + am2 x2 + · · · + amn xn = bm .

(8.99)

Part A 8.3

In these equations aij and bi form the set of known quantities, and the xi must be determined. The solution to these equations can found if they are linearly independent. Numerically, problems can arise due to truncation and roundoff errors that lead to an approximate linear dependence [8.31]. In this case the set of equations are approximately singular and special methods must be invoked. Much of the complexity of modern algorithms comes from minimizing the effects of such errors. For relatively small sets of nonsingular equations, direct methods in which the solution is obtained after a definite number of operations can work well. However, for very large systems iterative techniques are preferable [8.32]. A great many algorithms are available for solving (8.99), depending on the structure of the coefficients. For example, if the matrix of coefficients A is dense, using Gaussian elimination takes 2n 3 /3 operations; if A is also symmetric and positive definite, using the Cholesky algorithm takes a factor of two fewer operations. If A is triangular, i. e., either zero above the diagonal or zero below the diagonal, we can solve the above by simple substitution in only n 2 operations. For example, if A arises from solving certain elliptic partial differential

equations, such as Poisson’s equation, then Ax = b can be solved using multigrid methods in only n operations. We shall outline below how to solve (8.99) using elementary Gaussian elimination. More advanced methods, such as conjugate gradient, generalized minimum residuals, and the Lanczos method are treated elsewhere [8.33]. To solve Ax = b, we first use Gaussian elimination to factor the matrix A as PA = LU, where L is lower triangular, U is upper triangular, and P is a matrix which permutes the rows of A. Then we solve the triangular system Ly = Pb and Ux = y. These last two operations are easily performed using standard linear algebra libraries. The factorization PA = LU takes most of the time. Reordering the rows of A with P is called pivoting and is necessary for numerical stability. In the standard partial pivoting scheme, L has ones on its diagonal and other entries bounded in absolute value by one. The simplest version of Gaussian elimination involves adding multiples of one row of A to others to zero out subdiagonal entries, and overwriting A with L and U. We first describe the decomposition of PA into a product of upper and lower triangular matrices, A = LU ,

(8.100)

where the matrix A is defined by A = PA. A very nice algorithm for pivoting is given in [8.3] and will not be discussed further. Writing out the indices, Aij =

min(i,
j)

L ik Uk j .

(8.101)

k=1

We shall make the choice L ii = 1 .

(8.102)

These equations have the remarkable property that the elements Aij of each row can be scanned in turn, writing L ij and Uij into the locations Aij as we go. At each position (i, j), only the current Aij and alreadycalculated values of L i j and Ui j are required. To see how this works, consider the first few rows. If i = 1, A 1 j = U1 j ,

(8.103)

defining the first row of L and U. The U1 j are written over the A 1 j , which are no longer needed. If i = 2, A 21 = L 21 U11 ,

j =1

A 2 j

j ≥2.

= L 21 U1 j + U2 j ,

(8.104)

Computational Techniques

The first line gives L 21 , and the second U2 j , in terms of existing elements of L and U. The U2 j and L 21 are written over the A 2 j . (Remember that L ii = 1 by definition.) If i = 3, A 31 = L 31 U11 ,

j =1

A 32 A 3 j

j =2

= L 31 U12 + L 32 U22 ,

= L 31 U1 j + L 32 U2 j + U3 j , j ≥ 3 ,

(8.105)

yielding in turn L 31 ,L 32 , and U3 j , which are written over A 3 j . The algorithm should now be clear. At the ith row   j−1
 Aij − L ij = U −1 L ik Uk j  , j ≤ i −1 jj k=1

Uij = Aij −

i−1


L ik Uk j ,

j ≥i .

(8.106)

k=1

We observe from the first line of these equations that the algorithm may run into numerical inaccuracies if any U jj becomes very small. Now U11 = A 11 , while in general Uii = Aii − · · · . Thus the absolute values of the Uii are maximized if the rows are rearranged so that the absolutely largest elements of A in each column lie on the diagonal. A little thought shows that the solutions are unchanged by permuting the rows (same equations, different order).

8.4 Monte Carlo Methods

149

The LU decomposition can now be used to solve the system. This relies on the fact that the inversion of a triangular matrix is a simple process of back substitution. We replace ((8.99)) by two systems of equations. Written out in full, the equations for a typical column of y look like L 11 y1 = b 1 , L 21 y1 + L 22 y2 = b 2 , L 31 y1 + L 32 y2 + L 33 y3 = b 3 , .. (8.107) . , where the vector b is p = Pb. Thus from successive rows we obtain y1 , y2 , y3 , . . . in turn U11 x1 = y1 , U12 x1 + U22 x2 = y2 , U13 x1 + U23 x2 + U33 x3 = y3 , .. (8.108) . , and from successive rows of the latter we obtain x1 , x2 , x3 , . . . in turn. Library software also exists for evaluating all the error bounds for dense and band matrices (see discussion of Netlib in above). Gaussian elimination with pivoting is almost always numerically stable, so the error bound one expects from solving these equations is of the order of n, where  is related to the condition number of the matrix A. A good discussion of errors and conditioning is given in [8.3].

8.4 Monte Carlo Methods Here we summarize the basic tools needed in these methods, and how they may be used to produce specific distributions and make tractable the evaluation of multidimensional integrals with complicated boundaries. Detailed descriptions of these methods can be found in [8.3, 8, 34].

8.4.1 Random Numbers An essential ingredient of any Monte Carlo procedure is the availability of a computer-generated sequence of random numbers which is not periodic and is free of other significant statistical correlations. Often such numbers are termed pseudorandom or quasirandom, in distinction to truly random physical processes. While the quality of random number generators supplied with computers has greatly improved over time, it is impor-

Part A 8.4

Owing to the continuing rapid development of computational facilities and the ever-increasing desire to perform ab initio calcalutions, the use of Monte Carlo methods is becoming widespread as a means to evaluate previously intractable multidimensional integrals and to enable complex modeling and simulation. For example, a wide range of applications broadly classified as Quantum Monte Carlo have been used to compute, for example, the ground state eigenfunctions of simple molecules. Also, guided random walks have found application in the computation of Green functions, and variables chosen randomly, subject to particular constaints, have been used to mimic the electronic distribution of atoms. The latter application, used in the classical trajectory Monte Carlo technique described in Chapt. 58, allows the statistical quasiquantal representation of ion–atom collisions.

150

Part A

Mathematical Methods

tant to be aware of the potential dangers which can be present. For example, many systems are supplied with a random number generator based on the linear congruential method. Typically a sequence of integers n 1 , n 2 , n 3 , . . . is first produced between 0 and N − 1 by using the recurrence relation n i+1 = (an i + b) mod N ,

0 ≤ i < N −1 (8.109)

where a, b, N and the seed value n 0 are positive integers. Real numbers between 0 and (strictly) 1 are then obtained by dividing by N. The period of this sequence is at most N, and depends on the judicious choice of the constants, with N being limited by the wordsize of the computer. A user who is unsure that the character of the random numbers generated on a particular computer platform is proper can perform additional randomizing shuffles or use a portable random number generator, both procedures being described in detail by Knuth [8.5] and Press et al. [8.3], for example.

8.4.2 Distributions of Random Numbers Most distributions of random numbers begin with sequences generated uniformly between a lower and an upper limit, and are therefore called uniform deviates. However, it is often useful to draw the random numbers from other distributions, such as the Gaussian, Poisson, exponential, gamma, or binomial distributions. These are particularly useful in modeling data or supplying input for an event generator or simulator. In addition, as described below, choosing the random numbers according to some weighting function can signficantly improve the efficiency of integration schemes based on Monte Carlo sampling. Perhaps the most direct way to produce the required distribution is the transformation method. If we have a sequence of uniform deviates x on (0, 1) and wish to find a new sequence y which is distributed with probability given by some function f(y), it can be shown that the required transformation is given by  y −1  y(x) =  f(y) dy . (8.110)

Part A 8.4

0

Evidently, the indefinite integral must be both known and invertible, either analytically or numerically. Since this is seldom the case for distributions of interest, other less direct methods are most often applied. However, even these other methods often rely on the transformation

method as one “stage” of the procedure. The transformation method may also be generalized to more than one dimension [8.3]. A more widely applicable approach is the rejection method, also known as von Neumann rejection. In this case, if one wishes to find a sequence y distributed according to f(y), first choose another function f˜(y), called the comparison function, which is everywhere greater than f(y) on the desired interval. In addition, a way must exist to generate y according to the comparison function, such as use of the transformation method. Thus, the comparison function must be simpler or better known than the distribution to be found. One simple choice is a constant function which is larger than the maximum value of f(y), but choices which are “closer” to f(y) will be much more efficient. To proceed, y is generated uniformly according to f˜(y) and another deviate x is chosen uniformly on (0, 1). One then simply rejects or accepts y depending on whether x is greater than or less than the ratio f(y)/ f˜(y), respectively. The fraction of trial numbers accepted simply depends on the ratio of the area under the desired function to that under the comparison function. Clearly, the efficiency of this scheme depends on how few of the numbers initially generated must be rejected, and therefore on how closely the comparison function approximates the desired distribution. The Lorentzian distribution, for which the inverse definite integral is known (the tangent function), is a good comparison function for a variety of “bell-shaped” distributions such as the Gaussian (normal), Poisson, and gamma distributions. Especially for distributions which are functions of more than one variable and possess complicated boundaries, the rejection method is impractical and the transformation method simply inapplicable. In the 1950’s, a method to generate distributions for such situations was developed and applied in the study of statistical mechanics where multidimensional integrals (e.g., the partition function) must often be solved numerically, and is known as the Metropolis algorithm. This procedure, or its variants, has more recently been adopted to aid in the computation of eigenfunctions of complicated Hamiltonians and scattering operators. In essence, the Metropolis method generates a random walk through the space of the dependent variables, and in the limit of a large number of steps in the walk, the points visited approximate the desired distribution. In its simplest form, the Metropolis method generates this distribution of points by stepping through this space, most frequently taking a step “downhill” but

Computational Techniques

sometimes taking a step “uphill”. That is, given a set of coordinates q and a desired distribution function f(q), a trial step is taken from the ith configuration qi to the next, depending on whether the ratio f(qi + 1)/ f(qi ) is greater or less than one. If the ratio is greater than one, the step is accepted, but if it is less than one, the step is accepted with a probability given by the ratio.

8.4.3 Monte Carlo Integration The basic idea of Monte Carlo integration is that if a large number of points is generated uniformly randomly in some n-dimensional space, the number falling inside a given region is proportional to the volume, or definite integral, of the function defining that region. Though this idea is as true in one dimension as it is in n, unless there is a large number (“large” could be as little as three) of dimensions or the boundaries are quite complicated, the numerical quadrature schemes described previously are more accurate and efficient. However, since the Monte Carlo approach is based on just sampling the function at representative points rather than evaluating the function at a large number of finely spaced quadrature points, its advantage for very large problems is apparent. For simplicity, consider the Monte Carlo method for integrating a function of only one variable; the generalization to n dimensions being straightforward. If we generate N random points uniformly on (a, b), then in the limit of large N the integral is   b f 2 (x) − f(x) 2 1 , f(x) dx ≈ f(x) ± N N a

(8.111)

where f(x) ≡

N 1
f(xi ) N

(8.112)

i=1

is the arithmetic mean. The probable error given is appropriately a statistical one rather than a rigorous

References

151

error bound and is the one standard error limit. From this one can see that the error decreases as only N 1/2 , more slowly than the rate of decrease for the quadrature schemes based on interpolation. Also, the accuracy is greater for relatively smooth functions, since the Monte Carlo generation of points is unlikely to sample narrowly peaked features of the integrand well. To estimate the integral of a multidimensional function with complicated boundaries, simply find an enclosing volume and generate points uniformly randomly within it. Keeping the enclosing volume as close as possible to the volume of interest miminizes the number of points which fall outside, and therefore increases the efficiency of the procedure. The Monte Carlo integral is related to techniques for generating random numbers according to prescribed distributions described in Sect. 8.4.2. If we consider a normalized distribution w(x), known as the weight function, then with the change of variables defined by x y(x) =

w(x ) dx ,

(8.113)

a

the Monte Carlo estimate of the integral becomes b a

  1 f [x(y)] , f(x) dx ≈ N w [x(y)]

(8.114)

assuming that the transformation is invertible. Choosing w(x) to behave approximately as f(x) allows a more efficient generation of points within the boundaries of the integrand. This occurs since the uniform distribution of points y results in values of x distributed according to w and therefore “close” to f . This procedure, generally termed the reduction of variance of the Monte Carlo integration, improves the efficiency of the procedure to the extent that the transformed function f/w can be made smooth, and that the sampled region is as small as possible but still contains the volume to be estimated.

References

8.2 8.3

J. Stoer, R. Bulirsch: Introduction to Numerical Analysis (Springer, Berlin, Heidelberg 1980) R. L. Burden, J. D. Faires, A. C. Reynolds: Numerical Analysis (Prindle, Boston 1981) W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling: Numerical Recipes, the Art of Scientific

8.4

Computing (Cambridge Univ. Press, Cambridge 1992) M. Abramowitz, I. A. Stegun (Eds.): Handbook of Mathematical Functions, Applied Mathematics Series, Vol. 55 (National Bureau of Standards, Washington, Dover, New York 1968)

Part A 8

8.1

152

Part A

Mathematical Methods

8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13

8.14 8.15 8.16 8.17

8.18 8.19 8.20

D. E. Knuth: The Art of Computer Programming, Vol. 2 (Addison-Wesley, Reading 1981) G. Arfken: Mathematical Methods for Physicists (Academic Press, Orlando 1985) H. Jeffreys, B. S. Jeffreys: Methods of Mathematical Physics (Cambridge Univ. Press, Cambridge 1966) S. E. Koonin, D. C. Meredith: Computational Physics (Addison-Wesley, Redwood City 1990) P. M. Morse, H. Feshbach: Methods of Theoretical Physics (McGraw-Hill, New York 1953) R. Courant, D. Hilbert: Methods of Mathematical Physics (Interscience, New York 1953) P. J. Huber: Robust Statistics (Wiley, New York 1981) H. D. Young: Statistical Treatment of Experimental Data (McGraw-Hill, New York 1962) P. R. Bevington: Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York 1969) D. C. Champeney: Fourier Transforms and Their Physical Applications (Academic, New York 1973) R. W. Hamming: Numerical Methods for Scientists and Engineers (McGraw-Hill, New York 1973) D. F. Elliott, K. R. Rao: Fast Transforms: Algorithms, Analyses, Applications (Academic, New York 1982) D. Zwillinger: CRC Standard Mathematical Tables and Formulae, 31st edn. (Chapman & Hall/CRC, New York 2002) J. Lambert: Numerical Methods for Ordinary Differential Equations (Wiley, New York 1991) L. F. Shampine: Numerical Solution of Ordinary Differential Equations (Chapman Hall, New York 1994) J. Butcher: The Numerical Analysis of Ordinary Differential Equations: Runge–Kutta and General Linear Methods (Wiley, New York 1987)

8.21 8.22

8.23

8.24

8.25 8.26 8.27 8.28 8.29 8.30

8.31

8.32 8.33

8.34

G. Hall, J. Watt: Modern Numerical Methods for Ordinary Differential Equations (Clarendon, Oxford 1976) I. Gladwell, R. Wait (Eds.): A Survey of Numerical Methods for Partial Differential Equations (Clarendon, Oxford 1979) K. Rektorys: Variational Methods in Mathematics, Science, and Engineering, 2nd edn. (Reidel, Boston 1980) D. Cook, D. S. Malkus, M. E. Plesha: Concepts and Applications of Finite Element Analysis, 3rd edn. (Wiley, New York 1989) G. Nurnberger: Approximation by Spline Functions (Springer, Berlin, Heidelberg 1989) C. DeBoor: Practical Guide to Splines (Springer, New York 1978) C. J. Joachain: Quantum Collision Theory (Elsevier, New York 1983) L. S. Rodberg, R. M. Thaler: Introduction to the Quantum Theory of Scattering (Academic, New York 1967) M. L. Goldberger, K. M. Watson: Collision Theory (Wiley, New York 1964) P. G. Ciarlet: Introduction to Numerical Linear Algebra and Optimisation (Cambridge Univ. Press, Cambridge 1989) G. Golub, C. Van Loan: Matrix Computations, 2nd edn. (Johns Hopkins Univ. Press, Baltimore 1989) W. Hackbusch: Iterative Solution of Large Sparse Systems of Equations (Springer, New York 1994) A. George, J. Liu: Computer Solution of Large Sparse Positive Definite Systems (Prentice-Hall, Englewood Cliffs 1981) M. H. Kalos, P. A. Whitlock: The Basics of Monte Carlo Methods (Wiley, New York 1986)

Part A 8

153

This chapter summarizes the solutions of the one-electron nonrelativistic Schrödinger equation, and the one-electron relativistic Dirac equation, for the Coulomb potential. The standard notations and conventions used in the mathematics literature for special functions have been chosen in preference to the notations customarily used in the physics literature whenever there is a conflict. This has been done to facilitate the use of standard reference works such as Abramowitz and Stegun [9.1], the Bateman project [9.2, 3], Gradshteyn and Ryzhik [9.4], Jahnke and Emde [9.5], Luke [9.6, 7], Magnus, Oberhettinger, and Soni [9.8], Olver [9.9], Szego [9.10], and the new NIST Digital Library of Mathematical Functions project, which is preparing a hardcover update [9.11] of Abramowitz and Stegun [9.1] and an online digital library of mathematical functions [9.12]. The section on special functions contains many of the formulas which are needed to check the results quoted in the previous sections, together with a number of other useful formulas. It

9.1

Schrödinger Equation .......................... 9.1.1 Spherical Coordinates ................ 9.1.2 Parabolic Coordinates ................ 9.1.3 Momentum Space .....................

153 153 154 156

9.2

Dirac Equation .................................... 157

9.3

The Coulomb Green’s Function.............. 159 9.3.1 The Green’s Function for the Schrödinger Equation ...... 159 9.3.2 The Green’s Function for the Dirac Equation................ 161

9.4

Special Functions................................. 9.4.1 Confluent Hypergeometric Functions ................................. 9.4.2 Laguerre Polynomials ................ 9.4.3 Gegenbauer Polynomials............ 9.4.4 Legendre Functions ...................

162 162 166 169 169

References .................................................. 170 includes a brief introduction to asymptotic methods. References to the numerical evaluation of special functions are given.

9.1 Schrödinger Equation The nonrelativistic Schrödinger equation for a hydrogenic ion of nuclear charge Z in atomic units is 
1 Z ψ (r) = Eψ (r) . − ∇2 − (9.1) 2 r

9.1.1 Spherical Coordinates

 
 1 d2 Z  ( + 1) 2 d − − − R (r) + 2 dr 2 r dr r r2 = E R (r) .

(9.3)

The general solution to (9.3) is

The separable solutions of (9.1) in spherical coordinates are ψ (r) = Ym (θ, φ) R (r) ,

equation

R (r) = r  exp (ikr) [A 1 F1 (a; c; z) + BU (a, c, z)] ,

(9.2)

(9.4)

where Ym (θ, φ) is a spherical harmonic as defined by Edmonds [9.13] and R (r) is a solution of the radial

where 1 F1 and U are the regular and irregular solutions of the confluent hypergeometric equation defined

Part A 9

Hydrogenic Wa 9. Hydrogenic Wave Functions

154

Part A

Mathematical Methods

Part A 9.1

in (9.130) and (9.131) below, and √ k = 2E , a =  + 1 − ik−1 Z , c = 2 + 2 , z = − 2ikr .

(9.5) (9.6) (9.7)

The Rn, can be expanded in powers of 1/n [9.15] 1/2
2Z 2 (n + )! Rn, (r) = − (n −  − 1)!n 2+4 ∞    × r −1/2 gk() (8Zr)1/2 n −2k ,

(9.8)

k=0

(9.14)

A and B are arbitrary constants. The solution given in (9.4) has an r −−1 singularity at r = 0 unless B = 0 or a is a non-positive integer. The leading term for small r is proportional to r  when B = 0 and/or a is a nonpositive integer. The large r behavior of the solution for (9.4) follows from (9.134), (9.135), and (9.164) below. Bound state solutions, with energy 1 E = − Z 2 n −2 2

(9.9)

are obtained when a = −n +  + 1 where n >  is the principal quantum number. The properly normalized bound state solutions are 
 2Z Z(n −  − 1)! 2Zr  Rn, (r) = 2 (n + )! n n × exp (−Zr/n) L (2+1) n−−1 (2Zr/n) ,

where L (2+1) n−−1 is the generalized Laguerre polynomial defined in (9.187). The relation in (9.188) shows that 1 F1 and U are linearly dependent in this case, so that (9.4) is no longer the general solution of (9.3). A linearly independent solution for this case can be obtained by replacing the L (2+1) n−−1 (2Zr/n) in (9.10) by the second (2+1) (irregular) solution Mn−−1 (2Zr/n) of the Laguerre equation [see (9.194), (9.196), and (9.197)]. The first three Rn, are (9.11) R1,0 (r) = 2Z 3/2 exp (−Zr) , 
3/2
1 1 Z R2,0 (r) = (2 − Zr) exp − Zr , 2 2

R2,1 (r) =

where the functions are finite linear combinations of Bessel functions: k  () gk() (z) = z 3k ak,m J2+2m+k+1 (z) . (9.15) m=0 () The coefficients ak,m in (9.15) are calculated recursively from (2 + 2m + k + 1) () ak,m = 32 (2k + m) (2 + m + 2k + 1) 1 × (2 + 2m + k − 1)  () × (2 + 2m + k − 1) ak−1,m  () , + 32 (k − m + 1) (2 + m − k) ak,m−1

(9.16)

starting with the initial condition (9.10)




gk() (z)

(9.12)


 
1 1 3/2 1 1/2 Z Zr exp − Zr . 2 3 2

(9.13)

Additional explicit expressions, together with graphs of some of them, can be found in Pauling and Wilson [9.14].

() =1. a0,0

(9.17)

The expansion (9.14) converges uniformly in r for r in any bounded region of the complex r plane. However, it converges fast enough so that a few terms give a good description of Rn, only if r is small. The square root in (9.14) has not been expanded in inverse powers of n because it has a branch point at 1/n = 1/ which would reduce the radius of convergence of the expansion to 1/. In some cases, large n expansions of matrix elements can be obtained by inserting (9.14) for Rn, and integrating term by term; examples can be found in Drake and Hill [9.15]. An asymptotic expansion in powers of 1/n, which is valid from r equal to an arbitrary fixed positive number through the turning point at r = 2n 2 /Z out to r = ∞, can be assembled from (9.133), (9.166) – (9.181), and (9.188) below. The Rn, are not a complete set because the continuum has been left out. The Sturmian functions ρk, , given by  β 3 k! ρk, (β; r) = (βr) e−βr/2 Γ (k + 2 + 3) × L (2+2) (βr) , k

(9.18)

Hydrogenic Wave Functions

9.1.2 Parabolic Coordinates The Schrödinger equation (9.1) is separable in parabolic coordinates ξ, η, φ, which are related to spherical coordinates r, θ, φ via ξ = r + z = r [1 + cos (θ)] , η = r − z = r [1 − cos (θ)] , φ=φ.

(9.19)

where

= 4π

(9.22)




with 1 F1 and U defined in (9.130), (9.131) below, and √ k1 = ± k2 = ± 2E , (9.25) 1 a1 = (|m| + 1) − ik1−1 µ , (9.26) 2 1 a2 = (|m| + 1) − ik2−1 (Z − µ) , (9.27) 2 c = |m| + 1 . (9.28) A, B, C, and D are arbitrary constants; µ is the separation constant. An important special case is the well-known Coulomb function


1 −1 πk Z + ik · r ψC (r) = Γ 1 − ik−1 Z exp 2   −1 × 1 F1 ik Z; 1; i (kr − k · r) , (9.29) which is obtained by orienting the z-axis in the k direction and taking m = 0, −k1 = k2 = |k|, µ = Z + 12 i|k|.

  ∞   Γ  + 1 − ik−1 Z (2 + 1)! =0 m=−

−1

∗ × (−2ik) eπk Z/2 Ym (θk , φk )Ym (θ, φ)

 ikr −1 × r e 1 F1  + 1 − ik Z; 2 + 2; −2ikr ,

(9.21)

 1 ik1 η [A 1 F1 (a1 ; c; −ik1 η) 2 + BU (a1 , c, −ik1 η)] , (9.23)
 1 ik2 ξ [C 1 F1 (a2 ; c; −ik2 ξ) Ξ (ξ) = ξ |m|/2 exp 2 + DU (a2 , c, −ik2 ξ)] , (9.24) N (η) = η|m|/2 exp

ψC (r)

(9.20)

This separability in a second coordinate system is related to the existence of a “hidden” O(4) symmetry, which is also responsible for the degeneracy of the bound states [9.16, 17]. The solutions in parabolic coordinates are particularly convenient for derivations of the Stark effect and the Rutherford scattering cross section. The separable solutions of (9.1) in parabolic coordinates are ψ (r) = exp (imφ) N (η) Ξ (ξ) ,

ψC is normalized to unit incoming flux [see (9.34) below]. In applications, Z is often replaced by −Z 1 Z 2 , so that the Coulomb potential in (9.1) becomes +Z 1 Z 2 /r. Equation (9.232), the addition theorem for the spherical harmonics ([9.13] p. 63 Eq. 4.6.6), and the λ = c = 1 special case of (9.163) below can be used to expand ψC into an infinite sum of solutions of the form (9.2):

(9.30)

where k, θk , and φk are the spherical coordinates of k. ψC can be split into an incoming plane wave and an outgoing spherical wave with the aid of (9.134) below: ψC (r) = ψin (r) + ψout (r) ,

(9.31)

where


1 ψin (r) = exp ik · r − πk−1 Z 2   −1 (9.32) × U ik Z; 1; i (kr − k · r) , 
 Γ 1 − ik−1 Z 1  exp ikr − πk−1 Z ψout (r) = − 2 Γ ik−1 Z   −1 × U 1 − ik Z; 1; −i (kr − k · r) . (9.33)

The functions ψin and ψout can be expanded for kr − k· r large with the aid of (9.164). The result is   ψin (r) ∼ exp ik · r − ik−1 Z ln (kr − k · r)

 2 ∞  (−i)n Γ ik−1 Z + n  × n! Γ ik−1 Z n=0

× (kr − k · r)−n , (9.34)  −1 iΓ 1 − ik Z ψout (r) ∼ −  −1 Γ ik Z (kr − k · r)   × exp ikr − ik−1 Z ln (kr − k · r)

 2 ∞ n  Γ 1 − ik−1 Z + n i  × n! Γ 1 − ik−1 Z n=0

× (kr − k · r)−n .

(9.35)

155

Part A 9.1

do form a complete orthonormal set. The positive constant β, which is independent of k and , sets the length scale for the basis set (9.18).

9.1 Schrödinger Equation

156

Part A

Mathematical Methods

Part A 9.1

Because (9.1) is an elliptic partial differential equation, its solutions must be analytic functions of the Cartesian coordinates (except at r = 0, where the solutions have cusps). The n = 0 special case of (9.138) shows that ψin and ψout are logarithmically singular at k · r = kr. Thus ψin and ψout are not solutions to (9.1) at k · r = kr. The logarithmic singularity cancels when ψin and ψout are added to form ψC , which is a solution to (9.1). Bound state solutions, with energy

bound state solutions to (9.39). Let p, θ p , φ p and p , θ p , φp be the spherical coordinates of p and p . Change √ variables from p, p to χ, χ  via p = −2E tan (χ/2) √ and p = −2E tan χ  /2 . This brings (9.39) to the form   √ 2π 2 Z −1 −2E sec4 (χ/2)φ( p)   4      2     sec χ /2 φ p sin χ dχ sin θ p dθ p dφ p = , 2 − 2 [cos(χ)cos(χ  )+ sin(χ)sin(χ  )cos(γ  )]

1 E = − Z 2 (n 1 + n 2 + |m| + 1)−2 , (9.36) 2 are obtained when a1 = −n 1 and a2 = −n 2 where n 1 and n 2 are non-negative integers. The properly normalized bound state solutions, which can be put into one–one correspondence with the bound state solutions in spherical coordinates, are

(9.42)

ψn 1 ,n 2 ,m (η, ξ, φ)  β 2|m|+4 n 1 !n 2 ! = 2πZ (n 1 + |m|)! (n 2 + |m|)!   1 × exp imφ − β (η + ξ) 2 (|m|) × (ηξ)|m|/2 L (|m|) n 1 (βη) L n 2 (βξ) ,

(9.37)

where γ  is the angle between p and p . Equation (9.42) is solved by introducing spherical coordinates and spherical harmonics in four dimensions via a natural extension of the procedure used in three dimensions. Going to polar coordinates on x and y yields the cylindrical coordinates r2 , φ, z; the further step of going to polar coordinates on r2 and z yields spherical coordinates r3 , θ, φ. If there is a fourth coordinate w, spherical coordinates in four dimensions are obtained via the additional step of going to polar coordinates on r3 and w. The result is x = r4 sin (χ) sin (θ) cos (φ) ,

(9.43)

y = r4 sin (χ) sin (θ) sin (φ) ,

(9.44)

z = r4 sin (χ) cos (θ) ,

(9.45)

w = r4 cos (χ) .

where −1

β = Z (n 1 + n 2 + |m| + 1)

.

(9.38)

9.1.3 Momentum Space

(9.39)

in momentum space. Its solutions are related to the solutions in coordinate space via the Fourier transforms

−3/2

φ ( p) = (2π)

The volume element, which is easily obtained via the same series of transformations, is dV = r43 dr4 dΩ4 ,

The nonrelativistic Schrödinger equation (9.1) becomes the integral equation   φ p 1 2 Z p φ ( p) − 2 d3 p = Eφ ( p) 2 2π ( p − p )2

ψ (r) = (2π)−3/2

(9.46)

dΩ4 = sin2 (χ) dχ sin (θ) dθ dφ .

+1 × Cn−−1 [cos (χ)] Ym (θ, φ) ,

(9.49)

exp (i p · r) φ ( p) d3 p , (9.40) exp (−i p · r) ψ (r) d r . 3

(9.41)

A trick of Fock’s [9.16, 18] can be used to expose the “hidden” O(4) symmetry of hydrogen and construct the

(9.48)

The four-dimensional spherical harmonics [9.2, Vol. 2, Chap. XI] are  n (n −  − 1)!  sin (χ) Yn,,m (χ, θ, φ) = 2+1 ! 2π (n + )!

 

(9.47)

+1 where Cn−−1 is a Gegenbauer polynomial and n ≥  + 1 is an integer. They have the orthonormality property  ∗ Yn,,m (χ, θ, φ) Yn  , ,m  (χ, θ, φ) dΩ4

= δn,n  δ, δm,m  .

(9.50)

Hydrogenic Wave Functions

=

∞  n−1    2π 2 n−1 t Yn,,m (χ, θ, φ) n n=1 =0 m=−     ∗ × Yn,,m χ ,θ ,φ

(9.51)

 holds for |t| < 1, where γ  is the angle between    p and p .  Multiply both sides of (9.51) by Yn,,m χ , θ , φ dΩ4 (where dΩ4 is dΩ4 with χ, θ, φ replaced by χ  , θ  , φ ) and use the orthogonality relation (9.50). The result can be rearranged to the form

2π 2 n −1 t n−1 Yn,,m (χ, θ, φ)=    Yn,,m χ  , θ  , φ sin2 χ  dχ  sin(θ) dθ dφ . 1−2 [cos(χ)cos(χ  )+ sin(χ)sin(χ  )cos(γ  )] t+t 2

where the properly normalized radial functions are  n !

Fn, ( p) = 2

2+2 2

2(n −  − 1)! n p  πZ 3 (n + )! Z

Z 2+4 × +2 n 2 p2 + Z 2
2 2  n p − Z2 +1 × Cn−−1 . n 2 p2 + Z 2 (9.55)

The first three Fn, are 

2 Z4  , πZ 3 p2 + Z 2 2  32 Z 4 4 p2 − Z 2 F2,0 ( p) = √  , πZ 3 4 p2 + Z 2 3 F1,0 ( p) = 4

(9.56)

(9.57)

Z5 p 128 F2,1 ( p) = √  . 3πZ 3 4 p2 + Z 2 3

(9.58)

(9.52)

Analytic continuation can be used to show that (9.52) is valid for all complex t despite the fact that (9.51) is restricted to |t| < 1. Comparing the t = 1 case of (9.52) with (9.42) shows that E = − 12 Z 2 n −2 in agreement with (9.9), and that   normalizing cos4 (χ/2) φ (p ) = factor × Yn,,m (χ, θ, φ) .

(9.53)

Transforming from χ back to p brings these to the form  (9.54) φ ( p) = Ym θ p , φ p Fn, ( p) ,

The Fn, satisfy the integral equation 1 2 p Fn, ( p) 2  ∞
2  p + p2 Z Fn, p p d p Q − πp 2 p p 0

= E Fn, ( p) ,

(9.59)

which can be obtained by inserting (9.54) in (9.39). Here Q  is the Legendre function of the second kind, which is defined in (9.233) below.

9.2 Dirac Equation The relativistic Dirac equation for a hydrogenic ion of nuclear charge Z can be reduced to dimensionless form by using the Compton wavelength ~/ (mc) for the length scale and the rest mass energy mc2 for the energy scale. The result is
 Zα −iα · ∇ + β − ψ (r) = Eψ (r) , r

(9.60)

where α = e2 / (~c) is the fine structure constant, and α, β are the usual Dirac matrices:

α=

0 σ σ 0





,

β =

I 0 0 −I

.

(9.61)

Here σ is a vector whose components are the two by two Pauli matrices, and I is the two by two identity matrix

157

Part A 9.2

Equations (9.229) and (9.230) with λ = 1, equation (9.231), and the addition theorem for the three dimensional spherical harmonics Ym can be used to show that    1 − 2 cos (χ) cos χ  −1    + sin (χ) sin χ  cos γ  t + t 2

9.2 Dirac Equation

158

Part A

Mathematical Methods

Part A 9.2

given by





0 −i , i 0  

1 0 1 0 . , I= (9.62) σz = 0 1 0 −1 The solutions to (9.60) in spherical coordinates have the form   G (r) χκm (θ, φ)   , (9.63) ψ (r) =    m iF (r) χ−κ (θ, φ) σx =

0 1 1 0

,

σy =

where, for positive energy states, G (r) is the radial part of the large component and iF (r) is the radial part of the small component. For negative energy states, G (r) is the radial part of the small component and iF (r) is the radial part of the large component. χ is the two component spinor

1/2   κ κ + 12 − m Y|κ+ 1 |− 1 ,m− 1  − 2 2 2   |κ| 2κ + 1   m χκ =  . 1/2   1 κ + 2 +m   Y|κ+ 1 |− 1 ,m+ 1 2 2 2 2κ + 1 (9.64)

The relativistic quantum number κ is related to the total angular momentum quantum number j by 
1 . κ =± j+ (9.65) 2 Because j takes on the values 12 , 32 , 52 , . . . , κ is restricted to the values ±1, ±2, ±3, . . . . The spinor χκm obeys the useful relations m σ · rˆ χκm = − χ−κ ,

(9.66)

σ · L χκm = − (κ + 1) χκm ,

(9.67)

where rˆ = r/r and L = r × p with p = −i∇. Equations (9.66), (9.67), and the identity
  iσ · L σ · p = σ · rˆ rˆ · p + (9.68) r can be used to derive the radial equations, which are

 d 1+κ Zα + G(r) − 1 + E + F(r) = 0 , dr r r


(9.69)


 1−κ Zα d + F(r) − 1 − E − G(r) = 0 . dr r r

(9.70)

Equations (9.158), (9.159), (9.161), and (9.162) below can be used to show that the general solution to (9.69) and (9.70) is G (r) = r γ exp (−λr) (1 + E)1/2 {A [ f 2 (r) + f 1 (r)] +B [ f 4 (r) + f 3 (r)]} ,

(9.71)

F (r) = r γ exp (−λr) (1 − E)1/2 {A [ f 2 (r) − f 1 (r)] +B [ f 4 (r) − f 3 (r)]} ,

(9.72)

f 1 (r) = Zαλ−1 − κ 1 F1 (a; c; 2λr) ,

(9.73)

f 2 (r) = a 1 F1 (a + 1; c; 2λr) ,

(9.74)

f 3 (r) = U (a, c, 2λr) ,

f 4 (r) = Zαλ−1 + κ U (a + 1, c, 2λr) ,

(9.75)

where

λ = (1 + E)1/2 (1 − E)1/2 ,

1/2 γ = − 1 + κ 2 − Z 2 α2 ,

(9.76) (9.77) (9.78)

a = 1 + γ − λ−1 E Zα ,

(9.79)

c = 3 + 2γ .

(9.80)

A and B are arbitrary constants. Because γ is in general not an integer, the solutions have a branch point at r = 0, and become infinite at r = 0 when κ = ±1, which makes γ negative. The solutions for E < −1 and E > +1 are in the continuum, which implies that one of the factors (1 + E)1/2 , (1 − E)1/2 is real with the other imaginary. Square integrable solutions, with energy E n,κ =

−1/2
Z 2 α2 Z 1+ , |Z| (n + 1 + γ )2

(9.81)

are obtained when a = −n where n is a non-negative integer. The properly normalized square integrable solutions are 1/2  G n,κ (r) = Cn,κ (2λr)γ exp (−λr) 1 + E n,κ   (2) (1) × gn,κ (9.82) (r) + gn,κ (r) , 1/2  Fn,κ (r) = Cn,κ (2λr)γ exp (−λr) 1 − E n,κ   (2) (1) × gn,κ (9.83) (r) − gn,κ (r) ,

1/2 (2+2γ) (1) gn,κ Ln (r) = Zαλ−1 − κ (2λr) , (9.84)

Hydrogenic Wave Functions

(2+2γ)

× L n−1 

(2λr) ,

2λ4 n! . ZαΓ (n + 3 + 2γ )

Cn,κ =

(9.85)

2E 2 −2

(9.86)

When n = 0, |Zαλ−1 | = |κ|, and the value of κ whose sign is the same as the sign of Zαλ−1 is not permitted. (2+2γ) (2) Also, L −1 (2λr) is counted as zero, so that g0,κ (r) = 0. The eigenvalues and eigenfunctions for the first four states for Z > 0 will now be written out explicitly in terms of the variable ρ = Zαr. For the 1 S1/2 ground state, with n = 0, j = 12 , κ = −1, the formulae are  (9.87) E 0,−1 = 1 − Z 2 α2 ,   4Z 3 α3 1 + E 0,−1 (2ρ) E0,−1 −1 e−ρ ,  G 0,−1 (r) = Γ 1 + 2E 0,−1  F0,−1 (r) = −

 4Z 3 α3 1 − E 0,−1  Γ 1 + 2E 0,−1

× (2ρ) E0,−1 −1 e−ρ .

  Z 3 α3 2E − κ 1 + E  1,κ 1,κ

G 1,κ (r) =  2 2 2E 1,κ Γ 4E 1,κ + 1

(9.88)

(9.89)

The formulae for the 2 S1/2 excited state, with n = 1, j = 12 , κ = −1, and for the 2 P1/2 excited state, with n = 1, j = 12 , κ = 1, can be written together. They are 1/2
1 1 E 1,κ = + 1 − Z 2 α2 , (9.90) 2 2

× ρ1 1,κ e−ρ1 /2    × 2E 1,κ − κ − 1 2E 1,κ + κ − ρ1 , (9.91)

  Z 3 α3 2E − κ 1 − E  1,κ 1,κ

F1,κ (r) = −  2 Γ 4E 2 + 1 2E 1,κ 1,κ 2E 2 −2

× ρ1 1,κ e−ρ1 /2    × 2E 1,κ − κ + 1 2E 1,κ + κ − ρ1 , (9.92)

where ρ1 = ρ/E 1,κ . For the 2 P3/2 excited state, with n = 0, j = 12 , κ = −2, the formulae are  1 E 0,−2 = 1 − Z 2 α2 , (9.93) 4   Z 3 α3 1 + E 0,−2 2E0,−2 −1 −ρ/2 ρ  G 0,−2 (r) = e , 2Γ 1 + 4E 0,−2 (9.94)

 F0,−2 (r) = −

 Z 3 α3 1 − E 0,−2 2E0,−2 −1 −ρ/2 ρ  e . 2Γ 1 + 4E 0,−2 (9.95)

9.3 The Coulomb Green’s Function The abstract Green’s operator for a Hamiltonian H is the inverse G (E) = (H − E)−1 . It is used to write the solution to (H − E) |ξ = |η in the form |ξ = G|η. It has the spectral representation   1 G (E) = |e j  e j | . (9.96) Ej −E j

The sum over j in (9.96) runs over all of the spectrum of H, including the continuum. For the bound state part of the spectrum, the numbers E j and vectors |e j  are the eigenvalues and eigenvectors of H. For the continuous spectrum, |e j  e j | is a projection valued measure [9.19]. The representation (9.96) shows that G (E) has first order poles at the eigenvalues. The reduced Green’s operator (also known as the generalized Green’s operator),

which is the ordinary Green’s operator with the singular terms subtracted out, remains finite when E is at an eigenvalue. It can be calulated from  ∂ (red) G (E k ) = lim [(E − E k ) G (E)] . E→E k ∂E (9.97)

The coordinate and momentum space representatives of the abstract Green’s operator are the Green’s functions. The nonrelativistic Coulomb Green’s function has been discussed by Hostler and Schwinger [9.20, 21]. A unified treatment of the Coulomb Green’s functions for the Schrödinger and Dirac equations has been given by Swainson and Drake [9.22]. Reduced Green’s functions are discussed in the third of the Swainson–Drake papers, and in the paper of Hill and Huxtable [9.23].

159

Part A 9.3



−1/2 (2) gn,κ (r) = − (n + 2 + 2γ ) Zαλ−1 − κ

9.3 The Coulomb Green’s Function

160

Part A

Mathematical Methods

Part A 9.3

9.3.1 The Green’s Function for the Schrödinger Equation The Green’s function G (S) for the Schrödinger equation (9.98) is a solution of 
  Z 1 − ∇ 2 − − E G (S) r, r  ; E = δ r − r  . 2 r (9.98)

An explicit closed form expression for G (S) is Γ (1 − ν)  G (S) r, r  ; E = 2π|r − r  |  ∂ × Wν, 1 (z 2 ) M 1 (z 1 ) 2 ∂z 1 ν, 2  ∂ − Mν, 1 (z 1 ) Wν, 1 (z 2 ) , 2 2 ∂z 2 (9.99)

where Mν,1/2 and Wν,1/2 are the Whittaker functions defined in (9.132) and (9.133) below, and ν = Z (−2E)−1/2 ,  z 1 = (−2E)1/2 r + r  − |r − r  | ,  z 2 = (−2E)1/2 r + r  + |r − r  | .

(9.100) (9.101) (9.102)

The branch on which (−2E)1/2 is positive should be taken when E < 0. When E > 0, the branch which corresponds to incoming (or outgoing) waves at infinity can be selected with the aid of the asymptotic approximation

Γ (1 − ν) ν  1 , z G (S) r, r  ; E ≈ exp − z 2 2 2 2π|r − r  | (9.103)

which holds when z 2 z 1 . This approximation is obtained by using (9.130), (9.132), (9.133), and (9.164) in (9.99). A number of useful expansions for G (S) can be obtained from the integral representation  G (S) r, r  ; E
2ν ∞  2Z 1 coth ρ sinh (ρ) = ν 2 0 ! "  1/2 × I0 ν−1 Z sinh (ρ) 2rr  [1 + cos (Θ)]    (9.104) × exp −ν−1 Z r + r  cosh (ρ) dρ , where Θ is the angle between r and r  . These expansions, and other integral representations, can be found in [9.20–

22]. The partial wave expansion of G (S) is  G (S) r, r  ; E  (S)     ∗ = g r, r  ; ν Ym (θ, φ) Ym θ ,φ . ,m

(9.105) (S)

The radial Green’s function g is a solution of the radial equation 
  ( + 1) 1 d2 2 d − − + 2 dr 2 r dr r2   δ r − r Z (S)   − − E g r, r ; ν = . (9.106) r rr  The standard method for calculating the Green’s function of a second order ordinary differential equation ([9.24] pp. 354–355) yields (S)  g r, r  ; ν    (2Z)2+2 Γ ( + 1 − ν) exp −ν−1 Z r + r  = 2+1 (2 + 1)!ν

   × rr 1 F1  + 1 − ν; 2 + 2; 2ν−1 Zr< [3 pt]

× U  + 1 − ν, 2 + 2, 2ν−1 Zr> , (9.107) where r< is the smaller of the pair r, r  and r> is the (S) larger of the pair r, r  . Matrix elements of g can be calculated with the aid of the formula for the double Laplace transform, which is ∞

∞ dr

0

=

 +1 (S)   dr  rr  exp −λr − λr  g r, r  ; ν

0

2 (2 + 1)! ν 2+3  − ν + 1 2Z




4Z 2 (νλ + Z) (νλ + Z)

2+2

× 2 F1 (2 + 2,  − ν + 1;  − ν + 2; 1 − ζ ) , (9.108)

where

 2νZ λ + λ , ζ= (νλ + Z) (νλ + Z)

(9.109)

Matrix elements with respect to Slater orbitals can be calculated from (9.108) by taking derivatives with respect to λ and/or λ to bring down powers of r and r  . Matrix elements with respect to Laguerre polynomials can be calculated by using (9.108) to evaluate integrals

Hydrogenic Wave Functions

(9.110)

     G˜ (S) p, p ; E = (2π)−3 exp −i p · r − p · r   × G (S) r, r  ; E d3 r d3r  .

(9.111)

The
Green’s function G˜ (S) is a solution of   1 1 2 Z p − E G˜ (S) p, p ; E − 2 2π 2 ( p − p )2    3   (S)  p , p ; E d p = δ p − p . (9.112) × G˜ An explicit closed form expression for G˜ (S) is  G˜ (S) p, p ; E  δ p − p = 1 2 2p −E Z

  + 1 2π 2 | p − p |2 2 p2 − E 12 ( p )2 − E  νq × 1+ 1−ν 

  1−q 1−q × 1, 1 − ν; 2 − ν; F 2 1 1+q 1+q
 
1+q 1+q , − 2 F1 1, 1 − ν; 2 − ν; 1−q 1−q (9.113)

where  q=

2E | p − p |2 4E 2 − 4E p · p + ( p p )2

.

(9.114)

9.3.2 The Green’s Function for the Dirac Equation The Green’s function G D for the Dirac equation (9.60) is a 4
× 4 matrix valued solution  of  Zα − E G D r, r  ; E −iα · ∇ + β − r  = δ r − r  I4 , (9.115)

where I4 is the 4 × 4 identity matrix. The partial wave expansion of G D is

    G κ,m −iG κ,m 11 12 G D r, r ; E = , (9.116) iG κ,m G κ,m κ,m 21 22 where

  m m†  θ , φ g11 r, r  ; E , G κ,m 11 = χκ (θ, φ) χκ G κ,m 12

=

χκm

 m†  (θ, φ) χ−κ θ  , φ g12 r, r  ;

(9.117)

E ,

(9.118)

G κ,m 21 G κ,m 22

    θ , φ g21 r, r  ; E ,

=

m χ−κ

(θ, φ) χκm †

=

m χ−κ

 m†  (θ, φ) χ−κ θ  , φ g22 r, r  ;

(9.119)

E .

(9.120)

The identity

  δ r − r  δ r − r I4 = rr  

  χ m (θ, φ) χκm † θ  , φ 0 κ × m†  m 0 χ−κ (θ, φ) χ−κ θ  , φ κ,m (9.121)

can  be used to show that the radial functions g jk r, r  ; E satisfy the equation 

  1−κ Zα d − + 1 − E −   r dr r     

  d 1+κ Zα  + − 1+E+ dr r r

   g11 r, r  ; E g12 r, r  ; E ×   g21 r, r  ; E g22 r, r  ; E   δ r − r 1 0 . = rr  0 1

(9.122)

The solution to (9.122) is

   g11 r, r  ; E g12 r, r  ; E (2λ)1+2γ Γ (a)     = Γ (3 + 2γ ) g21 r, r ; E g22 r, r ; E #  

      G < (r) × Θ r −r G > r F> r F< (r) 

$ G > (r)       G < r F< r , +Θ r − r F> (r) (9.123)

161

Part A 9.3

over the generating function (9.199) for the Laguerre polynomial [9.23]. Other methods of calculating matrix elements are discussed in Swainson and Drake [9.22]. The Green’s function G˜ (S) in momentum space is related to the coordinate space Green’s function G (S) via the Fourier transforms      G (S) r, r  ; E = (2π)−3 exp i p · r − p · r   × G˜ (S) p, p ; E d3 p d3 p ,

9.3 The Coulomb Green’s Function

162

Part A

Mathematical Methods

Part A 9.4

where a is defined by (9.79), Θ is the Heaviside unit function, defined by  1, x > 0 ,    1 Θ (x) = 2, x = 0 ,    0, x < 0,

F< (r) = r γ exp (−λr) (1 − E)1/2 [ f 2 (r) − f 1 (r)] , (9.126) γ

G > (r) = r exp (−λr) (1 + E)

1/2

[ f 4 (r) + f 3 (r)] , (9.127)

(9.124)

F> (r) = r γ exp (−λr) (1 − E)1/2 [ f 4 (r) − f 3 (r)] . (9.128)

The functions G < (r) and F< (r) obey the boundary conditions at r = 0. The functions G > (r) and F> (r) obey the boundary conditions at r = ∞. Integral representations and expansions for the Dirac Green’s function can G < (r) = r γ exp (−λr) (1 + E)1/2 [ f 2 (r) + f 1 (r)] , be found in [9.22] and [9.25]. Matrix element evaluation (9.125) is discussed in [9.22].

and the functions G < , F< , G > , and F> are special cases of the homogeneous solutions (9.71 – 9.80):

9.4 Special Functions This section contains a brief list of formulae for the special functions which appear in the solutions discussed above. Derivations, and many additional formulae, can be found in the standard reference works listed in the bibliography. For numerically useful approximations and available software packages, see Olver et al. [9.12], and Lozier and Olver [9.26].

9.4.1 Confluent Hypergeometric Functions The confluent hypergeometric differential equation is   d2 d z 2 + (c − z) − a w (z) = 0 . (9.129) dz dz Equation (9.129) has a regular singular point at r = 0 with indices 0 and 1 − c and an irregular singular point at ∞. The regular solution to (9.129) is the confluent hypergeometric function, denoted by 1 F1 in generalized hypergeometric series notation. It can be defined by the series 1 F1 (a; c; z) =

Γ (c) Γ (a)

∞  n=0

Γ (a + n) z n . Γ (c + n) n!

(9.130)

The series (9.130) for 1 F1 converges for all finite z if c is not a negative integer or zero. It reduces to a polynomial of degree n in z if a = −n where n is a positive integer and c is not a negative integer or zero. The function 1 F1 (a; c; z) is denoted by the symbol M (a, c, z) in Abramowitz and Stegun [9.1], in Jahnke and Emde [9.5], and in Olver [9.9], by 1 F1 (a; c; z) in both of Luke’s books [9.6, 7] and in Magnus et al. [9.8], and by Φ (a, c; z) in the Bateman project [9.2, 3] and

Gradshteyn and Ryzhik [9.4]. The irregular solution to (9.129) is U (a, c, z) =

Γ (1 − c) 1 F1 (a; c; z) Γ (1 + a − c) Γ (c − 1) 1−c + z Γ (a) × 1 F1 (1 + a − c; 2 − c; z) .

(9.131)

The function U (a, c, z) is multiple-valued, with principal branch −π < arg z ≤ π. It is denoted by the symbol U (a, c, z) in Abramowitz and Stegun [9.1], in Magnus et al. [9.8], and in Olver [9.9], by ψ (a; c; z) in the first of Luke’s books [9.6], by U (a; c; z) in the second of Luke’s books [9.7], and by Ψ (a, c; z) in the Bateman project [9.2, 3] and Gradshteyn and Ryzhik [9.4]. The Whittaker functions Mκ,µ and Wκ,µ , which are related to 1 F1 and U via 
1 1 Mκ,µ (z) = exp − z z µ+ 2 2
 1 ×1 F1 µ + − κ; 2µ + 1; z , 2 
1 1 Wκ,µ (z) = exp − z z µ+ 2 2 
1 × U µ + − κ, 2µ + 1, z , 2

(9.132)

(9.133)

are sometimes used instead of 1 F1 and U. For numerical evaluation and a program, see [9.27, 28].

Hydrogenic Wave Functions

1 F1 (a; c; z) =

Γ (c) Γ (c) iπa e U (a, c, z) + Γ (c − a) Γ (a) × ez+iπ(a−c) U (c − a, c, −z) , 

=

(9.134)

+1 , Im z > 0 , −1 , Im z < 0 .

0

(9.135)

can also be obtained from U as the discontinuity across a branch cut: 1 F1

z c−1 exp (−z) 1 F1 (a; c; z) Γ (1 − a) Γ (c)  −iπ c−1  ze = U c − a, c, ze−iπ 2πi  iπ c−1   − ze U c − a, c, zeiπ . (9.136)

(9.137)

k=−n

(9.138)

[Ψ (k + a) − Ψ (k + 1)    − Ψ (k + n + 1)] /k! ,

−n ≤ k ≤ −1 ,

Here Ψ is the logarithmic derivative of the gamma function: Ψ (z) = Γ (z) /Γ (z) .

0

(9.142)

The basic transformation formulae for 1 F1 and U are 1 F1 (a; c; z) =

ez 1 F1 (c − a; c; −z) ,

U (a, c, z) = z

1−c

(9.143)

U (a − c + 1, 2 − c, z) .

(9.140)

(9.145)

(z + a − 1) 1 F1 (a; c; z) = (a − c) 1 F1 (a − 1; c; z) + (c − 1) 1 F1 (a; c − 1; z) ,

(9.146)

c 1 F1 (a; c; z) = c 1 F1 (a − 1; c; z) + z 1 F1 (a; c + 1; z) ,

(9.147)

(a + 1 − c) 1 F1 (a; c; z) = a 1 F1 (a + 1; c; z) + (1 − c) 1 F1 (a; c − 1; z) ,

(9.148)

c (z + a) 1 F1 (a; c; z) = a c 1 F1 (a + 1; c; z) + (c − a) z 1 F1 (a; c + 1; z) ,

(9.149)

c (z + c − 1) 1 F1 (a; c; z) = c (c − 1) × 1 F1 (a; c − 1; z) + (c − a) 1 F1 (a; c + 1; z) ,

(9.150)

(z + 2a − c) U (a; c; z) = U (a − 1; c; z) + a (a − c + 1) U (a + 1; c; z) ,

k≥0. (9.139)



1 Γ (a)

+ a 1 F1 (a + 1; c; z) ,

(−1)n+1 U (a, n + 1, z) = Γ (a − n)  1 × ln (z) 1 F1 (a; n + 1; z) n! $ ∞  Γ (a + k) ak z k + , Γ (a) (k + n)!

ak =

U (a, c, z) =

The recurrence relations among contiguous functions are (z + 2a − c) 1 F1 (a; c; z) = (a − c) 1 F1 (a − 1; c; z)

A formula for U (a, c, z) when c is the integer n + 1 can be obtained by taking the c → n + 1 limit of the right-hand side of (9.131) to obtain

 (−1)k+1 (−k − 1)! ,   

(9.141)

∞ e−zt t a−1 (1 + t)c−a−1 dt .

(9.144)

The Wronskian of the two solutions is d U (a, c, z) 1 F1 (a; c; z) dz d − U (a, c, z) 1 F1 (a; c; z) dz = −Γ (c) z −c exp (z) /Γ (a) .

where

n is a non-negative integer. When n = 0, the sum from k = −n to −1 is omitted. The basic integral representations for 1 F1 and U are Γ (c) 1 F1 (a; c; z) = Γ (a) Γ (c − a) 1 × ezt t a−1 (1 − t)c−a−1 dt ,

(9.151)

(z + a − 1) U (a; c; z) = U (a − 1; c; z) + (c − a − 1) U (a; c − 1; z) ,

(9.152)

(c − a) U (a; c; z) = − U (a − 1; c; z) + zU (a; c + 1; z) ,

(9.153)

163

Part A 9.4

The regular solution can be written as a linear combination of irregular solutions via

9.4 Special Functions

164

Part A

Mathematical Methods

Part A 9.4

(a + 1 − c) U (a; c; z) = a U (a + 1; c; z) + U (a; c − 1; z) , (9.154) (z + a) U (a; c; z) = a (a − c + 1) U (a + 1; c; z) + zU (a; c + 1; z) ,

(9.155)

(z + c − 1) U (a; c; z) = (c − a − 1) (c − 1) × U (a; c − 1; z) + zU (a; c + 1; z) .

The asymptotic expansion of 1 F1 for large z is obtained by using (9.164) and exp [z + iπ (a − c)] U (c − a, c, −z) ∼ ez z a−c

∞  Γ (c − a + n) Γ (1 − a + n) −n z , n!Γ (c − a) Γ (1 − a) n=0

(9.156)

5 5 − π < arg z < π , (9.165) 2 2

Useful differentiation formulae include d −1 1 F1 (a + 1; c + 1; z) , 1 F1 (a; c; z) = a c dz (9.157)

 d  a z 1 F1 (a; c; z) = a z a−1 1 F1 (a + 1; c; z) , dz (9.158)

 d  −z c−a−1 e z 1 F1 (a + 1; c; z) dz = (c − a − 1) e−z z c−a−2 1 F1 (a; c; z) , d U (a, c, z) = −a U (a + 1, c + 1, z) , dz  d  a z U (a, c, z) dz = a (a − c + 1) z a−1 U (a + 1, c, z) ,  d  −z c−a−1 e z U (a + 1, c, z) dz = − e−z z c−a−2 U (a, c, z) .

(9.159) (9.160)

(9.161)

(9.162)

An important multiplication theorem is 1 F1 (a; c; z 1 z 2 ) ∞  Γ (a + k) Γ (λ + 2k) = (−z 1 )k k!Γ (a) Γ (λ + k) k=0

× 2 F1 (−k, λ + k; c; z 1 ) × 1 F1 (a + k; λ + 2k + 1; z 2 ) .

(9.163)

The fundamental asymptotic expansion for large z is U (a, c, z) ∞  Γ (a + n) Γ (1 + a − c + n) ∼ z −a (−z)−n , n!Γ (a) Γ (1 + a − c)

which is a consequence of (9.164), on the right-hand side of (9.134). In the asymptotic expansion (9.165), and in the asymptotic expansion for 1 F1 , the change in the factor exp [iπ (a − c)] as arg z passes through zero is compensated by the phase change which comes from a factor (−z)a−c in the asymptotic expansion of U (c − a, c, −z). The change in the factor exp (iπa) in the first term of (9.134) as arg z passes through zero is not compensated by any other phase change. However, this discontinuity occurs in a region in which this first term is negligible compared to the second term. This is an example of the Stokes phenomenon [9.29], which occurs because the single-valued function 1 F1 is being approximated by multiple-valued functions. The large z asymptotic expansion of 1 F1 is valid for − 32 π < arg z < 3 2 π, which is the overlap of the domain of validity of the expansions (9.164) and (9.165). Uniform asymptotic expansions for the Whittaker functions Mκ,µ and Wκ,µ introduced in (9.132), (9.133) have been constructed via Olver’s method. The following result [9.9], (p. 412, Ex. 7.3), which holds for x positive, κ large and positive, and µ unrestricted, gives the flavor of these approximations: 
24/3 π 1/2 κ κ+(1/6) xζ 1/4 Wκ,µ (4κx) = φn (κ, µ) exp (κ) x − 1   n  As (ζ ) × Ai (4κ)2/3 ζ 2s (4κ) s=0   Ai (4κ)2/3 ζ + (4κ)2/3 ×

n  Bs (ζ ) s=0

(4κ)2s

+ 2n+1,2 (4κ, ζ ) . (9.166)

n=0

3 3 − π < arg z < π . (9.164) 2 2

Here Ai is the Airy function, and 2n+1,2 is an error term which tends to zero faster than the last term kept when

Hydrogenic Wave Functions





1/2 4 , (−ζ )3/2 = cos−1 x 1/2 − x − x 2 3 0<x≤1.

(9.167)

(9.168)

ζ is an analytic function of x in the neighborhood of x = 1; conversely, x is an analytic function of ζ in the neighborhood of ζ = 0. The differential form of the relations (9.167), (9.168) is 2ζ 1/2 dζ = x −1/2 (x − 1)1/2 dx

x≥1, (9.169)

1/2

2 (−ζ )

dζ = x

−1/2

(1 − x)

1/2

dx

0<x≤1.

The functions As (ζ ), Bs (ζ ), and ψ (ζ ), which appear to be singular at ζ = 0, are actually analytic functions of ζ at ζ = 0. The first few coefficient functions are: A0 (ζ ) = 1 ,

3/2 
 x 1 1 −1/2 2 8µ − B0 (ζ ) = ζ 4 x −1 2 $
2
 x x 5 5 × − + − ζ −2 , x −1 x −1 6 48

(9.177)

1 A1 (ζ ) = ζ −1 [B0 (ζ ) − ψ (ζ )] + ζ [B0 (ζ )]2 . 4 (9.178)

The function φn is calculated from φn (κ, µ) =

1 Bs (ζ ) = 1/2 2ζ

ζ



0

 dη ψ (η) As (η) − As (η) 1/2 , η x≥1,

Bs (ζ ) =

(9.172)

0

1 2 (−ζ )1/2 

−As (η)

ζ

dη

1 1 As+1 (ζ ) = − Bs (ζ ) + 2 2



,

(4κ)

2s

−

n−1  s=0

lim

ζ→∞

ζ 1/2 Bs (ζ ) (4κ)2s+1

.

(9.179)

The first two φn are φ0 (κ, µ) = 1 , φ1 (κ, µ) = 1 −









12µ2 − 1 12µ2 − 1 + 24κ 24κ

(9.180)

2

. (9.181)

A bound for the error term 2n+1,2 has been given by Olver ([9.9] p. 410, Eq. 7.13). The extension to the complex case can be found in Skovgarrd [9.30]. Skovgarrd’s expansions are in powers of (4κ)−1 instead of (4κ)−2 , because the factor 1/φn (κ, µ) has been expanded out in inverse powers of 4κ in his results; as a consequence, his coefficient functions As and Bs differ from Olver’s. The corresponding

asymptotic expansions for U µ + 12 − κ, 2µ + 1, 4κz and for the Laguerre poly(2µ)

[ψ (η) As (η)

(−η)1/2

n  As (∞) s=0

(9.170)

A Taylor series expansion of ζ about x = 1 is most easily constructed by expanding the x −1/2 in (9.169) or (9.170) about x = 1 and integrating term by term. The opening terms of such an expansion are  1 ζ = 2−2/3 (x − 1) − (x − 1)2 5    17 + (9.171) (x − 1)3 + O (x − 1)4 . 175 The coefficient functions As and Bs are calculated recursively from

(9.176)




0<x≤1, (9.173)

ψ (ζ ) Bs (ζ ) dζ , (9.174)

 2 4µ − 1 ζ 5 (3 − 8x) ζ + ψ (ζ ) = + . x (x − 1) 4x (x − 1)3 16ζ 2

(9.175)

nomial L κ−µ−1/2 (4κz) can be constructed with the aid of (9.133) and (9.188). Formulae for matrix element integrals can be obtained by inserting the integral representation (9.141) and interchanging the orders of integration. An example is ∞ z µ 1 F1 (a1 ; c1 ; λ1 z) 1 F1 (a2 ; c2 ; λ2 z) dz 0

= F2 (µ + 1, a1 , a2 , c1 , c2 ; −λ1 , −λ2 ) .

(9.182)

Here F2 is one of the hypergeometric functions of two variables introduced by Appell, which can be defined

165

Part A 9.4

κ → ∞ with n fixed. ζ is related to x by 

1/2  4 3/2 2 ζ = x −x − ln x 1/2 + (x − 1)1/2 , 3 x≥1,

9.4 Special Functions

166

Part A

Mathematical Methods

Part A 9.4

either by the integral representation ([9.2], Vol. 1, p. 230, Eq. 2) F2 (α, β, β  , γ, γ  ; x, y) Γ(γ)Γ(γ  ) =  Γ(β)Γ(β )Γ(γ − β)Γ(γ  − β  ) 1 1  × du dv u β−1 vβ −1 (1 − u)γ −β−1 0

0

× (1 − v)γ

 −β  −1

(1 − ux − vy)−α ,

(9.183)

or by the series expansion ([9.2], Vol. 1, p. 224, Eq. 7) 



F2 (α, β, β , γ, γ ; x, y) ∞ ∞   Γ (α + m + n) = m!n!Γ (α) m=0 n=0   Γ (β + m) Γ β  + n Γ (γ ) Γ γ  m n x y , × Γ (β) Γ (β  ) Γ (γ + m) Γ (γ  + n) (9.184)

which converges for |x| + |y| < 1. Numerical evaluation of F2 is particularly convenient in the special cases where it can be expressed in terms of ordinary hypergeometric functions 2 F1 ([9.2], Vol. 1, Chap. 2), which are easy to calculate, or in terms of elementary functions. The key is the formula ([9.2], Vol. 1, p. 238, Eq. 3) F2 (α, β, β  , α, α; x, y) 

= (1 − x)−β (1 − y)−β
 xy , × 2 F1 β, β  ; α; (1 − x)(1 − y)

9.4.2 Laguerre Polynomials The Laguerre polynomials L (α) n are the polynomial solutions of the differential equation  2  d d z 2 + (α + 1 − z) + n L (α) n (z) = 0 . (9.187) dz dz They are a special case of the confluent hypergeometric function: Γ (n + α + 1) L (α) 1 F1 (−n; α + 1; z) n (z) = n!Γ (α + 1) (−1)n U (−n, α + 1, z) , n! and are given explicitly by =

L (α) n (z) =

n  k=0

(9.188)

Γ (α + n + 1) (−z)k . k! (n − k)!Γ (α + k + 1) (9.189)

The L (α) n are sometimes called generalized Laguerre polynomials, because L (0) n , which is often denoted by L n , is the polynomial introduced by Laguerre. This Laguerre polynomial differs from the “associated Laguerre p function” for which the symbol L q (with p and q both integers) is often used in the physics literature. The relation between the two is  p  ( p) (9.190) L q (z) physics = (−1) p+q q!L q− p (z) . The first three L (α) n are

(9.185)

which shows that it is necessary to get Appell functions F2 in which the first, fourth, and fifth parameters are equal. This can be achieved by exploiting whatever freedom in the choice of a1 , a2 , c1 , and c2 is available, and by using identities such as F2 (α + 2, β, β  , α + 1, α + 1; x, y) −1 

= (α + 1) β y × F2 (α + 1, β, β  + 1, α + 1, α + 1; x, y)  + F2 (α + 1, β, β  , α + 1, α + 1; x, y) + 2(α + 1)−2 ββ  xy

 × F2 (α + 2, β + 1, β  + 1, α + 2, α + 2; x, y)

+ (α + 1)−1 βx × F2 (α + 1, β + 1, β  , α + 1, α + 1; x, y) , (9.186) to obtain Appell functions F2 for which the reduction (9.185) can be used.

L (α) 0 (z) = 1 , L (α) 1 (z) = α + 1 − z L (α) 2 (z) =

(9.191)

,

(9.192)

1 1 (α + 1) (α + 2) − (α + 2) z + z 2 . 2 2 (9.193)

Equation (9.188) shows that the irregular solution U does not supply a linearly independent second solution. A second solution which remains linearly independent and finite when α is a positive integer is  Mn(α) (z) = − Γ (α) z −α 1 F1 (−n − α; 1 − α; z)  − cos (πα) Γ (1 − α) L (α) n (z) . (9.194) The Wronskian of the two solutions is d (α) d (α) M (z) − Mn(α) (z) L (z) L (α) n (z) dz n dz n = Γ (n + α + 1) z −α−1 exp (z) /n! .

(9.195)

Hydrogenic Wave Functions

Mn(m) (z) =

∞  ln (z) L (m) + (z) n k=−m

(n + m)! k bk z , (k + m)!

Other useful integration formulae include ∞ (α) x α+1 exp (−x) L (α) n (x) L n  (x) dx 0

=

(9.196)

∞

where   − (−k − 1)!/ (n − k)!, −m ≤ k ≤ −1 ,        k    (−1) [Ψ (n + 1 − k) − Ψ (k + 1) bk = −Ψ (k + m + 1)] / [k! (n − k)!] ,     0≤k≤n,        (−1)n (k − n − 1)!/k! , k ≥ n + 1 . (9.197)

The Rodrigues formula is L (α) n (z) =

 dn  1 −α z exp (z) n z n+α exp (−z) . n! dz (9.198)

The generating function is  
∞ wz −α−1 n = exp − L (α) (1 − w) n (z) w . 1−w n=0

(9.199)

The Christoffel–Darboux formula is n 

k! L (α) (w) L (α) k (z) Γ (k + α + 1) k k=0   (α) (α) (α) (α) (n + 1)! L n (w)L n+1 (z)− L n+1 (w)L n (z) . = Γ (n + α + 1) (w − z) (9.200)

Γ (n + α + 1)  −nδn,n  +1 + (2n + α + 1) δn,n  n!  − (n + α + 1) δn,n  −1 , (9.202) (α) x α−1 exp (−x) L (α) n (x) L n  (x) dx

0

= ∞

Γ (n < + α + 1) , n < !α

0

=

  Γ (n < + α + 1) )  α (α + 1) n + n  + 4α + 5 n < !α (α + 2)  *  × (1 − n < ) + 2 (n + 1) n  + 1 + 2α n + n  ,  n < = min n, n  . (9.204)

The differentiation and recursion relations are d (α) z L (α) (z) = nL (α) n (z) − (n + α) L n−1 (z) , dz n (9.205)

 −1 L (α) = (n + 1) (2n + α + 1 − z) L (α) (z) n (z) n+1  (9.206) − (n + α) L (α) n−1 (z) . Additional relations can be obtained as special cases of the relations listed above for the confluent hypergeometric function by using the relation (9.188). The coefficients ck in a Laguerre polynomial expansion such as ∞  k! F (x) = ck Γ (k + α + 1) k=0 
1 × (βx)α/2 exp − βx L (α) k (βx) 2 (9.207)

are given by the integral 
∞ 1 ck = β F (x) (βx)α/2 exp − βx L (α) k (βx) dx . 2

(α) x α exp (−x) L (α) n (x) L n  (x) dx

0

= (n!)−1 Γ (n + α + 1) δn,n  .

(9.203)

(α) x α−2 exp (−x) L (α) n (x) L n  (x) dx

The orthonormality relation is ∞

 n < = min n, n  ,

0

(9.201)

(9.208)

167

Part A 9.4

A formula for Mn(α) (z) when α is a positive integer m can be obtained by taking the α → m limit of the right-hand side of (9.194) to obtain

9.4 Special Functions

168

Part A

Mathematical Methods

Part A 9.4

The rate of convergence of the expansion (9.207) is determined by the asymptotic behavior of the integral (9.208) for large k. A convenient way of extracting this asymptotic behavior will be described with the special case α=0,

(9.209)

F (x) = (x + c)ν exp (−γx) ,

(9.210)

as an example. Since the discussion is intended to be illustrative rather than exhaustive, only the case γ < 12 β will be considered. The method is based on the generating function (9.199). The generating function g (z) for the coefficients ck is given in general by g (z) =

∞ 

ck z n

k=0

= β (2+α)/2 (1 − z)−α−1 G



 β (1 + z) , 2 (1 − z)

where c(1) k is the contribution from a contour which runs clockwise around the branch cut from −∞ to − (β + 2γ ) / (β − 2γ ) on the negative real axis, and c(2) k is the contribution from a contour which runs clockwise around the branch cut from +1 to +∞ on the positive real axis. The asymptotic behavior of c(1) k can be extracted in straightforward fashion via the method of Darboux ([9.9], pp. 309–315, 321) ([9.32], pp. 116– 122). The result is 
ν
4β 2β = c(1) (−1)k k β + 2γ (β + 2γ ) (β − 2γ )
k   ν β − 2γ ×k (9.217) 1 + O k−1 . β + 2γ The contribution c(2) k requires a somewhat different strategy. The first step writes it as an integral along the real axis from +1 to +∞ of the jump across the branch cut. Evaluating the jump yields

(9.211)

where G is the Laplace transform ∞ G (λ) = x α/2 F (x) exp (−λx) dx .

c(2) k (9.212)

0

The asymptotic analysis of the expansion coefficient ck begins with the Cauchy integral for ck , which is  1 z −k−1 g (z) dz , (9.213) ck = 2πi C

where the contour C is a small circle which runs counterclockwise around the origin. The contour C is deformed to give integrals which can be evaluated via standard methods for the asymptotic evaluation of integrals [9.31, 32]. For the example (9.209), (9.210), g (z) = βcν+1 (1 − z)−1 U [1, ν + 2, t (z)] , (9.214) (β + 2γ ) c + (β − 2γ ) cz . (9.215) 2 (1 − z) This g (z) has a branch point on the negative real axis at z = − (β + 2γ ) / (β − 2γ ), and a combination of a branch point and an essential singularity on the positive real axis at z = 1. It is convenient to take the associated branch cuts to run from −∞ to − (β + 2γ ) / (β − 2γ ) on the negative real axis, and from +1 to +∞ on the positive real axis. The contour C can be deformed into the sum of two contours which run clockwise around the branch cuts. Then t (z) =

(2) ck = c(1) k + ck ,

(9.216)

 −ν−1 
∞ β 1 ν β −γ x = dxx β + Γ (−ν) 2 0  
1 β − γ c − (k + 1) ln (1 + x) × exp − 2  −1 −βcx . (9.218)

The integrand in (9.218) has a saddle point at x ≈ [βc/ (k + 1)]1/2 . The asymptotics for (k + 1) c large can be extracted via the method of steepest descent (also known as the saddle point method) ([9.9], pp. 136–138), ([9.32], pp. 85–103), [9.31], (see Chapt. 8). However, if k + 1 is large with (k + 1) c small or moderate, the steepest descent approximation breaks down because the saddle point is too close to the branch point of the integrand at x = 0. The breakdown can be cured by using the small x approximations [β + (1/2β − γ)x]−ν−1 ≈ β −ν−1 and ln(1 + x) ≈ x − 1/2(k + 1)−1 βc to obtain
(ν+1)/2 2β c (2) ck = exp (βc) Γ (−ν) (k + 1) β  

  × K ν+1 2 (k + 1) βc 1 + O k−1/2 . (9.219)

The function K ν+1 which appears in (9.219) is a modified Bessel function of the third kind in standard notation. The large z approximation K ν+1 (z) ≈ π 1/2 (2z)−1/2 exp (−z) can be used to recover the result of a steepest descent approximation to

Hydrogenic Wave Functions

and are given explicitly by

−(2ν+3)/4  1 βc k + = c(2) k 2 Γ (−ν)    
1 1/2 × exp γc − 2 βc k + 2   −1 × 1+ O k . π 1/2 βcν+1

Cnλ (z) =

k=0

[(β − 2γ ) / (β + 2γ )] for

(9.220)

The first three Cnλ are C0λ (z) = 1 ,

and  
  1 1/2 exp −2 βc k + 2

(9.222)

c(2) k ,

for the steepest descent approximation to is determined by the location of the associated singularity. The next most important part, which is kν for c(1) k , and (k + 1/2)−(2ν+3)/4 for the steepest descent approximation to c(2) k , is determined by the nature of the singularity (i. e., by the value of ν).

9.4.3 Gegenbauer Polynomials The Gegenbauer polynomials are a special case of the Jacobi polynomial, and of the hypergeometric function:

Γ λ + 12 Γ (2λ + n) (λ− 1 ,λ− 1 ) 2

Pn 2 Cnλ (z) = (z) Γ (2λ) Γ λ + n + 12 Γ (2λ + n) n!Γ (2λ)
 1 1 1 × 2 F1 −n, n + 2λ; λ + ; − 2 z , 2 2 (9.223)

(9.225)

C1λ (z) = 2λz , C2λ (z) = 2λ (λ + 1) z 2 − λ .

(9.226) (9.227)

Additional Cnλ can be obtained with the aid of the recursion relation  λ Cn+1 (z) = (n + 1)−1 2 (n + λ) zCnλ (z)  λ (9.228) − (n + 2λ − 1) Cn−1 (z) , which is valid for n ≥ 1. A generating function is ∞

−λ  1 − 2zt + t 2 = Cnλ (z) t n .

(9.229)

n=0

(9.221)

c(1) k ,

=

(−1)k Γ (λ + n − k) (2z)n−2k . k! (n − 2k)!Γ (λ) (9.224)

The approximation (9.219) remains valid as c → 0. The approximation (9.220) does not. The generating function method outlined above has one very nice feature: if the singularity in the complex z-plane which dominates the asymptotics is known, the analysis can be inverted to obtain a “convergence acceleration function” which builds in this singularity, and has no other singularities in the finite complex z plane. The difference between the original F (x) and this convergence acceleration function will have an expansion of the form (9.207) which converges faster than the expansion of F (x). Examples can be found in [9.33–36]. (2) The contributions c(1) k and ck exhibit two typical features. The most rapidly varying part, which is k

[n/2] 

An important addition theorem, which can be used to derive addition theorems for spherical harmonics in any number of dimensions, is Cnλ [cos (γ )] =

Γ (2λ − 1)  22 (2λ + 2 − 1) Γ (n + 2λ + ) [Γ (λ)]2 n

=0

× Γ (n −  + 1) [Γ (λ + )]2  × sin (χ) sin χ    λ+ λ+  cos χ  × Cn− [cos (χ)] Cn−   λ− 1  × C 2 cos γ  , (9.230)   cos (γ ) = cos (χ) cos χ   + sin (χ) sin χ  cos γ  . (9.231)

9.4.4 Legendre Functions The Legendre polynomials P (z) are a special case of the Gegenbauer polynomial, and of the hypergeometric function: 1/2

P (z) = Cn (z)
 1 1 = 2 F1 −n, n + 1; 1; − z . 2 2

(9.232)

169

Part A 9.4

(9.218), which is

9.4 Special Functions

170

Part A

Mathematical Methods

Part A 9

The second (irregular) solution to the Legendre equation, which appears in (9.43) above, can be defined by the integral representation 1

 Q  (z) = 2−−1 1 − t 2 (z − t)−−1 dt . −1

The first two Q  are
 z+1 1 Q 0 (z) = ln , 2 z−1

(9.233)

(9.234)


 z+1 1 Q 1 (z) = z ln −1 , 2 z−1

(9.235)

Additional Q  can be obtained with the aid of the recursion relation Q +1 (z) = [(2 + 1) z Q  (z)  −Q −1 (z) / ( + 1) ,

(9.236)

which is valid for  ≥ 1.

References 9.1 9.2

9.3

9.4

9.5 9.6 9.7 9.8

9.9

9.10 9.11

9.12

9.13

9.14

9.15

M. Abramowitz, I. A. Stegun: Handbook of Mathematical Functions (Dover, New York 1965) A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi: Higher Transcendental Functions, Vol. 1, 2, 3 (McGraw-Hill, New York 1955) p. 1953 A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi: Tables of Integral Transforms, Vol. 1, 2 (McGraw-Hill, New York 1954) I. S. Gradshteyn, I. W. Ryzhik: Tables of Integrals, Series, and Products, 4th edn. (Academic Press, New York 1965) E. Jahnke, F. Emde: Tables of Functions with Formulae and Curves, 4th edn. (Dover, New York 1945) Y. L. Luke: The Special Functions and Their Approximations, Vol. 1, 2 (Academic Press, New York 1969) Y. L. Luke: Mathematical Functions and Their Approximations (Academic Press, New York 1975) W. Magnus, F. Oberhettinger, R. P. Soni: Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd edn. (Springer, Berlin, Heidelberg 1966) F. W. J. Olver: Asymptotics and Special Functions (Academic Press, New York 1974) Reprinted A. K. Peters, Wellesley 1997 G. Szego: Orthogonal Polynomials, Vol. 23, 4th edn. (American Mathematical Society, Providence 1975) F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark: NIST Handbook of Mathematical Functions, in preparation F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark: NIST Digital Library of Mathematical Functions, in preparation (see http://dlmf.nist.gov/) A. R. Edmonds: Angular Momentum in Quantum Mechanics (Princeton Univ. Press, Princeton 1960) Sec. 2.5 L. Pauling, E. Bright Wilson: Introduction to Quantum Mechanics With Applications to Chemistry (McGraw-Hill, New York 1935) Sect. 21 G. W. F. Drake, R. N. Hill: J. Phys. B 26, 3159 (1993)

9.16 9.17

9.18

9.19

9.20 9.21 9.22 9.23 9.24 9.25 9.26

9.27 9.28 9.29 9.30

9.31

L. I. Schiff: Quantum Mechanics, 3rd edn. (McGrawHill, New York 1968) pp. 234–239 J. D. Talman: Special Functions, A Group Theoretic Approach (W. A. Benjamin, New York 1968) pp. 186– 188 V. Fock: Z. Physik 98, 145 (1935) The relation of the Runge–Lenz vector to the “hidden” O(4) symmetry is discussed in Section 3.6.3 M. Reed, B. Simon: Methods of Modern Mathematical Physics. I. Functional Analysis (Academic Press, New York, London 1972) pp. 263–264 Chapters VII The Spectral Theorem and VIII Unbounded Operators. See Theorem VIII.6 L. Hostler: J. Math. Phys. 5, 591, 1235 (1964) J. Schwinger: J. Math. Phys. 5, 1606 (1964) R. A. Swainson, G. W. F. Drake: J. Phys. A 24, 79, 95, 1801 (1991) R. N. Hill, B. D. Huxtable: J. Math. Phys. 23, 2365 (1982) R. Courant, D. Hilbert: Methods of Mathematical Physics, Vol. 1 (Interscience, New York 1953) D. J. Hylton: J. Math. Phys. 25, 1125 (1984) D. W. Lozier, F. W. J. Olver: Mathematics of Computation 1943–1993: A Half-Century of Computational Mathematics, Proceedings of Symposia in Applied Mathematics Vol. 48, ed. by W. Gautschi (American Mathematical Society, Providence 1994) I. J. Thompson, A. R. Barnett: J. Comput. Phys. 64, 490 (1986) I. J. Thompson, A. R. Barnett: Comp. Phys. Commun. 36, 363 (1985) R. E. Meyer: SIAM Rev. 31, 435 (1989) H. Skovgarrd: Uniform Asymptotic Expansions of Confluent Hypergeometric Functions and Whittaker Functions (Gjellerups, Copenhagen 1966) N. Bleistein, R. A. Handelsman: Asymptotic Expansions of Integrals (Holt, Rinehart, & Winston, New York 1975) Reprinted Dover, New York 1986

Hydrogenic Wave Functions

9.33

R. Wong: Asymptotic Approximations of Integrals (Academic Press, San Diego 1989) Reprinted SIAM, Philadelphia 2001 R. C. Forrey, R. N. Hill: Ann. Phys. 226, 88–157 (1993)

9.34 9.35 9.36

R. N. Hill: Phys. Rev. A 51, 4433 (1995) R. C. Forrey, R. N. Hill, R. D. Sharma: Phys. Rev. A 52, 2948 (1995) C. Krauthauser, R. N. Hill: Can. J. Phys. 80, 181 (2002)

171

Part A 9

9.32

References

173

Part B

Atoms

Part B Atoms

10 Atomic Spectroscopy William C. Martin, Gaithersburg, USA Wolfgang L. Wiese, Gaithersburg, USA

21 Atomic Structure: Multiconfiguration Hartree–Fock Theories Charlotte F. Fischer, Nashville, USA

11 High Precision Calculations for Helium Gordon W. F. Drake, Windsor, Canada

22 Relativistic Atomic Structure Ian P. Grant, Oxford, UK

12 Atomic Multipoles William E. Baylis, Windsor, Canada 13 Atoms in Strong Fields S. Pedro Goldman, London, Canada Mark M. Cassar, Windsor, Canada 14 Rydberg Atoms Thomas F. Gallagher, Charlottesville, USA 15 Rydberg Atoms in Strong Static Fields Thomas Bartsch, Atlanta, USA Turgay Uzer, Atlanta, USA 16 Hyperfine Structure Guy T. Emery, Brunswick, USA 17 Precision Oscillator Strength and Lifetime Measurements Lorenzo J. Curtis, Toledo, USA 18 Spectroscopy of Ions Using Fast Beams and Ion Traps Eric H. Pinnington, Edmonton, Canada Elmar Träbert, Bochum, Germany 19 Line Shapes and Radiation Transfer Alan Gallagher, Boulder, USA 20 Thomas–Fermi and Other Density-Functional Theories John D. Morgan III, Newark, USA

23 Many-Body Theory of Atomic Structure and Processes Miron Ya. Amusia, Jerusalem, Israel 24 Photoionization of Atoms Anthony F. Starace, Lincoln, USA 25 Autoionization Aaron Temkin, Greenbelt, USA Anand K. Bhatia, Greenbelt, USA 26 Green’s Functions of Field Theory Gordon Feldman, Baltimore, USA Thomas Fulton, Baltimore, USA 27 Quantum Electrodynamics Jonathan R. Sapirstein, Notre Dame, USA 28 Tests of Fundamental Physics Peter J. Mohr, Gaithersburg, USA Barry N. Taylor, Gaithersburg, USA 29 Parity Nonconserving Effects in Atoms Jonathan R. Sapirstein, Notre Dame, USA 30 Atomic Clocks and Constraints on Variations of Fundamental Constants Savely G. Karshenboim, St. Petersburg, Russia Victor Flambaum, Sydney, Australia Ekkehard Peik, Braunschweig, Germany

175

Atomic Spectro 10. Atomic Spectroscopy

10.1

Frequency, Wavenumber, Wavelength... 176

10.2 Atomic States, Shells, and Configurations 176 10.3 Hydrogen and Hydrogen-Like Ions ........ 176 10.4 Alkalis and Alkali-Like Spectra .............. 177 10.5 Helium and Helium-Like Ions; LS Coupling .................................................. 177 10.6 Hierarchy of Atomic Structure in LS Coupling ..................................... 177 10.7

Allowed Terms or Levels for Equivalent Electrons ....................... 178 10.7.1 LS Coupling .............................. 178 10.7.2 jj Coupling ............................... 178

10.8 Notations for Different Coupling Schemes 10.8.1 LS Coupling (Russell–Saunders Coupling) ................................. 10.8.2 jj Coupling of Equivalent Electrons .................................. 10.8.3 J1 j or J1 J2 Coupling ..................... 10.8.4 J1 l or J1 L2 Coupling (J1 K Coupling) ........................... 10.8.5 LS 1 Coupling (LK Coupling) .......... 10.8.6 Coupling Schemes and Term Symbols .....................

179 179 180 180 180 181 181

10.9 Eigenvector Composition of Levels......... 181 10.10 Ground Levels and Ionization Energies for the Neutral Atoms .......................... 182 10.11 Zeeman Effect ..................................... 183 10.12 Term Series, Quantum Defects, and Spectral-Line Series....................... 184 10.13 Sequences........................................... 185 10.13.1 Isoelectronic Sequence .............. 185 10.13.2 Isoionic, Isonuclear, and Homologous Sequences ....... 185 10.14 Spectral Wavelength Ranges, Dispersion of Air .................................. 185 10.15 Wavelength (Frequency) Standards ....... 186 10.16 Spectral Lines: Selection Rules, Intensities, Transition Probabilities, f Values, and Line Strengths ................ 10.16.1 Emission Intensities (Transition Probabilities) ............ 10.16.2 Absorption f Values ................... 10.16.3 Line Strengths........................... 10.16.4 Relationships Between A, f, and S ................... 10.16.5 Relationships Between Line and Multiplet Values ................. 10.16.6 Relative Strengths for Lines of Multiplets in LS Coupling ........

186 186 186 186 187 192 193

10.17 Atomic Lifetimes.................................. 194 10.18 Regularities and Scaling....................... 194 10.18.1 Transitions in Hydrogenic (One-Electron) Species ............... 194 10.18.2 Systematic Trends and Regularities in Atoms and Ions with Two or More Electrons .................................. 194 10.19 Spectral Line Shapes, Widths, and Shifts 195 10.19.1 Doppler Broadening .................. 195 10.19.2 Pressure Broadening ................. 195 10.20 Spectral Continuum Radiation .............. 196 10.20.1 Hydrogenic Species.................... 196 10.20.2 Many-Electron Systems .............. 196 10.21 Sources of Spectroscopic Data ............... 197 References .................................................. 197

Part B 10

This chapter outlines the main concepts of atomic structure, with some emphasis on terminology and notation. Atomic radiation is discussed, in particular the wavelengths, intensities, and shapes of spectral lines, and a few remarks are made regarding continuous spectra. We include updated tabulations of ionization energies for the neutral atoms and transition probabilities for persistent lines of selected neutral atoms. Some sources of additional atomic spectroscopic data are mentioned. Experimental techniques and the details of atomic theoretical methods are not covered in this chapter; these and a number of other subjects pertinent to atomic spectroscopy are treated in one or more of at least fifteen other chapters in this book.

176

Part B

Atoms

10.1 Frequency, Wavenumber, Wavelength The photon energy due to an electron transition between an upper atomic level k (of energy E k ) and a lower level i is ∆E = E k − E i = hν = hcσ = hc/λvac ,

(10.1)

Part B 10.3

where ν is the frequency, σ the wavenumber in vacuum, and λvac the wavelength in vacuum. The most accurate spectroscopic measurements are determinations of transition frequencies, the unit being the Hertz (1 Hz = 1 s−1 ) or one of its multiples. A measurement of any one

of the entities frequency, wavenumber, or wavelength (in vacuum) is an equally accurate determination of the others since the speed of light is exactly defined [10.1]. The most common wavelength units are the nanometer (nm), the Ångström (1 Å = 10−1 nm) and the micrometer (µm). The SI wavenumber unit is the inverse meter, but in practice wavenumbers are usually expressed in inverse centimeters: 1 cm−1 = 102 m−1 , equivalent to 2.997 924 58 × 104 MHz. Energy units and conversion factors are further discussed in Chapt. 1.

10.2 Atomic States, Shells, and Configurations A one-electron atomic state is defined by the quantum numbers nlm l m s or nl jm j , with n and l representing the principal quantum number and the orbital angular momentum quantum number, respectively. The allowed values of n are the positive integers, and l = 0, 1, . . . , n − 1. The quantum number j represents the angular momentum obtained by coupling the orbital and spin angular momenta of an electron, i. e., j = l + s, so that j = l ± 1/2. The magnetic quantum numbers m l , m s , and m j represent the projections of the corresponding angular momenta along a particular direction; thus, for example, m l = −l, −l + 1 · · · l and m s = ±1/2. The central field approximation for a many-electron atom leads to wave functions expressed in terms of products of such one-electron states [10.2,3]. Those electrons having the same principal quantum number n belong to the shell for that number. Electrons having both the same n value and l value belong to a subshell, all electrons in a particular subshell being equivalent. The notation for

a configuration of N equivalent electrons is nl N , the superscript usually being omitted for N = 1. A configuration of several subshells is written as nl N n
l
M · · · . The numerical values of l are replaced by letters in writing a configuration, according to the code s, p, d for l = 0, 1, 2 and f , g, h . . . for l = 3, 4, 5 . . . , the letter j being omitted. The Pauli exclusion principle prohibits atomic states having two electrons with all four quantum numbers the same. Thus the maximum number of equivalent electrons is 2(2l + 1). A subshell having this number of electrons is full, complete, or closed, and a subshell having a smaller number of electrons is unfilled, incomplete, or open. The 3p6 configuration thus represents a full subshell and 3s2 3p6 3d10 represents a full shell for n = 3. The parity of a configuration is even or odd according to whether Σi li is even or odd, the sum being taken over all electrons (in practice only those in open subshells need be considered).

10.3 Hydrogen and Hydrogen-Like Ions The quantum numbers n, l, and j are appropriate [10.4]. A particular level is denoted either by nl j or by nl 2 L J with L = l and J = j. The latter notation is somewhat redundant for one-electron spectra, but is useful for consistency with more complex structures. The L values are written with the same letter code used for l values, but with capital letters. The multiplicity of the L term is equal to 2S+1 = 2s+1 = 2. Written as a superscript, this number expresses the doublet character of the structure:

each term for L ≥ 1 has two levels, with J = L ± 1/2, respectively. The Coulomb interaction between the nucleus and the single electron is dominant, so that the largest energy separations are associated with levels having different n. The hyperfine splitting of the 1 H 1s ground level [1420.405 751 766 7(1 0)MHz] results from the interaction of the proton and electron magnetic moments and gives rise to the famous 21 cm line. The

Atomic Spectroscopy

separations of the 2n − 1 excited levels having the same n are largely determined by relativistic contributions, including the spin–orbit interaction, with the result that each of the n − 1 pairs of levels

10.6 Hierarchy of Atomic Structure in LS Coupling

having the same j value is almost degenerate; the separation of the two levels in each pair is mainly due to relatively small Lamb shifts (see Sects. 28.2 and 27.10).

a strong l-dependence of the energy levels [10.5]. The differing extent of “core penetration” for ns and np electrons can in some cases, for example, give an energy difference comparable to or exceeding the difference between the np and (n + 1)p levels. The spin–orbit fine-structure separation between the nl (l > 0) levels having j = l − 1/2 and l + 1/2, respectively, is relatively small.

10.5 Helium and Helium-Like Ions; LS Coupling The energy structure of the normal 1snl configurations is dominated by the electron–nucleus and electron– electron Coulomb contributions [10.4]. In helium and in helium-like ions of the lighter elements, the separations of levels having the same n and having l = s, p, or d are mainly determined by direct and exchange electrostatic interactions between the electrons – the spin–orbit, spin– other orbit, and other relativistic contributions are much smaller. This is the condition for L S coupling, in which: (a) The orbital angular momenta of the electrons are coupled to give a total orbital angular momentum L = Σi li . (b) The spins of the electrons are coupled to give a total spin S = Σi si . The combination of a particular S value with a particular L value comprises a spectroscopic term, the

notation for which is 2S+1L. The quantum number 2S + 1 is the multiplicity of the term. The S and L vectors are coupled to obtain the total angular momentum, J = S+ L, for a level of the term; the level is denoted as 2S+1L J . The parity is indicated by appended degree symbols on odd parity terms. For 1snl configurations, L = l and S = 0 or 1, i. e., the terms are singlets (S = 0) or triplets (S = 1). As examples of the He i structure, the ionization energy (energy required to remove one of the 1s electrons in the 1s2 ground configuration) is 24.5874 eV, the 1s2s 3 S −1 S separation is 0.7962 eV, the 1s2p 3P◦ −1P◦ separation is 0.2539 eV, and the 1s2p 3P◦2 −3 P◦0 fine-structure spread is only 1.32 × 10−4 eV.

10.6 Hierarchy of Atomic Structure in LS Coupling The centrality of L S coupling in the analysis and theoretical interpretation of atomic spectra has led to the acceptance of notations and nomenclature well adapted to discussions of particular structures and spectra [10.2]. The main elements of the nomenclature are shown in Table 10.1, most of the structural entities having already been defined in the above discussions of simple spectra. The quantum numbers in the table represent a full description for complex configurations, and the accepted names for transitions between the structural elements are also given.

As an example, the Ca i 3d4p 3D ◦2 level belongs to the 3D ◦ term which, in turn, belongs to the 3d4p 3 (P ◦ D ◦ F ◦ ) triplet triad. The 3d4p configuration also has a 1 (P ◦ D ◦ F ◦ ) singlet triad. The 3d4s configuration has only monads, one 1D and one 3D . The 3d4s 3D 2 –3d4p 3D ◦3 line belongs to the corresponding 3D –3D ◦ triplet multiplet, and this multiplet belongs to the great Ca i 3d4s 3D –3d4p 3 (P ◦ D ◦ F ◦ ) supermultiplet of three triplet multiplets discussed by Russell and Saunders in their classic paper on the alkaline-earth spectra [10.6]. The 3d4s–3d4p transition

Part B 10.6

10.4 Alkalis and Alkali-Like Spectra In the central field approximation there exists no angularmomentum coupling between a closed subshell and an electron outside the subshell, since the net spin and orbital angular momenta of the subshell are both zero. The nl j quantum numbers are, then, again appropriate for a single electron outside closed subshells. However, the electrostatic interactions of this electron with the core electrons and with the nucleus yield

177

178

Part B

Atoms

Table 10.1 Atomic structural hierarchy in L S coupling and

names for the groups of all transitions between structural entities

Part B 10.7

array includes both the singlet and triplet supermultiplets, as well as any (L S-forbidden) intercombination or intersystem lines arising from transitions between levels of the singlet system and those of the triplet system. The order of the two terms in the transitions as written above, with the lower-energy term on the left, is standard in atomic spectroscopy. Examples of notations for complex configurations are given in Sect. 10.8.

Structural entity

Quantum numbers a

Group of all transitions

Configuration

(n i li ) Ni

Transition array

Polyad

(n i li ) Ni γS1 L 1 nl SL, SL
· · ·

Supermultiplet

Term

(n i li ) Ni γSL

Multiplet

Level

(n i li ) Ni γSL J

Line

State

(n i li ) Ni γSL JM

Line component

a

The configuration may include several open subshells, as indicated by the i subscripts. The letter γ represents any additional quantum numbers, such as ancestral terms, necessary to specify a particular term

10.7 Allowed Terms or Levels for Equivalent Electrons 10.7.1 LS Coupling The allowed LS terms of a configuration consisting of two nonequivalent groups of electrons are obtained by coupling the S and L vectors of the groups in all possible ways, and the procedure may be extended to any number of such groups. Thus the allowed terms for any configuration can be obtained from a table of the allowed terms for groups of equivalent electrons. The configuration l N has more than one allowed term of certain LS types if l > 1 and 2 < N < 4l (d3 –d7 , f 3 –f 11 , etc.). The recurring terms of a particular LS term type from d N and f N configurations are assigned sequential index numbers in the tables of Nielson and Koster [10.7]; the index numbers stand for additional numbers having group-theoretical significance that serve to differentiate the recurring terms, except for a few terms of f 5 and f 9 , f 6 and f 8 , and f 7 . These remaining terms, which occur only in pairs, are further labeled A or B to indicate Racah’s separation of the two terms. The index numbers of Nielson and Koster are in practice the most frequently used labels for the recurring terms of f N configurations. Use of their index numbers for the recurring terms of d N configurations has perhaps the disadvantage of substituting an arbitrary number for a quantum number (the seniority) that itself distinguishes the recurring terms in all cases. The actual value of the seniority number is rarely needed, however, and a consistent notation for the d N and f N configurations is desirable. A table of the allowed LS terms of the l N electrons for l ≤ 3 is given in [10.8], with all recurring terms having the index numbers of Nielson and Koster as a following on-line integer. The theoretical group labels are

also listed. Thus the d3 2D term having seniority 3 is designated 2D 2, instead of 23 D, in this scheme; and the level having J = 3/2 is designated 2D 3/2 2.

10.7.2 j j Coupling The allowed J values for a group of N equivalent electrons having the same j value, l Nj , are given in Table 10.2 for j = 1/2, 3/2, 5/2, and 7/2 (sufficient Table 10.2 Allowed J values for l Nj equivalent electrons ( jj) coupling l Nj

Allowed J values

l1/2

1/2

2 l1/2

0

3 l3/2 and l3/2

3/2

2 l3/2

0, 2

4 l3/2

0

5 l5/2 and l5/2

5/2

2 l5/2

0, 2, 4

4 and l5/2

3 l5/2

3/2, 5/2, 9/2

6 l5/2

0

7 l7/2 and l7/2

7/2

2 and l 6 l7/2 7/2

0, 2, 4, 6

3 l7/2

3/2, 5/2, 7/2, 9/2, 11/2, 15/2

5 and l7/2

4 l7/2

0, 2 2, 4 2, 2 4, 4 4, 5, 6, 8

8 l7/2

0

Atomic Spectroscopy

or Q = 2l + 2, whichever is smaller. The (degenerate) levels for a set with both Q and R nonzero have wave functions defined by the quantum numbers (αJ1 , βJ2 )J, with J1 and J2 deriving from the Q and R groups, respectively. The symbols α and β represent any additional quantum numbers required to identify levels. The J values of the allowed levels for each (αJ1 , βJ2 ) subset are obtained by combining J1 and J2 in the usual way.

10.8 Notations for Different Coupling Schemes In this section we give enough examples to make clear the meaning of the different coupling-scheme notations. Not all the configurations in the examples have been identified experimentally, and some of the examples of a particular coupling scheme given for heuristic purposes may be physically inappropriate. Cowan [10.3] describes the physical conditions for the different coupling schemes and gives experimental examples.

10.8.1 LS Coupling (Russell–Saunders Coupling) Some of the examples given below indicate notations bearing on the order of coupling of the electrons 1. 3d7 4F 7/2

 2. 3d7 4F 4s4p 3P ◦ 6F ◦9/2

 3. 4f 7 8S ◦ 6s6p2 4P 11P ◦5

 4. 3p5 2P ◦ 3d2 1G 2F ◦7/2 

5. 4f 10 3K 2 6s6p 1P ◦ 3L ◦6

 6. 4f 7 8S ◦ 5d 7D ◦ 6p 8F 13/2


 7. 4f 7 8S ◦ 5d 9D ◦ 6s 8D ◦ 7s 9D ◦5


 8. 4f 7 8S ◦ 5d 9D ◦ 6s6p 3P ◦ 11F 8

 
9. 4f 7 8S ◦ 5d2 1G 8G ◦ 6p 7F 0


 10. 4f 2F ◦ 5d2 1G 6s 2G 1P ◦1 . In the second example, the seven 3d electrons couple to give a 4F term, and the 4s and 4p electrons couple to form the 3P ◦ term; the final 6F ◦ term is one of nine possible terms obtained by coupling the 4F grandparent and 3P ◦ parent terms. The next three examples are similar to the second. The meaning of the index number 2 following the 3K symbol in the fifth example is explained in Sect. 10.7.1.

The coupling in example 6 is appropriate if the interaction of the 5d and 4f electrons is sufficiently stronger than the 5d–6p interaction. The 7D ◦ parent term results from coupling the 5d electron to the 8S ◦ grandparent, and the 6p electron is then coupled to the 7D ◦ parent to form the final 8F term. A space is inserted between the 5d electron and the 7D ◦ parent to emphasize that the lat
 ter is formed by coupling a term 8S ◦ listed to the left of the space. Example 7 illustrates a similar coupling order carried to a further stage; the 8D ◦ parent term results from the coupling of the 6s electron to the 9D ◦ grandparent. Example 8 is similar to examples 2–5, but in 8 the first of the two terms that couple to form the final 11F term, i. e., the 9D ◦ term, is itself formed by the coupling of the 5d electron to the 8S ◦ core term. Example 9 shows an 8G ◦ parent term formed by coupling the 8S ◦ and 1G grandparent terms.
A space is again used to emphasize  that the following 8G ◦ term is formed by the coupling of terms listed before the space. A different order of coupling is indicated in the final example, the 5d2 1G term being coupled first to the external 6s electron
instead of directly to the 4f core by a space to electron. The 4 f 2F ◦ core term is
isolated  denote that it is coupled (to the 5d2 1G 6s 2G term) only after the other electrons have been coupled. The notation in this particular case (with a single 4f electron) could be simplified by writing the 4f electron after the 2G term to which it is coupled. It appears more important, however, to retain the convention of giving the core portion of the configuration first. The notations in examples 1–5 are in the form recommended by Russell et al. [10.10], and used in both the Atomic Energy States [10.11] and Atomic Energy Levels [10.8, 12] compilations. The spacings used in the remaining examples allow different orders of coupling of the electrons to be indicated without the use of additional parentheses, brackets, etc.

179

Part B 10.8

4 for l ≤ 3). The l7/2 group has two allowed levels for each of the J values 2 and 4. The subscripts distinguishing the two levels in each case are the seniority numbers [10.9]. The allowed levels of the configuration nl N may be obtained by dividing the electrons into sets of two Q R nll−1/2 , Q + R = N. The possible sets groups nll+1/2 run from Q = N − 2l (or zero if N < 2l) up to Q = N

10.8 Notations for Different Coupling Schemes

180

Part B

Atoms

Some authors assign a short name to each (final) term, so that the configuration can be omitted in tables of classified lines, etc. The most common scheme distinguishes the low terms of a particular SL type by the prefixes a, b, c, . . . , and the high terms by z, y, x, . . . [10.12].

Part B 10.8

10.8.2 j j Coupling of Equivalent Electrons This scheme is used, for example, in relativistic calculations. The lower-case j indicates the angular momentum of one electron ( j = l ± 1/2) or of each electron in an l Nj group. Various ways of indicating which of the two possible j values applies to such a group without writing the j-value subscript have been used by different authors; we give the j values explicitly in the examples below. We use the symbols Ji and j to represent total angular momenta. 
1. 6p21/2 0 ◦
2. 6p21/2 6p3/2 3/2
 3. 6p21/2 6p23/2 2
 4. 4d35/2 4d23/2 9/2, 2 11/2 The relatively large spin–orbit interaction of the 6p electrons produces jj-coupling structures for the 6p2 , 6p3 , and 6p4 ground configurations of neutral Pb, Bi, and Po, respectively; the notations for the ground levels of these atoms are given as the first three examples above. The configuration in the first example shows the notation for equivalent electrons having the same j value l Nj , in this case two 6p electrons each having j = 1/2. A convenient notation for a particular level (J = 0) of such a group is also indicated. The second example extends this notation to the case of a 6p3 configuration divided into two groups according to the two possible j values. A similar notation is shown for the 6p4 level in the third example; this level might also be designated (6p−2 3/2 )2 , the negative superscript indicating the two 6p holes. The (J1 , J2 ) J term and level notation shown on the right in the fourth example is convenient because each of the two electron groups 4d35/2 and 4d23/2 has more than one allowed total Ji value. The assumed convention is that J1 applies to the group on the left (J1 = 9/2 for the 4d35/2 group) and J2 to that on the right.

10.8.3 J1 j or J1 J2 Coupling



◦ 1. 3d9 2D 5/2 4p3/2 5/2, 3/2 3
 

2. 4f 11 2H ◦9/2 2 6s6p 3P ◦1 9/2, 1 7/2





 3. 4f 9 6H ◦ 5d 7H ◦8 6s6p 3P ◦0 8, 0 8

 
◦

4. 4f 12 3H 6 5d 2D 6s6p 3P ◦ 4F ◦3/2 6, 3/2 13/2

 ◦

5. 5f 4 5I 4 6d3/2 4, 3/2 11/2 7s7p 1P ◦1 11/2, 1 9/2
◦
 4 5f 5 6. 5f7/2 5/2 8, 5/2 21/2 7p3/2 21/2, 3/2 10
3 ◦ 

 ◦ 3 5f 3 7. 5f7/2 5/2 9/2, 9/2 9 7s7p P 2 9, 2 7 The first five examples all have core electrons in L S coupling, whereas jj coupling is indicated for the 5f core electrons in the last two examples. Since the J1 and J2 values in the final (J1 , J2 ) term have already been given as subscripts in the configuration, the (J1 , J2 ) term notations are redundant in all these examples. Unless separation of the configuration and final term designations is desired, as in some data tables, one may obtain a more concise notation by simply enclosing the entire configuration in brackets and adding the final J value as a subscript. Thus, the in the first example can be designated as  9level
◦ 3d 2D 5/2 4p3/2 3 . If the configuration and coupling order are assumed to be known, still shorter designations may be used; for example, 
 the
fourth 
level  above might then be given as 3H 6 3P ◦ 4F ◦3/2 13/2 or 
3 H 6 , 3P ◦ , 4F ◦3/2 13/2 . Similar economies of notation are of course possible, and often useful, in all coupling schemes.

10.8.4 J1l or J1 L2 Coupling (J1 K Coupling) 1. 2. 3. 4.


  ◦ 3p5 2P ◦1/2 5g 2 9/2 5
   4f 2 3H 4 5g 2 3 5/2

  ◦ 4f 13 2F ◦7/2 5d2 1D 1 7/2 7/2

  ◦ 4f 13 2F ◦5/2 5d6s 3D 3 9/2 11/2

The final terms in the first two examples result from coupling a parent-level J1 to the orbital angular momentum of a 5g electron to obtain a resultant K , the K value being enclosed in brackets. The spin of the external electron is then coupled with the K angular momentum to obtain a pair of J values, J = K ± 1/2 (for K = 0). The multiplicity (2) of such pair terms is usually omitted from the term symbol, but other multiplicities occur in the more general J1 L 2 coupling (examples 3 and 4). The last two examples are straightforward extensions of J1 l coupling, with the L2 and S2 momenta of the “external” term (1D and 3D in examples 3 and 4, respectively) replacing the l and s momenta of a single external electron.

Atomic Spectroscopy

10.8.5 LS 1 Coupling (LK Coupling)  
2 ◦  1. 3s2 3p P 4f
G 2 7/2 3 
 ◦ 2. 3d7 4P 4s4p 3P ◦ D ◦ 3 5/2 7/2

by the J value). For configurations of more than one unfilled subshell, the angular momenta involved in the final coupling derive from two groups of electrons (either group may consist of only one electron). These are often an inner group of coupled electrons and an outer group of coupled electrons, respectively. In any case the quantum numbers for the two groups can be distinguished by subscripts 1 and 2, so that quantum numbers represented by capital letters without subscripts are total quantum numbers for both groups. Thus, the quantum numbers for the two vectors that couple to give the final J are related to the term symbol as follows: Quantum numbers for vectors that couple Term to give J symbol

10.8.6 Coupling Schemes and Term Symbols

Coupling scheme

The coupling schemes outlined above include those now most frequently used in calculations of atomic structure [10.3]. Any term symbol gives the values of two angular momenta that may be coupled to give the total electronic angular momentum of a level (indicated

LS J1 J2 J1 L 2 (→ K ) L S1 (→ K )

L, S J1 , J2 K, S2 K, S2

2S+1 L

(J1 , J2 ) 2S2 +1 [K ]

2S2 +1 [K ]

10.9 Eigenvector Composition of Levels The wave functions of levels are often expressed as eigenvectors that are linear combinations of basis states in one of the standard coupling schemes. Thus, the wave function Ψ(αJ ) for a level labeled αJ might be expressed in terms of LS coupling basis states Φ(γSL J ):  Ψ(αJ ) = c(γSL J ) Φ(γSL J ) . (10.2) γSL

The c(γSL J ) are expansion coefficients, and 2    c(γSL J ) = 1 .

(10.3)

γSL

The squared expansion coefficients for the various γSL terms in the composition of the αJ level are conveniently expressed as percentages, whose sum is 100%. Thus the percentage contributed by the pure Russell–Saunders state γSL J is equal to 100 × |c(γSL J )|2 . The notation for Russell–Saunders basis states has been used only for concreteness; the eigenvectors may be expressed in any coupling scheme, and the coupling schemes may be different for different configurations included in a single calculation

(with configuration interaction). “Intermediate coupling” conditions for a configuration are such that calculations in both L S and jj coupling yield some eigenvectors representing significant mixtures of basis states. The largest percentage in the composition of a level is called the purity of the level in that coupling scheme. The coupling scheme (or combination of coupling schemes if more than one configuration is involved) that results in the largest average purity for all the levels in a calculation is usually best for naming the levels. With regard to any particular calculation, one does well to remember that, as with other calculated quantities, the resulting eigenvectors depend on a specific theoretical model and are subject to the inaccuracies of whatever approximations the model involves. Theoretical calculations of experimental energy level structures have yielded many eigenvectors having significantly less than 50% purity in any coupling scheme. Since many of the corresponding levels have nevertheless been assigned names by spectroscopists, some caution is advisable in the acceptance of level designations found in the literature.

181

Part B 10.9

The orbital angular momentum of the core is coupled with the orbital angular momentum of the external electron(s) to give the total orbital angular momentum L. The letter symbol for the final L value is listed with the configuration because this angular momentum is then coupled with the spin of the core (S1 ) to obtain the resultant K angular momentum of the final term (in brackets). The multiplicity of the [K ] term arises from the spin of the external electron(s).

10.9 Eigenvector Composition of Levels

182

Part B

Atoms

10.10 Ground Levels and Ionization Energies for the Neutral Atoms

Part B 10.10

Fortunately, the ground levels of the neutral atoms have reasonably meaningful L S-coupling names, the corresponding eigenvector percentages lying in the range from ≈ 55% to 100%. These names are listed in Table 10.3, except for Pa, U, and Np; the lowest few ground-configuration levels of these atoms comprise better 5f N (L 1 S1 J1 ) 6d j 7s2 (J1 j) terms than L S-coupling terms. As noted in Sect. 10.8.2, the jjcoupling names given there for the ground levels of Pb, Bi, and Po are more appropriate than the alternative L S-coupling designations in Table 10.3.

The ionization energies in the table are from recent compilations [10.13, 14]. The uncertainties are mainly in the range from less than one to several units in the last decimal place, but a few of the values may be in error by 20 or more units in the final place, i. e., the error could be greater than 0.2 eV. Although no more than four decimal places are given here, values for both the neutral and singly-ionized atoms are given to their full accuracies in [10.14].

Table 10.3 Ground levels and ionization energies for the neutral atoms Element Ground configuration a

Ground level

Ionization energy (eV)

Z

Element Ground configuration a

H

1s

2S

13.5984

27

Co

2

He

1s2

1S

3

Li

1s2

2s

2S

4

Be

1s2

2s2

1S

5

B

1s2

2s2

2p

6

C

1s2

2s2

2p2

7

N

1s2

2s2

2p3

8

O

1s2

2s2

2p4

9

F

1s2

2s2

2p5

10

Ne

1s2

2s2

2p6

11

Na

[Ne]

3s

12

Mg

[Ne]

3s2 3p

Z 1

1/2 0 1/2

16

S

[Ne]

3s2

17

Cl

[Ne]

3s2

3p5

3s2

3p6

0 2P ◦ 1/2 3P 0 4S ◦ 3/2 3P 2 2P ◦ 3/2 1S 0 2S 1/2 1S 0 2P ◦ 1/2 3P 0 4S ◦ 3/2 3P 2 2P ◦ 3/2 1S

13

Al

[Ne]

3s2

14

Si

[Ne]

3s2

3p2

15

P

[Ne]

3s2

3p3 3p4

18

Ar

[Ne]

19

K

[Ar]

4s

2S

4s2

1S

20

Ca

[Ar]

21

Sc

[Ar]

3d

4s2 4s2

22

Ti

[Ar]

3d2

23

V

[Ar]

3d3

4s2

24

Cr

[Ar]

3d5

4s 4s2 4s2

25

Mn

[Ar]

3d5

26

Fe

[Ar]

3d6

1/2

0 2D 3/2 3F 2 4F 3/2 7S 3 6S 5/2 5D 4

Ground level

Ionization energy (eV)

[Ar]

3d7

4s2

4F

9/2

7.8810

4s2

3F

4

7.6398

1/2

7.7264

24.5874

28

Ni

[Ar]

3d8

5.3917

29

Cu

[Ar]

3d10

4s

2S

4s2

1S

9.3227

30

Zn

[Ar]

3d10

8.2980

31

Ga

[Ar]

3d10

4s2

4p

4s2

4p2

11.2603

32

Ge

[Ar]

3d10

14.5341

33

As

[Ar]

3d10

4s2

4p3

4s2

4p4

13.6181

34

Se

[Ar]

3d10

17.4228

35

Br

[Ar]

3d10

4s2

4p5

21.5645

36

Kr

[Ar]

3d10

4s2

4p6

5.1391

37

Rb

[Kr]

5s

7.6462

38

Sr

[Kr]

5s2

5.9858

39

Y

[Kr]

4d

5s2

8.1517

40

Zr

[Kr]

4d2

5s2

10.4867

41

Nb

[Kr]

4d4

5s 5s

10.3600

42

Mo

[Kr]

4d5

12.9676

43

Tc

[Kr]

4d5

5s2 5s 5s

15.7596

44

Ru

[Kr]

4d7

4.3407

45

Rh

[Kr]

4d8

6.1132

46

Pd

[Kr]

4d10

6.5615

47

Ag

[Kr]

4d10

5s 5s2

6.8281

48

Cd

[Kr]

4d10

6.7462

49

In

[Kr]

4d10

5s2

5p

6.7665

50

Sn

[Kr]

4d10

5s2

5p2

5s2

5p3

5s2

5p4

7.4340

51

Sb

[Kr]

4d10

7.9024

52

Te

[Kr]

4d10

0 2P ◦ 1/2 3P 0 4S ◦ 3/2 3P 2 2P ◦ 3/2 1S 0 2S 1/2 1S 0 2D 3/2 3F 2 6D 1/2 7S 3 6S 5/2 5F 5 4F 9/2 1S 0 2S 1/2 1S 0 2P ◦ 1/2 3P 0 4S ◦ 3/2 3P 2

9.3942 5.9993 7.8994 9.7886 9.7524 11.8138 13.9996 4.1771 5.6949 6.2173 6.6339 6.7589 7.0924 7.28 7.3605 7.4589 8.3369 7.5762 8.9938 5.7864 7.3439 8.6084 9.0096

Atomic Spectroscopy

10.11 Zeeman Effect

183

Table 10.3 Ground levels and ionization energies for the neutral atoms, cont.

Z

Element Ground configuration a

Z

Element Ground configuration a

Ground level

[Kr]

5p5

2P ◦ 3/2

10.4513

79

Au

[Xe] 4f 14

5d10

6s

2S

54 Xe

[Kr] 4d10 5s2

5p6

1S

0

12.1298

80

Hg

[Xe] 4f 14

5d10

6s2

1S

55 Cs

[Xe]

6s

2S

1/2

3.8939

81

Tl

[Xe] 4f 14

5d10

6s2

6p

0 2P ◦ 1/2

56 Ba

[Xe]

6s2

1S

5.2117

82

Pb

[Xe] 4f 14

5d10

6s2

6p2

3P

57 La

[Xe]

5d

6s2

4f 14

5d10

6s2

6p3

0 4S ◦ 3/2

58 Ce

[Xe] 4f

5d

6s2

3P

4d10

0 2D 3/2

[Xe]

4f 3

6s2

1G ◦ 4 4I ◦ 9/2

[Xe]

4f 4

6s2

5I

[Xe]

4f 5

6s2

4 6H ◦ 5/2

62 Sm

[Xe]

4f 6

6s2

7F

63 Eu

[Xe] 4f 7

6s2

64 Gd

[Xe]

4f 7

6s2

65 Tb

[Xe] 4f 9

60 Nd 61 Pm

0 8S ◦ 7/2

83

Bi

[Xe]

5.5387

84

Po

[Xe] 4f 14

5d10

6s2

6p4

4f 14

5d10

6s2

6p5

2 2P ◦ 3/2

5d10

6s2

6p6

1S

0 1/2

5.473

85

At

[Xe]

5.5250

86

Rn

[Xe] 4f 14

5.582

87

Fr

[Rn]

7s

2S

7s2

1S

[Rn]

6d

7s2

0 2D 3/2

7s2

3F

6d

7s2

(4, 3/2)11/2

5.89

(4I ◦9/2 )

6d

7s2

(9/2, 3/2)◦6

6.1941

(5I 4 )

6d

7s2

(4, 3/2)11/2

6.2657

7F

6.0260

[Xe]

8

5.9389

[Xe] 4f 11

6s2

4I ◦ 15/2

6.0215

68 Er

[Xe]

4f 12

6s2

3H

69 Tm

[Xe] 4f 13

6s2

6 2F ◦ 7/2

70 Yb

[Xe]

4f 14

6s2

1S

71 Lu

[Xe] 4f 14

72 Hf

[Xe]

3F

73 Ta

[Xe] 4f 14

5d3

6s2

4F

74 W

[Xe]

4f 14

5d4

6s2

5D

75 Re

[Xe]

4f 14

5d5

6s2

6S

76 Os

[Xe] 4f 14

5d6

6s2

5D

77 Ir

[Xe]

4f 14

5d7

6s2

4F

78 Pt

[Xe] 4f 14

5d9

6s

3D

6.1498

90

Th

[Rn]

6d2

5.8638

91

Pa

[Rn] 5f 2

(3H 4 )

92

U

[Rn]

5f 3

93

Np

[Rn] 5f 4 5f 6

7s2

6.1077

94

Pu

[Rn]

6.1843

95

Am

[Rn] 5f 7

7s2

5f 7

7s2

0 8S ◦ 7/2

7s2

9D ◦ 2 6H ◦ 15/2

5f 10

7s2

5I

6.2542

96

Cm

[Rn]

5.4259

97

Bk

[Rn] 5f 9

6d

6.8251

98

Cf

[Rn]

3/2

7.5496

99

Es

[Rn] 5f 11

7s2

8 4I ◦ 15/2

[Rn]

5f 12

7s2

3H

5f 13

7s2

6 2F ◦ 7/2 1S

5/2 4

9/2 3

7.8640

100

Fm

7.8335

101

Md

[Rn]

8.28

102

No

[Rn] 5f 14

7s2

5f 14

7s2

7p?

0 2P ◦ ? 1/2

7s2

?

3F

9.02

103

Lr

[Rn]

8.9588

104

Rf

[Rn] 5f 14

6d2

5.2784 5.17 6.3067

2

2

0

4.0727

[Rn]

67 Ho

6s2

10.7485

Ac

66 Dy

5d2

8.414

Ra

5I

4f 14

7.2855

89

6s2

0 2D 3/2

7.4167

88

4f 10

6s2

6.1082

5.6704

6s2

5d

10.4375

5.6437

9D ◦ 2 6H ◦ 15/2

5d

9.2255

1/2

5.5769

Ionization energy (eV)

2?

5.9738 5.9914 6.1979 6.2817 6.42 6.50 6.58 6.65 4.9 ? 6.0 ?

An element symbol in brackets represents the electrons in the ground configuration of that element

10.11 Zeeman Effect The Zeeman effect for “weak” magnetic fields (the anomalous Zeeman effect) is of special interest because of the importance of Zeeman data in the analysis and theoretical interpretation of complex spectra. In a weak field, the J value remains a good quantum number although in general a level is split into magnetic sub-

levels [10.3]. The g value of such a level may be defined by the expression for the energy shift of its magnetic sublevel having magnetic quantum number M, which has one of the 2J + 1 values, −J, −J + 1, . . . , J: ∆E = gMµB B .

(10.4)

Part B 10.11

53 I

5s2

59 Pr

a

Ionization Ground energy level (eV)

184

Part B

Atoms

The magnetic flux density is B and µB is the Bohr magneton (µB = e~/2m e ). The wavenumber shift ∆σ corresponding to this energy shift is ∆σ = gM(0.466 86B cm−1 ) ,

(10.5)

Part B 10.12

with B representing the numerical value of the magnetic flux density in teslas. The quantity in parentheses, the Lorentz unit, is of the order of 1 or 2 cm−1 for typical flux densities used to obtain Zeeman-effect data with classical spectroscopic methods. Accurate data can be obtained with much smaller fields, of course, by using higher-resolution techniques such as laser spectroscopy. Most of the g values now available for atomic energy levels were derived by application of the above formula (for each of the two combining levels) to measurements of optical Zeeman patterns. A single transverse-Zeemaneffect pattern (two polarizations, resolved components, and sufficiently complete) can yield the J value and the g value for each of the two levels involved. Neglecting a number of higher-order effects, we can evaluate the g value of a level βJ belonging to a pure LS-coupling term using the formula  gβSL J = 1 + (ge − 1) (10.6)  J(J + 1) − L(L + 1) + S(S + 1) . × 2J(J + 1) The independence of this expression from any other quantum numbers (represented by β) such as the con-

figuration, etc., is important. The expression is derived from vector coupling formulas by assuming a g value of unity for a pure orbital angular momentum and writing the g value for a pure electron spin as ge [10.15]. A value of 2 for ge yields the Landé formula. If the anomalous magnetic moment of the electron is taken into account, the value of ge is 2.002 3193. “Schwinger” g values obtained with this more accurate value for ge are given for levels of SL terms in [10.8]. The usefulness of g SL J values is enhanced by their relation to the g values in intermediate coupling. In the notation used in (10.2) for the wave function of a level βJ in intermediate coupling, the corresponding g value is given by  g SL J |c(γSL J )|2 , (10.7) gβJ = γSL

where the summation is over the same set of quantum numbers as for the wave function. The gβJ value is thus a weighted average of the Landé g SL J values, the weighting factors being just the corresponding component percentages. Formulas for magnetic splitting factors in the J1 J2 and J1 L 2 coupling schemes are given in [10.8] and [10.15]. Some higher-order effects that must be included in more accurate Zeeman-effect calculations are treated by Bethe and Salpeter [10.4] and by Wybourne [10.15], for example. High precision calculations for helium are given in [10.16]. See also Chapt. 13 and Chapt. 15.

10.12 Term Series, Quantum Defects, and Spectral-Line Series The Bohr energy levels for hydrogen or for a hydrogenic (one-electron) ion are given by Z2 , (10.8) n2 in units of the Rydberg for the appropriate nuclear mass. For a multielectron atom, the deviations of a series of (core)nl levels from hydrogenic E n values may be due mainly to core penetration by the nl electron (low l-value series), or core polarization by the nl electron (high l-value series), or a combination of the two effects. In either case it can be shown that these deviations can be approximately represented by a constant quantum defect δl in the Rydberg formula, En = −

E nl = −

Z c2 Z2 = − ∗c 2 , 2 (n − δl ) (n )

(10.9)

where Z c is the charge of the core and n ∗ = n − δ is the effective principal quantum number. If the core includes only closed subshells, the E nl values are with respect to a value of zero for the (core)1 S0 level, i. e., the 1 S0 level is the limit of the (core)nl series. If the quantities in (10.9) are taken as positive, they represent term values or ionization energies; the term value of the ground level of an atom or ion with respect to the ground level of the next higher ion is thus the principal ionization energy. If the core has one or more open subshells, the series limit may be the baricenter of the entire core configuration, or any appropriate sub-structure of the core, down to and including a single level. The E nl values refer to the series of corresponding (core)nl structures built on the particular limit structure. The value of the quantum defect depends to some extent on which (core)nl structures are represented by the series formula.

Atomic Spectroscopy

The quantum defect in general also has an energy dependence that must be taken into account if lower members of a series are to be accurately represented by (10.9). For an unperturbed series, this dependence can be expressed by the extended Ritz formula

= δ0 +

a b + +··· , (n − δ0 )2 (n − δ0 )4

(10.10)

with δ0 , a, b . . . constants for the series (δ0 being the limit value of the quantum defect for high series members) [10.17]. The value of a is usually positive for core-penetration series and negative for

core-polarization series. A discussion of the foundations of the Ritz expansion and application to high precision calculations in helium is given in [10.18]. A spectral-line series results from either emission or absorption transitions involving a common lower level and a series of successive (core)nl upper levels differing only in their n values. The principal series of Na i, 3s 2S 1/2 − np 2P ◦1/2,3/2 (n ≥ 3), is an example. The regularity of successive upper term values with increasing n (10.9,10.10) is of course observed in line series; the intervals between successive lines decrease in a regular manner towards higher wavenumbers, and the series of increasing wavenumbers converges towards the term value of the lower level as a limit.

10.13 Sequences Several types of sequences of elements and/or ionization stages are useful because of regularities in the progressive values of parameters relating to structure and other properties along the sequences. All sequence names may refer either to the atoms and/or ions of the sequence or to their spectra.

10.13.1 Isoelectronic Sequence A neutral atom and those ions of other elements having the same number of electrons as the atom comprise an isoelectronic sequence. (Note that a negative ion having this number of electrons is a member of the sequence.) An isoelectronic sequence is named according to its neutral member; for example, the Na i isolectronic sequence.

10.13.2 Isoionic, Isonuclear, and Homologous Sequences An isoionic sequence comprises atoms or ions of different elements having the same charge. Such sequences have probably been most useful along the d- and f-shell rows of the periodic table. Isoionic analyses have also been carried out along p-shell rows, however, and a finestructure regularity covering spectra of the p-shell atoms throughout the periodic table is known [10.19]. The atom and successive ions of a particular element comprise the isonuclear sequence for that element. The elements of a particular column and subgroup in the periodic table are homologous. Thus the C, Si, Ge, Sn, and Pb atoms belong to a homologous sequence having np2 ground configurations (Table 10.3). The singly ionized atoms of these elements comprise another example of a homologous sequence.

10.14 Spectral Wavelength Ranges, Dispersion of Air The ranges of most interest for optical atomic spectroscopy are: ∼ 2–20 µm mid-infrared (ir) 700–2000 nm near ir 400–700 nm visible 200–400 nm near ultraviolet (uv) 100–200 nm vacuum uv or far uv 10–100 nm extreme uv (euv or xuv) < 10 nm soft X-ray, X-ray

The above correspondence of names to ranges should not be taken as exact; the variation as to the extent of some of the named ranges found in the literature is considerable. Wavelengths in standard air are often tabulated for the region longer than 200 nm. These wavelengths can be related to energy-level differences by conversion to the corresponding (vacuum) wavenumbers or frequencies [10.20, 21].

185

Part B 10.14

δ = n − n∗

10.14 Spectral Wavelength Ranges, Dispersion of Air

186

Part B

Atoms

10.15 Wavelength (Frequency) Standards

Part B 10.16

In 2001 the Comité International des Poids et Mesures recommended values for optical frequency standards from stabilized lasers using various absorbing atoms, atomic ions, and molecules [10.22]. These frequencies range from 29 054 057 446 579 Hz (10.318 436 884 460 µm; relative standard uncertainty 1.4 × 10−13 ) for a transition in OsO4 to 1 267 402 452 889.92 kHz (236.540 853 549 75 nm; relative standard uncertainty 3.6 × 10−13 ) for a transition in the 115 In+ ion [10.22]. Extensive tables of wavenumbers for molecular transitions in the mid-ir range of 2.3 to 20.5 µm are included in a calibration atlas published in 1991 [10.23]. Wavenumbers of Ar i [10.24] and Ar ii [10.25] emission lines having uncertainties as small as 0.0003 cm−1 are included in tables for these spectra covering a broad range from 222 nm to 5.865 µm. Measurements of U and Th lines (575 to 692 nm) suitable for wavenumber calibration at uncertainty levels of 0.0003 cm−1 or 0.0004 cm−1 were reported in [10.26]. Comprehensive tables of lines for U [10.27], Th [10.28], and I2 [10.29, 30] are useful for cali-

bration at uncertainty levels of 0.002 to 0.003 cm−1 , the atlas of the Th spectrum extending down to 278 nm. A 1974 compilation gives reference wavelengths for some 5400 lines of 38 elements covering the range 1.5 nm to 2.5 µm, with most uncertainties between 10−5 and 2 × 10−4 nm [10.31]. The wavelengths for some 1100 Fe lines selected from the Fe/Ne hollowcathode spectrum have been recommended for reference standards over the range 183 nm to 4.2 µm, with wavenumber uncertainties 0.001 to 0.002 cm−1 [10.32]. Wavelengths for about 3000 vuv and uv lines (110 to 400 nm) from a Pt/Ne hollow-cathode lamp have been determined with uncertainties of 0.0002 nm or less [10.13, 33]. More recent high-accuracy measurements of ultraviolet lines of Fe i, Ge i, Kr ii, and Pt i, ii include some wavelengths with uncertainties smaller than 10−5 nm [10.34]. The wavelengths tabulated for the Kr and Pt lines in [10.34] extend from 171 to 315 nm, and the accuracies of earlier measurements of a number of spectra useful for wavelength calibration are discussed.

10.16 Spectral Lines: Selection Rules, Intensities, Transition Probabilities, f Values, and Line Strengths The selection rules for discrete transitions are given in Table 10.4.

where I(λ) is the specific intensity at wavelength λ, and λ0 the wavelength at line center.

10.16.1 Emission Intensities (Transition Probabilities)

10.16.2 Absorption f Values

The total power  radiated in a spectral line of frequency ν per unit source volume and per unit solid angle is line = (4π)−1 hν Aki Nk ,

(10.11)

where Aki is the atomic transition probability and Nk the number per unit volume (number density) of excited atoms in the upper (initial) level k. For a homogeneous light source of length l and for the optically thin case, where all radiation escapes, the total emitted line intensity (SI quantity: radiance) is +∞ Iline = line l = I(λ)dλ 0

= (4π)−1 (hc/λ0 ) Aki Nk l ,

(10.12)

In absorption, the reduced absorption W(λ) = [I(λ) − I
(λ)]/I(λ)

(10.13)

is used, where I(λ) is the incident intensity at wavelength λ, e.g., from a source providing a continuous background, and I
(λ) the intensity after passage through the absorbing medium. The reduced line intensity from a homogeneous and optically thin absorbing medium of length l follows as +∞ Wik =

W(λ) dλ = 0

e2 λ2 Ni f ik l , 40 m e c2 0 (10.14)

where f ik is the atomic (absorption) oscillator strength (dimensionless).

Atomic Spectroscopy

10.16 Spectral Lines: Selection Rules, Intensities, Transition Probabilities, f Values, and Line Strengths

187

Table 10.4 Selection rules for discrete transitions Magnetic dipole (M1) (“forbidden”)

Electric quadrupole (E2) (“forbidden”)

Rigorous rules

1. ∆J = 0, ±1 (except 0  ↔ 0) 2. ∆M = 0, ±1 (except 0  ↔ 0 when ∆J = 0) 3. Parity change

∆J = 0, ±1 (except 0  ↔ 0) ∆M = 0, ±1 (except 0  ↔ 0 when ∆J = 0) No parity change

∆J = 0, ±1, ±2 (except 0  ↔ 0, 1/2 ↔ 1/2, 0  ↔ 1) ∆M = 0, ±1, ±2

With negligible configuration interaction

4. One electron jumping, with ∆l = ±1, ∆n arbitrary

No change in electron configuration; i. e., for all electrons, ∆l = 0, ∆n = 0

No change in electron configuration; or one electron jumping with ∆l = 0, ±2, ∆n arbitrary

For L S coupling only

5. ∆S = 0 6. ∆L = 0, ±1 (except 0  ↔ 0)

∆S = 0 ∆L = 0 ∆J = ±1

∆S = 0 ∆L = 0, ±1, ±2 (except 0  ↔ 0, 0  ↔ 1)

No parity change

Numerically, in customary units (A in s−1 , λ in Å, S in atomic units),

10.16.3 Line Strengths Aki and f ik are the principal atomic quantities related to line intensities. In theoretical work, the line strength S is also widely used (see Chapt. 21): S = S(i, k) = S(k, i) = |Rik |2 , Rik = ψk |P|ψi ,

Aki =

(10.18) (10.15)

and for S and ∆E in atomic units,

(10.16)

where ψi and ψk are the initial- and final-state wave functions and Rik is the transition matrix element of the appropriate multipole operator P (Rik involves an integration over spatial and spin coordinates of all N electrons of the atom or ion).

f ik =

2 (∆E/gi ) S . 3

(10.19)

gi and gk are the statistical weights, which are obtained from the appropriate angular momentum quantum numbers. Thus for the lower (upper) level of a spectral line, gi(k) = 2Ji(k) + 1 and for the lower (upper) term of a multiplet,  (2Ji(k) + 1) = (2L i(k) + 1) (2Si(k) + 1) . g¯ i(k) =

10.16.4 Relationships Between A, f, and S The relationships between A, f , and S for electric dipole (E1, or allowed) transitions in SI units (A in s−1 , λ in m, S in m2 C2 ) are 2π e2 gi 16π 3 Aki = f = S. ik m e c0 λ2 gk 3h 0 λ3 gk

6.6702 × 1015 gi 2.0261 × 1018 f = S, ik gk λ2 λ3 gk

Ji(k)

(10.20)

The Aki values for strong lines of selected elements are given in Table 10.5. For comprehensive numerical

(10.17)

Table 10.5 Wavelengths λ, upper energy levels E k , statistical weights gi and gk of lower and upper levels, and transition probabilities Aki for persistent spectral lines of neutral atoms. Many tabulated lines are resonance lines (marked “g”), where the lower energy level belongs to the ground term Spectrum Ag

λa (Å)

Ek (cm−1 )

3280.7g 3382.9g 5209.1 5465.5

30 473 29 552 48 744 48 764

gi

gk

2 2 2 4

4 2 4 6

Aki (108 s−1 )

Accuracy b

1.4 1.3 0.75 0.86

B B D D

Part B 10.16

Electric dipole (E1) (“allowed”)

188

Part B

Atoms

Table 10.5 Wavelengths λ, upper energy levels E k , statistical weights gi and gk of lower and upper levels, . . . , cont.

Part B 10.16

Spectrum

λa (Å)

Ek (cm−1 )

gi

Al

3082.2g 3092.7g 3944.0g 3961.5g 1048.2g 4158.6 3453.5 4259.4 7635.1 7948.2 8115.3 1890.4g 1937.6g 2288.1 2349.8 2428.0g 2676.0g 1825.9g 1826.4g 2496.8g 2497.7g 5535.5g 6498.8 7059.9 7280.3 2348.6g 2650.6 2228.3g 2898.0 2989.0 3067.7g 1488.5g 1540.7g 7348.5 1561.4g 1657.0g 1930.9 2478.6 4226.7g 4302.5 5588.8 6162.2 6439.1 2288.0g 3466.2 3610.5 5085.8

32 435 32 437 25 348 25 348 95 400 117 184 32 431 118 871 106 238 107 132 105 463 52 898 51 610 54 605 53 136 41 174 37 359 54 767 54 767 40 040 40 040 18 060 24 980 23 757 22 947 42 565 59 696 44 865 45 916 44 865 32 588 67 184 64 907 78 512 64 087 60 393 61 982 61 982 23 652 38 552 38 259 31 539 35 897 43 692 59 498 59 516 51 484

2 4 2 4 1 5 10 3 5 1 5 4 4 6 4 2 2 2 4 2 4 1 7 7 5 1 9 4 4 4 4 4 4 4 5 5 5 1 1 5 7 5 7 1 3 5 5

Ar

As

Au B

Ba

Be Bi

Br

C

Ca

Cd

Accuracy b

gk

Aki (108 s−1 )

4 6 2 2 3 5 12 1 5 3 7 6 4 4 2 4 2 4 6 2 2 3 7 9 7 3 9 4 2 4 2 4 4 6 7 5 3 3 3 5 7 3 9 3 5 7 3

0.63 0.74 0.493 0.98 5.32 0.0140 1.1 0.0398 0.245 0.186 0.331 2.0 2.0 2.8 3.1 1.99 1.64 1.76 2.11 0.864 1.73 1.19 0.54 0.50 0.32 5.547 4.24 0.89 1.53 0.55 2.07 1.2 1.4 0.12 1.18 2.52 3.51 0.340 2.18 1.36 0.49 0.354 0.53 5.3 1.2 1.3 0.56

C C C C B B C+ B C C C D D D D B+ B+ B B C C B D D D AA AA D C C C D D D A A B+ B+ B+ C+ D C D C D D C

Atomic Spectroscopy

10.16 Spectral Lines: Selection Rules, Intensities, Transition Probabilities, f Values, and Line Strengths

189

Table 10.5 Wavelengths λ, upper energy levels E k , statistical weights gi and gk of lower and upper levels, cont. λa (Å)

Ek (cm−1 )

gi

Cl

1347.2g 1351.7g 4526.2 7256.6 3405.1 3453.5 3502.3 3569.4 3578.7g 3593.5g 3605.3g 4254.3g 4274.8g 5208.4 3876.1g 4555.3g 4593.2g 8521.1g 8943.5g 2178.9g 3247.5g 3274.0g 5218.2 954.83g 6856.0 7398.7 7754.7 3581.2 3719.9g 3734.9 3745.6g 3859.9g 4045.8 2874.2g 2943.6g 4033.0g 4172.0g 2651.6g 2709.6g 2754.6g 3039.1 537.03g 584.33g 3888.6 4026.2 4471.5 5875.7

74 226 74 866 96 313 85 735 32 842 32 431 32 028 35 451 27 935 27 820 27 729 23 499 23 386 26 788 25 792 21 946 21 765 11 732 11 178 45 879 30 784 30 535 49 942 104 731 116 987 115 918 117 623 34 844 26 875 33 695 27 395 25 900 36 686 34 782 34 788 24 789 24 789 37 702 37 452 37 702 40 021 186 209 171 135 185 565 193 917 191 445 186 102

4 2 4 6 10 10 10 8 7 7 7 7 7 5 2 2 2 2 2 2 2 2 4 4 6 6 4 11 9 11 5 9 9 2 4 2 4 1 3 5 5 1 1 3 9 9 9

Co

Cr

Cs

Cu

F

Fe

Ga

Ge

He

Accuracy b

gk

Aki (108 s−1 )

4 2 4 4 10 12 8 8 9 7 5 9 7 7 4 4 2 4 2 4 4 2 6 4 8 6 6 13 11 11 7 9 9 4 6 2 2 3 1 3 3 3 3 9c 15 c 15 c 15 c

4.19 3.23 0.051 0.15 1.0 1.1 0.80 1.6 1.48 1.50 1.62 0.315 0.307 0.506 0.0038 0.0188 0.0080 0.3276 0.287 0.913 1.39 1.37 0.75 5.77 0.494 0.285 0.382 1.02 0.162 0.901 0.115 0.0969 0.862 1.2 1.4 0.49 0.92 0.85 2.8 1.1 2.8 5.663 17.99 0.09475 0.1160 0.2458 0.7070

C C C C C+ C+ C+ C B B B B B B C C C AA A B B B C C C C+ C+ B+ B+ B+ B+ B+ B+ C C C C C C C C AA AA AA AA AA AA

Part B 10.16

Spectrum

190

Part B

Atoms

Table 10.5 Wavelengths λ, upper energy levels E k , statistical weights gi and gk of lower and upper levels, cont.

Part B 10.16

Spectrum

λa (Å)

Ek (cm−1 )

gi

Hg

2536.5g 3125.7 4358.3 5460.7 1782.8g 1830.4g 3039.4g 3256.1g 4101.8g 4511.3g 4044.1g 4047.2g 7664.9g 7699.0g 5570.3 5870.9 7601.5 8112.9 3232.7g 4602.9 6103.6 6707.8g 2025.8g 2852.1g 4703.0 5183.6 2794.8g 2798.3g 2801.1g 4030.8g 4033.1g 4034.4g 1199.6g 1492.6 4935.1 7468.3 8216.3 5890.0g 5895.9g 5682.6 8183.3 735.90g 743.72g 5852.5 6402.2 6074.3

39 412 71 396 62 350 62 350 56 093 54 633 32 892 32 915 24 373 24 373 24 720 24 701 13 043 12 985 97 919 97 945 93 123 92 294 30 925 36 623 31 283 14 904 49 347 35 051 56 308 41 197 35 770 35 726 35 690 24 820 24 788 24 779 83 365 86 221 106 478 96 751 95 532 16 973 16 956 34 549 29 173 135 889 134 459 152 971 149 657 150 917

1 3 3 5 4 4 2 4 2 4 2 2 2 2 5 3 5 5 2 6 6 2 1 1 3 5 6 6 6 6 6 6 4 6 4 6 6 2 2 2 2 1 1 3 5 3

I In

K

Kr

Li

Mg

Mn

N

Na

Ne

Accuracy b

gk

Aki (108 s−1 )

3 5 3 3 4 6 6 4 2 2 4 2 4 2 3 5 5 7 6c 10 c 10 c 6c 3 3 5 3 8 6 4 8 6 4 6 4 2 4 6 4 2 4 4 3 3 1 7 1

0.0800 0.656 0.557 0.487 2.71 0.16 1.3 1.3 0.56 1.02 0.0124 0.0124 0.387 0.382 0.021 0.018 0.31 0.36 0.01002 0.233 0.6860 0.3691 0.84 4.95 0.255 0.575 3.7 3.6 3.7 0.17 0.165 0.158 4.01 3.13 0.0176 0.193 0.223 0.611 0.610 0.103 0.453 6.11 0.486 0.682 0.514 0.603

B B B B C D D D C C C C B+ B+ D D D D A B AA AA D B C B C C C C+ C+ C+ B+ B+ B B+ B+ AA AA C C B B B B B

Atomic Spectroscopy

10.16 Spectral Lines: Selection Rules, Intensities, Transition Probabilities, f Values, and Line Strengths

191

Table 10.5 Wavelengths λ, upper energy levels E k , statistical weights gi and gk of lower and upper levels, cont. λa (Å)

Ek (cm−1 )

gi

Ni

3101.6 3134.1 3369.6g 3414.8 3524.5 3619.4 1302.2g 4368.2 5436.9 7156.7 7771.9 1775.0g 1782.9g 2136.2 2535.6 2802.0g 2833.1g 3683.5g 4057.8g 4201.8g 4215.5g 7800.3g 7947.6g 1474.0g 1666.7 1807.3g 4694.1 3907.5g 3911.8g 4020.4g 4023.7g 2506.9g 2516.1g 2881.6 5006.1 5948.5 2840.0g 3034.1g 3175.1g 3262.3 2428.1g 4607.3g 3642.7g 3653.5g 3998.6g 4981.7 5210.4g

33 112 33 611 29 669 29 481 28 569 31 031 76 795 99 681 105 019 116 631 86 631 56 340 56 090 58 174 58 174 46 329 35 287 34 960 35 287 23 793 23 715 12 817 12 579 67 843 69 238 55 331 73 921 25 585 25 725 24 866 25 014 39 955 39 955 40 992 60 962 57 798 38 629 34 641 34 914 39 257 41 172 21 698 27 615 27 750 25 388 26 911 19 574

5 3 9 7 7 5 5 3 7 5 5 4 4 6 4 5 1 3 5 2 2 2 2 5 5 5 5 4 6 4 6 3 5 5 3 3 5 3 5 5 1 1 7 9 9 11 9

O

P

Pb

Rb

S

Sc

Si

Sn

Sr Ti

Accuracy b

gk

Aki (108 s−1 )

7 5 7 9 5 7 3 9c 5 5 7 6 4 4 4 7 3 1 3 4 2 4 2 7 5 3 7 6 8 4 6 5 5 3 5 5 5 1 3 3 3 3 9 11 9 13 9

0.63 0.73 0.18 0.55 1.0 0.66 3.41 0.007 58 0.0180 0.505 0.369 2.17 2.14 2.83 0.95 1.6 0.58 1.5 0.89 0.018 0.015 0.370 0.340 1.6 6.3 3.8 0.0067 1.28 1.37 1.65 1.44 0.466 1.21 1.89 0.028 0.022 1.7 2.0 1.0 2.7 0.17 2.01 0.774 0.754 0.408 0.660 0.0357

C+ C+ C C C C A B C+ B A C C C C D D D D C C B B D C C D C+ C+ C+ C+ C C C D D D D D D C B B C+ B C+ C+

Part B 10.16

Spectrum

192

Part B

Atoms

Table 10.5 Wavelengths λ, upper energy levels E k , statistical weights gi and gk of lower and upper levels, cont.

Part B 10.16

Spectrum

λa (Å)

Ek (cm−1 )

Tl

2767.9g 3519.2g 3775.7g

36 118 36 200 26 478

5350.5g

26 478

3566.6g

28 650

3571.6 3584.9g

38 338 27 887

3183.4g 4111.8 4379.2

U

V

Xe

Zn

a b c

Aki (108 s−1 )

Accuracy b

gi

gk

2 4 2

4 6 2

1.26 1.24 0.625

C C B

4

2

0.705

B

11

11

0.24

B

17 13

15 15

0.13 0.18

C B

31 541 26 738 25 254

6 10 10

8 10 12

2.4 1.01 1.1

C+ B C

4384.7

25 112

8

10

1.1

C

1192.0g 1295.6g 1469.6g

83 890 77 186 68 046

1 1 1

3 3 3

6.2 2.5 2.8

C C B

4671.2 7119.6

88 470 92 445

5 7

7 9

0.010 0.066

D D

2138.6g 3302.6

46 745 62 772

1 3

3 5

7.09 1.2

B B

3345.0 6362.3

62 777 62 459

5 3

7 5

1.7 0.474

B C

A “g” following the wavelength indicates that the lower level of the transition belongs to the ground term, i. e., the line is a resonance line. Wavelengths below 2000 Å are in vacuum, and those above 2000 Å are in air Accuracy estimates pertain to Aki values: AA, uncertainty within 1%; A, within 3%; B, within 10%; C, within 25%; D, within 50% The Aki , λ, gi , and gk are multiplet values; see (10.20) and Sect. 10.16.5

tables of A, f , and S, including forbidden lines, see Sect. 10.21. Experimental and theoretical methods to determine A, f , or S values as well as atomic lifetimes are discussed in Chapts. 17, 18, and 21. Conversion relations between S and Aki for the most common forbidden transitions are given in Table 10.6. Oscillator strengths f are not used for forbidden transitions, i. e., magnetic dipole (M1), electric quadrupole (E2), etc. [Numerical example: For the 1s2p 1P ◦1 − 1s3d 1 D2 (allowed) transition in He i at 6678.15 Å: gi = 3; gk = 5; Aki = 6.38 × 107 s−1 ; fik = 0.711; S = 46.9 a02 e2 .]

Table 10.6 Conversion relations between S and Aki for forbidden transitions SI units a

Numerically, in customary units b

Electric quadrupole

Aki =

16π 5 S 15h 0 λ5 gk

Aki =

1.1199 × 1018 S gk λ5

Magnetic dipole

Aki =

16π 3 µ0 S 3h λ3 gk

Aki =

2.697 × 1013 S gk λ3

a b

A in s−1 , λ in m. Electric quadrupole: S in m4 C2 . Magnetic dipole: S in J2 T−2 A in s−1 , λ in Å. S in atomic units: a04 e2 = 2.013 × 10−79 m4 C2 (electric quadrupole), e2 h 2 /16π 2 m 2e = µ2B = 8.601 × 10−47 J2 T−2 (magnetic dipole). µB is the Bohr magneton

Atomic Spectroscopy

10.16 Spectral Lines: Selection Rules, Intensities, Transition Probabilities, f Values, and Line Strengths

10.16.5 Relationships Between Line and Multiplet Values The relations between the total strength and f value of a multiplet (M) and the corresponding quantities for the lines of the multiplet (allowed transitions) are 

Sline ,
−1  ¯ gi f M = λ¯ gi λ(Ji , Jk ) f(Ji , Jk ) .

(10.21) (10.22)

Jk ,Ji

λ¯ is the weighted (“multiplet”) wavelength in vacuum: λ¯ = n λ¯ air = hc/ ∆E ,

(10.23)

where ∆E = E k − E i   = (¯gk )−1 gk E k − (¯gi )−1 gi E i Jk

Ji

(10.24)

and n is the refractive index of standard air.

10.16.6 Relative Strengths for Lines of Multiplets in LS Coupling Table 10.7 lists relative line strengths for frequently encountered symmetrical (P → P, D → D) and normal (S → P, P → D) multiplets in L S coupling. The strongest, or principal, lines are situated along the main diagonal of the table and are called x1 , x2 , etc. Their

Table 10.7 Relative strengths for lines of multiplets in L S coupling Jm − 4

Symmetrical multiplets P–P, D–D etc. Jm Jm − 1 Jm − 2

Jm − 3

z2 y3 x4

z3 y4

Jm Jm − 1 Jm − 2 Jm − 3

y3 x4

4

5

Multiplicity 1

12 6.00 4.00 2.00

15 7.00 5.00 3.00

SM = x1 x2 x3 x4

36 12.60 1.60 1.00

45 14.00 1.25 2.25

y1 y2 y3 y4

3.75 3.00

5.40 5.00

7.00 6.75

P–D 45 21.00 11.25 5.00

60 24.00 12.60 5.00

75 27.00 14.00 5.25

3.75 3.75

5.40 6.40 5.60

7.00 8.75 6.75

SM = x1 x2 x3 x4 x5 y1 y2 y3 y4

0.25

0.60 1.00

1.00 2.25 3.00

z1 z2 z3

Normal multiplets S–P, P–D, D–F, etc. Jm Jm − 1 Jm − 2 Jm − 3 Jm − 1 Jm − 2 Jm − 3 Jm − 4

x1

Multiplicity 1

y1 x2

2

z1 y2 x3

3

x1 y1

S–P SM = x1 y1 z1

3 3.00

6 4.00 2.00

SM = x1 x2 x3

9 9.00

18 10.00 4.00

y1 y2 SM = x1 x2 x3 y1 y2 y3 z1 z2 z3

2.00

15 15.00

30 18.00 10.00 2.00

9 5.00 3.00 1.00 P–P 27 11.25 2.25

25 25.00

y1 x2 y2

2

3 D–D 50 75 28.00 31.11 18.00 17.36 11.25 2.00

35 35.00

y2 x3 y3

70 40.00 28.00

2.00

3.89 3.75

D–F 105 45.00 31.11 21.00

4

5

100 34.29 17.29 8.00 5.00 5.71 7.00 5.00

125 37.50 17.50 6.25 1.25 7.50 10.00 8.75 5.00

140 50.00 34.29 22.40 14.00

175 55.00 37.50 24.00 14.00 7.00 7.50 10.50 10.00 7.00 0.50 1.00 1.00

3.89 3.89

5.71 7.31 5.60

0.11

0.29 0.40

Part B 10.16

SM =

193

194

Part B

Atoms

strengths normally diminish along the diagonal. The satellite lines yn and z n are usually weaker and deviate more from the L S values than the stronger diagonal lines when departures from L S coupling are encoun-

tered. The total multiplet strengths SM are also listed in Table 10.7. A discussion of their normalization as well as more extensive tables are given in [10.35].

10.17 Atomic Lifetimes Part B 10.18

The radiative lifetime τk of an atomic level k is related to the sum of transition probabilities to all levels i lower in energy than k:

−1  Aki . (10.25) τk = i

If only one branch (i
) exists (or if all other branches may be neglected), one obtains Aki
τk = 1, and τk = 1/Aki
.

(10.27)

Precision lifetime measurement techniques are discussed in Chapts. 17 and 18.

The branching fraction of a particular transition, say to state i
, is defined as  Aki
/ Aki = Aki
τk . (10.26) i

10.18 Regularities and Scaling 10.18.1 Transitions in Hydrogenic (One-Electron) Species

Data for some lines of the main spectral series of hydrogen are given in Table 10.8.

The nonrelativistic energy of a hydrogenic transition (10.1, 10) is

(∆E) Z = (E k − E i ) Z = R Z hc Z 2 1/n i2 − 1/n 2k .

10.18.2 Systematic Trends and Regularities in Atoms and Ions with Two or More Electrons

(10.28)

Hydrogenic Z Scaling The spectroscopic quantities for a hydrogenic ion of nuclear charge Z are related to the equivalent quantities in hydrogen (Z = 1) as follows (neglecting small differences in the values of R Z ):

(∆E) Z (λvac ) Z SZ fZ AZ

= Z 2 (∆E)H , = Z −2 (λvac )H , = Z −2 SH , = fH , = Z 4 AH .

Atomic quantities for a given state or transition in an isoelectronic sequence may be expressed as power series expansions in Z −1 : Z −2 E = E 0 + E 1 Z −1 + E 2 Z −2 + . . . ,

(10.35)

−1

+... ,

(10.36)

f = f 0 + f 1 Z −1 + f 2 Z −2 + . . . ,

(10.37)

Z S = S0 + S1 Z 2

+ S2 Z

−2

(10.29) (10.30) (10.31) (10.32) (10.33)

For large values of Z, roughly Z > 20, relativistic corrections become noticeable and must be taken into account. f -value Trends f values for high series members (large n
values) of hydrogenic ions decrease according to 
−3
. (10.34) f n, l → n
, l ± 1 ∝ n


where E 0 , f 0 , and S0 are hydrogenic quantities. For transitions in which n does not change (n i = n k ), f 0 = 0, since states i and k are degenerate. For equivalent transitions of homologous atoms, f values vary gradually. Transitions to be compared in the case of the “alkalis” are [10.36]
 
  nl − n
l
Li → (n + 1)l − n
+ 1 l
Na 
  → (n + 2)l − n
+ 2 l
Cu → . . . . Complex atomic structures, as well as cases involving strong cancellation in the integrand of the transition integral, generally do not adhere to this regular behavior.

Atomic Spectroscopy

10.19 Spectral Line Shapes, Widths, and Shifts

195

Table 10.8 Some transitions of the main spectral series of hydrogen Customary name a

1–2 1–3 1–4 1–5 1–6 2–3 2–4 2–5

(Lα ) (Lβ ) (Lγ ) (Lδ ) (L ) (Hα ) (Hβ ) (Hγ )

a b c d

λb (Å) 1215.67 1025.73 972.537 949.743 937.803 6562.80 4861.32 4340.46

gic 2 2 2 2 2 8 8 8

gk 8 18 32 50 72 18 32 50

Aki (108 s−1 )

Transition

Customary name a

λb (Å)

gic

gk

Aki (108 s−1 )

4.699 5.575 (−1)d 1.278 (−1) 4.125 (−2) 1.644 (−2) 4.410 (−1) 8.419 (−2) 2.530 (−2)

2–6 2–7 3–4 3–5 3–6 3–7 3–8

(Hδ ) (H ) (Pα ) (Pβ ) (Pγ ) (Pδ ) (P )

4101.73 3970.07 18 751.0 12 818.1 10 938.1 10 049.4 9545.97

8 8 18 18 18 18 18

72 98 32 50 72 98 128

9.732 (−3) 4.389 (−3) 8.986 (−2) 2.201 (−2) 7.783 (−3) 3.358 (−3) 1.651 (−3)

Lα is often called Lyman α, Hα = Balmer α, Pα = Paschen α, etc. Wavelengths below 2000 Å are in vacuum; values above 2000 Å are in air For transitions in hydrogen, gi(k) = 2(n i(k) )2 , where n i(k) , is the principal quantum number of the lower (upper) electron shell The number in parentheses indicates the power of 10 by which the value has to be multiplied

10.19 Spectral Line Shapes, Widths, and Shifts Observed spectral lines are always broadened, partly due to the finite resolution of the spectrometer and partly due to intrinsic physical causes. The principal physical causes of spectral line broadening are Doppler and pressure broadening. The theoretical foundations of line broadening are discussed in Chapts. 19 and 59.

Resonance broadening (self-broadening) occurs only between identical species and is confined to lines with the upper or lower level having an electric dipole transition (resonance line) to the ground state. The FWHM may be estimated as

10.19.1 Doppler Broadening

where λ is the wavelength of the observed line; f r and λr are the oscillator strength and wavelength of the resonance line; gk and gi are the statistical weights of its upper and lower levels. Ni is the ground state number density. For the 1s2p 1 P◦1 − 1s3d 1 D2 transition in He i [λ = 6678.15 Å; λr (1s2 1 S0 − 1s2p 1 P◦1 ) = 584.334 Å; gi = 1; gk = 3; f r = 0.2762] at Ni = 1 × 1018 cm−3 : ∆ λR 1/2 = 0.036 Å. Van der Waals broadening arises from the dipole interaction of an excited atom with the induced dipole of a ground state atom. (In the case of foreign gas broadening, both the perturber and the radiator may be in their respective ground states.) An approximate formula for the FWHM, strictly applicable to hydrogen and similar atomic structures only, is

Doppler broadening is due to the thermal motion of the emitting atoms or ions. For a Maxwellian velocity distribution, the line shape is Gaussian; the full width at half maximum intensity (FWHM) is, in Å, 
−7 λ (T/M )1/2 . ∆λD 1/2 = 7.16 × 10

(10.38)

T is the temperature of the emitters in K, and M the atomic weight in atomic mass units (amu).

10.19.2 Pressure Broadening Pressure broadening is due to collisions of the emitters with neighboring particles (see also Chapts. 19 and 59). Shapes are often approximately Lorentzian, i. e., I(λ) ∝ {1 + [(λ − λ0 )/∆λ1/2 ]2 }−1 . In the following formulas, all FWHMs and wavelengths are expressed in Å, particle densities N in cm−3 , temperatures T in K, and energies E or I in cm−1 .

−30 ∆λR (gi /gk )1/2 λ2 λr f r Ni , 1/2  8.6 × 10

2/5

16 2 3/10 N, ∆λW 1/2  3.0 × 10 λ C 6 (T/µ)

(10.39)

(10.40)

where µ is the atom-perturber reduced mass in units of u, N the perturber density, and C6 the inter-

Part B 10.19

Transition

196

Part B

Atoms

Part B 10.20

action constant. C6 may be roughly estimated as 2 follows: C6 = Ck − Ci , with Ci(k) = (9.8 × 1010 )αd Ri(k) 2 3 2 (αd in cm , R in a0 ). Mean atomic polarizability αd ≈ (6.7 × 10−25 ) (3IH /4E ∗ )2 cm3 , where IH is the ionization energy of hydrogen and E ∗ the energy of the first excited level of the perturber atom. 2 ≈ 2.5 [I /(I − E 2 Ri(k) H i(k) )] , where I is the ionization energy of the radiator. Van der Waals broadened lines are red shifted by about one-third the size of the FWHM. For the 1s2p 1 P◦1 − 1s3d 1 D2 transition in He i, and with He as perturber: λ = 6678.15 Å; I = 198 311 cm−1 ; E ∗ = E i = 171 135 cm−1 ; E k = 186 105 cm−1 ; µ = 2. At T = 15 000 K and N = 1 × 1018 cm−3 : ∆λW 1/2 = 0.044 Å. Stark broadening due to charged perturbers, i. e., ions and electrons, usually dominates resonance and van der Waals broadening in discharges and plasmas. The FWHM for hydrogen lines is 
−9 ∆λS,H α1/2 N 2/3 (10.41) e , 1/2 = 2.50 × 10

Table 10.9 Values of Stark-broadening parameter α1/2 of the Hβ line of hydrogen (4861 Å) for various temperatures and electron densities T(K) 5000 10 000 20 000 30 000

Ne (cm−3 ) 1017

1015

1016

0.0787 0.0803 0.0815 0.0814

0.0808 0.0840 0.0860 0.0860

0.0765 0.0851 0.0902 0.0919

1018 ... 0.0781 0.0896 0.0946

where Ne is the electron density. The half-width parameter α1/2 for the Hβ line at 4861 Å, widely used for plasma diagnostics, is tabulated in Table 10.9 for some typical temperatures and electron densities [10.35]. This reference also contains α1/2 parameters for other hydrogen lines, as well as Stark width and shift data for numerous lines of other elements, i. e., neutral atoms and singly charged ions (in the latter, Stark widths and shifts depend linearly on Ne ). Other tabulations of complete hydrogen Stark profiles exist.

10.20 Spectral Continuum Radiation 10.20.1 Hydrogenic Species Precise quantum-mechanical calculations exist only for hydrogenic species. The total power cont radiated (per unit source volume and per unit solid angle, and expressed in SI units) in the wavelength interval ∆λ is the sum of radiation due to the recombination of a free electron with a bare ion (free–bound transitions) and bremsstrahlung (free–free transitions): cont =

e6 Ne N Z Z 2 2π03 (6πm e )3/2   1 hc ∆λ × exp − (kT )1/2 λkT λ2   2  n
2 Z 2 I  Z IH γfb H × exp 3 2 kT  kT n n n≥(Z 2 IH λ/hc)1/2       Z 2 IH − 1 + γff + γ¯fb exp
2  (n + 1) kT (10.42)

where Ne is the electron density, N Z the number density of hydrogenic (bare) ions of nuclear charge Z, IH the

ionization energy of hydrogen, n
the principal quantum number of the lowest level for which adjacent levels are so close that they approach a continuum and summation over n may be replaced by an integral. (The choice of n
is rather arbitrary; n
as low as 6 is found in the literature.) γfb and γff are the Gaunt factors, which are generally close to unity. (For the higher freebound continua, starting with n
+ 1, an average Gaunt factor γ¯fb is used.) For neutral hydrogen, the recombination continuum forming H− becomes important, too [10.37]. In the equation above, the value of the constant factor is 6.065 × 10−55 W m4 J1/2 sr−1 . [Numerical example: For atomic hydrogen (Z = 1), the quantity cont has the value 2.9 W m−3 sr−1 under the following conditions: λ = 3 × 10−7 m; ∆λ = 1 × 10−10 m; N e (= N Z=1 ) = 1 × 1021 m−3 ; T = 12 000 K. The lower limit of the summation index n is 2; the upper limit n
has been taken to be 10. All Gaunt factors γfb , γ¯fb , and γff have been assumed to be unity.]

10.20.2 Many-Electron Systems For many-electron systems, only approximate theoretical treatments exist, based on the quantum-defect

Atomic Spectroscopy

method (for results of calculations for noble gases, see, e.g., [10.38]). Experimental work is centered on the noble gases [10.39]. Modifications of the continuum by autoionization processes must also be considered.

References

Near the ionization limit, the f values for boundbound transitions of a spectral series (n
→ ∞) make a smooth connection to the differential oscillator strength distribution d f/ d in the continuum [10.40].

transition probabilities are available from the Atomic Spectra Database (ASD) at the NIST site [10.13]. Section 10.15 includes additional references for wavelength tables.

References 10.1

10.2 10.3 10.4

10.5 10.6 10.7

10.8

10.9 10.10 10.11 10.12

10.13

B. N. Taylor (Ed.): The International System of Units (SI), NIST Spec. Publ. 330 (U.S. Government Printing Office, Washington 1991) p. 3 E. U. Condon, G. H. Shortley: The Theory of Atomic Spectra (Cambridge Univ. Press, Cambridge 1935) R. D. Cowan: The Theory of Atomic Structure and Spectra (Univ. of California Press, Berkeley 1981) H. A. Bethe, E. E. Salpeter: Quantum Mechanics of One- and Two-Electron Atoms (Plenum, New York 1977) B. Edlén: Encyclopedia of Physics, Vol. 27, ed. by S. Flügge (Springer, Berlin, Heidelberg 1964) H. N. Russell, F. A. Saunders: Astrophys. J 61, 38 (1925) C. W. Nielson, G. F. Koster: Spectroscopic Coefficients for the pn , dn , and f n Configurations (MIT Press, Cambridge 1963) W. C. Martin, R. Zalubas, L. Hagan: Atomic Energy Levels – The Rare-Earth Elements, Nat. Stand. Ref. Data Ser., Nat. Bur. Stand. No. 60 (United States Government Printing Office, Washington 1978) A. de-Shalit, I. Talmi: Nuclear Shell Theory (Academic, New York 1963) H. N. Russell, A. G. Shenstone, L. A. Turner: Phys. Rev. 33, 900 (1929) R. F. Bacher, S. Goudsmit: Atomic Energy States (McGraw-Hill, New York 1932) C. E. Moore: Atomic Energy Levels, Nat. Stand. Ref. Data Ser., Nat. Bur. Stand. No. 35 (United States Government Printing Office, Washington 1971) W. C. Martin, A. Musgrove, S. Kotochigova, J. E. Sansonetti: Ground Levels and Ionization Energies for the Neutral Atoms (version 1.3, 2003) This is one of several online NIST databases referred to in this chapter. The databases are accessible by selecting “Physical Reference Data” at the NIST Physics Laboratory website: http://physics.nist.gov.

10.14

10.15 10.16 10.17 10.18 10.19

10.20 10.21 10.22 10.23

10.24

10.25

10.26 10.27

J. E. Sansonetti, W. C. Martin: Handbook of Basic Atomic Spectroscopic Data, NIST online database, http://physics.nist.gov/PhysRefData/Handbook. These tables include selected data on wavelengths, energy levels, and transition probabilites for the neutral and singly-ionized atoms of all elements up through einsteinium (Z = 1 − 99) B. G. Wybourne: Spectroscopic Properties of Rare Earths (Wiley, New York 1965) Z. C. Yan, G. W. F. Drake: Phys. Rev. A 50, R1980 (1980) W. C. Martin: J. Opt. Soc. Am. 70, 784 (1980) G. W. F. Drake: Adv. At. Mol. Opt. Phys. 32, 93 (1994) U. Fano, W. C. Martin: Topics in Modern Physics, A Tribute to E. U. Condon, ed. by W. E. Brittin, H. Odabasi (Colorado Associated Univ. Press, Colorado 1971) pp. 147–152 B. Edlén: Metrologia 2, 71 (1966) E. R. Peck, K. Reeder: J. Opt. Soc. Amer. 62, 958 (1972) T. J. Quinn: Metrologia 40, 103 (2003) A. G. Maki, J. S. Wells: Wavenumber Calibration Tables from Heterodyne Frequency Measurements, NIST Spec. Publ. 821 (U. S. Government Printing Office, Washington 1991) W. Whaling, W. H. C. Anderson, M. T. Carle, J. W. Brault, H. A. Zarem: J. Res. Natl. Inst. Stand. Technol. 107, 149 (2002) W. Whaling, W. H. C. Anderson, M. T. Carle, J. W. Brault, H. A. Zarem: J. Quant. Spectrosc. Radiat. Transfer 53, 1 (1995) C. J. Sansonetti, K.-H. Weber: J. Opt. Soc. Am. B 1, 361 (1984) B. A. Palmer, R. A. Keller, R. Engleman Jr.: Los Alamos National Laboratory Report LA-8251-MS, UC-34a (1980)

Part B 10

10.21 Sources of Spectroscopic Data Access to most of the atomic spectroscopic databases currently online is given by links at the Plasma Gate server [10.41]. Extensive data from NIST compilations of atomic wavelengths, energy levels, and

197

198

Part B

Atoms

10.28 10.29

10.30 10.31

Part B 10

10.32

10.33

10.34

B. A. Palmer, R. Engleman Jr.: Los Alamos National Laboratory Report LA-9615-MS, UC-4 (1983) S. Gerstenkorn, P. Luc: Atlas du Spectre d’Absorption de la Molécule d’Iode entre 14 800–20 000 cm−1 (Editions du CNRS, Paris 1978) S.Gerstenkorn, P. Luc: Rev. Phys. Appl. 14, 791 (1979) V. Kaufman, B. Edlén: J. Phys. Chem. Ref. Data 3, 825 (1974) G. Nave, S. Johansson, R. C. M. Learner, A. P. Thorne, J. W. Brault: Astrophys. J. Suppl. Ser. 94, 221 (1994), and references therein J. E. Sansonetti, J. Reader, C. J. Sansonetti, N. Acquista: Atlas of the Spectrum of a Platinum/Neon Hollow-Cathode Lamp in the Region 1130–4330 Å, J. Res. Natl. Inst. Stand. Technol. 97, 1–212 (1992), online database G. Nave, C. J. Sansonetti: J. Opt. Soc. Amer. B 21, 442 (2004)

10.35

10.36 10.37 10.38 10.39

10.40

10.41

A. N. Cox (Ed.): Allen’s Astrophysical Quantities, 4th edn. (American Inst. Physics Press, Springer, New York 2000) A. W. Weiss: J. Quant. Spectrosc. Radiat. Transfer 18, 481 (1977) H. R. Griem: Spectral Line Broadening by Plasmas (Academic, New York 1974) J. R. Roberts, P. A. Voigt: J. Res. Natl. Bur. Stand. 75, 291 (1971) I. I. Sobelman: Atomic Spectra and Radiative Transitions, 2nd edn. (Springer, Berlin, Heidelberg 1992) A. T. M. Wilbers, G. M. W. Kroesen, C. J. Timmermans, D. C. Schram: J. Quant. Spectrosc. Radiat. Transfer 45, 1 (1991) Y. Ralchenko: Databases for Atomic and Plasma Physics; at site http://plasma-gate.weizmann.ac.il/DBfAPP.html

199

11. High Precision Calculations for Helium

High Precision 11.1

The Three-Body Schrödinger Equation .......................... 199 11.1.1 Formal Mathematical Properties ................................ 200

11.2

Computational Methods ....................... 11.2.1 Variational Methods .................. 11.2.2 Construction of Basis Sets ........... 11.2.3 Calculation of Matrix Elements .... 11.2.4 Other Computational Methods ....

200 200 201 202 205

11.3

Variational Eigenvalues........................ 205 11.3.1 Expectation Values of Operators and Sum Rules .......................... 205

11.4

Total Energies ..................................... 208 11.4.1 Quantum Defect Extrapolations ... 211 11.4.2 Asymptotic Expansions .............. 213

11.5

Radiative Transitions ........................... 215 11.5.1 Basic Formulation ..................... 215 11.5.2 Oscillator Strength Table............. 216

11.6

Future Perspectives.............................. 218

References .................................................. 218

11.1 The Three-Body Schrödinger Equation The Schrödinger equation for a three-body system consisting of a nucleus of charge Ze, and mass M, and two electrons of charge −e and mass m e is
 2 1 2 1  2 P + Pi + V(RN , Ri ) Ψ = EΨ , 2M N 2m e i=1

(11.1)

where P i = (~/i)∇i and V(RN , Ri ) = −

Ze2 e2 Ze2 − + |RN−R1 | |RN−R2 | |R1−R2 | (11.2)

depends only on the relative particle separations. Since the center of mass (c.m.) is then an ignorable coordinate, it can be eliminated by defining the relative particle coordinates ri = Ri − RN to obtain
 2 1  2 1 pi + p1 · p2 + V(r1 , r2 ) Ψ = EΨ , 2µ M i=1

(11.3)

where µ = m e M/(m e + M) is the electron reduced mass and the term Hmp = p1 · p2 /M is called the mass polarization operator. For computational purposes, it is usual to measure distance in units of aµ = (m e /µ)a0 and energies in units of e2 /aµ = 2(µ/m e )R∞ so that (11.3) assumes the dimensionless form
 2 1 2 µ − ∇ρi − ∇ρ1 ·∇ρ2 + V(ρ1 , ρ2 ) Ψ = εΨ , 2 M i=1

(11.4)

where ρi = ri /aµ , ε = E/(e2 /aµ ), and V(ρ1 , ρ2 ) = −

Z Z 1 . − + ρ1 ρ2 |ρ1 − ρ2 |

(11.5)

The limit µ/M → 0 defines the infinite nuclear mass problem with eigenvalue ε0 and eigenfunction Ψ0 . If the mass polarization term is treated as a small perturbation, then the total energy assumes the form    µ 2 µ µ e2 E = ε0 + ε1 + ε2 + · · · , M M m e a0 (11.6)

Part B 11

Exact analytic solutions to the Schrödinger equation are known only for atomic hydrogen, and other equivalent two-body systems (see Chapt. 9). However, very high precision approximations are now available for helium, which are essentially exact for all practical purposes. This chapter summarizes the computational methods and tabulates numerical results for the ground state and several singly excited states. Similar methods can be applied to other three-body problems.

200

Part B

Atoms

where ε1 = −Ψ0 |∇ρ1 · ∇ρ2 |Ψ0  determines the firstorder specific mass shift and ε2 is the second-order coefficient. The common (µ/m e )ε0 mass scaling of all eigenvalues determines the normal mass shift (isotope shift). Since µ/m = 1 − µ/M, the shift is −(µ/M)ε0 .

pansion has the form Ψ(r1 , r2 ) =

j/2] ∞ [ 

R j (ln R)k φ j,k ,

(11.8)

j=0 k=0

where [ ] denotes “greatest integer in”, and R = (r12 + r22 )1/2 is the hyperradius. The leading coefficients are

11.1.1 Formal Mathematical Properties

Part B 11.2

Two-Particle Coalescences The exact nonrelativistic wave function for any manybody system contains discontinuities or cusps in the spherically averaged radial derivative with respect to rij as rij → 0, where rij = |ri − r j | is any interparticle coordinate. If the masses and charges are m i and qi respectively, then the discontinuities are given by the Kato cusp condition [11.1]  ¯ 2 ∂Ψ ~ = µij qi q j Ψ(rij = 0) , (11.7) ∂rij rij =0

φ0,0 = 1 ,

 1 φ1,0 = − Zr1 + Zr2 − r12 R, 2 

π − 2 r1 · r2 . φ2,1 = − 2Z 3π R2

(11.9)

The next term φ2,0 is known in terms of a lengthy expression [11.7–9], but higher terms have not yet been obtained in closed form. The Fock expansion has been proved convergent for all R < 12 [11.10], and extended to pointwise convergence for all R [11.11, 12].

where µij = m i m j /(m i + m j ) and Ψ¯ denotes the wave function averaged over a sphere centered at rij = 0. If Ψ vanishes at rij = 0, then its leading dependence on rij is of the form rijl Ylm (rij ) for some integer l > 0 [11.2]. Equation (11.7) applies to any Coulombic system. The electron–nucleus cusp in the wave functions for hydrogen provides a simple example.

Asymptotic Form The long range behavior of many-electron wave functions has been studied from several points of view [11.13–15]. The basic result of [11.16] is that at large distances, the one-electron density behaves as

Three-Particle Coalescences Three-particle coalescences are described by the Fock expansion [11.3–6], as recently discussed by Myers et al. [11.7]. For the S-states of He-like ions, the ex-

where t = (2I1 )1/2 , I1 is the first ionization potential (in a.u.), and Z ∗ = Z − N + 1 is the screened nuclear charge seen by the outer most electron. For hydrogenic systems with principal quantum number n, I1 = (Z ∗ )2 /2n 2 .

ρ1/2 (r) ≈ r Z

∗ /t−1

e−tr ,

(11.10)

11.2 Computational Methods of H with eigenvalues E 1 < E 2 < E 3 < · · · , so that

11.2.1 Variational Methods Most high precision calculations for the bound states of three-body systems such as helium are based on the Rayleigh–Ritz variational principle. For any normalizable trial function Ψtr , the quantity E tr =

Ψtr |H|Ψtr  Ψtr |Ψtr 

(11.11)

satisfies the inequality E tr ≥ E 1 , where E 1 is the true ground state energy. Thus E tr is an upper bound to E 1 . The inequality is easily proved by expanding Ψtr in the complete basis set of eigenfunctions Ψ1 , Ψ2 , Ψ3 , · · ·

Ψtr =

∞ 

ci Ψi ,

(11.12)

i=1

where the ci are expansion coefficients. This can always be done in principle, even though the exact Ψi are not actually

∞known.2 If Ψtr is normalized so that Ψtr |Ψtr  = 1, then i=1 |ci | = 1 and E tr = |c1 |2 E 1 + |c2 |2 E 2 + |c3 |2 E 3 + · · · = E 1 + |c2 |2 (E 2 − E 1 ) + |c3 |2 (E 3 − E 1 ) + · · · (11.13) ≥ E1 , which proves the theorem.

High Precision Calculations for Helium

The basic idea of variational calculations then is to write Ψtr in some arbitrarily chosen mathematical form with variational parameters (subject to normalizability and boundary conditions at the origin and infinity), and then adjust the parameters to obtain the minimum value of E tr . The minimization problem for the case of linear variational coefficients can be solved algebraically. For example, let j k χ p (α, β) = r1i r2 r12

−αr1 −βr2

e

c p χ p (α, β) ,

(11.15)

p=1

then the solution to the system of equations ∂E tr /∂c p = 0, p = 1, . . . , N, is exactly equivalent to solving the N-dimensional generalized eigenvalue problem Hc = λO c ,

λ3

E3 λ2 E2

(11.16)

where c is a column vector of coefficients c p ; and H and O have matrix elements H pq = χ p |H|χq  and O pq = χ p |χq . There are N eigenvalues λ1 , λ2 , . . . λ N , of which the lowest is an upper bound to E 1 . Extension to Excited States By the Hylleraas–Undheim–MacDonald (HUM) theorem [11.17, 18], the remaining eigenvalues λ2 , λ3 , . . . are also upper bounds to the exact energies E 2 , E 3 , . . . , provided that the spectrum is bounded from below. The HUM theorem is a consequence of the matrix eigenvalue interleaving theorem, which states that as the dimensions of H and O are progressively increased by adding an extra row and column, the N old eigenvalues λ p fall between the N + 1 new ones. Consequently, as illustrated in Fig. 11.1, all eigenvalues numbered from the bottom up must move inexorably downward as N is increased. Since the exact spectrum of bound states is obtained in the limit N → ∞, no λ p can cross the corresponding exact E p on its way down. Thus λ p ≥ E p for every finite N.

11.2.2 Construction of Basis Sets Since the Schrödinger equation (11.4) is not separable in the electron coordinates, basis sets which incorporate

λ1 E1

1

2

3 N

4

5

Fig. 11.1 Diagram illustrating the Hylleraas–Undheim–

MacDonald Theorem. The λ p , p = 1, . . . , N are the variational eigenvalues for an N-dimensional basis set, and the Ei are the exact eigenvalues of H. The highest λ p lie in the continuous spectrum of H

the r12 = |r1 − r2 | interelectron coordinate are most efficient. The necessity for r12 terms also follows from the Fock expansion (11.8). A basis set constructed from terms of the form (11.14) is called a Hylleraas basis set [11.19, 20]. (The basis set is often expressed in terms of the equivalent variables s = r1 + r2 , t = r1 − r2 , u = r12 .) With χ p (α, β) defined as in (11.14), the general form for a state of total angular momentum L is Ψtr =

[L/2]  l1 =0

C p,l1 χ p (α, β)r1l1 r2l2 YlM1 L−l1 L (ˆr1 , rˆ2 )

p

± exchange ,

(11.17)

where YlM1 l2 L (ˆr1 , rˆ2 ) =



Yl1 m 1 (ˆr1 )Yl2 m 2 (ˆr2 )

m 1 ,m 2

× l1l2 m 1 m 2 |LM

(11.18)

is the vector coupled product of angular momenta l1 , l2 for the two electrons. The sum over p in (11.17) typically includes all terms in (11.14) with i + j + k ≤ Ω, where Ω is an integer determining a so-called Pekeris shell of terms, and the exchange term denotes the interchange of r1 and r2 with (+) for singlet states and (−) for triplet states. Convergence is studied by progressively

Part B 11.2

N 

λ5

λ4

E∞ E5 E4

201

(11.14)

denote the members of a basis set, where p is an index labeling distinct triplets of nonnegative integer values for the powers {i, j, k}, and α, β are (for the moment) fixed constants determining the distance scale. If Ψtr is expanded in the form Ψtr =

11.2 Computational Methods

202

Part B

Atoms

increasing Ω. The number of terms is

transformation is 

1 N = (Ω + 1)(Ω + 2)(Ω + 3) . 6

Part B 11.2

Basis sets of this type were used by many authors, culminating in the extensive high precision calculations of Pekeris and coworkers [11.21] for low-lying states, using as many as 1078 terms. Their accuracy is not easily surpassed because of the rapid growth of N with Ω, and because of numerical linear dependence in the basis set for large Ω. Recently, their accuracy has been surpassed by two principal methods. The first explicitly includes powers of logarithmic and half-integral terms in χ p , as suggested by the Fock expansion [11.22–25]. This is particularly effective for S-states. The second focuses directly on the multiple distance scales required for an accurate representation of the wave function by writing the trial function in terms of the double basis set [11.26] Ψtr =

[L/2]  

 (2) C (1) χ (α , β ) + C χ (α , β ) p 1 1 p 2 2 p,l1 p,l1

l1 =0 p × r1l1 r2l2 Yl1 l2 L (r1 , r2 ) ±

exchange ,

2π dr1 dr2 =

(11.19)

where each χ p (α, β) is of the form (11.14), but with different values for the distance scales α1 , β1 and α2 , β2 in the two sets of terms. They are determined by a complete minimization of E tr with respect to all four parameters, producing a natural division of the basis set into an asymptotic sector and a close-range correlation sector. The method produces a dramatic improvement in accuracy for higher-lying Rydberg states (where variational methods typically deteriorate rapidly in accuracy) and is also effective for low-lying S-states [11.27–29]. Nonrelativistic energies accurate to 1 part in 1016 are obtainable with modest computing resources. Another version of the variational method is the quasi-random (or stochastic) method in which nonlinear exponential parameters for all three of r1 , r2 , and r12 are chosen at random from certain specified intervals [11.30, 31]. The method is remarkably accurate and efficient for low-lying states, but subject to severe roundoff error.

11.2.3 Calculation of Matrix Elements The three-body problem has the unique advantage that the full six-dimensional volume element (in the c.m. frame) can be transformed to the product of a three-dimensional angular integral (ang) and a threedimensional radial integral (rad) over r1 , r2 , and r12 . The

2π dφ

0

π

0

∞ ×

sin θ1 dθ1

dϕ1 0

r 1 +r2

∞ r1 dr1

0

r2 dr2 0

r12 dr12 ,

|r1 −r2 |

(11.20)

where θ1 , ϕ1 are the polar angles of r 1 and φ is the angle of rotation of the triangle formed by r 1 , r 2 , and r 12 about the r 1 direction. The polar angles θ2 , ϕ2 are then dependent variables. The basic angular integral is Yl∗1 m 1 (θ1 , ϕ1 )Yl2 m 2 (θ2 , ϕ2 )ang = 2πδl1 l2 δm 1 m 2 Pl1 (cos θ) ,

(11.21)

where cos θ ≡ rˆ1 · rˆ2 denotes the radial function cos θ =

2 r12 + r22 − r12 , 2r1r2

(11.22)

and Pl (cos θ) is a Legendre polynomial. The angular integral over vector-coupled spherical harmonics is [11.32]  M ∗  Yl  l  L  (ˆr1 , rˆ2 ) YlM1 l2 L (ˆr1 , rˆ2 ) ang 1 2  = δ L,L  δ M,M  CΛ PΛ (cos θ) , (11.23) Λ

where 1 CΛ = [(2l1 + 1)(2l1 + 1)(2l2 + 1)(2l2 + 1)]1/2 2 × (−1) L+Λ (2Λ + 1)     l2 l2 Λ L l1 l2 l1 l1 Λ , × 0 0 0 0 0 0 Λ l2 l1 (11.24)

and the sum over Λ includes all nonvanishing terms. This can be extended to general matrix elements of tensor operators by further vector coupling [11.32]. Radial Integrals Table 11.1 lists formulas for the radial integrals arising from matrix elements of H, as well as those from the Breit interaction (see Sect. 21.1). Although they can all be written in closed form, some have been expressed as infinite series in order to achieve good numerical stability. The exceptions are formulas 5 and 10 in the Table, which became unstable as α → β. More elaborate

High Precision Calculations for Helium

11.2 Computational Methods

203

c c Table 11.1 Formulas for the radial integrals I0 (a, b, c; α, β) = r1a r2b r12 e−αr1 −βr2 rad and I0 (a, b, c; α, β) = r1a r2b r12

n−1

ln r12 e−αr1 −βr2 rad ;

log

k−1

ψ(n) = −γ + k=1 is the digamma function, 2 F1 (a, b; c; z) is the hypergeometric function, and s = a + b + c + 5. Except as noted, the formulas apply for a ≥ −1, b ≥ −1, c ≥ −1 1.

I0 (−2, −2, −1; α, β)

=



 α+β α+β 2 2 ln + ln α β β α

2.

I0 (a, b, c; α, β)

=

2 c+2

I0 (a, b, c; α, α)

i=0

 c+2 [Fa+2i+2, b+c−2i+2 (α, β) + Fb+2i+2, a+c−2i+2 (β, α)] 2i + 1

(c ≥ −1, s ≥ 0) 

j q  q! β ( p + j)!    q ≥ 0, p ≥ 0  p+1 q+1  j! α+β   (α + β) β j=0

 j+1 ∞  where F p,q (α, β) = p! j! α  q < 0, p ≥ 0    α p+q+2 ( j − q)! α + β   j= p+q+1  a p 104 T), such as those encountered at the surface of neutron stars, is also called the quadratic Zeeman effect, as the last term in (13.16) is dominant. In this range, perturbation calculations fail to yield good results as the field is too large, and even at fields of the order B ∼ 107 T the Landau high B approximation of (13.8) and (13.9) is not adequate. Very accurate calculations have been performed using variational finite basis set techniques for both the relativistic Dirac and nonrelativistic Schrödinger Hamiltonians. The calculations use the following relativistic basis set [13.7] that includes nuclear size effects (R is the nuclear size) and contains both asymptotic limits, the Coulomb limit for B = 0 and the Landau limit for B → ∞:  (k) 2 2 r qk −1+2n e−anν r −βρ Ωk r ≤ R (k,ν) ψnl = γν −1+n e−λr−βρ2 Ω r > R b(k) k nν r (13.27)

with Ωk = (cos θ)l−|m k | (sin θ)|m k | eim k φ ωk ,

(13.28)

Part B 13.3

where n = 0, 1, . . . , Nr , q1 = q2 = k 0 , m k = µ − σk /2,

k = 1, 2, 3, 4,

ν = 1, 2, 3, 4 ,

q3 = q4 = k0
, σ1 = σ3 = 1,

σ2 = σ4 = −1 .

Here, k refers to the component ψ (k) in (13.11), and λ and β are variational parameters. The power of r at the origin is given by  |κ| if κ < 0 k0 = (13.29) |κ| + 1 if κ > 0 , 
|κ| + 1 if κ < 0 k0 = (13.30) |κ| if κ > 0 , The index ν refers to the two regular and two irregular solutions for r > R that match the corresponding powers at the origin k0 and k0
. γ1 = γ0 , γ2 = γ0 + 1, γ3 = −γ0 , γ4 = −γ0 + 1 , (13.31)

γ0 =

 κ 2 − (αZ)2 ,


1 1 + , κ = ∓ ν± 2 2

(13.32)

    ϑ1 ϑ−1 , ω2 = , ω1 = 0 0     0 0 ω3 = , ω4 = , iϑ−1 iϑ−1

(13.33)

where ϑk is a two-component Pauli spinor: σz ϑk = k ϑi . For even (odd) parity states, the value of l for the large components (k = 1, 2) is an even (odd) number greater than or equal to |m k | up to 2Nθ (for even parity) or 2Nθ + 1 (for odd parity), while for the small components (k = 3, 4) it is an odd (even) number greater than or equal to |m k | up to 2Nθ + 1 (for even parity) or 2Nθ (for odd parity), since the small component has a different nonrelativistic parity than the large component. (k) The coefficients anν and b(k) nν are determined by the continuity condition of the basis functions and their first derivatives at R. For a point nucleus, the section r ≤ R is omitted; for a nonrelativistic calculation, take α = 0 in the basis set. Table 13.1 presents relativistic (Dirac) energies for the ground state of one-electron atoms. Values for a point nucleus and finite nuclear size corrections are given. Table 13.2 presents the relativistic energies for n = 2 Table 13.1 Relativistic ground state binding energy −E gs /Z 2 and finite nuclear size correction δE nuc /Z 2 (in a.u.) of hydrogenic atoms for various magnetic fields B (in units of 2.35 × 105 T). δE nuc should be added to E gs Z

B

−Egs /Z2

δEnuc /Z2

1

0

0.500 006 656 597 483 75

1.557 86 × 10−10

1

10−5

0.500 011 656 4837

1.5579 × 10−10

1

10−2

0.504 981 572 360

1.5580 × 10−10

1

10−1

0.547 532 408 3429

1.5718 × 10−10

1

2

1.022 218 0290

3.23 × 10−10

1

10

1.747 800 68

1.182 × 10−9

1

20

2.215 400 91

2.360 × 10−9

1

200

4.727 1233

3.032 × 10−8

1

500

6.257 0326

8.778 × 10−8

20

0

0.502 691 308 407 5098

1.3372 × 10−6

20

1

0.503 930 867 05

1.34 × 10−6

20

10

0.514 950 248

1.3 × 10−6

20

100

0.612 377 94

1.4 × 10−6

40

0

0.511 129 686 143

1.1878 × 10−5

92

0

0.574 338 140 7377

8.4155 × 10−4

92

1

0.574 386 987

8.4155 × 10−4

Atoms in Strong Fields

Table 13.2 Relativistic binding energy −E 2S,−1/2 for the   

2S1/2 mj = − 12 and −E 2P,−1/2 for the 2P1/2 m j = − 12 excited states of hydrogen (in a.u.) in an intense magnetic field B (in units of 2.35 × 105 T)

13.4 Atoms in Electric Fields

Table 13.3 Relativistic corrections δE = (E − E NR )/|E R |

to the nonrelativistic energies E NR for the ground state and n = 2 excited states of hydrogen in an intense magnetic field B (in units of 2.35 × 105 T). The numbers in brackets denote powers of 10

B

−E2S,−1/2

−E2P,−1/2

10−6

0.125 002 580 164

0.125 002 283 074

B

δEgs

B

δE2S,−1/2

δE2P,−1/2

0.125 006 104 950

0.1 1 2 3 20 200 500 2000 5000

−1.08[−5] −5.21[−6] −4.03[−6] −3.48[−6] −1.09[−6] 4.61[−6] 8.81[−6] 1.85[−5] 2.78[−5]

1[−6] 1[−4] 1[−3] 1[−2] 0.05 0.1 1 10 100

−1.66[−5] −1.66[−5]

−1.43[−5] −7.86[−6] −7.72[−6] −7.30[−6]

10−5 10−4

0.125 052 044 95

10−3

0.125 050 967 92 0.125 499 4694

10−2

0.129 653 6428

0.05

0.142 018 956

0.1

0.148 091 7386

0.2

0.148 989 58

0.5

0.150 810 15

1

0.160 471 07

0.260 009 34

10

0.208 955 91

0.382 663 18

100

0.256 191

0.463 6641

0.129 851 3642 0.162 411 0524

excited states of hydrogen with the (negligible) finite nuclear size correction included.

231

−1.60[−5] −1.57[−5] −1.74[−6] −1.3[−5] −2.0[−5] −3.9[−5]

−6.00[−6] −1.05[−5] −3.48[−5] −1.0[−4]

Table 13.3, which displays the relativistic corrections of the energies of the previous two tables, presents one of the most interesting relativistic results: the change in sign of the relativistic correction of the energy of the ground state at B ∼ 107 T.

13.4 Atoms in Electric Fields An external electric field F introduces the perturbing potential V = −d · F , where d=



qi ri

(13.34)

(13.35)

13.4.2 Linear Stark Effect

i

is the dipole moment of the atom, and i runs over all electrons in the atom. In the case of strong external electric fields, bound states do not exist because the atom ionizes. Consider a hydrogenic atom in a static electric field F = F zˆ .

(13.36)

e2 Z 1 + eFz . 4π0 r

The electric field (13.36) produces a dipole potential  4π Y10 (ˆr ) , VF = eFz = eF r (13.38) 3 which does not preserve parity. A first-order perturbation calculation for the energy E n(1) = n |VF | n

The total potential acting on the electron is then Vtot (r) = −

number of bound states, now v(±∞, ρ) = ±∞ and v has a local maximum. On the z axis, this maximum occurs √ at z max = − Z|e|/(4π0 F ) for which v(z max , 0) = 0. There is then a potential barrier through which the electron can tunnel, i. e., there are no bound states any longer but resonances. The potential barrier is shallower the stronger the field; the well can contain a smaller number of bound states and ionization occurs.

(13.37)

Consider the z-dependence of this potential. Call ρ =  x 2 + y2 and v(z, ρ) = V(x, y, z). Unlike the Coulomb case in which vCoul (±∞, ρ) = 0 resulting in an infinite

(13.39)

yields null results unless the unperturbed states are degenerate with states of opposite parity. In the remainder of this chapter, atomic units will be used. Final results for energies can be multiplied by 2R∞ hc to translate to SI or other units. The calculation can be carried out in detail for the case of hydrogenic

Part B 13.4

13.4.1 Stark Ionization

232

Part B

Atoms

atoms [13.8]. In this case it is convenient to work in parabolic coordinates: ϕ denotes the usual angle in the xy-plane, and ξ =r +z , η =r −z .

(13.40)

The Hamiltonian for a hydrogenic atom with a field VF = 12 F (ξ − η) from (13.38) is (ξ + η)H = ξh + (ξ) + ηh − (η) .

(13.41)

The wave function is written in the form 1 Ψ(ξ, η, ϕ) = √ ψ+ (ξ)ψ− (η)eiml ϕ , 2πZ with the ψ± satisfying h ± (x)ψ± (x) = Eψ± (x) ,

(13.42)

where x = ξ for ψ+ and x = η for ψ− , and
 d 2Z ± m l2 1 2 d x − + 2 ∓ Fx , h ± (x) = − x dx dx x 2 2x (13.44)

with (13.45)

Using the notation

Part B 13.4

√  = −2E , 1 n ± = Z ± / − (|m l | + 1) , 2 n = n + + n − + |m l | + 1 , n + , N− = 0, 1, . . . , n − |m l | + 1 , |m l | = 0, 1, 2, . . . , n − 1 , δn = n + − n − ,

(n ± + |m l |)!

1

1 2

(13.46)

1

l |) (x) , e− 2 x (x) 2 |ml | L n(|m ±

(13.47)

L b(a)

δn

,

and to first order in F, 1 E = − 2 ≈ E (0) + E (1) , 2 1 Z2 (0) , E =− 2 n2 3 F n δn . E (1) = 2 Z

(13.50)

(13.51)

A perturbation linear in the field F yields no contribution to nondegenerate states (e.g., the ground state n + = n − = m = 0; n = 1). In this case, the lowest order contribution comes from the quadratic Stark effect, the contribution of order F 2 . The quadratic perturbation to a level E n(0) caused by a general electric field F can be written in terms of the symmetric tensor αijn as 1 E n(2) = − αijn Fi F j , 2 with      n |di | m m d j  n n αij = −2 , En − Em m

(13.52)

(13.53)

where di is defined in (13.35). For a field (13.36),

1

 n±! 2

n Z

m =n

where n is the principal quantum number, the unperturbed eigenfunctions are ψ± (x) =

2F

3 2

13.4.3 Quadratic Stark Effect (13.43)

Z = Z+ + Z− .

From these Z = − n

where the are generalized Laguerre polynomials (Sect. 9.4.2). The zero-order eigenvalues are 
ml + 1 (0) . (13.48) Z ± = n± + 2 The first-order perturbation yields 1 F (1) = ± 2 6n±(n ± Z± 4  + m l + 1) + m l (m l + 3) + 2 . (13.49)

1 ∆E n = − αn F 2 , 2 where  | n |z| m |2 . αn ≡ αnzz = −2e2 En − Em m

(13.54)

(13.55)

m =n

In terms of (13.46), a general nonrelativistic expression for the dipole polarizability of hydrogenic ions is [13.9]

a3 n 4 (13.56) αn = 0 4 17n 2 − 3 δn2 − 9m l2 + 19 . 8Z For the ground state of hydrogenic atoms, 9a03 . (13.57) 2Z 4 Table 13.4 lists the relativistic values for the ground n=1 , obtained by calculating (13.55) state polarizability αrel αn=1 =

Atoms in Strong Fields

using relativistic variational basis sets [13.10]. The values are interpolated by   a3 9 14 − (αZ)2 + 0.53983(αZ)4 . αn=1 = 04 2 3 Z (13.58)

13.4.4 Other Stark Corrections Third Order Corrections For the energy correction cubic in the external field (13.36), one obtains [13.9] 3 3 n 7 F E (3) = 32 Z

× δn 23n 2 − δn2 + 11m l2 + 39 . (13.59) Relativistic Linear Stark-Shift of the Fine Structure of Hydrogen For a Stark effect small relative to the fine structure, the degenerate levels corresponding to the same value of j split according to  
3 2 1 2 nm F . (13.60) δm E n j = n − j+ 4 2 j( j + 1)

2π 1s|δ(r)|1s = 2 − 31F 2 .

(13.61)

Table 13.4 Relativistic dipole polarizabilities for the ground state of hydrogenic atoms Z


 n=1 4 αrel Z / a03

1 5 10 20 30 40 50 60 70 80 90 100

4.499 7515 4.493 7883 4.475 1644 4.400 8376 4.277 5621 4.106 2474 3.888 1792 3.625 0295 3.318 8659 2.972 1524 2.587 7205 2.168 6483

For the Bethe logarithm β defined by  | 1s | p| n |2 (E n − E 1s ) ln |E n − E 1s | β1s = n  , 2 n | 1s | p| n | (E n − E 1s ) (13.62)

the result is [13.12] β1s = 2.290 981 375 205 552 301 + 0.316 205(6) F 2 .

(13.63)

These results are useful in calculating an asymptotic expansion for the two-electron Bethe logarithm [13.13].

13.5 Recent Developments The drastic change of an atom’s internal structure in the presence of external electric and magnetic fields is shown most clearly through the changes induced in its spectral features. Of these features, avoided crossings are a distinctive example. Recent work in this area by Férez and Dehesa [13.14] has suggested the use of Shannon’s information entropy [13.15], defined by  S = − ρ(r) ln ρ(r)dr , (13.64) where ρ(r) = |ψ(r)|2 , as an indicator or predictor of such irregular features of atomic spectra. By studying some excited states of hydrogen in parallel fields it was shown that, for the states involved, a marked confinement of the electron cloud and an information-theoretic

233

exchange occurs when the magnetic field strength is adjusted adiabatically through the region of an avoided crossing. The field strengths studied are characteristic of compact astronomical objects, such as white dwarfs and neutron stars. Although the effects of strong magnetic fields on the structure and dynamics of hydrogen have been known for some time, knowledge of the helium atom in such fields has only recently become sufficient for comparison with astrophysical observations [13.16–18]. As one example of their importance, such studies have proven critical in showing the presence of helium in the atmospheres of certain magnetic white dwarfs [13.19]. In recent years, the increased sophistication and resolution of observation techniques has not only in-

Part B 13.5

Other Stark Corrections in Hydrogen The expectation value of the delta function, is, in a.u. [13.11],

13.5 Recent Developments

234

Part B

Atoms

creased the number of known astronomical objects, but also motivated the study of the effects of strong fields on heavier atoms [13.20]. Another interesting area of current research concerns the relationship between quantum mechanics and classically chaotic systems. For these studies, Rubidium Rydberg atoms are an ideal system since laboratory

fields can easily push the atom to the strong-field limit [13.21–23]. For a very useful review of various topics up to 1998 see [13.24]; a more concise review, concerning the electronic structure of atoms, molecules, and bulk matter, including some properties of dense plasma, in strong fields, is given in [13.25].

References 13.1

13.2

13.3 13.4 13.5

13.6

Part B 13

13.7 13.8

13.9

13.10 13.11 13.12

L. D. Landau, E. M. Lifshitz: Quantum Mechanics (Course of Theoretical Physics), Vol. 3 (Pergamon, Oxford 1977) p. 456 L. D. Landau, E. M. Lifshitz: The Classical Theory of Fields (Course of Theoretical Physics), Vol. 2 (Pergamon, Oxford 1975) p. 49 A. Messiah: Quantum Mechanics (Wiley, New York 1999) p. 491 C. Itzykson, J.-B. Zuber: Quantum Field Theory (McGraw-Hill, New York 1980) p. 67 H. A. Bethe, E. Salpeter: Quantum Mechanics of One- and Two-electron Atoms (Plenum, New York 1977) p. 208 H. A. Bethe, E. Salpeter: Quantum Mechanics of One- and Two-electron Atoms (Plenum, New York 1977) p. 211 Z. Chen, S. P. Goldman: Phys. Rev. A 48, 1107 (1993) H. A. Bethe, E. Salpeter: Quantum Mechanics of One- and Two-electron Atoms (Plenum, New York 1977) p. 229 H. A. Bethe, E. Salpeter: Quantum Mechanics of One- and Two-electron Atoms (Plenum, New York 1977) p. 233 G. W. F. Drake, S. P. Goldman: Phys. Rev. A 23, 2093 (1981) G. W. F. Drake: Phys. Rev. A 45, 70 (1992) S. P. Goldman: Phys. Rev. A 50, 3039 (1994)

13.13 13.14 13.15 13.16 13.17 13.18 13.19 13.20 13.21

13.22

13.23 13.24

13.25

S. P. Goldman, G. W. F. Drake: Phys. Rev. Lett. 68, 1683 (1992) R. González-Férez, J. S. Dehesa: Phys. Rev. Lett. 91, 113001 (2003) C. E. Shannon: Bell Syst. Tech. J. 27, 623 (1948) W. Becken, P. Schmelcher, F. K. Diakonos: J. Phys. B. 32, 1557 (1999) W. Becken, P. Schmelcher: Phys. Rev. A 63, 053412 (2001) W. Becken, P. Schmelcher: Phys. Rev. A 65, 033416 (2002) S. Jordan, P. Schmelcher, W. Becken, W. Schweizer: Astron. Astrophys. 336, 33 (1998) P. Schmelcher: private communication J. von Milczewski, T. Uzer: Atoms and Molecules in Strong External Fields, edited by P. Schmelcher and W. Schweizer (Springer, Berlin, Heidelberg 1998) p. 199 J. Main, G. Wunner: Atoms and Molecules in Strong External Fields, edited by P. Schmelcher and W. Schweizer (Springer, Berlin, Heidelberg 1998) p. 223 J. R. Guest, G. Raithel: Phys. Rev. A 68, 052502 (2003) P. Schmelcher, W. Schweizer (Eds.): Atoms and Molecules in Strong External Fields (Springer, Berlin 1998) D. Lai: Ref. Mod. Phys 73, 629 (2001)

235

Rydberg Atom 14. Rydberg Atoms

Rydberg atoms are those in which the valence electron is in a state of high principal quantum number n. They are of historical interest since the observation of Rydberg series helped in the initial unraveling of atomic spectroscopy [14.1]. Since the 1970s, these atoms have been studied mostly for two reasons. First, Rydberg states are at the border between bound states and the continuum, and any process which can result in either excited bound states or ions and free electrons usually leads to the production of Rydberg states. Second, the exaggerated properties of Rydberg atoms allow experiments to be done which would be difficult or impossible with normal atoms.

14.1

Wave Functions and Quantum Defect Theory ................. 235

14.2

Optical Excitation and Radiative Lifetimes ....................... 237

14.3

Electric Fields ...................................... 238

14.4 Magnetic Fields ................................... 241 14.5 Microwave Fields ................................. 242 14.6 Collisions ............................................ 243 14.7

Autoionizing Rydberg States ................. 244

References .................................................. 245

14.1 Wave Functions and Quantum Defect Theory

where E is the energy, r is the distance between the electron and the proton, and θ and φ are the polar and azimuthal angles of the electron’s position. Equation (14.1) can be separated, and its solution expressed as the product Ψ(r, θ, φ) = R(r)Ym (θ, φ) ,

(14.2)

where  and m are the orbital and azimuthal-orbital angular momentum (i. e., magnetic) quantum numbers and Ym (θ, φ) is a normalized spherical harmonic. R(r) satisfies the radial equation 2R(r) ( + 1)R d2 R(r) 2 dR(r) +2E R(r)+ = + , r dr r dr 2 r2 (14.3)

which has the two physically interesting solutions f(, E, r) , (14.4) r g(, E, r) . R(r) = (14.5) r The f and g functions are the regular and irregular Coulomb functions which are the solutions to a variant of (14.3). As r → 0 they have the forms [14.3] R(r) =

f (, E, r) ∝ r +1 , g(, E, r) ∝ r

−

(14.6)

,

(14.7)

irrespective of whether E is positive or negative. As r → ∞, for E > 0 the f and g functions are sine and cosine waves, i. e., there is a phase shift of π/2 between them. For E < 0 it is useful to introduce ν, defined by E = −1/2ν2 , and for E < 0 as r → ∞ f = u(, ν, r) sin πν − v(, ν, r)eiπν , g = −u(, ν, r) cos πν + v(, ν, r)e

iπ(ν+1/2)

(14.8)

, (14.9)

where u and v are exponentially increasing and decreasing functions of r. As r → ∞, u → ∞ and v → 0.

Part B 14

Many of the properties of Rydberg atoms can be calculated accurately using quantum defect theory, which is easily understood by starting with the H atom [14.2]. We shall use atomic units, as discussed in Sect. 1.2. The Schrödinger equation for the motion of the electron in a H atom in spherical co-ordinates is
 1 1 − ∇2 − Ψ(r, θ, φ) = EΨ(r, θ, φ) , (14.1) 2 r

236

Part B

Atoms

Requiring that the wave function be square integrable means that as r → 0 only the f function is allowed. Equation (14.8) shows that the r → ∞ boundary condition requires that sin πν be zero or ν an integer n, leading to the hydrogenic Bohr formula for the energies: E=−

1 . 2n 2

Hydrogen r Sodium r

(14.10)

The classical turning point of an s wave occurs at r = 2n 2 , and the expectation values of positive powers of r reflect the location of the outer turning point, i. e.,  k r ≈ n 2k . (14.11)

Part B 14.1

The expectation values of negative powers of r are determined by the properties of the wave function at small r. The normalization constant of the radial wave function scales as n −3/2 , so that R(r) ∝ n −3/2r +1 for small r. Accordingly, the expectation values of negative powers of r, except r −1 , and any properties which depend on the small r part of the wave function, scale as n −3 . Using the properties of the wave function and the energies, the nscaling of the properties of Rydberg atoms can be determined. The primary reason for introducing the Coulomb waves instead of the more common Hermite polynominal solution for the radial function is to set the stage for single channel quantum defect theory, which enables us to calculate the wave functions and properties of one valence electron atoms such as Na. The simplest picture of an Na Rydberg atom is an electron orbiting a positively charged Na+ core consisting of 10 electrons and a nucleus of charge +11. The ten electrons are assumed to be frozen in place with spherical symmetry about the nucleus, so their charge cloud is not polarized by the outer valence electron, although the valence electron can penetrate the tenelectron cloud. When the electron penetrates the charge cloud of the core electrons, it sees a potential well deeper than −1/r due to the decreased shielding of the +11 nuclear charge. For Na and other alkali atoms, we assume that there is a radius rc such that for r < rc the potential is deeper than −1/r, and for r > rc it is equal to −1/r. As a result of the deeper potential at r < rc , the radial wave function is pulled into the core in Na, relative to H, as shown in Fig. 14.1. For r ≥ rc , the potential is a Coulomb −1/r potential, and R(r) is a solution of (14.3) which can be expressed

Fig. 14.1 Radial wave functions for H and Na showing that

the Na wave function is pulled in toward the ionic core

as

  R(r) = f (, ν, r) cos τ − g(, ν, r) sin τ r , (14.12)

where τ is the radial phase shift. Near the ionization limit, E ∼ 0, and as a result, the kinetic energy of the Rydberg electron is greater than 1/rc (∼ 10 eV) when r < rc . As a result, changes in E of 0.10 eV, the n = 10 binding energy, do not appreciably alter the phase shift τ , and we can assume τ to be independent of E. The  dependence of τ arises because the centrifugal ( + 1)/r term in (14.3) excludes the Rydberg electron from the region of the core in states of high . Applying the r → ∞ boundary condition to the wave function of (14.12) leads to the requirement that the coefficient of u vanish, i. e., cos τ sin(πν) + sin τ cos(πν) = 0 ,

(14.13)

which implies that sin(πν + τ ) = 0 or ν = n − τ /π. Usually τ /π is written as δ and termed the quantum defect, and the energies of members of the n series are written as 1 1 = − ∗2 , (14.14) E=− 2(n − δ )2 2n where n ∗ = n − δ is often termed the effective quantum number (see also Sect. 11.4.1). Knowledge of the quantum defect δ of a series of  states determines their energies, and it is a straightforward matter to calculate the Coulomb wave function specified in (14.12) using a Numerov algorithm [14.4, 5]. This procedure gives wave functions valid for r ≥ rc , which can be used to calculate many of the properties of Rydberg atoms with great accuracy. The effect of core penetration on the energies is easily seen in the energy level diagram of Fig. 14.2. The Na  ≥ 2 states have the same energies as hydro-

Rydberg Atoms

E (× 1000 cm–1) 0

8s

8p

4p

Hydrogen

8d

8f

4d

4f

8g

n=8

gen, while the s and p states, with quantum defects of 1.35 and 0.85 respectively, lie far below the hydrogenic energies. Although it is impossible to discern in Fig. 14.2, the Na  ≥ 2 states also lie below the hydrogenic energies. For these states it is not core penetration, but core polarization which is responsible for the shift to lower energy. Contrary to our earlier assumption that the outer electron does not affect the inner electrons if r > rc , the outer electron polarizes the inner electron cloud even when r > rc , and the energies of even the high  states fall below the hydrogenic energies. The leading term in the polarization energy is due to the dipole polarizability of the core, αd . For high  states it gives a quantum defect of [14.6]

n=4

3d

n=3

4s

–20 3p n=2

–30

–40

237

Fig. 14.2 Energy levels of Na and H Sodium

5s

–10

14.2 Optical Excitation and Radiative Lifetimes

3αd . (14.15) 45 Quantum defects due primarily to core polarization rarely exceed 10−2 , while those due to core penetration are often greater than one.

3s

δ =

14.2 Optical Excitation and Radiative Lifetimes

σn =

σPI . n 3 ∆ω

(14.16)

A typical value for σPI is 10−18 cm2 . For a resolution ∆ω = 1 cm−1 (6 × 10−6 a.u.) the cross section for exciting an n = 20 atom is 3 × 10−17 cm2 . From the wave functions of the Rydberg states, we can also derive the n −3 dependence of the photoexcitation cross section. The dipole matrix element from the ground state to a Rydberg state only involves the part of the Rydberg state wave function near the core. At small r, the Rydberg wave function only depends on n through the n −3 normalization factor, and as a result, the squared dipole matrix element between the ground state and the Rydberg state and the cross section both have an n −3 dependence. Radiative decay, which is covered in Chapt. 17, is, to some extent, the reverse of optical excitation. The general expression for the spontaneous transition rate from the n state to the n  state is the Einstein A coefficient, given by [14.2] 4 α3 g> , An,n  = µ2n,n  ω3n,n  3 2gn + 1

(14.17)

Part B 14.2

Optical excitation of the Rydberg states from the ground state or any other low lying state is the continuation of the photoionization cross section σPI below the ionization limit. The photoionization cross section, discussed more extensively in Chapt. 24, is approximately constant at the limit. Above and below the limit the average photoabsorption cross section is the same, as evidenced by the fact that a discontinuity is not evident in an absorption spectrum, i. e., it is not possible to see where the unresolved Rydberg states end and the continuum begins. Nonetheless, below the limit the cross section is structured by the ∆n spacing of 1/n 3 between adjacent members of the Rydberg series. In any experiment, there is a finite resolution ∆ω with which the Rydberg states can be excited. It can arise, for example, from the Doppler width or a laser linewidth. This resolution determines the cross section σn for exciting the Rydberg state of principal quantum number n. Explicitly, σn is given by

238

Part B

Atoms

where µn,n  and ωn,n , are the electric dipole matrix elements and frequencies of the n → n  transitions, gn and gn are the degeneracies of the n and n  states, and g> is the greater of gn and gn . The lifetime τn of the n state is obtained by summing the decay rates to all possible lower energy states. Explicitly,  1 = An,n  . (14.18) τn n

Due to the ω3 factor in (14.17), the highest frequency transition usually contributes most heavily to the total radiative decay rate, and the dominant decay is likely to be the lowest lying state possible. For low- Rydberg states, the lowest lying  states are bound by orders of magnitude more than the Rydberg states, and the frequency of the decay is nearly independent of n. Only the squared dipole moment depends on n, as n −3 , because of the normalization of the Rydberg wave function at the core. Consequently, for low- states, τn ∝ n 3 .

n¯ =

1 . eω/kT − 1

(14.21)

The stimulated emission or absorption rate K n,n  from state n to state n  is given by 4 α3 ng ¯ > K n,n  = µn,n  ω3n,n  . 3 2gn + 1

(14.22)

Summing these rates over n and  gives the total blackbody decay rate 1/τnbb . Explicitly,  1 = K n,n  . bb τn n 

(14.23)

T at any given temperature is The resulting lifetime τn given by

(14.19)

As a typical example, the 10f state in H has a lifetime of 1.08 µs [14.7]. The highest  states, with  = n − 1, have radiative lifetimes with a completely different n dependence. The only possible transitions are n → n − 1, with frequency 1/n 3 . In this case the dipole moments reflect the large size of both the n and n − 1 states and have the n 2 scaling of the orbital radius. Using (14.17) for  = n − 1 leads to τn(n−1) ∝ n 5 .

states for n ≥ 10, and these photons drive transitions to higher and lower states [14.9]. A convenient way of describing blackbody radiation is in terms of the photon occupation number n, ¯ given by

(14.20)

Part B 14.3

Another useful lifetime, τn , is that corresponding to the average decay rate of all , m states of the same n. It scales as n 4.5 [14.2, 8]. Equation (14.17) describes spontaneous decay to lower lying states driven by the vacuum. At room temperature, 300 K, there are many thermal photons at the frequencies of the n → n ± 1 transitions of Rydberg

1 bb = 1/τn + 1/τn . T τn

(14.24)

For low- states with 10 < n < 20, blackbody radiation produces a 10% decrease in the lifetimes, but for high- states of the same n, it reduces the lifetimes by a factor of bb ∝ n −2 , this term must dominate normal ten. Since 1/τn spontaneous emission at high n. The above discussion of spontaneous and stimulated transitions is based on the implicit assumption that the atoms are in free space. If the atoms are in a cavity, which introduces structure into the blackbody and vacuum fields, the transition rates are significantly altered [14.10]. These alterations are described in Chapt. 79. If the cavity is tuned to a resonance, it increases the transition rate by the finesse of the cavity (approximately the Q for low-order modes). On the other hand, if the cavity is tuned between resonances, the transition rate is suppressed by a similar factor.

14.3 Electric Fields As a starting point, consider the H atom in a static electric field E in the z-direction, and focus on the states of principal quantum number n. The field couples  and  ± 1 states of the same m by the electric dipole matrix elements. Since the states all have a common zero field energy of −1/2n 2 , and the off-diagonal Hamiltonian matrix elements are all proportional to E, the eigenstates

are field-independent linear combinations of the zero field  states of the same m, and the energy shifts from −1/2n 2 are linear in E. In this first-order approximation, the energies are given by [14.2] E=−

1 3 + (n 1 − n 2 )nE , 2n 2 2

(14.25)

Rydberg Atoms

where n 1 and n 2 are parabolic quantum numbers (see Sect. 9.1.2) which satisfy n 1 + n 2 + |m| + 1 = n .

0.001

0.0

–0.001

B R

–0.002

–0.003 –2800

–1600

0

1600

2800 z (a 0)

Fig. 14.3 Combined Coulomb–Stark potential along the

z-axis when a field of 5 × 10−7 a.u. (2700 V/cm) is applied in the z-direction (solid). The extreme red state (R) is near the saddlepoint, and the extreme blue (B) state is held on the upfield side of the atom by an effective potential (dashed) roughly analogous to a centrifugal potential

Part B 14.3

Energy (cm–1) –250 22s

–260 21p

–270 n = 20

–280

21s

–290 20p

–300 n = 19

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0 ε (kV / cm)

Fig. 14.4 Energies of Na m = 0 levels of n ≈ 20 as a function of electric field. The shaded region is above the classical

ionization limit

239

E (arb. units)

(14.26)

Consider the m = 0 states as an example. The n 1 − n 2 = n − 1 state is shifted up in energy by 32 n(n − 1)E and is called the extreme blue Stark state, and the n 2 − n 1 = n − 1 state is shifted down in energy by 32 n(n − 1)E and is called the extreme red Stark state. These two states have large permanent dipole moments, and in the red (blue) state the electron spends most of its time on the downfield (upfield) side of the proton as shown in Fig. 14.3, a plot of the potential along the z-axis. We have here ignored the electric dipole couplings to other n states, which introduce small second order Stark shifts to lower energy. As implied by (14.26), states of higher m have smaller shifts. In particular, the circular m =  = n − 1 state has no first order shift since there are no degenerate states to which it is coupled by the field. The Stark effect in other atoms is similar, but not identical to that observed in H. This point is shown by Fig. 14.4, a plot of the energies of the Na m = 0 levels near n = 20. The energy levels are similar to those of H in that most of the levels exhibit apparently linear Stark shifts from the zero field energy of the high- states. The differences, however, are twofold. First, the levels from s and p states with nonzero quantum defects join the manifold of Stark states at some nonzero field, given

14.3 Electric Fields

240

Part B

Atoms

approximately by [14.4] 2δ (14.27) E = 5 , 3n where δ is the magnitude of the difference between δ and the nearest integer. Second, there are avoided crossings between the blue n = 20 and red n = 21 Stark states. In H these states would cross, but in Na they do not because of the finite sized Na+ core, which also leads to the nonzero quantum defects of the ns and np states. This point, and other related points, are described in Chapt. 15. Field ionization is both intrinsically interesting and of great practical importance for the detection of Rydberg atoms [14.11]. The simplest picture of field ionization can be understood with the help of Fig. 14.4. The potential along the z-axis of an atom in a field E in the z-direction is given by 1 V = − −Ez . (14.28) r If an atom has an energy E relative to the zero field limit, it can ionize classically if the energy E lies above the saddle point in the potential. The required field is given by E2 . (14.29) 4 Ignoring the Stark shifts and using E = −1/2n 2 yields the expression 1 E= . (14.30) 16n 4 E=

Part B 14.3

The H atom ionizes classically as described above, or by quantum mechanical tunneling which occurs at slightly lower fields. Since the tunneling rates increase exponentially with field strength, typically an order of magnitude for a 3% change in the field, specifying the classical ionization field is a good approximation to the field which gives an ionization rate of practical interest. The red and blue states of H ionize at very different fields, as shown by Fig. 14.5, a plot of the m = 0 Stark states out to the fields at which the ionization rates are 106 s−1 [14.12]. First, note the crossing of the levels of different n mentioned earlier. Second, note that the red states ionize at lower fields than do the blue states, in spite of the fact that they are lower in energy. In the red states, the electron is close to the saddle point of the potential of Fig. 14.3, and it ionizes according to (14.29). If the Stark shift of the extreme red state to lower energy is taken into account, (14.30) becomes 1 E= 4. (14.31) 9n

E (× 1000 cm–1) –500

–1000

–1500

0

50

100 ε (kV / cm)

Fig. 14.5 Energies of H m = 0 levels of n = 9, 10, and 11 as functions of electric field. The widths of the levels due to ionization broaden exponentially with fields, and the onset of the broadening indicated is at an ionization rate of 106 s−1 . The broken line indicating the classical ionization limit, E = E 2 /4 passes near the points at which the extreme red states ionize

In the blue state the electron is held on the upfield side of the atom by an effective potential roughly analogous to a centrifugal potential, as shown by Fig. 14.3. At the same field the blue state’s energy is lower relative to the saddle point of its potential, shown by the broken line of Fig. 14.3, than is the energy of the red state relative to the saddle point of its potential, given by (14.28) and shown by the solid line of Fig. 14.3. As shown by the broken line of Fig. 14.5, the classical ionization limit of (14.29) is simply a line connecting the ionization fields of the extreme red Stark states. All other states are stable above the classical ionization limit. In the Na atom, ionization of m = 0 states occurs in a qualitatively different fashion [14.12]. Due to the finite size of the Na+ core, there are avoided crossings between the blue and red Stark states of different n, as is shown by Fig. 14.4. In the region above the classical ionization limit, shown by the shaded region of Fig. 14.4, the same coupling between hydrogenically stable blue states and the degenerate red continua leads to autoionization of the blue states [14.13]. As a result, all states above the classical ionization limit ionize at experimentally significant rates. In higher m states, the core coupling is smaller, and the behavior is more similar to H.

Rydberg Atoms

Field ionization is commonly used to detect Rydberg atoms in a state selective manner. Experiments are most often conducted at or near zero field, and afterwards the field is increased in order to ionize the atoms. Exactly how the atoms pass from the low field to the high ionizing field is quite important. The passage can be adiabatic, diabatic or anything in between. The selectivity is best if the passage is purely adiabatic or purely diabatic, for in these two cases unique paths are followed. In zero field, optical excitation from a ground s state leads only to final np states. In the presence of an electric field, all the Stark states are optically accessible, because they all have some p character. The fact that all the Stark states are optically accessible from the ground state allows the population of arbitrary  states of nonhydrogenic atoms by a technique called Stark switching [14.6, 14]. In any atom other than H, the  states are nondegenerate in zero field, and each of them is adiabatically connected to one, and only one, high field Stark state, as shown by Fig. 14.4. If one of the Stark states is excited with a laser and the field reduced to zero adiabatically, the atoms are left in a single zero field  state. In zero field, the photoionization cross section is structureless. However, in an electric field, it exhibits obvious structure, sometimes termed strong field mixing resonances. Specifically, when ground state s atoms are exposed to light polarized parallel to the static field, an oscillatory structure is observed in the cross section, even above the zero field ionization limit [14.15]. The

14.4 Magnetic Fields

origin of the structure can be understood with the aid of a simple classical picture [14.16, 17]. The electrons ejected in the downfield direction can simply leave the atom, while the electrons ejected in the upfield direction are reflected back across the ionic core and also leave the atom in the downfield direction. The wave packets corresponding to these two classical trajectories are added, and they can interfere constructively or destructively at the ionic core depending on the phase accumulation of the reflected wave packet. Since the phase depends on the energy, there is an oscillation in the photoexcitation spectrum. This model suggests that no oscillations should be observed for light polarized perpendicular to the static field, and none are. The oscillations can also be thought of as arising from the remnants of quasistable extreme blue Stark states which have been shifted above the ionization limit, and, using this approach or a WKB approach, one can show that the spacing between the oscillations at the zero field limit is ∆E = E 3/4 [14.18, 19]. The initial photoexcitation experiments were done using narrow bandwidth lasers, so that the time dependence of the classical pictures was not explicitly observed. Using mode locked lasers it has been possible to create a variety of Rydberg wave packets [14.20, 21] and observe, in effect, the classical motion of an electron in an atom. Of particular interest, it has been possible to directly observe the time delay of the ejection of electrons subsequent to excitation in an electric field [14.22].

spaced by ∆E = 3~ B/2, in the photoionization cross section above the ionization limit [14.24]. The origin of this structure is similar to the origin of the strong field mixing resonances observed in electric fields. An electron ejected in the plane perpendicular to the B fields is launched into a circular orbit and returns to the ionic core. The returning wave packet can be in or out of phase with the one leaving the ionic core, and thus, can interfere constructively or destructively with it. While the electron motion in the xy-plane is bound, motion in the z-direction is unaffected by the magnetic field and is unbounded above the ionization limit, leading to resonances of substantial width. The Coulomb potential does provide some binding in the z-direction and allows the existence of quasistable three-dimensional orbits [14.25].

Part B 14.4

14.4 Magnetic Fields To first order, the energy shift of a Rydberg atom due to a magnetic field B (the Zeeman effect) is proportional to the angular momentum of the atom. Since the states optically accessible from the ground state have low angular momenta, the energy shifts are the same as those of low-lying atomic states. In contrast, the second order diamagnetic energy shifts are proportional to the area of the Rydberg electron’s orbit and scale as B 2 n 4 [14.23]. The diamagnetic interaction mixes the  states, allowing all to be excited from the ground state, and produces large shifts to higher energies. The energy levels as a function of magnetic field are reminiscent of the Stark energy levels shown in Fig. 14.5, differing in that the energy shifts are quadratic in the magnetic field. One of the most striking phenomena in magnetic fields is the existence of quasi-Landau resonances,

241

242

Part B

Atoms

14.5 Microwave Fields

Part B 14.5

Strong microwave fields have been used to drive multiphoton transitions between Rydberg states and to ionize them. Here we restrict our attention to ionization. Ionization by both linearly and circularly polarized fields has been explored with both H and other atoms. Hydrogen atoms have been studied with linearly polarized fields of frequencies up to 36 GHz [14.26]. When the microwave frequency ω 1/n 3 , ionization of m = 0 states occurs at a field of E = 1/9n 4 (E 2 /4), which is the field at which the extreme red Stark state is ionized by a static field. Due to the second-order Stark effect, the blue and red shifted states are not quite mirror images of each other, and when the microwave field reverses, transitions between Stark states occur. There is a rapid mixing of the Stark states of the same n and m by a microwave field, and all of them are ionized at the same microwave field amplitude, E = 1/9n 4 . Important points are that no change in n occurs and the ionization field is the same as the static field required for ionization of the extreme red Stark state. As ω approaches 1/n 3 , the field falls below 1/9n 4 due to ∆n transitions to higher lying states, allowing ionization at lower fields. This form of ionization can be well described as the transition to the classically chaotic regime [14.27]. For ω > 1/n 3 the ionization field is more or less constant, and for ω > 1/2n 2 the process becomes photoionization. The ionization of nonhydrogenic atoms by linearly polarized fields has also been investigated at frequencies of up to 30 GHz, but the result is very different from the hydrogenic result. For ω 1/n 3 and low m, ionization occurs at a field of E ≈ 1/3n 5 [14.28]. This is the field at which the m = 0 extreme blue and red Stark states of principal quantum number n and n + 1 have their avoided crossing. For n = 20 this field is ≈ 500 V/cm, as shown by Fig. 14.4. How ionization occurs can be understood with a simple model based on a time-varying electric field. As the microwave field oscillates in time, atoms follow the Stark states of Na shown in Fig. 14.4. Even with very small field amplitudes, transitions between the Stark states of the same n are quite rapid because of the zero field avoided crossings. If the field reaches 1/3n 5 , the avoided crossing between the extreme red n and blue n + 1 state is reached, and an atom in the blue n Stark state can make a Landau–Zener transition to the red n + 1 Stark state. Since the analogous red–blue avoided crossings between higher lying states occur at lower fields, once an atom has made the n → n + 1 transi-

tion it rapidly makes a succession of transitions through higher n states to a state which is itself ionized by the field. The Landau–Zener description given above is somewhat oversimplified in that we have ignored the coherence between field cycles. When it is included, we see that the transitions between levels are resonant multiphoton transitions. While the resonant character is obscured by the presence of many overlapping resonances, the coherence substantially increases the n → n + 1 transition probability even when E < 1/3n 5 . The fields required for ionization calculated using this model are lower than 1/3n 5 , in agreement with the experimental observations. Nonhydrogenic Na states of high m behave like H, because no states with significant quantum defects are included, and the n → n + 1 avoided crossings are vanishingly small. Experiments on ionization of alkali atoms by circularly polarized fields of frequency ω show that for ω 1/n 3 , a field amplitude of E = 1/16n 4 is required for ionization [14.29]. This field is the same as the static field required. In a frame rotating with frequency ω, the circularly polarized field is stationary and cannot induce transitions, so this result is not surprising. On the other hand, when the problem is transformed to the rotating frame, the potential of (14.28) is replaced by 1 ω2 ρ2 V = − −Ex − , r 2

(14.32)

where ρ2 = x 2 + y2 , and we have assumed the field to be in the x-direction in the rotating frame. This potential has a saddle point below E = 1/16n 4 [14.30]. As n or ω is raised so that ω → 1/n 3 , the experimentally observed field falls below 1/16n 4 , but not so fast as implied by (14.32). Equation (14.32) is based solely on energy considerations, and ionization at the threshold field implied by (14.32) requires that the electron escape over the saddle point in the rotating frame at nearly zero velocity. For this to happen, when ω approaches 1/n 3 , more than n units of angular momentum must be transferred to the electron, which is unlikely. Models based on a restriction of the angular momentum transferred from the field to the Rydberg electron are in better agreement with the experimental results. Small deviations of a few percent from circular polarization allow ionization at fields as low as E = 1/3n 5 . This sensitivity can be understood as follows. In the rotating frame, a field with slightly elliptical polarization appears to be

Rydberg Atoms

a large static field with a superimposed oscillating field at frequency 2ω. The oscillating field drives transitions to states of higher energy, allowing ionization at fields less than E = 1/16n 4 . In the regime in which ω > 1/n 3 , microwave ionization of nonhydrogenic atoms is essentially the same

14.6 Collisions

243

as it is in H [14.31]. In this regime, the microwave field couples states differing in n by more than one, and the pressure or absence of quantum defects is not so important. Consequently, only for ω > 1/n 3 is the microwave ionization of H and other atoms different.

14.6 Collisions Since Rydberg atoms are large, with geometric cross sections proportional to n 4 , one might expect the cross sections for collisions to be correspondingly large. In fact, such is often not the case. A useful way of understanding collisions of neutral atoms and molecules with Rydberg atoms is to imagine an atom or molecule M passing through the electron cloud of an Na Rydberg atom. There are three interactions e− −Na+ ,

e− −M ,

M−Na+ .

(14.33)

Vd =

µ1 µ2 . R3

(14.34)

Here µ1 and µ2 are the dipole matrix elements of the upward and downward transitions in the two atoms, and R is their separation. At room temperature, this process leads to enormous cross sections, substantially in excess of the geometric cross sections. At the low temperatures (300 µK) attainable using cold atoms, the atoms do not move, and therefore cannot collide. However, resonant dipole–dipole energy transfer is still observed due to the static dipole–dipole interactions of not two, but many atoms [14.36, 37].

Part B 14.6

The long range e− –Na+ interaction determines the energy levels of the Na atom. The short range of the e− –M and M–Na+ interactions makes it likely that only one will be important at any given time. This approximation, termed the binary encounter approximation, is described in Chapt. 56. The M–Na+ interaction can only lead to cross sections of ≈ 10–100 Å2 . On the other hand, since the electron can be anywhere in the cloud, the cross sections due to the e− –M interaction can be as large as the geometric cross section of the Rydberg atom. Accordingly, we focus on the e− –M interaction. Consider a thermal collision between M and an Na Rydberg atom. Typically, M passes through the electron cloud slowly compared with the velocity of the Rydberg electron, and it is the e− –M scattering which determines what happens in the M–Na collision, as first pointed out by Fermi [14.32]. First consider the case where M is an atom. There are no energetically accessible states of atom M which can be excited by the low energy electron, so the scattering must be elastic. The electron can transfer very little kinetic energy to M, but the direction of the electron’s motion can change. With this thought in mind, we can see that only the collisional mixing of nearly degenerate  states of the same n has very large cross sections. The -mixing cross sections are approximately geometric at low n [14.33]. If the M atom comes anywhere into the Rydberg orbit, scattering into a different  state occurs. At high n,

the cross section decreases, because the probability distribution of the Rydberg electron becomes too dilute, and it becomes increasingly likely that the M atom will pass through the Rydberg electron’s orbit without encountering the electron. The n at which the peak -mixing cross section occurs increases with the electron scattering length of the atom. While -mixing cross sections are large, n changing cross sections are small ≈ 100 Å2 since they cannot occur when the Rydberg electron is anywhere close to the outer turning point of its orbit [14.34]. If M is a molecule, there are likely to be energetically accessible vibrational and rotational transitions which can provide energy to or accept energy from the Rydberg electron, and this possibility increases the likelihood of n changing collisions with Rydberg atoms [14.11]. Electronic energy from the Rydberg atom must be resonantly transferred to rotation or vibration in the molecule. In heavy or complex molecules, the presence of many rotational-vibrational states tends to obscure the resonant character of the transfer, but in several light systems the collisional resonances have been observed clearly [14.11]. Using the large Stark shifts of Rydberg atoms it is possible to tune the levels so that resonant energy transfer between two colliding atoms can occur [14.35] by the resonant dipole–dipole coupling,

244

Part B

Atoms

Since Rydberg atoms are easily perturbed by electric fields, it is hardly a surprise that collisions of charged particles with Rydberg atoms have large cross sections. In cold Rydberg atom samples, these large cross sections

can lead to the spontaneous evolution to a plasma, since the macroscopic positive charge of the cold ions can trap any liberated electrons, leading to impact ionization for a large part of the Rydberg atom sample [14.38, 39].

14.7 Autoionizing Rydberg States The bound Rydberg atoms considered thus far are formed by adding the Rydberg electron to the ground state of the ionic core. It could equally well be added to an excited state of the core [14.40]. Figure 14.6 shows the energy levels of the ground 5s state of Sr+ and the excited 5p state. Adding an n electron to the 5s state yields the bound Sr 5sn state, and adding it to the excited 5p state gives the doubly excited 5pn state, which is coupled by the Coulomb interaction to the degenerate 5s  continuum. The 5pn state autoionizes at the rate Γn given by [14.41] 2

Γn = 2π|5pn|V |5s  | ,

(14.35)

where V denotes the Coulomb coupling between the nominally bound 5pn state and the 5s  continuum. A more general description of autoionization can be found in Chapt. 25. A simple picture, based on superelastic electron scattering from the Sr+ 5p state, gives the scaling of the autoionization rates of (14.35) with n and . The n Rydberg electron is in an elliptical orbit, and each time

Part B 14.7 5p

5s

Fig. 14.6 Sr+ 5s and 5p states (—), the Rydberg states of

Sr converging to these two ionic states are shown by (—), the continuum above the two ionic levels (///). The 5pn states are coupled to the 5s  continua and autoionize

it comes near the core it has an n-independent probability γ of scattering superelastically from the Sr+ 5p ion, leaving the core in the 5s state and gaining enough energy to escape from the Coulomb potential of the Sr+ core. The autoionization rate of the 5pn state is obtained by multiplying γ by the orbital frequency of the n state, 1/n 3 to obtain Γn =

γ . n3

(14.36)

Equation (14.36) displays the n dependence of the autoionization rate explicitly and the  dependence through γ . As  increases, the closest approach of the Rydberg electron to the Sr+ is at a larger orbital radius, so that superelastic scattering becomes progressively less probable, and γ decreases rapidly with increasing . The simple picture of autoionization given above implies a finite probability of autoionization each time the n electron passes the ionic core, so the probability of an atom’s remaining in the autoionizing state should resemble stair steps [14.42], which can be directly observed using mode locked laser excitation and detection [14.43]. To a first approximation, the Sr 5pn states can be described by the independent electron picture used above, but in states converging to higher lying states of Sr+ , the independent electron picture fails. Consider the Sr+  ≥ 4 states of n > 5. They are essentially degenerate, and the field due to an outer Rydberg electron converts the zero field  states to superpositions much like Stark states. The outer electron polarizes the Sr+ core, so that the outer electron is in a potential due to a charge and a dipole, and the resulting dipole states of the outer electron display a qualitatively different excitation spectrum than do states such as the 5pn states, which are well described by an independent particle picture [14.44]. When both electrons are excited to very high-lying states, with the outer electron in a state of relatively low , the classical orbits of the two electrons cross. Time domain measurements, made using wave packets, show that in this case autoionization is likely to occur in the first orbit of the outer electron [14.45].

Rydberg Atoms

References

245

References 14.1 14.2

14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10

14.11

14.12 14.13 14.14 14.15

14.18 14.19 14.20 14.21 14.22 14.23

14.24 14.25 14.26 14.27 14.28

14.29 14.30 14.31 14.32 14.33 14.34

14.35 14.36

14.37 14.38

14.39

14.40 14.41 14.42 14.43 14.44 14.45

of Atoms and Molecules, ed. by R. F. Stebbings, F. B. Dunning (Cambridge Univ. Press, Cambridge 1983) W. R. S. Garton, F. S. Tomkins: Astrophys. J. 158, 839 (1969) A. Holle, J. Main, G. Wiebusch, H. Rottke, K. H. Welge: Phys. Rev. Lett. 61, 161 (1988) B. E. Sauer, M. R. W. Bellerman, P. M. Koch: Phys. Rev. Lett. 68, 1633 (1992) R. V. Jensen, S. M. Susskind, M. M. Sanders: Phys. Rept. 201, 1 (1991) P. Pillet, H. B. van Linden van der Heuvell, W. W. Smith, R. Kachru, N. H. Tran, T. F. Gallagher: Phys. Rev. A 30, 280 (1984) P. Fu, T. J. Scholz, J. M. Hettema, T. F. Gallagher: Phys. Rev. Lett. 64, 511 (1990) M. Nauenberg: Phys. Rev. Lett. 64, 2731 (1990) A. Krug, A. Buchleitner: Phys. Rev. A 66, 053416 (2002) E. Fermi: Nuovo Cimento 11, 157 (1934) T. F. Gallagher, S. A. Edelstein, R. M. Hill: Phys. Rev. A 15, 1945 (1977) F. Gounand, J. Berlande: Experimental studies of the interaction of Rydberg atoms with atomic species at thermal energies. In: Rydberg States of Atoms and Molecules, ed. by R. F. Stebbings, F. B. Dunning (Cambridge Univ. Press, Cambridge 1983) T. F. Gallagher: Phys. Rept. 210, 319 (1992) I. Mourachko, D. Comparat, F. de Tomasi, A. Fioretti, P. Nosbaum, V. M. Akulin, P. Pillet: Phys. Rev. Lett. 80, 253 (1998) W. R. Anderson, J. R. Veale, T. F. Gallagher: Phys. Rev. Lett. 80, 249 (1998) M. P. Robinson, B. Laburthe-Tolra, M. W. Noel, T. F. Gallagher, P. Pillet: Phys. Rev. Lett. 85, 4466 (2000) S. K. Dutta, J. R. Guest, D. Feldbaum, A. WalzFlannigan, G. Raithel: Phys. Rev. Lett. 86, 3993 (2001) T. F. Gallagher: J. Opt. Soc. Am. B 4, 794 (1987) U. Fano: Phys. Rev. 124, 1866 (1961) X. Wang, W. E. Cooke: Phys. Rev. Lett. 67, 696 (1991) S. N. Pisharody, R. R. Jones: Phys. Rev. A 65, 033418 (2002) U. Eichmann, V. Lange, W. Sandner: Phys. Rev. Lett. 68, 21 (1992) S. N. Pisharody, R. R. Jones: Science 303, 813 (2004)

Part B 14

14.16 14.17

H. E. White: Introduction to Atomic Spectra (McGraw-Hill, New York 1934) H. A. Bethe, E. A. Salpeter: Quantum Mechanics of One and Two Electron Atoms (Academic, New York 1975) U. Fano: Phys. Rev. A 2, 353 (1970) M. L. Zimmerman, M. G. Littman, M. M. Kash, D. Kleppner: Phys. Rev. A 20, 2251 (1979) S. A. Bhatti, C. L. Cromer, W. E. Cooke: Phys. Rev. A 24, 161 (1981) R. R. Freeman, D. Kleppner: Phys. Rev. A 14, 1614 (1976) A. Lindgard, S. E. Nielsen: At. Data Nucl. Data Tables 19, 534 (1977) E. S. Chang: Phys. Rev. A 31, 495 (1985) W. E. Cooke, T. F. Gallagher: Phys. Rev. A 21, 588 (1980) S. Haroche, J. M. Raimond: Radiative properties of Rydberg states in resonant cavities. In: Advances in Atomic and Molecular Physics, Vol. 20, ed. by D. Bates, B. Bederson (Academic, New York 1985) F. B. Dunning, R. F. Stebbings: Experimental studies of thermal-energy collisions of Rydberg atoms with molecules. In: Rydberg States of Atoms with Molecules, ed. by R. F. Stebbings, F. B. Dunning (Cambridge Univ. Press, Cambridge 1983) D. S. Bailey, J. R. Hiskes, A. C. Riviere: Nucl. Fusion 5, 41 (1965) M. G. Littman, M. M. Kash, D. Kleppner: Phys. Rev. Lett. 41, 103 (1978) R. R. Jones, T. F. Gallagher: Phys. Rev. A 38, 2946 (1988) R. R. Freeman, N. P. Economou, G. C. Bjorklund, K. T. Lu: Phys. Rev. Lett. 41, 1463 (1978) W. P. Reinhardt: J. Phys. B 16, 635 (1983) J. Gao, J. B. Delos, M. C. Baruch: Phys. Rev. A 46, 1449 (1992) T. F. Gallagher: Rydberg Atoms (Cambridge Univ. Press, Cambridge 1994) A. R. P. Rau: J. Phys. B 12, L193 (1979) R. R. Jones, L. D. Noordam: Adv. At. Mol. Opt. Phys. 38, 1 (1997) G. Alber, P. Zoller: Phys. Rep. 199, 231 (1991) J. B. M. Warntjes, C. Wesdorp, F. Robicheaux, L. D. Noordam: Phys. Rev. Lett. 83, 512 (1999) D. Kleppner, M. G. Littman, M. L. Zimmerman: Rydberg atoms in strong fields. In: Rydberg States

247

Rydberg Atom 15. Rydberg Atoms in Strong Static Fields

Confronting classical and quantum mechanics in systems whose classical motion is chaotic is one of the fundamental problems of physics, as evidenced by the enormous outpouring of research during the last three decades [15.1, 2]. Highly excited Rydberg atoms in external fields [15.3] play a prominent role in this quest because they are the best known examples of quantum systems whose classical counterpart is chaotic. For a wide variety of field configurations and field strengths, their spectra can be measured to high precision. At the same time, since their Hamiltonians are known analytically, they are equally amenable to accurate theoretical investigations using either classical or quantum mechanics. This chapter is restricted to a description of Rydberg atoms in strong static fields. Related

Scaled-Energy Spectroscopy.................. 248

15.2

Closed-Orbit Theory ............................. 248

15.3

Classical and Quantum Chaos ................ 15.3.1 Magnetic Field .......................... 15.3.2 Parallel Electric and Magnetic Fields .................. 15.3.3 Crossed Electric and Magnetic Fields ..................

249 249 250 250

15.4 Nuclear-Mass Effects ............................ 251 References .................................................. 251

information on atoms in strong fields can be found in Chapt. 13 of this Handbook, on Rydberg atoms in Chapt. 14, and on the interaction of atoms with strong laser fields in Chapt. 74.

to that of the nuclear Coulomb field when the field strengths are in the order of the atomic units of electric field strength, F0 = e/(4πε0 a02 ) = 5.142 206 42  ×
(44) 1011 V/m, or magnetic field strength, B0 = ~/ ea02 = 2.350 517 42 (20) × 105 T, which is far beyond experimental reach. However, the relative importance of the external fields scales with the principal quantum number n as n 4 F and n 3 B, so that for highly excited atoms, laboratory fields can easily be “strong”. Atomic units will be used throughout this chapter. In a non-hydrogenic atom, the influence of the inner-shell electrons can be summarized by means of a short-range effective core potential or a set of quantum defects [15.22]. For laboratory field strengths, the core is too small to be appreciably influenced by the external fields. For this reason, the field-free quantum defects can be used to model core effects even in the presence of external fields [15.23].

Part B 15

Different configurations of external fields have been studied: (i) an electric field, which in hydrogen leads to integrable classical dynamics [15.4, 5] (ii) a magnetic field, which produces a transition from regular to chaotic classical dynamics and which sparked the interest in Rydberg atoms as prototype examples for the study of the quantum-classical correspondence [15.6– 15] and references therein (iii) parallel electric and magnetic fields [15.16, 17] (iv) crossed electric and magnetic fields which break all continuous symmetries of the unperturbed atom and thus allow one to study the transition from regularity to chaos in three coupled degrees of freedom [15.18–21] and references therein. The hydrogen atom is the prototype example for states with a single highly excited electron under the influence of strong external fields. For an electron in the hydrogen ground state, the influence of external electric or magnetic fields becomes comparable

15.1

248

Part B

Atoms

15.1 Scaled-Energy Spectroscopy The Hamiltonian for a hydrogen atom in a zˆ -directed magnetic field and an electric field of arbitrary orientation is H=

p2 1 1 1 − + BL z + B 2 ρ2 + r · F = E , 2 r 2 8 (15.1)

where ρ2 = x 2 + y2 and L z is the angular momentum component along the magnetic field axis. The dynamics depends on three parameters: the field strengths F and B and the energy E. We can reduce the number of independent parameters to two if we exploit a scaling property of the Hamiltonian: In terms of the scaled quantities r˜ = w−2 r , E˜ = w2 E ,

p˜ = w p F˜ = w4 F

(15.2)

with the scaling parameter w = B −1/3 ,

(15.3)

the scaled Hamiltonian reads 1 p˜ 2 1 1 − + L˜ z + ρ˜ 2 + r˜ · F˜ = E˜ . (15.4) H˜ = r˜ 2 2 8 The scaled dynamics thus depends only on two parameters, the scaled energy E˜ and the scaled electric field ˜ Instead of the above scaling with the magstrength F. netic field strength, which is the most common one, equivalent scaling prescriptions with the electric field strength or the energy can be used [15.24].

The way of recording an atomic spectrum that is most suitable for the investigation of quantum-classical correspondence is scaled-energy spectroscopy. A scaled spectrum consists of a list of eigenvalues wn of the scaling parameter (15.3) characterizing the quantum states for a given scaled energy E˜ and scaled electric field ˜ It offers the advantage that the underlying strength F. classical dynamics does not change across the spectrum, which makes the spectrum more easily accessible to a semiclassical interpretation (Sect. 15.2). For this reason, scaled-energy spectroscopy has been adopted in numerous experimental [15.4, 5, 8, 18] and theoretical investigations. To obtain a theoretical description of a scaled spectrum, the Schrödinger equation must be rewritten in terms of the scaling parameter w. In the case of a single external field, either electric or magnetic, this procedure leads to a generalized eigenvalue equation for the scaling parameter w [15.25]. In the presence of both electric and magnetic fields, the scaled spectrum is described by a quadratic eigenvalue equation that has become tractable only recently [15.26]. In a non-hydrogenic atom, the extent of the core imposes an absolute length scale and thus breaks the scaling symmetry. However, if the extent of the Rydberg electron’s orbital is large, the size of the core can be neglected and the scaling behavior is restored. This renders scaled-energy spectroscopy a useful concept also for non-hydrogenic atoms [15.4, 5, 18].

15.2 Closed-Orbit Theory

Part B 15.2

Among the most remarkable effects strong external fields produce in Rydberg atoms are the Quasi-Landau oscillations: Close to the ionization limit, the photoabsorption spectrum of Ba I in a magnetic field shows regular oscillations [15.6]. This phenomenon was given a convincing interpretation by Starace [15.27] and embedded by Du and Delos [15.28, 29] and Bogomolny [15.30] into the general framework of closed-orbit theory, which has since become the central interpretative tool for a description of Rydberg spectra in external fields [15.4, 5, 8, 18]). Recently, it has also been used for the computation of Rydberg spectra [15.31, 32]. Closed-orbit theory represents an atomic photoabsorption spectrum as a superposition of regular oscillations, each of which is related to a closed orbit

of the underlying classical dynamics, i. e., to an orbit that starts and ends at the position of the nucleus. The period of an oscillation is given by the return time of the associated closed orbit (divided by ~), the amplitude is determined on the one hand by the initial state and the polarization of the exciting photon, and on the other hand by the stability of the closed orbit. Once the initial state is specified, both can therefore be calculated within classical mechanics. A spectrum that contains contributions from many closed orbits can look enormously complicated. Its Fourier transform, on the other hand, consists of a series of isolated peaks that can be identified with the contributions of individual closed orbits. The Fourier transform thus provides the means to identify the crucial dynam-

Rydberg Atoms in Strong Static Fields

ics underlying a complicated spectrum. If a spectrum is recorded at constant external field strength, however, this analysis is inhibited by the fact that the oscillations are not strictly harmonic because both the return time of a closed orbit and the recurrence amplitude associated with it vary across the spectrum. This is the principal reason why scaled-energy spectroscopy (see Sect. 15.1) is customarily used. In a scaled spectrum, the period of an oscillation is given by the scaled action of the corresponding orbit and is fixed across the spectrum. Although initially devised for atoms in magnetic fields [15.28–30], closed-orbit theory is equally applicable to atoms in electric [15.33] as well as parallel [15.34] or crossed [15.34, 35] electric and magnetic fields. In the case of non-hydrogenic atoms, the influence of the ionic core can be modelled either by means of an effective classical potential [15.35, 36] or in terms of quantum defects [15.37]. Recently, closed-orbit theory

15.3 Classical and Quantum Chaos

249

has even been applied to the spectra of simple molecules in external fields [15.38]. Since a non-hydrogenic core is much smaller than the extent of a closed orbit (which is comparable to the size of the atomic Rydberg state), it does not appreciably modify the shape of the orbit. The peaks observed in a hydrogen spectrum are therefore also observed in the corresponding spectrum of a non-hydrogenic atom, although their strengths may be altered considerably (core shadowing) [15.37]. The principal effect of a core is to scatter the electron returning along one closed orbit into the initial direction of another, so that concatenations of hydrogenic closed orbits appear in the spectrum [15.37]. For this reason, the closed orbits of the hydrogen atom in external fields are the crucial ingredient for the interpretation of any Rydberg spectrum. They have been systematically studied for hydrogen in magnetic [15.39, 40] as well as electric [15.41] and crossed [15.42, 43] fields.

15.3 Classical and Quantum Chaos

P(S) =

π −πS/4 Se 2

(15.5)

that restricts small spacings. On the other hand, integrable systems possess a complete set of quantum numbers, so that levels are allowed to cross. This gives rise to a Poissonian NNS distribution P(S) = e−S

(15.6)

that favors small spacings. For mixed regular-chaotic systems, a transition from a Poisson to a Wigner NNS distribution is found [15.14].

15.3.1 Magnetic Field An atom exposed to a magnetic field possesses rotational symmetry around the magnetic field axis, which leads in classical mechanics to the conservation of the angular-momentum component L z along the field axis and in quantum mechanics to a good magnetic quantum number m = L z . The dynamics of the rotation coordinate can thus be separated, leaving two coupled degrees of freedom. The quantum numbers l and n that characterize the pure hydrogen states both break down in a magnetic field. However, the field affects them differently: Whereas l breaks down extensively even for small fields (l-mixing region), the breakdown of n is only achieved through considerably stronger fields or higher energies (n-mixing region). Since chaotic dynamics can only exist in at least two degrees of freedom, a single n-manifold of electronic energy levels does not have enough degrees of freedom to support chaos. Chaos can develop only when different n-manifolds mix (intermanifold chaos). The regular dynamics that prevails as long as n is approximately conserved is reflected in the existence of a second adiabatic con-

Part B 15.3

In the absence of external electric and magnetic fields, the classical dynamics described by the atomic Hamiltonian (15.1) is integrable and completely degenerate [15.44]. When external fields are present, a transition to classical chaos can be observed whose details depend on the precise field configuration (see below). It is characterized by the break-up of invariant tori and the appearance of irregular regions in the classical phase space. Chaos, as understood in classical mechanics, does not exist in closed quantum systems [15.45]. Nevertheless, in the dynamics of a quantum system clear indications of regularity or chaos in the underlying classical system can be found [15.2]. Most prominent among them is the statistical distribution of nearest-neighbor energy level spacings (NNS). In a classically chaotic system, energy levels show avoided crossings. Level repulsion is statistically reflected by NNS following a Wigner distribution

250

Part B

Atoms

stant of motion (apart from the energy), which is given by [15.46, 47] Λ = 4A2 − 5A2z , in terms of the Runge–Lenz vector   r 1 1 , ( p × L − L × p) − A= √ r −2E 2

(15.7)

(15.8)

and is conserved to second order in the magnetic field strength. As the magnetic field strength increases, corresponding to an increase in the scaled energy from −∞ toward zero, the classical dynamics changes from regular to almost entirely chaotic [15.13, 14]. For positive scaled energies above E˜ c = 0.328 782 . . . , completely hyperbolic dynamics is reached [15.48]. In step with the onset of classical chaos, the quantum NNS distribution changes from a Poisson to a Wigner distribution [15.13, 14].

15.3.2 Parallel Electric and Magnetic Fields In parallel fields, as in a pure magnetic field, an atom retains rotational symmetry around the field axis. It therefore shows a similar transition from regular dynamics to intermanifold chaos at scaled energies E˜ ≈ 0. At small field strengths, a second-order adiabatic invariant akin to (15.7) is given by [15.49, 50] Λβ = 4A2 − 5(A z − β)2 + 5β 2 ,

(15.9)

where the parameter β=

12 F 5 n2 B2

(15.10)

measures the relative strengths of the electric and magnetic fields.

Part B 15.3

15.3.3 Crossed Electric and Magnetic Fields In nonaligned electric and magnetic fields, the rotational symmetry of the field-free atom is broken completely, so that all three degrees of freedom are coupled. The angular momentum quantum numbers l and m break down extensively even at small field strengths; the principal quantum number n follows only gradually. Even when n is approximately conserved, however, in the crossedfields atom two coupled degrees of freedom remain. They allow the occurrence of chaotic dynamics within a single n-manifold (intramanifold chaos) [15.51, 52].

The intramanifold dynamics can conveniently be described in terms of the vectors [15.53] 1 I1 = (L + A) , 2 1 I2 = (L − A) , (15.11) 2 that obey independent angular momentum Poisson bracket (or, in quantum mechanics, commutator) relations. For fixed n, I1 and I2 are restricted to the spheres n2 . (15.12) 4 They span a four-dimensional space that is a convenient representation of the intramanifold phase space. Within a given n manifold, the position vector r can be replaced with − 32 n A [15.54]. Using this replacement, we can rewrite the contributions to the Hamiltonian (15.1) that are linear in the field strengths as I12 = I22 =

Hlin = ω1 · I1 + ω2 · I2

(15.13)

with the constant vectors 1 ω1 = (B − 3n F) , 2 1 ω2 = (B + 3n F) . (15.14) 2 The first-order Hamiltonian Hlin describes a precession of the vectors I1 and I2 around ω1 and ω2 , respectively, and preserves the integrability of the dynamics. Intramanifold chaos arises only if the quadratic contribution to the Hamiltonian (15.1) is taken into account. It can be detected either in classical mechanics [15.51, 52] or in quantum mechanics via its imprint on the intramanifold NNS distribution [15.52]. The properties of the crossed-fields hydrogen atom above the ionization threshold provide an example of a chaotic scattering system. Classically, chaotic ionization manifests itself in a fractal dependence of the electron escape time on the initial conditions [15.55]. Experimentally, a distinction has been made between “prompt” electrons that ionize fast, and “delayed” electrons that ionize only after more than 100 ns [15.20]. The latter can be interpreted as electrons that undergo chaotic scattering and circle the nucleus many times before they escape. A detailed classical model of chaotic scattering was presented in [15.56]. In quantum mechanics, chaotic scattering can be identified through the occurrence of Ericson fluctuations in the above-threshold spectrum [15.55].

Rydberg Atoms in Strong Static Fields

References

251

15.4 Nuclear-Mass Effects So far, only the relative motion of the electron with respect to the ionic core has been described. This is appropriate if the nucleus can be assumed to be infinitely heavy and thus not to take part in the motion. To include the effects of a finite nuclear mass, the description must start from the coupled two-body Hamiltonian and then work toward a separation of the internal dynamics from the center-of-mass (CM) motion. It turns out that in the presence of a magnetic field, unlike the field-free two-body problem, a complete separation of the relative and CM motions is impossible. Instead, only a pseudo-separation can be achieved, where the relative and CM motions remain coupled through a new constant of motion called the pseudomomentum K [15.57]. This coupling introduces a number of novel effects into the dynamics (see [15.58] for a detailed discussion).

The influence of the CM motion on the internal dynamics is twofold: on the one hand, the motion of the atom in the magnetic field causes an induced electric field (motional Stark effect). On the other hand, the kinetic energy of the CM motion gives rise to an additional confining potential for the internal motion that could, in principle, locate the electron at a large distance from the nucleus, and produce atomic states with a huge dipole moment. Conversely, the motion of the CM is driven by the internal motion, most strongly so in the case of vanishing pseudomomentum. It thus reflects the transition from regular to chaotic internal dynamics: A regular internal motion leads to a regular CM motion, whereas chaotic internal dynamics, for K = 0, give rise to a classical diffusion of the CM.

References 15.1 15.2 15.3 15.4 15.5

15.6 15.7 15.8

15.10 15.11

15.12

15.13

15.14 15.15

15.16 15.17 15.18 15.19 15.20

15.21 15.22 15.23 15.24

15.25 15.26 15.27 15.28

H. Hasegawa, M. Robnik, G. Wunner: Prog. Theor. Phys. Suppl. 98, 198 (1989) D. Delande: Chaos in atomic and molecular physics. In: Chaos and Quantum Physics, Session LII of Les Houches, ed. by M. J. Giannoni, A. Voros, J. ZinnJustin (North-Holland, Amsterdam 1991) pp. 665– 726 M. Courtney, H. Jiao, N. Spellmeyer, D. Kleppner, D. Gao, J. B. Delos: Phys. Rev. Lett. 74, 1538 (1995) I. Seipp, K. T. Taylor, W. Schweizer: J. Phys. B 29, 1 (1996) G. Raithel, M. Fauth, H. Walther: Phys. Rev. A 44, 1898 (1991) J.-P. Connerade, M.-S. Zhan, J. Rao, K. T. Taylor: J. Phys. B 32, 2351 (1999) S. Freund, R. Ubert, E. Flöthmann, K. Welge, D. M Wang, J. B Delos: Phys. Rev. A 65, 053408 (2002) T. Uzer: Phys. Scr. 90, 176 (2001) H. Friedrich: Theoretical Atomic Physics (Springer, Berlin, Heidelberg 1998) J. Rao, K. T. Taylor: J. Phys. B 30, 3627 (1997) H. Friedrich: Scaling properties for atoms in external fields. In: Atoms and Molecules in Strong External Fields, ed. by P. Schmelcher, W. Schweizer (Plenum Press, New York 1998) pp. 153–167 T. S. Monteiro, G. Wunner: Phys. Rev. Lett. 65, 1100 (1990) J. Rao, K. T. Taylor: J. Phys. B 35, 2627 (2002) A. F. Starace: J. Phys. B 6, 585 (1973) M. L. Du, J. B. Delos: Phys. Rev. A 38, 1896 (1988)

Part B 15

15.9

M. C. Gutzwiller: Chaos in Classical and Quantum Mechanics (Springer, Berlin, Heidelberg 1990) F. Haake: Quantum Signatures of Chaos, 2nd edn. (Springer, Berlin, Heidelberg 2000) T. F. Gallagher: Rydberg Atoms (Cambridge Univ. Press, Cambridge 1994) A. Kips, W. Vassen, W. Hogervorst: Phys. Rev. A 59, 2948 (1999) R. V. Jensen, H. Flores-Rueda, J. D. Wright, M. L. Keeler, T. J. Morgan: Phys. Rev. A 62, 053410 (2000) W. R. S. Garton, F. S. Tomkins: Astroph. J. 185, 839 (1969) K. T. Lu, F. S. Tomkins, W. R. S. Garton: Proc. Roy. Soc. London Ser. A 362, 421 (1978) J. Main, G. Wiebusch, K. Welge, J. Shaw, J. B. Delos: Phys. Rev. A 49, 847 (1994) R. J. Elliott, G. Droungas, J.-P. Connerade: J. Phys. B 28, L537 (1995) R. J. Elliott, G. Droungas, J.-P. Connerade, X.-H. He, K. T. Taylor: J. Phys. B 29, 3341 (1996) J. C. Gay: The structure of Rydberg atoms in strong static fields. In: NATO Advanced Study Institute Series B: Physics, Vol. 143, ed. by J. P. Briand (Plenum Press, New York 1986) pp. 107–152 J. C. Gay: Hydrogenic systems in electric and magnetic fields. In: The spectrum of atomic hydrogen: Advances, ed. by G. W. Series (World Scientific, Singapore 1988) pp. 367–446 H. Friedrich, D. Wintgen: Phys. Rep. 183, 37 (1989)

252

Part B

Atoms

15.29 15.30 15.31 15.32 15.33 15.34 15.35 15.36 15.37 15.38 15.39 15.40 15.41 15.42 15.43

M. L. Du, J. B. Delos: Phys. Rev. A 38, 1913 (1988) E. B. Bogomolny: Sov. Phys. JETP 69, 275 (1989) J. Main, G. Wunner: Phys. Rev. A 59, R2548 (1999) T. Bartsch, J. Main, G. Wunner: J. Phys. B 36, 1231 (2003) J. Gao, J. B. Delos, M. Baruch: Phys. Rev. A 46, 1449 (1992) J.-M. Mao, K. A. Rapelje, S. J. Blodgett-Ford, J. B. Delos: Phys. Rev. A 48, 2117 (1996) K. Weibert, J. Main, G. Wunner: Ann. Phys. (NY) 268, 172 (1998) B. Hüpper, J. Main, G. Wunner: Phys. Rev. A 53, 744 (1996) P. A. Dando, T. S. Monteiro, D. Delande, K. T. Taylor: Phys. Rev. A 54, 127 (1996) A. Matzkin, T. S. Monteiro: Phys. Rev. Lett. 87, 143002 (2001) M. A. Al-Laithy, P. F. O’Mahony, K. T. Taylor: J. Phys. B 19, L773 (1986) J. Main, A. Holle, G. Wiebusch, K. H. Welge: Z. Phys. D 6, 295 (1987) J. Gao, J. B. Delos: Phys. Rev. A 49, 869 (1994) D. M. Wang, J. B. Delos: Phys. Rev. A 63, 043409 (2001) T. Bartsch, J. Main, G. Wunner: Phys. Rev. A 67, 063410 (2003)

15.44 15.45 15.46 15.47 15.48 15.49

15.50 15.51 15.52 15.53 15.54 15.55 15.56 15.57 15.58

H. Goldstein: Classical Mechanics (Addison-Wesley, Reading 1965) M. V. Berry: Phys. Scr. 40, 335 (1989) E. A. Solov’ev: Sov. Phys. JETP 55, 1017 (1982) D. R. Herrick: Phys. Rev. A 26, 232 (1982) K. T. Hansen: Phys. Rev. E 51, 1838 (1995) P. Cacciani, E. Luc-Koenig, J. Pinard, C. Thomas, S. Liberman: J. Phys. B 21, 3499 (1988) and references therein P. A. Braun: Rev. Mod. Phys. 65, 115 (1993) and references therein J. von Milczewski, T. Uzer: Phys. Rev. E 55, 6540 (1997) J. Main, M. Schwacke, G. Wunner: Phys. Rev. A 57, 1149 (1998) M. J. Englefield: Group Theory and the Coulomb Problem (Wiley-Interscience, New York 1972) W. Pauli: Z. Phys. 36, 339 (1926) J. Main, G. Wunner: J. Phys. B 27, 2835 (1994) C. Jaffé, D. Farrelly, T. Uzer: Phys. Rev. Lett. 84, 610 (2000) J. E. Avron, I. W. Herbst, B. Simon: Ann. Phys. (NY) 114, 431 (1978) P. Schmelcher, L. S. Cederbaum: Two interacting charged particles in strong static fields. A variety of two-body phenomenon. In: Structure and Bonding, Vol. 86, ed. by L. S. Cederbaum, K. C. Kulander, N. H. March (Springer, Berlin 1997) pp. 27–62

Part B 15

253

16. Hyperfine Structure

Hyperfine Stru Hyperfine structure in atomic and molecular spectra is a result of the interaction between electronic degrees of freedom and nuclear properties other than the dominant one, the nuclear Coulomb field. It includes splittings of energy levels (and thus of spectral lines) from magnetic dipole and electric quadrupole interactions (and higher multipoles, on occasion). Isotope shifts are experimentally entangled with hyperfine structure, and the so-called field effect in the isotope shift can be naturally included as part of hyperfine structure. Studies of hyperfine structure can be used to probe nuclear properties, but they are an equally important probe of the structure of atomic systems, providing especially good tests of atomic wave functions near the nucleus. There are also isotope shifts owing to the mass differences between different nuclear species, and the study of these shifts provides useful atomic information, especially about correlations between electrons. Hyperfine effects are usually small and often, but not always, it is sufficient to consider only

Splittings and Intensities ..................... 16.1.1 Angular Momentum Coupling ..... 16.1.2 Energy Splittings ....................... 16.1.3 Intensities ................................

254 254 254 255

16.2 Isotope Shifts ...................................... 16.2.1 Normal Mass Shift ..................... 16.2.2 Specific Mass Shift ..................... 16.2.3 Field Shift ................................ 16.2.4 Separation of Mass Shift and Field Shift ..........................

256 256 256 256

16.3 Hyperfine Structure.............................. 16.3.1 Electric Multipoles ..................... 16.3.2 Magnetic Multipoles .................. 16.3.3 Hyperfine Anomalies .................

258 258 258 259

257

References .................................................. 259 diagonal matrix elements for the atomic or molecular system and for the nuclear system. In some cases, however, matrix elements offdiagonal in the atomic space, even though small, can be of importance; one possible result is to cause admixtures sufficient to make normally forbidden transitions possible.

The study of hyperfine structure in free atoms, ions, and molecules is part of the more extensive research area of hyperfine interactions, which includes the study of atoms and molecules in matter, both at rest, for example as part of the structure of a solid, and moving, such as ions moving through condensed or gaseous matter. This more general subject also includes the ways in which atomic electrons shield the nucleus, or antishield it, from external or collective fields. Thus nuclear magnetic resonance, nuclear quadrupole resonance, electron-nuclear double resonance, recoilless nuclear absorption and emission, nuclear orientation, production of polarized beams, and many other widely used techniques, are intimately connected with hyperfine effects. Though hyperfine effects are ordinarily small in electronic systems, they can become much larger in “exotic” atoms: those with a heavier lepton or hadron as the

Part B 16

In the diagonal case, one can picture each electron undergoing elastic scattering from the nucleus and returning to its initial bound state. As pointed out by Casimir [16.1, 2], however, the internal conversion of nuclear gamma-ray transitions involves the inelastic down-scattering from an excited nuclear state to a lower one as an electron goes from an initial bound state to the continuum. By further conversion of bound to continuum states, one sees the connection with electron scattering from the nucleus – elastic, inelastic, and break-up. Hyperfine structure of outer-shell electronic states is at the low momentum-transfer end of this chain of related processes. Some of the standard textbooks which discuss hyperfine structure are [16.3–10] and a few newer texts [16.11–14]. Especially relevant are [16.15–19] and the conference proceedings [16.20–22].

16.1

254

Part B

Atoms

“light” particle. Hyperfine effects are typically related to light particle density at the nucleus, or to expectation values of r −3 , and thus scale as the cube of the light particle mass. The study of muonic atoms has contributed importantly to knowledge of the nuclear charge distribution [16.23–26]. There has been considerable interest in pionic atoms, where the strong interaction also con-

tributes to hyperfine structure (e.g., [16.27, 28]), and also in kaonic, antiprotonic, and other hadronic “atoms” [16.29, 30]. See especially [16.31] for recent work on antiprotonic helium. Some other examples of interaction between atomic and nuclear degrees of freedom are discussed in Chapt. 90.

16.1 Splittings and Intensities 16.1.1 Angular Momentum Coupling

16.1.2 Energy Splittings

When the nuclear system is in an isotropic environment, each nuclear state β has a definite value of nuclear angular momentum Iβ ~, where the possible values of Iβ are related to the number of nucleons (protons plus neutrons) in the same way as those for Jα are related to the number of electrons in electronic state |α
. The nuclear operators, eigenstates, and eigenvalues are related to each other in the same way as for atomic angular momentum by

Electromagnetic interactions between atomic electrons and the nucleus can be expanded in a multipole series
HeN = T k (N) · T k (e) ,

I 2 |β
= Iβ (Iβ + 1)|β
, Iz |β
= Mβ |β
,

(16.1)

in units with ~ = 1. Shift operators move the system from one M-value to another, as for the atomic system (see Chapt. 2), and the operator I is the generator of rotations. When the combined atomic-nuclear system is considered, in an isotropic environment, it is the total angular momentum of the combined system defined by F = J+I ,

(16.2)

that has definite values. The state of the combined system can be labeled by γ , so that F 2 |γ
= Fγ (Fγ + 1)|γ
, Fz |γ
= Mγ |γ
.

(16.3)

Part B 16.1

The shift operators are defined as before, and it is now F that is the generator of rotations of the (combined) system, or of the coordinate frame to which the system is referred. By the rules of combining angular momenta, the possible values of the quantum number F are separated by integer steps and run from an upper limit of Jα + Iβ to a lower limit of |Jα − Iβ |. The number of possible eigenvalues F is the smaller of 2Jα + 1 and 2Iβ + 1. Experimental values of the nuclear quantum number I may be found in a number of compilations [16.32–34].

k


k (−1)i Tqk (N)T−q (e) ≡

(16.4)

k,q

where T k (N) is an irreducible tensor operator of rank k operating in the nuclear space, and similarly T k (e) operates in the space of the electrons. Since one is taking diagonal matrix elements (in the nuclear space, at least) in states that are to a very good approximation eigenstates of the parity operator, only even electric multipoles (E0, E2, etc.) and odd magnetic multipoles (M1, M3, etc.) contribute to the series. The effects of the parity nonconserving weak interaction are considered in Chapt. 29. The term with k = 0 contributes directly to the structure (and fine structure) of atomic systems, and its dominant contributions come from the external r −1 electrostatic field of the nucleus. The hyperfine Hamiltonian is defined by subtracting that term to obtain
   T 0 (N)T 0 (i) − − Ze2 /ri Hhfs = i

+




T k (N) · T k (e) ,

(16.5)

k=1

where Z is the nuclear charge number. The difference between the Ze2 /r term(s) and the full monopole term is called the field effect or finite nuclear size effect in the isotope shift, and the remaining terms contribute dipole (k = 1), quadrupole (k = 2), and higher multipoles in hyperfine structure. Since the hyperfine Hamiltonian can be expressed as a multipole expansion, its contributions to the pattern of energy levels for the various F values in a given

Hyperfine Structure

Jα , Iβ multiplet in first-order perturbation theory can be described relatively simply in terms of 3– j and 6– j symbols. The contribution of the term which is the scalar product of electron and nuclear operators of multipole k is ∆E k (JIF, JIF )   J J k J+I +F = (−) I I F 



−1 J k J I k I Ak , (16.6) × J 0 −J I 0 −I where for k ≥ 1,



 Ak = JJ T k (e) JJ · II T k (N) II .

(16.7)

The commonly used hfs coefficients A, B, etc., are related to the Ak by A = A1 /I J,

B = 4A2 ,

C = A3 ,

D = A4 . (16.8)

The isotope shift A0 is the matrix element of the reduced monopole operator. The pattern of the splitting depends on the total angular momentum F wholly through the 6– j symbol. Since for k = 0 the value of the 6– j symbol is independent of F, the monopole term shifts all levels of the hyperfine multiplet equally, independent of the value of F. The F-dependence of the dipole contribution can be found from the fact that the same 6– j symbol would appear for any scalar product of k = 1 operators, for example J · I. But in this product space, with J, I, and F all good quantum numbers, the diagonal matrix elements of J · I are just 1 J · I
= [F(F + 1) − J(J + 1) − I(I + 1)] , 2 (16.9)

A is the magnetic dipole hyperfine structure constant for the atomic level J and nuclear state I. M1 hfs shows the same pattern of splittings as spin-orbit fine structure, described sometimes as the Landé interval rule.

255

Electric quadrupole hfs is described by the quadrupole hyperfine structure constant B. If we define the quantity K = [F(F + 1) − J(J + 1) − I(I + 1)], then ∆E 2 (JIF, JIF ) 1  B [3K(K + 1)/2 − 2J(J + 1)I(I + 1)] . = 4 J(2J − 1)I(2I − 1) (16.12)

The constant B is related to the tensor operators by 1 B = [J(2J − 1)/(J + 1)(2J + 3)]−1/2 4    × J T 2 (e) J × [I(2I − 1)/(I + 1)(2I + 3)]−1/2    × I T 2 (N) I ,

(16.13)

For higher multipoles, see [16.36]. The multipole expansion is important because it is valid for relativistic as well as nonrelativistic situations, and for nuclear penetration effects (hyperfine anomalies discussed in Sect. 16.3.3) as well as for normal hyperfine structure. Its limitation comes from its nature as a first-order diagonal perturbation. Off-diagonal contributions, even when small, can perturb the pattern, but, more importantly, can lead to misleading values for the Ak coefficients, including the isotope shift.

16.1.3 Intensities When hyperfine structure is observed as a splitting in an optical transition between different atomic levels, there are relations between the intensities of the components. The general rule for reduced matrix elements of a tensor operator operating in the first part of a coupled space is ([16.35, p. 152])    JIF  Q λ (e) J  IF    F F λ λ+I +F  +J  1/2 = (−1) (2F + 1) J J I    × (2J + 1)1/2 J  Q λ (e) J  . (16.14) For a dipole transition (λ = 1) connecting atomic states J and J  , with fixed nuclear spin I, the line strength S FF  of the hyperfine component connecting F and F  is related to the line strength S JJ  by 2  F F 1  S FF  = (2F + 1)(2F + 1) S JJ  . J J I (16.15)

Part B 16.1

so that ∆E 1 (JIF, JIF ) 1 = A[F(F + 1) − J(J + 1) − I(I + 1)] , (16.10) 2 where, in terms of reduced matrix elements according to the convention of Brink and Satchler ([16.35, p. 152]), (the first version given in Sect. 2.8.4)    A = [J(J + 1)]−1/2 J T 1 (e) J [I(I + 1)]−1/2    × I T 1 (N) I . (16.11)

16.1 Splittings and Intensities

256

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16.2 Isotope Shifts Two distinct mechanisms contribute to isotope shifts in atomic energy levels and transition energies. First, there are shifts due to the different mass values of the isotopes; these mass shifts can again be separated into two kinds, the normal mass shift and the specific mass shift. Second, there are shifts due to different nuclear charge distributions in different isotopes. Shifts of this sort are called field shifts, and can be considered to be the monopole part of the hyperfine interaction. The usual convention is to describe an isotope shift in a transition as positive when the line frequency is greater for the heavier isotope.

16.2.1 Normal Mass Shift

remains a set of mass polarization terms 1
p j · pk , Hmp = M

(16.18)

j 1/τc the static wing approaches the quasistatic limit described in Sect. 19.1.7. If ∆V(R) changes monotonically with decreasing R, only one side of the line has a static contribution and the other, antistatic side falls off exponentially at |∆ω| > τc [19.11]. However, this situation is seldom observed, as more than one difference potential generally contributes and there is usually a static contribution on both sides of the line. Another factor that produces a small divergence from the Voigt profile is the velocity dependence of the

shift and width. When combined with the higher velocities of atoms emitting or absorbing in the Doppler wing, this produces an asymmetry in the Doppler wings [19.26, 27].

19.1.5 Examples: Line Core It is possible to deconvolve a Voigt line shape to separate the Doppler and Lorentzian components, and thereby deduce broadenings of considerably less than the Doppler width ([19.28] and references therein). However, the broadening is most easily observed at perturber densities where the collisional broadening exceeds the Doppler broadening. Such a pressure-dependent line shape is shown in Fig. 19.2, for a range of perturber density n P such that the broadening exceeds the Doppler width and hyperfine structure, yet the line core is described by the impact theory [19.29]. In Fig. 19.2 the normalized line intensity has been divided by n P ; as the line wings are proportional to n P they are constant in such a plot, while the line center broadens and shifts with increasing n P . For this case of fairly heavy atoms, (2πcτc )−1 = ∆kc corresponds to ≈ 0.5 cm−1 , and the line becomes asymmetric and non-Lorentzian beyond ≈ 1 cm−1 (Fig. 19.3a); the red wing intensity falls more slowly and the blue wing more rapidly than (∆ω)−2 . This behavior is typical for most heavy perturbers, and is I (λ) / N兰 I(λ) d λ(cm3/Å) 10 –18 Kr

10 –19

10 –20

–1.5

–1.0

–0.5

0

0.5

1.0 1.5 ∆ k(cm–1)

Fig. 19.2 Normalized line shape of the Rb 5P 3/2 − 5S 1/2

transition broadened by Kr, for Kr densities of 4.5, 9, 18, and 27 × 1018 cm−3 (top to bottom). Hyperfine structure and instrumental resolution cause ≈ 0.3 cm−1 of the broadening shown

Line Shapes and Radiation Transfer

a) 10 –20

I(λ)/N兰I(λ) dλ(m3/Å)

10 –21 10 –22 10 –23 10 –24 10 –25

b)

R(ao)

B–X

15 10 A–X 7

B–X

5 –1000

–100

–10

–1 1

10

for transitions to higher states, as the interactions have a longer range. In addition, nearby intensity peaks or satellites often occur, and strongly affect the line as pressure increases. An example calculation, based on an interpretation of measured spectra [19.16], is shown in Fig. 19.4. This shows how a line with a satellite feature progressively broadens and finally blends with the satellite as n P increases. With the advent of saturated-absorption (Doppler free) spectroscopy, collisional line broadening can be measured at much lower densities, where 2γc  ∆ωD . In principle, this can allow measurement of line broadenings and shifts, although with a complication that affects the line shape; the same collisions that produce optical phase shifts also change the atomic velocity. These velocity changes have a minor effect outside the Doppler envelope where high pressure measurements are normally made, but they are quite important in saturated absorption line shapes. This affects primarily the low intensity wings of the line, so it does not prevent measuring the broadening and shift of the nearly Lorentzian core.

100 1000 k – ko

Fig. 19.3a,b Normalized intensity in the wings of the Rb 5P 3/2 − 5S 1/2 transition broadened by Kr, in frequency units of k = 1/λ. The measured spectrum in (a) is from [19.29–31]. The solid line is at 310 K and the dashed line at 540 K. The difference potentials corresponding to the A, B and X states of Rb − Kr, taken from [19.32], are shown in (b)

Intensity

1 × 1018

2.5 × 1018

attributed to a long range attractive Vu which dominates Vu − Vg . For the lowest n P shown in Fig. 19.2, a convolution with the Doppler, hyperfine and instrumental broadenings showed that the line is essentially a symmetric Lorentzian for |∆k| < ∆kc [19.29]. However, at the highest density, the half-height point is beginning to fall outside of ∆kc ; the impact approximation is marginally valid for describing the half width of the line at this density. Most early experiments were done at more than 10 times this density [19.15]; most of the linecore was at |∆k| > ∆kc and describable by the static theory (Sect. 19.1.7) rather than the impact approximation. The impact approximation was also not valid under these conditions because collisions overlap in time. These very broad lines are well represented by the multiple-perturber, static theories that assume scalarly additive perturber interactions [19.5, 8]. This transition between an impact and quasistatic line core, and to multiple perturber interactions, occurs at lower pressures

5 × 1018

1 × 1019 1.5 × 1019 2 × 1019 –46

–38

–30

–22

–14

–6 +2 Frequency (cm–1)

Fig. 19.4 Calculated line shapes of the Cs(9P 1/2 − 6S 1/2 ) line broadened by Xe at the densities indicated (from [19.16]). The assumed interaction is based on measured line shapes, but data corresponding to the calculated conditions are not available

283

Part B 19.1

30 25 20

19.1 Collisional Line Shapes

284

Part B

Atoms

Details can be found in [19.33] and references therein. Two-photon absorption yields Doppler free lines that are not affected by velocity changing collisions. This provides the most exacting test of line shapes. These narrow lines are precisely Lorentzian, with a broadening that reflects the upper state interaction since this is usually much stronger than that of the ground state. The technique has been used to measure the broadening of two photon transitions to many excited states [19.34,35].

19.1.6 ∆ and γc Characteristics Part B 19.1

Since 1970, neutral broadening has generally been measured in the |∆ω| < 1/τc region where the impact approximation and (19.7) is valid. Measurements through 1982 are tabulated in [19.16], and through 1992 in [19.36]. More recent measurements are tabulated in the NIST Reference Data bibliography, which is accessible (free) at the web site http://physics.nist.gov/PhysRefData. These involve primarily metal vapor resonance lines broadened by noble gases. For collisions with the heavier (more polarizable) gases, the sizes of these measured broadening rate coefficients generally fall within a factor of 10 range, and approximately fit the prediction of (19.7) with ∆Vu (R) − ∆Vg (R) = C6 R−6 [19.37] and C6 given by a simple effective quantum number formula. This occurs because the potentials are fairly close to van der Waals for R ≥ RW and the broadening is insensitive to details of the potentials at R < RW since cos ∆θ in (19.7) averages to ≈ 0 for the closer (strong) collisions. It also occurs because the full quantum solution for broadening by a van der Waals interaction, with Zeeman degeneracies, yields nearly the same result as the above single-level theory with an average C6 [19.38]. For the heavy, more polarizable perturbers, the excited state interactions are attractive and red shifts occur, but the measured shifts have a very poor correlation with the van der Waals prediction. As b decreases and ∆θ(b, v) increases, sin ∆θ oscillates and major cancellations occur in the average of sin ∆θ in (19.7). The shift is therefore only a fraction of the broadening and is very sensitive to the interaction throughout the region R ≈ RW . This often differs considerably from the van der Waals form, at the typical ≈ 5 cm−1 interaction energy at RW . The shape of the red wing just beyond ∆ωc also frequently fails to fit that expected for a van der Waals interaction [19.29]. Thus, the often good agreement of γc with van der Waals numbers is not a reliable indicator of the actual V(R) in the relevant R region, even for heavy noble gas perturbers.

For He and sometimes Ne perturbers, a repulsive interaction due to charge overlap normally dominates at R ≈ RW , causing a blue shift as well as a larger broadening than the van der Waals prediction.

19.1.7 Quasistatic Approximation The impact approximation is valid for |∆ω|  1/τc , where the 1/τc is typically 1–10 cm−1 . For larger |∆ω| the line shape becomes asymmetric, with higher intensity on the wing corresponding to the long-range Vu (R) − Vg (R). At large detunings where ∆ω
1/τc a major simplification occurs. The COA describes the interacting  atom pair as an oscillator of frequency ωc (R) = Vu (R) − Vg (R) /~ when at separation R. Since R is time dependent during the classical orbit, ωc is as well and the Fourier spectrum is broadened relative to the simple distribution of ωc (t) that occurs during the orbit. But if the motion is sufficiently slow, the intensity at ω reduces to the probability of finding the atom pair at the appropriate R(ω = ωc ). The spectrum then reduces, at low pressure, to the probability distribution of pair separations R, subject to (19.5) between R and ω. This is the binary quasistatic, static, or statistical spectrum, which accurately describes most line wings for |∆ω| > 1/τc . When the pressure is large enough to yield a significant probability of one perturber at R ≤ Rc , multiple-perturber interactions must also be considered as in [19.5]. This intuitive deduction of the statistical spectrum from the COA [19.4] can also be obtained more formally from (19.6) by expanding the exponent about the time during a collision when ωc (t) = ω. Alternatively, it follows directly from (19.2) using WKB wave functions to evaluate free–free molecular Franck– Condon factors [19.6, 7]. This result is identical to the classical Franck–Condon principle (CFCP), originally established in the context of bound–bound molecular radiation. The CFCP yields important insights for all molecular radiation. Again consider (19.2) with the substitution of the WKB wave functions φq , given below it. Examples of φu and φg are given in Fig. 19.1. For large detunings ω − ω0 , as shown in Fig. 19.1, the integrand on the right side oscillates rapidly everywhere except at the stationary phase point Rc , where ku = kg . As a consequence, the dominant contribution to the integral occurs at Rc and one can consider the transition to be localized at Rc . Since ku (Rc ) = kg (Rc ), Tu (Rc ) = Tg (Rc ) = Tc also holds, and as can be seen in Fig. 19.1 it then follows that ~ω = Vu (Rc ) − Vg (Rc ). Thus, radiation at

Line Shapes and Radiation Transfer

I(ω) = Nn P Γ 4πR(ω)2 × exp[−Vi (ω)/kB T ]/[ dω(R)/ dR] ,

(19.8)

where N is the radiator density and I(ω) the radiation per unit volume and frequency interval. Figure 19.3a gives an example of far wing emission line shapes versus photon energy in units of cm−1 , for the Kr broadened Rb D2 line for which ∆kc ≈ 0.7 cm−1 . These data are normalized by dividing by perturber density, so they are independent of perturber density for the density region of the experiment. The excited state produces two Vu (R), called the A and B states, while the ground state produces one Vg (R), called the X state. Each of these potentials has a single minimum at long range and is strongly repulsive at close range, but the well depths and positions are very different [19.32]. This causes the complex forms of Vu (R) − Vg (R) that are shown in Fig. 19.3b. There I have plotted ln(R) vertically and ln(∆V(R)/hc) horizontally, where ∆V refers to VA − VX and VB − VX . The right side of (19.8) can also be written as the exponential and constant factors times dR3 (ω)/ dω. Since ln[R3 (ω)] ∝ ln[R(ω)], the static spectrum at ∆k = ∆ω/2πc is proportional to the slopes of the curves in Fig. 19.3b, divided by |∆k| due to the ln(∆k) horizontal axis. One can see qualitatively that the overall spectrum follows such a relation to the lines in Fig. 19.3b; in fact in most spectral regions this relation is quantitatively accurate.

The temperature dependence in Fig. 19.3a corresponds to the exponential factors in (19.8) [19.30]. At large R, both ∆V(R) are attractive, and this causes a large intensity on the negative ∆k (red) wing. However, once |∆k| exceeds ≈ 20 cm−1 , where VB − VX reverses direction, the red wing intensity drops rapidly. This extremum in ∆V(R) causes a satellite at ≈ −20 cm−1 , although it is spread out by the finite collision speed and does not cause a distinct peak in the spectrum. Satellite features are discussed in more detail in the next paragraph. The antistatic blue wing drops rapidly for several decades, then suddenly flattens beyond ≈ 10 cm−1 due to the positive portion of VB − VX at small R. The remaining blue wing is the B–X band, and has a satellite at ≈ 350 cm−1 as VB − VX passes through another extremum. The theory predicts this at 800 cm−1 , but clearly represents all the basic aspects correctly. This satellite is also spread out by finite collision speed, but a definite intensity peak remains. The red wing beyond ≈ 50 cm−1 is the A–X band. The feature near −1000 cm−1 is due to the exponential factor in (19.8), not an extremum in ∆V(R); the feature essentially disappears if the normalized intensity is extrapolated to infinite temperature.

19.1.8 Satellites In regions of the wing where the intensity falls slowly with increasing frequency, motional broadening of the static spectrum is not noticeable and the static spectrum is a good approximation. However, if ∆V(R), or equivalently ωc (R), has an extremum at some RS , the denominator of (19.8) is zero at ω(RS ) = ωS . This produces a local maximum, or satellite, in the far wing intensity, as seen in Fig. 19.3a at 350 cm−1 . If one expands ω(R) in a Taylor series about R = RS this produces in (19.8) a square root divergence of finite area, with no intensity beyond ωS . The area under this feature is meaningful, but not its shape; the quasistatic assumption is clearly not valid for such sharp features. The more accurate satellite shape is obtained by returning to (19.2) and expanding Vu (R), Vg (R) and the WKB wave functions about RS , or using (19.6) with ωc (t) expanded about t(RS ). Sando and Wormhoudt used the former method to obtain a universal satellite shape [19.39]. Szudy and Baylis improved the expansion to yield a smooth transition to the quasistatic spectrum at smaller detunings [19.37]. This result is nearly the same as Sando et al. in the spectral region of the satellite, but it more accurately connects to the adjacent static line wing. Intensity undulations between the satellite and the line occur in this calculation; these arise from alter-

285

Part B 19.1

frequency ω “occurs” when the atoms are near Rc , where the electronic state energies differ by ~ω. Note that this holds for all initial kinetic energies and angular momenta, as long as the conditions for validity of the Born–Oppenheimer and WKB approximations hold for the initial and final nuclear motions. This is the CFCP, which is equivalent to the classical-oscillator model for radiation at large detunings from the atomic transition. Another insight evident from Fig. 19.1 is that the photon energy associated with the frequency difference ω − ωc is supplied by nuclear kinetic energy ~(ω − ω0 ) = Ti − Tf . This transformation of nuclear into electronic energy takes place as the nuclei move from large R to Rc on one V(R) and back to large R on the other. If an absorbing or emitting atom interacts as Vi (R) with a density n P of perturbers in a vapor of temperature T , the probability of a perturber at separation R → R + dR is n P 4πR2 exp[−Vi (R)/kB T ] dR if the interatomic motion is in equilibrium. Inverting (19.5) for R(ω) yields dR = dω/( dω/ dR), and this pair of relations yields the (single perturber) quasistatic (QS) spectrum

19.1 Collisional Line Shapes

286

Part B

Atoms

Part B 19.1

nating constructive and destructive interference between two contributions to the same frequency from R > RS and R < RS . This can not be seen in the low resolution of Fig. 19.3a, but such undulations are seen near the 350 cm−1 satellite [19.31]. At antistatic detunings beyond ωS , which are not quasistatically allowed, the calculated intensities decay exponentially. This is also observed experimentally [19.31] and is the same behavior predicted for the antistatic wing of a line [19.11]. At higher perturber densities and closer to the line core, corresponding to larger R interactions, the multiple perturber probability distribution must be included. If the interactions are additive, this leads to a secondary satellite at twice the detuning of the low pressure satellite, as seen in Fig. 19.4. The wings of a collisionally broadened atomic line are molecular radiation. In the context of molecular bound state spectroscopy, a satellite is a “head of band heads,” corresponding to a frequency region where bound–bound band heads congregate. This occurs, of course, at the classical satellite frequency and when Vu (R) − Vg (R) has an extremum [19.40]. An extremum in ∆V(R) is the most common cause of satellites, but similar looking features can occur for other reasons. Forbidden bands often appear in the wings of forbidden atomic transitions, due to an increase in the transition dipole moment µ(R) resulting from the collisional interaction. These are described, in the QS approximation, by (19.8) with Γ replaced by Γ (R). If Γ (R) increases rapidly with decreasing R, the intensity increases as ω moves into the far wing until the dR3 and exponential factors cause a net decrease at small R. This leads to forbidden bands far from the atomic frequency, such as those in [19.41]. In some cases, a collisioninduced feature also appears at the frequency of the forbidden transition. The shapes of such features, which also include radiative collisions, in which both atoms change state, are calculated and reviewed in [19.42]. A variety of related line shape phenomena has been investigated, including the relation between absorbed and emitted wavelengths (spectral redistribution), the dependence of fluorescence polarization on absorbed wavelength (polarization redistribution), and high power effects. Some references regarding these phenomena are [19.43–47].

19.1.9 Bound States and Other Quantum Effects The validity of the QS spectrum requires the validity of the WKB approximation in the initial and final state,

but it is not restricted to free–free molecular transitions. In fact, the equilibrium probability distribution in (19.8) must include bound states in an attractive Vi (R). The QS spectrum describes the average behavior of bound– free and bound–bound molecular bands, as well as the free–free radiation implied by the above method of derivation. The quantum character is expressed in the discrete bound–bound lines that make up this average, and in Condon oscillations, where the intensity oscillates about the average IQS (ω). The latter occur as oscillations in Franck–Condon factors in the bound–bound case, and as smooth oscillations in bound–free spectra and low resolution bound–bound spectra. An additional quantum feature occurs in regions of the spectrum dominated by classical turning points, usually at the far edge of a line wing where the intensity is dropping rapidly. There, quantum tunneling past the edge of the classically allowed region spreads the spectrum. Yet another is the energy hω0 /2 of the ground vibrational state, which effectively adds to kB T in (19.8) for attractive Vi . All of these quantum features become more pronounced as the reduced mass decreases; examples and details can be found in [19.40, 48–50].

19.1.10 Einstein A and B Coefficients The relationship between absorption coefficient B12 (ω), stimulated emission coefficient B21 (ω) and spontaneous emission coefficient A21 (ω) are given by the Einstein relations; A21 /B21 = 8πhλ−3 and B21 /B12 = g1 /g2 . These relations are most familiar for atomic lines, but if they are referred to the density of absorbers dNg / dω and emitters dNu / dω that emit or absorb at ω, then they also apply to the wings of lines, i. e.,

gu dNg 2πh λ gg dω 1 2 gu dNg = λ A21 (ω) , 4 gg dω 1 dNu , g(ω) = λ2 A21 (ω) 4 dω dNu . I(ω) = ~ωA21 (ω) dω

k(ω) = B12 (ω)

(19.9) (19.10) (19.11)

Here k(ω) is the absorption coefficient due to lower state atoms, g(ω) is the stimulated emission coefficient and I(ω) the spontaneous emission due to excited state population, and gu and gg are the statistical weights of the atomic states. For absorbing atoms of density Ng and perturber density n P , the QS approximation with

Line Shapes and Radiation Transfer



Vg dNg dR = Ng n P 4πR2 exp − , dω dω kB T

I(ω) dω = A21 Nu (19.12)

and equivalently for a radiating atom density Nu with perturber interaction Vu . Normally most of the radiation, and dNg / dω, is concentrated at the atomic line, so integrating over dω near the line leads to the relations ∞



gu gg

,

(19.13)

hω 2π

, etc.

(19.14)

0

Note that

g(ω)/Nu ~ω ; ∝ exp − (19.15) k(ω)/Ng kB T if Nu /Ng is also in equilibrium at T , this yields the correct equilibrium relation between k(ω), g(ω), I(ω), and a black body spectrum. While these relations are much more general than the QS theory, the latter provides a helpful conceptual basis. The above expressions in terms of spontaneous emission thus cover all cases.

19.2 Radiation Trapping Atoms and ions efficiently absorb their own resonance radiation, and their emission can be reabsorbed before escaping a vapor. Molecules are less efficient absorbers, since each electronic transition branches into multiple-line bands, but interesting effects result if such reabsorption occurs. This emission and reabsorption process is fundamental to the formation of stellar lines, where it is called radiation transfer, and to confined vapors and plasmas where it is also called radiation diffusion or trapping. Fraunhofer’s observation of dark lines in the stellar spectrum result from this radiation transfer process. Highly sophisticated treatments of line formation in inhomogeneous and nonequilibrium plasmas containing many species [19.19, 20] also apply to laboratory plasmas, but the simplifications inherent in a one- or two-element, confined plasma with cylindrical or planar symmetry leads to easier treatments. This sections discusses only a uniform density and temperature, confined atomic vapor. The flourescent lamp in which 254 nm mercury radiation diffuses to the walls and excites a phosphor, provides a prime example of radiation trapping. Its improvement motivated the seminal Biberman [19.51, 52] and Holstein [19.53, 54] theories, continuing through modern theory and experiment that is particularly relevant to electrodeless and compact lamps. Dense clouds of cold, trapped atoms are also influenced by radiation trapping. Reference [19.55] provides and excellent overview of this topic, which we will not discuss here. The effect of radiation trapping on the polarization of flourescent radiation played a major role in developing a correct understanding of the coherent response of atoms to radiation. This is reviewed in [19.44], and will

287

Part B 19.2

0

1 k(ω) dω = λ2 A21 Ng 2



∞

equilibrated internuclear motion sets

19.2 Radiation Trapping

not be covered here. Molisch and Oehry [19.56] have provided a detailed discussion of research on radiation transport up to 1998.

19.2.1 Holstein–Biberman Theory An atom in a dense vapor may be excited by externally applied radiation plus the fluorescence from other excited atoms within the vapor, and it will decay by spontaneous emission (neglecting stimulated emission). This is expressed by the Holstein–Biberman equation  dn(r, t)/ dt = S(r, t) + γ K(r − r  )n(r  , t) d3r  vol

− γn(r, t) ,

(19.16)

where n(r, t) is the excited state density at position r, S(r, t) is the excitation rate due to externally applied radiation, γ is the spontaneous emission rate, the kernel K(r − r  ) is the probability of a reabsorption at r due to fluorescence by an atom at r  , and the integral is over the vapor filled volume [19.51–54]. Since K(r, r  ) is assumed the same for all excited atoms, this contains an implicit assumption that all atoms emit the same fully redistributed spectrum. The solution of this linear integral equation, subject to boundary values at the vapor boundary, can be expressed as a sum over an orthogonal set of solutions n(r, t)i = n(r)i exp(−gi γt) of the homogeneous equation n(r, t) =

∞  i=1

a(t)i n(r)i ,

(19.17)

288

Part B

Atoms

Part B 19.2

t where, if S(r, t) = S(r) f(t), then a(t)i = a¯i −∞ f (t  ) × exp[−gi γ(t − t  )] dt  and a¯i = S(r)n(r)i d3r. Here n(r, t)i is the ith normal mode and gi γ is the decay rate of this mode, as it would decay without change in its shape n(r)i from a pulse of excitation. Two shapes of vapor regions have been studied in detail: an infinitely long cylinder of radius R and the region between two infinite parallel plates with separation L. The first three symmetric modes of the latter slab geometry are shown with unit height in Fig. 19.5. A spatial integration over the normalized i = 1 or fundamental mode yields 1 and all others integrate to zero, so a(t)1 equals the total excited state population. g1 is the escape probability; i. e., the probability of photon escape averaged over the fundamental mode distribution of emitters n(r)1 . Since n(r, t) must be everywhere positive, the negative contributions of the higher order modes only reduce the density in some regions. The gi can vary from 0 to 1 and increase with increasing i, so that higher order modes die out faster after pulsed excitation. The ratios of decay rates is gi : g3 : g5 = 1 : 3.7 : 6.4 for the symmetric slab modes shown in Fig. 19.5. For steady state excitation, (19.17) yields a(t)i = a¯i /gi γ , so the lower order modes are more heavily weighted because they decay more slowly. The fundamental mode decay rate g1 γ is of primary interest in most situations, and we will now discuss its properties.

1.0

The kernel K(x) is the probability of fluorescence transport over a distance x followed by reabsorption, averaged over the emitted frequency distribution. It is conceptually useful to express it in terms of the spectrally averaged transmission T (x) 1 dT (x) K(x) = , (19.18) 4πx 2 dx ∞ T (x) = L(ω) exp[−k(ω)x] dω , 0

where L(ω) is the emission line shape normalized to unit area, and x = |r − r  |. If one assumes that the fluorescence frequency of an atom does not depend on the frequency it absorbed (i. e., complete spectral redistribution), this leads to k(ω) = κL(ω), where κ = (λ2 /8π) (gu /gg )nΓ and gu and gg are statistical weights. This simplification applies under most conditions and will be used here; its range of validity and more accurate treatments are discussed below. The transmission factor L(ω) and the integrand of (19.18) are shown in Fig. 19.6, for a Gaussian line shape and several values of k0 x, where k0 is the line center absorption coefficient. At small k0 x, the transmitted spectrum is similar to L(ω); for these conditions T (x)  exp(−kav x), where kav  0.7k0 is the average attenuation. For k0 x > 5, the transmission is small except at the edges of the line. The transmitted radiation is then predominately in the ω region near ω1 , defined by k0 xL(ω1 ) = 1. Since the integrand is sharply peaked near ω1 , this leads to simple analytic forms for T (x). In

1 0.5

1

3

0

5 –0.5

×5 –1.0 –1.0

–0.5

0

0.5

1.0

Fig. 19.5 The first three symmetric eigenfunctions ( j =

0.2, 4) of radiation trapping between slab windows, for a Doppler line profile, from [19.57–59]. The windows are at ±1

0 –3

–2

–1

0

1

2 3 (ω – ωo) / 0.6 ∆ωD

Fig. 19.6 Gaussian emission spectrum L(ω) (short-dash

line), transmissions T(ω) (long-dash lines), and transmitted intensities (solid lines) for k0 x = 2, 10, and 50

Line Shapes and Radiation Transfer

with G 1 = 1.03 and G 1  0.65. For an infinite cylinder, the same equations hold with L/2 → R and slightly larger G i values. Exact G i and G 1 values can be found in [19.57–59].

19.2.2 Additional Factors As noted above, the line shape of a two-level atom in a thermal vapor is a Voigt shape; a convolution of a Lorentzian of width Γ + 2γc with a Gaussian of width ∆ωD . In most cases, ∆ωD
Γ , so in the absence of a buffer gas the line shape is nearly Gaussian at low density (n). As a result, ω1 is in the Gaussian region of the line at low density and g1 behaves similarly to the Gaussian transmission in Fig. 19.7 with x replaced by the confinement dimension. k0 is proportional to n, so from (19.19a) g1 is approximately inversely proportional to n for k0 L > 5. As n increases, ω1 moves further into the wing of the line, and when ω1 reaches the Lorentzian tail of the Voigt line profile a transition to (19.19b) occurs, where k0 corresponds to a purely Lorentzian line. (That the core of the line does not have a Lorentzian shape does not matter, since the fraction of emission well into the Lorentzian wing is nearly the same as that of a pure Lorentzian line.) In the absence of a collision, a two level atom reradiates in its rest frame the same frequency it absorbed.

0

289

log transmission Holstein Gauss approximation Lorentzian Holstein Lorentzian approximation

Gaussian

–1

a = 0.1

a= 0.01

–2

a= 0.001

a=0 –3 –1

0

1

2

3 log (k0x)

Fig. 19.7 Transmission T(x) versus distance in units of k0 x, for Voigt line shapes with the a parameters indicated, where a = (ln 2)1/2 ∆ωLor /∆ωGauss . The Gaussian limit corresponds to a = 0 and the Lorentzian limit to a = 1. The Holstein, large k0 x, approximations are also indicated

Thermal motion redistributes this coherent scattering frequency within the Doppler envelope when the emission and absorption are in different directions, but it does not transfer it into the natural Lorentzian wing outside the Doppler envelope. This leads to the property that an atomic vapor will scatter frequencies in the natural wing, but will not emit in this wing unless it absorbed there or is excited by or during a collision. With line broadening collisions, a fraction Γ/(Γ + 2γc ) of optical attenuation is coherently scattered and a fraction 2γc /(Γ + 2γc ) is redistributed into “incoherent” emission with a Lorentzian spectrum of width Γ + 2γc centered at ω0 + δ in the reference frame of the moving atom. This redistributed emission can escape in the Lorentzian wing of the Voigt line. In this radiation transport problem, the consequence is that (19.19b) with k0 = n(λ2 /2π)(Γ/γc )(gu /gg ), corresponding to a Lorentzian with ΓTotal = 2γc not Γ + 2γc , provides the best approximation to g1 in the density region where ω1 is in the Lorentzian wing of the line. Since k0 ∝ n/γc and in the absence of a buffer gas γc = kc n, where kc is the rate coefficient for self broadening collisions, g1 becomes independent of n. In fact, kc ∝ Γ as well, so g1 is also independent of Γ . For the case of a J = 0 ground state and a J = 1 excited state, g1 = 0.21(λ/L)1/2 ; the broadening coefficient for other cases can be found in [19.60]. If the

Part B 19.2

the large k0 x limit, T (x)  [k0 x(π ln k0 x)1/2 ]−1 in the Gaussian case, and T (x)  (πk0 x)−1/2 for a Lorentzian line shape. These asymptotic forms of T (x) are compared with the exact T (x) in Fig. 19.7; T (x) follows the asymptotic formulas for k0 x > 5 and 10, respectively. T (x) for several Voigt line shapes is also shown in Fig. 19.7; these follow the Gaussian T (x) at smaller k0 x, then rise above as ω1 moves into the Lorentzian wing. For radiative escape from a cell, transmission over distances near the cell dimension (R or L) is most important, since transport over this distance often escapes the vapor and transport over much smaller distances does not have much effect. The escape probability g1 , averaged over the fundamental mode distribution, is close to T (R) or T (L/2), while the higher order modes are related to the same asymptotic forms of T (x) at smaller distances. Thus, in the large k0 L slab case,  Gi     k L( 1 ln k L)1/2 Gaussian line (19.19a) 0 0 2 gi =   G1    Lorentzian line, (19.19b) (k0 L)1/2

19.2 Radiation Trapping

290

Part B

Atoms

broadening is due to a buffer gas, γc = kc n B in (19.19b) yields  n 1/2 B g1 ∝ ; (19.20) n

Part B 19.2

this has been studied in [19.64]. Post et al. have numerically evaluated g1 for all values of k0 L for slab and cylinder geometries, by integrating the radiative escape probability g(z) over the fundamental mode distribution N(z), where z is the position between the windows [19.65]. To obtain g(z) they integrate over the angular distribution of the emission, using T (x) from the exact line shape. Thus all features of the calculation correspond to the Holstein–Biberman theory for an isolated line without approximation. As will now be discussed real atomic vapors are generally not that simple. Many atomic “lines” have multiple components due to hyperfine structure and isotope shifts; some components are isolated while others are separated by less than a Doppler line width and overlap. The absorption line shape then becomes a weighted sum over components, each with an equivalent Voigt shape. In a high density vapor or a plasma, collisions will usually distribute Sodium density (cm ) 1014 1015 –3

g1eff

1011

100

1012

1013

R = Γeff

ΓN = 2γc

10 –1

H(3P3/2)

10 –2

H(3P1/2)

10 –3 P(3P3/2) 10 –4

10 –1

100

101

10 2

10 3 10 4 Optical depth (k 0 L / 2)

Fig. 19.8 Radiative escape probability g1 for Na vapor excited to

the 3P 3/2 state, for a slab geometry. The Holstein approximation for the 3P 3/2 − 3S 1/2 (D2) line and the 3P 1/2 − 3S 1/2 (D1) line are indicated as dashed lines. The Post-type calculation of [19.61] for the D2 line is indicated as a solid line. Solid squares are data from [19.62], and open circles are data from [19.63]. The effective escape probability corresponds to the D2 line rate at low densities but a combination of D1 and D2 at high densities

the excited state population between the isotopes and hyperfine states in proportion to their isotopic fraction and statistical weight. The emission line shape L(ω) is then a similarly weighted distribution over components. Since radiation only escapes in the wings of a line component at high k0 L, overlapping components act almost as a single component. If the line has M isolated components, the right-hand side of (19.19a) and (19.19b) become sums over the fraction f j of the intensity in the j component times the escape probability for that component. The latter is obtained, for large k0 L, by replacing k0 with k0 f j in (19.19a) and (19.19b). The net result, after summing over components, is an increase in gi by a factor of ≈ M in the Gaussian case and ≈ M 1/2 in the Lorentzian. This approximation was obtained by Holstein in the context of the Hg 254 nm radiation under conditions appropriate to the fluorescent lamp [19.66]. Walsh made a more detailed study of these overlapping components [19.67], and the dependence of g1 on the ratio of line separation to Doppler width is also given in [19.63].

19.2.3 Measurements The overall behavior of g1 versus n is shown in Fig. 19.8 for the Na(3P 3/2 ) or D2 resonance line in pure Na vapor [19.62, 63]. In this type of experiment the fundamental mode decay rate is established by a combination of optimally exciting that spatial mode and of waiting until the fluorescence decay is exponential in time after termination of the excitation. A transition to approximately 1/n behavior, corresponding to (19.19a), is seen to occur at k0 L/2 ≈ 5. At k0 L/2 ≈ 100 the transition to n 0 behavior, corresponding to a selfbroadened Lorentzian line in (19.19b), can be seen. The behavior at k0 L < 5 fits the Milne diffusion theory [19.68] as well as the Post et al. theory shown as a solid line; this is also similar to T (L/2), as seen in Fig. 19.7. For 5 < k0 L/2 < 100, the behavior is similar to (19.19a) (dashed line), but the Post et al. theory (solid line) is ≈ 20% higher due to the inclusion of the Na hyperfine structure (hfs splitting  Doppler width). For k0 L/2 > 1000, the Post theory converges to the Holstein–Lorentzian-line result with ΓTotal = 2γc . The experiment is complicated in the 50 < k0 L/2 < 500 region by fine structure mixing [19.62]. The 3P 3/2 state was excited, but at high densities, collisions populate the 3P 1/2 state, which has a smaller g1 than the 3P 3/2 state (Fig. 19.8). At low densities, g1eff = g1 (3P 3/2 ), and at high densities these states are statistically populated

Line Shapes and Radiation Transfer

g1 values that compared favorably with the measurements. By extending the simulations over a large range of a parameter space, they constructed an analytic formula for g1 of a single-component line in cylindrical geometry [19.76]. This formula includes effects of incomplete frequency redistribution and varying ratios of Doppler broadening, radiative broadening and collisional broadening, so that it can be applied to any resonance line. Payne et al. [19.71] did not observe the predicted dip for the Ar resonance line; again a minor isotope with an isolated line occurs and could be very important at these high optical depths. Phelps [19.70] reported such a dip for the Ne 74.3 nm resonance line, but with rather large uncertainties due to the necessity of correcting for other collisional effects. Again there are isotopes with isolated lines that may have effected the data. Thus, experiments have verified the essential aspects of the above theories, but quantitative agreement in all aspects has not yet been achieved. The fact that the escaping radiation is concentrated in the wings of the line, near the unity optical depth point ω1 , is reflected in the emitted spectrum. Calculated examples are shown in [19.74]; the Gaussian case looks somewhat like the transmitted spectra in Fig. 19.6 for x ≈ L/4. These spectra, and all results described so far, are calculated assuming no motion of the atoms. This is appropriate in the central region of the vapor, because the distance moved in an excited state lifetime (L v = v/Γ ) is much smaller than L. In fact, resonant collisions between excited and ground state atoms further limits the distance an excited atom moves in one direction before transfer of excitation. However, near the window or wall of the container, atomic motion will cause wall collisions of excited atoms and loss of radiation. This loss will be primarily within the Doppler core of the line, since these frequencies can only escape if emitted near the vapor edge. This loss depends on the excited state density in the neighborhood of the wall, and can be significant if L v > 1/k0 . The excited atom density near the wall must be self consistent with the radiation transport and wall quenching. This situation has been modeled and studied experimentally ([19.77] and references therein). Additional aspects of radiation trapping, such as higher-order spatial modes and non-uniform absorber distributions, can be significant in lighting plasmas (and trapped atom clouds). Propagator function techniques have been developed for modeling radiation transport when the excitation has unusual temporal or spatial character [19.78, 79]. Non-uniform absorber spatial distributions can be particularly important at high power densities, and have been considered in [19.80].

291

Part B 19.2

and g1eff = 13 g1 (3P 1/2 ) + 23 g1 (3P 3/2 ). The transition density where the fine structure mixing rate R equals Γeff is indicated in Fig. 19.8. The theory is also complicated in this intermediate k0 L region by the necessity of including incomplete frequency redistribution [19.65]; this leads to the dip in g1 near k0 L ≈ 500. While the overall behavior of the data in Fig. 19.8 is consistent with the Post et al. theory, there is ≈ 30% systematic discrepancy at k0 L/2 = 10−100 and the dip near 500 is not seen. Part of this difference probably results from the experimental geometry, which was between a slab and a cylinder of radius R = L/2; g1 for the cylinder is 17% larger than the slab value used in Fig. 19.8. The fundamental mode decay rate has also been measured for the Hg 254 nm [19.69] and 185 nm [19.65] lines, for the Ne resonance line [19.70] and for the Ar resonance line [19.71]. The Hg measurements are complicated by multiple isotopes and hyperfine structure, producing a mixture of partially overlapping and isolated lines combined with density-dependent uncertainties in excited state populations of the various isotopes. Serious efforts to model and measure these effects have been made [19.65, 67, 69, 72]. The Ne and Ar measurements have similar complications, as will now be discussed. In essence, g1 behaves like the Gaussian T (x = L/2) in Fig. 19.7 until n is large enough for ω1 to approach the collision induced Lorentzian wing of the Voigt line. g1 then decreases more slowly since the line wing does not fall off as rapidly as a Gaussian. With continued increase in n, ω1 moves further into the Lorentzian wing, a broader spectral region escapes and g1 reaches a minimum. Finally, when the entire escaping spectral region is Lorentzian, g1 reaches the constant value described above. Independent and detailed treatments of this density region, including incomplete frequency redistribution, predict a dip in g1 as seen in Fig. 19.8 [19.65, 71, 73, 74]. However, this has not been clearly confirmed experimentally. In Fig. 19.8 this dip occurs where fine structure mixing also occurs, and in addition the data are higher than the calculations throughout this n region. Post et al. [19.65] did observe such a dip for the Hg (149 nm) resonance line, but the data do not fit the calculation at other densities; hyperfine and isotopic structure within the line cause major complications. This long-standing issue has finally been clarified by Menningen and Lawler [19.75], who measured the decay of the Hg (185 nm) resonance line following laser excitation. They observed a clear dip in g1 due to incomplete redistribution. They also carried out sophisticated Monte Carlo simulations, obtaining

19.2 Radiation Trapping

292

Part B

Atoms

References 19.1 19.2 19.3 19.4 19.5

Part B 19

19.6 19.7 19.8 19.9 19.10 19.11 19.12 19.13 19.14 19.15 19.16 19.17 19.18 19.19 19.20 19.21 19.22 19.23 19.24 19.25 19.26 19.27 19.28 19.29 19.30 19.31 19.32 19.33 19.34 19.35 19.36

H. A. Lorentz: Proc. Akad. Wet. (Amsterdam) 8, 591 (1906) V. Weisskopf: Phys. Z. 34, 1 (1933) J. Holtsmark: Ann. Phys. (Leipzig) 58, 577 (1919) H. G. Kuhn: Philos. Mag. 18, 987 (1934) H. Margenau, W. W. Watson: Rev. Mod. Phys. 8, 22 (1936) A. Jablonski: Z. Physik 70, 723 (1931) A. Jablonski: Acta Phys. Polon. 27, 49 (1965) E. Lindholm: Ark. Fys. A 32, 1 (1945) H. M. Foley: Phys. Rev. 69, 616 (1946) H. M. Foley: Phys. Rev. 73, 259 (1948) T. Holstein: Phys. Rev. 79, 744 (1950) P. W. Anderson: Phys. Rev. 76, 647 (1949) P. W. Anderson: Phys. Rev. 86, 809 (1952) M. Baranger: Phys. Rev. 112, 855 (1958) S-Y. Chen, M. Takeo: Rev. Mod. Phys. 29, 20 (1957) N. Allard, J. Kielkopf: Rev. Mod. Phys. 54, 1103 (1982) H. Griem: Plasma Spectroscopy (McGraw Hill, New York 1964) H. Griem: Spectral Line Broadening by Plasmas (Academic, New York 1974) J. Jeffries: Spectral Line Formation (Blaisdell, Waltham 1968) D. Mihalas: Stellar Atmospheres (Freeman, San Francisco 1970) J. van Kranendonk: Cnd. J. Phys. 46, 1173 (1968) J. Cooper: Rep. Prog. Phys. 29, 35 (1966) J. Simons: Energetic Principles of Chemical Reactions (Jones Bartlett, Boston 1983) J. Szudy, W. E. Baylis: J. Quant. Spectrosc. Radiat. Trans. 17, 681 (1977) R. E. Walkup, A. Spielfiedel, D. E. Pritchard: Phys. Rev. Lett. 45, 986 (1980) J. Ward, J. Cooper, E. W. Smith: J. Quant. Spectrosc. Radiat. Trans. 14, 555 (1974) D. G. McCarten, N. Lwin: J. Phys. B 10, 17 (1977) D. N. Stacey, R. C. Thompson: Acta Phys. Polon. A 54, 833 (1978) Ch. Ottinger, R. Scheps, G. W. York, A. Gallagher: Phys. Rev. A 11, 1815 (1975) D. Drummond, A. Gallagher: J. Chem. Phys. 60, 3426 (1974) C. G. Carrington, A. Gallagher: Phys. Rev. A 10, 1464 (1974) J. Pascale, J. Vandeplanque: J. Chem. Phys. 60, 2278 (1974) M. J. O’Callaghan, J. Cooper: Phys. Rev. A 39, 6206 (1989) B. P. Stoicheff, E. Weinberger: Phys. Rev. Lett. 44, 733 (1980) K. H. Weber, K. Niemax: Z. Phys. A 307, 13 (1982) J. R. Fuhr, A. Lesage: Bibliography of Atomic Line Shapes and Shifts, NIST Special Publication 366

19.37 19.38 19.39 19.40 19.41 19.42 19.43 19.44 19.45 19.46 19.47 19.48 19.49 19.50 19.51 19.52 19.53 19.54 19.55 19.56 19.57 19.58 19.59 19.60 19.61 19.62 19.63 19.64 19.65 19.66 19.67 19.68 19.69 19.70

(U. S. Gov’t. Printing Office, Washington 1993), Suppl. 4 J. Szudy, W. E. Baylis: J. Quant. Spectrosc. Radiat. Trans. 15, 641 (1975) D. N. Stacey, J. Cooper: J. Quant. Spectrosc. Radiat. Trans. 11, 1271 (1971) K. M. Sando, J. Wormhoudt: Phys. Rev. A 7, 1889 (1973) L. K. Lam, M. M. Hessel, A. Gallagher: J. Chem. Phys. 66, 3550 (1977) A. Tam, G. Moe, W. Park, W. Happer: Phys. Rev. Lett. 35, 85 (1975) T. Holstein, A. Gallagher: Phys. Rev. A 16, 2413 (1977) J. L. Carlston, A. Szoke: J. Phys. B 9, L231 (1976) A. Omont: Prog. Quant. Electr. 5, 69 (1977) J. Light, A. Szoke: Phys. Rev. A 15, 1029 (1977) K. Burnett, J. Cooper, R. J. Ballagh, E. W. Smith: Phys. Rev. A 22, 2005 (1980) A. Streater, J. Cooper, W. J. Sandle: J. Quant. Spectrosc. Radiat. Trans. 37, 151 (1987) J. Tellinghausen: J. Mol. Spectrosc. 103, 455 (1984) F. H. Mies: J. Chem. Phys. 48, 482 (1968) C. G. Carrington, D. Drummond, A. V. Phelps, A. Gallagher: Chem. Phys. Lett. 22, 511 (1973) L. M. Biberman: J. Exp. Theor. Phys. U. S. S. R. 17, 416 (1947) L. M. Biberman: J. Exp. Theor. Phys. U. S. S. R. 59, 659 (1948) T. Holstein: Phys. Rev. 72, 1212 (1947) T. Holstein: Phys. Rev. 83, 1159 (1951) A. Fioretti, A. F. Molisch, J. H. Müller, P. Verkerk, M. Allegrini: Opt. Commun. 149, 415 (1998) A. F. Molisch, B. P. Oehry: Radiation Trapping in Atomic Vapours (Clarendon, Oxford 1998) C. van Trigt: Phys. Rev. 181, 97 (1969) C. van Trigt: Phys. Rev. A 4, 1303 (1971) C. van Trigt: Phys. Rev. 13, 726 (1976) C. G. Carrington, D. N. Stacey, J. Cooper: J. Phys. B 6, 417 (1973) J. Huennekins, T. Colbert: J. Quant. Spectrosc. Radiat. Trans. 41, 439 (1989) J. Huennekins, A. Gallagher: Phys. Rev. A 28, 238 (1983) T. Colbert, J. Huennekens: Phys. Rev. 41, 6145 (1990) J. Huennekins, H. J. Park, T. Colbert, S. C. McClain: Phys. Rev. 35, 2892 (1987) H. A. Post, P. van de Weijer, R. M. M. Cremers: Phys. Rev. A 33, 2003 (1986) T. Holstein: Phys. Rev. 83, 1159 (1951) P. J. Walsh: Phys. Rev. 116, 511 (1959) E. Milne: J. Math. Soc. (London) 1, 40 (1926) T. Holstein, D. Alpert, A. O. McCoubrey: Phys. Rev. 85, 985 (1952) A. V. Phelps: Phys. Rev. 114, 1011 (1959)

Line Shapes and Radiation Transfer

19.71 19.72

19.73 19.74 19.75

M. G. Payne, J. E. Talmage, G. S. Hurst, E. B. Wagner: Phys. Rev. A 9, 1050 (1974) J. B. Anderson, J. Maya, M. W. Grossman, R. Lagushenko, J. F. Weymouth: Phys. Rev. A 3, 2986 (1985) G. J. Parker, W. N. G. Hitchon, J. E. Lawler: J. Phys. B 26, 4643 (1993) C. van Trigt: Phys. Rev. A 1, 1298 (1970) K. L. Menningen, J. E. Lawler: J. Appl. Phys. 88, 3190 (2000)

19.76 19.77 19.78 19.79 19.80

References

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J. E. Lawler, J. J. Curry, G. G. Lister: J. Phys. D 33, 252 (2000) A. Zajonc, A. V. Phelps: Phys. Rev. A 23, 2479 (1981) J. E. Lawler, G. J. Parker, W. N. G. Hitchon: J. Quant. Spectrosc. Radiat. Trans. 49, 627 (1993) G. J. Parker, W. N. G. Hitchon, J. E. Lawler: J. Phys. B 26, 4643 (1993) J. J. Curry, J. E. Lawler, G. G. Lister: J. Appl. Phys. 86, 731 (1999)

Part B 19

295

Thomas–Ferm 20. Thomas–Fermi and Other Density-Functional Theories

The key idea in Thomas–Fermi theory and its generalizations is the replacement of complicated terms in the kinetic energy and electron–electron repulsion energy contributions to the total energy by relatively simple functionals of the electron density ρ. This chapter first describes Thomas– Fermi theory, and then its various generalizations which attempt to correct, with varying success, some of its deficiencies. It concludes with an overview of the Hohenberg-Kohn and Kohn-Sham density functional theories.

296 296 298 299

In the early years of quantum physics, Thomas [20.1] and Fermi [20.2–5] independently invented a simplified theory, subsequently known as Thomas–Fermi theory, to describe nonrelativistically an atom or atomic ion with a large nuclear charge Z and a large number of electrons N. Many qualitative features of this model can be studied analytically, and the precise solution can be found by solving numerically a nonlinear ordinary differential equation. Lenz [20.6] demonstrated that this equation for the electrostatic potential could be derived from a variational expression for the energy as a functional of the electron density. Refinements to Thomas–Fermi theory include a term in the energy functional to account for electron exchange effects introduced by Dirac [20.7], and nonlocal gradient corrections to the kinetic energy introduced by von Weizsäcker [20.8]. Although the Hartree–Fock method or other more elaborate techniques for calculating electronic structure now provide much more accurate results (Chapts. 21, 22, and 23), Thomas–Fermi theory provides quick estimates and global insight into the total energy and other properties of a heavy atom or ion. A rigorous analysis

299 300 300 301 301

302 303

303 303 304

of Thomas–Fermi theory by Lieb and Simon [20.9, 10] showed that it is asymptotically exact in that it yields the correct leading asymptotic behavior, for both the total nonrelativistic energy and the electronic density, in the limit as both Z and N tend to infinity, with the ratio Z/N fixed. (In a real atom, of course, relativistic and other effects become increasingly important as Z increases.) However, Thomas–Fermi theory has the property that molecules do not bind, as first noted by Sheldon [20.11] and proved by Teller [20.12]. That the interatomic potential energy curve for a homonuclear diatomic molecule is purely repulsive was demonstrated by Balàzs [20.13]. This ‘no binding’ property of clusters of atoms was used by Lieb and Thirring [20.14] to prove the stability of matter, in the sense that as the number of particles increases, the total nonrelativistic energy decreases only linearly rather than as a higher power of the number of particles, as it would if electrons were bosons rather than fermions. Lieb went on to explore the mathematical structure of the modifications of the Thomas–Fermi model when gradient terms (von Weizsäcker) and/or exchange (Dirac) terms are included [20.15, 16]. A review article by Spruch [20.17] explicates the linkage between long-developed physi-

Part B 20

20.1 Thomas–Fermi Theory and Its Extensions ............................... 20.1.1 Thomas–Fermi Theory................ 20.1.2 Thomas–Fermi–von Weizsäcker Theory ..................................... 20.1.3 Thomas–Fermi–Dirac Theory.......

20.1.4 Thomas–Fermi–von Weizsäcker– Dirac Theory ............................. 20.1.5 Thomas–Fermi Theory with Different Spin Densities ...... 20.2 Nonrelativistic Energies of Heavy Atoms 20.3 General Density Functional Theory ........ 20.3.1 The Hohenberg–Kohn Theorem for the One-Electron Density ...... 20.3.2 The Kohn–Sham Method for Including Exchange and Correlation Corrections ........ 20.3.3 Density Functional Theory for Excited States ...................... 20.3.4 Relativistic and Quantum Field Theoretic Density Functional Theory ..................................... 20.4 Recent Developments........................... References ..................................................

296

Part B

Atoms

cal intuition and the mathematically rigorous results obtained in the 1970’s and 1980’s. The older literature was reviewed by Gombás [20.18, 19] and by March [20.20].

An outgrowth of Thomas–Fermi theory is the general density functional theory initiated by Hohenberg and Kohn [20.21] and by Kohn and Sham [20.22], as discussed in Sect. 20.3 of this chapter.

20.1 Thomas–Fermi Theory 20.1.1 Thomas–Fermi Theory In a D-dimensional Euclidean space, the expectation value of the electronic kinetic energy operator in a quantum state ψ can be approximated by 
 D 2/D ~2 2 D ρ(D+2)/D (r) d D r , 2π me D + 2 2Ω D (20.1)

where

Part B 20.1

Ω D = D π D/2 /Γ (1 + D/2)

(20.2)

is the surface area of a unit hypersphere in D dimensions [20.17, p. 176]. These expressions can easily be derived by considering the energy levels of a system of a large number of noninteracting fermions confined to a D-dimensional box. Specialization to the physically interesting case of D = 3 yields the well-known expression  
~2 3 2/3 2 3 2π (20.3) ρ5/3 (r) d r , me 5 2Ω3 where Ω3 = 3π 3/2 /Γ (1 + 3/2) = 4π .

(20.4)

The electron–nucleus attraction energy in a threedimensional space is given exactly by  (20.5) ρ(r)V(r) d r , where V(r) is the Coulomb potential due to a single nucleus  (V(r) = −Z/r) or to several nuclei [V(r) = − i Z i /|r − Ri |]. The electron–electron Coulomb repulsion energy in a three-dimensional space is approximated by  ρ(r)ρ(r
) 1 d r d r
, (20.6) 2 |r − r
| which tends to overestimate the actual repulsion energy because it includes the self-energy of the densities of individual electrons. This is, however, a higher-order

effect for a system with a large number of electrons concentrated in a small region of space. As was suggested by Fermi and Amaldi [20.23], this overestimation can be approximately corrected for an atom with N electrons by multiplying this term by the ratio of the number of ordered pairs of different electrons to the total number of ordered pairs N(N − 1) 1 . =1 − (20.7) N N2 This is approximately correct for an atom, with many electrons concentrated close together, but it would still be an overestimate for a diffuse system, such as one composed of N electrons and N protons separated by large distances of O(R), for which the ground-state electron–electron repulsion term should be proportional to 12 N(N − 1)/R rather than to N times a constant of O(1). For this reason the Fermi–Amaldi correction, which complicates the mathematical analysis without eliminating the unphysical overestimation of the electron–electron repulsion term, is not usually included. It is evident that the treatment of both the electronic kinetic energy term and the electron–electron repulsion energy term depends on the assumption that the number N of electrons (actually, the number of electrons per atom) is large. Hence the Thomas–Fermi model is sometimes called the statistical model of an atom. The three contributions to the total energy are now added together and one seeks to minimize their sum, the Lenz functional [20.6]  
3 2/3 ~2 2 3 ρ5/3 (r) d r E[ρ] = 2π me 5 2Ω3   1 ρ(r)ρ(r
) d r d r
, + ρ(r)V(r) d r + 2 |r − r
| (20.8)

over all admissible densities ρ. The mathematical question now arises: what is an admissible density? The answer was provided by Lieb and Simon [20.9, 10]: a density for which both  ρ(r) d r , (20.9)

Thomas–Fermi and Other Density-Functional Theories

the total number of electrons, and  ρ5/3 (r) d r ,

(20.10)

which is proportional to the estimate of their kinetic energy, are finite, automatically yields finite values of the other terms in the expression for the energy. As Lieb and Simon proved, the minimization of this functional over all such densities yields a well-determined result. Carrying out the variation of E[ρ] with respect to ρ yields the Euler–Lagrange equation
 3 2/3 2/3 ~2 2 0= 2π ρ (r) me 2Ω3  ρ(r
) d r
. (20.11) + V(r) + |r − r
|

(20.12)

so that γp ρ2/3 (r) = φ(r) .

(20.13)

By Poisson’s theorem,   2 (3) −∇ φ = 4π Z i δ (r − Ri ) − ρ(r) ,

recognized by Milne [20.24]. The numerical solution of this equation with the appropriate boundary conditions at r = 0 and r = ∞ was extensively discussed by Baker [20.25], and accurate solutions tabulated by Tal and Levy [20.26]. The numerical solution determines that the total energy of a neutral atom is E = − 3.678 745 21 . . . γp−1 Z 7/3 = − 1.537 490 24 . . . Z 7/3 Ry .

(20.14) −3/2

and from (20.13) one has ρ = γp φ3/2 , so from the integral equation for the electronic density ρ one obtains the differential equation   −3/2 3/2 2 (3) Z i δ (r − Ri ) − γp φ −∇ φ = 4π

(20.17)

Another possibility is to do the constrained minization over all densities which obey  (20.18) ρ(r) d r = λ , where λ is the number of electrons, which for purposes of mathematical analysis is allowed to be nonintegral. One then introduces a Lagrange multiplier −µ, the chemical potential, to correspond with this constraint, and thereby obtains the Euler–Lagrange equation
 ~2 2 3 2/3 2/3 0= 2π ρ (r) me 2Ω3  ρ(r
) d r
+ µ , (20.19) + V(r) + |r − r
| which holds wherever ρ is positive. As was shown by Lieb and Simon [20.9, 10], this procedure too is well-defined. The analogue of (20.13), the relationship between the density and the electrostatic potential for the neutral atom, is now γp ρ2/3 (r) = [φ(r) − µ]+ ,

i

(20.20)

where [ f ]+ = max( f, 0). The corresponding differential equation for the potential φ is   2 −∇ φ = 4π Z i δ(3) (r − Ri ) i −3/2 3/2 [φ(r) − µ]+ −γp

.

(20.21)

i

(20.15)

for the potential φ. In the case of a single nucleus, the usual separation of variables in spherical polar coordinates yields for φ the ordinary differential equation 1 d2 −3/2 3/2 (rφ) = 4π γp φ , (20.16) r dr 2 whose similarity to Emden’s equation, which Eddington had used to study the internal constitution of stars, was

297

Lieb and Simon rigorously proved a large number of results concerning the solution of the Thomas–Fermi model. When V(r) is a sum of Coulomb potentials arising  from a set of nuclei of positive charges Z i , with i Z i = Z, then the energy E(λ) is a continuous, monotonically decreasing function of λ for 0 ≤ λ ≤ Z, and its derivative dE/ dλ is the chemical potential −µ(λ), which vanishes at λ = Z. For λ in this range, there is a unique minimizing density ρ, whereas for λ > Z there

Part B 20.1

The sum of the last two terms is of course the negative of the total electrostatic potential φ(r), so one sees that in Thomas–Fermi theory the density is proportional to the 3/2-power of the potential. To simplify subsequent manipulations, let
 ~2 2 ~2  2 2/3 3 2/3 3π 2π = = γp , me 2Ω3 2m e

20.1 Thomas–Fermi Theory

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is no unique minimizing ρ, since one can place arbitrarily large clumps of charge with arbitrarily low energy arbitrarily far away from the nuclei. In the atomic case, with a single nucleus, ρ(r) is a spherically symmetric monotonically decreasing function of r. Moreover, for an atom or atomic ion, the Thomas– Fermi density obeys the virial theorem 2T  = −V  = −2E ,

(20.22)

Part B 20.1

and for a neutral atom the electronic kinetic energy, electron–nucleus attraction energy, and electron– electron repulsion energy terms in the expression for the total energy satisfy the ratios 3 : −7 : 1. It is straightforward to examine the behavior of the Thomas–Fermi density ρ in the limit as either r → 0 or r → ∞. For large r, the electron density vanishes identically outside a sphere of finite radius for a positive ion. For a neutral atom, the ordinary differential equation (20.16) for the potential φ can be analyzed to show that φ(r)  γp (3γp /π)2 r −4 ,

(20.23)

distance set by Z −1/3 , within which most of the electron density is located, there resides in the ‘mantle’ region a fraction of electrons proportional to Z 2/3 /Z = Z −1/3 , and almost all of these are concentrated within a sphere of radius of about 10 a0 . Moving deeper into the core and approaching the nucleus, the −Z/r singularity in the electron–nucleus Coulomb potential dominates the smeared-out electron– electron potential, so one readily finds that
 Z 3/2 ρ(r)  . (20.28) γp r This singularity is integrable but unphysical, since it arises from the approximation of the local kinetic energy by ρ5/3 , which breaks down where ρ is rapidly varying on a length scale proportional to 1/Z. In a ‘real’ nonrelativistic heavy atom governed by the Schrödinger equation, the actual electron density at the nucleus is finite, being proportional to Z 3 . This unphysical singularity in the electron density in Thomas–Fermi theory can be eliminated by adding a gradient correction to the Thomas–Fermi kinetic energy term.

from which it follows that ρ(r)  (3γp /π)3 r −6 ,

(20.24)

independent of Z. This implies that as Z → ∞, a neutral atom described by the Thomas–Fermi model has a finite size defined in terms of a radius within which all but a fixed amount of electronic probability density is located. For example, if one defines the size of an atom as that value of ra for which  1 ρ(r) d r = , (20.25) 2 |r|≥ra

one finds that in the large-Z limit
1/3 3γp 8π . ra = 3 π

(20.26)

In atomic units, γp = 12 (3π 2 )2/3 , and ra = (9π)2/3 a0  9.3 a0 ,

(20.27)

which is about what one would expect for a ‘real’ nonrelativistic atom with a large nuclear charge Z. On the other hand, the characteristic distance scale in Thomas– Fermi theory, defined as the ‘average’ value of r, or in terms of a radius within which a fixed fraction of electronic probability density is located, is proportional to Z −1/3 , which shrinks to 0 as Z → ∞. The resolution of this paradox is that outside the typical ‘core’ scale of

20.1.2 Thomas–Fermi–von Weizsäcker Theory The semiclassical approximation (20.3) for the quantum kinetic energy in terms of a power of the density is capable of improvement, particularly in regions of space where the density is rapidly varying. The incorporation of such corrections leads to a gradient expansion for the kinetic energy [20.27]. The leading correction is of the form 

~2

 1/2  2 (r) d r . (20.29)

∇ρ 2m Addition of such a term to the Thomas–Fermi expression for the kinetic energy yields a theory which avoids many of the unphysical features of ordinary Thomas–Fermi theory at very short and moderately large distances. The more important points, as rigorously proved in Lieb’s review article [20.15, 16], are as follows. The leading features of the energy are unchanged; for large Z the energy E(Z) of a neutral atom or atomic ion is still proportional to Z 7/3 , but now there enter higherorder corrections arising from the gradient terms of order Z 7/3 Z −1/3 = Z 2 and higher powers of Z −1/3 . The maximum number of electrons which can be bound by an atom of nuclear charge Z is no longer exactly Z, but a slightly larger number; thus Thomas–Fermi–von Weizsäcker theory allows for the formation of negatively charged atomic ions. It was further proven by

Thomas–Fermi and Other Density-Functional Theories

299

cated nature of the universal density functional, which must include terms which rigorously suppress an unphysical feature like spontaneous ionization of a distant pair of heteronuclear atoms [20.32, 33]. It is evident from the mathematical properties of Thomas–Fermi– von Weizsäcker theory and related models that a density functional which ‘fixes up’ the Thomas–Fermi expression simply by adding a few gradient terms and/or simple exchange terms and the like must still differ in important ways from the universal density functional, particularly for properties of extended systems.

20.1.3 Thomas–Fermi–Dirac Theory The effect of the exchange of electrons can be approximated, following Dirac [20.7], by including in the Thomas–Fermi energy functional an expression of the form  4/3 1  ρ4/3 (r) d r . − 3 3π 2 (20.30) 4π Minimization of the resulting Thomas–Fermi–Dirac energy functional over all admissible densities ρ whose integral is λ yields a well-defined E(λ), which has the correct behavior for λ ≤ Z, and it has been shown that for an atom the exchange correction to the energy is of order Z 5/3 . However, this model exhibits unphysical behavior for λ > Z, because one can obtain a completely artificial lowering of the energy by placing many small clumps of electronsa large distance from the nucleus, for which the negative ρ4/3 d r term dominates the energy expression [20.15, 16]. At the conclusion of his original article, Dirac clearly stated that the correction he had derived, although giving a better approximation in the interior of an atom, gives “a meaningless result for the outside of the atom” [20.7]. It is therefore clear that any physically reasonable theory must somehow profoundly modify this correction in the region where the electronic density is very small.

20.1.4 Thomas–Fermi–von Weizsäcker–Dirac Theory One can also include the Dirac exchange correction in the Thomas–Fermi–von Weizsäcker energy functional. In this case, however, the mathematical foundations of the theory are still incomplete ([20.15, 16, pp. 638–9]). Nonetheless, it is clear that this theory too suffers from the unphysical lowering of the energy by small clumps of electrons at large distances from the nucleus. In summary, one can say that the inclusion of Dirac’s exchange correction in its most straightforward form

Part B 20.1

Benguria and Lieb [20.28] that in the Thomas–Fermi– von Weizsäcker model a neutral atom can bind at most one extra electron, and that a neutral molecule can bind at most as many extra electrons as it has nuclei. The effect on the electronic density ρ is more profound. While the general shape and properties of ρ in the ‘core’ and ‘mantle’ regions is unchanged, the fact that ∇ρ1/2 need not, and in general does not, vanish when ρ vanishes on some surface implies that for a positive ion ρ no longer vanishes outside of a sphere, as it does in the case of Thomas–Fermi theory, but instead extends over all space. For positive ions, neutral atoms, and negative ions alike, ‘differential inequality’ techniques [20.29] can be used to show that ρ(r) decays exponentially for large r, with the constant in the exponential proportional to µ1/2 (λ). For small r, the gradient terms dominate the energy expression, so one finds that the electronic density no longer diverges as r → 0, but instead tends to a finite limit, with a first derivative which obeys a relation analogous to the Kato cusp condition [20.30] (see Sect. 11.1.1). The study of molecules within the Thomas–Fermi– von Weizsäcker model involves several subtleties and pitfalls, which can lead to physical absurdities. Since two neutral atoms with different nuclear charges will in general have different chemical potentials, a pair of such atoms placed a long distance apart will spontaneously ionize, with a small amount of electric charge being transferred from one to the other until the chemical potentials of the positively charged ion and the negatively charged ion become equal. The result is a long-range Coulomb attraction between them [20.31]. This phenomenon does not occur in the real world, since the amount of electric charge which can be transferred is quantized in units of −e, and it is empirically true that the smallest atomic ionization potential exceeds the largest atomic electron affinity. For two neutral atoms with the same nuclear charges, the situation is more subtle. Nonetheless, a careful analysis shows that in this case too, though no spontaneous ionization occurs, there is a long-range attractive interaction between them arising from the overlap of the exponentially small tails of the electron clouds. Since electron correlation is not included in this model, it could not be expected to describe attractive van der Waals forces. In summary, the Thomas–Fermi–von Weizsäcker model yields a more realistic picture of a single atom than does the Thomas–Fermi model. However, it does not provide a useful picture for understanding the interaction between atoms at large distances. These kinds of unphysical features provide a glimpse into the compli-

20.1 Thomas–Fermi Theory

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leads to an improvement of energies for positive ions or neutral atoms, but to unphysical behavior for systems where the charge of the electrons exceeds the nuclear charge, in line with Dirac’s own observations on the limitations of his correction [20.7]. We see here again a manifestation of how complicated must be the behavior of the true universal density functional.

20.1.5 Thomas–Fermi Theory with Different Spin Densities As was remarked by Lieb and Simon [20.10], it is possible to consider a variant of Thomas–Fermi theory with a pair of spin densities ρα and ρβ for the spin-up and spin-down electrons, with the two adding together to produce the total electronic density ρ. This theory has been rigorously formulated and analyzed by Goldstein

and Rieder [20.34]. Because the problem is nonlinear, the mathematical complications are substantial, and the theory is not a trivial extension of ordinary Thomas– Fermi theory. Goldstein and Rieder first considered the case where the total number of electrons of each type of spin is specified in advance [20.35]. There is no mathematical obstacle to constructing such a spin-polarized Thomas–Fermi theory, but it does not yield the kind of spontaneous spin-polarization that one observes in the ground states of many real quantum mechanical atoms and molecules, which is not surprising in view of the fact that such spin-polarization in accord with Hund’s first rule arises from exchange and correlation effects not included in this simple functional. However, in the case where the electronic spins (but not their currents) are coupled to an external magnetic field, the ground state is naturally spin-polarized [20.34].

Part B 20.2

20.2 Nonrelativistic Energies of Heavy Atoms Thomas–Fermi theory suggests that (20.17) provides the leading term in a power series expansion for the nonrelativistic energy of a neutral atom of the form

E(Z) = − c7 Z 7/3 + c6 Z 6/3 + c5 Z 5/3 + · · · (20.31)

with c7 = 1.537 490 24 . . . Ry, c6 = −1 Ry, and c5  0.5398 Ry. The c6 term was first calculated by Scott [20.36] from the observation that it arises from the energy of the innermost electrons for which the electron–electron interaction can be neglected. The difference between the exact and Thomas–Fermi energies for this case of noninteracting electrons yields the correct c6 [20.17, 37]. A mathematically nonrigorous but physically insightful justification of the Scott correction was provided in 1980 by Schwinger [20.38]. This result has now been rigorously proved, with upper and lower bounds coinciding [20.39–43]. The c5 term is much more subtle, since it arises from a combination of effects from the exchange interaction and from the bulk motion of electrons in the Thomas– Fermi potential. A general analytic procedure devised by Schwinger [20.44] yields the above value, in good agreement with a much earlier estimate by March and Plaskett [20.45]. A numerical check of these results, based on a fit to Hartree–Fock calculations for Z up to 290 with correlation corrections, yielded the values [20.46]

c5 = 0.55 ± 0.02 Ry and c4  0. It seems likely that, because of shell structure, the terms c4 and beyond have an oscillatory dependence on Z [20.47]. The oscillatory structure and other refinements of Thomas–Fermi theory are considered in a series of papers by Englert and Schwinger [20.48–50]. Iantchenko, Lieb, and Siedentop [20.51] have proven Lieb’s ‘strong Scott conjecture’ that for small r, the rescaled density for the exact quantum system converges to the sum of the densities of the bound noninteracting hydrogenic orbitals; the properties of this function were explored by Heilman and Lieb [20.52]. Fefferman and Seco [20.53] have rigorously proved the correctness of Schwinger’s procedure for calculating not just the O(Z 6/3 ) Scott correction but also the O(Z 5/3 ) exchange term. Their full proof includes a demonstration that the Hartree–Fock energy agrees with the exact quantum energy through O(Z 5/3 ), with an error of smaller order [20.54]. Numerous auxiliary theorems and lemmas are published in [20.55–58]. Progress toward obtaining higher-order oscillatory terms is described in [20.59–63]. The analytical evaluation of accurate approximations to the energy of a heavy atom, or at least of the contributions to that energy of all but the few outermost electronic orbitals, would be of particularly great value if it led to the construction of more accurate and better justified pseudopotentials [20.64–67] for describing the valence orbitals.

Thomas–Fermi and Other Density-Functional Theories

20.3 General Density Functional Theory

301

20.3 General Density Functional Theory The literature on general density functional theory and its applications is enormous, so any bibliography must be selective. The reader interested in learning more could begin by consulting a number of review articles [20.68–73], collections of articles [20.74–76], and conference proceedings [20.77–87], and the recent textbooks by Parr and Yang [20.88] and by Dreizler and Gross [20.89].

20.3.1 The Hohenberg–Kohn Theorem for the One-Electron Density

subject to the constraint  ρ(r) d r = N (a positive integer) ,

(20.33)

yields the ground state energy of a quantum-mechanical N-electron system moving in this external potential. However, Hohenberg and Kohn’s ‘theorem’ is like a mathematical ‘existence theorem’; no procedure exists to calculate explicitly this unknown universal functional, which surely is extremely complicated if it can be written down at all in closed form. (E. Bright Wilson, however, defined it, implicitly and whimsically, as follows: “Take the ground-state density and integrate it to find the total number of electrons. Find the cusps in the density to locate all nuclei, and then use the cusp condition – that the radial derivative of the density at the cusp is minus twice the nuclear charge density at each cusp – to determine the charges on each nucleus. Finally solve Schrödinger’s equation for the ground-state density or any other property that is desired” (paraphrased by B. I. Dunlap, in [20.83, p. 3], from J. W. D. Connolly).) Moreover, Hohenberg and Kohn glossed over two problems: it is not clear a priori that every wellbehaved ρ is derivable from a well-behaved properly antisymmetric many-electron wave function (the socalled n-representability problem, since n was used by Hohenberg and Kohn to represent the density of electrons), and it is also not clear a priori that every well-behaved density ρ can be derived from a quantummechanical many-electron wave function ψ which is the properly antisymmetric ground-state wave function for

F[ρ] = min [ψ, (T + V )ψ] ,

(20.34)

with the minimum being taken over all properly antisymmetric normalized ψ’s which yield that ρ. A great deal of effort has been devoted to trying to find approximate representations of the universal functional F[ρ]. One route is mathematical, and features a careful exploration of the abstract properties which F[ρ] must have. Another route is numerical, and can be characterized as involving the guessing of some ansatz with a general resemblance to Thomas– Fermi–von Weizsäcker–Dirac theory, with some flexible parameters which are determined by least-squares fitting of the energies resulting from insertion of Hartree– Fock densities into the trial functional to theoretical Hartree–Fock energies, or the like. If, however, the basic ansatz exhibits unphysical features in the case of negatively charged ions or heteronuclear molecules, it is not likely that the optimization of parameters in that ansatz will get one closer to the true universal density functional. In the opinion of this writer, significant progress in density functional theory based solely upon the one-electron density is likely to require a major revolution in our mathematical understanding of this field, with a useful procedure made explicit for constructing progressively better approximations to the universal density functional, which, like π or other transcendental numbers, probably will never be written down exactly in closed form. Moreover, the numerical solution of the highly nonlinear Euler–Lagrange

Part B 20.3

In 1964 Hohenberg and Kohn [20.21] argued that there exists a universal density functional F[ρ], independent of the external potential V(r), such that minimization of the sum  F[ρ] + ρ(r)V(r) d r , (20.32)

a system of electrons moving in some external potential V(r) (the so-called v-representability problem, since Hohenberg and Kohn used v in place of V ). The n-representability problem was solved by Gilbert [20.90] and by Harriman [20.91], who gave a prescription for starting from an arbitrary wellbehaved ρ and from it constructing a many-electron wave function ψ which generated that ρ [20.33,92]. The v-representability problem is much more formidable, as demonstrated by the discovery that there are wellbehaved densities ρ which are not the ground-state densities for any fermionic system in an external potential V [20.92, 93]. Following Percus’ definition of a universal kinetic energy functional for independent fermion systems [20.94], Levy [20.95] proposed to circumvent this v-representability problem by modifying the definition of F[ρ] so that instead of being defined in terms of densities which might not be v-representable, it is defined as

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equations for a very complicated density functional is likely to require large amounts of computer time, as well as problems with landing in local minima of the energy.

20.3.2 The Kohn–Sham Method for Including Exchange and Correlation Corrections

Part B 20.3

Density functional theory posed solely in terms of the one-electron density and based upon the Hohenberg–Kohn variational principle provides no general procedure for accurately calculating relatively small energy differences such as excitation energies, ionization potentials, electron affinities, or the binding energies of molecules. There is, however, a powerful method inspired by the Hohenberg–Kohn variational principle, which has been used with great success in the calculation of such quantities. This is the Kohn–Sham variational method [20.22]. The key idea in the Kohn–Sham variational method is to replace the nonlocal exchange term in the Hartree– Fock equations with an exchange-correlation potential, which at least in principle can be used to determine energies exactly. The oldest, simplest, and most common ansatz used for the exchange-correlation potential involves the local density approximation (LDA), in which one assumes that the exchange-correlation potential for the actual system under study has the same functional form as does the exchange-correlation potential for a uniform interacting gas of electrons. If the density is not too small or not too rapidly varying, the exchange part of this potential can be approximated by ρ1/3 , which appears in Dirac’s first-order approximation for exchange energies, with a systematic procedure for deriving higher-order corrections in a gradient expansion. The correlation part of this potential is accurately known from Ceperley and Alder’s quantum Monte Carlo calculation of the properties of the uniform electron gas [20.96]. One therefore retains the important features of the quantum theory based on wave functions, with a determinantal approximation to ψ, while approximately including exchange and correlation effects through a simply computable effective potential. Higher corrections, which are important for quantitative accuracy, can be incorporated by taking account of the variation of ρ by means of a gradient expansion [20.27] involving ∇ρ and higher derivatives [20.89, Chapt. 7], thus yielding a generalized gradient approximation (GGA) for the exchange-correlation potential.

The Kohn–Sham procedure has become the backbone for the vast majority of accurate calculations of the electronic structure of solids [20.72, 86]. In the 1990’s, motivated by Becke’s work on constructing simple gradient-corrected exchange potentials [20.97–103], and incorporating the Lee, Yang, and Parr (LYP) expression for the correlation potential [20.104] derived from Colle and Salvetti’s correlation-energy formula [20.105– 107], the Kohn–Sham method is finding increasing application in efficiently estimating relatively small energy differences of relevance to chemistry [20.108–110] (However, Becke’s gradient-corrected exchange potential does not have the correct 1/r behavior at large r, as was observed by several authors [20.111–113]). For definitive results, however, one must still resort to an ab initio theory which at least in principle converges toward the correct result. The generation of improved generalized gradient approximations has recently become a growth industry, with increasingly many proposals of increasingly greater complexity [20.97, 104, 113–125]. Inevitably, some expressions work better for some properties than for others. It is found that usually most of the errors in the long-range tails of the exchange and correlation potentials tend to cancel each other, thus leading to better overall energies than one could reasonably expect [20.126]. Under these circumstances, it is important to have benchmarks for testing the accuracy of the various approximations. Such comparisons have been carried out for two important sets of two-electron systems [20.127–129]: 1. a pair of electrons moving in harmonic potential wells and coupled by the Coulomb repulsion, which yields an exactly solvable system; 2. helium-like ions of variable nuclear charge Z, for which extremely accurate energies and wave functions are available which take account of the behavior of the exact but unknown wave function in the vicinity of all two-particle coalescences and the three-particle coalescence. The results indicate that the approximate exchangecorrelation potentials differ quite considerably from the true exchange-correlation potentials, thus indicating the need for further analytical work in understanding how to design accurate exchange-correlation potentials, and for devising tests of exchange-correlation potentials for larger atoms and for molecules. Another important way of testing the validity of various approximate exchange and correlation potentials is checking whether they obey inequalities

Thomas–Fermi and Other Density-Functional Theories

imposed by such general properties as scaling and the Hellmann–Feynman theorem. Such general tests have been devised by Levy and his co-workers [20.130– 139], who have found that many of the commonly used approximate potentials violate general inequalities which must be obeyed by the exact potential. These abstract results are helpful in designing potentials which should be better approximations to the true potential.

20.3.3 Density Functional Theory for Excited States The Hohenberg–Kohn theorem and the Kohn–Sham method were originally formulated in terms of the ground electronic state. These techniques can be extended to calculate the ground state of a given symmetry [20.140], but that leaves unresolved the issue of using density functional theory to calculate the en-

20.4 Recent Developments

ergies of excited states for a given symmetry. Using the Rayleigh–Ritz principle for ensembles, general abstract procedures for generalizing density functional theory to excited state calculations have been formulated by Theophilou [20.141] and by several other workers [20.142–151]. Unfortunately, the errors typically seem to be much larger than for ground-state density functional theory.

20.3.4 Relativistic and Quantum Field Theoretic Density Functional Theory At a formal level, one can discuss the development of density functional theory for a relativistic system of electrons. For an overview of this challenging subject, see the discussions by Dreizler and Gross [20.89, Chapt. 8] and by Dreizler [20.152]. Much of the formalism carries over, but no good way has yet been found of incorporating vacuum polarization corrections.

tional theory for chemical systems have been surveyed in two very recent review articles [20.162, 163]. Although the locality of DFT was proved for a large class of functionals [20.164–166], this issue has come under recent dispute. The question that has been raised is whether there exists an exact Thomas-Fermi model for non-interacting electrons. If such an exact model does not exist, as it is a direct consequence of the HohenbergKohn theorem, then DFT would be incomplete. Nesbet [20.167–171] has argued that such a theory would be inconsistent with the Pauli exclusion principle for atoms of more than two electrons (or for a two electron atom where both electrons are in the same spin state). The contention is that if only the total electron density were normalized (which corresponds to only one Lagrange multiplier), as in the TF model, then no shell structure can exist; hence such a system would violate the exclusion principle. A counter-example has recently been constructed by Lindgren and Salomonson [20.172] showing that shell structure can indeed be generated through a single Lagrange multiplier. In addition, they have verified numerically that a local Kohn-Sham potential can reproduce to high accuracy the many-body electron density and the 2s eigenvalue for the 1s2s 3 S state of neutral helium.

Part B 20.4

20.4 Recent Developments During the last eight years there has continued to be exponentially growing interest in applications of density functional theory of the Kohn-Sham variety to atoms and molecules, especially those of chemical relevance which are too large for accurate ab initio electronic structure calculations. The awarding of the 1998 Nobel Prize in Chemistry to Walter Kohn and John A. Pople recognised their individual contributions to this increasingly important field. Their Nobel lectures were published the following year in the Reviews of Modern Physics [20.153, 154]. Since a comprehensive summary of the wideranging developments in density functional theory during the past decade is not feasible within the limited space available for this supplementary section, I will briefly cite some of the most extensive surveys of various aspects of this field that have appeared since 1995. Many aspects of density functional theory were reviewed in four consecutive volumes of Topics in Current Chemistry published in 1996 [20.155], and in 1999 an entire volume of Advances in Quantum Chemistry was devoted to density functional theory [20.156]. This has also been the subject of several conference proceedings [20.157–159] and introductory textbooks [20.160, 161]. Developments in time-dependent density func-

303

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References 20.1 20.2 20.3 20.4 20.5 20.6 20.7 20.8 20.9 20.10 20.11 20.12 20.13 20.14 20.15 20.16 20.17 20.18

Part B 20

20.19

20.20 20.21 20.22 20.23 20.24 20.25 20.26 20.27 20.28 20.29

20.30 20.31 20.32

20.33

20.34 20.35 20.36

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20.37 20.38 20.39 20.40 20.41 20.42 20.43 20.44 20.45 20.46 20.47 20.48 20.49 20.50 20.51 20.52 20.53 20.54 20.55 20.56 20.57 20.58 20.59 20.60 20.61 20.62 20.63 20.64 20.65 20.66 20.67 20.68 20.69 20.70 20.71 20.72 20.73

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20.78 20.79

20.80

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20.85

20.86

20.87 20.88

20.89

20.90 20.91 20.92

20.93 20.94 20.95 20.96 20.97

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References

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Atoms

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Vol. 21, ed. by S. B. Trickey (Academic, New York 1990) pp. 69–95 M. Levy: In: Density Functional Methods in Chemistry, ed. by J. K. Labanowski, J. W. Andzelm (Springer, New York 1991) pp. 175–193 A. Görling, M. Levy: Phys. Rev. A 45, 1509 (1992) Q. Zhao, M. Levy, R. G. Parr: Phys. Rev. A 47, 918 (1993) A. Görling, M. Levy: Phys. Rev. B 47, 13105 (1993) M. Levy, J. P. Perdew: Phys. Rev. B 48, 11638 (1993) A. Görling, M. Levy: Phys. Rev. A 50, 196 (1994) M. Levy, A. Görling: Philos. Mag. B 69, 763 (1994) O. Gunnarson, B. J. Lundqvist: Phys. Rev. B 13, 4274 (1976) A. K. Theophilou: J. Phys. C 12, 5419 (1979) N. Hadjisavvas, A. Theophilou: Phys. Rev. A 30, 2183 (1984) N. Hadjisavvas, A. K. Theophilou: Philos. Mag. B 69, 771 (1994) J. Katriel: J. Phys. C 13, L375 (1980) A. Theophilou: Phys. Rev. A 32, 720 (1985) W. Kohn: Phys. Rev. A 34, 737 (1986) E. K. U. Gross, L. N. Oliveira, W. Kohn: Phys. Rev. A 37, 2805, 2809 (1988) L. N. Oliveira, E. K. U. Gross, W. Kohn: Phys. Rev. A 37, 2821 (1988) and earlier references therein H. Englisch, H. Fieseler, A. Haufe: Phys. Rev. A 37, 4570 (1988) L. N. Oliveira: In: Density Functional Theory of Many-Fermion Systems, Advances in Quantum Chemistry, Vol. 21, ed. by S. B. Trickey (Academic, New York 1990) pp. 135–154 L. N. Oliveira, E. K. U. Gross, W. Kohn: Int. J. Quantum Chem. Symp. 24, 707 (1990) R. M. Dreizler: Phys. Scr. T46, 167 (1993) W. Kohn: Rev. Mod. Phys. 71, 1253 (1999) J. A. Pople: Rev. Mod. Phys. 71, 1267 (1999) R. F. Nalewajski (Ed.): Functionals and Effective Potentials I; Relativistic and Time Dependent Ex-

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307

21. Atomic Structure: Multiconfiguration Hartree–Fock Theories

Atomic Structu This chapter outlines variational methods for the determination of wave functions either in nonrelativistic LS or relativistic LSJ theory. The emphasis is on Hartree–Fock and multiconfiguration Hartree–Fock theory though configuration interaction methods are also mentioned. Some results from the application of these methods to a number of atomic properties are presented.

Hamiltonians: Schrödinger and Breit–Pauli................. 307

21.2

Wave Functions: LS and LSJ Coupling ..... 308

21.3

Variational Principle ............................ 309

21.4

Hartree–Fock Theory ............................ 21.4.1 Diagonal Energy Parameters and Koopmans’ Theorem ........... 21.4.2 The Fixed-Core Hartree–Fock Approximation.......................... 21.4.3 Brillouin’s Theorem ...................

309 311 311 311

21.5

Multiconfiguration Hartree–Fock Theory ............................ 21.5.1 Z-Dependent Theory ................. 21.5.2 The MCHF Approximation............ 21.5.3 Systematic Methods................... 21.5.4 Excited States ........................... 21.5.5 Autoionizing States ...................

313 313 314 315 316 316

21.6 Configuration Interaction Methods........ 316 21.7

Atomic Properties ................................ 21.7.1 Isotope Effects .......................... 21.7.2 Hyperfine Effects ....................... 21.7.3 Metastable States and Lifetimes ........................... 21.7.4 Transition Probabilities .............. 21.7.5 Electron Affinities......................

318 318 319 320 321 321

21.8 Summary ............................................ 322 References .................................................. 322

21.1 Hamiltonians: Schrödinger and Breit–Pauli The state of a many-electron system is described by a wave function Ψ that is the solution of a partial differential equation (called the wave equation), (H − E)Ψ = 0 ,

(21.1)

where H is the Hamiltonian operator for the system and E the total energy. The operator H depends on the system (atomic, molecular, solid-state, etc.) as well as the underlying quantum mechanical formalism (nonrelativistic, Breit–Pauli, Dirac–Coulomb, or Dirac–Breit, etc.). In atomic systems, the Hamiltonian of the nonrelativistic Schrödinger equation is (in atomic units) Hnr = −

1 2

N 


∇i2 +

i=1

2Z ri

 +


1 . rij

(21.2)

i< j

Here Z is the nuclear charge of the atom with N electrons, ri is the distance of electron i from the nucleus,

and rij is the distance between electron i and electron j. This equation was derived under the assumption of a point-nucleus of infinite mass. The term 2Z/r represents the nuclear attraction and 1/rij the inter-electron repulsion. The operator Hnr has both a discrete and continuous spectrum: for the former, Ψ(r1 , r2 , . . . , r N ) has a probability interpretation and consequently must be square integrable. In the Breit–Pauli approximation, the Hamiltonian is extended to include relativistic corrections up to relative order (αZ)2 . It is convenient to write the Breit–Pauli Hamiltonian as the sum [21.1] HBP = Hnr + Hrel ,

(21.3)

where Hrel represents the relativistic contributions. The latter may again be subdivided into nonfine-structure (NFS) and fine structure (FS) contributions: Hrel = HNFS + HFS .

(21.4)

Part B 21

21.1

21.4.4 Properties of Hartree–Fock Functions ................................. 312

308

Part B

Atoms

The NFS contributions HNFS = Hmass + HD + Hssc + Hoo

(21.5)

shift nonrelativistic energy levels without splitting the levels. The mass-velocity term α2
4 ∇i (21.6) Hmass = − 8 i

corrects for the variation of mass with velocity; the oneand two-body Darwin terms HD = −

α2 Z
2 −1 α2
2 −1 ∇i ri + ∇i rij 8 4 i

(21.7)

i< j

are the corrections of the one-electron Dirac equation due to the retardation of the electromagnetic field produced by an electron; the spin–spin contact term 8πα2
Hssc = − (si · s j )δ(rij ) (21.8) 3 i< j

Part B 21.2

accounts for the interaction of the spin magnetic moments of two electrons occupying the same space; the orbit–orbit interaction   α2
pi · p j rij (rij · pi ) · p j Hoo = − + (21.9) 2 rij rij3 i< j

accounts for the interaction of two orbital moments. The FS contributions HFS = Hso + Hsoo + Hss ,

(21.10)

split the nonrelativistic energy levels into a series of closely-spaced fine structure levels. The nuclear spin– orbit interaction Hso =

α2 Z
1 (li · si ) , 2 ri3 i

(21.11)

represents the interaction of the spin and angular magnetic moments of an electron in the field of the nucleus. The spin–other-orbit term   α2
rij Hsoo = − × pi · (si + 2s j ) , (21.12) 2 rij3 i= j and the spin–spin term  
1 3 2 si · s j − 2 (si · rij )(s j · rij ) , Hss = α rij r3 i< j ij (21.13)

arise from spin-dependent interactions with the other electrons in the system.

21.2 Wave Functions: LS and LSJ Coupling In the configuration interaction model, the approximate wave function Ψ for a many-electron system is expanded in terms of configuration state functions (CSF). The assignment of nl quantum numbers to electrons specifies a configuration, often written as (n 1l1 )q1 (n 2l2 )q2 · · · (n m lm )qm , where qi is the occupation of subshell (n i li ). Associated with each subshell are one-electron spin-orbitals φ(r, θ, ϕ, σ) = (1/r)Pnl (r)Ylml (θ, ϕ)χm s (σ) , where Pnl (r) is the radial function, Ylml (θ, ϕ) a spherical harmonic, and χm s (σ) a spinor. Each CSF is a linear combination of products of one-electron spin-orbitals, one for each electron in the system, such that the sum is an eigenfunction of the total angular momenta operators L 2 , L z and the total spin operators S2 , Sz . It can be considered to be a product of radial factors, one for each electron, an angular and a spin factor obtained

by vector coupling methods. It also is required to be antisymmetric with respect to the interchange of any pair of electron co-ordinates. Often, the specification of the configuration and the final L S quantum numbers is sufficient to define the configuration state, but this is not always the case. Additional information about the order of coupling or the seniority of a subshell of equivalent electrons may be needed. Let γ specify the configuration information and any additional information about coupling to uniquely specify the configuration state function denoted by Φ(γL S). The wave function for a many-electron system is usually labeled in the same manner as a CSF and generally designates the largest component. Thus, in the L S approximation, Ψ(γL S) =

M
α=1

cα Φ(γα L S) .

Atomic Structure: Multiconfiguration Hartree–Fock Theories

However, cases are known where the configuration states are so highly mixed that no dominant component can be found. Then the assignment is made using other criteria. Clearly no two states should have the same label. In the L SJ scheme, the angular and spin momenta are coupled to form an eigenstate of the total mo-

21.4 Hartree–Fock Theory

309

menta J 2 , Jz . The label often still includes an L S designation, as in 2p 23 P 2 , but only the subscript J is a good quantum number. Thus, Ψ(γL SJ ) =

M


cα Φ(γα L α Sα J ) .

(21.14)

α=1

21.3 Variational Principle Variational theory shows the equivalence between solutions of the wave equation, (H − E)ψ = 0, and stationary solutions of a functional. For bound states where approximate solutions Ψ are restricted to a square ˜ the best solutions are those integrable subspace, say H, for which the energy functional E(Ψ) = Ψ |H|Ψ /Ψ |Ψ 

(21.15)

is stationary. The condition δE(Ψ ) = 0 leads to ˜ E = E(Ψ) . δΨ |H − E|Ψ  = 0, ∀ δΨ ∈ H, (21.16)

Several results readily follow. The eigenvalues of H are bounded from below. Let E 0 ≤ E 1 ≤ · · · . Then E 0 ≤ E(Ψ), ∀ Ψ ∈ H˜ .

(21.17)

Consequently, for any approximate wave function, the computed energy is an upper bound to the exact lowest eigenvalue. By the Hylleraas–Undheim– MacDonald theorem (see Sect. 11.3.1) the computed excited states are also upper bounds to the exact eigenvalues, provided that the correct number of states lies below.

In the Hartree–Fock (HF) approximation, the approximate wave function consists of only one configuration state function. The radial function of each spin–orbital is assumed to depend only on the nl quantum numbers. These are determined using the variational principle and the nonrelativistic Schrödinger Hamiltonian. The energy functional can be written as an energy expression for the matrix element Φ(γL S)|H|Φ(γL S). Racah algebra may be used to evaluate the spin– angular contributions, resulting in two types of radial integrals: One-Body Let L be the differential operator

L=

d2 2Z ( + 1) − + . r dr 2 r2

×

0 k r< k+1 r>

P(c; r1 )P(d; r2 ) dr1 dr2 , (21.21)

∞

which is called a Slater integral. It has the symmetries

P(nl; r) 0

where θ is the angle between the vectors r1 and r2 , and r< , r> are the lesser and greater of r1 , r2 , respectively. In general, let a, b, c, d be four nl quantum numbers, two from the left (bra) and two from the right (ket) CSF. Then ∞ ∞ k R (ab, cd) = P(a; r1 )P(b; r2 ) 0

(21.18)

Then, 1 I(nl, n l  ) = − 2

Two-Body The other integrals arise from the multipole expansion of the two-electron part
rk 1 < = P k (cos θ) , (21.20) k+1 r12 r > k

Rk (ad, cb) ≡ Rk (cb, ad) ≡ Rk (cd, ab)  

× LP(n l ; r) dr .

(21.19)

≡ Rk (ab, cd) .

Part B 21.4

21.4 Hartree–Fock Theory

310

Part B

Atoms

In the Hartree–Fock approximation, the Slater integrals that occur depend on only two sets of quantum numbers. These special cases are denoted separately as

function ∞ Y (ab; r) = r

F (a, b) ≡ R (ab, ab) and k

k

G k (a, b) ≡ Rk (ab, ba) .

0 r

(21.22)

=

m
qi I(n i li , n i li )

+

qi q j

P(a; s)P(b; s) ds

r

∞ F (a, b) = k

k=0 2 min(li ,l j )

j

r

so that

2li qi − 1
k f k (i, i)F (n i li , n i li ) + 2


 s k

P(a; s)P(b; s) ds

∞  k+1 r P(a; s)P(b; s) ds (21.25) + s

i




k+1 r>

0

The former is the direct interaction between a pair of orbitals whereas the latter arises from the exchange operator. The energy expression may be written as E(γL S) =

k r
0 =− 4li + 1 0 0 0 f k (i, j ) = 1, i = j and k = 0 , = 0, i = j and k > 0 , 2   2li + 1 li k l j gk (i, j ) = − . 4li + 1 0 0 0

(21.26)

0

Then the variations are 1 δI(a, b) = − (1 + δa,b ) 2

∞ δP(b; r)LP(b; r) dr , 0

∞ k δF (a, b) = 2(1 + δa,b ) δP(a; r)P(a; r) 0   1 k Y (bb; r) dr , × r   ∞ 1 k Y k (ab; r) dr . δG (a, b) = 2 δP(a; r)P(b; r) r 0

(21.28)

(21.24)

Let (Pa ) be an integral that depends on Pa . Then δ is defined as the first-order term of (Pa + δPa ) − (Pa ). To derive the first-order variation of the F k and G k integrals (and Rk in general) it is convenient to replace the variables (r1 , r2 ) by (r, s) and introduce the

The part of the expression that depends on P(n i li ; r), for example, is the negative of the removal energy of the entire (n i li )qi subshell, say −E¯ [(n i li )qi ]. The stationary condition for a Hartree–Fock solution applies to this expression, but since the variations must be constrained in order to satisfy orthonormality assumptions, Lagrange multipliers λij need to be introduced. The stationary condition applies to the functional 

F [P(n i li )] = − E¯ (n i li )qi
δli ,l j λij P(n i li )|P(n j l j ) . + j

(21.29)

Atomic Structure: Multiconfiguration Hartree–Fock Theories

Applying the variational conditions to each of the integrals, and dividing by −qi , we get the equation  2  d 2 l(l + 1) [Z − Y(n + l ; r)] − − ε i i ii dr 2 r r2 × P(n i li ; r)
2 δli ,l j εij P(n j l j ; r) , (21.30) = X(n i li ; r) + r j=i

where Y(n i li ; r) = (qi − 1) +







qj

j=i

f k (i, i)Y k (n i li n i li ; r)

k k

q j Y (n j l j n j l j ; r) ,

j=i

X(n i li ; r) =







gk (i, j )Y k (n i li n j l j ; r)

k

× P(n j l j ; r) .

(21.31)

21.4.1 Diagonal Energy Parameters and Koopmans’ Theorem

k

¯ i li )qi ] is the Hartree–Fock value of the rewhere E[(n moval energy functional E¯ [(n i li )qi ]. In the special case where qi = 1, εii is twice the removal energy, or ionization energy. This is often referred to as Koopmans’ Theorem; but, as discussed in Sect. 21.4.3, if a rotation of the radial basis leaves the wave function unchanged while transforming the matrix of energy parameters (εij ), the removal energies are extreme values obtained by setting the off-diagonal energy parameters to zero. For multiply occupied shells, εii /2 can be interpreted as an average removal energy, with a correction arising from the self-interaction.

21.4.2 The Fixed-Core Hartree–Fock Approximation The above derivation has assumed that the solution is stationary with respect to all allowed variations. In practice, it may be convenient to assume that certain radial functions are “fixed” or “frozen”. In other words, these

311

radial functions are assumed to be given. Such approximations are often made for core orbitals and so, this is called a fixed-core HF approximation.

21.4.3 Brillouin’s Theorem The Hartree–Fock approximation has some special properties not possessed by other single configuration approximations. One such property is referred to as satisfying Brillouin’s theorem, though, in complex systems with multiple open shells of the same symmetry, Brillouin’s theorem is not always obeyed. Let Φ HF (γ L S) be a Hartree–Fock configuration state, where γ denotes the configuration and coupling scheme. With Φ HF (γ L S) are associated the m Hartree– Fock radial functions P HF (n 1l1 ; r), P HF (n 2l2 ; r), . . . , P HF (n m lm ; r). These radial functions define the occupied orbitals. To this set may be added virtual orbitals that maintain the necessary orthonormality conditions. Let one of the radial functions (nl) be replaced by another (n l), either occupied or virtual, without any change in the coupling of the spin–angular factor. Let the resulting function be denoted by F(nl → n l). The perturbation of the Hartree–Fock radial function, P(nl; r) → P HF (nl; r) + P(n l; r) induces a perturbation Φ HF (γ L S) → Φ HF (γ L S) + F(nl → n l). But the Hartree–Fock energy is stationary with respect to such variations and so, Φ HF (γ L S) | H | F(nl → n  l) = 0 .

(21.33)

If the function F(nl → n l) is a CSF for a configuration γ ∗ , or proportional to one, then Brillouin’s theorem is said to hold between the two configuration states. When n l is a virtual orbital, it may happen that F(nl → n  l) is a linear combination of CSFs, as in the 2p → 3p replacement from 2p 32 P, yielding a linear combination of {2p 2 (1 S)3p, 2p 2 (3 P)3p, 2p(1 D)3p}, the linear combination being determined by coefficients of fractional parentage. Thus, Brillouin’s theorem will not hold for any of the three individual configuration states in the above equation, only for the linear combination. When perturbations are constrained by orthogonality conditions between occupied orbitals, the perturbation is of the form of a rotation, where both are perturbed simultaneously, P(nl; r) → P(nl; r) + P(n l; r) , P(n l; r) → P(n l; r) − P(nl; r) .

(21.34)

Then the perturbation has the form F(nl → n l, n l → −nl). For 1s 2 2s 2 S, the simultaneous perturbations,

Part B 21.4

The diagonal (εii ) and off-diagonal (εij ) energy parameters are related to the Lagrange multipliers by εii = 2λii /qi and εij = λij /qi . In fact,  2 εii = E¯ (n i li )qi qi
− (qi − 1) f k (i, i)F k (n i li , n i li ) (21.32)

21.4 Hartree–Fock Theory

312

Part B

Atoms

F(1s → 2s, 2s → −1s), lead to a linear combination of {1s2s 22 S, 1s 32 S}. The CSF for the 1s 32 S is identically zero by antisymmetry, and so Brillouin’s theorem holds for the lithium-like ground state. In the 1s2s 3 S state, neither the 1s → 2s nor the 2s → 1s substitutions are allowed; in fact, it can be shown that for these states, Brillouin’s theorem holds for all mono-excited configurations. The same is not true for 1s2s 1 S where the simultaneous perturbations lead to the condition      Φ HF (1s2s 1 S)|H| Φ 1s 2 − Φ 2s 2 =0. (21.35)

Thus,  Brillouin’s   theorem is not obeyed for either the Φ 1s 2 or Φ 2s 2 CSF in an HF calculation for 1s2s. The importance of Brillouin’s theorem lies in the fact that certain interactions have already been included to first order. This has the consequence that certain classes of diagrams can be omitted in many-body perturbation theory [21.2].

21.4.4 Properties of Hartree–Fock Functions

Part B 21.4

Term Dependence The radial distribution for a given nl orbital may depend significantly on the L S term. A well known example is the 1s 2 2s2p configuration in Be which may couple to form either a 3 P or 1 P term. The energy expression differs only in the exchange interaction, ±(1/3)G 1 (2s, 2p), where the + refers to 1 P and the − to 3 P. Clearly, the energies of these two terms differ. What is not quite as obvious is the extent to which the P(2p) radial functions differ for the two states. The most affected orbital is the one that is least tightly bound, which in this case is the 2p orbital. Figure 21.1 shows the two radial functions. The 1 P orbital is far more diffuse (not as localized) as the one for 3 P. Such a change in an orbital is called L S term dependence. Orbital Collapse Another phenomenon, called orbital collapse, occurs when an orbital rapidly contracts as a function of the energy. This could be an L S dependent effect, but it can also occur along an isoelectronic sequence. This effect is most noticeable in the high-l orbitals. In hydrogen, the mean radius of an orbital is r = (1/2)[3n 2 − l(l + 1)] a0 . Thus, the higher-l orbitals are more contracted; but in neutral systems, the high-l orbitals have a higher energy and are more diffuse. This is due, in part, to the l(l + 1)/r 2 angular momentum barrier that appears in the definition of the L operator.

P(r) 3

P

0.5 0.4 0.3 1

P

0.2 0.1 0 0

5

10

15

20 Radius (a. u.)

Fig. 21.1 A comparison of the 2p Hartree–Fock radial functions for the 1s2p 1,3 P states of Be

In the Hartree model, it is possible for V(r) + l(l + 1)/r 2 to have two wells: an inner well and an outer shallow well. As the lowest eigenfunction changes rapidly from the outer well to the inner well, as Z changes, orbital collapse is said to occur. Quantum Defects and Rydberg Series Spectra of atoms often exhibit phenomena associated with a Rydberg series of states where one of the electrons is in an nl orbital, with n assuming a sequence of values. An example is the 1s 2 2snd 3 D series in Be, n = 3, 4, 5, . . . . For such a series, an useful concept is that of a quantum defect parameter δ. In hydrogen, the ionization energy (IP ) in atomic units is 1/(2n 2 ). In complex neutral systems, the effective charge would be the same at large r. As n increases, the mean radius becomes larger and the probability of the electron being in the hydrogen-like potential increases. Thus, one could define an effective quantum number, n ∗ = n − δ, such that (21.36) IP (nl) = (1/2)/(n − δ)2 . Table 21.1 The effective quantum number and quantum defect parameters of the 2snd Rydberg series in Be 3D

1D

n

n∗

δ(nl)

n∗

δ(nl)

3 4 5 6

2.968 3.960 4.957 5.955

0.032 0.040 0.043 0.045

3.014 4.012 5.013 6.013

−0.014 −0.012 −0.013 −0.013

Atomic Structure: Multiconfiguration Hartree–Fock Theories

For ionized systems, the equation must be modified to IP (nl) = (1/2)[(Z − N + 1)/(n − δ)]2 . Often, this parameter is defined with respect to observed data, but it can also be used to evaluate Hartree–Fock energies, where IP = εnl,nl /2, so that εnl,nl = [(Z − N + 1)/(n − δ)]2 . Table 21.1 shows the effective quantum number and quantum defect for the

21.5 Multiconfiguration Hartree–Fock Theory

313

Hartree–Fock 2snd 3 D and 1 D orbitals in Be as a function of n. For the triplet part the quantum defect is positive whereas for the singlet it is negative. This is the effect of exchange. Note that as n increases the quantum defect becomes constant. This observation is often the basis for determining ionization potentials from observed data.

21.5 Multiconfiguration Hartree–Fock Theory

E corr = E exact − E HF .

(21.37)

In this definition, E exact is not the observed energy – it is the exact solution of Schrödinger’s equation which itself is based on a number of assumptions.

  1
2 2 H0 = − ∇i + , 2 ρ i
1 V= , ρij

(21.39)

(21.40)

i> j

and Schrödinger’s equation becomes   H0 + Z −1 V ψ = Z −2 E ψ .

(21.41)

With Z −1 V regarded as a perturbation, the expansions of ψ and E in the powers of Z −1 are ψ = ψ0 + Z −1 ψ1 + Z −2 ψ2 + · · · ,

(21.42)

in the ρ unit of length, and  E = Z 2 E 0 + Z −1 E 1 + Z −2 E 2 + Z −3 E 3 + · · · . (21.43)

The zero-order equation is H0 ψ0 = E 0 ψ0 .

(21.44)

The solutions of this equation are products of hydrogenic orbitals. Table 21.2 Observed and Hartree–Fock ionization potentials for the ground states of neutral atoms, in eV. (See also Table 10.3.)

21.5.1 Z-Dependent Theory An indication of the important correlation corrections can be obtained from a perturbation theory study of the exact wave function. In the following section, we follow closely the approach taken by Layzer et al. [21.4] in the study of the Z-dependent structure of the total energy. Let us introduce a new scaled length, ρ = Zr. Then the Hamiltonian becomes   H = Z 2 H0 + Z −1 V ,

where

(21.38)

Atom

Obs.

HF

Diff.

Li Be B C N O F Ne

5.39 9.32 8.30 11.26 14.53 13.62 17.42 21.56

5.34 8.42 8.43 11.79 15.44 14.45 18.62 23.14

0.05 0.90 −0.13 −0.53 −0.91 −0.85 −1.20 −1.58

Part B 21.5

The Hartree–Fock method predicts many atomic properties remarkably well; but when analyzed carefully, systematic discrepancies can be observed. Consider the ionization potentials tabulated in Table 21.2 compared with the observed values. In these calculations, the energy of the ion was computed using the same radial functions as for the atom. Thus, no “relaxation” effects were included. The observed data include other effects as well, such as relativistic effects, finite mass and volume of the nucleus, but these are small for light atoms. For these systems, the largest source of discrepancy arises from the fact that the Hartree–Fock solution is an independent particle approximation to the exact solution of Schrödinger’s equation. Neglected entirely is the notion of “correlation in the motion of the electrons”; each electron is assumed to move independently in a field determined by the other electrons. For this reason, the error in the energy was defined by Löwdin [21.3], to be the correlation energy, that is,

314

Part B

Atoms

Let |(nl)νL S be a configuration state function constructed by vector coupling methods from products of hydrogenic orbitals. Here (nl) represents a set of N quantum numbers (n 1l1 , n 2 l2 , . . . , n N l N ) and ν any additional quantum numbers such as the coupling scheme or seniority needed to distinguish the different configuration states. Then, H0 |(nl)νL S = E 0 |(nl)νL S , 1
1 E0 = − . 2 n i2 i

(21.45) (21.46)

Part B 21.5

Since E 0 is independent of the li , different configurations may have the same E 0 ; that is, E 0 is degenerate. According to first-order perturbation theory for degenerate states, ψ0 then is a linear combination of the degenerate configuration state functions |(nl  )ν L S with the same set of principal quantum numbers n i and parity π. The coefficients are components of an eigenvector of the interaction matrix (nl  )ν L S|V |(nl)νL S, and E 1 is the corresponding eigenvalue. This is the set of configurations referred to as the complex by Layzer [21.5] and denoted by the quantum numbers (n)πL S. The zero-order wave function ψ0 describes the many-electron system in a general way. It can be shown that the square of the expansion coefficients of ψ0 over the degenerate set of configuration states can be interpreted as a probability that the many-electron system is in that configuration state, that ψ1 is then a weighted linear combination of first-order corrections to each such configuration state. Let us now assume the nondegenerate case where ψ0 = Φ (γ L S). The configurations interacting with γ L S are of two types: those that differ by a single electron (single substitution S) and those that differ by two electrons (double substitution D). The former can be further subdivided into three categories: (i) Those that differ from γ L S by one principal quantum number but retain the same spin–angular coupling. These configuration states are part of radial correlation. (ii) Those that differ by one principal quantum number but differ in their coupling. If the only change is the coupling of the spins, the configuration states are part of spin-polarization. (iii) Those that differ in the angular momentum of one electron and are accompanied by a change in angular coupling of the configuration state and possibly also the spin coupling. The latter represent orbital-polarization. The sums over intermediate states involve infinite sums. In practice, the set of orbitals is finite. In the nondegenerate case, these orbitals can be divided into occupied orbitals and unoccupied, or virtual, orbitals

depending on whether or not they occur in the reference configuration that defines γ L S. Single and double (SD) replacements of occupied orbitals by other occupied or virtual orbitals generate the set of configurations that interact with ψ0 . Consider the 1s 2 2s ground state of Li and the {1s, 2s, 3s, 4s, 2p, 3p, 4p, 3d, 4d, 4f } set of orbitals. The 1s and 2s orbitals are occupied and all the other orbitals are virtual orbitals, vl. The set of replacements can then be classified as follows: Replacement

Configuration

Type of correlation

1s → 2s

1s2s 2

2s → vs 1s → vs

1s 2 vs 1svs(1 S)2s 1svs(3 S)2s 1svlv l 2svlv l

Radial and spin-polarization Radial Radial Spin-polarization Core-polarization Core

1s2s → vlv l 1s 2 → vlv l

The above discussion has considered only the Z-dependence of the wave function, but the notion can readily be extended to other properties. For example, in transition studies, the dipole transition matrix element decreases as 1/Z, whereas the transition energy increases linearly with Z for ∆n = 0 transitions, and quadratically as Z 2 otherwise. A first-order theory for oscillator strengths (FOTOS) [21.6] is based on similar concepts.

21.5.2 The MCHF Approximation In the multiconfiguration Hartree–Fock (MCHF) method, the wave function is approximated by a linear combination of orthogonal configuration states so that m
Ψ(γ L S) = ci Φ (γi L S) , (21.47) i

where m


ci2 = 1 .

i

Then the energy expression becomes m
m 


ci c j Hij E Ψ (γ L S) = i

(21.48)

j

where

     Hij = Φ (γi L S) H Φ γ j L S .

(21.49)

Because Hij = H ji , the sum over i, j may be limited to the diagonals and the lower part of the matrix H = (Hij ),

Atomic Structure: Multiconfiguration Hartree–Fock Theories

called the interaction matrix. Let c = (ci ) be a column vector of the expansion coefficients, also called mixing coefficients. Then the energy of the system is E = cT Hc .

(21.50)

Let P be the column vector of radial functions, (Pa , Pb , . . . )T . Since the interaction matrix elements depend on the radial functions, it is clear that the energy functional depends on both P and c. In deriving the MCHF equations, the energy needs to be expressed in terms of the radial functions and c. From the theory of angular momenta, it follows that
ij
ij Hij = qab I(a, b) + vabcd;k Rk (ab, cd) , ab

abcd;k

(21.51)

where the sum over ab or abcd covers the occupied orbitals of configuration states i and j. Substituting into the energy expression (21.48), and interchanging the order of summation, we get

E(Ψ) = qab I(a, b) + vabcd;k Rk (ab, cd) , ab

abcd;k

(21.52)

where



i

vabcd;k =

ij

ci c j qab and

j



i

ij

ci c j vabcd;k .

j

In this form, the energy is expressed as a list of integrals and their contribution to the energy – a form suitable for the derivation of the MCHF radial equations. As in the derivation of the Hartree–Fock equations, the stationary principle must be applied to a functional that includes Lagrange multipliers for all the constraints. Thus,

δ a , b λab a|b − E ci2 . F (P, c) = E(Ψ) + a 10) it becomes necessary to include a relativistic correction factor. Numerical values of this correction factor can be found in [21.23]. Table 21.4 shows the convergence of an MCHF calculation for the specific mass shift parameter and the electron density at the nucleus of the ground state of Boron II as the active set increases. The calculated 11 B −10 B isotope shift is 13.3 mÅ with an estimated uncertainty of 1%. The size of the isotope shift is similar to the limit of resolution of the Goddard High Resolution Spectrograph aboard the Hubble Space Telescope [21.24]. E fs =

21.7.2 Hyperfine Effects

The nuclear operator M(1) is related to the scalar magnetic dipole moment, µ I , according to   γ I II | M0(1) | γ I II = µ I . (21.63) Table 21.4 Specific mass shift parameter and electron density at the nucleus as a function of the active set Active set

S (a.u.)

|ψ(0)|2

HF 2s 1p 3s 2p 1d 4s 3p 2d 1f 5s 4p 3d 2f 1g 6s 5p 4d 3f 2g 1h 7s 6p 5d 4f 3g 2h 1i

0.000 00 −0.020 17 0.625 18 0.624 81 0.601 69 0.598 03 0.597 09

72.629 72.452 72.490 72.497 72.501 72.503 72.504

319

The magnetic dipole moments are known quantities, obtained with high accuracy from experiments. For a recent tabulation see [21.26]. The electronic operator is T (1) =

N  α2
(1) 2l (i)ri−3 2 i=1

√ (1) − gs 10 C (2) (i) × s(1) (i) ri−3  8 (1) + gs πδ(ri )s (i) , 3

(21.64)

where gs = 2.002 3193 is the electron spin g-factor, δ(r) √ the three-dimensional delta function and Cq(k) = 4π/(2k + 1)Ykq , with Ykq being a normalized spherical harmonic. The first term of the electronic operator represents the magnetic field generated by the orbiting electric charges and is called the orbital term. The second term represents the field generated by the orbiting magnetic dipole moments, which are coupled to the spin of the electrons. This is the spin-dipole term. The last term represents the contact interaction between the nuclear magnetic dipole moment and the electron magnetic moment. It is called the Fermi contact term and contributes only for s-electrons. By recoupling I and J, the interaction energy can be rewritten as W M1 (J ) = (−1) I +J−F W(I JI J; F1) × γ J JT (1) γ J Jγ I IM(1) γ I I , (21.65)

where W(I JI J; F1) is a W coefficient of Racah. When the magnetic dipole interaction constant AJ =

1 µI γ J JT (1) γ J J I [J(J + 1)(2J + 1)] 21 (21.66)

is introduced, the energy is given by W M1 (J ) =

1 AJC , 2

(21.67)

where C = F(F + 1) − J(J + 1) − I(I + 1). In theoretical studies, the A-factor is often given as a linear combination of the hyperfine parameters
(1) l0 (i)ri−3 | γL SM L M S  , al = γL SM L M S | i

(21.68)

Part B 21.7

The magnetic hyperfine structure (hfs) is due to an interaction between the magnetic field generated by the electrons and the nuclear magnetic dipole moment. The interaction couples the nuclear and electronic angular momenta to a total momentum F = I + J, and the interaction energy can be written as the expectation value of a scalar product between an electronic and a nuclear tensor operator [21.25] (see Chapt. 16)  W M1 (J ) = γ I γ J I JFM F | T (1)  · M(1) | γ I γ J I JFM F . (21.62)

21.7 Atomic Properties

320

Part B

Atoms

ad = γL SM L M S |
(2) 2C0 (i)s0(1) (i)ri−3 | γL SM L M S  , (21.69) i

ac = γL SM L M S |
(1) 2s0 (i)ri−2 δ(ri ) | γL SM L M S  ,

(21.70)

i

where M L = L and M S = S. These parameters correspond to the orbital, spin-dipole and Fermi contact term of the electronic operator. The electric hyperfine structure is due to the interaction between the electric field gradient produced by the electrons and the nonspherical charge distribution of the nucleus. The interaction energy is W E2 (J ) = γ I γ J I JFM F | T (2) · M(2) | γ I γ J I JFM F  ,

(21.71)

where the nuclear operator M(2) is related to the scalar electric quadrupole moment, Q, according to Q . (21.72) γ I II | M0(2) | γ I II = 2 The electronic operator is T (2) = −

N


C (2) (i)ri−3 ,

(21.73)

Part B 21.7

i=1

and represents the electric field gradient. By introducing the electric quadrupole interaction constant B, 1  2 J(2J − 1) B J = 2Q (J + 1)(2J + 1)(2J + 3) × γ J JT (2) γ J J , (21.74) the interaction energy can be written as 3 C(C + 1) − I(I + 1)J(J + 1) . W E2 (J ) = B J 4 2I(2I − 1)J(2J − 1) (21.75)

In many cases the electric hyperfine interaction is weaker than the magnetic and manifests itself as a small deviation from the Landé interval rule for the magnetic hfs. If the electronic part of the interaction can be calculated accurately, a value of the electric quadrupole moment Q, which is a difficult quantity to measure with direct nuclear techniques, can be deduced from the measured B-factor. A recent tabulation of nuclear quadrupole moments is given in [21.27]. Table 21.5 shows the convergence of an MCHF active space calculation for two different isotopes for the 1s 2 2s2p 1 P state of B II [21.24]. Some oscillations are

Table 21.5 MCHF Hyperfine constants (in MHz) for the

1s 2 2s2p 1 P state of B II 10 B

11 B

Active set

A1

B1

A1

B1

HF 2s 1p 3s 2p 1d 4s 3p 2d 1f 5s 4p 3d 2f 1g 6s 5p 4d 3f 2g 1h 7s 6p 5d 4f 3g 2h 1i

60.06 60.22 60.98 60.05 60.48 60.85 60.81

8.338 8.360 8.193 7.677 7.764 8.052 8.002

179.36 179.83 182.11 179.34 180.62 181.71 181.60

4.001 4.011 3.932 3.684 3.725 3.864 3.840

observed since each new “layer” of orbitals may localize in different regions of space.

21.7.3 Metastable States and Lifetimes States above an ionization threshold may decay via a radiationless transition to a continuum. When the interaction with the continuum is spin-forbidden the state is metastable. The nsnp 24 P of negative ions, for example, decay through Breit–Pauli interactions with nsnp 2 doublets. The latter, in turn interact with continuum states, thus opening a decay channel. Such metastable states may be treated as bound states. The foundation for the theory of autoionization was laid down by Fano [21.28], where he developed a configuration interaction (CI) theory for autoionization. Let Ψb (N; γL S) be a normalized, multiconfiguration component of a discrete perturbor for an N-electron system, in which all orbitals are bound orbitals, decaying exponentially for large r. Let Ψk be an asymptotically normalized continuum component of the wave function, also for an N-electron system, at energy E, of the form,   Ψk (N; Eγ  L S) = |Ψb N − 1; β L˜ S˜ · φ(kl)L S , 

(21.76)

 where Ψb N − 1; β L˜ S˜ is a bound MCHF wave function for the (N − 1)-electron target system. Then, in the L Scoupling scheme, the width of the autoionizing state is given by the “Golden Rule” Γ = 2πVE20 ,

(21.77)

where VE0 = Ψb (N; γL S)|H − E 0 |Ψk (N; E 0 γ  L S) . (21.78)

A similar formula can be derived for the L SJ-scheme and the Breit–Pauli Hamiltonian. In the above equation,

Atomic Structure: Multiconfiguration Hartree–Fock Theories

21.7.4 Transition Probabilities The most fundamental quantity for the probability of a transition from an initial state i to a final state f is the reduced matrix element related to the line strength by S1/2 = Ψi ||O||Ψ f  ,

(21.79)

where O is the transition operator. In the case of the electric dipole transition, there are two frequently used forms: the  length form O = j r j , and the velocity form, O = j ∇ j /E i f , where E i f = E f − E i . For exact non-relativistic wave functions, the two forms are equivalent, but for approximate wave functions, the ma-

321

trix elements in general differ. Thus, the computation of the line strength and the oscillator strength, or f -value, where f = (2/3)E i f S/[(2Si + 1)(2L i + 1)] , forms a critical test of the wave function in nonrelativistic theory and also the model describing a many-electron system. The same operators are often also used in Breit–Pauli calculations. In this case, the velocity form of the operator has neglected some terms of order (αZ)2 and the length value is preferred. Some well-known discrepancies between theory and experiment existed for more than a decade for the resonance transitions of Li and Na. For the nonrelativistic 2s–2p transition in Li, a full-core CI [21.35] calculation produced f -values of (0.747 04, 0.747 04, 0.753 78) for the length-, velocity-, and acceleration form, respectively. When relativistic corrections were included, the value changed to 0.747 15, in agreement with a number of theories, tabulated to only four decimal places. A fast beam-laser experiment by Gaupp et al. [21.36], yielded a value of 0.7416 ± 0.0012 but this value was revised in 1996 by a beam-gas-laser experiment in perfect agreement with theory [21.37]. In the case of the resonance transition in Na, when theory included correlation in the core as well as core-polarization and some relativistic effects, results were in agreement with an almost simultaneous cascade of new experimental values [21.38]. For MCHF calculations of transition data, an important consideration is that the matrix element is between two different states. For independently optimized wave functions, the orbitals of the initial and final states are not orthonormal, as assumed when Racah algebra techniques are used to evaluate the transition matrix element. Through the use of biorthogonal transformations, the orbitals and the coefficients of expansion of the wave functions of the initial and final state can be transformed efficiently so that standard Racah algebra techniques may be applied [21.39]. Table 21.6 shows the convergence of an MCHF calculation for the ground state of Boron from independently optimized wave functions.

21.7.5 Electron Affinities By definition, the electron affinity is Ae = E(A− ) − E(A). Thus it is the energy difference between Hamiltonians differing by one electron. Correlation plays a very important role in the binding of the extra electron in the negative ion. It has been known for a long time that the alkali metals have a positive electron affinity, but

Part B 21.7

E 0 = Ψb (N; γL S)|H|Ψb (N; of   the  γL S). The energy core, ortarget, is E target = Ψb N − 1; β L˜ S˜ |H|Ψb N − 1; β L˜ S˜ , where H in the latter equation is the Hamiltonian for an (N − 1)-electron system. The wave function for the continuum component is assumed to have only one unknown, namely φ(kl) = Pkl (r)|ls, the one-particle continuum function, where |ls is the known spin–angular part. The radial equation for Pkl (r) has exactly the same form as (21.26), except that εnl,nl = −k2 , where E 0 = E target + k2 /2. The radial function can be obtained iteratively using an SCF procedure from outward integration. One of the more accurate calculations of a lifetime is that for He− 1s2s2p 4 P 5/2 by Miecznik et al. [21.29]. In this case, a lifetime of (345±10) µs was found and compared with a recent experimental value of (350±15) µs [21.30]. In this L SJ state, the 4 P interacts with the 1s 2 kf 2 F. It was found that correlation in the target of the continuum orbital modified the lifetime. In calculations like these, it is always a question whether orthogonality conditions should be applied [as in projection operator formalism (see Chapt. 25)]. Some theorems relating to this question have been published by Brage et al. [21.31]. The position and widths of autoionizing resonances can also be determined from the study of photoionization or photodetachment using a spline Galerkin method together with inverse iteration. No boundary condition need be applied nor is there an inner and outer region. Resonance properties are obtained from a fitting of the cross-section [21.32]. Non-orthogonal, spline-based R-matrix methods with an inner and outer region have also been developed where there is no need of the “Buttle correction” and, at the same time, the non-orthogonality eliminates the need of certain pseudo-states [21.33]. For an extensive review of the application of splines in atomic and molecular physics, see [21.34].

21.7 Atomic Properties

322

Part B

Atoms

Part B 21

only recently has it been found, theoretically [21.40] and experimentally [21.41], that some of the alkaline earths may also be able to bind an extra electron. The d-electrons need to play a strong role, so Be and Mg, do not have a positive Ae , but according to the most recent experimental measurement, the electron affinity for Ca is 18 meV [21.42]. A calculation based on the spline methods and using a model potential with adjustable parameters to describe the core, obtained a value of 17.7 meV [21.43] in close agreement with experiment. Atomic systems such as Ca are often thought of as two-electron systems and indeed, for qualitative descriptions, many observations can be explained. A number of physical effects need to be considered when predicting such electron affinities: (i) Valence correlation is crucial for obtaining binding. (ii) Intershell correlation between the valence electrons and the core (core polarization) is also important. The first electron may polarize the core considerably, but this is reduced by the second electron since the two avoid each other dynamically and prefer to be on opposite sides of the core. (iii) Core rearrangement, which occurs because of the presence of one or more outer electrons, and is particularly large if any of these penetrate the core. In the case of Ca+ , the fixed-core Hartree–Fock energy of 3d 2 D state is 300 meV higher if computed in the fixed potential the Ca+2 ion compared with a fully variational calculation! (iv) Intracore exclusion effects due to the presence of an

Table 21.6 Convergence

of transition data for the 1s 2 2s 2 2p 2 P o → 1s 2 2s2p 2 2 D transition in Boron with increasing active set N

g fl

g fv

Sl

∆E (cm−1 )

3 4 5 6 7 Expt. a

0.6876 0.2456 0.2625 0.2891 0.2928 0.28(02)

0.8156 0.2696 0.2695 0.2866 0.2900

2.5534 0.9959 1.0705 1.1868 1.2036

53 197 48 720 48 440 48 125 48 051 47 857

a

[21.44]

extra valence electron which reduces the correlation of the core. (v) Relativistic shift effects, which are present in observed levels and are particularly important for s-electrons. Model potential methods attempt to capture all but valence correlation in a potential so that calculations for Calcium, for example, can proceed as though for a two-electron system. A review of various theoretical approaches, many of which include different effects, may be found in [21.45]. For small systems such as Li, the electron affinity has been computed [21.46] to experimental accuracy [21.47] of (0.6176 ± 0.0002) eV. In neutral oxygen, it has been found that there is an isotope effect on the electron affinity [21.48].

21.8 Summary More comprehensive treatments of atomic structure may be found [21.49, 50]. An atomic structure package is available for many of the calculations described

here [21.51]. A review of the application of systematic procedures to the prediction of atomic properties has been published [21.52, 53].

References 21.1 21.2 21.3 21.4 21.5 21.6 21.7

R. Glass, A. Hibbert: Comput. Phys. Commun. 16, 19 (1978) I. Lindgren, J. Morrison: Atomic Many-Body Theory (Springer, Berlin, Heidelberg, New York 1982) P.-O. Löwdin: Phys. Rev. 97, 1509 (1955) D. Layzer, Z. Horák, M. N. Lewis, D. P. Thompson: Ann. Phys. (N. Y.) 29, 101 (1964) D. Layzer: Ann. Phys. (N. Y.) 8, 271 (1959) C. A. Nicolaides, D. R. Beck: Chem. Phys. Lett. 35, 79 (1975) C. Froese Fischer: Comput. Phys. Rep. 3, 273 (1986)

21.8 21.9 21.10 21.11

21.12 21.13

C. Froese Fischer: J. Comput. Phys. 13, 502 (1973) M. R. Godefroid, J. Lievin, J.-Y. Metz: Int. J. Quantum Chem. XL, 243 (1991) G. Tachiev, C. Froese Fischer: J. Phys. B 32, 5805 (1999) S. A. Alexander, J. Olsen, P. Öster, H. M. Quiney, S. Salomonson, D. Sundholm: Phys. Rev. A 43, 3355 (1991) C. Froese Fischer: J. Phys. B 26, 855 (1993) K. T. Chung, X.-W. Zhu, Z.-W. Wang: Phys. Rev. A 47, 1740 (1993)

Atomic Structure: Multiconfiguration Hartree–Fock Theories

21.14

21.15 21.16

21.17 21.18 21.19

21.20 21.21 21.22 21.23 21.24 21.25 21.26 21.27 21.28 21.29

21.31 21.32 21.33 21.34

21.35

21.36 21.37 21.38

21.39 21.40 21.41 21.42 21.43 21.44 21.45 21.46 21.47 21.48 21.49 21.50 21.51 21.52 21.53

K. T. Chung: Proceedings of the international conference on highly charged ions, Manhattan Kansas, 1992. In: AIP Conference Proceedings #274 (AIP, New York 1993) A. Gaupp, P. Kuske, H. J. Andrä: Phys. Rev. A 26, 3351 (1982) U. Volz, H. Schmoranzer: Phys. Scripta T65, 48 (1996) P. Jönsson, A. Ynnerman, C. Froese Fischer, M. R. Godefroid, J. Olsesn: Phys. Rev. A 53, 4021 (1996) J. Olsen, M. R. Godefroid, P. Jönsson, P. Å. Malmqvist, C. Froese Fischer: Phys. Rev. E 52, 449 (1995) C. Froese Fischer, J. B. Lagowski, S. H. Vosko: Phys. Rev. Lett. 59, 2263 (1987) D. J. Pegg, J. S. Thompson, R. N. Compton, G. D. Alton: Phys. Rev. Lett. 59, 2267 (1989) K. W. McLaughlin, D. W. Duquette: Phys. Rev. Lett. 72, 1176 (1994) H. W. van der Hart, C. Laughlin, J. E. Hansen: Phys. Rev. Lett. 71, 1506 (1993) T. R. O’Brian, J. E. Lawler: Astron. Astrophys. 255, 420 (1992) C. Froese Fischer, T. Brage: Can. J. Phys. 70, 1283 (1992) C. Froese Fischer: J. Phys. B 26, 855 (1993) J. Dellwo, Y. Liu, D. J. Pegg, G. D. Alton: Phys. Rev. A 45, 1544 (1992) M. R. Godefroid, C. Froese Fischer: Phys. Rev. A 60, R2637 (1999) I. I. Sobel’man: Introduction to the Theory of Atomic Spectra (Pergamon, Oxford 1972) R. D. Cowan: The Theory of Atomic Structure and Spectra (Univ. of California Press, Berkeley 1981) C. Froese Fischer: Computer Phys. Commun. 64, 369 (1991) C. Froese Fischer, P. Jönsson: Comput. Phys. Commun. 84, 37 (1994) C. Froese Fischer, T. Brage, P. Jönsson: Computational Atomic Structure: An MCHF approach (IOP, Bristol 1997)

323

Part B 21

21.30

E. Lindroth, H. Persson, S. Salomonson, A.-M. Mårtensson-Pendrill: Phys. Rev. A 45, 1493 (1992) R. L. Kelly: J. Phys. Chem. Ref. Data 16, 1371–1678 (1987), Suppl. 1 K. T. Chung: Many-Body Theory of Atomic Structure and Photoionization, ed. by T. N. Chang (World Scientific, Singapore 1993) p. 83 E. Clementi, C. Roetti: At. Data Nucl. Data Tables 14, 177 (1974) A. Hibbert: Comput. Phys. Commun. 9, 141 (1975) M. Bentley, H. W. van der Hart, M. Landtman, G. M. S. Lister, Y.-T. Shen, N. Vaeck: Phys. Scr. T47, 7 (1993) T. Brage, C. Froese Fischer: Phys. Scr. 49, 651 (1994) O. Zatsarinny, C. Froese Fischer: J. Phys. B 35, 4669 (2002) D. S. Hughes, C. Eckart: Phys. Rev. 36, 694 (1930) W. H. King: Isotope Shifts in Atomic Spectra (Plenum, New York 1984) P. Jönsson, S. G. Johansson, C. Froese Fischer: Astrophys. J. 429, L45 (1994) I. Lindgren, A. Rosén: Case Stud. At. Phys. 1772 4, 93 (1974) P. Raghavan: At. Data Nucl. Data Tables 42, 189 (1989) P. Pyykkö: Z. Naturforsch. 47a, 189 (1992) U. Fano: Phys. Rev. 129, 1866 (1961) G. Miecznik, T. Brage, C. Froese Fischer: Phys. Rev. A 47, 3718 (1993) T. Andersen, L. H. Andersen, P. Balling, H. K. Haugen, P. Hvelplund, W. W. Smith, K. Taulbjerg: Phys. Rev. A 47, 890 (1993) T. Brage, C. Froese Fischer, N. Vaeck: J. Phys. B 26, 621 (1993) J. Xi, C. Froese Fischer: Phys. Rev. A 53, 3169 (1996) O. Zatsarinny, C. Froese Fischer: J. Phys. B 35, 4161 (2002) H. Bachau, E. Cormier, J. E. Hansen, F. Martin: Rep. Prog. Phys. 64, 1815 (2001)

References

325

Relativistic Ato 22. Relativistic Atomic Structure

light, c, has the numerical value α−1 = 137.035 999 11(46), where α is the fine structure constant.

22.1

Mathematical Preliminaries .................. 22.1.1 Relativistic Notation: Minkowski Space-Time .............. 22.1.2 Lorentz Transformations............. 22.1.3 Classification of Lorentz Transformations ........................ 22.1.4 Contravariant and Covariant Vectors..................................... 22.1.5 Poincaré Transformations ...........

326 326 326 326 327 327

22.2 Dirac’s Equation .................................. 328 22.2.1 Characterization of Dirac States ... 328 22.2.2 The Charge-Current 4-Vector ...... 328 22.3 QED: Relativistic Atomic and Molecular Structure ....................... 22.3.1 The QED Equations of Motion ...... 22.3.2 The Quantized Electron–Positron Field........................................ 22.3.3 Quantized Electromagnetic Field . 22.3.4 QED Perturbation Theory ............ 22.3.5 Propagators .............................. 22.3.6 Effective Interaction of Electrons .

329 329 329 330 331 333 333

22.4 Many-Body Theory For Atoms ............... 22.4.1 Effective Hamiltonians ............... 22.4.2 Nonrelativistic Limit: Breit–Pauli Hamiltonian ............ 22.4.3 Perturbation Theory: Nondegenerate Case .................. 22.4.4 Perturbation Theory: Open-Shell Case ........................ 22.4.5 Perturbation Theory: Algorithms .

334 335

22.5 Spherical Symmetry ............................. 22.5.1 Eigenstates of Angular Momentum .............................. 22.5.2 Eigenstates of Dirac Hamiltonian in Spherical Coordinates ............ 22.5.3 Radial Amplitudes ..................... 22.5.4 Square Integrable Solutions........ 22.5.5 Hydrogenic Solutions ................. 22.5.6 The Free Electron Problem in Spherical Coordinates ............

337

335 335 336 337

337 338 340 341 342 343

Part B 22

Relativistic quantum mechanics is required for the description of atoms and molecules whenever their orbital electrons probe regions of space with high potential energy near the atomic nuclei. Primary effects of a relativistic description include changes to spatial and momentum distributions; spin–orbit interactions; quantum electrodynamic corrections such as the Lamb shift; and vacuum polarization. Secondary effects in many-electron systems arise from shielding of the outer electrons by the distributions of electrons in penetrating orbitals; they change orbital binding energies and dimensions and so modify the order in which atomic shells are filled in the lower rows of the Periodic Table. Relativistic atomic and molecular structure theory can be regarded as a simplification of the fundamental description provided by quantum electrodynamics (QED). This treats the atom or molecule as an assembly of electrons and atomic nuclei interacting primarily by exchanging photons. This model is far too difficult and general for most purposes, and simplifications are required. The most important of these is the representation of the nuclei as classical charge distributions, or even as point particles. Since the motion of the nuclei is usually slow relative to the electrons, it is often adequate to treat the nuclear motion nonrelativistically, or even to start from nuclei in fixed positions, correcting subsequently for nuclear motion. The emphasis in this chapter is on relativistic methods for the calculation of atomic structure for general many-electron atoms based on an effective Hamiltonian derived from QED in the manner sketched in Sect. 22.2 below. An understanding of the Dirac equation, its solutions and their numerical approximation, is essential material for studying many-electron systems, just as the corresponding properties of the Schrödinger equation underpin Chapt. 21. We shall use atomic units throughout. Where it aids interpretation we shall, however, insert factors of c, m e and ~. In these units, the velocity of

326

Part B

Atoms

22.6 Numerical Approximation of Central Field Dirac Equations ............ 22.6.1 Finite Differences ...................... 22.6.2 Expansion Methods ................... 22.6.3 Catalogue of Basis Sets for Atomic Calculations .............. 22.7

Many-Body Calculations ....................... 22.7.1 Atomic States............................ 22.7.2 Slater Determinants................... 22.7.3 Configurational States................ 22.7.4 CSF Expansion ...........................

344 344 345 347 350 350 350 350 350

22.7.5 Matrix Element Construction ....... 22.7.6 Dirac–Hartree–Fock and Other Theories .................... 22.7.7 Radiative Corrections ................. 22.7.8 Radiative Processes ...................

350 351 353 353

22.8 Recent Developments........................... 354 22.8.1 Technical Advances ................... 354 22.8.2 Software for Relativistic Atomic Structure and Properties ............ 354 References .................................................. 355

22.1 Mathematical Preliminaries 22.1.1 Relativistic Notation: Minkowski Space-Time

22.1.3 Classification of Lorentz Transformations

An event in Minkowski space-time is defined by a 4-vector x = {x µ } (µ = 0, 1, 2, 3) where x 0 = ct is the time coordinate and x 1 , x 2 , x 3 are Cartesian coordinates in 3-space. The bilinear form (The Einstein suffix convention, in which repeated pairs of Greek subscripts are assumed to be summed over all values 0, 1, 2, 3, will be used where necessary in this chapter.)

Rotations Lorentz transformations with matrices of the form

1 0
(22.5) , Λ= 0 R

(x, y) = x µ gµν yν ,

(22.1)

in which

Part B 22.1

  1 0 0 0 

  0 −1 0 0 g = gµν = gµν =   0 0 −1 0 0 0 0 −1

(22.2)

are called metric coefficients, defines the metric of Minkowski space.

22.1.2 Lorentz Transformations Lorentz transformations are defined as linear mappings Λ such that (Λx, Λy) = (x, y)

(22.3)

so that gµν = Λρ µ gρσ Λσ ν .

(22.4)

This furnishes 10 equations connecting the 16 components of Λ; at most 6 components can be regarded as independent parameters. The (infinite) set of Λ matrices forms a regular matrix group (with respect to matrix multiplication) called the Lorentz group, L, designated O(3,1) [22.1, 2].

where R ∈ SO(3) is an orthogonal 3 × 3 matrix with determinant +1, and 0 is a null three dimensional column vector, correspond to three-dimensional space rotations. They form a group isomorphic to SO(3). Boosts Lorentz transformations with matrices of the form

γ(v) γ(v)v
, (22.6) Λ= γ(v)v I3 + (γ(v) − 1)nn


with v = vn a three dimensional column vector, |n| = 1, v = |v| and γ(v) = (1 − v2 /c2 )−1/2 , are called boosts. The matrix Λ describes an ‘active’ transformation from an inertial frame in which a free classical particle is at rest to another inertial frame in which its velocity is v. Boosts form a submanifold of L though they do not in general form a subgroup. However, the set of boosts in a fixed direction n form a one-parameter subgroup. Discrete Transformations The matrices



1 0
−1 0
P= , T= with PT = −I4 0 −I3 0 I3 (22.7)

are called discrete Lorentz transformations and form a subgroup of the Lorentz group along with the iden-

Relativistic Atomic Structure

tity I4 . The matrix P performs space or parity inversion; the matrix T performs time reversal. Infinitesimal Lorentz Transformations The proper Lorentz transformations close to the identity are of particular importance: they have the form µ µ Λµ ν = δν + ων + · · · ,

µ µ (Λ−1 )µ ν = δν − ων + · · · ,

(22.8)

where

and  is infinitesimal. The infinitesimal generators, components ωµν , can be treated as quantum mechanical observables: see Sect. 22.2.1. The Lorentz Group The Lorentz group L is a Lie group with a sixdimensional group manifold which has four connected components, namely  ↑ L+ ≡ Λ ∈ L |Λ0 0 ≥ 1, detΛ = +1 , (22.9)  ↑ ↑ L− ≡ Λ ∈ L |Λ0 0 ≥ 1, detΛ = −1 = PL+ , (22.10)

↓ L+

 ↑ ≡ Λ ∈ L |Λ0 0 ≤ 1, detΛ = −1 = T L+ ,

↓ L+

 ↑ ≡ Λ ∈ L |Λ0 0 ≤ 1, detΛ = +1 = PTL+ .

(22.11)

(22.12)

aµ = gµν aν ,

(22.14)

so that aµ aµ = aµ gµν aν = (a, a)

containing the identity is a Lie subgroup of L called the proper Lorentz group. All its group elements can be obtained from boosts and rotations. It is not simply connected because the subgroup of rotations is not simply connected. The group is also noncompact as the subset of boosts is homeomorphic to R3 . ↑ These topological properties of L+ are essential for understanding the properties of relativistic wave equations. In particular the multiple connectedness forces the introduction of spinor representations, and to the appearance of half-integer angular momenta or spin.

22.1.4 Contravariant and Covariant Vectors Contravariant 4-vectors (such as events x) transform according to the rule (22.13)

(22.15)

is invariant with respect to Lorentz transformations. Similarly, we can construct a contravariant 4-vector from a covariant one by writing (22.16)

The transformation law for covariant vectors is therefore

aµ → aµ = [Λ−1 ]ν µ aν .

(22.17)

The most important example of a covariant vector is the 4-momentum operator pµ = i

∂ ∂x ν

µ = 0, 1, 2, 3 .

(22.18)

From this we derive the contravariant 4-momentum operator with components pµ by writing pµ = gµν pν 

∂ ∂ ∂ ∂ = i 0 , −i 1 , −i 2 , −i 3 , ∂x ∂x ∂x ∂x

(22.19)

in agreement with nonrelativistic expressions.

22.1.5 Poincaré Transformations More generally, a Poincaré transformation is obtained by combining Lorentz transformations and space-time translations: Π(x) = Λx + a .

(22.20)

The set of all Poincaré transformations, Π = (a, Λ), with the composition law (a1 , Λ1 )(a2 , Λ2 ) = (a1 + Λ1 a2 , Λ1 Λ2 ) ,

(22.21)

also forms a group, P . Properties of the Lorentz and Poincaré groups will be introduced as needed. For a concise account of their properties see [22.3]. For more detail on relativistic quantum mechanics in general see textbooks such as [22.3, 4].

Part B 22.1

↑ The connected component L+

327

Covariant 4-vectors can be formed by writing

aµ = gµν aν .

ωµν = −ωνµ

aµ → aµ = Λµ ν aν .

22.1 Mathematical Preliminaries

328

Part B

Atoms

22.2 Dirac’s Equation We present Dirac’s equation for an electron in a classical electromagnetic field defined by the 4-potential Aµ (x): Covariant Form  µ   γ pµ − eAµ (x) − m e c ψ(x) = 0 .

γ µ (µ = 0, 1, 2, 3), are 4 × 4 matrices. ψ(x) is a 4-component spinor wave function. Here, and elsewhere in this chapter, identity matrices are omitted when it is safe to do so. Dirac Gamma Matrices

defines the Dirac Hamiltonian. The  matrices α, with
Cartesian components α1 , α2 , α3 , and β, have the standard representation

1 0 0 β=γ = (22.26) 0 −1

0 σi i 0 i i = 1, 2, 3 . α =γ γ = (22.27) σi 0

Anticommutation relations:

22.2.1 Characterization of Dirac States

γ µ γ ν + γ ν γ µ = 2gµν .

•

    hˆ D = cα · p − eA(x, t) + eφ(x, t) + βm e c2 (22.25)

(22.22)

where

•

and

Standard representation:

1 0 0 γ = 0 −1

0 σi i γ = i = 1, 2, 3 , −σ i 0

where σi (i = 1, 2, 3) are Pauli matrices [22.1–4].

Part B 22.2

Noncovariant Form In the majority of atomic structure calculations, a frame of reference is chosen in which the nuclear center is taken to be fixed at the origin. In this case it is convenient to write Dirac’s equation in noncovariant form. Then functions of

x = (x , x) , 0

where x 0 = ct, can be regarded as functions of the time t and the position 3-vector x, so that (22.22) is replaced by i

∂ ψ(x, t) = hˆ D ψ(x, t) ∂t

(22.23)

where the scalar and 3-vector potentials are defined by φ(x, t) = cA0 (x) ,   A(x, t) = A1 (x), A2 (x), A3 (x) ,

(22.24)

The solutions of Dirac’s equation span representations of the Lorentz and Poincaré groups, whose infinitesimal generators can be identified with physical observables. The Lorentz group algebra has 10 independent selfadjoint infinitesimal generators: these can be taken to be the components pµ of the four-momentum (which generate displacements in each of the four coordinate directions); the three generators, Ji , of rotations about each coordinate axis in space; and the pseudovector wµ . The irreducible representations can be characterized by invariants ( p, p) = m 2e c2 ,

(22.28)

3 (w, w) = −m 2e c2 s2 = − m 2e c2 , 4

(22.29)

where p is the momentum four-vector and s is a 3-vector defined in terms of Pauli matrices by 1 si = σi , 2

i = 1, 2, 3 .

which can be interpreted as the electronic angular momentum (intrinsic spin) in its rest frame. For more detail see [22.3] and the original papers [22.5, 6].

22.2.2 The Charge-Current 4-Vector Dirac’s equation (22.22) is covariant with respect to Lorentz (22.3) and Poincaré (22.20) transformations, provided that there exists a nonsingular 4 × 4 matrix S(Λ)

Relativistic Atomic Structure

with the property

  ψ (x) = S(Λ)ψ Λ−1 (x − a) .

(22.30)

The matrices S(Λ) are characterized by the equation S

−1

λ

λ

(Λ)γ S(Λ) = Λ

µγ

µ

.

(22.31)

The most important observable expression required in this chapter is the charge–current four-vector j µ = ecψ(x)γ µ ψ(x) ,

(22.32)

329

by virtue of (22.31). The component j 0 (x) can be interpreted as a multiple of the charge density ρ(x), j 0 (x) = ecρ(x) = ecψ(x)γ 0 ψ(x) = ecψ † (x)ψ(x) (22.34)

and the space-like components as the current density j i (x) = ecψ(x)γ i ψ(x) = ecψ † (x)αi ψ(x) . (22.35) The charge–current density satisfies a continuity equation, which in noncovariant form reads ∂ρ(x)  ∂ j i (x) + =0, ∂t ∂x i 3

where the Dirac adjoint is defined by ψ(x) = ψ † (x)γ 0 ,

22.3 QED: Relativistic Atomic and Molecular Structure

(22.33)

and the dagger denotes spinor conjugation and transposition. Since   ψ (x) = ψ Λ−1 (x − a) γ 0 S(Λ)† γ 0   = ψ Λ−1 (x − a) S−1 (Λ) , j µ (x) transforms as a 4-vector j µ (x) = Λµ ν j ν (x)

i=1

or, in covariant notation, ∂µ j µ = 0 .

(22.36)

This is readily proved by using the Dirac equation (22.22) and its Dirac adjoint. Equation (22.36) is clearly invariant under Poincaré transformations, and this yields the important property that electric charge is conserved in Dirac theory.

22.3 QED: Relativistic Atomic and Molecular Structure µ

22.3.1 The QED Equations of Motion The conventional starting point [22.7–10] for deriving equations of motion in quantum electrodynamics (QED) is a Lagrangian density of the form (22.37)

The first term is the Lagrangian density for the free electromagnetic field, F µν (x), 1 (22.38) Lem (x) = − F µν Fµν , 4 the second term is the Lagrangian density for the electron–positron field in the presence of the external µ potential Aext (x),   µ   µ ¯ Le (x) = ψ(x) γµ p − eAext (x) − m e c ψ(x) . (22.39)

We assume that the electromagnetic fields are expressible in terms of the four-potentials by µ

F µν = ∂ µ Aνtot − ∂ ν Atot ,

Lint (x) = − jµ (x)Aµ (x) ,

(22.40)

accounts for the interaction between the uncoupled electrons and the radiation field. The field equations deduced from (22.37) are     µ µ γµ p − Aext (x) − m e c ψ(x) = γµ (x)ψ(x)Aµ (x) ∂µ F µν (x) = j ν (x) , (22.41) and clearly exhibit the coupling between the fields. Quantum electrodynamics requires the solution of the system (22.41) when Aµ (x), ψ(x) and its adjoint ¯ ψ(x) are quantized fields. This formulation is purely formal: it ignores all questions of zero-point energies, normal ordering of operators, choice of gauge associated with the quantized photon field, or the need to include (infinite) counterterms to render the theory finite.

22.3.2 The Quantized Electron–Positron Field

where µ

µ

Atot (x) = Aext (x) + Aµ (x)

Furry’s bound interaction picture of QED [22.7, 11] exploits the fact that a one-electron model is often a good

Part B 22.3

L(x) = Lem (x) + Le (x) + Lint (x) .

is the sum of a four-potential Aext (x) describing the fields generated by classical external charge–current distributions, and a quantized field Aµ (x) which through

330

Part B

Atoms

starting point for a more accurate calculation of atomic or molecular properties. The electrons are described by a field operator   am ψm (x) + b†n ψn (x) , ψ(x) = E m >E F

E n <E F

(22.42)

where E F ≥ −mc2 is a “Fermi level” separating the states describing electrons (bound and continuum) from the positron states (lower continuum) in the chosen timeindependent model potential V(r). Equation (22.42) is written as if the spectrum were entirely discrete, as in finite matrix models; more generally, this must be replaced by integrals over the continuum states together with a sum over the bound states. We assume that the amplitudes ψm (x) are orthonormalized (which can be achieved, for example, by enclosing the system in a finite † box). The operators am and am respectively annihilate † and create electrons, and bn and bn perform the same role for vacancies in the “negative energy” states, which we interpret as antiparticles (positrons). These operators satisfy the anticommutation rules (see Sect. 6.1.1)   † am , am = δm,m ,   † bn , bn = δn,n , (22.43) where [a, b] = ab + ba. All other anticommutators vanish. The operator representing the number of electrons in state m is then † am , Nm = am

(22.44)

Part B 22.3

having the eigenvalues 0 or 1; the states of a system of noninteracting electrons and positrons can therefore be labeled by listing the occupation numbers, 0 or 1 of the one-electron states participating. We define the vacuum state as the (reference) state |0 in which Nm = Nn = 0 for all m, n, so that am |0 = bn |0 = 0 .

(22.45)

The operator representing the total number of particles is given by    Nm + (1 − Nn ) . N = ψ † (x)ψ(x) dx = E m >E F



E n <E F

This is not quite satisfactory: N = En <EF 1 is infinite for the vacuum state, as are the total charge and energy of the vacuum. These infinite “zero-point” values can be eliminated by introducing normal ordered operators. A product of

annihilation and creation operators is in normal order if it is rearranged so that all annihilation operators are to the right of all creation operators. Such a product has a null value in the vacuum state. In performing the rearrangement, each anticommutator is treated as if it were zero. We denote normal ordering by placing the op† † erators between colons. Thus : am am : = am am whilst † † : bn bn : = −bn bn . This means that if we redefine N by    Nm − Nn , N = : ψ † (x)ψ(x) : dx = E m >E F

E n <E F

(22.46)

then 0|N|0 = 0. The same trick eliminates the infinity from the total energy of the vacuum;  H0 = : ψ † (x)hˆ D ψ(x) : dx   Nm E m − Nn E n , (22.47) = E m >E F

E n <E F

so that 0|H0 |0 = 0. The current density operator is given by the commutator of two field variables  1  µ ¯ , ψ(x) , (22.48) j µ (x) = − ec ψ(x)γ 2 ¯ where the Dirac adjoint, ψ(x) is defined by (22.33). The definition (22.48) differs from (22.32) by expressing the total current as the difference between the electron (negatively charged) and positron (positively charged) currents. We can write   j µ (x) =: j µ (x) : + ec Tr γ µ SF (x, x) , (22.49) where SF (x, y) is the Feynman causal propagator, defined below. Since 0| : j µ (x) : |0 = 0, the last term in (22.49) is the vacuum polarization current due to the asymmetry between positive and negative energy states induced by the external field. From this, the net charge of the system is  1 Q= j 0 (x) d3 x (22.50) c     = −e Nm − Nn  + Q vac . E m >E F

E n <E F

Q vac is the total charge of the vacuum, which vanishes for free electrons, but is finite in the presence of an external field (the phenomenon of vacuum polarization). Note that whilst Q is conserved for all processes, the total number of particles need not be; it is always possible to

Relativistic Atomic Structure

add virtual states incorporating electron–positron pairs without changing Q.

22.3.3 Quantized Electromagnetic Field The four-potential of the quantized electromagnetic field can be expressed in terms of a spectral expansion over the field modes in, for example, plane waves  3  d3 k  (λ) q (k)µ(λ) (k) e−ik·x Aµ (x) = 2(2π)3 k0 λ=0  + q (λ)† (k)µ(λ)∗ (k) eik·x . (22.51) µ(λ) (k)

The vectors describe the polarization modes; there are four linearly independent vectors, which may be assumed real, for each k on the positive light cone. Two of these (λ = 1, 2) can be chosen perpendicular to the photon momentum k (transverse polarization); one component (λ = 3) along k (longitudinal polarization); and the final component (λ = 0) is time-like (scalar polarization). The operators q (λ) (k) and q (λ)† (k) describe respectively photon absorption and emission. They satisfy commutation relations  

q (λ) (k), q (λ )† (k ) = δλ,λ δk,k , λ, λ = 1, 2, 3   q (0) (k), q (0)† (k ) = − δk,k ; (22.52) all other commutators vanish. The photon vacuum state, |0γ , is such that

In the interaction representation, this gives an equation of motion i~∂t |Φ(t) = HI (t)|Φ(t) ,

22.3.4 QED Perturbation Theory The textbook perturbation theory of QED, see for example [22.7,8,10,12] and other works, has been adapted for applications to relativistic atomic and molecular structure and is also the source of methods of nonrelativistic many-body perturbation theory (MBPT). We offer a brief sketch emphasizing details not found in the standard texts. The Perturbation Expansion The Lagrangian approach leads to an interaction Hamiltonian  (22.54) HI = − j µ (x)Aµ (x) dx .

(22.55)

where |Φ(t) is the QED state vector, and HI (t) = exp(iH0 t/~)HI exp(−iH0 t/~) , where H0 = Hem + He is the Hamiltonian for the uncoupled photon and electron–positron fields. If S(t, t ) is the time development operator such that |Φ(t) = S(t, t )|Φ(t ) , then i~∂t S(t, t ) = HI (t)S(t, t ) . The equivalent integral equation, incorporating the inital condition S(t, t) = 1, i S(t, t0 ) = 1 − ~

t HI (t1 )S(t1 , t0 ) d1 t ,

(22.56)

t0

can be solved iteratively, giving S(t, t0 ) = 1 +

∞ 

S(n) (t, t0 ) ,

(22.57)

n=1

where S

(n)

t (t, t0 ) = (−i/~)

n

t1

tn−1 dt2 · · ·

dt1 t0

t0

dtn t0

× HI (t1 )HI (t2 ) · · · HI (tn ) .

(22.53)

Further details may be found in the texts [22.7–10].

331

This can be put into a more symmetric form by using time-ordered operators. Define the T -product of two operators by    A(t1 )B(t2 ) , t1 > t2 T A(t1 )B(t2 ) = (22.58) ±B(t2 )A(t1 ) , t2 > t1 where the positive sign refers to the product of photon operators and the negative sign to electrons. Then S

(n)

t t t (−i/~)n (t, t0 ) = dt1 dt2 · · · dtn n! t0 t0 t0   × T HI (t1 )HI (t2 ) · · · HI (tn ) . (22.59)

The operator S(t, t ) relates the state vector at time t to the state vector at some earlier time t < t. Its ma-

Part B 22.3

q (λ) (k)|0γ = 0 .

22.3 QED: Relativistic Atomic and Molecular Structure

332

Part B

Atoms

trix elements therefore give the transition amplitudes for different processes, for example the emission or absorption of radiation by a system, or the outcome of scattering of a projectile from a target. The techniques for extracting cross-sections and other observable quantities from the S-operator are described at length in the texts [22.7, 8, 10, 12]. Although the use of normal ordering means that the charge and mass of the reference state, the vacuum, is zero, it fails to remove other infinities due to the occurrence of divergent integrals. The method of extracting finite quantities from this theory involves renormalization of the charge and mass of the electron. We shall refer especially to [22.10, Chapt. 8] for a detailed discussion. The most difficult technical problems are posed by mass renormalization. Formally, we modify the interaction Hamiltonian to read

T [φ(t1 )φ(t2 ) · · · φ(tn )] which is done using Wick’s Theorem [22.10, p. 25]. In the simplest case, T [φ(t1 )φ(t2 )] = : φ(t1 )φ(t2 ) : + 0 | T [φ(t1 )φ(t2 )] | 0 . (22.62) The vacuum expectation value is called a contraction. More generally, we have

j µ (x)Aµ (x) − δM(x) , where δM(x) is the mass renormalization operator

Part B 22.3

 1  ¯ ψ(x) δM(x) = δm ψ(x), 2 where δm is infinite. A further problem is that electrons in a manyelectron atom or molecule move in a potential which is quite unlike that of the bare nucleus. It is therefore useful to introduce a local mean field potential, say U(x), representing some sort of average interaction with the rest of the electron charge distribution, so that the zero-order orbitals satisfy   cα · p + βc2 + Vnuc (x) + U(x) − E m ψm (x) = 0 . (22.60)

With this starting point, the interaction Hamiltonian becomes HI (x) = HI(1) (x) + HI(2) (x) ,

Effective Interactions Although the S-matrix formalism provides in principle a complete computational scheme for many-electron systems, it is generally too cumbersome for practical use, and approximations are necessary. Usually, this is a matter of selecting a subset of dominant contributions to the perturbation series depending on the application. We are faced with the evaluation of T -products of the form

(22.61)

where HI(1) (x) = −U(x), HI(2) (x) = j µ (x)Aµ (x) − δM(x) , and the electron current is defined in terms of the mean field orbitals of (22.60). The expression HI(2) (x) is sometimes referred to as a fluctuation potential. The term j µ (x)Aµ (x) is proportional to the electron charge, e, which serves as an ordering parameter for perturbation expansions.

T [φ(t1 )φ(t2 ) · · · φ(tn )] = : φ(t1 )φ(t2 ) · · · φ(tn ) :  + 0|T [φ(t1 )φ(t2 )]|0 : φ(t3 ) · · · φ(tn ) :  + permutations  + 0|T [φ(t1 )φ(t2 )] |0 0|T [φ(t3 )φ(t4 )]|0  × : φ(t5 ) · · · φ(tn ) : + permutations · · · . This result has the effect that a T -product with an odd number of factors vanishes. A rigorous statement can be found in all standard texts; each term in the expansion gives rise to a Feynman diagram which can be interpreted as the amplitude of a physical process. As an example, consider the simple but important case S(2) =

 (ie)2  µ T j (x)Aµ (x). j ν (y)Aν (y) . 2!

(22.63)

One of the terms (there are others) found by using Wick’s Theorem is   j µ (x) j ν (y) 0|T Aµ (x)Aν (y) |0 . We see that this involves the contraction of two photon amplitudes   1 − D Fµν (x − y) = 0 | T Aµ (x)Aν (y) | 0 , 2 which plays the role of a propagator (22.69): it relates the photon amplitudes at two space-time points x, y.

Relativistic Atomic Structure

With the introduction of a spectral expansion for the electron current (22.48), the contribution to the energy of the system becomes 1 † † : a a as ar : pq|V |rs , 2 pqrs p q

(22.64)

which can be interpreted, in the familiar language of ordinary quantum mechanics, as the energy of two electrons due to the electron–electron interaction V which is directly related to the photon propagator.

22.3.5 Propagators Propagators relate field variables at different space-time points. Here we briefly define those most often needed in atomic and molecular physics.

22.3 QED: Relativistic Atomic and Molecular Structure

333

potential of the external field aµ (x) has only a scalar time-independent part, Vnuc (x), satisfies   cα · p + βc2 + (Vnuc x) − z G(x, y, z) = δ(3) (x2 − x1 ) .

(22.68)

G(x2 , x1 , z) is a meromorphic function of the com2 plex variable z with
branch  , and cuts  points
at z = ±c 2 2 along the real axis
c , ∞ and  − ∞, −c . The poles lie on the segment − c2 , c2 at the bound eigenvalues of the Dirac Hamiltonian for this potential. Photons The photon propagator DFµν (x2 − x1 ) is constructed in a similar manner:

   ! 1 − DFµν (x2 − x1 ) = 0 T Aµ (x2 )Aν (x1 ) 0 , 2 (22.69)

Electrons Define Feynman’s causal propagator for the electron– positron field by the contraction   ¯ 1 ) |0 . (22.65) SF (x2 , x1 ) = 0|T ψ(x2 )ψ(x

This has a spectral decomposition of the form   ¯ t1 > t2 , E m >E F ψm (x 2 )ψm (x 1 ) SF (x2 , x1 ) = −  ¯ E n <E F ψn (x 2 )ψn (x 1 ) t1 < t2 , (22.66)

SF (x2 , x1 ) =

1 2πi

∞

−∞ n

1 = 2πi

∞

ψn (x2 )ψ¯ n (x1 ) −iz(t2 −t1 ) e dz E n − z(1 + iδ)

G(x2 , x1 , z)γ 0 e−iz(t2 −t1 ) dz ,

−∞

(22.67)

where δ is a small positive number, the sum over n includes the whole spectrum, and where the Green’s function G(x2 , x1 , z), in the specific case in which the

DFµν (x2 − x1 ) = gµν DF (x2 − x1 ) 
 i d4 q D q 2 , = − gµν (2π)4 (22.70)

where
 D q2 =

1 , q 2 + iδ

and δ is a small positive number. This is not unique, as the four-potentials depend on the choice of gauge; for details see [22.8, Sect. 77]. The various forms for the electron–electron interaction given below express such gauge choices.

22.3.6 Effective Interaction of Electrons The expression (22.63) can be viewed in several ways: it is the interaction of the current density j µ (x) at the space-time point x with the four-potential due to the current j µ (y); the interaction of the current density j µ (y) with the four-potential due to the current j µ (x); or, as is commonly assumed in nonrelativistic atomic theory, the effective interaction between two charge density distributions, as represented by (22.64). In terms of the corresponding Feynman diagram, it can be thought of as the energy due to the exchange of a virtual photon.

Part B 22.3

which ensures that positive energy solutions are propagated forwards in time, and negative energy solutions backwards in time in accordance with the antiparticle interpretation of the negative energy states. By noting that the stationary state solutions ψm (x) have time dependence exp(−iE m t), we can write (22.66) in the form

where µ0 is the permeability of the vacuum. This has the integral representation

334

Part B

Atoms

The form of V depends on the choice of gauge potential, as follows. Feynman Gauge

pq|V |rs   F (x, y)ψq† (y)ψs (y) d3 x d3 y = ψ †p (x)ψr (x)vsq (22.71)

where F (x, y) = vsq

eiωsq R (1 − αx · α y ) , R

(22.72)

with Es − Eq . c This interaction gives both a real and an imaginary contribution to the energy; only the former is usually taken into account in structure calculations. Since the orbital indices are dummy variables, it is usual to symmetrize the interaction kernel by writing   1 F v¯ F (x, y) = vsq (x, y) + vrFp (x, y) , 2 R = x − y,

R = |R|,

ωsq =

which places the orbitals on an equal footing.

Part B 22.4

Coulomb Gauge Here the Feynman propagator is replaced by that for the Coulomb gauge, giving  eiωsq R eiωsq R T vsq − αx · α y (x, y) = R R 
 eiωr p R − 1 + (αx ·∇) α y ·∇ (22.73) ωr2p R

in which the operator ∇ involves differentiation with respect to R. Symmetrization is also used with this interaction.

Breit Operator The low frequency limit, ωr p → 0, ωsq → 0, is known as the Breit interaction: 1 T lim vsq (x, y) = + vB (R) , (22.74) ωr p ,ωsq →0 R

where



αx · R α y · R 1 . αx · α y + v (R) = − 2R R2 B

Gaunt Operator This is a further approximation in which vB (R) is replaced by αx · α y vG (R) = − , (22.75) R the residual part of the Breit interaction being neglected. Comments The choice of gauge should not influence the predictions of QED for atomic and molecular structure when the perturbation series is summed to convergence, so that it should not matter if the unapproximated effective operators are taken in Feynman or Coulomb gauge. However, this need not be true at each order of perturbation. It has been shown that the results are equivalent, order by order, if the orbitals have been defined in a local potential, but not otherwise. There have also been suggestions that the Feynman operator introduces spurious terms in lower orders of perturbation that are canceled in higher orders [22.13]. For this reason, most structure calculations have used Coulomb gauge. It is often argued, following Bethe and Salpeter [22.14, Sect. 38], that the Breit interaction should only be used in first order perturbation theory. The reason is the approximation ω → 0; however, this approximation is quite adequate for many applications in which the dominant interactions involve only small energy differences.

22.4 Many-Body Theory For Atoms The relativistic theory of atomic structure can be viewed as a simplification of the QED approach using an effective Hamiltonian operator in which the Dirac electrons interact through the effective electron–electron interaction of Sect. 22.3.6. This approach retains the dominant terms from the perturbation solution; those that are omitted are small and can, with sufficient trouble, be

taken into account perturbatively [22.10,15]. In particular, radiative correction terms requiring renormalization are explicitly omitted, and their effects incorporated at a later stage. Once a model has been chosen, the techniques and methods used for practical calculations acquire a close resemblance to those of the nonrelativistic theory described, for example in Chapt. 21.

Relativistic Atomic Structure

The models which are closest to QED are those in which the full electron–electron interaction is included, usually in Coulomb gauge. We define a Fock space Hamiltonian

where, as in (22.47),   H0 = Nm E m − Nn E n , E m >E F

(22.76)

H1 = −



The nonrelativistic limit of the Dirac–Coulomb–Breit Hamiltonian is described in Chapt. 21. The derivation is given in many texts, for example [22.8, 10, 14], and in principle involves the following steps: 1. Express the relativistic 4-spinor in terms of nonrelativistic Pauli 2-spinors of the form (see Sect. 21.2)

(22.77)

E n <E F

in which the states are those determined with respect to a mean-field central potential U(x) as in (22.60)   cα · p + βc2 + Vnuc (x) + U(x) − E m ψm (x) = 0 , and

335

22.4.2 Nonrelativistic Limit: Breit–Pauli Hamiltonian

22.4.1 Effective Hamiltonians

HDCB = H0 + H1 + H2

22.4 Many-Body Theory For Atoms

: a†p aq : p|U(x)|q

pq

1 † † : a a ar as : pq|V |sr . H2 = 2 pqrs p q Here the sums run only over states p with E p > E F ; this means that states with E p < E F are treated as inert. The models are named according to the choice of V from Sect. 22.5.3.

Dirac–Coulomb Models The electron–electron interaction is simply taken to be the static 1/R potential. Note that although the equations are relativistic, the choices of electron–nucleus interaction all implicitly restrict these models to a frame in which the nuclei are fixed in space. The full electron– electron interaction is gauge invariant; however, it is common to start from the Dirac–Coulomb operator, in which case the gauge invariance is lost. Since radiative transition rates are sensitive to loss of gauge invariance [22.16] the choice of potential in (22.76) can make a big difference. Such choices may also affect the rate of convergence in correlation calculations in which the relativistic parts of the electron–electron interaction are treated as a second, independent, perturbation.

Thus the Breit–Pauli Hamiltonian is written as the sum of terms of Sect. 21.2 which can be correlated with specific parts of the parent relativistic operator: 1. One-body terms originate from the Dirac Hamiltonian: they are Hmass (21.5), the one-body part of HDarwin (21.7) and the spin–orbit couplings Hso (21.11) and Hsoo (21.12). The forms given in these equations assume that the electron interacts with a point-charge nucleus and only require the Coulomb part of the electron–electron interaction. 2. Two-body terms, including the two-body parts of HDarwin (21.7), the spin–spin contact term Hssc (21.8), the orbit–orbit term Hoo (21.9) and the spin–spin term Hss (21.13) originate from the Breit interaction.

22.4.3 Perturbation Theory: Nondegenerate Case We give a brief resumé of the Rayleigh–Schrödinger perturbation theory following Lindgren [22.17]. The material presented here supplements the general discussion of perturbation theory in Chapt. 5. First consider the simplest case with a nondegenerate reference state Φ belonging to the Hilbert space H satisfying H0 |Φ = E 0 |Φ ,

(22.78)

which is a first approximation to the solution of the full problem H|Ψ  = E|Ψ ,

H = H0 + V .

(22.79)

Next, introduce a projection operator P such that P = |Φ Φ|,

P|Φ = |Φ ,

Part B 22.4

Dirac–Coulomb–Breit Models These incorporate the full Coulomb gauge operator (22.73) or the less accurate Breit operator (22.74). The fully retarded operator is usually taken in the symmetrized form. The Gaunt operator (22.75) is sometimes considered as an approximation to the Breit operator.

Pnl (r) Ylml (θ, φ)χm s (σ) , r where χm s is a 2-component eigenvector of the spin operator s to lowest order in 1/c. 2. Extract effective operators to order 1/c2 . φnlml ,m s (x) = const.

336

Part B

Atoms

and its complement Q = 1 − P, projecting onto the complementary subspace H \ {Φ}. With the intermediate normalization

Consider now the case ! in which there are several unperturbed states, Φ (a) , a = 1, 2, . . . , d, having the same energy E 0 , which span a d-dimensional linear subspace (the model space) M ⊂ H, so that ! ! H0 Φ (a) = E 0 Φ (a) , a = 1, 2, . . . , d .

Φ|Ψ  = Φ|Φ = 1 , it follows that P|Ψ  = |Φ Φ|Ψ  = |Φ , so that the perturbed wave function can be decomposed into two parts: |Ψ  = (P + Q)|Ψ  = |Φ + Q|Ψ  . Thus, with intermediate normalization, E = Φ|H|Ψ  = E 0 + Φ|V |Ψ  . We now use this decomposition to write (22.79) in the form (E 0 − H0 )|Ψ  = (V − ∆E)|Ψ  ,

(22.80)

where ∆E = Φ|V |Ψ . Thus (E 0 − H0 )Q|Ψ  = Q(V − ∆E)|Ψ  .

Q , E 0 − H0

Let P be the projector onto M, and Q onto the orthogonal subspace M⊥ . ! The perturbed states Ψ (a) , a = 1, 2, . . . , d are related to the unperturbed states by the wave operator Ω, ! ! Ψ (a) = Ω Φ (a) , a = 1, 2, . . . , d . The effective Hamiltonian, Heff , is defined so that ! ! Heff Φ (a) = E (a) Φ (a) , and thus ! ! ! ΩHeff Φ (a) = E (a) Ψ (a) = HΩ Φ (a) . Thus on the domain M we can write an operator equation

(22.81)

Introduce the resolvent operator R=

22.4.4 Perturbation Theory: Open-Shell Case

ΩHeff P = HΩP ,

(22.83)

known as the Bloch equation. We now partition Heff so that (22.82)

Part B 22.4

which is well-defined except on {Φ}. Then the perturbation contribution to the wave function is Q|Ψ  = R(V −∆E)|Ψ  = R(V |Ψ −|Ψ  Φ|V |Ψ ) . The Rayleigh–Schrödinger perturbation expansion can now be written ! ! |Ψ  = |Φ + Ψ (1) + Ψ (2) + · · · E = E 0 + E (1) + E (2) + · · · The contributions are ordered by the number of occurrences of V , the leading terms being ! Ψ (1) = RV |Φ , !
 Ψ (2) = RVRV − R2 VPV |Φ , and so on. The corresponding contribution to the energy can then be found from ! E (n) = Φ|V Ψ (n−1) .

Heff P = (H0 + Veff )P , enabling a reformulation of (22.83) as the commutator equation [Ω, H0 ]P = (VΩ − ΩVeff )P .

(22.84)

With the intermediate normalization convention of Sect. 22.4.3, this becomes ! ! Ψ (a) = P Φ (a) so that PΩP = P and Heff P = PHΩP ,

Veff P = PVΩP .

Then (22.84) can be put in the final form [Ω, H0 ]P = (VΩ − ΩPVΩ)P .

(22.85)

The general Rayleigh–Schrödinger perturbation expansion can now be generated by expanding the wave operator order by order Ω = 1 + Ω (1) + Ω (2) + · · · ,

Relativistic Atomic Structure

and inserting into (22.85), resulting in a hierarchy of equations  (1)  Ω , H0 P = (V − PV )P = QVP , 
 (2)  Ω , H0 P = QVΩ (1) − Ω (1) PV P , (n) = PVΩ (n−1) . and so on, with Heff

22.4.5 Perturbation Theory: Algorithms The techniques of QED perturbation theory of Sect. 22.3.4 can be utilized to give computable expressions for perturbation calculations order by order. They exploit the second quantized representation of operators of Sect. 22.4.1 along with the use of diagrams to express the contributions to the wave operator and the energy as sums over virtual states. The use of Wick’s

22.5 Spherical Symmetry

337

theorem to reduce products of normally-ordered operators, and the linked-diagram or linked-cluster theorem are explained in Lindgren’s article [22.17] and Chapt. 5. Further references and discussion of features which can exploit vector-processing and parallel-processing computer architectures may be found in [22.18]. The theory can also be recast so as to sum certain classes of terms to completion. This depends on the possibility of expressing the wave operator as a normally ordered exponential operator Ω = {exp S} = 1 + {S} +

1 2 {S } + · · · , 2!

where the normally ordered operator S is known as the cluster operator. Expanding S order by order leads to the coupled cluster expansion (see also Chapts. 5 and 27).

22.5 Spherical Symmetry A popular starting point for most calculations in atomic and molecular structure is the independent particle central field approximation. This assumes that the electrons move independently in a potential field of the form 1 A0 (x) = φ(r) , r = |x| ; c Ai (x) = 0 , i = 1, 2, 3 .

(22.86)

Consider stationary solutions with energy E satisfying hˆ D ψ E (x) = Eψ E (x) . Since hˆ D is invariant with respect to rotation about r = 0, it commutes with the generators J1 , J2 , J3 mentioned in Sect. 22.1.1, corresponding to components of the total angular momentum j of the electron, usually decomposed into an orbital part l and a spin part s, j = l +s

(22.88)

l j = i jkl xk ∂l , 1 s j =  jkl σkl , 2

j = 1, 2, 3 j = 1, 2, 3 .

22.5.1 Eigenstates of Angular Momentum We can construct simultaneous eigenstates of the operators j 2 and j3 by using the product representation D (l) × D (1/2) of the rotation group SO(3), which is reducible to the Clebsch–Gordan sum of two irreps D (l+1/2) ⊕ D (l−1/2) .

(22.89)

We construct a basis for each irrep from products of basis vectors for D (1/2) and D (l) respectively. D (1/2) is a 2-dimensional representation spanned by the simultaneous eigenstates φσ of s2 and s3 3 s2 φσ = φσ , 4

s3 φσ = σφσ ,

1 σ =± , 2

for which we can use 2-rowed vectors



1 0 , φ−1/2 = . φ1/2 = 0 1 The representation D (l) is (2l + 1)-dimensional; its basis vectors can be taken to be the spherical harmonics  m  Yl (θ, ϕ) | m = −l, −l + 1, . . . , l ,

Part B 22.5

Clearly φ(r) is left unchanged by any rotation about the origin, r = 0, but transforms as the component A0 (x) of a 4-vector under other types of Lorentz and Poincaré transformation such as boosts or translations. However, solutions in central potentials of this form have a simple form which is convenient for further calculation. With this restriction on the 4-potential, Dirac’s Hamiltonian becomes   hˆ D = cα · p + eφ(r) + βm e c2 . (22.87)

where

338

Part B

Atoms

so that

The vectors (22.92) satisfy

l 2 Ylm (θ, ϕ) = l(l + 1)~2 Ylm (θ, ϕ) , l3 Ylm (θ, ϕ) = m ~ Ylm (θ, ϕ) .

j 2 χ j,m,a = j( j + 1)χ j,m,a ,

We shall assume that spherical harmonics satisfy the standard relations l± Ylm (θ, ϕ) = [l(l+1)−m(m ± 1)]1/2 ~Ylm±1 (θ, ϕ) , where l± = l1 ± l2 , so that

(22.93)

The parity of the angular part is given by (−1)l , with the two possibilities distinguished by means of the operator K = −( j 2 − l 2 − s2 + 1) = −(2s · l + 1)

1/2

2l + 1 Clm (θ, ϕ) , 4π

1/2 m m (l − m)! Cl (θ, ϕ) = (−1) Plm (θ) eimϕ , (l + m)! if m ≥ 0 , Ylm (θ, ϕ) =

Cl−m (θ, ϕ) = (−1)m Clm (θ, ϕ)∗ .

l 2 χ j,m,a = l(l + 1)χ j,m,a ,

3 s2 χ j,m,a = χ j,m,a , 4 1 l = j − a, a = ±1 . 2

(22.90)

Basis functions for the representations D j with j = l ± 12 have the form (The order of coupling is significant, and great confusion results from a mixing of conventions. Here we couple in the order l, s, j. The same spin-angle functions are obtained if we use the order s, l, j but there is a phase difference (−1)l− j+1/2 = (−1)(1−a)/2 . You have been warned!) χ j,m,a (θ, ϕ) # " 1 1 l, m − σ, , σ l, , j, m Ylm−σ (θ, ϕ)φσ = 2 2 σ

Part B 22.5

(22.91)

where l, m − σ, 12 , σ | l, 12 , j, m is a Clebsch–Gordan coefficient with 1 l = j − a, a = ±1, 2 m = − j, − j + 1, . . . , j − 1, j . Inserting explicit expressions for the Clebsch–Gordan coefficients gives   $ %1/2 m−1/2 j+1−m Y (θ, ϕ) − j+1/2   2 j+2 χ j,m,−1 (θ, ϕ) =  $ %1/2 , m+1/2 j+1+m Y (θ, ϕ) j+1/2 2 j+2  $ %1/2 m−1/2 j+m Y j−1/2 (θ, ϕ)  2j χ j,m,1 (θ, ϕ) = $ %1/2 . m+1/2 j−m Y j−1/2 (θ, ϕ) 2j (22.92)

so that K χ j,m,a = k χ j,m,a ,



1 a, k = − j + 2

(22.94)

a = ±1 .

The basis vectors are orthonormal on the unit sphere with respect to the inner product (χ j ,m ,a | χ jma )   † = χ j ,m ,a (θ, ϕ)χ j,m,a (θ, ϕ) sin θ dθ dϕ = δ j , j δm ,m δa ,a .

(22.95)

22.5.2 Eigenstates of Dirac Hamiltonian in Spherical Coordinates Eigenstates of Dirac’s Hamiltonian (22.87) in spherical coordinates with a spherically symmetric potential V(r) = eφ(r), hˆ D ψ E (r) = Eψ E (r) ,

(22.96)

are also simultaneous eigenstates of j 2 , of j3 and of the operator

K 0 , K= (22.97) 0 −K

where K is defined in (22.94) above. Denote the corresponding eigenvalues by j, m and κ, where 

1 . (22.98) κ =± j+ 2 Then the simultaneous eigenstates take the form

1 PEκ (r)χκ,m (θ, ϕ) , ψ Eκm (r) = (22.99) r iQ Eκ (r)χ−κ,m (θ, ϕ) where κ = −( j + 1/2)a is the eigenvalue of K , and the notation χκ,m replaces the notation χ j,m,a used previously in (22.91). The factor i in the lower component

Relativistic Atomic Structure

ensures that, at least for bound states, the radial amplitudes PEκ (r), Q Eκ (r) can be chosen to be real. This decomposition into radial and angular factors exploits the identity   F(r) χκ,m (θ, ϕ) σ·p r

 1 dF κF = i~ + χ−κ,m (θ, ϕ) (22.100) r dr r and gives a reduced eigenvalue equation  %  $

mc2 − E + V −c drd − κr P (r) Eκ  $ %  =0. Q Eκ (r) c drd + κr −mc2 − E + V (22.101)

Angular Density Distributions It is a remarkable fact that the angular density distribution

Aκ,m (θ, ϕ) = χκ,m (θ, ϕ)† χκ,m (θ, ϕ) ,

and 1 ( j − m)! A−( j+1/2),m (θ, ϕ) = 4π ( j + m)!  2 2 m+1/2 2 m−1/2 × ( j + m) P j−1/2 (µ) + P j−1/2 (µ) ,

Al,m (θ, ϕ)nr = Ylm (θ, ϕ)

|κ|

|m|

4π A|κ|,m (θ,ϕ)

1

1 2 3 2 1 2 5 2 3 2 1 2

1

2 3

3 2 2 sin θ 1 2 2 (1 + 3 cos θ) 15 4 8 sin θ 3 2 2 8 sin θ(1 + 15 cos θ) 3 2 2 2 4 (3 cos θ − 1) + 3 sin

2

2 2l + 1 (l − m)! |m| P (µ) ; = 4π (l + m)! l are listed in Table 22.2.

Radial Density Distributions The probability density distribution ρ Eκm (r) associated with the stationary state (22.99) is given by  1 ρ E,κ,m (r) = 2 |PE,κ (r)|2 Aκ,m (θ, ϕ) r  2 + |Q E,κ (r)| A−κ,m (θ, ϕ) . (22.103)

θ cos2 θ

Table 22.2 Nonrelativistic angular density functions l

|m|

4π Al,m (θ,ϕ)nr

0

0

1

1

1 0

3 2 2 sin θ 3 cos2 θ

2

2 1 0

15 4 8 sin θ 15 2 θ cos2 θ sin 2 5 2 θ − 1)2 (3 cos 4

Since Aκ,m does not depend on the sign of κ, the angular part can be factored so that ρ E,κ,m (r) =

D E,κ (r) A|κ|,m (θ, ϕ) , r2

where

  D E,κ (r) = |PE,κ (r)|2 + |Q E,κ (r)|2

(22.104)

defines the radial density distribution. Subshells in j–j Coupling The notion of a subshell depends on the observation that the set {ψ E,κ,m , m = − j, . . . , j} have a common radial density distribution. The simplest atomic model is one in which the electrons move independently in a mean field central potential. Since j  m=− j

ρ E,κ,m (r) =

2 j + 1 D E,κ (r) , 4π r2

(22.105)

a state of 2 j + 1 independent electrons, with one in each member of the set {ψ E,κ,m , m = − j, . . . , j}, has a spherically symmetric probability density. If E belongs to the point spectrum of the Hamiltonian, then (22.105) gives a distribution localized in r, and we refer to the states {E, κ, m}, m = − j, . . . , j as belonging to the subshell {E, κ}. The notations in use for Dirac central field states are set out in Table 22.3. Here l is associated with the orbital angular quantum number of the upper pair of

Part B 22.5

where µ = cos θ, was demonstrated by Hartree [22.19]. Angular densities for the lowest |κ| values are given in Table 22.1. The corresponding nonrelativistic angular densities

339

Table 22.1 Relativistic angular density functions

(22.102)

where m = − j, − j + 1, . . . , j − 1, j, is independent of the sign of κ; the equivalence of 1 ( j − m)! A j+1/2,m (θ, ϕ) = 4π ( j + m)!  2 2 m+1/2 2 m−1/2 × ( j − m + 1) P j+1/2 (µ) + P j+1/2 (µ) ,

22.5 Spherical Symmetry

340

Part B

Atoms

Table 22.3 Spectroscopic labels and angular quantum num-

bers Label:
 κ = − j + 12 a

s

p

p

d

d

f

f

−1

+1

−2

+2

−3

+3

−4

j = l + 12 a

1 2

1 2

3 2

3 2

5 2

5 2

7 2

a

1

−1

1

−1

1

−1

1

0

1

1

2

2

3

3

1

0

2

1

3

2

4

l= j− l¯ = j +

1 2a 1 2a

components and l¯ with the lower pair. Note the useful equivalence κ(κ + 1) = l(l + 1) . Defining κ¯ := −κ we have also κ( ¯ κ¯ + 1) = l¯(l¯ + 1).

22.5.3 Radial Amplitudes

Part B 22.5

Textbooks on quantum electrodynamics usually contain extensive discussions of the formalism associated with the Dirac equation but rarely go beyond the treatment of the hydrogen atom Chapt. 10. Greiner’s textbook [22.4] is an honorable exception, with many worked examples. A more exhaustive list of problems in which exact solutions are known is contained in [22.20]; it is particularly rich in detail about equations of motion and Green’s functions in external electromagnetic fields of various configurations; coherent states of relativistic particles; charged particles in quantized plane wave fields. It also incorporates discussion of extensions of the Dirac equations due to Pauli which include explicit interaction terms arising from anomalous magnetic or electric moments. Atoms and molecules with more than one electron are not soluble analytically so that numerical models are needed to make predictions. The solutions are sensitive to boundary conditions on which we focus in this section. For large r, solutions of (22.101) can be found proportional to exp(±λr), where & λ = + c2 − E 2 /c2 . (22.106) Thus λ is real when −c2 ≤ E ≤ c2 , and pure imaginary otherwise. Singular Point at r = 0 Singularities of the nuclear potential near r = 0 have a major influence on the nature of solutions of the Dirac equation. Suppose that the potential has the form Z(r) , V(r) = − (22.107) r

so that Z(r) is the effective charge seen by an electron at radius r from the nuclear center. The dependence of Z(r) on r may reflect the finite size of the nuclear charge distribution, so far treated as a point, or the screening due to the environment. Assume that Z(r) can be expanded in a power series of the form Z(r) = Z 0 + Z 1 r + Z 2r 2 + · · ·

(22.108)

in a neighborhood of r = 0. This property characterizes a number of well-used models 1. Point nucleus: Z 0 = 0; Z n = 0, n > 0. 2. Uniform nuclear charge distribution: 

 3Z r2  − 1− 2 , 0 ≤ r ≤ a , 2a 3a V(r) =  − Z , r>a. r (22.109)

This gives the expansion Z 0 = −3Z/2a, Z 1 = 0, Z 2 = +Z/2a3 , Z n = 0 for n > 2 when r ≤ a. 3. Fermi distribution: The nuclear charge density has the form ρ0 ρnuc (r) = , 1 + exp[(r − a)/d] where ρ0 is chosen so that the total charge on the nucleus is Z. Other nuclear models, reflecting the density distributions deduced from nuclear scattering experiments, can be found in the literature. Series Solutions Near r = 0 Any solution for the radial amplitudes of Dirac’s equation in a central potential

P(r) , u(r) = (22.110) Q(r)

with radial density D(r) = P 2 (r) + Q 2 (r) , can be expanded in a power series near the singular point at r = 0 in the form
 u(r) = r γ u 0 + u 1 r + u 2 r 2 + · · · , (22.111) where



pk uk = , qk

k = 1, 2, . . .

and γ, pk , qk are constants which depend on the nuclear potential model.

Relativistic Atomic Structure

Point Nuclear Models For a Coulomb singularity, Z 0 = 0, the leading coefficients satisfy

22.5 Spherical Symmetry

341

Finite nuclear models.

The behavior is entirely regular:

 P(r) = O r l+1 , Q(r) = O c−1 → 0 .

−Z 0 p0 + c(κ − γ)q0 = 0 , c(κ + γ) p0 − Z 0 q0 = 0 , so that

(22.112)

γ = ± κ2 −

(22.113)

Finite Nuclear Models Finite nuclear models, for which Z 0 = 0, have no singularity in the potential at r = 0. The indicial equation (22.113) reduces to γ = ±|κ|, so that for κ < 0, 
P(r) = p0r l+1 + O r l+3 , (22.114)
l+4  l+2 , Q(r) = q1r + O r (22.115)

with )
q1 / p0 = E − mc2 + Z 1 [c(2l + 3)] , q0 = p1 = 0 , and for κ ≥ 1, (22.116) (22.117)

with )
p1 /q0 = − E − mc2 + Z 1 [c(2l + 1)] , p0 = q1 = 0 . In both cases the solutions consist of either purely even powers or purely odd powers of r, contrasting strongly with the point nucleus case, where both even and odd powers are present in the series expansion. The Nonrelativistic Limit For a solution linked to a nonrelativistic state with orbital angular momentum l, one expects the nonrelativistic limit
 P(r) = O r l+1 , c → ∞ .

The limiting behavior reveals some significant features.

Z2 +··· , 2c2 |κ|

(22.113) shows that the leading coefficient p0 vanishes in the limit so that, 
 P(r) ≈ p1 r l+1 1 + O r 2 , when κ ≥ 1, l = κ . (22.118)

All higher powers of odd relative order vanish in the limit for both components. The behavior in the case κ < 0 is entirely regular.

22.5.4 Square Integrable Solutions * Square integrable solutions require D E,κ (r) dr to be finite; since the solutions are smooth, except possibly near the singular endpoints r → 0 and r → ∞, we focus on the behavior at the endpoints: r →∞ For real values of λ the condition ∞ D E,κ (r) dr < ∞ , 0 < R < ∞ , R

requires that PEκ (r), Q Eκ (r) are proportional to exp(−λr) with λ > 0. This means that bound states can only exist when E lies in the interval −c2 ≤ E ≤ c2 . Outside this interval solutions are necessarily of scattering type and so ∞ D E,κ (r) dr R

diverges when |E| > c2 . r →0 This limit requires

R

D E,κ (r) dr < ∞, 0

R > 0 .

Part B 22.5


P(r) = p1r l+1 + O r l+3 , 
Q(r) = q0r l + O r l+2 ,

Since γ = |κ| −

(

Z 02 , c2 c(κ + γ) q0 Z0 = = . p0 c(κ − γ) Z0

Point nuclear models.

342

Part B

Atoms

Since D E,κ (r) ∼ r ±2γ as r → 0, this condition holds when ±γ > − 12 . Only the solution with γ+> 0 satisfies the condition when |γ | > 12 , or Z < α−1 κ 2 − 1/4, and the solution with γ < 0 must be disregarded. This corresponds to the limit point case of a second-order differential operator [22.21]. In the special √ case |κ| = 1 or j = 12 this √ limits Z to be smaller than c 3/2 ≈ 118.6. For Z > c 3/2, both solutions are square integrable near the origin (the limit circle case) and the differential operator is no longer essentially self-adjoint. The Coulomb potential must have a finite expectation for any physically acceptable solution, so that we also require R

D E,κ (r)

dr < ∞, r

Substitute for E from (22.120) to get 1/2  N = (n r + γ)2 + α2 Z 2  1/2 = n 2 − 2n r (|κ| − γ) ,

where n = n r + |κ| is the principal quantum number, the exact equivalent of the principal quantum number of the nonrelativistic state to which the Dirac solution reduces in the limit c → ∞. With this notation, the radial amplitudes for bound hydrogenic states are PEκ (r)

 = N Eκ (c + E/c)1/2 ργ e−ρ/2 − n r M(−n r + 1,  2γ + 1; ρ) + (N − κ)M(−n r , 2γ + 1; ρ) ,

R > 0 .

0

This is always satisfied by the solution with γ > 0 for all |Z| < α−1 |κ|, but not by the solution with γ < 0. Imposing this condition restores essential self-adjointness (on a restricted domain) for 118 < Z ≤ 137.

(22.122)

Q Eκ (r)

 = N Eκ (c − E/c)1/2 ργ e−ρ/2 − n r M(−n r + 1,  2γ + 1; ρ) − (N − κ)M(−n r , 2γ + 1; ρ) ,

22.5.5 Hydrogenic Solutions

(22.123)

The wave functions for hydrogenic solutions of Dirac’s equation are presented in Sect. 22.8.2. Here we note some properties of hydrogenic solutions that reveal dynamical effects of relativity in the absence of screening by orbital electrons. In this case Z 0 = Z, Z n = 0, n > 0. When −c2 < E < c2 we have bound states. The parameter λ, (22.106), can conveniently be written

Part B 22.5

λ = Z/N ,

(22.119)

so that rearranging (22.106) gives ( Z2 E = +c2 1 − 2 2 , N c

where

N Eκ =

1/2

Table 22.4 Radial moments ρs  (22.120)

Z2 + O(1/c2 ) 2N 2 so that N is closely related to the principal quantum number, n, appearing in the Rydberg formula. As in Sect. 22.8.2, we write ρ = 2λr. Define the inner quantum number E = c2 −

NE , c2

Γ (2γ + n r + 1) αZ · 2 2N (N − κ) n r ! [Γ(2γ + 1)]2

is the normalization constant. For definitions of the confluent hypergeometric functions M(a, b; c; z) see [22.22, Sect. 13.1]. Table 22.4 lists expectation values of simple powers of the radial variable ρ = 2Zr/N from [22.23]

n r = 0, 1, 2, . . . .

−3l(l + 1)]

Relativistic a  2 N 2 (5N 2 − 2κ 2 )R2 (N )  +N 2 (1−γ 2 ) − 3κN 2 R(N )

1

3n 2 − l(l + 1)

−κ + (3N 2 − κ 2 )R(N )

0

1

1 nγ + (|κ| − γ)|κ| 2γN 3

s

Nonrelativistic 2

essentially equivalent to Sommerfeld’s fine structure formula. In the formal nonrelativistic limit, c → ∞, we have

n r = −a = −γ +

(22.121)

−1 −2 −3 a

2n 2 [5n 2 + 1

1 2n 2 1 2n 3 (2l + 1) 1 4n 3 l(l+1)(2l+1)

R(N ) =

+ 1 − Z 2 /N 2 c2

κ 2 R(N ) − sgnκ)

2γ 2 N 3 (2γ

N 2 + 2γ 2 κ 2 − 3N 2 κR(N ) 4N 5 γ(γ 2 − 1)(4γ 2 − 1)

Relativistic Atomic Structure

and [22.24]. Simple algebra, using the inequalities γ < |κ| and N < n, yields the inequality ρs nκ < ρs nl ,

s>0;

the inequality is reversed for s < 0. In the same way, it is easy to deduce that relativistic hydrogenic eigenvalues lie below the nonrelativistic eigenvalues nκ < nl . Thus, in the absence of screening, Dirac orbitals both contract and are stabilized with respect to their nonrelativistic counterparts. The relativistic and nonrelativistic expectation values approach each other as the relativistic coupling constant, Z/c = αZ → 0. This formal nonrelativistic limit is approached as α → 0 or c → ∞, in which the speed of light is regarded as infinite.

22.5.6 The Free Electron Problem in Spherical Coordinates The radial equation (22.101) for the free electron (V(r) = 0) gives a pair of first order ordinary differential equations 

κ d 2 − Q Eκ (r) , (mc − E)PEκ (r) = c dr r 


 κ d + PEκ (r) = mc2 + E Q Eκ (r) , (22.124) c dr r

(22.125)

where p2 = m 2 c2 − E 2 /c2 = p. p and the angular quantum numbers κ and κ¯ are associated respectively with the upper and lower components. These are defining equations of Riccati–Bessel functions [22.22, Sect. 10.1.1] of orders l and l¯ respectively, where κ(κ + 1) = l(l + 1),

κ( ¯ κ¯ + 1) = l¯(l¯ + 1) .

Thus the solutions of (22.125) are functions of the variable x = pr of the form PEκ (r) = Ax fl (x),

Q Eκ (r) = Bx fl¯(x) ,

where the ratio of A and B is determined by (22.124) and where fl (x) is a spherical Bessel function of the

343

first, second or third kind [22.22, Sect. 10.1.1]. Thus 1/2

E + mc2 x fl (x) , PEκ (r) = N πE 1/2

E − mc2 Q Eκ (r) = N sgn(κ) x fl¯(x) . πE (22.126)

Equations (22.124) require that Riccati–Bessel solutions of the same type be chosen for both components. The possibilities are: Standing Waves The two solutions of the same type are fl (x) = jl (x), fl (x) = yl (x). The jl (x) are bounded everywhere, including the singular points x = 0, x → ∞ and have zeros of order l at x = 0. The yl (x) are bounded at infinity but have poles of order l + 1 at x = 0. Progressive Waves The spherical Hankel functions (functions of the third kind) are linear combinations

h l(1) (x) = jl (x) + iyl (x),

h l(2) (x) = jl (x) − iyl (x) .

Recalling that p is real if and only if |E| > mc2 , we see that h l(1) (x), h l(2) (x) are bounded as x → ∞ and have poles of order l + 1 at x = 0. Notice that when |E| < mc2 , which does not occur for a free particle, p becomes pure imaginary and no solution exists which is finite at both singular points. The normalization constant N can be determined by using the well-known result ∞ π jl ( pr) jl ( p r)r 2 dr = 2 δ( p − p ) . 2p 0

The choice N = 1 ensures that ∞  † † PEκ (r)PE κ (r)+ Q Eκ (r)Q E κ (r) dr = δ( p− p ) . 0

Noting that δ(E − E ) =

dp δ( p − p ) , dE

and d p/ dE = c2 p/E gives ∞  † † PEκ (r)PE κ (r)+ Q Eκ (r)Q E κ (r) dr = δ(E− E ) . 0


1/2 when N = |E|/c2 p .

Part B 22.5

from which we deduce that

 κ(κ + 1) d2 PEκ (r) 2 + p − PEκ (r) = 0 , dr 2 r2 

κ( d2 Q Eκ (r) ¯ κ¯ + 1) 2 Q Eκ (r) = 0 , + p − dr 2 r2

22.5 Spherical Symmetry

344

Part B

Atoms

22.6 Numerical Approximation of Central Field Dirac Equations The main drive for understanding methods of numerical approximation of solutions of Dirac’s equation comes from their application to many-electron systems. Approximate wave functions for atomic or molecular states are usually constructed from products of one-electron orbitals, and their determination exploits knowledge gained from the treatment of one-electron problems. Whilst the numerical methods described here are strictly one-electron in character, extension to many-electron problems is relatively straightforward.

22.6.1 Finite Differences The numerical approximation of eigensolutions of the first order system of differential equations (22.101)

PEκ (r) E Q Eκ (r) %  $

mc2 + V(r) − c drd − κr P (r) Eκ  =  $ d κ% Q Eκ (r) c dr + r − mc2 + V(r) (22.127)

Part B 22.6

can be achieved by more or less standard finite difference methods given in texts such as [22.25]. For states in either continuum, E > mc2 or E < −mc2 , the calculation is completely specified as an initial value problem for a prescribed value of E starting from power series solutions in the neighborhood of r = 0. Solutions of this sort exist for all values of (complex) E except at the bound eigensolutions in the gap −mc2 < E < mc2 . For bound states, the calculation becomes that of a two-point boundary value problem in which the eigenvalue E has to be determined iteratively along with the numerical solution. We concentrate on the latter, which is more involved. It is convenient to write nκ = E nκ − mc , 2

(22.128)

so that  approaches the nonrelativistic eigenvalue in the limit c → ∞. For the one-electron problem, (22.101) can be written in the general form   dr du 1 dr + r + W(s) u(s) = χ(s) , (22.129) J ds c ds ds where u(s) and χ(s) are two-component vectors, such that



u(s) =

P(s) Q(s)

,J=

0 1 −1 0

,



−rV(r) −cκ W(s) = , −cκ 2rc2 − rV(r) and r(s) is a smooth differentiable function of a new independent variable s. This facilitates the use of a uniform grid for s mapping onto a suitable nonuniform grid for r. Common choices are rn = r0 esn ,

sn = nh,

n = 0, 1, 2, . . . , N ,

for suitable values of the parameters r0 and h, and 

rn = sn , n = 0, 1, 2, . . . , N , A rn + log 1 + r0 where A is a constant, chosen so that the spacing in rn increases exponentially for small values of n and approaches a constant for large values of n. The exponentially increasing spacing is appropriate for tightly bound solutions, but a nearly linear spacing is advisable to ensure numerical stability in the tails of extended and continuum solutions. The most convenient numerical algorithm involves double shooting from s0 = 0 and s N = Nh towards an intermediate join point s = Jh, adjusting  until the trial solutions have the right number of nodes and have left- and right-limits at s = Jh which agree to a pre-set tolerance (commonly about 1 part in 108 ). The deferred correction method [22.26, 27] allows the precision of the numerical approximation to be improved as the iteration converges. Consider the simplest implicit linear difference scheme for the first order system dy = F [y(s), s] , ds based on the trapezoidal rule of quadrature, is 1 z j+1 − z j = h(F j+1 + F j ) , 2

(22.130)

which has a local truncation error O(h 2 ). The precision can be improved, at the expense of increasing the computational cost per iterative cycle, by adding higher order difference terms to the right-hand side in (22.130). Use of the trial solution from the previous cycle leaves the stability properties of (22.130) are unaltered, but the converged solution has much higher accuracy. To apply this to the Dirac system, write f(s) = dr/ ds and h f(s j )W(s j ) . A±j = J ± 2c

Relativistic Atomic Structure

Also consider a slightly generalized problem in which that V(r) is replaced by a discretized potential U (ν) j may change from one iteration to the next as in a selfconsistent field calculation. The first iteration is (1) −(0) (1) A+(0) Uj j+1 U j+1 − A j   h + (0) + r j f(s j )U (1) r j+1 f(s j+1 )U (1) j j+1 2c  1  = h f(s j+1 )χ(s j+1 )(0) + f(s j )χ(s j )(0) , 2

(22.131)

where superscript 0 refers to initial estimates and superscript 1 to the result of the first iteration. On the (ν + 1)-th iteration, we solve (ν+1) −(ν) (ν+1) Uj A+(ν) j+1 U j+1 − A j   h r j+1 f(s j+1 )U (ν+1) + (ν) + r j f(s j )U (ν+1) j j+1 2c  1  = h f(s j+1 )χ(s j+1 )(ν) + f(s j )χ(s j )(ν) 2 1 (22.132) + δ3 U (ν) j+1/2 + · · · , 12 (ν) where δ3 U j+1/2 is the central-difference correction of order 3 [22.22, Sect. 25.1.2]. Higher order difference corrections (at least to order 5) are included in modern codes to improve the accuracy and numerical stability of weakly bound solutions. This deferred correction algorithm can be shown to converge asymptotically to the required solution of the differential system with a lo
cal truncation error of order O h 2 p+2 when difference corrections of order 2 p + 1 are employed [22.28].

Methods of solving the Dirac equation which represent the one-electron wave function as a linear combination of sets of square integrable functions (basis sets) have become popular in the last 10 years. Simple and rigorous criteria for choosing effective basis sets for this purpose are now available, and classes of functions that satisfy these criteria are known. Consequently, cheap and accurate calculations of the electronic structure of atoms and molecules are now a practical possibility. Finite difference algorithms generate eigensolutions one at a time. Basis set methods replace the differential operator hˆ D of (22.87) with a finite symmetric (in some cases complex Hermitian) matrix of dimension 2N. The spectrum of this operator, which is of course a pure point spectrum, consists of three pieces: N eigensolutions with E < −mc2 ( < −2mc2 ) representing the

345

eigenstates of the lower continuum; Nb < N eigensolutions in the gap −mc2 < E < mc2 (−2mc2 <  < 0) corresponding to bound states; and N − Nb eigensolutions with E > mc2 ( > 0) representing the eigenstates of the upper continuum. For properly chosen basis sets, the approximation properties of bound state eigensolutions are similar to those of the equivalent nonrelativistic eigensolutions. Solutions at continuum energies have the correct behavior near r = 0, but their amplitudes decrease exponentially like bound state solutions at large values of r. The criteria on which this description rests are as follows: A. The eigenstates of hˆ D are 4-component central field spinors whose components are coupled. The basis functions should therefore also consist of 4-component spinors of the form , 1 f κL (r)χκ,m (θ, ϕ) Φκm (r) = . (22.133) r i f κS (r)χ−κ,m (θ, ϕ) B. The spinor basis functions should, as far as practicable, satisfy the boundary conditions near r = 0 of Sect. 22.5.3. They should also be square integrable at infinity. C. Acceptable spinor basis functions should satisfy the relation i~

f L (r) f κS (r) χ−κ,m (θ, ϕ) → σ · p κ χκ,m (θ, ϕ) r r (22.134)

in the nonrelativistic limit, c → ∞. D. Acceptable spinor basis functions must have finite expectation values of component operators of hˆ D , namely α · p, β and V(r). Finite Basis Set Formalism Assume that each solution of the target problem is approximated as a linear combination 

L L 1 j cκ j f κ j (r)χκ,m (θ, ϕ) , ψκm (r) =  r i j cκS j f κSj (r)χ−κ,m (θ, ϕ) (22.135)

cκL j , cκS j

where j = 1 · · · N, are arbitrary constants, so that each j-term on the right-hand side has the form (22.133). This enables us to construct a Rayleigh quotient W[ψ] =

ψ|hˆ D |ψ , ψ|ψ

(22.136)

Part B 22.6

22.6.2 Expansion Methods

22.6 Numerical Approximation of Central Field Dirac Equations

346

Part B

Atoms

where both ψ|hˆ D |ψ and ψ|ψ are quadratic expressions in the expansion coefficients c Lj , c Sj . By requiring that W[ψ] shall be stationary with respect to arbitrary variations in the expansion coefficients, we arrive at the matrix eigenvalue equation





cκL SκLL 0 cκL , (22.137) fκ S =  SS 0 Sκ cκ cκS where the matrix Hamiltonian is denoted by

VκLL cΠκL S fκ = , cΠκSL VκSS − 2mc2 SκSS cκL , cκS are N-vectors, and VκLL , VκSS , SκLL , SκSS , Π κL S and Π κSL are all N × N matrices. Using superscripts T to denote either of the letters L, S, the elements of the matrices are defined by ∞ TT Sκij

=

T f iκT ∗ (r) f jκ (r) dr ,

(22.138)

0

∞ TT Vκij =

T f iκT ∗ (r)V(r) f jκ (r) dr ,

approximate trial solutions should have the correct analytic character as defined in Sects. 22.5.3 and 22.5.4. An expansion of f iκL (r) and f iκS (r) at r = 0 must reproduce this analytic behavior exactly if the approximation is to be physically reliable. The boundary conditions are part of the definition of a quantum mechanical operator; changing them gives a different operator with a different eigenvalue spectrum, so that trial functions which do not satisfy the boundary conditions of the physical problem cannot reproduce the physical solution. The behavior as r → ∞ is less crucial. Provided a bound wavefunction is well approximated over the region containing most of the electron density, the results are insensitive to many choices. C. The correct reduction of the Dirac equation to Schrödinger’s equation in the nonrelativistic limit (for example see [22.4, p. 97]) depends upon the operator identity p2 = (σ · p)(σ · p) . In the basis set formalism, the matrix equivalent of this equation is

(22.139)

1  L S SL Πκik Πκk j , 2 N

Tlij =

0

and

(22.142)

k=1

∞ LS Πκij

=



κ d S f jκ f iκL∗ (r) − + (r) dr , dr r

where ∞ Tlij =

0

(22.140)

Part B 22.6

∞ SL Πκij =

f iκS∗ (r)

κ d + dr r

 L f jκ (r) dr .

(22.141)

0

If f iκL (r) and f iκS (r) vanish at both r = 0 and r → ∞, then a simple integration by parts shows that ΠκL S and ΠκSL are Hermitian conjugate matrices. Physically Acceptable Basis Sets The four criteria described above are exploited in the following way:

A. The structure of (22.133) ensures (i) that the upper and lower components have properly matched angular behavior. It also emphasizes that the radial parts are part of a spinor structure which should be kept intact when making approximations. B. The nuclear singularity drives the dynamics of the electronic motion. It is therefore important that



d2 l(l + 1) L f jκ − 2+ (r) dr 2 dr r2

1 f iκL∗ (r)

0

is the ij-element of the nonrelativistic radial kinetic energy matrix. This is not true in general unless criterion C holds [22.29, 30]. The criterion can only be satisfied by matched pairs of functions f iκL (r), f iκS (r), ruling out all choices of basis set in which large and small components are not matched in pairs. Another way of viewing this result is to observe that for a general basis set, the sum over intermediate states in (22.142) is necessarily incomplete. The HermiL S = Π SL ensures that tian conjugacy property, Πκij κ ji the omitted terms give real and non-negative contributions. Thus all other choices of basis set cause (22.142) to underestimate the nonrelativistic kinetic energy [22.29] and to give spuriously large relativistic energy corrections. We emphasize that (22.134) need only be true in the limit c → ∞; however, basis sets used for finite values of c should be smooth functions of c−1 as c → ∞

Relativistic Atomic Structure

so that the equality i~

f κS (r) r

χ−κ,m (θ, ϕ) = σ · p

f κL (r) r

347

22.6.3 Catalogue of Basis Sets for Atomic Calculations χκ,m (θ, ϕ) (22.143)

holds in the limit. D. This ensures that the basis functions are in the domain of the Dirac operator; the meaning of this statement can be made precise in a functional analytic discussion such as in [22.3]. Some implications for the finite basis set approach are given in the author’s paper [22.15, pp. 235–253], which discusses the convergence of expectation values of operators for approximate Dirac wavefunctions obtained by this method. Here the main importance is that a (possibly singular) multiplicative operator V(r) (say, −Z/r) has N × N matrices VκLL , VκSS with finite elements. This must be true both for exact solutions and for approximations if the wave functions are to represent physical states. In particular, both matrices (N) must have a lowest eigenvalue Vmin say. Consider now the quantity (ψ|hˆ D (λ)|ψ), where 
hˆ D (λ) = cα · p + βm e c2 + λV(r) . With λ = 0 we have a free Dirac particle with a two-branched continuous spectrum E > mc2 and E < −mc2 . A negative definite V(r) has always (ψ|V(r)|ψ) > Vmin ; clearly, (N) Vmin ≥ Vmin > −2mc2 ,

22.6 Numerical Approximation of Central Field Dirac Equations

(22.144)

(22.145)

where x = 2λr is a scaled radial coordinate, with fixed λ which can be related to an energy parameter + E 0R = c2 1 − λ2 /c2 , and µ2 , a root of the equation µ4 − 2cµ2 /λ + 1 = 0, is given by % c$ 1 + E 0R /c2 . (22.146) µ2 = λ This choice ensures that f nLr κ (x) tends smoothly to the corresponding Coulomb Sturmian in the nonrelativistic limit c → ∞ [22.31]. L 2 boundary conditions are satisfied if αn r κ = Nn r κ λ/Z; when αn r κ = 1, then f nLr κ (x), f nSr κ (x) respectively coincide with the Dirac– Coulomb eigenfunctions Pnκ (r) and Q nκ (r) having principal quantum number n = n r + |κ|. The explicit form for L-spinors, in terms of Laguerre polynomials (2γ) (see Sect. 9.3.2), L nr (x), is  (2γ) (L) γ −x/2 f κ,n (x) = N x e − (1 − δn r ,0 )L n r −1 (x) n r ,κ r  (Nn r ,κ − κ) (2γ) L n r (x) , (22.147) + (n r + 2γ)  (2γ) (S) γ −x/2 f κ,n (x) = N x e − (1 − δn r ,0 )L n r −1 (x) n ,κ r r  (Nn r ,κ − κ) (2γ) L n r (x) , (22.148) − (n r + 2γ) where

Nn r κ =

n r ! (2γ + n r ) 2Nn r κ (Nn r κ − κ) Γ(2γ + n r )

1/2 (22.149)

is chosen so that the diagonal elements gnκ r ,n r of the Gram (or overlap) matrix are unity for both large and

Part B 22.6

for all values of N. So if we increase λ from 0 to 1, the eigenvalues of trial solutions corresponding to eigenvalues in the upper continuum at λ = 0 will be smoothly decreasing functions of λ bounded below by Vmin for all values of N. It follows that the upper set of
eigenvalues has a fixed lower bound in the gap − mc2 , mc2 for each finite matrix approximation. If the basis set satisfies suitable completeness criteria in an appropriate Hilbert space as N → ∞ (see [22.15, pp. 235–253], [22.31] for more details) we see that, if (22.144) holds for all values of N, the infinite ) sequence {E (N N+i , N = N0 , N0 + 1, . . . } of eigenvalues approximating the ith bound state has a finite lower bound, and therefore, by the completeness of the real numbers,
it must havea limit point E i in the bound state gap − mc2 , mc2 . Thus Rayleigh–Ritz approximations for Dirac’s Hamiltonian converge in the same fashion as the corresponding nonrelativistic Rayleigh–Ritz approximations [22.30, 31].

A. L-Spinors: L-spinors [22.31] are related to Dirac hydrogenic functions in much the same way as Sturmian functions [22.32, 33] are related to Schrödinger hydrogenic functions (Sect. 22.3). They are solutions of the differential equation system     1 αn r κ Zµ2 d κ L − +  −  f n r κ (x) cx dx x  2    µ  = 0 ,   κ 1 Z d S µ f n r κ (x) + − − dx x 2 αn r κ µ2 cx

348

Part B

Atoms

small components. Both Gram matrices are tri-diagonal with non-zero off-diagonal elements

where T = L, S, gm (θ, r) = r θ e−λm r , A L = A S = 1, B L = B S = 0 for κ < 0 ,  (κ + 1 − N1,κ )(2γ + 1)   AL =    2(N1,κ − κ)  (κ − 1 − N1,κ )(2γ + 1) for κ > 0 , S A =   2(N1,κ − κ)     BL = BS = 1

(κ) = g(n gn(κ) r ,(n r +1) r +1),n r

 ηT (n r + 1)(2γ + n r + 1)(Nn r κ − κ) 1/2 = , 2 Nn r κ N(n r +1),κ (N(n r +1),κ − κ)

(22.150)

ηL

ηS

where T = L, S, = −1 and = +1. This convention facilitates the construction of the blocks of the matrix Hamiltonian (22.137), which are banded when the operators are the powers r n , n > −1. The properties of Laguerre polynomials ensure that the matrix of the Coulomb potential is diagonal. For a full discussion of L-spinors, their orthogonality and completeness properties, and applications to hydrogenic atoms see [22.31]. L-spinors are most useful for hydrogenic problems, either for isolated atoms or for atoms in strong electromagnetic fields (see Chapt. 13). The equivalent nonrelativistic Coulomb Sturmians have for a long time been used to study the Zeeman effect on high Rydberg levels, especially in the region where chaotic behavior is expected [22.34] (see Chapt. 15). B. S-Spinors: S-spinors have the functional form of the most nearly nodeless L-spinors characterized by the minimal value of n r , and can be viewed as the relativistic analogues of Slater functions (STOs). When κ is negative, take n r = 0, so that (L) (S) (x) = − f κ,0 (x) f κ,0

Part B 22.6

N0,κ − κ (2γ) L 0 (x) . 2γ When κ is positive, we must take n r = 1, and then = Nκ,0 x γ exp(−x/2)

(L) f κ,1 (x) = N1,κ x γ e−x/2   N1,κ − κ (2γ) (2γ) L (x) , × −L 0 (x) + 1 + 2γ 1 (S) f κ,1 (x) = N1,κ x γ e−x/2   N1,κ − κ (2γ) (2γ) L 1 (x) . × −L 0 (x) − 1 + 2γ These can be simplified by inserting the ex(2γ) (2γ) plicit expressions L 0 (x) = 1, L 1 (x) = 2γ + 1 − x. We define a set of S-spinors with exponents {λm , m = 1, 2, . . . , N} by rewriting the above in the form

f m(T ) (r) = A T gm (γ, r) + B T gm (γ + 1, r) , (22.151)

(22.152)

and γ=

& κ 2 − Z 2 /c2 ,

N1,κ =

& κ 2 + 2γ + 1 .

The choice of the set of positive real exponents {λm , m = 1, 2, . . . , N}, must be such as to assure Rayleigh–Ritz convergence [22.15, pp. 235–253)] and to maximize the rate at which it is attained. In particular, if one particular exponent is chosen to have the value λm = Z/Nn r ,κ , then the corresponding S-spinor is a true hydrogenic solution. In this case the trial solution is exact. Clearly, S-spinors inherit desirable properties of L-spinors and, in particular, satisfy criteria A–D. All elements of the matrix Hamiltonian of the Dirac hydrogenic problem can be expressed in terms of Euler’s integral for the gamma function [22.22, Sect. 6.1.1]: ∞ Γ(z) = k

z

t z−1 e−kt dt ,

(Rz > 0 , Rk > 0)

0

and are therefore readily written down and evaluated. The effectiveness of this method depends upon the choice of exponent set: see D below. We refer to calculations using this scheme for many-electron systems in Sect. 22.7. C. G-Spinors: The G-spinors are the relativistic analogues of nonrelativistic spherical Gaussians (SGTO), popular in quantum chemistry for studying the electronic structure of atoms and molecules. They satisfy the relativistic boundary conditions for a finite size nuclear charge density distribution at r = 0, and are therefore the most convenient for relativistic molecular electronic structure calculations. They are defined so that (22.143) holds for finite c as well as in the nonrelativistic limit, which is equivalent to 

κ d f m(S) (r) = const. + f m(L) (r) . (22.153) dr r

Relativistic Atomic Structure

Thus, if (L) l+1 −λm r r e , (22.154) f m(L) (r) = Nl,m   2 (S) (κ + l + 1)r l − 2λm r l+2 e−λm r . f m(S) (r) = Nl,m 2

(22.155)

Note that the leading term in (22.155) vanishes when κ < 0, so that the radial amplitude r −1 f m(S) (r) is never singular, even in the s-state case when κ = −1, l = 0. D. Exponent Sets for S- and G-Spinors: Quantum chemists are familiar with the use of nonrelativistic STO and GTO basis sets, and there are extensive collections of optimized exponents which permit economical calculations for atomic and molecular calculations [22.35–37]. These sets are a good starting point for relativistic calculations also. By and large, the compilations ignore mathematical completeness, which although desirable is unattainable in practice. However, basis sets can almost always be constructed to give adequate numerical precision for most purposes. An effective alternative to optimization, especially for atoms, is to use geometrical sequences {λm } of the form

λm = α N β m−1 N ,

m = 1, 2, . . . , N ,

(22.156)

where a, b are positive constants. Experience shows that no linear dependence problems (caused by illconditioning of the ST matrices) are encountered when β N > 1.2 for S-spinors, with N up to about 30, or β N > 1.5 for G-spinors with N up to about 50. E. Other Types of Analytic Basis Sets; Variational Collapse: The earliest work with atoms [22.38,39] used STO functions of the form {r γ exp(−λm r), m = 1, . . . N} for both large and small components, whilst Kagawa [22.40, 41] used integer powers instead of the noninteger γ . Drake and Goldman [22.42] used functions of the form

{r γ +i exp(−λr), i = 0, . . . N − 1}. For hydrogenic problems, these worked well for negative κ states, but gave a single spurious eigenvalue for positive κ, which could be simply deleted from the basis set. Various test calculations are included in the review article [22.43, Sect. IV]. Other attempts to use GTOs in the early 1980’s led to problems interpreted as a failure of the Rayleigh–Ritz method because of the presence of “negative energy states” with a spectrum unbounded below: so-called “variational collapse”. It is clear that all these approaches fail to observe three, and sometimes all, of the four criteria for acceptable basis sets. They are incapable of satisfying the physical boundary conditions, and it is therefore hardly surprising that they give unphysical spectra. Several procedures have been advocated to overcome the problem, of which the two most popular are “kinetic balance” and projection operators. Kinetic balance, suggested by Lee and McLean [22.44], advocates augmenting a GTO basis, common to both large and small components, with additional functions to “balance the set kinetically”. This appears to “fix up” the problem for the upper spectrum, but introduces spurious states, mainly in the lower part of the spectrum, as well as increasing the size of the small component basis set. There is no rigorous nonrelativistic limit, and no mathematical proof of convergence such as that guaranteed by criteria A–D. A model with spurious negative energy states cannot furnish a consistent physical interpretation of negative energy solutions as positron states, expected of a proper relativistic theory. If “variational collapse” is attributed to the absence of a lower bound to the Dirac spectrum as a whole, the idea of introducing a projection operator to eliminate collapse seems attractive. This is easy to do for free electrons, where the operators hˆ D ± E , 2|E| select positive/negative energy solutions. Unfortunately, this cannot be done in the presence of a potential except by an approximation which complicates calculations and reduces the efficiency of algorithms. The “negative energy sea” also depends upon the choice of potential; perturbing the potential (as long as it does not change the domain of the Hamiltonian) induces a unitary transformation taking one set of eigenstates into another which inevitably mixes the old positive and negative energy states. For example a relativistic calculation on a hydrogenic atom in which the nuclear Λ± =

349

Part B 22.6

which depend upon just two parameters α N , β N . A convenient way to do this is to find a pair α N0 , β N0 for small N0 , say N0 = 9, in a cheap and simple nonrelativistic calculation and then to increase N systematically using relations such as 

b αN ln β N βN − 1 a N0 = , or = , α N0 β N0 − 1 ln β N0 N

22.6 Numerical Approximation of Central Field Dirac Equations

350

Part B

Atoms

charge is perturbed gives incorrect answers if the negative energy contribution to the perturbation series is omitted [22.45]. In any event, the finite matrix eigensolutions include both positive energy and negative energy states. It is therefore a simple matter to exclude the negative energy states if their contribution is expected to be negligible; this is the no virtual-pair approximation. The negative energy solutions are inert spectators for most atomic processes, just as are those positive energy solutions which lie deep in the atomic core. It is easy to go beyond the no virtual-pair approximation if the physical problem demands it.

Finite Element Methods: Johnson et al. [22.46, 47], following earlier work on relativistic ion–ion collisions by Bottcher and Strayer [22.48], popularized the use of a basis of B-splines in relativistic atomic calculations. See Sect. 8.1.1. The method has mainly been of use in relativistic many-body calculations on the spectra of heavy ions. See Sect. 21.6 for spline-Galerkin representations in nonrelativistic atomic structure, such as [22.49]. Parpia and Fischer explored the spline-Galerkin approach for the Dirac equation [22.50], but this method has not been extended so far to relativistic many-electron atoms.

22.7 Many-Body Calculations 22.7.1 Atomic States The construction of atomic many-electron wavefunctions from products of central field Dirac orbitals is employed to simplify the algorithms for calculating electronic structures and properties. This can be either in the context of expansions in Slater determinants of the traditional type, or by use of Racah algebra. A complete description of the methods of the latter sort used in popular computer codes is found in [22.27, Sect. 2].

22.7.2 Slater Determinants An antisymmetric state of N independent electrons in configuration space can be constructed in the form

Part B 22.7

{α1 , α2 , . . . , α N } = x1 , x2 , . . . , xn |aα† 1 aα† 2 · · · aα† N |0

(22.157)

ψα1 (x1 ) ψα2 (x1 ) · · · ψα N (x1 ) ψα1 (x2 ) ψα2 (x2 ) · · · ψα N (x2 ) ··························· ψα1 (xN ) ψα2 (xN ) · · · ψα N (xN ) This Slater determinant is an antisymmetric  eigenfunction of H0 corresponding to the energy E αn and of the angular momentum projection  J3 = j3,αn corresponding to the eigenvalue M3 = m 3,αn . Defining the parity of a Dirac electron orbital as that of its upper component, (−1)lαn , we see that this has parity Π(−1)lαn . 1 = N!

22.7.3 Configurational States Configurational state functions (CSF) having specified total angular momentum J and parity Π can be con-

structed by vector  addition of the individual angular momenta: J = jαn . We write such states as  γJM|m α1 , m α2 . . . , m α N  φ(γJM ) = {m αn }

× {α1 , α2 , . . . , α N } ,

(22.158)

where γJM|m α1 , m α2 . . . , m α N  is a generalized Clebsch–Gordon coefficient, and γ defines the angular momentum coupling scheme. A list of orbital quantum numbers, {α1 , α2 , . . . , α N } defines an electron configuration. If the configuration belongs to a single subshell, then the states share a common set of labels {n, κ} where n is the principal quantum number. In j − j coupling, the α-subshell states of Nα equivalent electrons can therefore be identified (we can suppress the projection Mα and the parity Πα ) by the labeling α Nα , γα , Jα , where γα distinguishes degenerate states of the same Jα . For j − j coupling, such labels are needed only for j ≥ 52 ; the seniority scheme, [22.27, Sects. 2.3, 2.4)], provides a complete classification for j < 92 . A list of states of configurations j N , classified in terms of the seniority number v and of total angular momentum J, appears in Table 22.5.

22.7.4 CSF Expansion Atomic state functions (ASF) are linear superpositions of CSF’s, of the form Ψ(γΠJ ) =

N 

cα φ(γα J ) ,

(22.159)

α=1

where cα are a set of (normally) real coefficients. These coefficients are usually chosen so that Ψ(γΠJ ) is an

Relativistic Atomic Structure

Table 22.5 j N configurational states in the seniority

scheme. The multiplicity of each unresolved degenerate state is indicated by a superscript j

N

v

J

1 2

0, 2 1 0, 4 1, 3 2 2 0, 6 1, 5 2, 4

0 0 0 1 0 2 0 1 0 2 1 3 0 1 0 2 1 3 0 2 4 0 1 0 2 1 3 0 2 4 1 3 5

0

3 2

5 2

3 7 2

0, 8 1, 7 2, 6 3, 5 4

9 2

0, 10 1, 9 2, 8 3, 7

5 7 2

0 3 2

0 2 0 5 2

0 2, 4 5 2 3 9 2, 2

351

underlying the first are straightforward and may be found in atomic physics texts and review articles such as [22.26, 27]. The use of second quantization and diagrammatic methods of the quantum theory of angular momentum provides a powerful means of reducing matrix elements between atomic CSF’s to a linear combination of radial integrals in a systematic way. The method, which is fully explained in [22.27], leads to a complete classification of matrix element expressions for all the one- and two-electron operators treated in this chapter. A full implementation within the j − j coupling seniority scheme is available in various versions of the GRASP code [22.51–53].

22.7.6 Dirac–Hartree–Fock and Other Theories

0

The notation above echoes that of the nonrelativistic theory of Chapt. 21, and it is possible to proceed along similar lines.

0 2, 4, 6 7 2 3 5 9 11 15 2, 2, 2, 2 , 2

0 2, 4, 6 2, 4, 5, 8 0 9 2

0 2, 4, 6, 8 9 2 3 5 7 9 11 13 15 17 21 2, 2, 2, 2, 2 , 2 , 2 , 2 , 2

0 2, 4, 6, 8 0, 2, 3, 42 , 5, 62 , 7, 8, 9, 10, 12 9 2 3 2, 3 2,

5 2, 5 2,

7 2, 7 2,

9 2, 9 2,

11 2 , 11 2 ,

13 2 , 13 2 ,

15 2 , 15 2 ,

17 2 , 17 2 ,

21 2 19 25 2 , 2

eigenstate of the many-electron Hamiltonian matrix in a finite subspace of CSF’s.

22.7.5 Matrix Element Construction A full presentation of the reduction of matrix elements between CSF’s to computable form is beyond the scope of this chapter. There are two approaches: one is based on expanding all CSF’s and ASF’s in Slater determinants, whilst the other exploits the properties of central field orbital spinors. The principles

Dirac–Hartree–Fock Theory Dirac–Hartree–Fock theory works exactly as described in Sect. 21.4; relativistic counterparts of Koopmans’ theorem, fixed-core approximations, Brillouin’s theorem are easy to obtain. The properties of Dirac–Hartree– Fock functions closely resemble those of Hartree–Fock functions, though allowance must
be made for the fact  that, for example, n p orbitals with κ = −2, j = 32
and n p¯ orbitals with κ = +1, j = 12 have different spatial distributions as a consequence of the dynamical and indirect effects of relativity. For further insight see [22.23, 26]. Most such calculations are currently made with updated versions of the codes of Desclaux [22.54] or Grant [22.51–53] which rely on finite difference methods resting on the techniques of Sect. 22.6.1. Further details may be found in the code descriptions. Finite Matrix Methods for Atoms and Molecules In view of the rapid pace of development of finite matrix methods, especially for the treatment of relativistic molecular electronic structure in the Born–Oppenheimer (fixed nucleus) approximation, it seems appropriate to give a brief outline of the extension of the oneelectron equations of Sect. 22.4.2 to the many-electron case. The method of approximation generalizes the onebody approximation scheme of Sect. 22.6.2 to the manybody problem based on the effective Hamiltonian of

Part B 22.7

4, 6

1 2

22.7 Many-Body Calculations

352

Part B

Atoms

Sect. 22.4.1. This leads to an energy functional of the form E = E0 + E1

(22.160)

where E 0 is the expected value of H0 (22.76) and E 1 the expected value of H1 for the finite basis many-body trial function. This leads to matrix Dirac–Fock equations of the form FX = ESX .

where cκTp are the expansion coefficients. The Breit interaction matrices have the similar form  ¯¯ T¯ T¯ (2 j + 1)eν ( jj )DκT Trs K κν,TT BκTTpq = pq,κ rs , κ rs

(22.169)

and ¯

T BκTpq =

(22.161)

 ¯ (2 j + 1)DκT Trs ν

κ rs

  T¯ T¯ T ν,T T¯ T¯ T

× dν (κκ )K κν,T pq,κ rs + gν (κκ )Mκ pq,κ rs .

In general, the Fock matrix F is a sum of several matrices F = f +g+b ,

ν

(22.162)

(22.170)

where, for each symmetry κ and nuclear center, A, of the molecule, f can be partitioned into blocks

LS LL cΠ Aκ VAκ f Aκ = (22.163) . cΠ SL V SS − 2mc2 SSS

The matrix elements are constructed from standard radial integrals

Aκ

Aκ

Aκ

The matrix

J LL − K LL −K L S g= −K SL J SS − K SS

(22.164)

Part B 22.7

is the matrix of the Breit interaction. In the atomic (one nuclear center) case, following [22.55], these matrices can also be blocked by symmetry κ. Using superscripts T to label the L or S components, and the notation T¯ to denote the complementary label: T¯ = S when T = L or T¯ = L when T = S, then the direct Coulomb part JκTT has matrix elements % $  T¯ T¯ 0,TT T¯ T¯ + D J JκTTpq = (2 j +1) DκTT rs Jκ0,TTTT



κ rs pq,κ rs κ pq,κ rs , κ rs

(22.166) TT

whilst the exchange part K κ has the form 

TT



K κTTpq = (2 j + 1)bν ( jj )DκTT rs K κν,TT pq,κ rs , ν

(22.167)

where TT denotes any combination of component la

bels. Here DκTT is a density matrix with elements



T DκTTpq = cκTp∗ cκq ,

∞ ∞ T f κTp (r1 ) f κq (r1 )Uν (r1 , r2 ) 0

×

is the matrix of the Coulomb repulsion part of the electron–electron interaction and

BLL BL S . b= (22.165) BSL BSS

κ rs



T Jκν,TTT pq,κ rs =

(22.168)

0



f κT r (r2 ) f κT s (r2 ) dr2

dr1

(22.171)

where

 r ν /r ν+1 Uν (r1 , r2 ) = 1 2 r ν /r ν+1 2

1

for r1 < r2 , for r1 > r2 .

Similarly







TT ν,TTT T K κν,TT pq,κ rs = Jκ p,κ r,κq,κ s

(22.172)

and T¯ T¯ T Mκν,T pq,κ rs

∞ ∞ =

¯

f κTp (r1 ) f κT r (r1 )Uν (r1 , r2 ) 0

r1 ¯

T (r2 ) f κT s (r2 ) dr2 dr1 . × f κq

(22.173)

Further details about the coefficients bν ( jj ), eν ( jj ), dν (κκ ) and gν (κκ ) may be found in [22.55]. This formalism has been implemented for closed shell atoms with both S-spinors and G-spinors [22.55]. Computational aspects of calculating the radial integrals using S-spinors are discussed in [22.56, 57], and can be adapted with relatively small modifications to G-spinor basis sets. As yet, there have been relatively few applications by comparison with codes based on finite difference methods, but the potential can be gauged from papers such as [22.55, 58–61], which deal with Dirac– Fock and Dirac–Fock–Breit calculations, many-body perturbation theory and coupled-cluster schemes. G-spinor basis sets provide the most promising technique for application to the electronic struc-

Relativistic Atomic Structure

ture of molecules; computer codes are under active development. Electron Correlation in Atomic Calculations Here we use the term correlation to denote methods which go beyond the single determinant approximation of Dirac–Hartree–Fock theory. These include configuration interaction schemes, in which each ASF is represented as a linear combination of CSF’s built from previously determined orbital spinors and multiconfiguration Dirac–Fock calculations in which the orbitals are optimized simultaneously. Calculations representative of state of the art techniques will be found in [22.62, 63]. Many-body perturbation theory calculations and coupled-cluster calculations are not well suited to calculations with finite difference codes, because of the expense of calculating more than a limited orbital basis and all the matrix elements required. Calculations based on finite matrix methods enable this sort of calculation to be done more economically. Some justification for the use of finite matrix methods in relativistic many-body theory is given in [22.15, pp. 235–253]. The relativistic version of quantum defect theory [22.64, 65] also gives insight into the competing roles of relativistic dynamics and screening in atoms. Compared with nonrelativistic quantum defect theory, it has been under-used.

22.7.7 Radiative Corrections

in Feynman gauge is   ∆E a = lim R − iαπ.mc2 ψ¯ a (x2 )γ µ SF (x2 , x1 ) Λ→∞

× γ ν ψa (x1 )gµν DFΛ (x2 − x1 ) d3 x2 d3 x1  × d(t2 − t1 ) − δm(Λ) ψa |β|ψa  , (22.174)

where δm(Λ) =

  α 2 3 3 mc ln(Λ2 ) + . π 4 8

This represents the contribution from virtual processes involving the exchange of a single photon. The photon propagator has been modified to give the photon
 and effective mass Λ, so that the denominator of D q 2 (22.69) becomes q 2 − Λ2 + iδ. The two parts of this formula diverge as Λ → ∞, though the limit of their difference is finite. This makes calculation difficult and expensive. There are several approaches: 1. For atomic number Z
20, an expansion in powers of the electron–nucleus coupling parameter αZ = Z/c is satisfactory. 2. At larger atomic numbers an expansion in αZ evidently fails to converge, and nonperturbative methods must be sought. This too is computationally difficult and expensive. The results for hydrogenic ions have been tabulated [22.66] for atomic numbers in the range 1 ≤ Z ≤ 100. (See [22.10, Chapt. 2] for an up-to-date summary biased towards applications to the spectroscopy of highly-ionized atoms.) 3. Processes involving more than one virtual photon are hard to calculate, and have mostly been ignored. See [22.10] for references. Vacuum Polarization The contribution of vacuum polarization is next in order of importance in the list of radiative corrections in atoms. As shown by (22.49), the nuclear potential generates a current in the vacuum that is responsible for a short-range screening of the nuclear charge. This can be represented as a local perturbing potential which is easy to take into account [22.67–69].

22.7.8 Radiative Processes Electron Self-Energy For a one-electron system, the renormalized expression for the self-energy of an electron in the state a

353

The operator j µ (x)Aµ (x) which occurs in the interaction Hamiltonian (22.61) describes processes in

Part B 22.7

The term “radiative corrections” is usually interpreted to mean QED contributions to energies, expectation values or rates of atomic or molecular processes that arise from interaction between the electron–positron and photon fields, apart from those directly attributable to the nonrelativistic Coulomb interaction. This includes the relativistic and retardation effects embodied in the effective interaction between electrons as well as contributions from processes that are not so included. We consider two such processes, the electron self-energy and the vacuum polarization, which involve interactions of the same formal order as those giving rise to the covariant electron–electron interaction discussed above, but which are formally infinite. These are the lowest order processes requiring renormalization. See [22.7, 8, 10, 15] for more details.

22.7 Many-Body Calculations

354

Part B

Atoms

which the number of photons present can increase or decrease by one. The Fock space operator may be written    (ρ)† (ρ) aa† ab qρ† Mab (t) + aa† ab qρ Mab (t) , Hint = a,b ρ

(22.175)

where the first set of terms in the sum represents emission of a photon in the mode labeled ρ and the second to absorption of a photon by the same initial state. † The operators aa and aa are anticommuting annihilation and creation operators of electrons, whilst qρ and † qρ are commuting annihilation and creation operators of

photons. If ω denotes photon frequency, then (ρ)†

(ρ)

(ρ)

(ρ)

Mab (t) = Mab ei(Ea −Eb +ω)t , Mab (t) = Mab ei(Ea −Eb −ω)t , where (ρ)

Mab =

$ ω %1/2   ψa† (x) Φ (ρ) (x) + cα · A(ρ) (x) πc × ψb (x) d3 x

is the transition amplitude. For a discussion of this expression including the effect of gauge transformations on the computed amplitudes, the elimination of angular coordinates for atomic central field orbitals and connection with the nonrelativistic limit, see [22.10, 16, 27].

22.8 Recent Developments 22.8.1 Technical Advances

Part B 22.8

Relativistic atomic structure continues to develop to meet modern demands for high quality calculations on many-electron atoms. The computing power now available makes it possible to carry out multi-configurational Dirac–Hartree–Fock (MCDHF) or configuration interaction (CI) calculations on a scale unimaginable when this chapter was first drafted. Some of the software currently available is surveyed below. On the theoretical side, there have been new technical applications of tensor operator theory. Whilst the approach initiated by Fano [22.27, 70] continues to be the basis on which many relativistic and nonrelativistic calculations are based, recent work aims to simplify the calculation, not only by exploiting second quantization techniques and the coupling of tensor operators, but by better utilization of quasispin methods [22.71–74]. A new jj-coupling package along these lines [22.75] has been constructed for evaluation of fractional parentage coefficients, reduced fractional parentage coefficients (in which the dependence on particle number is extracted as a quasispin 3 j-symbol), matrix elements of unit tensors T k and double tensor operators W kq k j , from which to construct many-particle matrix elements of physical operators. Fritzsche et al. [22.76–80] have recently published utilities which exploit the capabilities of the Maple computer algebra system to evaluate Racah algebra expressions.

22.8.2 Software for Relativistic Atomic Structure and Properties Many software packages for relativistic atomic physics calculations can now be downloaded from the internet. The earliest codes, which generate many-electron wavefunctions and bound energy levels, taking account of the full relativistic electron–electron interaction and QED corrections, of Desclaux [22.54] and Grant et al. [22.51], though now much modified, are still in use, as is the code of Chernysheva and Yakhontov [22.81]. These codes can use various (MC)DHF and CI procedures, albeit with not more than a few hundred CSF. A more recent version of Grant et al.’s code appeared in 1989 [22.52] and GRASP92 embodied major changes to the user interface and to file-handling to permit calculations with very large CSF sets [22.53]. Most earlier calculations were of the AL or EAL type, in which a large number of states are treated together using a common orbital set. These are cheap and work well for highly ionized, few-electron systems but the results only have modest accuracy. More accurate treatment of electron correlation requires MCDHF calculations on single levels (OL calculations) or small groups of fine structure levels (EOL calculations). The CSF sets are chosen through some active space (AS) procedure as in nonrelativistic MCHF [22.82]; complete active spaces (CAS) are often too large for practical use, so that the AS must be restricted in some way, for example by using only SD (single and double) replacements from the reference CSF set. With such large CSF basis sets it is not

Relativistic Atomic Structure

practical or desirable to diagonalize the complete Hamiltonian matrix, and Davidson’s version [22.83, 84] of the Lanczos algorithm, as implemented by Stathopoulos and Fischer [22.85], is used in GRASP92 to construct the small number of eigenvalues and eigenvectors of physical importance. This approach generally gives highly accurate wavefunctions and energy levels for a small number of atomic states. Each state is determined in a separate SCF calculation, and therefore has its own set of orbitals. The GRASP software for calculating radiative transition probabilities was based on the assumption that initial and final states of a transition are described by the same orbital set. Most if this machinery can still be used by way of a procedure to express sets of non-orthogonal orbitals as a biorthonormal system [22.86]. An adaptation for GRASP92 was used, for example, to calculate radiative transition probabilities for lines of the C III spectrum [22.87] and the oscillator strengths of the n d 2 D3/2 − (n + 1)p 2 P01/2,3/2 lines in Lu (n = 5) and Lw (n = 6) which are very sensitive to correlation effects [22.88]. These two calculations involved CSF sets of order 300,000. Desclaux’s code, which uses an expansion of the many-electron wavefunction in determinantal wavefunctions rather than the Fano approach using jj-coupled CSFs, has simi-

References

355

larly been modernized [22.89]; its capabilities are rather similar to those of GRASP. There is no published description. GRASP92 has been enhanced recently with new utilities to calculate hyperfine interactions [22.90–92] and isotope shifts [22.93]. Fritzsche et al. have developed a new suite of programs, RATIP (an acronym for Relativistic Atomic Transition and Ionization Properties), which uses MCDHF wavefunctions from GRASP92 to study a range of atomic properties [22.94, 95]. Like Desclaux’s package, this expresses jj-coupled symmetry functions in terms of Slater determinants [22.96] and also provides the relevant utilities for coefficients of fractional parentage and the calculation of angular coefficients. The package supports CI calculations of ASF and energy levels taking account of the Breit interaction and QED estimates. A new utility [22.97] permits calculation of relaxed orbital radiative transition probabilities and lifetimes within the RATIP framework. The code generates continuum orbitals, which enable calculation of Auger energies, relative intensities and angular distributions, and should also enable calculation of photoionization cross-sections and angular distributions. The papers cited contain information on how to obtain the programs, many of which are also obtainable from the Computer Physics Communications International Program Library [22.98].

References

22.2 22.3 22.4 22.5 22.6 22.7 22.8 22.9 22.10

B. G. Wybourne: Classical Groups for Physicists (Wiley, New York 1974) J. P. Elliott, P. G. Dawber: Symmetry in Physics (Macmillan, Basingstoke, London 1979) B. Thaller: The Dirac Equation (Springer, Berlin, Heidelberg 1992) W. Greiner: Relativistic Quantum Mechanics (Springer, Berlin, Heidelberg 1990) L. L. Foldy: Phys. Rev. 102, 568 (1956) Yu. M. Shirokov: Sov. Phys. JETP 6, 568,919,929 (1958) S. S. Schweber: Introduction to Relativistic Quantum Field Theory (Harper Row, New York 1964) V. B. Berestetskii, E. M. Lifshitz, L. P. Pitaevskii: Relativistic Quantum Theory (Pergamon, Oxford 1971) C. Itzykson, J.-B. Zuber: Quantum Field Theory (McGraw-Hill, New York 1980) L. N. Labzowsky, G. L. Klimchitskaya, Yu. Yu. Dmitriev: Relativistic Effects in the Spectra of Atomic Systems (Institute of Physics Publishing, Bristol 1993)

22.11 22.12 22.13 22.14

22.15

22.16 22.17 22.18

22.19

W. H. Furry: Phys. Rev. 81, 115 (1951) I. P. Grant, H. M. Quiney: Adv. At. Mol. Phys. 23, 37 (1988) I. Lindgren: J. Phys. B 23, 1085 (1990) H. A. Bethe, E. E. Salpeter: Quantum Mechanics of One-, and Two-Electron Systems (Springer, Berlin, Heidelberg 1957) W. R. Johnson, P. J. Mohr, J. Sucher (Eds.): Relativistic, Quantum Electrodynamic, and Weak Interaction Effects in Atoms (American Insitute of Physics, New York 1989) I. P. Grant: J. Phys. B 7, 1458 (1974) I. Lindgren in 22.15, pp. 3–27 D. J. Baker, D. Moncrieff, S. Wilson: Vector processing and parallel processing in many-body perturbation theory calculations for electron correlation effects in atoms and molecules. In: Supercomputational Science, ed. by R. G. Evans, S. Wilson (Plenum Press, New York 1990) pp. 201– 209 D. R. Hartree: Proc. Camb. Phil. Soc. 25, 225 (1929)

Part B 22

22.1

356

Part B

Atoms

22.20

22.21

22.22 22.23 22.24 22.25

22.26 22.27 22.28 22.29 22.30 22.31 22.32 22.33 22.34 22.35

22.36

22.37

Part B 22

22.38 22.39 22.40 22.41 22.42 22.43 22.44 22.45 22.46 22.47 22.48

22.49

V. G. Bagrov, D. M. Gitman: Exact Solutions of Relativistic Wave Equations (Kluwer Academic, Dordrecht 1990) E. A. Coddington, N. Levinson: Theory of Ordinary Differential Equations (McGraw-Hill, New York 1955) M. Abramowitz, I. A. Stegun: Handbook of Mathematical Functions (Dover, New York 1965) V. M. Burke, I. P. Grant: Proc. Phys. Soc. 90, 297 (1967) J. Kobus, J. Karwowski, W. Jaskolski: Phys. Rev. A 20, 3347 (1987) G. Hall, J. M. Watt: Modern Numerical Methods for Ordinary Differential Equations (Clarendon Press, Oxford 1976) I. P. Grant: Adv. Phys. 19, 747 (1970) I. P. Grant: Methods in Computational Chemistry, Vol. 2 (Clarendon Press, Oxford 1976) I. P. Grant: Phys. Scr. 21, 443 (1980) K. G. Dyall, I. P. Grant, S. Wilson: J. Phys. B 17, 493 (1984) I. P. Grant: J. Phys. B 19, 3187 (1986) I. P. Grant, H. M. Quiney: Phys. Rev. A 62, 022508 (2000) M. Rotenberg: Ann. Phys. 19, 262 (1962) M. Rotenberg: Adv. At. Mol. Phys. 6, 233 (1970) C. W. Clark, K. T. Taylor: J. Phys. B 15, 1175 (1982) P. ˘Carsky, M. Urban: Ab Initio Calculations. Methods, and Applications in Chemistry (Springer, Berlin, Heidelberg 1980) S. Huzinaga, J. Andzelm, M. Klobukowski, E. Radzio-Andselm, Y. Sakai, H. Tatewaki (Eds.): Gaussian Basis Sets for Molecular Calculations (Elsevier, Amsterdam 1984) R. Poirier, R. Kari, I. G. Csizmadia: Handbook of Gaussian Basis Sets (Elsevier, Amsterdam 1985) Y.-K. Kim: Phys. Rev. 154, 17 (1967) Y.-K. Kim: Phys. Rev. A 159, 190 (1967) T. Kagawa: Phys. Rev. A 12, 2245 (1975) T. Kagawa: Phys. Rev. 22, 2340 (1980) G. W. F. Drake, S. P. Goldman: Phys. Rev. A 23, 2093 (1981) G. W. F. Drake, S. P. Goldman: Adv. At. Mol. Phys. 25, 393 (1988) Y. S. Lee, A. D. McLean: J. Chem. Phys. 76, 735 (1982) H. M. Quiney, I. P. Grant, S. Wilson: J. Phys. B 18, 2805 (1985) W. R. Johnson, S. A. Blundell, J. Sapirstein: Phys. Rev. A 37, 307 (1988) W. R. Johnson, S. A. Blundell, J. Sapirstein: Phys. Rev. A 40, 2233 (1988) C. Bottcher, M. R. Strayer: Pair production at GeV/u energies. In: Atomic Theory Workshop on Relativistic and QED Effects in Heavy Atoms, AIP Conference Proceedings No 136 (AIP, New York 1985) pp. 268– 298 T. Brage, C. F. Fischer: Phys. Scr. 49, 651 (1994)

22.50 22.51

22.52

22.53 22.54 22.55 22.56

22.57

22.58

22.59

22.60

22.61 22.62 22.63 22.64 22.65 22.66 22.67 22.68 22.69 22.70 22.71 22.72 22.73

C. F. Fischer, F. A. Parpia: Phys. Lett. A 179, 198 (1993) I. P. Grant, B. J. McKenzie, P. H. Norrington, D. F. Mayers, N. C. Pyper: Comput. Phys. Commun. 21, 207 (1980) K. G. Dyall, I. P. Grant, C. T. Johnson, F. A. Parpia, E. P. Plummer: Comput. Phys. Commun. 55, 425 (1989) F. A. Parpia, C. F. Fischer, I. P. Grant: Comput. Phys. Commun. 94, 249 (1996) J. P. Desclaux: Comput. Phys. Commun. 9, 31 (1975) see also 13, 71 (1977) H. M. Quiney, I. P. Grant, S. Wilson: J. Phys. B 20, 1413 (1987) H. M. Quiney: Relativistic atomic structure calculations I: Basic theory and the finite basis set approximation. In: Supercomputational Science, ed. by R. G. Evans, S. Wilson (Plenum Press, New York 1990) pp. 159–184 H. M. Quiney: Relativistic atomic structure calculations II: Computational aspects of the finite basis set method. In: Supercomputational Science, ed. by R. G. Evans, S. Wilson (Plenum Press, New York 1990) pp. 185–200 S. Wilson: Relativistic molecular structure calculations. In: Methods in Computational Chemistry, Vol. 2, ed. by S. Wilson (Plenum Press, New York 1988) p. 73 H. M. Quiney: Relativistic many-body perturbation theory. In: Methods in Computational Chemistry, Vol. 2, ed. by S. Wilson (Plenum Press, New York 1988) p. 227 H. M. Quiney, I. P. Grant, S. Wilson: On the relativistic many-body perturbation theory of atomic and molecular electronic structure. In: Many-Body Methods in Quantum Chemistry, Lecture Notes in Chemistry, Vol. 52, ed. by U. Kaldor (Springer, Berlin, Heidelberg 1989) pp. 307–344 H. M. Quiney, I. P. Grant, S. Wilson: J. Phys. B 23, L271 (1990) S. Fritzsche, I. P. Grant: Phys. Scr. 50, 473 (1994) A. Ynnerman, C. F. Fischer: Phys. Rev. A 51, 2020 (1995) W. R. Johnson, K. T. Cheng: J. Phys. B 12, 863 (1979) C. M. Lee, W. R. Johnson: Phys. Rev. A 22, 979 (1980) W. R. Johnson, G. Soff: At. Data Nucl. Data Tables 33, 405 (1985) E. A. Uehling: Phys. Rev. 48, 55 (1935) E. H. Wichmann, N. M. Kroll: Phys. Rev. 101, 843 (1956) L. W. Fullerton, G. A. Rinker: Phys. Rev. A 13, 1283 (1976) U. Fano: Phys. Rev. A 140, A67 (1965) Z. B. Rudzikas: Theoretical Atomic Spectroscopy (Cambridge Univ. Press, Cambridge 1997) G. Gaigalas, Z. B. Rudzikas, C. Froese Fischer: J. Phys. B 30, 3747 (1997) G. Gaigalas: Lithuanian J. Phys. 39, 80 (1999)

Relativistic Atomic Structure

22.74 22.75 22.76 22.77 22.78 22.79 22.80 22.81 22.82

22.83 22.84 22.85

J. Kaniauskas, Z. B. Rudzikas: J. Phys. B 13, 3521 (1980) G. Gaigalas, S. Fritzsche: Comput. Phys. Commun. 134, 86 (2001) S. Fritzsche: Comput. Phys. Commun. 103, 51 (1997) S. Fritzsche, S. Varga, D. Geschke, B. Fricke: Comput. Phys. Commun. 111, 167 (1998) G. Gaigalas, S. Fritzsche, B. Fricke: Comput. Phys. Commun. 135, 219 (2001) T. Inghoff, S. Fritzsche, B. Fricke: Comput. Phys. Commun. 139, 297 (2001) S. Fritzsche, T. Inghoff, T. Bastug, M. Tomaselli: Comput. Phys. Commun. 139, 314 (2001) L. V. Chernysheva, V. L. Yakhontov: Comput. Phys. Commun. 119, 232 (1999) C. F. Fischer, T. Brage, P. Jönsson: Computational Atomic Structure. An MCHF Approach (Institute of Physics Publishing, Bristol, Philadelphia 1997) E. R. Davidson: J. Comput. Phys. 17, 87 (1975) E. R. Davidson: Comput. Phys. Commun. 53, 49 (1989) A. Stathopoulos, C. F. Fischer: Comput. Phys. Commun. 79, 268 (1994)

22.86 22.87 22.88 22.89 22.90 22.91 22.92 22.93 22.94 22.95 22.96 22.97 22.98

References

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J. Olsen, M. R. Godefroid, P. Jönsson, P. Å. Malmqvist, C. F. Fischer: Phys. Rev. E 52, 4499 (1995) P. Jönsson, C. F. Fischer: Phys. Rev. A 57, 4967 (1998) Y Zou, C. F. Fischer: Phys. Rev. Lett. 88, 183001 (2002) J. P. Desclaux, private communication, December 2003 ´ , P. Jönsson, C. F. Fischer: Phys. Rev. A 53, J. Bieron 1 (1995) ´ , I. P. Grant, C. F. Fischer: Phys. Rev. A 58, J. Bieron 4401 (1998) ´ , C. F. Fischer, I. P. Grant: Phys. Rev. A 59, J. Bieron 4295 (1999) P. Jönsson, C. F. Fischer: Comput. Phys. Commun. 100, 81 (1997) S. Fritzsche: J. Elec. Spec. Rel. Phenom. 114–116, 1155 (2001) S. Fritzsche: Phys. Scr. T 100, 37 (2002) S. Fritzsche, J. Anton: Comput. Phys. Commun. 124, 353 (2000) S. Fritzsche, C. F. Fischer, C. Z. Dong: Comput. Phys. Commun. 124, 340 (2000) The CPC Program Library home page is at http://www.cpc.cs.qub.ac.uk/cpc/

Part B 22

359

23. Many-Body Theory of Atomic Structure and Processes

Many-Body Th All atoms except hydrogen are many-body systems, in which the interelectron interaction plays an important or even decisive role. The aim of this chapter is to describe a consistent method for calculating the structure of atoms and the characteristics of different atomic processes, by applying perturbation theory to take into account the interelectron interaction. This method involves drawing a characteristic diagram based on the structure or process. This is then used to create an analytical expression to the lowest order in the interelectron interaction. Higher-order corrections are subsequently generated. This technique was invented about half a century ago in quantum electrodynamics by Feynman [23.1], then modified and adjusted for multiparticle systems by a number of authors. Its application to atomic structure and atomic processes required further modifications, which were initiated at the end of the fifties (see, e.g., [23.2]) and later. The corresponding technique was successfully applied to the calculation of a wide variety of characteristics and processes in many papers and several review articles [23.3, 4]. The increasing amount of experimental data available has led to improved accuracy for this technique, so that it can be applied to current problems considering not only atoms and ions, both positive and negative [23.5–8], but also molecules [23.9], clusters [23.10] and fullerenes.

Diagrammatic Technique...................... 23.1.1 Basic Elements.......................... 23.1.2 Construction Principles for Diagrams............................. 23.1.3 Correspondence Rules ................ 23.1.4 Higher-Order Corrections and Summation of Sequences.....

23.2 Calculation of Atomic Properties ........... 23.2.1 Electron Correlations in Ground State Properties ......... 23.2.2 Characteristics of One-Particle States ................ 23.2.3 Electron Scattering .................... 23.2.4 Two-Electron and Two-Vacancy States ...................................... 23.2.5 Electron–Vacancy States............. 23.2.6 Photoionization in RPAE and Beyond .................. 23.2.7 Photon Emission and Bremsstrahlung ..................

360 360 360 362 363 365 365 366 367 369 370 371 374

23.3 Concluding Remarks ............................ 375 References .................................................. 376

The elements of the diagrammatic technique, which form a convenient and simple “language”, are given together with the rules for creating “sentences” using basic “words”. A kind of “dictionary” helps to translate diagrammatic “sentences” into analytical expressions suitable for calculations.

be taken into account. The sum of all many-body diagrams is completely equivalent to the many-particle Schrödinger equation. Therefore, taking all of them into account is just as complicated as solving the corresponding equation. Compared with other approaches, the diagrammatic technique can easily uncover hidden approximations and transparently demonstrate possible sources of corrections.

Part B 23

An essential part of the program is to learn how the simplest approximation can be improved, and what are the mechanisms and processes connected with, and responsible for, higher-order corrections. When the diagrammatic technique of many-body theory is used, it is unnecessary to be restricted to a finite number of lowest-order terms in the interelectron interaction. On the contrary, some infinite sequences may

23.1

360

Part B

Atoms

23.1 Diagrammatic Technique 23.1.1 Basic Elements

diagrams:

Each physical atomic process (or a process with participation of a molecule, cluster or fullerenes) involves an electronic interaction with a projectile or external field, in general time-dependent, or a mutual interelectronic interaction. By convention, the ground state of the atom (if it is not degenerate) is regarded as the vacuum state. Then the simplest process in this target is excitation of an electron to an unoccupied level, leaving behind a vacancy. The basic elements of a diagram are a)

b)

c)

d)

e)

a)

b)

c)

(23.3)

Just as for (23.2), diagrams (23.3) have their timereversed counterparts. Inclusion of interelectron interaction leads to a number of processes of which some examples are

f) a)

b)

c)

d)

e)

f)

(23.1)

where (a) with an arrow directed to the right represents an electron excited to a vacant level; (b) with an arrow directed to the left represents a vacancy; (c) with a cross represents the static Coulomb interaction; (d) represents the interelectron Coulomb interaction; (e) represents interaction with a time-dependent external field, usually electromagnetic; and (f ) represents the very act of interaction. The elements (23.1a–f ) in combination can describe the following real or virtual basic processes a)

b)

c)

(23.2)

Part B 23.1

which represent (a) photon absorption by the vacuum with electron–vacancy pair creation; (b) electron excitation; and (c) vacancy excitation. Diagrams (23.2) depict processes as developing in time, shown increasing from left to right. A vacancy can be thought of as an antiparticle to the electron, moving backward in time. The time-reverse of processes (23.2) represent processes of photon emission due to annihilation of an electron–vacancy pair, vacancy transition, and electron inelastic scattering, respectively. A static [for example, Coulomb (23.1c)] field can virtually create an electron–vacancy pair, or affect the moving electron or vacancy, as shown in the following

(23.4)

Here, (23.4a) describes creation of two electron– vacancy pairs, (23.4b) represents the simplest picture of electron inelastic scattering, (23.4c) depicts vacancy decay with electron–vacancy pair creation, (23.4d) stands for electron–electron scattering, (23.4e) represents a process which can be called electron–vacancy annihilation and creation, while (23.4f ) shows electron– vacancy scattering.

23.1.2 Construction Principles for Diagrams The foundations of diagrammatic techniques are discussed in a number of books such as [23.11]. This chapter presents recipes for the construction and evaluation of diagrams corresponding to various atomic processes [23.12]. The basic procedure is to connect the initial and final states of the atom, drawn at the left and right sides of the diagram, using any of the elements in (23.1). In doing so, the following rules apply: 1. At each dot (23.1f ), only three lines can meet: wavy (or dashed) and electron–vacancy. 2. A vacancy cannot be transformed into an electron or vice versa.

Many-Body Theory of Atomic Structure and Processes

3. Electrons and vacancies can be created only pairwise from the vacuum. 4. Only linked diagrams are allowed; i. e. only those having no parts entirely disconnected from one another. The simplest or initial approximation to a process is represented by a diagram which includes the lowest possible number of elements (23.1–23.4). Higher-order corrections can be derived by including additional elements of interaction with the static field of the nucleus (23.1c) and between electrons and/or vacancies (23.1d). As an illustration of the method, consider the following three processes: 1. one-electron photoionization – the initial state is a photon while the simplest final state is an electron– vacancy pair. They can be combined together giving the basic diagram (23.2a). 2. elastic electron scattering – the initial and final states are single electrons. To describe the simplest scattering process, the interaction with the Coulomb field must be taken into account, leading to (23.3b). To account for interelectron interaction, the simplest element

(23.5)

a)

or b)

3. inelastic electron scattering – the initial state is a single electron. For the final state we choose one with two electrons and therefore a single vacancy. The simplest diagram in this case is given by (23.4b). To illustrate the description of the ground state characteristics, consider the contributions to the ground state energy of an atom. If this state is not degenerate, its potential energy is given by vacuum diagrams which have no free lines in the initial or final states. The simplest vacuum diagrams are a)

b)

c)

(23.7)

Higher-order corrections to all these diagrams can be obtained by adding elements such as a static external field (23.3) or interelectron interaction (23.4) without changing the initial and final states of the processes. The lowest-order processes are represented by (23.2a), (23.5), (23.6), (23.4b), and (23.7). There are many corrections even in the next order of interaction, either with an external field or with electrons or vacancies. To illustrate, only one correction to each process will be presented: 1. Simple photoionization (23.2a) may be combined with (23.4e) to obtain

(23.8)

This describes the effect of the creation of another electron–vacancy pair, after annihilation of the first one formed by absorption of the initial photon. 2. Simple elastic electron scattering (23.5) can be combined with an extra interaction term (23.4f ) between the incoming electron and the vacancy of the loop (23.5), to obtain

(23.6)

Diagram (23.6a) is obtained from (23.5) by permutation of the electron lines on one side of the interelectron interaction. Diagram (23.6b) is equivalent to (23.6a), but is simpler to draw.

361

(23.9)

Part B 23.1

must be introduced. It is a modification of (23.4b) accounting for the interaction of an incoming electron with all target electrons individually, not altering their states. This is emphasized by the loop in which the same vacancy leaving the lower dot reenters it. Indistinguishability of all electrons as fermions is taken into account by permutation of the electron (vacancy) line ends, as illustrated in the following diagrams:

23.1 Diagrammatic Technique

362

Part B

Atoms

3. Simple inelastic electron scattering (23.4b) can also be combined with (23.4f ), accounting for the interaction of an electron and vacancy created in the lowest-order process, to obtain

(23.10)

4. The ground state energy term (23.7b) can be combined with (23.7b) and the element (23.3c) of the interaction between the vacancy in the (23.7b) loops and the static field to obtain

Coulomb interelectron interaction. Each of the letters p, q, t, s represents a full set of n,
, m
, σ quantum numbers. Vacancy states are below (and include) the highest occupied energy level, called the Fermi level, so that p ≤ F. Electron states are above the Fermi level so that q > F. Thus diagram (23.2a) is represented by  p|W|q with p ≤ F and q > F. Apart from initial and final states, each diagram can have sections, i. e., intervals between successive interactions. For instance (23.9) and (23.10) each have one section. Each section is represented by an inverse energy −1 denominator
εd . It includes the sum over all vacancy energies vac εi minus the sum of the electron energies
ε to which the entrance energy E of the diagram n el (e.g. ~ω for a time-dependent field) must be added: ε−1 d

=

  vac

(23.11)

Higher-order corrections can be constructed step by step by introducing further elements of interaction. In some cases, classes of diagrams may be taken into account up to infinite order by solving closed systems of integral or differential equations.

Part B 23.1

Diagram (23.2) →  p|W|q , Diagram (23.3) →  p|U|q , Diagram (23.4) →  pt|V |qs ,

−1 εn + E

.

(23.13)

el

The second correspondence rule is to identify sections and write down their energy denominators. After attributing to each electron (vacancy) line a letter, denoting its state, the analytical expression for a diagram is given by Analytical Expression = (the product of all interaction matrix elements) × (all energy denominators)−1 × (−1) L summed over all intermediate electron and vacancy states,

23.1.3 Correspondence Rules These rules describe how to obtain an analytical expression corresponding to a given diagram. One starts by choosing a zero-order approximation which can be that of independent electrons moving in the Coulomb field of an atomic nucleus. Atoms with completely occupied shells, or subshells having a non-degenerate ground state, can be chosen as the vacuum. Electron (vacancy) states are characterized in this case by the quantum numbers n,
, m
, and σ = ±1/2. The first correspondence rule is to substitute a matrix element for each interaction:

εi −



(23.14)

where L is equal to the sum of the total number of vacancy lines and closed vacancy or electron–vacancy loops. Although electrons are fermions, the summation in (23.14) has no additional restrictions caused by the Pauli principle. It runs over all electron (> F ) and vacancy (≤ F ) states, including those where two or more electrons (or vacancies) are in the same state. The correspondence rules (23.12), (23.13), and (23.14) can be illustrated by giving as examples the analytical expressions of two diagrams (23.8) and (23.9). Attributing letters denoting electron and vacancy states, diagram (23.8) becomes

(23.12)

where W is the interaction potential of an electron with the external time-dependent field, U is the interaction potential of an electron (vacancy) with an external static field, for example that of the nucleus, and V is the

ω

r t

f i

(23.15)

Many-Body Theory of Atomic Structure and Processes

According to (23.12–23.14), the analytical formula   t|W|rri|V |t f  Ai f (ω) = (−1)2+1 εt − εr + ω r>F,t≤F

(23.16)




is obtained. The symbol includes summation over discrete levels and integration. In (23.16), the intermediate state is r > F, t ≤ F and the diagram has two vacancies (t and i) and one loop rt. Integration must be performed over those states r which belong to the continuum. Assigning letters denoting states, (23.9) appears as p

q

p'

r t

(23.17)

where p, q, p , r > F, while t ≤ F. According to (23.14),    pt|V |qrqr|V | p t ∆E = (−1)1+1 , εt − εq − εr + ε p r,q>F;t≤F

(23.18)

where the intermediate states are q, r > F and t ≤ F. It has one vacancy and one electron–vacancy loop rt. An intermediate state in a diagram can be real or virtual. It is real if the energy conservation law can be fulfilled, i. e. if for some values of the section energy the following relation holds:   E= εn − εi . (23.19)

363

An intermediate state is virtual if the energy conservation law (23.19) is violated for all values of the section energy. In general, the
bigger the virtuality,
i. e. the difference E − el εn + vac εi , the smaller the contribution to the amplitude of the process.

23.1.4 Higher-Order Corrections and Summation of Sequences An important feature of the diagrammatic technique is the convenience in constructing higher-order corrections and in the summation of infinite sequences of diagrams. According to (23.13), each new interaction line leads to an additional interaction matrix element, extra energy denominator and summation over new intermediate states. An important example of infinite summation is that of determining the one-electron states. The interaction with the nucleus (23.3b) and with atomic electrons (23.5) and (23.6) is not small and must be taken into account nonperturbative; i. e., these elements must be iterated infinitely. To simplify the drawing, only the element (23.5) is repeated, leading to the diagrammatic equation: p

p =

p

q

+

p

q

+ ... 1

p =

q'

+

p

i

q

2

i

i'

+

vac

el

η→0

vac

el

 =P E − 



εn +

−1

εi

vac

el

− iπδ E −



 el

εn +



 εi

,

(23.20)

vac

for Here P denotes that the principal value is to be taken on integration over intermediate state energies. The result of (23.20) can thus be complex.

i (23.21)

Indeed, everything in the infinite sum which is in front of the dashed line repeats the infinite sum itself, thus leading to a closed equation of the form

 p| ˜ =  p| +

  q>F

 pi|V ˜ |qi

1 q| . −εq + ε p

(23.22)

The two interactions leading to (23.21) can be permuted, so that the interaction 1 can be after 2. This leads to

Part B 23.1

If (23.19) can be fulfilled, a prescription for avoiding the singularity in (23.13) is to substitute the expression ε−1 d Q, where −1    εn + εi + iη Q = lim E −

ε−1 d .

23.1 Diagrammatic Technique

364

Part B

Atoms

extension of the sum to include states with q ≤ F. As a result, the summation in (23.22) must be performed over all states q. Interaction with the nucleus and the other electrons affects also the occupied (or vacancy) states i in (23.21) and therefore the latter must be modified by inserting the elements (23.3b), (23.5), and (23.6) into them. Here again, the vacancy line in (23.5) and (23.6) must be modified by including the corrections (23.3b), (23.5), (23.6) and so on. Finally, the diagrammatic equation p

p

p

=

q

p

+

q

+

p + i

i i p

p

p

i

+

+ q

q

When a perturbative approach is used, it is essential to define the zero-order approximation. In this chapter, and very often in the literature, the Hartree–Fock approximation is used in this role. To simplify the drawing of diagrams, from now on single (rather than double) lines will represent electrons (vacancies), whose wave functions are determined in the HF approximation by (23.24). Obviously, in this case elements (23.3a), (23.5) and (23.6) should not be added to any other diagrams. The procedure used in deriving (23.21) and (23.23) is in fact more general. Let us separate all diagrams describing elastic scattering which do not include a single one-electron or one-vacancy state as intermediate. Depicting their total contribution by a square, the precise one-particle state is determined by an infinite sequence of iterative diagrams which can be summed, similarly to (23.21), by p

+ q

p

p

=

+

+…

+

q (23.23)

q Σˆ

p

q' Σˆ

+

q Σˆ

p Σˆ

+… q

is obtained. The doubled line for i emphasizes that the vacancy wave function is determined by an equation similar to (23.21). The corresponding analytical equation looks like (23.22), but includes also the Coulomb interaction with the nucleus and the exchange interaction with other atomic electrons. The summation over q in this equation is extended over all q, not only q > F. Multiplying the corresponding equation by ( Hˆ 0 − ε p ) from the right (atomic units are used in this chapter: 2 e = m e = ~ = 1), where Hˆ 0 = −∇  /2, and using the
 completeness of the functions q |qq| = δ(r − r ), results in the equation  −

Part B 23.1

  dr  ∇2 Z   2 φ p (r) − + |φ (r )| − ε i p 2 r |r  − r| i≤F  dr  φ∗ (r  )φ p (r  )φi (r) (23.24) = |r  − r| i i≤F

for the electron wave function φ p (r). Here φi (r) are wave functions determined by equations similar to (23.24). These are the Hartree–Fock (HF) equations. HF includes a part for interelectron interaction matrix elements, namely that given by (23.5) and (23.6). The rest is called the residual interaction, and its inclusion leads beyond the HF frame, accounting for correlations.

p =

p +

Σˆ

q

p +

Σˆ q (23.25)

Here the single line stands for an HF state. Using the correspondence rule (23.14), an analytical equation similar to the Schrödinger equation can be derived with the operator
ˆ playing the role of an external potential. The essential difference is, however, that this “potential” depends in principle upon the energy and state of the particle. The same kind of iterative procedure leading to (23.21) or (23.23) will be used several times in this chapter. Other zero-order approximations can be chosen. But then diagrams with corrections of the type (23.3a) must be included, with the external static field potential equal to the difference between the HF and the chosen one. To calculate the numerical value of a given diagram or a sequence of diagrams one needs to know, according to the description given above, the matrix elements of external fields and interelectron interactions obtained with the help of one-electron HF wave functions. The required calculational procedures are described in [23.13].

Many-Body Theory of Atomic Structure and Processes

23.2 Calculation of Atomic Properties

365

23.2 Calculation of Atomic Properties 23.2.1 Electron Correlations in Ground State Properties

vector in the direction of R. Substituted into (23.26), this potential leads to the expression

A major advantage of the diagrammatic technique in many-body theory is that it is usually unnecessary to know the total wave function of the atom. On the contrary, only actively participating electrons or vacancies appear in a diagram. The HF zero-order approximation for one-electron and one-vacancy wave functions is used in what follows. All atomic characteristics and cross sections for atomic processes calculated with HF form the one-electron approximation. Everything beyond the HF frame, i. e., caused by residual interaction, are called correlation corrections or correlations. They can be calculated using the many-body perturbation theory (MBPT) [23.3], random phase approximation (RPA) [23.14] and random phase approximation with exchange (RPAE) or its generalized version GRPAE [23.12, 13]. The simplest diagrammatic expression for the correlation energy is given by the two diagrams a)

b)

k i

k j

i j n

n (23.26)

(2) The analytical expression ∆E corr for (23.26a) is   ij|V |knkn|V |ij (2) = . (23.27) ∆E corr εi + ε j − εk − εn k,n>F;i, j≤F

C6 R6

(23.28)

for the interatomic potential [23.15], where   |i|r|k|2 | j|r|n|2 . C6 ≈ (εi + ε j − εk − εn )

(23.29)

k,n>F;i, j≤F

Calculations [23.16] show that the inclusion of higher-order corrections is important for obtaining accurate values for ∆E corr and C6 . However, to improve accuracy by taking into account the corrections to diagrams (23.26) requires considerable effort. Indeed, there are several types of corrections to (23.26) such as (i) screening of the Coulomb interelectron interaction by the electron–vacancy excitations; (ii) interaction between vacancies ij; (iii) interaction between electrons and vacancies ki(n j ) (k j(ni)); and (iv) interaction between electrons kn. Corrections to the HF field itself which acts upon electrons k, n and vacancies i, j are discussed in Sect. 23.2.3. Screening of the Coulomb interelectron interaction is very important, and in many cases must be taken into account non-perturbative. The simplest way to do this is to use RPA, which defines the effective interelectron interaction Γ˜ as a solution of an integral equation, shown diagrammatically by k =

~ Γ

+

~ Γ

i

(23.30)

If V in (23.27) is replaced by Γ˜ , an expression for ∆E corr in RPA can be derived. Exchange is very important in atoms and molecules, so diagram (23.30) can be modified to include this effect, thus leading to the effective interaction Γ in RPAE [23.12, 13, 16]: k

k ~ Γ

=

+ i

~ Γ

+ i

~ Γ (23.31)

Replacing V in (23.27) by Γ gives a rather accurate expression for ∆E corr in RPAE. Taking into account

Part B 23.2

The analytical expression for (23.26b) differs from (23.27) by the sign and an exchange matrix element kn|V | ji instead of a direct kn|V |ij one. The contribution (23.26) overestimates the correlation energy by ≈10%. Diagrams (23.26) can also be used to describe the interaction potential of two atoms, designated A and B. Let the ki states belong to A and n j to B. At large distances R between the atoms, the contribution of (23.26b) is exponentially small. Because the vacancies i and j are located inside atoms A and B respectively, the interelectron potential V = |rA − rB + R|−1 at large distances R RA,B , (RA,B are atomic radii), can be expanded as a series in powers of R−1 . The first term giving a non-vanishing contribution to (23.27) is V R−3 [(rA · rB ) − 3(rA · n)(rB · n)], n being the unit

U(R) = −

366

Part B

Atoms

screening also affects the long-range interatomic interaction considerably by altering the constant C6 in (23.28). The ground state energy of an atom or molecule is modified by an external field. For a not too intense electromagnetic field, the simplest correction to the ground state energy is given by the diagrams k + i

k

i (23.32)

Considering a dipole external field, its
interaction with the atomic electrons is given by W = i≤F E · ri , E being the strength of the field. The ground state energy shift is given by ∆E = −α(ω)E 2 /2, where α(ω) is the dynamical dipole polarizability and ω is the frequency of the field. According to (23.32), α(ω) is determined by   2|i|z|k|2 (εk − εi ) α(ω) = , (23.33) (εk − εi )2 − ω2 k>F;i≤F

where z is a component of the vector r. RPAE corrections to α(ω) are discussed in Sect. 23.2.5 in connection with the photoionization process. Non-dipole polarizabilities of other multipolarities can be obtained in the lowest order of interelectron interaction using (23.32) with a properly chosen interaction operator between the
electromagnetic field and an electron, instead of W = i≤F E · ri .

23.2.2 Characteristics of One-Particle States

Part B 23.2

A single vacancy or electron can propagate from one instant of interaction to another, as described to zero order by elements (23.1b) [or (23.1a)] with dots (23.1f ) at the ends. This line represents an HF one-particle state with a given angular momentum, spin, and total momentum. Accounting for virtual or real atomic excitations leads, for a vacancy, to a diagram similar to (23.25) but with oppositely directed arrows. Because the interaction with these excitations is usually much smaller than the energy distance between shells, in the sum over q only the term q = i, i being the considered vacancy state, need be taken into account. Interaction with the vacuum leaves the angular momentum, spin,and total momentum unaltered. It can however change the energy, and lead to a finite lifetime for a vacancy state. Analytically, the vacancy propagation in the HF approximation is described by the one-particle HF Green’s

function G HF : G iHF (E) = 1/(εi − E) .

(23.34)

Solving (23.25) for a vacancy i with only Σii terms included gives G i (E) = 1/[εi + Σii (E) − E] .

(23.35)

The pole in G(E) which determines the vacancy energy is shifted from E = εi to E i = εi + Σii (E i ). The quantity Σii (E) is called the self-energy, and is in general a complex function of energy, its imaginary part determining the lifetime of the vacancy i. Near E i , (23.35) can be written in the form G i (E) ≈ Fi / [εi + Σii (E i ) − E] = Fi /(E i − E) , (23.36)

where

−1 ∂Σii (E) Fi = 1 − ∂E E=Ei 

(23.37)

is called the spectroscopic factor. It characterizes the probability for more complicated configurations to be admixed into a single vacancy state i [23.12]. An important problem is to calculate the self-energy part Σ(E). The first nonzero contributions are a)

b) i'

j' j n

i

+

i'

i j

k

+ exchange terms.

n (23.38)

Specific calculations [23.16] demonstrate that if the intermediate electron states n [in (23.38a)] and kn [in (23.38b)] are found in the field of vacancies jj  and ii  j, the diagrams (23.38) are able to reproduce the values of the correlation energy shift with about 5% accuracy. For outer subshell vacancies, the contributions (23.38a) and (23.38b) are almost equally important, to a large extent cancelling each other. For example, (23.38a) shifts the outer 3p vacancy in Ar to lower binding energies by 0.1 Ry, while the contribution of (23.38b) is −0.074 Ry. The total value 0.026 Ry is small and close to the experimental one, which is 0.01 Ry. For inner vacancies, (23.38a) is dominant because the intermediate states in (23.38b) have large virtualities and are therefore small. The main contribution to the sum over j  comes from the term j  = i  = i, which gives for the energy

Many-Body Theory of Atomic Structure and Processes

shift of level i

 

∆εi = Σii(2) (εi ) =

n>F; j≤F

|n|r −1 | j|2 . εn − ε j

(23.39)

The value (23.39) is positive. Most important higherorder corrections will be included if V in (23.38a) is replaced by Γ from (23.31). The physical meaning of diagram (23.38a) is transparent: it accounts for configuration mixing of one vacancy i and “two vacancies jj  – one electron n” states in the lowest order in the interelectron interaction. Diagram (23.38b) is not as transparent, and for i = i  its intermediate state appears to violate the Pauli principle. However, as noted in connection with (23.14), the Pauli principle should not be considered as a restriction in constructing intermediate states. Diagram (23.38a) and its exchange can have an imaginary part, which gives the probability of Auger decay γi(A) , calculated to the lowest order in the interelectron interaction. For (23.38a) one has γi(A)

= Im [Σii (E i )]   |in|V | jj  |2 δ(ε j + ε j  − εn − εi ) . =2π j, j  ≤F;n>F

(23.40)

γi(A)

j, j  ≤F;n>F

(23.41)

367

Generally, for any Fermi particle, Fi ≤ 1 [23.17] because there cannot be more than one particle in a given state. Note that the integrand in (23.41) is the lowest-order admixture of the jj  n state to a pure one-vacancy state i. A small F value means strong mixing. For atoms, Fi is usually close to 1, but there are exceptions where F is small. For example, F5s in Xe is about 0.33 [23.12]. The operator Σij(n) (ε) has non-diagonal matrix elements, which leads to admixture of other onevacancy j or one-electron n states to the vacancy i. A measure of this admixture is given by the ratio Σij(n) (εi )/(ε j(n) − εi ). In higher orders, decay processes more complex than those described by the imaginary part of (23.38a) become possible. For example, this could be a two-electron Auger decay in which the transition energy is distributed between two outgoing electrons. An example of the lowest-order diagram for this process is i

j j1

ε1 ε2 j2

(23.42)

This is one of those diagrams which describe the mixing of a pure one-vacancy state with a quite complex configuration jj1 j2 ε1 ε2 .

23.2.3 Electron Scattering Propagation of an electron in a discrete level or in a scattering state can be described in the same way as for a vacancy. The electron wave function is determined by (23.25). Using the correspondence rule (23.14), (23.25) can be expressed analytically in the form    ∇2 Z  dr   2 − − + φ (r ) − ε ψε (r) i 2 r |r  − r| i≤F  dr  φ∗ (r  )ψε (r  )φi (r) = |r  − r| i i≤F  ˆ + Σ(r, r  , ε)ψε (r  )dr  . (23.43)

The terms with the Coulomb interelectron interaction |r  − r|−1 determine the Hartree–Fock self-consistent potential. The last term in (23.43) represents the nonlocal energy dependent polarization interaction of the continuous spectrum electron with the target atom. Although

Part B 23.2

The width is usually much smaller than Re[Σii (E i )], but there are several exceptional cases with abnormally large Auger widths, among which the most impressive is the 4p-vacancy in Xe with its γ4p ≈ 10 eV. Higher-order corrections include those which are taken into account when V in (23.40) is replaced by Γ from (23.31). The others include jj  vacancy–vacancy interaction, the interaction between vacancies jj  and the electron n and so on. As noted above, all of these can be obtained step by step by inserting the elements (23.4) into (23.38). To select the most important corrections, a physical idea and/or experience are necessary. For instance, if the energy transferred in the decay process ∆ε = ε j  − ε j is close to some threshold energies of atomic intermediate or outer shells, corrections which include virtual excitation of this shell must be taken into account. The contribution to the spectroscopic factor from (23.38a) is given according to (23.37) by  −1   2  |in|V | j j|   Fi(2) ≡  1 +  . (ε j + ε j  − εn − εi )2

23.2 Calculation of Atomic Properties

368

Part B

Atoms

(23.43) resembles the ordinary Schrödinger equation, because of the energy dependence of the self-energy ˆ part Σ(r, r  , ε), it is not the same. Consequently, the wave function ψε (r), often called a Dyson orbital, must be normalized according to the condition (ψε |ψε ) = Fε δ(ε − ε) , where

(23.44)

  −1 ∂ Σ(r, r  , E) ˆ Fε = 1 − ε . (23.45) ε ∂E E=ε 

This is different from that for ordinary wave functions by the factor Fε < 1. In Fε , the matrix element of the ˆ operator ∂ Σ(r, r  , E)/∂E is calculated between states ψε (r). It is seen that Fε is the same spectroscopic factor as determined by (23.37), but for a continuous spectrum or an excited electron state. The object described by the wave function ψε (r) differs from an individual electron because it can be unstable, and its state is mixed with those of more complicated configurations, such as “two electrons” k k – one vacancy j  . This object is called a quasi-electron. Equation (23.43) also determines the energies of electrons in discrete levels which are shifted from their HF values. Contrary to the case of deep vacancies, it is impossible to predict the sign of the energy shift without detailed calculations. For incoming electron energies ε higher than the ˆ target ionization threshold, the operator Σ(r, r  , ε) acquires an imaginary part, thus becoming an optical potential for the projectile. The additional elastic scattering phase shifts from their HF values can be expressed ˆ r  , ε) between the wave via matrix elements of Σ(r, ∗ functions φε (r) and ψε (r); but to find numbers for these phase shifts, the self-energy part or polarization interˆ action Σ(r, r  , ε) must be calculated. It appears that the second-order projectile–target interactions

Part B 23.2

a)

ε

ε

1 k' 2

b) +

ε

ε

k''

j'

2'

1' j

+ exchange terms

j

gives Σ(r, r  ,ε) = −δ(r − r  )

αHF (0) , 2r 4

r, r  → ∞ (23.47)

αHF (0)

where is the static (ω = 0) dipole polarizability determined by (23.33). Accounting for each additional interaction between the projectile and target increases the power of r in the denominator of (23.47). Most important is the interaction between target electrons. By including this, the asymptotic expression (23.47) is also modified, where instead of αHF (0), the RPAE polarizability αRPAE (0) appears. If the target–projectile interelectron interaction is taken into account to second order as in (23.46) while all ˆ the rest is included exactly, the expression for Σ(r, r  , ε) for r, r  → ∞ is still given by (23.46), but with the exact static polarizability. Many experimental results for low energy electron scattering by noble gases, alkalis, and alkaline earths agree well with calculations of elastic scattering phase shifts obtained by solving equation (23.42) in which ˆ Σ(r, r  , ε) is given by (23.46) with RPAE corrections taken into account. Total cross sections are reproduced with an accuracy as high as several percent, including the Ramsauer minimum region. This is illustrated in Fig. 23.1 for the e− + Xe case [23.18]. This approximation is also reasonably good in describing the angular distributions. As the projectile energy ε increases, the contribution of (23.46) decreases rapidly. The approach presented here applies to other incoming particles, such as for instance positrons e+ . σ (E) (arb. units)

150

100

50 (23.46)

provide a reasonably good approximation [23.10, 12]. ˆ The expression for Σ(r, r  , ε) simplifies at distances far from the target. Only (23.46a) contributes in this region, while other terms are exponentially small. Expanding the Coulomb interelectron interactions in (23.46a) in powers of r1 /r1 1, r2 /r2 1

0

0

2

4

6

8

10 E (eV)

Fig. 23.1 Electron–Xe atom elastic scattering cross section [23.12]. Solid line: including polarization interaction; dashed line: HF; dash-dotted line: experiment

Many-Body Theory of Atomic Structure and Processes

369

23.2.4 Two-Electron and Two-Vacancy States One can construct a diagrammatic equation for the wave function of a two-electron or a two-vacancy state by separating all diagrams describing two-electron (twovacancy) scattering which do not include these states as intermediate, and denoting their total contribution by a circle. Then the exact two-electron (vacancy) state is determined by the infinite sequence of diagrams k

kk' =

k

k' k ˆ Π k'

k'1

k1

k2 ˆ Π

+

k'

+

k

k1 ˆ Π

+

ˆ Π

+… k'1

k'2

k'

+…

i i'

k =

kk' +

k'

k1 ˆ Π

kk'

ˆ Π

+ k'1

i i' (23.48)

The analytic equation for two electrons in an atom interacting with each other can be written in the form   ˆ 1 + Hˆ 2HF + Σ ˆ 2 + Qˆ Π ˆ 12 − ε Hˆ 1HF + Σ × ψ12 (r1 · r2 ) = 0 .

(23.49)

HF is the HF part of the one-particle Hamiltonian Here Hˆ 1(2) ˆ 12 is the effective interelectron interaction in (23.24), Π ˆ and Q is the projection operator

Qˆ = 1 − n 1 − n 2 ,

(23.50)

with n 1(2) being the Fermi step function, n 1(2) = 1 for 1(2) ≤ F and n 1(2) = 0 for 1(2) > F. The function n 1(2) thus eliminates contributions of vacant states. The operator Qˆ takes into account the fact that propagation of two electrons (or, more precisely, quasi-electrons due to ˆ takes place in a system of other parthe presence of Σ) ticles which occupy all levels up to the Fermi level. The presence of Qˆ makes (23.49) essentially different from a simple two-electron Schrödinger equation: Qˆ requires that after each act of interaction described ˆ 12 , both participants either remain electrons or by Π become vacancies. Diagrams (23.48) describe the states of two electrons outside the closed shell core, or electron scattering by an atom with one electron outside the closed shells. ˆ 12 is nonlocal, dependent upon ε, and can In general, Π

Part B 23.2

The exchange between e+ and target electrons must be omitted by discarding (23.6) and all but (23.46a) terms ˆ in the polarization interaction Σ(r, r  , ε). The incoming positron in its intermediate state k [see (23.46a)] interacts strongly with the virtually excited electron k , forming a positronium-like object. Corresponding diagrams are obtained by inserting elements (23.4d) for the e+ –e− interaction into (23.46a). Summation of the infinite sequence of such diagrams corresponds to substituting the product φk (r1 )φk (r2 ) in the intermediate state of (23.46a) by the exact e+ e− wave function in the field of a target with a vacancy j. Such a program is very complicated, and a simplification has proved to be satisfactory. Only the positronium (Ps) binding is taken into account by subtracting its binding energy in the denominator of (23.13). This is equivalent to adding the Ps ionization potential IPs to ε in (23.46) [23.19]. It enhances the polarization interaction and leads to an interesting qualitative feature: the possibility of alteration of the sign of the interaction (23.46). Indeed, instead of α(0), the corresponding expression for e+ -atom collision includes α(IPs ). For alkalis, the binding energy is less than IPs , and for the energy region ~ω > IPs the polarizability (23.33) is negative, while α(0) > 0. This leads to a repulsive polarization interaction, rather than the usual attractive one. This difference affects the cross section qualitatively [23.20]. Another, more complicated, approximation substitutes φk (r1 )φk (r2 ) by the product of the precise Ps wave function ψPs (|r1 − r2 |) and the wave function of the free motion of the Ps center of gravity [23.21] According to the diagrams (23.23), (23.43) describes the target with an additional electron. If the target is a neutral atom, solution of (23.43) with discrete energy values describes negative ion states; both ground and excited. Again, as in Sect. 23.2.2, diagrams (23.46) with RPAE corrections form a reasonably good starting point for calculating the negative ion binding energies, even in cases when this binding is comparatively small, as in alkaline earth negative ions [23.22]. The inclusion of only the outer shell polarizability ( j is a vacancy in the outer shell) leads to overbinding of the additional electron forming the negative ion. Only the inclusion of screening due to inner shell excitations yields good agreement with experiment. For instance, recent measurements of Ca− affinity [23.23] give about 20 meV, while the calculations without inner shell excitations give about 50 meV [23.22]. Their inclusion must considerably reduce the theoretical value.

23.2 Calculation of Atomic Properties

370

Part B

Atoms

have an imaginary part. The lowest-order approximation ˆ 12 is V = |r1 − r2 |−1 . Equation (23.49) must be to Π solved in order to obtain, for example, the excitation spectrum of two electrons in atoms with two electrons outside closed subshells, such as Ca. Instead of V , the interaction Γ from (23.31) can be used, which would account for screening due to virtual excitation of inner shell electrons. For low level two-electron excitations, ˆ 12 can be neglected. the energy dependence of Π The same type of equation can be obtained for twovacancy states, describing their energies, decay widths, and structure due to configuration mixing with more complex states. For inner and intermediate shell vacancies, however, the corresponding corrections can be taken into account perturbatively, and the screening by outer shells is not essential. The admixture of outer shell excitation with inner vacancies is important, leading to satellites of the main spectral lines. The interaction between vacancies leads to correlation two-vacancy decay processes in which the energy is carried away by a single electron or photon. Some diagrams exemplifying these processes in the lowest possible order of interaction are a)

i1

b)

j1

i

levels of atoms with three or more electrons (or vacancies) outside of closed shells. This is a very complicated calculation because even two electrons in the Coulomb field of the nucleus is a difficult three-body problem.

23.2.5 Electron–Vacancy States The one-electron–one-vacancy state is the simplest excitation of a closed shell system under the action of an external time-dependent field, represented by (23.2a). Beginning with electrons and vacancies described in the HF approximation, the result of residual interactions leads to excitations of more complex states, including those with two or more electron–vacancy pairs. The interaction can also lead to a single electron–vacancy pair. Let us concentrate on the latter case and separate all diagrams describing electron–vacancy interaction which do not include these states as intermediate, and denote them by a circle. Then the exact electron–vacancy state is determined by the infinite sequence of diagrams ki

k + i

i R k'

+ i2

j2

q

j3

ε

i2

i

j2

q

Part B 23.2

Even in this order, there are several diagrams giving together the amplitude of the correlation Auger (23.51a) and radiative (23.51b) decay. For inner shells, a specific feature of such processes is that the released energy is about twice that for a single vacancy decay. Of course, in these cases the decay probability is relatively small. It is not, however, necessarily much smaller than that of individual vacancies if the energies of the intermediate and initial states are close to each other. The presence of residual two-body forces leads to effective multiparticle interactions. The simplest diagram presenting a three-electron interaction is given by n'1

n2

n'2

n3

n'3

k' +

k

i'

i

k'' R

k' +…

R i''

i'

(23.52)

The role of multi-particle interactions in atomic structure is far from being clear. Diagrams similar to (23.52) are important if it is of interest to calculate the energy

+… i'

ω

k

ki +

=

(23.51)

n1

R

k

j1

%1

k

=

k' +

R

i

i'

ki R k' i' (23.53)

Contrary to the electron–electron case in Sect. 23.2.4, the analytical expression corresponding to (23.53) cannot be represented as a Schrödinger-type equation. Indeed, being symmetric under time reversal, (23.53) leads to an equation depending, unlike (23.49), upon the second power of the electron–vacancy energy ˆ in (23.46) are ω. Note that R in (23.53) and  different. It is necessary to solve (23.53) when calculating the photoabsorption amplitude, which can then be represented as k'

k

i'

i

ω (23.54)

The amplitude for elastic photon scattering is also expressed via the exact electron–vacancy state, determined

Many-Body Theory of Atomic Structure and Processes

by (23.53) to be k'

D(ω) = d + D(ω)χ(ω)U ,

ω i'

i

371

that creates the possibility of qualitative analyzes of its solutions:

k

ω

23.2 Calculation of Atomic Properties

(23.55)

The other case where it is necessary to solve (23.53) is the scattering of electrons, both elastic and inelastic, by atoms with a vacancy in their outer shell, such as the halogens. The simplest approximation to R is given by the Coulomb interaction to lowest order (23.4e) and (23.4f ). With such R, the sequence of diagrams (23.53) is the same as (23.31) and thus forms the RPAE, which is often used to describe photoionization and other atomic processes. Both terms (23.4e) and (23.4f ) contribute to the electron–vacancy in (23.53) only if the external field is spin independent, such as an ordinary photon. For a magnetic interaction, which is proportional to spin, the term (23.4e) does not contribute.

23.2.6 Photoionization in RPAE and Beyond In RPAE, the photoionization amplitude k|D(ω)|i is determined by solving an integral equation obtained from (23.53) and (23.54) using the correspondence rule (23.14) [23.12]: k|D(ω)|i = k|d|i     k |D(ω)|i  ki  |V |ik − k i + ω − εk + εi  + iη k >F;i  ≤F  i  |D(ω)|k kk |V |ii  − i  i , − (23.56) ω − εk − εi  − iη

where U is a combination of the direct Vd and exchange Ve Coulomb interelectron potentials, U = Vd − Ve , χ(ω) = χ1 (ω) + χ2 (ω), χ1 (ω) = 1/(ω − ω + iη) and χ2 (ω) = 1/(ω + ω ), with ω being the excitation energy of the virtual electron–vacancy state. Using Γ from (23.31), one can present D(ω) as D(ω) = d + dχ(ω)Γ(ω) .

(23.58)

Equation (23.57) allows a rather simple, also symbolic, solution D(ω) = d/[1 − χ(ω)U] .

(23.59)

If the denominator in (23.59) has a solution Ω determined by the equation 1 − χ(Ω)U = 0 ,

(23.60)

at Ω > I, where I is the atomic ionization potential, then the cross section has a powerful maximum called a giant resonance with energy Ω. A giant resonance is of a collective nature, in the sense that it appears to be due to coherent virtual excitation of all electrons of at least one considered multi-electron subshell. These intra-shell correlations are most important for multielectron shells with large photoionization cross sections. Their inclusion leads to a quantitative description of the above mentioned giant resonances – huge maxima in the photoionization cross sections. An example is the 4d10 photoionization cross section of Xe shown in Fig. 23.2 [23.12], where satisfactory agreement with experiment is demonstrated. It appears that all RPAE intra-shell time-forward diagrams, such as that on the first line in (23.53), and the first term in brackets in (23.56) with εi  = εi , may be taken into account by the matrix element k˜ |d|i. This is the one-electron approximation, but with the function φ˜ k (r) calculated in the term-dependent HF approximation [23.12]. Term-dependency means that only the total angular momentum and spin and their projections for the electron–vacancy pair are conserved, being equal to that of the incoming photon. The individual values for the electron and vacancy angular momentum and spin are not considered to be good quantum numbers. Thus the term-dependent HF includes a large fraction of RPAE correlations.

Part B 23.2

where d is the dipole operator describing the photon– electron interaction and V = |r1 − r2 |−1 . To obtain the RPAE photoionization cross section, the usual expression (24.19) must be multiplied by the square modulus of the ratio k|D(ω)|i/k|d|i, with ω = εk + Ii . The RPAE corrections described by the term in square brackets in (23.56) are very large for outer and intermediate electron shells. Through the sum over i  , if only terms with the same energies εi  = εi are included, (23.56) accounts for intra-shell correlations. By adding terms with εi  = εi , the effect of inter-shell correlations is taken into account. For atoms, (23.56) has to be solved numerically, but can be presented in a symbolical operator form

(23.57)

372

Part B

Atoms

of a powerful discrete excitation into a continuum, with which the excitation interacts strongly), continuous spectrum autoionization (modification of a broad continuous spectrum excitation due to its strong interaction with a narrow continuum that happens in negative ions [23.22]) and quadrupole giant resonances [23.24]. Above we concentrated on dipole Giant resonances. Quadrupole amplitudes in RPAE are determined by an equation similar to (23.57):

σγ (Mb)

30

20

10

Q(ω) = q + Q(ω)χ(ω)U , 0

4

5

6

7

8

9

10

11 ω (Ry)

Fig. 23.2 Photoabsorption in the vicinity of the 4d10 sub-

shell threshold in Xe [23.7]. Solid line: RPAE; dashed line: experiment

RPAE permitted to the prediction of interference resonances. To describe them, let us consider a situation in which the direct HF amplitude ds is small, while there are other electrons with large photoionization amplitude Db , Db (ω) ds . Then, from (23.57) one has Ds (ω) ≈ ds + Db (ω)χ(ω) Ubs ≈ Db (ω)χ(ω)Ubs ds

(23.61)

Part B 23.2

if the inter-transition interaction Ubs is not too small. The enhancement of the photoionization amplitude described by (23.61) manifests itself as a resonance in the partial cross section of s electrons photoionization. Very often the term Db (ω)χ(ω)Ubs and ds are of opposite sign, so that the total amplitude acquires two minima, along with an extra maximum, thus forming a rather complicated structure in the partial cross section that was named interference or correlation resonance. Usually, these resonances are manifestations of inter-shell correlations. These are taken into account if the sum over i  in (23.56) includes terms with εi  = εi . An example is the 5s 2 subshell in Xe, which is strongly affected by the outer 5p 6 and inner 4d 10 neighboring electrons. Due to this interaction, the 5s 2 cross section is completely altered, as illustrated in Fig. 23.3 [23.12]. The RPAE results predict a qualitative feature of the experimental data, namely the formation of a maximum and two minima in the cross section. The second minimum is not seen in Fig. 23.3, since it lies at considerably higher ω. RPAE is able to describe a number of other effects, such as giant autoionizational resonance (decay

(23.62)

where q is the quadrupole amplitude in HF approximation. Giant quadrupole resonance was found in excitations of 4d6 electrons in Xe [23.25]. Its direct observation in photoabsorption is almost impossible, since the corresponding cross section is very small due to the inclusion of the extra factor α2 = 1/c2 ≈ 10−4 as compared to the dipole cross section. However, the quadrupole amplitude leads to noticeable corrections to the angular distributions of photoelectrons where their relative contribution is considerably bigger. Note that the amplitude of electron elastic scattering on an atom with a vacancy is expressed in RPAE via Γ given by (23.31) [23.26]. For the inner or deep intermediate shells, RPAE proves to be insufficient. First, screening of the Coulomb interaction between the outgoing or virtually excited electron and the vacancy [see (23.4f )] must be taken into account. This can be done by replacing V by Γ σ (Mb) 1.0

0.5

0

2

4

6

8

10 ω (Ry)

Fig. 23.3 Photoionization of 5s2 electrons in Xe [23.7].

Solid line: RPAE with effects of 5p6 and 4d10 included; dashed line: 5s2 electrons only; dash-dotted line: with effect of 5p6 electrons; dash-double-dotted line: with effect of 4d10 electrons; dotted line: experiment

Many-Body Theory of Atomic Structure and Processes

from (23.31). The ionization potential (or the energy of the vacancy i) must also be corrected, which requires inclusion of at least the contribution from the first term of (23.38). It has been demonstrated [23.27] that the screening of the electron–vacancy interaction can be taken into account by calculating the wave function of the virtually excited or outgoing electron in the selfconsistent HF field of an ion instead of that of a neutral atom. A method which uses only these one-particle wave functions in (23.54) (23.56) is called the generalized RPAE or GRPAE. The use of this approximation considerably improves the agreement with experiment near the intermediate shell thresholds, decreasing there the cross section value and shifting its maximum to higher energies. GRPAE permitted to disclose intra-doublet resonances that results from interaction of electrons belonging to two components of the spin-orbit doublet, e.g., 3d 3/2 and 3d 5/2 in Xe, Cs and Ba atoms [23.24, 28]. RPAE and GRPAE corrections affect not only the cross sections but also characteristics of the photoelectron angular distribution, i. e., dipole and non-dipole angular anisotropy parameters [23.12, 16, 24]. As an example, Fig. 23.4a presents the partial cross sections [23.28] while Fig. 23.4b depicts the dipole anisotropy parameter β [23.24] for 3d5/2 and 3d3/2 electrons in Cs. The effect of intra-doublet resonance – an additional maximum in the 3d5/2 cross section under the action of 3d3/2 electrons – is clearly seen. Figure 23.5 depicts the non-dipole angular anisotropy parameter γ5s for 5s electrons in Xe [23.29]. The parameter γns (in Fig. 23.5, n = 5) is given by the simple formula [23.16] γns (ω) = 6[|Q ns (ω)|/|Dns (ω)|] cos(∆q − ∆d ) , (23.63)

373

a) Cross section, Mb 20

Cs3d

15

10

5

0 720

730

740

750

760 770 Photon energy, eV

730

740

750

760 770 Photon energy, eV

b) β 2

1

0

–1

720

Fig. 23.4a,b Intra-doublet resonance in 3d10 Cs. (a) Partial photoionization cross sections σ5/2 and σ3/2 [23.28], (b) Dipole angular anisotropy parameters β5/2 and β3/2 . Solid line: data for 5/2 with account of 3/2; Dash-dotted line: data for 5/2 without account of 3/2; Dotted line: data for 3/2 with account of 5/2; Dashed line: data for 3/2 without account of 5/2

to considerable growth of the threshold cross section. Diagrammatically, the effect of decay may be described by k' i

j1 j2 k2

(23.64)

Here, the double line emphasizes that starting from the instant of decay, the photoelectron moves in the field j1 j2 of double instead of a single i vacancy. For inner vacancies, this is a strong effect which can lead even to recapture of the photoelectron into some of the discrete levels in the field of the double vacancy j1 j2 .

Part B 23.2

where Q ns (Dns ) are the RPAE (GRPAE) quadrupole (dipole) photoionization amplitudes and ∆q (∆d ) are their phases. Thus, γns (ω) is sensitive to the presence of interference, dipole, and quadrupole resonances. The latter is presented by a small but noticeable maximum on the high energy slope of the huge maximum, caused by the presence of the giant dipole resonance. The variation of γ5s near the 5s threshold is determined by the resonant behavior of cos(∆q − ∆d ), called phase resonance [23.24]. Close to inner shell thresholds, the Auger decay of a deep vacancy must be taken into account. Due to decay, the photoelectron instantly finds itself in the field of at least two vacancies instead of one, leading

23.2 Calculation of Atomic Properties

374

Part B

Atoms

γ

1

0.5

0 HF –0.5 RRPA RPAE –1 50

100

150

200 Photon energy, eV

Fig. 23.5 Nondipole anisotropy parameter γ5s (ω) of 5s2 electrons in Xe [23.29]

So-called Post-collision interactions in photoionization can be taken into account by this diagram when the Auger electron is much faster than the photoelectron [23.30]. If their speed is of the same order, their mutual Coulomb repulsion must be accounted for, leading to additional alteration of energy and angular redistributions. A photoelectron can excite or knock out another atomic electron. To lowest order in the residual interelectron interaction, this process can be represented by

k

k2 i1 k1 i1

(23.65)

Part B 23.2

While the formation of an initial electron–vacancy ki 1 pair requires RPAE or GRPAE for its description, the second step of (23.65) can be reasonably well reproduced by the lowest-order term in V . It appears that process (23.65) has a high probability for not too fast photoelectrons, changing considerably the cross section [23.31] for ionization by creating a vacancy i 1 . A comparatively simple diagram k i

i j n

(23.66)

describes the photoionization process in which a more complicated state is created in the ion than a single vacancy, e.g. a state with two vacancies and one electron. Here the doubled arrow indicates that the electron is in a discrete level n. Diagrams (23.64–23.66) present some corrections which mix electron–vacancy and two-electron–twovacancy configurations. Each additional interaction line increases the number of possible physical processes considerably. With growth of the number of particles actively participating in a process, the calculational difficulties increase enormously. However, this is not a shortcoming of the diagrammatic approach, but a specific feature of more and more complex physical processes.

23.2.7 Photon Emission and Bremsstrahlung The amplitude of photon emission in lowest order is given by the time-reverse of (23.2c): k

k' ω

(23.67)

This diagram represents ordinary Bremsstrahlung; i. e. a process of projectile deceleration in the field of the target. If the target has internal structure as atoms do, it can be really or virtually excited during the collision process. The simplest excitation means creation of an

Many-Body Theory of Atomic Structure and Processes

electron–vacancy pair. The annihilation of this pair results via the time-reverse of (23.2a) in photon emission. The process thus looks like k

k' k'' i

ω

23.3 Concluding Remarks

375

The analytical expression for the total Bremsstrahlung amplitude, including RPAE corrections to (23.68), is given by the expression   k|A(ω)|k  = k|d|k  + ki|V |k k  k >F;i≤F

(23.68)

To obtain the total Bremsstrahlung amplitude, the terms (23.67) and (23.68) must be summed. The polarization radiation (PR) created by the mechanism (23.68) has a number of features which are different from the ordinary Bremsstrahlung (OB) represented by (23.67). The intensity is proportional to 1/Mp2 for OB, where Mp is the projectile mass, and the spectrum, at least for high εk , is proportional to 1/ω. On the other hand, the PR intensity is almost completely independent of Mp and its frequency dependence is quite complex, being determined by the target polarizability α(ω) [23.32]. PR is most important for frequencies ω of the order of and higher than the target’s ionization potential. At sufficiently large distances and for neutral targets, PR starts to predominate over OB. Close to discrete excitations of the k electron, the contribution (23.68) becomes resonantly enhanced. Higher-order corrections are important in the PR amplitude. First, the Coulomb interaction V in (23.68) must be replaced by Γ from (23.31).

2(εk − εi ) i|D(ω)|k  . 2 η→+0 ω − (εk − εi )2 + iωη

× lim

(23.69)

To derive the Bremsstrahlung spectrum, the usual general expression must be multiplied by the square modulus of the ratio k|A(ω)|k /k|d|k . If the incoming electron is slow, corrections (23.46) also become important. The intermediate state in (23.68) includes two electrons k and k, and a vacancy i. The extent of interaction between them could be considerable. An important feature of PR is that it is nonzero even if the projectile is neutral, but is able to polarize the target. For example, it leads to emission of continuous spectrum radiation in atom–atom collisions, whose intensity for frequencies of the order of the ionization potentials is close to that in electron–atom collisions. The second and higher orders in the residual interaction involve processes more complicated than (23.68), for instance those which include simultaneous photon emission and target excitation (ionization) [23.33].

23.3 Concluding Remarks retardation and spin-dependence in the interparticle interaction makes the calculations much more complicated. These parts of the interaction appear as relativistic corrections. They are comparatively small in all but the heaviest atoms, and can be taken into account perturbatively. Beyond lowest order, these additional interactions are strongly altered when virtual excitations of electron–vacancy pairs and the Coulomb interaction between them is taken into account. An example is given by the sequence of diagrams k1

k'1

k2

k'2

+

k1

k'1

+

k1

k'1

k'2 k2

+…

k'2 k2 (23.70)

Part B 23.3

It is most convenient to apply diagrammatic techniques to closed shell atoms whose ground state is nondegenerate. Degeneracy means that some of the energy denominators (23.13) become zero with nonzero statistical weight. All such contributions must be summed to eliminate this degeneracy. This leads to strong mixing of some states. For example, the energy required for electron–vacancy transitions jn [see (23.38)] within an open shell is zero, thus leading to strong mixing of i and ijn states. If a pair with zero excitation energy has nonzero angular momentum, taking into account the mixing within such a pair destroys angular momentum as a characteristic of a one-vacancy state. This makes all calculations much more complicated, reflecting a specific feature of the degenerate physical system. In using the diagrams and formulas presented above, it is essential that the interelectron and electron– nucleus interactions be purely potential. Inclusion of

376

Part B

Atoms

where the heavy dashed line stands for the spindependent interelectron interaction. Note that here there are no electron–vacancy loops as in (23.28) because the Coulomb interaction is unable to affect the electron spin and thus to transfer spin excitations. The same kind of diagram describes the one-particle field acting upon an electron or vacancy due to the presence of spin-orbit interaction or weak interaction between electrons and the nucleus. For instance, the effective weak potential includes contributions from the

sequence wi

k1

wi

k'

+ k

wi k' +

k' + … k (23.71)

This is another example demonstrating how flexible and convenient the many-body approach is for considering different processes and interactions.

References 23.1 23.2 23.3

23.4

23.5

23.6 23.7 23.8

23.9

23.10

Part B 23

23.11

23.12 23.13

23.14 23.15

R. P. Feynman: Quantum Electrodynamics (Benjamin, New York 1961) H. A. Bethe, J. Goldstone: Proc. R. Soc. London A 238, 551 (1957) H. P. Kelly: Advances in Theoretical Physics, Vol. 2, ed. by K. A. Brueckner (Academic Press, New York 1968) pp. 75–169 M. Ya. Amusia: Many-body Effects in Electron Atomic Shells (A. F. Ioffe Physical-Technical Institute Publications, Leningrad 1968) pp. 1–144 in Russian M. Ya. Amusia: X-Ray and Inner-Shell Processes, AIP Conf. Proc. 389, ed. by R. L. Johnson, H. Schmidt-Böking, B. F. Sonntag (AIP Press, Woodbury, New York 1997) pp. 415–430 M. Ya. Amusia, J.-P. Connerade: Rep. Prog. Phys. 63, 41 (2000) M. Ya. Amusia: Phys. Essays 13, 444 (2000) M. Ya. Amusia: The Physics of Ionized Gases, ed. by N. Konjevich, Z. L. Petrovich, G. Malovich (Institute of Physics, Belgrade, Yugoslavia 2001) pp. 19–40 N. A. Cherepkov, S. K. Semenov, Y. Hikosaka, K. Ito, S. Motoki, A. Yagishita: Phys. Rev. Lett. 84, 250 (2000) A. N. Ipatov, V. K. Ivanov, B. D. Agap’ev, W. Ekardt: W. J. Phys. B: At. Mol. Opt. Phys. 31, 925 (1998) N. H. March, W. H. Young, S. Sampanthar: The Many-Body Problem in Quantum Mechanics (Cambridge Univ. Press, Cambridge 1967) M. Ya. Amusia: Atomic Photoeffect (Plenum Press, New York 1990) M. Ya. Amusia, L. V. Chernysheva: Computation of Atomic Processes (Institute of Physics Publishing, Bristol-Philadelphia 1997) D. Pines: The Many-Body Problem (Benjamin, New York 1961) L. D. Landau, E. M. Lifshits: Quantum Mechanics (Pergamon Press, Oxford 1965) pp. 319–322

23.16 23.17

23.18 23.19 23.20

23.21 23.22 23.23 23.24 23.25 23.26

23.27

23.28

23.29

23.30

M. Ya. Amusia, N. A. Cherepkov: Case Studies At. Phys. 5, 47 (1975) A. B. Migdal: Theory of Finite Fermi-Systems and Applications to Atomic Nuclei (Interscience, New York 1967) W. R. Johnson, C. Guet: Phys. Rev. A 49, 1041 (1994) M. Ya. Amusia, N. A. Cherepkov, L. V. Chernysheva: JETP 124, 1 (2003) S. Zhou, S. P. Parikh, W. E. Kauppila, C. K. Kwan, D. Lin, A. Surdutovich, T. S. Stein: Phys. Rev. Lett. 73, 236 (1994) G. F. Gribakin, W. A. King: Can. J. Phys. 74, 449 (1996) V. K. Ivanov: J. Phys. B: At. Mol. Opt. Phys. 32, R67 (1999) C. W. Walter, J. R. Peterson: Phys. Rev. Lett. 68, 2281 (1992) M. Ya. Amusia: Radiation Physics and Chemistry 70, 237 (2004) W. Johnson, K. Cheng: Phys. Rev. A 63, 022504 (2001) M. Ya. Amusia, V. A. Sosnivker, N. A. Cherepkov, L. V. Chernysheva, S. I. Sheftel: J. Tech. Phys. (USSR Acad. Sci.) 60, 1 (1990) in Russian M. Ya. Amusia: Photoionization in VUV and Soft X-Ray Frequency Regions, ed. by U. Becker, D. Shirley (Plenum Press, New York 1996) pp. 1–46 M. Ya. Amusia, L. V. Chernysheva, S. T. Manson, A. Z. Msezane, V. Radoevich: Phys. Rev. Lett. 88, 093002 (2002) O. Hemmers, R. Guillemin, E. P. Kanter, B. Krassig, D. W. Lindle, S. H. Southworth , R. Wehlitz, J. Baker, A. Hudson, M. Lotrakul, D. Rolles, W. C. Stolte, I. C. Tran, A. Wolska, S. W. Yu, M. Y. Amusia, K. T. Cheng, L. V. Chernysheva, W. R. Johnson, S. T. Manson: Phys. Rev. Lett. 91, 053002 (2003) M. Ya. Amusia, M. Yu. Kuchiev, S. A. Sheinerman: J. Exp. Theor. Phys. 76(2), 470 (1979)

Many-Body Theory of Atomic Structure and Processes

23.31 23.32

M. Ya. Amusia, G. F. Gribakin, K. L. Tsemekhaman, V. L. Tsemekhaman: J. Phys. B 23, 393 (1990) M. Ya. Amusia: Phys. Rep. 162, 249 (1988)

23.33

References

377

V. N. Tsitovich, I. M. Oiringel (Eds.): Polarizational Radiation of Particles and Atoms (Plenum Press, New York 1992)

Part B 23

379

24. Photoionization of Atoms

Photoionizatio This chapter outlines the theory of atomic photoionization, and the dynamics of the photon–atom collision process. Those kinds of electron correlation that are most important in photoionization are emphasized, although many qualitative features can be understood within a central field model. The particle–hole type of electron correlations are discussed, as they are by far the most important for describing the single photoionization of atoms near ionization thresholds. Detailed reviews of atomic photoionization are presented in [24.1] and [24.2]. Current activities and interests are well-described in two recent books [24.3, 4]. Other related topics covered in this volume are experimental studies of photon interactions at both low and high energies in Chapts. 61 and 62, photodetachment in Chapt. 60, theoretical descriptions of electron correlations in Chapt. 23, autoionization in Chapt. 25, and multiphoton processes in Chapt. 74.

24.1

General Considerations ........................ 379 24.1.1 The Interaction Hamiltonian ....... 379

24.1.2

Alternative Forms for the Transition Matrix Element 24.1.3 Selection Rules for Electric Dipole Transitions...... 24.1.4 Boundary Conditions on the Final State Wave Function 24.1.5 Photoionization Cross Sections ....

380 381 381 382

24.2 An Independent Electron Model ............ 24.2.1 Central Potential Model.............. 24.2.2 High Energy Behavior ................ 24.2.3 Near Threshold Behavior ............

382 382 383 383

24.3 Particle–Hole Interaction Effects ........... 24.3.1 Intrachannel Interactions........... 24.3.2 Virtual Double Excitations .......... 24.3.3 Interchannel Interactions........... 24.3.4 Photoionization of Ar.................

384 384 384 385 385

24.4 Theoretical Methods for Photoionization ............................. 386 24.4.1 Calculational Methods ............... 386 24.4.2 Other Interaction Effects............. 387 24.5 Recent Developments........................... 387 24.6 Future Directions ................................. 388 References .................................................. 388

24.1 General Considerations 24.1.1 The Interaction Hamiltonian Consider an N-electron atom with nuclear charge Z. In the nonrelativistic approximation, it is described by the Hamiltonian 
N  2 N
pi Ze2 e2 H= + − . (24.1) 2m ri |ri − r j | i=1

i> j=1

Under the most common circumstance of single-photon ionization of an outer-subshell electron, the interaction Hamiltonian in (24.2) may be simplified considerably. First, the third term in (24.2) may be dropped, as it introduces two-photon processes (since it is of second order in A). In any case, it is small compared with single photon processes since it is of second order in the coupling constant |e|/c. Second, we choose the Coulomb gauge

Part B 24

The one-electron terms in brackets describe the kinetic and potential energy of each electron in the Coulomb field of the nucleus; the second set of terms describe the repulsive electrostatic potential energy between electron pairs. The interaction of this atom with external electromagnetic radiation is described by the additional terms obtained upon replacing pi by pi + (|e|/c)A(ri , t),

where A(ri , t) is the vector potential for the radiation. The interaction Hamiltonian is thus N 
+|e| [ pi · A(ri , t) + A(ri , t) · pi ] Hint = 2mc i=1  e2 2 . |A(r , t)| (24.2) + i 2mc2

380

Part B

Atoms

for A, which fixes the divergence of A as ∇ · A = 0. A thus describes a transverse radiation field. Furthermore p and A now commute and hence the first and second terms in (24.2) may be combined. Third, we introduce the following form for A:  A(ri , t) =

2πc2 ~ ωV

 12

i(k·ri −ωt)


ˆ e

.

(24.3)

This classical expression for A may be shown [24.5] to give photoabsorption transition rates that are in agreement with those obtained using the quantum theory of radiation. Here k and ω are the wave vector and angular frequency of the incident radiation,
ˆ is its polarization unit vector, and V is the spatial volume. Fourth, the electric dipole (E1) approximation, in which exp[i(k · ri )] is replaced by unity, is usually appropriate. The radii ri of the atomic electrons are usually of order 1 Å. Thus for λ
100 Å, |k·ri |  1. Now λ
100 Å corresponds to photon energies ~ω  124 eV. For outer atomic subshells, most of the photoabsorption occurs for much smaller photon energies, thus validating the use of the E1 approximation. (This approximation cannot be used uncritically, however. For example, photoionization of excited atoms (which have large radii), photoionization of inner subshells (which requires the use of short wavelength radiation), and calculation of differential cross sections or other measurable quantities that are sensitive to the overlap of electric dipole and higher multipole amplitudes all require that the validity of the electric dipole approximation be checked.) Use of all of the above conventions and approximations allows the reduction of Hint in (24.2) to the simplified form Hint =

+|e| mc



2πc2 ~ ωV

 12
N


ˆ · pi exp(−iωt) .

i=1

(24.4)

Hint thus has the form of a harmonically time-dependent perturbation. According to time-dependent perturbation theory, the photoionization cross section is proportional to the absolute square of the matrix element of (24.4) between the initial and final electronic states described by the atomic Hamiltonian in (24.1). Atomic units, in which |e| = m = ~ = 1, are used in what follows.

Part B 24.1

24.1.2 Alternative Forms for the Transition Matrix Element The matrix element of (24.4) is proportional to the matrix  element of the momentum operator i pi . Alternative

expressions for this matrix element may be obtained from the following operator equations involving commutators of the exact atomic Hamiltonian in (24.1):  N  N

pi = −i ri , H , (24.5)  N


i=1



pi , H = −i

i=1

i=1 N
i=1

Zri . ri3

(24.6)

Matrix elements of (24.5) and (24.6) between eigenstates ψ0 | and |ψ f  of H having energies E 0 and E f respectively give ψ0 |

N


pi |ψ f  = −iωψ0 |

i=1

N


ri |ψ f  ,

i=1

(24.7)

ψ0 |

N
i=1

pi |ψ f  =

−i ψ0 | ω

N
Zri i=1

ri3

|ψ f  , (24.8)

N

where ω = E f − E 0 . Matrix elements of i=1 pi , N N 3 are known as the “velocr , and Zr /r i i i=1 i=1 i ity,” “length,” and “acceleration” forms of the E1 matrix element. Equality of the matrix elements in (24.7) and (24.8) does not hold when approximate eigenstates of H are used [24.6]. In such a case, qualitative considerations may help to determine which form is most reliable. For example, the length form tends to emphasize the large r part of the approximate wave functions, the acceleration form tends to emphasize the small r part of the wave functions, and the velocity form tends to emphasize intermediate values of r. If instead of employing approximate eigenstates of the exact H, one employs exact eigenstates of an approximate N-electron Hamiltonian, then inequality of the matrix elements in (24.7) and (24.8) is a measure of the nonlocality of the potential in the approximate Hamiltonian [24.7, 8]. The exchange part of the Hartree–Fock potential is an example of such a nonlocal potential. Nonlocal potentials are also implicitly introduced in configuration interaction calculations employing a finite number of configurations [24.7, 8]. One may eliminate the ambiguity of which form of the E1 transition operator to use by requiring that the Schrödinger equation be gauge invariant. Only the length form is consistent with such gauge invariance [24.7, 8]. However, equality of the alternative forms of the transition operator does not necessarily imply high accuracy. For example, they are exactly equal when one

Photoionization of Atoms

uses an approximate local potential to describe the N-electron atom, as in a central potential model, even though the accuracy is often poor. The length and velocity forms are also exactly equal in the random phase approximation [24.9], which does generally give accurate cross sections for single photoionization of closed shell atoms. No general prescription exists, however, for ensuring that the length and velocity matrix elements are equal at each level of approximation to the N-electron Hamiltonian.

24.1.3 Selection Rules for Electric Dipole Transitions

(24.9)

Here the atom A is ionized by the photon γ to produce a photoelectron with kinetic energy ε and orbital angular momentum . The photoelectron is coupled to the ion A+ with total orbital and spin angular momenta L  and S . In the electric dipole approximation, the photon may be regarded as having odd parity, i. e., πγ = −1, and unit angular momentum, i. e., γ = 1. This is obvious from (24.7) and (24.8), where the E1 operator is seen to be a vector operator. The component m γ of the photon in the E1 approximation is ±1 for right or left circularly polarized light and 0 for linearly polarized light. (The z axis is taken as kˆ in the case of circularly polarized light and as
ˆ in the case of linearly polarized light, where k and
ˆ are defined in (24.3).) Angular momentum and parity selection rules for the E1 transition in (24.9) imply the following relations between the initial and final state quantum numbers: (24.10) L  = L ⊕ 1 = L¯ ⊕  , M L  = M L + m γ = M L¯ + m  , 1 S = S = S¯ ⊕ , 2 M S = M S = M S¯ + m s ,

πA πA+ = (−1)+1 .

381

needed to specify uniquely the state of the ion A+ ) define a final state channel. All final states that differ only in the photoelectron energy ε belong to the same channel. The quantum numbers L  , S , M L  , M S , and πtot = (−1) πA+ are the only good quantum numbers for the final states. Thus the Hamiltonian (24.1) mixes final state channels having the same angular momentum and parity quantum numbers but differing quantum numbers for the ion and the photoelectron; i. e., differ¯ S, ¯ π A+ , and  but the same L  , S , M L  , M S and ing L,  (−1) πA+ .

24.1.4 Boundary Conditions on the Final State Wave Function

If one ignores relativistic interactions, then a general atomic photoionization process may be described in L Scoupling as follows: A(L, S, M L , M S , πA ) + γ(πγ , γ , m γ ) ¯ A+ )ε(L  , S , M L  , M S ) . −→ A+ ( L¯ Sπ

24.1 General Considerations

(24.11) (24.12) (24.13) (24.14)

− ψαE (r1 s1 , . . . , r N s N )

1

1 i∆α e r i(2πkα ) N
1 1 −i∆α † − θα (r1 s1 , . . . , rˆ N s N ) e Sα α , 1 r i(2πkα ) 2 N 

−→ θ (r s , . . . r N →∞ α 1 1

, rˆ N s N )

1 2

α

(24.15)

where the phase appropriate for a Coulomb field is 1 1 ∆α ≡ kα r N − πα + log 2kα r N + σα . (24.16) 2 kα The minus superscript on the wave function in (24.15) indicates an “incoming wave” normalization: i. e., − asymptotically ψαE has outgoing spherical Coulomb waves only in channel α, while there are incom† ing spherical Coulomb waves in all channels. Sα α is the Hermitian conjugate of the S-matrix of scattering theory, θα indicates the coupled wave function of the ion and the angular and spin parts of the photoelectron wave function, kα is the photoelectron momentum in channel α and α is its orbital angular momentum, and σα in (24.16) is the Coulomb phase shift. − While one calculates channel functions ψαE , experimentally one measures photoelectrons which asymptotically have well-defined linear momenta kα and well-defined spin states m 1 , and ions in well-defined 2 ¯ ¯ M ¯ . The wave states α¯ ≡ L¯ SM function appropriate for L S this experimental situation is related to the channel functions by uncoupling the ionic and electronic orbital and

Part B 24.1

Equation (24.14) follows from the parity (−1) of the photoelectron. The direct sum symbol ⊕ denotes the vector addition of A and B i.e, A⊕ B = A + B, A + B − 1, . . . , |A − B|. ¯ S, ¯ πA+ , In (24.9), the quantum numbers α ≡ L, , L  , S , M L  , M S (plus any other quantum numbers

Photoionization calculations obtain final state wave functions satisfying the asymptotic boundary condition that the photoelectron is ionized in channel α. This boundary condition is expressed as

382

Part B

Atoms

spin angular momenta and projecting the photoelectron angular momentum states α , m α onto the direction kˆ α by means of the spherical harmonic Y α m α (kα ). This relation is [24.1]: − (r s , . . . , r N s N ) ψαk ¯ α 1 1
iα exp(−iσ ) α = Y α m α (kˆ α ) 1 α m α kα2
L¯ M ¯α m α |L M L ×

L L M L SM S ¯ ¯ 1 m 1 |SM Sψ − (r1 s1 , . . . × SM αE S2 2

and inserting Hint (0), the differential photoionization cross section is 2 N
4π 2 kα dσα¯ − = pi |ψαk  . (24.20)
ˆ · ψ0 | ¯ α dΩ c ω i=1

, r N s N ) , (24.17)

where the coefficients in brackets are Clebsch–Gordon coefficients. This wave function is normalized to a delta function in momentum space, i. e.,

− † − ψα¯  k  d3r = δα¯ α¯  δ(kα − kα ) . (24.18) ψαk ¯ α α

1

The factors iα exp(−iσα )kα− 2 ensure that for large r N (24.17) represents a Coulomb wave (with momentum kα ) times the ionic wave function for the state α¯ plus a sum of terms representing incoming spherical waves. Thus only the ionic term α¯ has an outgoing wave. One uses the wave function in (24.17) to calculate the angular distribution of photoelectrons.

24.1.5 Photoionization Cross Sections If one writes Hint in (24.4) as Hint (t) = Hint (0) e−iωt , then from first order time-dependent perturbation theory, the transition rate for transition from an initial state with energy E 0 and wave function ψ0 to a final state with − total energy E f and wave function ψαk is ¯ α − dWkα = 2π|ψ0 |Hint (0)|ψαk |2 ¯ α

× δ(E f − E 0 − ω)kα2 dkα dΩ(kˆ α ) .

(24.19)

The delta function expresses energy conservation and the last factors on the right are the phase space factors for the photoelectron. Dividing the transition rate by the incident photon current density c/V , integrating over dkα ,

Implicit in (24.19) and (24.20) is an average over initial magnetic quantum numbers M L 0 M S0 and a sum over final magnetic quantum numbers M L¯ M S¯ m 1 . The length 2 form of (24.20) is obtained by replacing each pi by ωri (24.7). Substitution of the final state wave function (24.17) in (24.20) permits one to carry out the numerous summations over magnetic quantum numbers and obtain the form dσα¯ σα = ¯ [1 + βP2 (cos θ)] (24.21) dΩ 4π for the differential cross section [24.10]. Here σα¯ is the partial cross section for leaving the ion in the state α, ¯ β is the asymmetry parameter [24.11], P2 (cos θ) = 32 cos2 θ − 12 , and θ indicates the direction of the outgoing photoelectron with respect to the polarization vector
ˆ of the incident light. The form of (24.21) follows in the electric dipole approximation from general symmetry principles, provided that the target atom is unpolarized [24.12]. The partial cross section is given in terms of reduced E1 matrix elements involving the channel functions in (24.15) by 2 N

4π 2 − σα¯ = ω[L]−1 ri [1] ψαE  . ψ0

3c  α L

i=1

(24.22)

The β parameter has a much more complicated expression involving interference between different reduced dipole amplitudes [24.1]. Thus measurement of β provides information on the relative phases of the alternative final state channel wave functions, whereas the partial cross-section in (24.22) does not. From the requirement that the differential cross section in (24.21) be positive, one sees that −1 ≤ β ≤ +2.

Part B 24.2

24.2 An Independent Electron Model The many-body wave functions ψ0 and ψ − αE are usually expressed in terms of a basis of independent electron wave functions. Key qualitative features of photoionization cross sections can often be interpreted in terms

of the overlaps of initial and final state one electron radial wave functions [24.1, 13]. The simplest independent electron representation of the atom, the central potential model, proves useful for this purpose.

Photoionization of Atoms

24.2.1 Central Potential Model In the central potential (CP) model the exact H in (24.1) is approximated by a sum of single-particle terms describing the independent motion of each electron in a central potential V(r):  N 2
pi HCP = + V(ri ) . (24.23) 2m i=1

The potential V(r) must describe the nuclear attraction and the electron–electron repulsion as well as possible and must satisfy the boundary conditions V(r)−→ − Z/r and V(r)−→ − 1/r r→∞

(24.24)

r→0

in the case of a neutral atom. HCP is separable in spherical coordinates and its eigenstates can be written as Slater determinants of one-electron orbitals of the form r −1 Pn Ym (Ω) for bound orbitals and of the form r −1 Pε (r)Ym (Ω) for continuum orbitals. The oneelectron radial wave functions satisfy

 d2 Pε (r) ( + 1) + 2 ε − V(r) − Pε (r) = 0 , dr 2 2r 2 (24.25)

subject to the boundary condition Pε (0) = 0, and similarly for the discrete orbitals Pn (r). Hermann and Skillman [24.14] have tabulated a widely used central potential for each element in the periodic table as well as radial wave functions for each occupied orbital in the ground state of each element.

24.2.2 High Energy Behavior

383

values of r where Pn (r) is greatest. (3) Thus it is only necessary to approximate the atomic potential locally, e.g., by means of a screened Coulomb potential   Z − sn + Vn o Vn (r) = − (24.26) r appropriate for the n orbital. Here sn is the “innerscreening” parameter, which accounts for the screening of the nuclear charge by the other atomic electrons, o is the “outer-screening” parameter, which acand Vn counts for the lowering of the n electrons’ binding energy due to repulsion between the outer electrons and the photoelectron as the latter leaves the atom. The potential in (24.26) predicts hydrogen-like photoionization cross sections for inner-shell electrons with onsets determined by the outer-screening paramo. eters Vn Use of more accurate atomic central potentials in place of the screened hydrogenic potential in (24.26) generally enables one to obtain photoionization cross sections below the keV photon energy region to within 10% of the experimental results [24.15]. For  > 0 subshells and photon energies in the keV region and above, the independent particle model becomes increasingly inadequate owing to coupling with nearby ns-subshells, which generally have larger partial cross sections at high photon energies [24.16]. For high, but still nonrelativistic photon energies, i. e., ω  mc2 , the energy dependence of the cross section for the n subshell within the independent particle model is [24.17] 7

σn ∼ ω−− 2 .

(24.27)

However, when interchannel interactions are taken into account, the asymptotic energy dependence for subshells having  > 0 becomes independent of  [24.18]: 9

σn ∼ ω− 2 ( > 0) .

(24.28)

This result stems from coupling of the  > 0 photoionization channels with nearby s-subshell channels.

24.2.3 Near Threshold Behavior For photons in the vuv energy region, i. e., near the outer-subshell ionization thresholds, the photoionization cross sections for subshells with  ≥ 1 frequently have distinctly nonhydrogenic behavior. The cross section, instead of decreasing monotonically as for hydrogen, rises above threshold to a maximum (the so called delayed maximum above threshold). Then it decreases to a minimum (the Cooper minimum [24.19, 20]) and rises to

Part B 24.2

The hydrogen atom cross section, which is nonzero at threshold and decreases monotonically with increasing photon energy, serves as a model for inner-shell photoionization cross sections in the X-ray photon energy range. A sharp onset at threshold followed by a monotonic decrease above threshold is precisely the behavior seen in X-ray photoabsorption measurements. A simple hydrogenic approximation at high energies may be justified theoretically as follows: (1) Since a free electron cannot absorb a photon (because of kinematical considerations), at high photon energies one expects the more strongly bound inner electrons to be preferentially ionized as compared with the outer electrons. (2) Since the Pn (r) for an inner electron is concentrated in a very small range of r, one expects the integrand of the radial dipole matrix element to be negligible except for those

24.2 An Independent Electron Model

384

Part B

Atoms

a second maximum. Finally the cross section decreases monotonically at high energies in accordance with hydrogenic behavior. Such nonhydrogenic behavior may be interpreted as due either to an effective potential barrier or to a zero in the radial dipole matrix element. We examine each of these effects in turn. The delayed maximum above outer subshell ionization thresholds of heavy atoms (i. e., Z
18) is due to an effective potential barrier seen by  = 2 and  = 3 photoelectrons in the region of the outer edge of the atom (24.25). This effective potential lowers the probability of photoelectron escape until the photoelectrons have enough excess energy to surmount the barrier. Such behavior is nonhydrogenic. Furthermore, in cases where an inner subshell with  = 2 or 3 is being filled as Z increases (as in the transition metals, the lanthanides and the actinides) there is a double well potential. This double well has profound effects on the 3p-subshell spectra of the transition metals, the 4d-subshell spectra of the lanthanides, and the 5d-subshell spectra of the actinides, as well as on atoms with Z just below those of these series of elements [24.1, 21, 22].

Cross section minima arise due to a change in sign of the radial dipole transition matrix element in a particular channel [24.23, 24]. Rules for predicting their occurrence were developed by Cooper [24.19,20]. Studies of their occurrence in photoionization from excited states [24.25], in high Z atoms [24.26], and in relativistic approximation [24.27] have been carried out. Only recently has a proof been given [24.28] that such minima do not occur in atomic hydrogen spectra. For other elements, there are further rules on when and how many minima may occur [24.29–31]. Often within such minima, one can observe effects of weak interactions that are otherwise obscured. Relativistic and weak correlation effects on the asymmetry parameter β for s-subshells is a notable example [24.32]. Wang et al. [24.33] have also emphasized that near such minima in the E1 amplitudes, one cannot ignore the effects of quadrupole and higher corrections to the differential cross section. Central potential model calculations [24.33] show that quadrupole corrections can be as large as 10% of the E1 cross section at such cross section minima, even for low photon energies.

24.3 Particle–Hole Interaction Effects

Part B 24.3

The experimental photoionization cross sections for the outer subshells of the noble gases (The noble gases have played a prominent role in the development of the theory of photoionization for two reasons. These were among the first elements studied by experimentalists with synchrotron radiation beginning in the 1960’s. Also, their closed-shell, spherically symmetric ground states simplified the theoretical analysis of their cross sections.) near the ionization thresholds can be understood in terms of interactions between the photoelectron, the residual ion, and the photon field which are called, in many-body theory language, “particle–hole” interactions (see Chapt. 47). These may be described as interactions in which two electrons either excite or deexcite each other out of or into their initial subshell locations in the unexcited atom. To analyze the effects of these interactions on the cross section, it is convenient to classify them into three categories: intrachannel, virtual double excitation, and interchannel. These alternative kinds of particle–hole interactions are illustrated in Fig. 24.1 using both many-body perturbation theory (MBPT) diagrams and more “physical” scattering pictures. We discuss each of these types of interaction in turn.

24.3.1 Intrachannel Interactions The MBPT diagram for this interaction is shown on the left in Fig. 24.1a; on the right a slightly more pictorial description of this interaction is shown. The wiggly line indicates a photon, which is absorbed by the atom in such a way that an electron is excited out of the n subshell. During the escape of this excited electron, it collides or interacts with another electron from the same subshell in such a way that the second electron absorbs all the energy imparted to the atom by the photon; the first electron is de-excited back to its original location in the n subshell. For closed-shell atoms, the photoionization process leads to a 1 P 1 final state in which the intrachannel interaction is strongly repulsive. This interaction tends to broaden cross section maxima and push them to higher photon energies as compared with the results of central potential model calculations. Intrachannel interaction effects are taken into account automatically when the correct Hartree-Fock (HF) basis set is employed in which the photoelectron sees a net Coulomb field due to the residual ion and is coupled to the ion to form the appropriate total orbital L and

Photoionization of Atoms

a)

nl

e–

εl⬘ nl e–

ε⬘l⬘

nl

nl nl

e–

nl

εl⬘

ε⬘l⬘

nl –

e

c)

n1l1

ε⬘⬘l⬘⬘

e– n1l1 e– n0 l0

n0 l0

385

that are in very good agreement with experiment with the exception that resonance features are not predicted.

24.3.3 Interchannel Interactions

nl

b)

24.3 Particle–Hole Interaction Effects

ε⬘l⬘

Fig. 24.1a–c MBPT diagrams (left) and scattering pictures (right) for three kinds of particle–hole interaction: (a) intrachannel scattering following photoabsorption; (b) photoabsorption by a virtual doubly-excited state; (c) interchannel scattering following photoabsorption

spin S angular momenta. Any other basis set requires explicit treatment of intrachannel interactions.

24.3.2 Virtual Double Excitations

24.3.4 Photoionization of Ar An example of both the qualitative features exhibited by photoionization cross sections in the vuv energy region and of the ability of theory to calculate photoionization cross reactions is provided by photoionization of the n = 3 subshell of argon, i. e., +

Ar3s 2 3p 6 + γ → Ar 3s 2 3p 5 + e− +

→ Ar 3s3p 6 + e− .

(24.29)

Figure 24.2 shows the MBPT calculation of Kelly and Simons [24.34], which includes both intrachannel

Part B 24.3

The MBPT diagram for this type of interaction is shown on the left in Fig. 24.1b. Topologically, this diagram is the same as that on the left in Fig. 24.1a. In fact, the radial parts of the two matrix elements are identical; only the angular factors differ. A more pictorial description of this interaction is shown on the right of Fig. 24.1b. The ground state of the atom before photoabsorption is shown to have two electrons virtually excited out of the n subshell. In absorbing the photon, one of these electrons is deexcited to its original location in the n subshell, while the other electron in ionized. These virtual double excitations imply a more diffuse atom than in central-potential or HF models, with the effect that the overly repulsive intrachannel interactions are weakened, leading to cross sections for noble gas atoms

The interchannel interaction shown in Fig. 24.1c is important, particularly for s subshells. This interaction has the same form as the intrachannel interaction shown in Fig. 24.1a, except now when an electron is photoexcited out of the n 0 0 subshell, it collides or interacts with an electron in a different subshell – the n 1 1 subshell. This interaction causes the second electron to be ionized, and the first electron to fall back into its original location in the n 0 0 subshell. Interchannel interaction effects are usually very conspicuous features of photoionization cross sections. When the interacting channels have partial photoionization cross sections which differ greatly in magnitude, one finds that the calculated cross section for the weaker channel is completely dominated by its interaction with the stronger channel. At the same time, it is often a safe approximation to ignore the effect of weak channels on stronger channels. In addition, when the interacting channels have differing binding energies, their interchannel interactions lead to resonance structure in the channel with lower binding energy (arising from its coupling to the Rydberg series in the channel with higher binding energy). At high photon energies, s-subshell partial cross sections dominate over  > 0 subshell partial cross sections [(24.27), (24.28)]. Hence interchannel interactions of  > 0 subshells with nearby s-subshells change independent particle model predictions significantly. In particular, as noted in Sect. 24.3.2, such interactions can drastically change the magnitudes of the  > 0 partial cross sections [24.16] as well as their asymptotic energy behavior [24.18].

386

Part B

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and interchannel interactions as well as the effect of virtual double excitations. The cross section is in excellent agreement with experiment [24.35, 36], even to the extent of describing the resonance behavior due to discrete members of the 3s → εp channel. Figure 24.2 illustrates most of the features of photoionization cross sections described so far. First, the cross section rises to a delayed maximum just above the threshold because of the potential barrier seen by photoelectrons from the 3p subshell having  = 2. For photon energies in the range of 45–50 eV, the calculated cross section goes through a minimum because of a change in sign of the 3p → εd radial dipole amplitude. The HFL and HFV calculations include the strongly repulsive intrachannel interactions in the 1 P final-state channels and calculate the transition amplitude using the length (L) and velocity (V) form respectively for the electric dipole transition operator (24.7). With respect to the results of central potential model calculations, the HFL and HFV results have lower and broader maxima at higher energies. They also disagree with each other by a factor of two! Inclusion of virtual double excitations results in length and velocity results that agree to within 10% with each other and with experiment, except that the resonance structures are not reproduced. Finally, taking into account the interchannel interactions, one obtains the length and velocity form results shown in Fig. 24.2 by dash-dot and dashed curves re-

50

σ (10–18 cm2)

45

HFL

40 35 30 25 20

HFV

15 10 5 0 16

20

25

30

35

40

45 50 54 Photon energy (eV)

Fig. 24.2 Photoionization cross section for the 3p and 3s subshells of Ar. HFL and HFV indicate the length and velocity results obtained using HF orbitals calculated in a 1 P 1 potential. Dot-dash and dashed lines represent the length and velocity results of the MBPT calculation of Kelly and Simons [24.34]. Only the four lowest 3s → np resonances are shown; the series converges to the 3s threshold at 29.24 eV. Experimental results are those of Samson [24.35] above 37 eV and of Madden et al. [24.36] below 37 eV (After [24.34])

spectively. Agreement with experiment is excellent and the observed resonances are well-reproduced.

24.4 Theoretical Methods for Photoionization 24.4.1 Calculational Methods

Part B 24.4

Most of the ab initio methods for the calculation of photoionization cross sections (e.g., the MBPT method [24.37], the close-coupling (CC) method [24.38], the R-matrix method [24.39, 40], the random phase approximation (RPA) method [24.9], the relativistic RPA method [24.41], the transition matrix method [24.42,43], the multiconfiguration Hartree-Fock (MCHF) method [24.44–46], etc.) have successfully calculated outer p-subshell photoionization cross sections of the noble gases by treating in their alternative ways the key interactions described above, i. e., the particle– hole interactions. In general, these methods all treat both intrachannel and interchannel interactions to infinite order and differ only in their treatment of ground state correlations. (The exception is MBPT, which often treats interchannel interactions between weak and strong

channels only to first or second order.) These methods therefore stand in contrast to central potential model calculations, which do not treat any of the particle–hole interactions, and single-channel term-dependent HF calculations, which treat only the intrachannel interactions. The key point is that selection of the interactions that are included in a particular calculation is more important than the method by which such interactions are handled. Treatment of photoionization of atoms other than the noble gases presents additional challenges for theory. For example, elements such as the alkaline earths, which have s2 outer subshells, require careful treatment of electron pair excitations in both initial and final states. Open shell atoms have many more ionization thresholds than do the noble gases. Treatment of the resultant rich resonance structures typically relies heavily on quantum defect theory [24.46] (see Chapt. 32). All the methods

Photoionization of Atoms

listed above can be used to treat elements other than the noble gases, but a method which has come to prominence because of the excellent results it obtains for both alkaline earth and open-shell atoms is the eigenchannel R-matrix method [24.47].

24.4.2 Other Interaction Effects A number of interactions, not of the particle–hole type, lead to conspicuous effects in localized energy regions. When treating photoionization in such energy regions, one must be careful to choose a theoretical method which is appropriate. Among the interactions which may be important are the following: Relativistic and Spin-Dependent Interactions The fact that j =  − 12 electrons are contracted more than j =  + 12 electrons at small distances has an enormous effect on the location of cross section minima in heavy elements [24.15, 48]. It may explain the large observed differences in the profiles of a resonance decaying to final states that differ only in their fine structure quantum numbers [24.49]. Inner-Shell Vacancy Rearrangement Inner-shell vacancies often result in significant production of satellite structures in photoelectron spectra. Calculations for inner subshell partial photoionization cross sections are often substantially larger than results of photoelectron measurements [24.50–52]. This differ-

24.5 Recent Developments

387

ence is attributed to such satellite production, which is often not treated in theoretical calculations. Polarization and Relaxation Effects Negative ion photodetachment cross sections often exhibit strong effects of core polarization near threshold. These effects can be treated semi-empirically, resulting in excellent agreement between theory and experiment [24.53]. Even for inner shell photoionization cross sections of heavy elements, ab initio theories do not reproduce measurements near threshold without the inclusion of polarization and relaxation effects [24.54, 55]. An Example The calculation of the energy dependence of the asymmetry parameter β for the 5s subshell of xenon requires the theoretical treatment of all of the above effects. In the absence of relativistic interactions, β for Xe 5s would have the energy-independent value of two. Deviations of β from two are therefore an indication of the presence of these relativistic interactions. The greatest deviation of β from two occurs in the localized energy region where the partial photoionization cross section for the 5s subshell has a minimum. In this region, however, relativistic calculations show larger deviations from two than are observed experimentally. Inner shell rearrangement and relaxation effects play an important role [24.56, 57] and must be included to achieve good agreement with experiment.

24.5 Recent Developments see [24.64]). Besides asymmetries in the photoelectron angular distributions, non-dipole effects lead also to new features for spin-resolved measurements [24.65,66], and for the case of polarized atoms [24.67]. Recently, nondipole effects have been predicted to be significant also in double photoionization of helium at relatively low photon energies [24.68]. Finally, both experimental and theoretical studies of ionic species have flourished over the past decade. In particular, photodetachment of negative ions near excited atomic thresholds provides an opportunity to study correlated, three-body Coulomb states unencumbered by Rydberg series. Only with the advent of powerful computer workstations have theorists been able to carry out numerical calculations for such high, doubly excited states with spectroscopic accuracy. Following experiments for photodetachment of H− with excita-

Part B 24.5

One of the most intensively studied areas in atomic photoionization in recent years has been the double photoionization of the helium atom. Extensive sets of experimental measurements for the two electron angular distributions (i. e., the triply differential cross sections) have provided stringent tests for various theoretical models and their treatments of electron correlations. A number of excellent reviews of this field have been published recently [24.58–60]. Another intensively studied area has been the analysis and measurement of non-dipole effects in photoionization, which were first observed in the X-ray region [24.61] but have been found to be significant even in the vuv photon energy region [24.62, 63]. In general, these effects stem from interference between electric quadrupole and the (usual) electric dipole transition amplitudes in differential cross sections (for a recent review,

388

Part B

Atoms

tions of atomic levels in H(n > 2), theorists developed propensity rules for identifying and characterizing the dominant photodetachment channels [24.69–71]. More recently, experimental and theoretical interest shifted to the negative alkali ions (e.g., Li− and Na− ), which for low photon energies have outer electron detachment spectra grossly resembling that of H− . However, the negative alkali spectra contain clear signatures of propensity-rule-forbidden states that become increasingly prominent as the atomic number increases (owing to the nonhydrogenic inner electron cores). A brief review of low energy negative alkali photodetachment is given in [24.72]. Among the more general features brought to light by these studies is the mirroring of resonance profiles in alternative partial cross sections, which appears to be a very general phenomenon common to photodetachment and photoionization processes involving highly excited residual atoms or ions [24.73]. Recently, high energy (K -shell) photodetachment of the negative ions Li− and He− (resulting in two elec-

tron ionization) has been studied both experimentally and theoretically [24.74, 75]. These studies represent the first results for inner shell photodetachment. There is general agreement between theory and experiment well above the K edge, but the theoretical cross sections at the K edge are significantly higher than the experimental measurements. The latter discrepancy is now understood as arising from recapture of the lowenergy detached electron following Auger decay of the inner-shell vacancy, which when taken into account theoretically has been shown to provide results that agree with experiment [24.76, 77]. Also, the first experimental data together with theoretical analyses were recently presented for photoionization of ground and metastable positive ions (O+ and Sc++ ) [24.78, 79]. With the advent of data for photoionization of positive ions it now becomes possible (using the principle of detailed balance) to make connections to data for electron–ion photo-recombination cross sections [24.79].

24.6 Future Directions The construction of high brightness synchrotron light sources and the increasing use of lasers are providing the means to study atomic photoionization processes at an unparalleled level of detail. The synchrotrons generally produce photons in the soft X-ray and X-ray regions. Thus, inner shell vacancy production and decay, satellite production, and multiple ionization phenomena are all being increasingly studied. Laser sources are allowing production of atoms in tailored initial states. Studies of ions, both negativeand positive, in

well-specified states are also increasingly being carried out. Thus, photoionization of excited atoms and ions and, in particular, complete measurements of particular photoionization processes, are now possible. Recent collections of short review papers provide references to these topics [24.3, 4]. In addition, two recent reviews of experimental results for noble gas atom photoionization [24.80] and for metal atom photoionization [24.81] also provide valuable information on the current state of the correspondingtheoretical results.

References 24.1

24.2 24.3

Part B 24

24.4 24.5 24.6

A. F. Starace: Handbuch der Physik, Vol. XXXI, ed. by W. Mehlhorn (Springer, Berlin, Heidelberg 1982) pp. 1–121 M. Ya. Amusia: Atomic Photoeffect (Plenum, New York 1990) T. N. Chang (Ed.): Many-Body Theory of Atomic Structure and Photoionization (World Scientific, Singapore 1993) U. Becker, D. A. Shirley: VUV and Soft X-Ray Photoionization Studies (Plenum, New York 1994) J. J. Sakurai: Advanced Quantum Mechanics (Addison-Wesley, Reading 1967) p. 39 S. Chandrasekhar: Astrophys. J. 102, 223 (1945)

24.7 24.8 24.9 24.10 24.11 24.12 24.13

A. F. Starace: Phys. Rev. A 3, 1242 (1971) A. F. Starace: Phys. Rev. A 8, 1141 (1973) M. Ya. Amusia, N. A. Cherepkov: Case Studies in At. Phys. 5, 47 (1975) J. M. Blatt, L. C. Biedenharn: Rev. Mod. Phys. 24, 258 (1952) See Sect. 7 of [24.1] C. N. Yang: Phys. Rev. 74, 764 (1948) See, e.g., (9.6)–(9.15) of for expressions for the reduced matrix elements in (24.22) in which all angular integrations have been carried out and the results are expressed in terms of one-electron radial dipole matrix elements

Photoionization of Atoms

24.14 24.15 24.16

24.17

24.18

24.19 24.20 24.21 24.22

24.23 24.24 24.25 24.26 24.27 24.28 24.29 24.30 24.31 24.32 24.33 24.34 24.35 24.36 24.37

24.38 24.39

24.41 24.42

24.43 24.44 24.45 24.46 24.47

24.48 24.49 24.50 24.51 24.52 24.53 24.54

24.55 24.56 24.57 24.58 24.59 24.60 24.61

24.62

24.63

24.64 24.65 24.66 24.67 24.68 24.69 24.70

T. N. Chang, U. Fano: Phys. Rev. A 13, 282 (1976) J. R. Swanson, L. Armstrong, Jr.: Phys. Rev. A 15, 661 (1977) J. R. Swanson, L. Armstrong, Jr.: Phys. Rev. A 16, 1117 (1977) M. J. Seaton: Rep. Prog. Phys. 46, 167 (1983) C. H. Greene: Fundamental Processes of Atomic Dynamics, ed. by J. S. Briggs, H. Kleinpoppen, H. O. Lutz (Plenum, New York 1988) pp. 105–127 and references therein D. W. Lindle, T. A. Ferrett, P. A. Heiman, D. A. Shirley: Phys. Rev. A 37, 3808 (1988) M. Krause, F. Cerrina, A. Fahlman, T. A. Carlson: Phys. Rev. Lett. 51, 2093 (1983) J. B. West, P. R. Woodruff, K. Codling, R. G. Houlgate: J. Phys. B 9, 407 (1976) M. Y. Adam, F. Wuilleumier, N. Sandner, V. Schmidt, G. Wendin: J. Phys. (Paris) 39, 129 (1978) U. Becker, T. Prescher, E. Schmidt, B. Sonntag, H.E. Wetzel: Phys. Rev. A 33, 3891 (1986) K. T. Taylor, D. W. Norcross: Phys. Rev. A 34, 3878 (1986) M. Ya. Amusia: Atomic Physics 5, ed. by R. Marrus, M. Prior, H. Shugart (Plenum, New York 1977) pp. 537–565 W. Jitschin, U. Werner, G. Materlik, G. D. Doolen: Phys. Rev. A 35, 5038 (1987) G. Wendin, A. F. Starace: Phys. Rev. A 28, 3143 (1983) J. Tulkki: Phys. Rev. Lett. 62, 2817 (1989) J. S. Briggs, V. Schmidt: J. Phys. B 33, R1 (2000) G. C. King, L. Avaldi: J. Phys. B 33, R215 (2000) J. Berakdar, H. Klar: Phys. Rep. 340, 474 (2001) B. Krässig, M. Jung, M. S. Gemmel, E. P. Kanter, T. LeBrun, S. H. Southworth, L. Young: Phys. Rev. Lett. 75, 4736 (1995) O. Hemmers, R. Guillemin, E. P. Kanter, B. Krässig, D. W. Lindle, S. H. Southworth, R. Wehlitz, J. Baker, A. Hudson, M. Lotrakul, D. Rolles, W. C. Stolte, I. C. Tran, A. Wolska, S. W. Yu, M. Ya. Amusia, K. T. Cheng, L. V. Chernysheva, W. R. Johnson, S. T. Manson: Phys. Rev. Lett. 91, 053002 (2003) E. P. Kanter, B. Krässig, S. H. Southworth, R. Guillemin, O. Hemmers, D. W. Lindle, R. Wehlitz, M. Ya. Amusia, L. V. Chernysheva, N. L. S. Martin: Phys. Rev. A 68, 012714 (2003) D. W. Lindle, O. Hemmers: J. Elec. Spectros. Rel. Phen. 100, 297 (1999)(Special Issue, October) N. A. Cherepkov, S. K. Semenov: J. Phys. B 34, L211 (2001) T. Khalil, B. Schmidtke, M. Drescher, N. Müller, U. Heinzmann: Phys. Rev. Lett. 89, 053001 (2002) A. N. Grum-Grzhimailo: J. Phys. B 34, L359 (2001) A. Y. Istomin, N. L. Manakov, A. V. Meremianin, A. F. Starace: Phys. Rev. Lett. 92, 062002 (2004) H. R. Sadeghpour, C. H. Greene: Phys. Rev. Lett. 65, 313 (1990) J. M. Rost, J. S. Briggs, J. M. Feagin: Phys. Rev. Lett. 66, 1642 (1991)

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F. Hermann, S. Skillman: Atomic Structure Calculations (Prentice-Hall, Englewood Cliffs 1963) R. H. Pratt, A. Ron, H. K. Tseng: Rev. Mod. Phys. 45, 273 (1973) E. W. B. Dias, H. S. Chakraborty, P. C. Deshmukh, S. T. Manson, O. Hemmers, P. Glans, D. L. Hansen, H. Wang, S. B. Whitfield, D. W. Lindle, R. Wehlitz, J. C. Levin, I. A. Sellin, R. C. C. Perera: Phys. Rev. Lett. 78, 4553 (1997) H. A. Bethe, E. E. Salpeter: Quantum Mechanics of One- and Two-Electron Atoms (Springer, Berlin, Heidelberg 1957), Sects. 69-71 M. Ya. Amusia, N. B. Avdonina, E. G. Drukarev, S. T. Manson, R. H. Pratt: Phys. Rev. Lett. 85, 4703 (2000) J. W. Cooper: Phys. Rev. 128, 681 (1962) U. Fano, J. W. Cooper: Rev. Mod. Phys. 40, 441 (1968) , pp. 50–55, and references therein J. P. Connerade, J. M. Estiva, R. C. Karnatak (Eds.): Giant Resonances in Atoms, Molecules, and Solids (Plenum, New York 1987) and references therein D. R. Bates: Mon. Not. R. Astron. Soc. 106, 432 (1946) M. J. Seaton: Proc. Roy. Soc. A 208, 418 (1951) A. Z. Msezane, S. T. Manson: Phys. Rev. Lett. 48, 473 (1982) Y. S. Kim, R. H. Pratt, A. Ron: Phys. Rev. A 24, 1626 (1981) Y. S. Kim, A. Ron, R. H. Pratt, B. R. Tambe, S. T. Manson: Phys. Rev. Lett. 46, 1326 (1981) S. D. Oh, R. H. Pratt: Phys. Rev. A 34, 2486 (1986) R. H. Pratt, R. Y. Yin, X. Liang: Phys. Rev. A 35, 1450 (1987) R. Y. Yin, R. H. Pratt: 35, 1149 (1987) R. Y. Yin, R. H. Pratt: 35, 1154 (1987) S. T. Manson, A. F. Starace: Rev. Mod. Phys. 54, 389 (1982) M. S. Wang, Y. S. Kim, R. H. Pratt, A. Ron: Phys. Rev. A 25, 857 (1982) H. P. Kelly, R. L. Simons: Phys. Rev. Lett. 30, 529 (1973) J. A. R. Samson: Adv. At. Mol. Phys. 2, 177 (1966) R. P. Madden, D. L. Ederer, K. Codling: Phys. Rev. 177, 136 (1969) H. P. Kelly: Photoionization and Other Probes of Many-Electron Interactions, ed. by F. J. Wuilleumier (Plenum, New York 1976) pp. 83–109 P. G. Burke, M. J. Seaton: Methods Comput. Phys. 10, 1 (1971) P. G. Burke, W. D. Robb: Adv. At. Mol. Phys. 11, 143 (1975) P. G. Burke, W. D. Robb: Electronic and Atomic Collisions: Invited Papers and Progress Reports, ed. by G. Watel (North Holland, Amsterdam 1978) pp. 201– 280 W. R. Johnson, C. D. Lin, K. T. Cheng, C. M. Lee: Phys. Scr. 21, 409 (1980) T. N. Chang, U. Fano: Phys. Rev. A 13, 263 (1976)

References

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24.71 24.72

24.73 24.74

24.75

24.76

H. R. Sadeghpour, C. H. Greene, M. Cavagnero: Phys. Rev. A 45, 1587 (1992) A. F. Starace: Novel Doubly Excited States Produced in Negative Ion Photodetachment, ed. by F. Aumayr, H. Winter (World Scientific, Singapore 1998) pp. 107–116 C. N. Liu, A. F. Starace: Phys. Rev. A 59, R1731 (1999) N. Berrah, J. D. Bozek, A. A. Wills, G. Turri, L. L. Zhou, S. T. Manson, G. Akerman, B. Rude, N. D. Gibson, C. W. Walter, L. VoKy, A. Hibbert, S. M. Ferguson: Phys. Rev. Lett. 87, 253002 (2001) N. Berrah, J. D. Bozek, G. Turri, G. Akerman, B. Rude, H. L. Zhou, S. T. Manson: Phys. Rev. Lett. 88, 093001 (2002) J. L. Sanz-Vicario, E. Lindroth, N. Brandefelt: Phys. Rev. A 66, 052713 (2002)

24.77

24.78

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24.80 24.81

T. W. Gorczyca, O. Zatsarinny, H. L. Zhou, S. T. Manson, Z. Felfli, A. Z. Msezane: Phys. Rev. A 68, 050703 (R) (2003) A. M. Covington, A. Aguilar, I. R. Covington, M. Gharaibeh, C. A. Shirley, R. A. Phaneuf, I. Alvarez, C. Cisneros, G. Hinojosa, J. D. Bozek, I. Dominguez, M. M. Sant’Anna, A. S. Schlachter, N. Berrah, S. N. Nahar, B. M. McLaughlin: Phys. Rev. Lett. 87, 243002 (2001) S. Schippers, A. Müller, S. Ricz, M. E. Bannister, G. H. Dunn, J. Bozek, A. S. Schlacter, G. Hinojosa, C. Cisneros, A. Aguilar, A. M. Covington, M. F. Gharaibeh, R. A. Phaneuf: Phys. Rev. Lett. 89, 193002 (2002) V. Schmidt: Rep. Prog. Phys. 55, 1483 (1992) B. Sonntag, P. Zimmerman: Rep. Prog. Phys. 55, 911 (1992)

Part B 24

391

The phenomenon of autoionization, or more particularly the autoionization state itself, is treated for the most part in this chapter as a bound state. The process is rigorously a part of the scattering continuum (Chapt. 47), but, due mostly to the work of Feshbach [25.1], a rigorous formulation can be established whereby the main element of the theory can be made into a bound state problem with the scattering elements built around it. The major constituent of both these features is accomplished with projection operators. A brief description of the above elements of the theory, centered around projection operators, is the aim of this chapter [25.2], although some additional methods are discussed in Sect. 25.5. Rydberg units are used unless otherwise noted.

25.1

Introduction ....................................... 391 25.1.1 Auger Effect .............................. 391 25.1.2 Autoionization, Autodetachment, and Radiative Decay .................. 391

25.1.3

Formation, Scattering, and Resonances ........................ 391

25.2 The Projection Operator Formalism ....... 392 25.2.1 The Optical Potential ................. 392 25.2.2 Expansion of Vop : The QHQ Problem....................... 392 25.3 Forms of P and Q ................................. 25.3.1 The Feshbach Form.................... 25.3.2 Reduction for the N = 1 Target ..... 25.3.3 Alternative Projection and Projection-Like Operators ....

393 393 394

25.4 Width, Shift, and Shape Parameter ....... 25.4.1 Width and Shift ........................ 25.4.2 Shape Parameter....................... 25.4.3 Relation to Breit–Wigner Parameters...............................

394 394 395

394

396

25.5 Other Calculational Methods ................. 396 25.5.1 Complex Rotation Method .......... 396 25.5.2 Pseudopotential Method ............ 397 25.6 Related Topics ..................................... 398 References .................................................. 399

25.1 Introduction 25.1.1 Auger Effect Autoionization falls within the general class of phenomena known as the Auger effect. In the Auger effect an atomic system “seemingly” (Sect. 25.1.3) spontaneously decays into a partition of its constituent parts.

25.1.2 Autoionization, Autodetachment, and Radiative Decay If the initial, composite system is neutral, or positively charged, and its constituent decay particles are an electron and the residual ion, then the process is called autoionization. If the original system is a negative ion, so that the residual heavy particle system is neutral, then the process is technically called autodetachment; for the

most part, the physics and the mathematical treatment are the same. It is also possible, before electron emission takes place, that the system will alternatively decay radiatively to an autoionization state of lower energy, or a true bound state of the composite system. The latter process is called radiative stabilization (which is a basic part of dielectronic recombination (Chapt. 55)).

25.1.3 Formation, Scattering, and Resonances Autoionizing states are formed by scattering processes and photoabsorption. These are the inverses of the autoionization and photon emission processes by which they can decay. In the scattering process, formation of the autoionization state corresponds to a resonance

Part B 25

Autoionization 25. Autoionization

392

Part B

Atoms

Part B 25.2

in the scattering cross section (Chapt. 47). Autoionization is the process which corresponds to the decay of the resonance. The decay of the resonance (autoionization) is then seen to be the last half of the resonant scattering process. Strictly speaking therefore, the resonant or autoionization state, although it may be long-lived, is not completely stationary, and

that is the reason that the word “seemingly” was used to describe the Auger process. The compound system can also be formed by absorption of photons impinging on a bound state (usually the ground state) of the compound system, in which case the autionization state shows up as a “line” in the absorption spectrum.

25.2 The Projection Operator Formalism In the energy domain where the Schrödinger equation (SE) HΨ = EΨ

The most significant part of (25.7) is the optical potential Vop given by

(25.1)

describes scattering, the wave function does not vanish at infinity, i. e., lim Ψ = 0 ,

(25.2)

ri →∞

where ri is the radial coordinate of ith electron. The basic idea of the projection operator formalism [25.1] is to define projection operators P and Q which separate Ψ into scattering-like (PΨ ) and quadratically integrable (QΨ ) parts: Ψ = PΨ + QΨ .

Vop = PHQ(E − Q HQ)−1 Q HP .

(25.8)

Vop is a nonlocal potential; the most incisive way to give it meaning is to the define the Q HQ problem.

25.2.2 Expansion of Vop : The QHQ Problem The following eigenvalue problem constitutes the heart of the projection operator formalism:

(25.3)

Q HQΦn = En Φn .

Implicit in (25.3) are

(25.9)

completeness: P + Q = 1 , idempotency: P 2 = P , orthogonality: PQ = 0 ,

Q2 = Q ,

and the asymptotic properties   PΨ = lim Ψ ri →∞ lim . ri →∞  QΨ = 0

(25.4)

 δ (25.5)

25.2.1 The Optical Potential Straightforward manipulation of (25.1) and (25.3) leads to an important relation between QΨ and PΨ : QΨ = (E − Q HQ)−1 Q HPΨ .

(25.6)

From (25.6) a basic equation for PΨ (which, by virtue of (25.5), contains all the scattering information) emerges: (PHP + Vop − E )PΨ = 0 .

For calculational purposes, it is best to recast (25.9) in the variational form

(25.7)

ΦQ HQΦ ΦQΦ

 =0.

(25.10)

This equation may yield a discrete plus a continuous spectrum in an energy domain where the SE has only a continuous spectrum. Moreover, if the Q operator is appropriately chosen, then the discrete eigenvalues En are close to the desired class of many-body resonances, called variously Feshbach resonances, core-excited [25.3], or doubly-excited states. In terms of these solutions, Vop has the expansion

Vop =

 

PHQ|Φn (E − En )−1 Φn |Q HP . (25.11)

Autoionization

25.3 Forms of P and Q

25.3.1 The Feshbach Form Projection operators are not unique. Feshbach [25.1] has sketched a derivation of a “robust” projection operator for the general N-electron target system. Robust means that QΦ is devoid of open channels. The complete expression for P, including inelastic channels has been derived in [25.5] N+1   

ψ r (i) · ψ r (i) P= i=1



 vα (ri ) · ψ r (i) vα (ri ) · ψ r (i) + , λα − 1 λα

(25.12)

where the prime on the second summation means that terms with λα = 1 are to be omitted, ri denotes the radial coordinate of electron i, and r (i) stands for the angular and spin coordinates of electron i plus all coordinates of the remaining N electrons. Thus r (i) indicates the totality of all coordinates of the (N + 1) electrons except the radial coordinate of the ith electron ri . ψ(r (i) ) is the vector channel wave function in which the angular momentum and spin of the electron i are coupled to the target in state ν (ν = 0, . . . , νmax ). A component of a vector labels the inelastic channel, and dot products represent sums over channels. The Q operator is then made explicit by completeness: Q = 1 − P, (25.4). The α-indexed quantities in (25.12) arise from exchange; they are the eigensolutions of the integral eigenvalue problem vα (ri ) = λα K(ri |r j ) · vα (r j )r j . Here, K(ri |r j ) is a matrix with components  K µν (ri |r j ) ∝ N ψµ r (i) ψν r ( j) r (ij) ,

(25.13)

α=1

ν=0

Table 25.1 Test of sum rule (25.15) for the lowest

He− (1s2s 2 2 S) autodetachment state [25.4]. Projection operators are based on a 4 term Hylleraas φ0 and the variational form of vα given below. Values of other constants are given in [25.4] φ0 ∝ (1 + C1 r1 + C2 r2 + C12 r12 ) exp(−γ1 r1 − γ2 r2 ) +(r1 ↔ r2 ) ,

vα ∝ c(α) + c(α) r exp(−γ1 r) 11(α) 12(α)

+ c21 + c22 r exp(−γ2 r) , and the variational eigenvalues are [25.4] λ1 = 1.009 453 λ2 = 232.8540 λ4 = 4 817 341 λ3 = 80 101.08 Summation

Value (Ry−1 )

4  (λα )−1

0.994 9425

α=1

K(r|r)r

(25.14)

ergy of the lowest He− (1s2s 2 2 S) autodetachment state. The Q HQ results are denoted (Eˆ − E 0 )Quasi for the quasiprojection method and (E − E 0 )Complete for the complete projection method [25.4]. The entries labeled “Other results” give the full resonant energy. Units are eV (Eˆ − E0 )Quasi

a (E − E0 )Complete

Closed shell 19.366 19.593 Open shell 19.385 19.666 |1s1s  | + 2p 2 19.388 19.615 4 term Hylleraas 19.381 19.496 10 term Hylleraas 19.379 19.504 Other results 19.402 b , 19.376 c , 19.367 d , 19.367 ± 0.007 e a

b c d

(25.15)

0.994 9514

Table 25.2 Comparison of methods for calculating the en-

Target

and r (ij) indicates that all variables, except ri and r j are integrated over. In the inelastic regime, therefore, the [vα , λα ] are not associated with specific channels, but rather with the totality of open channels. This means that every component of vα is associated with all inelastic channels [25.5]. The vα are orthogonal, and can be normalized so that vα · vβ  = δαβ . The λα obey several sum rules [25.4], of which the most useful is ν nλ max  (λα )−1 = K νν (r|r)r ,

where n λ is the number of eigenvalues of (25.13). The vα can be accurately calculated by use of a variational principle [25.4]. A test of (25.15) for the lowest He−( 2 S) resonance, using Hylleraas functions to construct Q in the evaluation of Q HQ, is shown in Table 25.1 [25.4], and results for the resonance position are compared in Table 25.2.

e

Values obtained using R∞ = 13.605 698 eV and E 0 = −79.0151 eV [25.6] Complex rotation method; Junker and Huang [25.7] Hole-projection complex-rotation; Davis and Chung [25.8] Hermitian-representation complex-rotation; [25.9] Experiment; Brunt et al. [25.10]

Part B 25.3

25.3 Forms of P and Q

393

394

Part B

Atoms

Part B 25.4

25.3.2 Reduction for the N = 1 Target Explicit rigorous P and Q operators of the above type are only possible for N = 1 (i. e. hydrogenic) targets. In that case, spin can be easily eliminated by using spatially symmetric or antisymmetric wave functions. In the elastic region, P and Q reduce to [25.11, 12] P = P1 + P2 − P1 P2 ;

Q = 1 − P1 − P2 + P1 P2 . (25.16)

Here the Pi = φ(ri )φ(ri ) are purely spatial projectors. Forms for the inelastic continuum are easily generalized [25.13]. There have been many calculations of Q HQ for two-electron systems (N = 1), starting with fundamental work of O’Malley and Geltman [25.13]. A small sample is given in Table 25.3. They are given for their historical importance, demonstrating for the first time the convergence of eigenvalues to well defined values in the continuum. All eigenvalues (below the n = 2 threshold, in this case) correspond to resonances. Table 25.3 Energies Es of the He(2s2p 1P 0 ) autoionization

states below He+ (n = 2) threshold from the variational calculations of O’Malley and Geltman [25.13]. Units are Ry. N is the number of terms in the trial function N

s=1

s=2

s=3

9 15 20 25

− 1.377 08 − 1.380 44 − 1.382 16 − 1.383 16

− 1.178 92 − 1.183 12 − 1.183 48 − 1.190 00

− 1.097 16 − 1.104 32 − 1.108 28 − 1.111 88

25.3.3 Alternative Projection and Projection-Like Operators Two alternative methods based on the idea of projection are available: quasi-projectors and hole projection operators. Quasi-projectors [25.14] relax the condition of idempotency, but still maintain a discrete spectrum, which is in a one-to-one correspondence with resonances, with a predeterminable number of exceptions [25.2]. Hole projection operators have proven to be a more practical and effective approach [25.15]. The method uses one-particle (say, hydrogenic) orbitals, φn (q; r),  to build holes via projectors, [1 − n φn φn ], operating on the (N + 1)-particle wave function. The Rayleigh–Ritz functional is minimized with respect to the parameters in Φ, but it is maximized with respect to q, the nonlinear parameter in all the φn . In a model case, this minimax procedure has been shown [25.15] to optimize the eigenvalues to describe resonance energies; many calculations since then [25.16] have verified the minimax criteria in many-electron systems. More recently, the technique has been combined with complex rotation, so as to enable calculation of other resonant quantities [25.17]. Remarkably accurate results have been obtained. Finally, hole projectors are ideally suited for innershell vacancy states of many-electron systems if high accuracy is required [25.2]. Effectively, this amounts to a reliable method for optimizing parameters of a hole orbital to be used in an Auger transition integral for filling such a vacancy, although that method has apparently never been used (Chapt. 62).

25.4 Width, Shift, and Shape Parameter 25.4.1 Width and Shift Here one requires PΨ as well as QΨ . The former is obtained from a “nonresonant continuum,” defined as the scattering solution of   PHP + Vs(nr) − E PΨs(nr) = 0 , (25.17) where the nonresonant potential, Vs(nr) = Vop − PHQΦs (E − Es )−1 Φs Q HP , (25.18)

excludes the resonant state s from the optical potential. In terms of PΨs(nr) , whose phase shift, η0 , is smooth in the vicinity of E ≈ Es , a solution of the complete

problem, (25.7), can be constructed [25.13] with a phase shift η0 + ηr , where the additional phase shift   Γ/2 ηr = arctan (25.19) (Es + ∆s ) − E exhibits typical resonant behavior (0 < ηr < π). From (25.19) it is clear that the “true” position of the resonance is E s (E ) = Es + ∆s (E ) .

(25.20)

The width Γs and shift ∆s are given by [25.13]: Γs (E ) = 2k|Ψs(nr) (E ) PHQ Φs |2 , ∆s (E ) = Φs Q HP G P (E ) PHQΦs  ,

(25.21) (25.22)

Autoionization

25.4.2 Shape Parameter

(es + qs )2 |PΨ + QΨ |T |i|2 = , 2 1 + es |PΨs(nr) + QΨs(nr) |T |i|2

(25.23)

where es is the scaled energy es = (E − E s )/(Γs /2)

(25.24)

and T is a radiative transition operator (25.25). To analyze (25.23) in the Feshbach formalism, the key point [25.13] is to recognize that the bras on the right-hand side must include P as well as Q parts of the respective wave functions, as is indicated in (25.23). That is because the T operator is a perturbation and not part of the dynamical problem. With T in length form N+1 

zj ,

n=s

E − En

1+ E

9.5574

0.04707 ΓF

9.55736

0.04705

1 + EF

.

(25.26)

0.04703

k2F

9.55728 9.52

9.53

9.54

9.55

9.56

9.57

0.04701 9.58 9.59 2 k (eV)

Fig. 25.1 Precision calculation of H− (1 S) resonance. Solid curve:

E + 1 vs. k2 , where E = E + ∆, (25.20). kF2 is that value of k2 at which E + 1 = k2 . Dashed curve is Γ vs. k2 from (25.21), and ΓF = Γ kF2 . Curves are from calculations of [25.18]; results are E F = 9.557 35 eV and ΓF = 0.047 0605 eV. Applying corrections, (25.31) gives finally E BW = E F + O(10−6 ), ΓBW = 0.047 17 eV

With use of these relations in the rhs of (25.23) an explicit formula for qs was derived in [25.2]; it is given by qs =

˜ |i Φs Q|T (nr) ki|T |Ψs Ψs(nr) |PHQΦs 

where Q˜ = Q + Q HPG s +

,

(25.27)

 Q HPG s HQ|Φn Φn . E − En n=s

(25.28)

The Green’s function G s in (25.28) is the one associated with (25.17). It can be expanded in terms of the eigensolutions of (25.17), but its spectrum may have a discrete as well as continuous part, in which case, Gs =

the rhs of (25.23) can be calculated by noting that PΨs and PΨs(nr) can be, in principle, determined from (25.7) and (25.17). QΨs is then derived from PΨs using (25.6). But QΨs(nr) excludes the sth term from the right-hand side of (25.6): QΨs(nr) =

Γ

(25.25)

j=1

 Φn Φn Q HPΨ (nr) 

Γ (k2) 0.04709

9.55732

The shape of an isolated radiative transition between an autoionization state and some other state can be described by Γ , E, and an additional parameter qs , often called the shape parameter [25.19]. The ratio of transition probabality (in, say, absorption from an initial state labeled i) through the resonant state to its nonresonant value, parametrized in its Fano form [25.19] on the left-hand side of (25.23), can be equated to its meaning in Feshbach terms on the right-hand side of (25.23):

T∝

1 + E (k2) 9.55744

 PΨν(nr) PΨν(nr) ν

℘ + π



Es − Eν

√ PΨs(nr) (E  )PΨs(nr) (E  ) E  dE  . Es − E  (25.29)

The sum over ν refers to the discrete part of the spectrum of (25.17) (if there is one), and ℘ denotes a principal value integral over the continuum solutions. The latter are always assumed to be normalized as

395

Part B 25.4

where k is the scattering momentum (i. e. k2 = E − E 0 ), and G P is the Green’s function associated with (25.17); G P can be simplified from the form given in [25.13] [(2.28) of first article of [25.2]]. Equation (25.20) is an implicit equation for the energy at which the resonance occurs. It can be solved graphically [25.18], and that energy defines the Feshbach resonant energy E F , which differs (very slightly) from the Breit–Wigner energy (Sect. 25.4.3 and Fig. 25.1).

25.4 Width, Shift, and Shape Parameter

396

Part B

Atoms

Part B 25.5

Table 25.4 Comparison of high precision calculations with experiment for the resonance parameters of the He(1P 0 )

resonances below the n = 2 threshold. For photoabsorption, the appropriate Rydberg constant is R M = R∞ /(1 + m/M ) [25.20]. The value used here is R M = 13.603 833 eV, and E 0 = − 79.0151 eV [25.6] Calculation a s=1

Quantity

Units

Es

Ry

−1.385 7895

∆

eV

−0.007 13

Es − E0

eV

60.1444

Γ

eV

0.0369 d −2.849 e

qs

Experiment s=1

s=2

s=2

−1.194 182 0.000 6202 62.7587 0.000 1165 −4.606 e

60.133 ± 0.015 b 60.151 ± 0.0103 c

62.756 ± 0.01 b

0.038 ± 0.004 b 0.038 ± 0.002 c − 2.80 ± 0.25 b − 2.55 ± 0.16 c

a b c d e

Bhatia and Temkin [25.20], except as noted Madden and Codling [25.21] Morgan and Ederer [25.22] Bhatia, Burke, and Temkin [25.23] Bhatia and Temkin [25.24]

plane waves (not energy normalized) throughout this chapter. Equation (25.27) is a nontrivial example of what can be done with the projection operator technique. Not only does it allow very accurate calculations ([25.17, 20] and footnote e of Table 25.4), but it provides a theoretical incisiveness which far exceeds all previous resonance formalisms. A formula for the resonant scattering cross section can be derived which is of the same form as the lefthand side of (25.23); however, in that case, the parameter corresponding to qs is related to the nonresonant phase shift [25.25], and has no quantitative relationship to the above shape parameter (qs ).

25.4.3 Relation to Breit–Wigner Parameters Inferring resonance parameters from experimental data is generally done by fitting to resonance formulae in

which the resonance parameters are assumed to be energy independent. A phase shift for example would be inferred by assuming   ΓBW /2 . η(E ) = δ0 (E ) + arctan (25.30) E BW − E The relation between the above Breit–Wigner parameters and those of the Feshbach theory has been derived in lowest order by Drachman [25.26]: E BW = E F − (1/4)ΓF ( dΓF / dE ) E=EF , ΓBW = ΓF (1 + d∆F / dE ) E=EF , δ0 (E ) = η0 (E ) − (1/2) dΓF / dE .

(25.31)

In only one precision calculation (for the lowest 1 S resonance in electron–hydrogen scattering) have these differences, thus far, been evaluated [25.18]. A précis is given in Fig. 25.1.

25.5 Other Calculational Methods We now briefly review two calculational methods used for basic applications in autoionization of few body systems: (a) complex rotation and (b) a pseudopotential method.

25.5.1 Complex Rotation Method Complex rotation, which is based on a theorem of Balslev and Combes [25.27], has been extensively applied

Autoionization

ri → ri eiθ .

(25.32)

Under this replacement the Hamiltonian undergoes the transformation H → T e−2iθ + V e−iθ

(25.33)

(only Coulomb interactions are assumed). A real variational wave function Φ is used (for the applications here, they are of Hylleraas form, multiplied by rotational harmonics of symmetric Euler angles of the desired angular momentum, parity, and spin [25.31]). The functional [E] =

Φ|H|Φ Φ|T |Φ e−2iθ + Φ|V |Φ e−iθ = Φ|Φ Φ|Φ (25.34)

is then evaluated. Minimization of (25.34) with respect to the linear parameters, for a given value of θ, is carried out in the usual way, but by virtue of the complex dependence on rotation angle the matrix elements Hij in the matrix eigenvalue equation det |Hij (θ) − E∆ij | = 0

(25.35)

are complex. Thus the solution of (25.35) gives rise to complex eigenvalues E λ = E λ (θ). For a given λ, the optimum θ is the one for which E λ (θ) is effectively stationary as a function of θ [25.29]. Note that no projection operators are used: the real part of E λ corresponds to the Breit–Wigner (i. e., experimental) position of the resonance, and Im(E λ ) = ΓBW /2, where ΓBW corresponds to the Breit–Wigner width of the resonance. These parameters thus include the full Feshbach values plus corrections (25.31). Using Hylleraas wave functions with up to 1230 terms in the complex rotation method, resonance parameters have been obtained for resonance states of H− below the n = 2 and 3 thresholds of H which

compare very well with those obtained using the projection-operator, R-matrix and close-coupling methods. Results for the 1De states of H− are given in the Table 25.5. Similar calculations for Ps− have been carried out [25.32]. The complex rotation method has been applied to the autoionization states of many different systems including muonic systems [25.33], as well as to study the combined effect of electric field and spin-orbit interaction on resonance parameters [25.34].

25.5.2 Pseudopotential Method The second method that is included in this section is done so for the reason that it represents a rather different idea for the calculation of autoionization rather than being a more elaborate application of methodologies that are already known, with results too numerous to be referenced here. The method, described as a pseudopotential approach, was introduced by Martin et al. [25.35]. An effective Hamiltonian Heff is defined as Heff = H + MP ,

(25.36)

where M is a scalar parameter (i. e., a number), which will be taken to be very large, multiplying the P operator, (25.16). [Applications have thus far been restricted to one-electron targets and resonances below n = 3 excited state.] In practice, one minimizes the expectation value of Heff , i. e.,   Ψv |Heff |Ψv  =0, δ (25.37) Ψv Ψv  using an arbitrary, quadratically integrable, variational function Ψv . In order to understand the nature of the spectrum that arises from this variation, we imagine Ψv divided into its P and Q space components: Ψv = QΨv + PΨv = ΨvQ + ΨvP .

(25.38)

Table 25.5 Comparison of resonance parameters (in eV) obtained from different methods for calculating 1 D e states in H− Threshold n

Complex-cordinate rotation [25.29]

2 3

E 10.124 36 11.811 02

a

Close coupling (18-state), [25.30]

Γ 0.008 62 0.045 12

R-matrix [25.28] E 10.1252 11.810 97 a

Feshbach projection [25.20] (25.31) Γ 0.008 81 0.044 49 a

EF 10.1244

ΓF 0.010

397

Part B 25.5

with great accuracy. Two additional basic systems to be mentioned here are H− and Ps− (Ps = positronium). In complex rotation the particle distances are multiplied by a common phase factor

25.5 Other Calculational Methods

398

Part B

Atoms

Part B 25.6

The expectation value Ψv |Heff |Ψv  is written in matrix form as

where χE is the solution of the exchange approximation

Ψv |Heff |Ψv      Q P HQ Q HQ P ΨvQ = Ψv Ψv . H PQ H PP + M ΨvP

It is emphasized that this method only calculates the Feshbach energy; thus the shifts are not included. On the other hand the method uses no projection operators in calculating the matrix elements of H, and only the matrix elements of P by itself occur. This is much easier than a standard Q HQ calculation (Sect. 25.2.2). In practice, the matrix in (25.39) will expand to an N × N matrix, where N is the number of linear parameters in Ψv , and (if one uses a Hylleraas form of Ψv , for example) the matrix in (25.39) will not overtly divide itself into the simple form of this heuristic exposition pictured in (25.39) or (25.40). Nevertheless, the conclusion holds; in detail, the eigenvalue spectrum will span a range of values with those below the threshold, appropriate to the P operator being used, corresponding to real resonances, and the largest eigenvalue will approach the value of M used in the specific calculation. A sample of results for the He(1P) resonances below the n = 2 threshold of He+ taken from [25.35], with some comparisons, is given in Table 25.6. Note that the value of EF of the second resonance in the Martin et al. [25.35] calculation is lower than the rigorous Q HQ calculation [25.20]. It is believed that this may be due to the residual M dependence of Heff .

(25.39)

The eigenvalue problem resulting from (25.37) reduces to finding the eigenvalues of the determinant   HQ Q  − λ HQ P  = 0 . (25.40) det H PQ  H PP  + M − λ Note that only the bottom right component contains the term M. As a result, in the limit of large M, the eigenvalues, which can readily be solved for from (25.39), are   M + H  PP lim λ = . (25.41) H  M→large QQ

The lower eigenvalue is the desired Feshbach resonant energy EF = HQ Q . The width is calculated from [with our normalization, (25.21)] Γ = 2k|Ψλ |Heff |χE |2

(25.42)

(H PP − E )χE = 0

(25.43)

Table 25.6 Resonance energies EF (Ry) and widths (eV) for 1 P states of He below n = 2 threshold (−1 Ry) of He+ State

Martin et al. [25.35]

Lipsky and Conneely [25.36]

Bhatia and Temkin [25.20, 24]

1 2 3

Position −1.384 00 −1.194 60 −1.127 52

Position −1.376 72 −1.193 12 −1.125 84

Position −1.385 79 −1.194 18 −1.127 72

Width 0.0382 0.000 146 0.000 860

Width 0.0341 0.000 131 0.007 27

Width 0.0363 0.000 106 0.0090

25.6 Related Topics This chapter is necessarily of limited scope. Within the projection operator formalism, overlapping resonance theory [25.37] is discussed in Sect. 47.1.3. Recent calculations [25.38] have shown that such effects, when present, can induce significant alteration of isolated resonance results. Other prominent items not included are stabilization methods [25.39] and hyperspherical coordinate methods [25.40, 41]. The latter methods have the appealing property of presenting as  2 energies 1/2 a function of the hyperradius, R = r , which i i

look like potential energy curves of diatomic molecules as a function of the intermolecular separation, which is also usually denoted by R. The molecular structure analogy has also been used to uncover additional (approximate) symmetries with corresponding quantum labels [25.40, 41]. They thus have a global character not present in the foregoing methods. On a purely quantitative level, however, they are not generally as accurate as methods based on the projection operator or complex rotation formalism.

Autoionization

nificant application of the phenomena associated with autoionization to diagnostics of astrophysical [25.45] and fusion [25.46] plasmas, for example, shows that autoionization has considerable applied utility.

References 25.1 25.2

25.3 25.4 25.5 25.6 25.7 25.8 25.9 25.10 25.11 25.12 25.13 25.14 25.15 25.16

25.17 25.18 25.19 25.20 25.21 25.22 25.23 25.24 25.25 25.26 25.27

H. Feshbach: Ann. Phys. (N. Y.) 19, 287 (1962) For a more complete review of much of this material: A. Temkin, A. K. Bhatia: Autoionization, Recent Developments and Applications, ed. by A. Temkin (Plenum, New York, 1985), pp. 1 ff, 35 ff H. S. Taylor: Adv. Chem. Phys. 18, 91 (1970) A. Berk, A. Temkin: Phys. Rev. A 32, 3196 (1985) A. Temkin, A. K. Bhatia: Phys. Rev. A 31, 1259 (1985) C. L. Pekeris: Phys. Rev. 112, 1649 (1958) B. R. Junker, C. L. Huang: Phys. Rev. A 18, 313 (1978) B. F. Davis, K. Chung: Phys. Rev. A 29, 313, 2437 (1978) M. Bylicki: J. Phys. B 24, 413 (1991) J. Brunt, G. King, F. Read: J. Phys. B 10, 433 (1977) Y. Hahn, T. F. O’Malley, L. Spruch: Phys. Rev. 128, 932 (1962) Y. Hahn: Ann. Phys. (N. Y.) 58, 137 (1960) T. F. O’Malley, S. Geltman: Phys. Rev. A 137, 1344 (1965) A. Temkin, A. K. Bhatia, J. N. Bardsley: Phys. Rev. A 7, 1633 (1972) K. T. Chung: Phys. Rev. A 20, 1743 (1979) K. T. Chung, B. F. Davis: Hole-projection method for calculating Feshbach resonances and inner shell vacancies. In: Autoionization, Recent Developments and Applications, ed. by A. Temkin (Plenum, New York 1985) p. 73 K. T. Chung, B. F. Davis: Phys. Rev. A 26, 3278 (1982) Y.-K. Ho, A. K. Bhatia, A. Temkin: Phys. Rev. A 15, 1432 (1977) U. Fano: Phys. Rev. 124, 1866 (1960) A. K. Bhatia, A. Temkin: Phys. Rev. A 11, 2018 (1975) R. P. Madden, K. Codling: Astrophs. J. 141, 364 (1965) H. Morgan, D. Ederer: Phys. Rev. A 29, 1901 (1984) A. K. Bhatia, P. G. Burke, A. Temkin: Phys. Rev. A 8, 21 (1973) A. K. Bhatia, A. Temkin: Phys. Rev. A 29, 1895 (1984) K. Smith: The Calculation of Atomic Collision Processes (Wiley-Interscience, New York 1971) p. 47 R. J. Drachman: Phys. Rev. A 15, 1432 (1977) E. B. Balslev, J. W. Combes: Comm. Math. Phys. 22, 280 (1971)

25.28 25.29 25.30 25.31 25.32 25.33 25.34 25.35 25.36 25.37 25.38

25.39 25.40

25.41 25.42 25.43

25.44 25.45

25.46

T. Scholz, P. Scott, P. G. Burke: J. Phys. B 21, L139 (1988) A. K. Bhatia, Y. K. Ho: Phys. Rev. A 41, 504 (1990) J. Callaway: Phys. Rev. A 26, 199 (1982) A. K. Bhatia, A. Temkin: Rev. Mod. Phys. 36, 1050 (1964) A. K. Bhatia, Y. K. Ho: Phys. Rev. A 42, 1119 (1990) C.-Y. Hu, A. K. Bhatia: Muon Catalyzed Fusion 5/6, 439 (1990/1991) I. A. Ivanov, Y. K. Ho: Phys. Rev. A 68, 033410 (2003) F. Martin, O. Mo, A. Riera, M. Yan˜ ez: Europhys. Lett. 4, 799 (1987) L. Lipsky, M. J. Conneely: Phys. Rev. A 14, 2193 (1976) H. Feshbach: Ann. Phys. (N. Y.) 43, 410 (1967) D. C. Griffin, M. S. Pindzola, F. Robicheaux, T. W. Gorczyca, N. R. Badnell: Phys. Rev. Lett. 72, 3491 (1994) V. A. Mandelshtam, T. R. Ravuri, H. S. Taylor: J. Chem. Phys. 101, 8792 (1994) C.-D. Lin (Ed.): Classifications and properties of doubly excited states of atoms. In: Review of Fundamental Processes and Applications of Atoms and Ions (World Scientific, Singapore 1993) p. 24 D. R. Herrick, O. Sinanoglu: Phys. A 11, 97 (1975) F. Bely-Debau, A. H. Gabriel, S. Valonte: Mon. Not. Roy. Astron. Soc. 186, 305 (1979) C. J. Powell: Inner shell ionization cross sections. In: Electron Impact Ionization, ed. by T. D. Mark, G. H. Dunn (Springer, Berlin, Heidelberg 1985) Chap. 6 R. D. Cowan: The Theory of Atomic Structure and Spectra (Univ. of California Press, Berkeley 1981) G. Doschek: Diagnostics of solar and astrophysical plasmas dependent on autoionization. In: Autoionization, Recent Developments and Applications, ed. by A. Temkin, A. K. Bhatia (Plenum, New York 1985) p. 171, op. cit. in [25.2] M. Finkenthal: Atomic processes responsible for XUV emisssion. In: AIP Conference Proceedings # 206; Atomic Processes in Plasmas, 1990, ed. by Y.K. Kim, R. E. Elton (American Institute of Physics, New York 1990) p. 95

399

Part B 25

There are many other areas in which autoionization can play an important role, such as satellite line formation [25.42], inner-shell ionization [25.43], to mention only a few (Chapt. 62 and [25.44]). In addition, sig-

References

401

26. Green’s Functions of Field Theory

Green’s Functi

It subsequently became apparent that it was possible to formulate the problem more generally, with the full quantum electrodynamic (QED) Hamiltonian as a starting point [26.3, 4]. Thus, the theory contains both relativity, and virtual and real transverse photons, as well as the Coulomb interaction between the electrons. The relativistic (R) approximate equations, such as the Dirac–Fock (DF) and relativistic random phase approximation (RRPA) are the natural outcomes of the formalism, and the NR results can be viewed as approximations to these R cases. Moreover, the Green’s function approach (GFA) now provides a means of carrying out programs involving systematic approximations of successively higher and higher accuracy. The GFA provides a framework which allows one to make corrections to results obtained in the DF approximation or in the coupled cluster approximation (CCA) and manybody perturbation theory (MBPT), including magnetic (Breit type) interactions [26.5, 6]. It ensures that there is neither double counting nor omission of contributions. For radiative transitions, the formalism allows for a systematic treatment of such effects which is gauge independent that at any given level of approximation [26.7–10]. (It should be noted that there is a subtle difference between gauge invariance and gauge independence. The first refers to the transition amplitude and the second to quantities which are directly observable experimentally.) Finally, the GFA is numerically implementable.

26.1 The Two-Point Green’s Function ........... 402 26.2 The Four-Point Green’s Function .......... 405 26.3 Radiative Transitions ........................... 406 26.4 Radiative Corrections ........................... 408 References .................................................. 411 and random phase approximation (RPA). The starting point for the derivations was a nonrelativistic (NR) field-theoretical effective Hamiltonian for the system, which involved the nucleus–electron potential and only Coulomb interactions between electrons.

Renormalization can be carried out for radiative corrections, resulting in finite and calculable expressions [26.11–13]. The integro-differential equations (i. e., the Dyson equations) needed to calculate energies or transition amplitudes in nontrivial approximations are also soluble [26.8, 9, 14]. In quantum field theory, Green’s functions are defined in terms of vacuum expectation values of products of field operators. While this restriction can be relaxed, expectation values must still be taken for a nondegenerate state. As a practical matter, this requirement ultimately restricts one to consider atoms with electron numbers associated only with closed shells or subshells, and those with closed shells or subshells plus or minus one or two electrons. The corresponding Green’s functions considered here are the two- and four-point functions for energy levels, and the three- and five-point functions for transition amplitudes (leading to oscillator strengths). The restriction in electron numbers is clearly not a severe one. It allows one to cover many atomic species. Starting from relativistic QED, the electron field operator ψ(r, t) written in the Heisenberg picture, satisfies equal-time anticommutation relations

 ψ(r, t), ψ(r
, t) = ψ † (r, t), ψ † (r
, t) = 0 , (26.1)
 ψ(r, t), ψ † (r
, t) = δ3 (r − r
) . (26.2) (Spinor labels are suppressed.)

Part B 26

The discussion in this chapter is restricted to Green’s function techniques as applied to problems in atomic physics, specifically to the calculation of higher order (correlation, Breit, as well as radiative) corrections to energy levels, and also of transition amplitudes for radiative transitions of atoms which are gauge invariant (GI) at every level of approximation. Green’s function techniques were first applied to many-electron atoms in 1971 as specific instances of the use of field theory techniques in manyparticle problems [26.1, 2]. They initially provided alternative derivations of known approximations such as the Hartree–Fock (HF) approximation

402

Part B

Atoms

The time-translation operator, acting on any Heisenberg operator O(r, t) = {ψ(r, t), ψ † (r, t), jµ (r, t), . . . }, is O(r, t) = eiHt O(r, 0) e−iHt ,

(26.3)

Part B 26.1

where H is the full QED Hamiltonian. Notations and Definitions For brevity, plainface numbers are used to denote both a coordinate and time, while boldface numbers denote a coordinate vector alone. For example,

1 ≡ (1, t1 ) ≡ (r1 , t1 ) , d3 1 ≡ d3r1 ,

(26.4)

d4 1 ≡ d3r1 dt1 .

with u µ (12; r)    eαµ δ3 (1 − r)δ3 (2 − r) , for R       = e 1, 1 (∇ − ∇ )δ3 (1 − r)δ3 (2 − r) , 2 1   2im     for NR , (26.7)

where αµ = (α, 1), and the components of α are the usual Dirac matrices. The corresponding charge operators are defined as 

 1 3 †   e d 1 ψ (1, 0), ψ(1, 0) , for R 2 Q=    e d3 1 ψ † (1, 0)ψ(1, 0) , for NR .




For radiative transitions, M Nfi (k0 ) denotes the transition amplitude for the emission of a single photon of energy k0 for an N
-electron atom from

an initial energy E iN to a final energy E Nf , where 



M Nfi (k0 ) ≡ (2π)4 δ E Nf − E iN − k0 M Nfi (k0 ) . (26.5)

The notation on the right-hand side of (26.5) is some
what redundant, since k0 is taken to be k0 = E Nf −
E iN . The current density operators at t = 0 are given respectively by jµ (r, 0) 

 1  3 3 † R  d 1 d 2 ψ (1, 0), u (12; r)ψ(2, 0) ,  µ  2     for R =  3 3 † NR   d 1 d 2ψ (1, 0)u µ (12; r)ψ(2, 0) ,      for NR (26.6)

26.1 The Two-Point Green’s Function The two-point Green’s function, or one-body propagator [26.3, 4], for a system of lepton charge Ne is defined as the expectation value of a time-ordered product  

    G N (1, 1
) ≡ −i 0N T ψ(1)ψ † (1
)  0N , (26.13)

(26.8)

The transition amplitude for a photon of polarization four-vector µ (k), momentum k, energy k0 (k0 = |k|), and photon attachment point r is eik·r N

M (r; k0 ) . (26.9) M Nfi (k0 ) = d3 r √ 2k0 fi


In terms of the current density operator, M Nfi (r; k0 ) can be written as 
 

  M Nfi (r; k0 ) = Nf  µ (k) jµ (r, 0)  Ni 
 
   ≡ Nf  j k (r, 0)  Ni , (26.10)  N
 where  n is a state of leptonic charge number N
, with N
corresponding to an atom of N
electrons, with total
energy E nN . The term lepton charge is used to refer to the charge of electrons and positrons. In the dipole approximation, eik·r ≈ 1, µ jµ contains the quantity µ (k)u µ (12; r) ≡ λk (r)δ(1 − r)δ(2 − r) , where, in the radiation gauge,  e(k) · α , velocity form λk (r) = iek0 (k) · r , length form .

(26.11)

(26.12)

  where  N0 is the ground state of leptonic charge number N, with N corresponding to an atom with electron number N in a filled shell or subshell (N = 2, 4, 8, . . . ). The relative time (t1 − t1
) Fourier transform of (26.13) yields the spectral representation of

Green’s Functions of Field Theory

G N (1, 1
): G ωN =

 j   j   u u j

ω − ε j + iη

+

 ζ   ζ   v v ζ

ω − εζ + iη

, η = 0+ , (26.14)

    G ωN (1, 1
) ≡ 1|G ωN |1
, u j (r) ≡ r u j , etc. ,

 

(26.15)

and where the symbol denotes a summation over discrete and integration over continuous states. The two terms in (26.14) are obtained by counting each timeordering separately in (26.13), introducing a complete set of intermediate states and using the time translation operator (26.3). The functions u j (r) and vζ (r) are defined by       u j (r) ≡ N0 ψ(r, 0) N+1 , j     N+1 N  N+1 N = ei( E0 −E j )t u j (r) , (26.16) 0 ψ(r, t) j and

    N  0) vζ (r) ≡ N−1 0 , ψ(r, ζ     N−1 N N i( E ζ −E 0 )t ζ N−1  v (r) . ζ ψ(r, t) 0 = e

(26.17)

  Here,  N0 is the ground state of an atom of lepton charge number N (corresponding to a nondegenerate  state  – a closed shell or subshell) of energy E 0N ,  N+1 an j N+1 atomic state of total energy E and leptonic charge j  number N + 1, and  N−1 is the state of energy E ζN−1 ζ and leptonic charge number N − 1. These several states satisfy the eigenvalue equations       H  N0 = E 0N 0N ,       N+1 H  N+1 = E N+1 ,  j j j       = E ζN−1  N−1 . H  N−1 (26.18) ζ ζ The energy parameters ε j and εζ are defined by ε j ≡ E N+1 − E 0N , j ζ

ε ≡

E 0N

− E ζN−1

.

(26.19) (26.20)

Equations (26.19) and (26.20) are generalizations of Koopmans’s theorem [26.15] (see Sect. 21.4.1). In the DF approximation, the state  N+1 can be thought of as j N effective particles (electrons) making up the core,

plus one valence electron, with energy label j. The atom can also be an isoelectronic ion with nuclear charge number Z. In this (DF) approximation, the N−1  state  ζ is one of two possible types. It can have N independent electrons making up the core, with one of the core electrons missing, or equivalently the N electron core with one electron–hole, with energy label ζ . There is a finite number N of such hole states, which shall be labeled a. There are no other states in the NR (HF) case.  N−1 In the R (DF) case, the second group of states  ζ can also describe an atom with a core of N electrons, plus one positron. Its continuum energy label, will be taken as ζ = ¯ . This energy will appear with a negative sign in (26.14). The second step in leading to an explicit G N (1, 1
) is the generation of a Dyson equation which it satisfies. In both the DF and HF approximations, the Dyson equation can be obtained through a successive series of steps [26.11]. Working in the Coulomb gauge (see Sect. 27.2), begin with the Coulomb interaction between electrons and neglect the exchange of transverse virtual photons. The Dyson equation for the resulting two-point function is then expanded as a power series in the electron–electron (ee) interaction α Vee ≡ V = , (26.21) |x − y| resulting in an infinite set of Feynman diagrams (α = e2 /4π in rationalized mks units). As a next approximation, consider only diagrams involving single Coulomb exchanges and their iterates (“ladders”). That is, set aside for later consideration nonladder Feynman diagrams of two or more Coulomb photons and their iterates (e.g., two crossed Coulomb photon lines). A summation of the infinite set of these remaining terms N generates an equation for a propagator labeled G ω,Coul , which contains Coulomb radiative corrections in its kernel. Finally, modify the kernel by isolating these radiative corrections (self energy and vacuum polarization) by constructing a spectral representation of G “0” ω which mimics that for the usual QED propagator, which is a vacuum (rather than an N-lepton ground state) expectation value. This requires shifting the poles corresponding to core energies of the atom from the upper to the lower complex ω plane. The shifting of poles is accomplished by means of the equation 1 1 − = 2πiδ(x) , x − iη x + iη

(26.22)

403

Part B 26.1

where

26.1 The Two-Point Green’s Function

404

Part B

Atoms

which, when used to shift the poles of core electrons in the ω plane, gives N = 2πi G ω,Coul

states N       N  d3 x d3 y m ΣDF n = α

N   a  a v v  δ(ω − εa ) + G “0” . ω

a=1

1 m|x a|y |x − y|

× (x|n y|a − x|a y|n) N 
m  m  n a ≡ a |V |a − a |V |n

a=1

Part B 26.1

(26.23)

a=1

It is the first term on the right-hand side of this equation which occurs in the kernel of (26.24) below and generates the DF approximation. Some of the terms set aside in the course of this sequence of approximations are reconsidered in Sect. 26.4 to obtain more accurate energy results. What remains at the end of this sequence is an approximation to G ωN N of (26.14), designated as G ω,DF , and which satisfies the self-consistent Dyson equation N N N = gω + gω ΣDF G ω,DF , G ω,DF

(26.24)

N where DF is the kernel defined in (26.27). For the R (DF) case, gω−1

= ω − h (r), h (r) = α · p + mβ − Zα/r , R

R

(26.25)

≡

Zα p2 − . 2m r

|V |an



(26.27)

for arbitrary states m and n. The first term in brackets is the Hartree term and the second is the electron exchange term in the DF (HF) approximation. The DF (HF) equation is the homogeneous equation corresponding to the inhomogeneous (26.24). Thus,   
 N en − h − ΣDF |n = 0, n|n = δnn
. (26.28) The states |n (valence, core, and negative energy states) are orthonormal and complete. In the coordinate basis, (26.28) takes the more familiar form      N  (26.29) [en − h(x)] x|n − x ΣDF n = 0 , where  N     
  N x|m am |V |an x ΣDF n = m a=1 N 
x  n ≡ a |V |a a=1

≡

(26.26)

The function gω in (26.25) is the R or NR Coulomb Green’s function. It is the solution of the corresponding inhomogeneous c-number Schrödinger or Dirac equation. It is a single-particle equation which has had a long history of specific treatments (see Chapt. 9 and [26.16–19]). It is the semiclassical limit of the two-point propagator we consider here when only the c-number nuclear Coulomb potential is kept and all other (q-number) interactions are turned off. In the DF and HF in (26.14), we re approximations  place ε j by e j > 0, u j by | j (the valence energies and states of an atom with N + 1 electrons and a frozen relaxed core of N electrons), εζ by ea > 0, |u ζ  by |a, for the N discrete electron core states, and εζ by e¯ < 0, |u ζ  by |¯ , for the continuum of negative energy states (which do not appear in the HF approximation). The N contains only core states |a, in both the DF kernel ΣDF and HF approximations, and is given by the sum over

m a

a=1

and the corresponding h NR , used in the HF case, is h NR (r) =

N 


VDF (x, y) d3 yy|n .

(26.30)

One can also generate equations corresponding to higher approximations than DF using the same approach. For example, one can obtain a Brueckner equation [26.20],

 N en − h − ΣDF − ΣBN (en ) |n = 0 , (26.31) and the states now satisfy the orthonormality condition   d3 x d3 y n|x y|n
lim ω→en

      (ω − en )δ3 (x − y) − x  ΣBN (ω) − ΣBN (en )  y × ω − en

= δnn
,

(26.32)

where the energy-dependent kernel ΣB (en ) arises from irreducible Feynman diagrams involving two Coulomb photons. The kernel is given by       m ΣBN (en ) n

       1/2   m  i = V  j ij V  an en + ea − ei − e j a a,i, j

Green’s Functions of Field Theory

+

 a,b,i

      1/2   m  a V  b ab V  in . en + ei − ea − eb i (26.33)

26.2 The Four-Point Green’s Function

 

involves summations over core states (a, b), and summation and integration over valence states (i, j). In perturbation theory, the lowest order contribution of ΣB (en ) is an ee correlation term.

G N (12, 1
2
)  

 N    ≡ − N0 T ψ(1)ψ † (1
)ψ(2)ψ † (2
)  . (26.34) 0

In order to avoid unnecessary complication, only simple ladders of Vee ≡ V are considered. There are 4! possible time orderings. Of these, there are four groups of four with t1 , t1
> t2 , t2
, t1 , t1
< t2 , t2
, t1 , t2 > t1
, t2
, and t1 , t2 < t1
, t2
, corresponding to the particle–hole (PH/HP) and the two-particle/two-hole (2P/2H) cases for the first eight and last eight time orderings, respectively. For each of these cases, introduce a total time and relative time variable defined by 1 1 T = (t1 + t1
), T
= (t2 + t2
) , 2 2 t = t1 − t1
, t
= t2 − t2
,

(26.36)

for the last eight cases (2P/2H). For a particular set of eight time orderings, a time translation with respect to the relevant c.m. time t or t
, followed by a separate Fourier transformation for each case with respect to T − T
(with integration variable dΩ), yields contributions with poles in the separately defined Ω-planes at N − E 0N , for PH ω(iα) = E (iα)

−ω(iα) =

E 0N

N − E (iα) , for

HP

N+2 ω(ij) = E (ij) − E 0N , for 2P

ω(ab) =

E 0N

N−2 − E (ab) , for

2H .

(ia)

for the PH/HP case and ϕ(ij) (12) ≡ 12| ϕ(ij) (t) e−iω(ij) T       , = N0 T [ψ(1)ψ(2)]  N+2 (ij)   
γ(ab) (1
2
) ≡ 1
2
 γ(ab) (t
) e−iω(ab) T 
   

N ψ(1 = N−2 )ψ(2 ) T 0 , (ab)

(26.40)

(26.41)

for the 2P and 2H cases, respectively. The antisymmetry under exchange follows from the definitions (26.40) and (26.41): ζ(τ) (12) = −ζ(τ) (21) ,

(26.35)

for the first eight cases (PH/HP) and 1 1 T = (t1 + t2 ), T
= (t1
+ t2
) , 2 2 t = t1 − t2 , t
= t1
− t2
,

representation is of a form similar to (26.14), with wave functions corresponding to (26.16) and (26.17) given by     χ(ia) (11
) ≡ 1 χ(ia) (t) 1
e−iω(ia) T  

N  N †
 = 0 T ψ(1)ψ (1 )  (26.39)

ζ(τ) = ϕ(ij) , γ(ab) , (26.42)

where it is understood that the suppressed spinor indices are interchanged as well as the coordinate and time variables. The three amplitudes defined above satisfy Bethe– Salpeter (BS) equations. The PH/HP case is the analog of the positronium atom and the 2P case is analogous to He. For the case of Coulomb ladder exchanges, to which we have restricted ourselves, these BS equations can be reduced to simpler ones, with the relative time set equal to zero (the Salpeter equation [26.21]). The corresponding BS wave functions in the DF (HF) basis (rather than the coordinate basis), are for PH/HP,     m χ¯ (ia)  n ≡ d3 1 d3 1
m|11|χ(ia) (0)|1
1
|n ,

(26.37a)

(26.43)

(26.37b)

with m = k and n = c, or m = c and n = k, and for 2P/2H,        mn ζ¯(τ) ≡ d3 1 d3 2m|1n|2 12 ζ(τ) (0) ,

(26.38a) (26.38b)

(26.44)

Equations (26.37a,b) parallel (26.19) and (26.20) as generalizations of Koopmans’ theorem. The spectral

where ζ(τ) = ϕ(ij) , γ(ab) . The states |i, | j, |a, |b, |m, and |n label one-particle DF (HF) eigenkets and

Part B 26.2

26.2 The Four-Point Green’s Function The four-point Green’s function [26.4, 8, 11], or twobody propagator, is defined as

405

406

Part B

Atoms

Part B 26.3

ei , e j , ea , eb , em , and en , the corresponding eigenvalues. In the PH/HP case, the BS equation is     − (ω(ia) − ek + ec ) i χ¯ (ia)  a          k j j χ¯ (ia)  b = b V  c

        + kj V  bc b χ¯ (ia)  j , (26.45)       (ω(ia) − ec + ek ) a χ¯ (ia) i          c j j χ¯ (ia)  b = b V  k j,b

        + cj V  bk b χ¯ (ia)  j .



PH/HP (no 2P/2H terms)       ck ϕ¯ (ij) = ck γ¯(ab) = 0 , 2P/2H (no PH/HP terms)

(26.47)

(26.48)

The single indices a, b, c, d refer to core and i, j, k,  to valence DF (HF) states. The BS wave functions satisfy the orthonormality conditions         k χ¯ (ia)  c c χ¯ (ia)
 k k,c

       + c χ¯ (ia)  k k χ¯ (ia)
 c = δ(ia)(ia)
,

In the 2P/2H case, the coupled pairs of BS equations are

The wave functions also satisfy the additional conditions:         c χ¯ (ia)  d = k χ¯ (ia)   = 0 ,

  − (ω(τ) − ec − ed ) cd|ζ¯(τ)       ¯ c  m ζ mn , =   V (τ) n d

(26.49)

for the PH/HP case, and

       ± ζ¯(τ)  cd cd ζ¯(τ
) c>d

(26.46)

−



      ζ¯(τ)  k k ζ¯(τ
) = δ(τ)(τ
) ,

(26.50)

k>

m>n

  (ω(τ) − ek − e ) k|ζ¯(τ)        k  m mn ζ¯(τ) , =  V  n m>n

with m and n in these equations labeling either both core, or both valence states.

where +(−) corresponds to the 2H(2P) case and (τ) = (ab) or (ij). The PH/HP case in the GFA, involving Coulomb ladders for the ee interaction, is just the R and NR RPA [26.2]. The labels c, d should also refer to antiparticles, but the contributions of the integrals from these terms to (26.45) and (26.46) are negligible.

26.3 Radiative Transitions For the majority of applications, one begins with the function Γ N (12; 3), the reducible three-point vertex:  

 N   N N k † Γ (12; 3) = − 0 T ψ(1) j (3)ψ (2)  . 0

(26.51)

The usual strategy is followed. The spectral representation serves to identify the functions of ultimate interest. (Energies are not relevant in this case.) One then generates Dyson equations in the chosen approximation by summing an infinite series of perturbation terms. There are 3! time orderings in (26.51). As with the four-point function, not all of them are subsequently useful. The two useful cases are t1 > t3 > t2 , and t2 > t3 > t1 . To obtain a spectral representation for these two cases, one first carries out a time translation of t3 , using the operator exp(iHt3 ) of (26.3), so that t1 → τ1 = t1 − t3 , t2 → τ2 = t2 − t3 . One then

introduces complete sets of intermediate states. The functions defined in connection with the two-point function in (26.16) and (26.17) will now appear, as well as the radiative transition amplitude, defined in (26.10). If one next carries out a separate time translation of τ1 and τ2 and Fourier-transforms the resulting expressions, one obtains (with 3 replaced by r) Γ N (r; ω1 ω2 )   |u j M N+1 j (r)u | = (ω1 − ε j + iη)(ω2 − ε + iη) j

+

 ζχ

N−1 |vζ Mζχ (r)vχ |

(ω1 − εζ − iη)(ω2 − εχ − iη)

+··· . (26.52)

Green’s Functions of Field Theory

and pick out the residues of the ω1 and ω2 poles at specific energies εm and εn . [Λ N (r; ω1 ω2 ) has no such poles.] With the equivalent of the DF(HF) approximation to the Dyson equations (26.57) below, the corresponding kets form an orthonormal set, and scalar products are calculated with respect to m| and |n. The second term on the right-hand side of (26.52) refers to hole states and is of less interest than the first term describing 1P–1P transitions. From the first term, the transition matrix element in terms of Λ N is      N  (r; k ) = f (r; e e ) (26.54) M N+1 Λ i . 0 f i fi Generation of a Dyson Equation The approximation of only Coulomb “ladder” ee interactions in the two-point Green’s function produces an infinite set of Feynman diagrams to which one end of a single transverse photon line is attached in all possible ways. A resummation of these diagrams generates N the G ω,Coul functions, which are approximated by the DF (HF) propagators written in their spectral form. The scalar products leading to (26.54) can then be taken. A further simplification results because of the Coulomb ladder approximation, which is the same as in the four-point Green’s function case: the relative time can be set equal to zero, which corresponds to integrating over the relative frequency, ω = ω1 − ω2 . With the total frequency defined as Ω = 12 (ω1 + ω2 ), and the definitions

 1 1 1 N N ¯ dωΛ r; ω + Ω, ω − Ω , Λ (r, Ω) = 2π 2 2      ¯N  m Λ (Ω) n   eik·r   N  m Λ¯ (r, Ω) n , = d3r √ 2k0 Ωmn = en − em = k0 , (26.55)

with a relation similar to (26.55) for the quantity λk (r) defined in (26.11), i. e., eik·r m |λ(k0 )| n = d3r √ m|λk (r)|n , (26.56) 2k0

407

the matrix elements in the DF (HF) basis are [26.14, 20]       m Λ¯ N (k0 ) n = m|λ(k0 )|n   

     1  ¯N  m  n Λ a + (k )   V 0  j a ea − e j + k0 j aj

+

 

    1   m  n j Λ¯ N (k0 ) a . a V  j ea − e j − k0 (26.57)

As discussed in Sect. 26.1, the label a should include not just hole states but also negative energy states, which have been neglected in (26.57). Note also that only PH or HP matrix elements appear on the right-hand side of (26.57). Therefore, a closed set of inhomogeneous linear algebraic equations for b|Λ¯ N (k0 )| and |Λ¯ N (k0 )|b results from setting m = b, n =  or m = , n = b, respectively, in (26.57). These equations  N can  be Λ¯ (k0 ) j and solved numerically, and the resulting a     N j Λ¯ (k0 )a substituted in (26.57) to obtain the final result      ¯N  Λ M N+1 (k ) ≡ f (k ) (26.58)  i . 0 0 fi An integro-differential equation form, which provides the option of choosing alternative numerical techniques [26.14], is obtained from (26.57), after some rearrangement and passage to a coordinate basis. Defining m fi (k0 ) from λ(k0 ) in analogy with the definition of MfiN+1 (k0 ) from Λ¯ N (k0 ) in (26.58), one has  M N+1 (a|λ(k0 )|A−  fi (k0 ) = m fi (k0 ) + a

+ A+ |λ(k0 )|a) ,

(26.59)

where

    r   A±    [h(r) ∓ k0 − ea ] r  A± + a V a a  
r| j j| v± + V ± |a , =−

(26.60)

j

  

   j  j    j|v+ |a = i V  af ,  j|v− |a = f V  ia ,   j    j    a a .  j|V ± |a = b V  B± + B∓ V  b

(26.61)

b

(26.62)

The three-point Green’s function, as can be seen from this summary, describes transition amplitudes

Part B 26.3

We next define the three-point irreducible electron vertex Λ N from the reducible vertex Γ N (the electron “legs” in Γ N are missing in Λ N ) as       1 Γ N (r; ω1 ω2 ) 2            ≡ d3 1
d3 2
1 G ωN1  1
1
Λ N (r; ω1 ω2 ) 2
      × 2
G ωN2  2 , (26.53)

26.3 Radiative Transitions

408

Part B

Atoms

Part B 26.4

for radiative transitions between two valence states of atoms with closed shells (subshells) plus one electron (the 1P case). In the Coulomb ladder approximation, related to   the closely  DF equation, the photon vertex 1 Λ N (r; ω1 ω2 ) 2 of (26.53), is nonlocal in space, rather than the local vertex 1 |λk (r)| 2 of (26.11) [as follows from the factor of δ3 (1 − r)δ3 (2 − r)]. The presence of these additional nonlocal contributions to M N+1 fi (k0 ) is made apparent in (26.59). The m fi (k0 ) term is the contribution of the local vertex. Of course, if one knew the exact N + 1 electron wave functions and energies, only the local vertex would enter and a GI result would be obtained. However, the length and velocity versions of (26.12) are equal and GI even in the approximation just discussed [26.3, 7]. Gauge invariance is an essential constraint on radiative transition amplitudes. It has been proven for general gauges not only in the present approximation, but also in the ones discussed below [26.8, 10], all of them arising in the GFA. Since the effective potential in the DF (or HF) approximation is nonlocal, the effective current must also be nonlocal in order to maintain the GI. The somewhat more complicated Dyson equation satisfied by the nonlocal vertex corresponding to the Brueckner approximation has also been generated [26.22], and put in a numerically implementable form. An alternative approximation [26.3, 7] for transition matrix elements, proposed earlier [26.2] than the one just discussed, is based on the RPA [our PH/HP case, with wave function χ¯ ( fa) the solution of (26.45)]. Reference to (26.6), (26.7), (26.10) and (26.11) gives immediately     M Nfa (r; ω( fa) ) = −λk (r) r χ¯ ( fa)  r , (26.63) where k0 = k = ω( fa) and N corresponds to atoms with a closed shell or subshell of electrons. Aside from being a different approximation to transition amplitudes than (26.54), (26.63) covers a different set of cases than does (26.54), since the corresponding

initial states i in (26.54) are restricted to core states a in (26.63). Thus, for example, the case N = 2 in (26.54) can describe transitions between any two valence states of Li. The corresponding case for (26.63) is N = 4, but only transitions to a higher level, originating in the 1s or 2s level of Li, can be described by the formalism. Finally, we shall discuss radiative transitions [26.8] for 2P atoms (closed shell/subshell plus two valence electrons). The general case involves a five-point nonlocal vertex. However, in the ladder approximation, this reduces to transition matrix elements which contain combinations of DF, 2P BS wave functions, and three-point functions. The final expressions are M N+2 fi (r; k0 ) " $ # !  Λ¯ N (r, k ) Λ¯ N (r, k0 )   0 ¯ ¯ V −V = ζ( f p)   ζ(iq) .  k0 − ∆H k0 − ∆H  (26.64)

The expressions appearing in this equation are, in more detail, in the DF basis:    $   N !  ¯ (r, k0 ) q δns  Λ¯ N (r, k )  Λ m  m  0  q ≡ ,   k0 + eq − em n  k0 − ∆H  s (26.65)

q  where m, n, q, s label DF states (P or H), s ≡ |qs ≡ |q|s; (26.65) serves to define ∆H, as a difference of two DF Hamiltonians, the argument r of Λ¯ refers to the electron which emits the photon, and m     m  c     V ζ¯(τ) =   ¯ n V d cd ζ(τ) n cd

 m  k       ¯ + n V  k ζ(τ) .

(26.66)

k

In (26.66) we used the fact that, in the DF basis, |ζ¯(τ)  only has 2P or 2H components (26.46, 26.48).

26.4 Radiative Corrections This section summarizes radiative corrections for 1P/1H atoms, starting from the two-point Green’s function. The Dyson equations are generated and solved perturbatively for the energy to order α5 m (α3 a.u.). The perturbation theory starts from the DF solution as the zero-order one. As done for the three-point function in Sect. 26.3, the Dyson equation for radiative corrections involving

a single transverse photon is generated by expanding the two-point propagator to all orders in the Coulomb ladder approximation, inserting a transverse virtual photon in all possible ways, and then resumming. The resulting integral equation [26.23] is quite complicated, containing even the three-point vertex in its inhomogeneous term, as well as a mass counter term to eliminate divergences.

Green’s Functions of Field Theory

ee Coulomb potentials. The ee CL terms arising from core and valence self-energy terms are canceled exactly by those coming from Coulomb and transverse photon exchange. Thus, the numerous and rather complicated corresponding ee BL terms, which also are individually small compared to their associated CL terms, should additionally almost cancel, and are therefore neglected. One finally obtains (with all state labels denoting principal and orbital quantum numbers as well as spin indices: n ≡ (n, ; m s )) ∆E(n, ) = Z|0|n|2 F(n, ) ,

(26.67)

in a.u., where 0|n ≡ 0|n, 0 is the s-state HF wave function at the origin of coordinates, and serves to define an effective (shielded) nuclear charge Z n,eff : 1 r|n,  ≡ Ψn, (r), |0|n|2 ≡ 3 (Z n,eff α)3 . πn (26.68)

(Just as in hydrogen, for which the final expressions for the radiative corrections require NR and not R wave functions and energies, so in this case we use HF (NR) and not DF (R) quantities.) F(n, ) consists of a valence or hole contribution Fv,h , a core term Fcore (n, ), and one due to photon exchange between electrons Fee (n, ): F(n, ) = Fv,h (n, ) + Fcore (n, ) + Fee (n, ) , (26.69)

Fv,h (n, ) =

 4
so N(Z)δ0 + L(n) + Uv,h (n) , 3

(26.70)

4
N(Z)ρ(n) + L core (n) Fcore (n, ) = 3Z  so + U(n) + Ucore (n) , (26.71)

 4 C L E(Z)K (n) + K (n) , Fee (n, ) = 3Z (26.72)

with

so (n) Uv,h

and

so (n) Ucore

being the spin-orbit terms,

1 19 + ZαC5 , N(Z) = ln + 2 30 (Zα) 

1 11 − ln 2 , C5 = 3π 1 + 128 2 1 59 9 7 + − π, E(Z) = − ln 2 Zα 20 8 

 i/4π r  n,   p · LB (n) 3 L(n) = |0|n|2 r   r  − 3 LB (n) · p  n,  , r Z2 , LB (n) ≡ ln 2|en − H|

(26.73) (26.74) (26.75)

(26.76) (26.77)

409

Part B 26.4

It is sufficient for the present purposes to expand these vertices through first order matrix elements  ee ≡ V ,
of V but to exclude matrix elements of type am |V |an (where a is the label for core states), since these are already included in the DF approximation [see (26.27)]. Other radiative corrections are also generated, which include Coulomb photons. Two of these corrections, involving only such photons, have already been referred to in the text between (26.21) and (26.23). Finally, to order α5 m, corrections involving two transverse photons must be included. Systematic application of the pole-shifting process in (26.23) to the one- and two-transverse-photon and the Coulomb-photon expressions yields two types of terms: first, photon exchange (of one transverse photon or of two-photons of either kind, Coulomb or transverse) between core and valence states; and second, self-energy and vacuum polarization contributions. Among the photon exchange terms, one can identify those contributions included in other approaches (at least for a few special cases, such as those arising from a single transverse photon interaction, and from two Coulomb interactions and Coulomb–Breit interactions with positive energy intermediate electron states). These are electron correlation terms which are included as part of a MBPT calculation [26.5], or one which involves consideration of an infinite subset of MBPT terms [26.6]. They need not be re-evaluated. The terms not covered by calculations of the type in [26.5, 6], which are of O(mα5 c2 ), involve retardation in Coulomb-transverse photon exchange, negative energy intermediate electron states for two Coulomb and Coulomb-transverse photon exchanges, two transverse photon exchange, self-energy and vacuum polarization terms, as well as anomalous magnetic moment corrections. After lengthy calculation, one obtains final results in finite analytic form, which are numerically implementable. The results given below are obtained after further approximations. First, the remnants of the original integral equation are solved iteratively. Second, the self-energy terms are calculated in a joint expansion [26.24–26] in α and αZ, and are thus only valid for low Z in isoelectronic sequences. Finally, there are characteristic logarithmic terms generated by low virtual photon momenta in the self-energy contributions. A standard approach is to scale these with a factor of (αZ)2 to obtain Bethe log (BL) terms as constants independent of α, together with constant log (CL) terms of the form ln(αZ)2 . The BL terms are independent of Z for hydrogenic ions, and remain nearly so for other atoms. BL and CL terms are associated with both nuclear and

26.4 Radiative Corrections

410

Part B

Atoms

H the HF Hamiltonian, en ≡ e(n) the HF energy,     1 3/16π so n  3  n C , Uv,h (n) = (26.78) 2 |0|n| r where

Part B 26.4



C =

 j =  + 12 , −( + 1) j =  − 12

ρ(n) =

(26.79)

 i/4π   (ek − e j )(ea − 2ek + e j )
n  n  a V j  j|r|k · k| p|a + c.c. ea − e j |0|n|2 a

,

(26.80)

jk

 Z2 i/4π   (ek − e j )(ea − 2ek + e j )
n  n   j|r|k · k| p|a + c.c. L core (n) = a V j ln 2 ea − e j 2|ea − ek | |0|n| a

,

jk

&  1/2π    2  
n  n  n   n  % (b)
  a  a  p  a a V a − (a)
 V (a)
− a  V (b)
+ c.c. δnb , U(1) (n) = 2 |0|n| a  

   ea − eq  i/2π   (2ea − ek − eq )
n  n   + 2 a| p|q · k| p|a + c.c.  V ln U(2) (n) = q k e −e  eq − ek |0|n|2 a a k

(26.81) (26.82)

,

(26.83)

k R0

719

Born and Distorted Wave Methods The integral expression (47.107) for the direct ionization amplitude provides a starting point for the calculation of cross sections at higher energies. Both the Born series methods and distorted wave methods, which were described in Sect. 47.1.6 when we considered intermediate and high energy elastic scattering and excitation, have been used to obtain ionization amplitudes. Recently an important development of the distorted wave method has been made by Jones and Madison [47.81–83]. This new approach entitled the continuum distorted wave with eikonal initial state (CDW-EIS), commences from the two-potential expression for the transition amplitude given by Gell-Mann and Goldberger [47.84]  +  −    +    T fi = χ −f W + f Ψi + χ f Vi − W f βi . (47.126)

In the first term in this equation Ψi+ is the exact scattering wave function developed from the initial state satisfying outgoing wave boundary conditions, χ −f is a distorted wave corresponding to the final state satisfying incoming wave boundary conditions and the corresponding perturbation W + f is the adjoint of the operator W f and operates to the left. In the second term, βi is the unperturbed initial state which in the case of electron hydrogen atom scattering is βi = (2π)−3/2 exp(ik0 r1 )ψi (r2 ) ,

(47.127)

and Vi is the initial state interaction potential given in this case by Vi = −

1 1 + . r1 r12

(47.128)

An eikonal approximation is made for the initial state wave function Ψi+ in (47.126) and the final state wave function χ −f is represented by a CDW wave function [47.85]. In this way distortion effects are included in both initial and final state wave functions. Results using the CDC-EIS approximation have been compared with ECS calculations for electron hydrogen atom triple ionization cross sections at 54.4 eV [47.86]. The two calculations are generally in very good agreement for equal energy sharing between the outgoing electrons which was considered in this work. A further development of the distorted wave method for ionization has been made when the incident electron is fast and interacts weakly with the target atom or ion and the ejected electron is slow and interacts strongly with the residual ion [47.87, 88]. In this case (47.107) is applicable where the ionizing electron is represented

Part D 47.1

that is real for r ≤ R0 but is rotated into the upper half of the complex plane for r > R0 . This transformation has the desirable property that any outgoing wave evaluated on this contour dies exponentially as the coordinate becomes large. Thus the ECS procedure transforms any outgoing wave into a function that falls off exponentially outside R0 but is equal to the infinite range wave over the finite region of space where the coordinates are real. Producing meaningful ionization cross sections at energies several eV above the ionization threshold requires R0 to be at least 100 a0 . The grid must extend beyond R0 far enough to allow the complex scaled radial function to decay effectively to zero at the edge of the grid requiring grids that extend an additional 25 a0 beyond R0 . A detailed discussion of this method is given by McCurdy et al. [47.80].

47.1 Electron–Atom and Electron–Ion Collisions

720

Part D

Scattering Theory

by plane waves or distorted waves while the initial target state and the ejected electron and residual ion state are both represented by the close coupling expansion (47.8). This approximation is particularly useful when the ejected electron can be captured into an autoionizing state of the target atom or ion which then decays giving rise to the following excitation-autoionization (EA) process  



e− + Aq+ → Aq+ + e− → A(q+1)+ + 2e− . (47.129)

In this equation the bracket indicates a resonance state, while q is the charge on the atom A. This process together with related resonant excitation double autoionization (REDA) process 



e− + Aq+ → A(q−1)+ → Aq+ + e−

(q+1)+  + 2e− , → A (47.130) and the resonant excitation auto-double ionization (READI) process 



e− + Aq+ → A(q−1)+ → A(q+1)+ + 2e− , (47.131)

have attracted considerable experimental and theoretical interest [47.89–93]. However, while the EA process can be accurately treated using a distorted wave method if the incident electron is fast, both the REDA and READI processes involve capture of the incident electron into a resonant state and a strong coupling approach is required. An Example and Conclusions We conclude this section by mentioning a recent comparison that has been made between theory and ex-

periment for electron impact ionization of hydrogen at 17.6 eV [47.94]. At this energy, which is only 4 eV above the ionization threshold, strong coupling effects between the incident and ejected electrons and the residual proton are important and hence Born series and distorted wave methods are not applicable. This comparison thus provides a stringent test of theoretical calculations. In this work triple-differential cross section measurements with coplanar outgoing electrons both having 2 eV energy were compared with exterior complex scaling (ECS) and convergent close coupling (CCC) calculations. The two calculations show excellent overall agreement both with the shape and the magnitude of the experiment for a wide range of scattering angles. It is clear that a detailed theoretical understanding of electron hydrogen atom ionization has now been obtained over a wide range of energies. Although further work is required to predict accurate cross sections involving highly excited states of interest in plasma physics and astrophysical applications, for example in astrophysical H II regions [47.95], methods have been developed which should enable these cross sections to be accurately determined. Good progress has also been made in the study of electron impact ionization of multi-electron atoms and ions. However major problems still remain both due to the need to obtain accurate target states, which also applies to elastic scattering and excitation, and due to fundamental difficulties in carrying out accurate calculations for REDA and READI processes defined by (47.130) and (47.131) respectively. In the latter case major theoretical difficulties still arise in the accurate treatment of resonance states which decay with the emission of more than one electron.

47.2 Electron–Molecule Collisions Part D 47.2

47.2.1 Laboratory Frame Representation The processes that occur in electron collisions with molecules are more varied than those that occur in electron collisions with atoms and atomic ions because of the possibility of exciting degrees of freedom associated with the motion of the nuclei. In addition, the multicenter and nonspherical nature of the electron molecule interaction considerably complicates the solution of the collision problem by reducing its symmetry and by introducing multicenter integrals that are more difficult to calculate than those occurring for atoms and ions.

We first consider the derivation of the equations describing the collision in the laboratory frame of reference discussed by Arthurs and Dalgarno [47.96]. The Schrödinger equation describing the electron–molecule system is (Hm + Te + V ) Ψ = EΨ ,

(47.132)

where Hm is the molecular Hamiltonian, Te is the kinetic energy operator of the scattered electron and V is the electron–molecule interaction potential
1
Zi V(R, rm , r) = − . |r − ri | |r − Ri | i

i

(47.133)

Electron–Atom, Electron–Ion, and Electron–Molecule Collisions

Here, R represents the coordinates Ri of all the nuclei, rm represents the coordinates ri of the electrons in the target molecule, and r represents the coordinates of the scattered electron. The total energy E in (47.132) refers to the frame of reference where the c.m. of the whole system is at rest. As in the case of electron–atom and electron–ion collisions, introduce target eigenstates, and possibly pseudostates Φi , by the equation Φi |Hm | Φ j  = wi δij ,

(47.134)

and then expand the total wave function Ψ , in analogy with (47.8), in the form
Φi (R, rm ) Fij (r) Ψj = A i

+




χi (R, rm , r) bij ,

(47.135)

i

where the spin variables have been suppressed for notation simplicity, and we have not carried out a partial-wave decomposition of the wave function Fij representing the scattered electron. The subscripts i and j now represent the rotational and vibrational states of the molecule as well as its electronic states. Coupled equations for the functions Fij can be obtained by substituting expansion (47.135) into (47.132), and projecting onto the target states Φi and onto the square integrable functions χ j . After eliminating the coefficients bij , the coupled integrodifferential equations
 2  ∇ + ki2 Fij (r) = 2 (Vi + Wi + X i ) F j (r) 

(47.136)

47.2.2 Molecular Frame Representation The theory described in the previous section is completely general, and has been the basis of a number of early calculations for simple diatomic molecules such as H2 . However, major computational difficulties arise

721

because of the very large number of rovibrational channels that need to be retained in expansion (47.135) for all but the simplest low-energy calculations. This difficulty can be overcome by making a Born– Oppenheimer separation of the electronic and nuclear motion. The electronic motion is first determined with the nuclei held fixed. This is referred to as the fixednuclei approximation. The molecular rotational and vibrational motion is then included in a second step of the calculation. This procedure owes its validity to the large ratio of the nuclear mass to the electronic mass, and can be adopted when the collision time is much shorter than the periods of molecular rotation and vibration. Thus it is expected to be valid when the scattered electron energy is not close to a threshold, or when the energy does not coincide with that of a narrow resonance. In these cases, further developments described below are needed to obtain reliable cross sections. In order to formulate the collision process in this representation, adopt a frame of reference that is rigidly attached to the molecule. The fixed-nuclei approximation then starts from the Schrödinger equation (Hel + Te + V ) ψ = Eψ ,

(47.137)

where Hel is the electronic part of the target Hamiltonian obtained by assuming that the target nuclei have fixed coordinates denoted collectively by R. It follows that Hel is related to Hm in (47.132) by Hm = Hel + TR ,

(47.138)

where TR is the kinetic energy operator for the rotational and vibrational motion of the nuclei. The remaining quantities Te and V are the same as in (47.132). The solution of (47.137) proceeds in an analogous way to the solution of (47.2) for electron collisions with atoms and ions. We adopt an expansion similar to (47.8), where we now expand the function representing the motion of the scattered electron in terms of symmetry-adapted angular functions that transform as an appropriate irreducable representation (IRR) of the molecular point group (Burke et al. [47.97]). Substituting this expansion into (47.137), and projecting onto the corresponding channel functions and onto the square integrable functions, yields a set of coupled integrodifferential equations with the form given by (47.10), where now the channel indices i, j and  represent the component of the IRR, as well as the electronic state of the target, and where Γ represents the conserved quantum numbers that now include the IRR and the total spin. The final step is to solve these coupled integrodifferential equations for each set of nuclear coordinates R

Part D 47.2

are obtained, where ki2 = 2 (E − wi ) and where Vi , Wi , and X i are the direct, nonlocal exchange, and nonlocal correlation potentials. By expanding Fij in partial waves, a set of coupled radial integrodifferential equations result, analagous to (47.10) for atoms and ions. The scattering amplitude and cross section for a transition from an initial state |i = |ki , Φi , χ 1 m i  to 2 a final state | j = |k j , Φ j , χ 1 m j  is then given by (47.7), 2 where now the subscripts i and j refer collectively to the ro-vibrational and electronic states of the molecule.

47.2 Electron–Molecule Collisions

722

Part D

Scattering Theory

of importance in the collision, using one of the methods discussed in Sect. 47.1.5. This yields the K -matrices, S-matrices and cross sections for fixed R. For scattering calculations where only the ground electronic state has been included in the expansion, a number of approaches have been developed that replace the nonlocal exchange and correlation potentials by local potentials [47.2]. These approaches have proved particularly important in describing electronically elastic collisions of electrons with polyatomic molecules.

47.2.3 Inclusion of the Nuclear Motion This section discusses how observables involving the nuclear motion, such as rotational and vibrational excitation cross sections, and dissociative attachment cross sections, can be obtained from the solutions of the fixed-nuclei equations. The most widely used approach is the adiabaticnuclei approximation [47.98–100]. In the case of diatomic molecules in a 1Σ state, the scattering amplitude for a transition between electronic, vibrational and rotational states represented by iv jm j and i v j m j is given by   f i v j m j ,iv jm j kˆ · rˆ =        ˆ  f i i kˆ · rˆ ; R χiv (R)Y jm j R ˆ , χi v (R)Y j m j R (47.139)

Part D 47.2

  where f i i kˆ · rˆ ; R is the fixed-nuclei scattering amplitude, which depends parametrically on the nuclear coordinates R, and χiv and Y jm j are the molecular vibrational and rotational eigenfunctions, respectively. This approximation is valid provided that the collision time is short compared with the vibration and/or rotation times, and is widely used in such situations. The cross section is usually averaged over the degenerate sublevels m j and summed over m j , giving the cross section for the transition iv j to i v j . This leads to the relation

j+ j


 2 dσi v jt ,iv0 dσi v j ,iv j = , j0 jt 0 | jjt j 0 dΩ dΩ jt =| j− j |

(47.140)

provided that the small differences in the wave numbers for the different rotational channels can be neglected. A similar relation holds for symmetric top molecules such as NH3 . For spherical top molecules such as CH4 ,

the equivalent relation is [47.101, 102] dσi v j ,iv j 2 j + 1 = dΩ 2j +1



j+ j


jt =| j− j |

1 dσi v jt ,iv0 . 2 jt + 1 dΩ (47.141)

The sum in (47.140) or (47.141) over the final rotational state j is independent of the initial state j. Also, if the cross section is multiplied by the transition energy and then sum over j , the result, which is in the mean energy loss by the incident electron, is still independent of j. The adiabatic-nuclei approximation breaks down close to threshold or in the neighborhood of narrow resonances [47.103]. A straightforward way of including nonadiabatic effects that arise in vibrational excitation is to retain the vibrational terms in the Hamiltonian, but still to treat the rotational motion adiabatically. Hence, instead of (47.137), the equation # = Eψ # (Hel + Tvib + Te + V ) ψ

(47.142)

is solved, where Tvib is the kinetic energy operator for the nuclear vibrational motion, and where the other quantities have the same meaning as in (47.137). Adopting a frame of reference in which the molecule has fixed spacial orientation, and separating out the angular variables of the scattered electron, coupled integrodifferential equations coupling the target vibrational states as well the electronic states can be obtained. This approach has been adopted with success [47.104, 105], but is computationally demanding since the number of coupled channels can become very large. Vibrational excitation and dissociative attachment are particularly important in resonance regions when the scattered electron spends an appreciable time in the neighborhood of the molecule. As a result, a number of approaches has been developed describing these processes based on electron molecule resonance theories (e.g., [47.106–110]). The basic idea is that a series of fixed-nuclei resonance states ψn(r) are introduced for a range of values of R, either by imposing Siegert outgoing wave boundary conditions [47.111], or by introducing Feshbach projection operators [47.15,16]. The amplitude for a transition from an initial electronic– vibrational state iv to a final state i v is then given by
       Ti v ,iv = χi v R ζni R  G (r) n R,R n   × ζni (R) χiv (R) ,

(47.143)

where χiv are the vibrational eigenfunctions, ζni are the “entry amplitudes” from the initial or final electronic

Electron–Atom, Electron–Ion, and Electron–Molecule Collisions

states into the resonance states ψn(r) , and G (r) n are the Green’s functions that describe the propagation in the intermediate resonance states ψn(r) . Dissociative attachment can be described by a straightforward extension of this theory. The fixed-nuclei R-matrix method has also been extended to treat vibrational excitation and dissociative attachment [47.112]. A generalized R-matrix is introduced by an equation which, in analogy with(47.143), can be written as       1
 Ri v ,iv = χi v R wiΓ k R G kRM R , R 2a  Γk  × wik (R) χiv (R) , (47.144) Γ (R) are defined by where the surface amplitudes wik (47.76), and the Green’s function G kRM now describes the propagation in the intermediate R-matrix states defined by (47.73). Once the generalized R-matrix has been determined, the final step in the calculation is to solve the collision problem in the external region which, for diatomic molecules, is defined by the condition that the scattered electron coordinate r is greater than some given radius a, and the internuclear coordinate R is greater than some given radius A. Another approach that includes nonadiabatic effects is the energy-modified adiabatic approximation introduced by Nesbet [47.113]. In this approach, the S-matrix elements connecting the vibrational states are defined by

Si v ,iv = χi v | Si i (E − Hi ; R) | χiv  ,

(47.145)

47.2.4 Electron Collisions with Polyatomic Molecules In the last few years computer programs based on the ab initio methods described in Sect. 47.1.5 have been developed and used to calculate cross sections for electron collisions with polyatomic molecules of importance in many applications. Recent work includes electron collisions with nitrous oxide which is an important species in the upper atmosphere where it plays a role in ozone destruction. N2 O lasers are also of importance. Fixed-nuclei total cross sections, calculated using independent Bonn [47.116] and UK [47.117] polyatomic R-matrix programs are in good agreement with experiment [47.118], showing a 2Π shape resonance near 2 eV. The UK R-matrix program has also been used to study electron collisions with the open-shell radical OClO [47.119]. An important process in the stratospheric polar vortex involves the coupling of chlorine and bromine chemistry, in which OClO plays a key role. In this case OClO is formed by the reaction BrO + ClO → Br + OClO. Fixed-nuclei total cross sections were calculated including eight electronic states in the R-matrix expansion, giving good agreement with experiment [47.120], reproducing a shoulder in the cross section between 2 and 6 eV. Important advances have also been made in the theoretical treatment of vibrational excitation in polyatomic molecules. Electron collisions with CO2 molecules were calculated in the fixed-nuclei approximation using an electron–polyatomic molecular scattering program based on the complex Kohn variational method [47.121]. At low energies the cross section is dominated by a virtual state at threshold and a 2Π u shape resonance at 3.8 eV. As the molecule bends from its ground state linear configuration, the shape resonance splits into nondegenerate 2A1 and 2B1 configurations. The fixed-nuclei complex resonance energy surfaces are parametrized and motion on these surfaces is computed using a generalization of the Boomerang model [47.122]. The results reproduce the oscillations resulting from the interference between the nuclear and electronic motion seen experimentally by Allen [47.123, 124].

47.3 Electron–Atom Collisions in a Laser Field Electron–atom collisions in the presence of an intense laser field have recently attracted considerable attention because of the importance of these processes in applica-

723

tions such as laser plasma interactions, and also because of their fundamental interest in atomic collision theory. This section summarizes the basic theory, commenc-

Part D 47.3

where Si i (E − Hi ; R) is the S-matrix calculated in the fixed nuclei approximation at the internuclear separation R at an energy defined by the operator Hi = E i (R) + Tvib . This has the effect of including the internal energy of the target into the S-matrix elements, giving the correct threshold energies. Finally, we mention an off-shell T -matrix approach for including nonadiabatic effects discussed first by Shugard and Hazi [47.114]. This approach can also extend the range of validity of the adiabatic-nuclei approximation, while retaining much of its inherent simplicity. This has recently been applied with success to low-energy vibrational excitation of H2 and CH4 [47.115].

47.3 Electron–Atom Collisions in a Laser Field

724

Part D

Scattering Theory

ing with the scattering of electrons by a potential in the presence of a laser field. The discussion is then generalized to the scattering of electrons by complex atoms and ions, where ‘dressing’ of the atomic eigenstates by the laser field must be considered, and where simultaneous electron photon excitation (SEPE) processes, defined by −

−

nhν + e + Ai → e + A j ,

(47.146)

can occur. Recent reviews of aspects of this subject have been given by Mason [47.3], Newell [47.125] and Mittleman [47.126].

where ΨV satisfies the Schrödinger equation in the velocity gauge given by ! i 1 ∂ i ΨV (r, t) = − ∇ 2 − A (t) · ∇ + V (r) ∂t 2 c × ΨV (r, t) . (47.150) The corresponding equation for the free electron with V = 0 is readily solved to give the Volkov wave function [47.127] χk (r, t) 3

= (2π)− 2 exp [i (k · r − k · α0 sin ωt − Et)] , (47.151)

47.3.1 Potential Scattering We adopt a semiclassical description of the collision process in which the electrons and the target atom are described by the nonrelativistic Schrödinger equation, and the laser field is described classically. This is valid for most high intensity fields of current interest, where a typical coherence volume of the field contains a very large number of photons [47.126]. The time-dependent Schrödinger equation describing an electron scattered by a potential V (r) in the presence of an external electromagnetic (laser) field is then i

1 ∂ i Ψ (r, t) = − ∇ 2 − A (r, t) · ∇ ∂t 2 c ! 1 + 2 A2 (r, t) + V (r) Ψ (r, t) , 2c (47.147)

Part D 47.3

in the Coulomb gauge such that the vector potential satisfies ∇ · A = 0. We also assume that the laser field is monochromatic, monomode, linearly polarized, and spacially homogeneous (i. e., its wavelength is large compared with the range of the potential, or more generally with the size of the atom). Hence we can write A (r, t) = A (t) = ˆ A0 cos ωt ,

(47.148)

where ˆ is a unit vector along the field polarization direction and ω is the angular frequency. The A2 term in (47.147) can be removed by the unitary transformation   t   i A2 t dt  ΨV (r, t) , Ψ (r, t) = exp − 2 2c (47.149)

where k is the wave vector and E = energy. The quantity α(t) is defined by 1 α (t) = c

t

1 2 2k

  A t dt = α0 sin ωt ,

the kinetic

(47.152)

where α0 = E 0 /ω2 , E 0 being the electric field strength. To solve  (47.150),  introduce the causal Green’s func , t satisfying the equation tion G (+) r, t; r 0     ∂ 1 2 i i + ∇ + A (t) · ∇ G (+) r, t; r , t 0 ∂t 2 c     = δ r − r δ t − t . (47.153) This Green’s function is given by   G (+) r, t; r , t 0   χk (r, t) χk∗ (r , t ) dk , (47.154) = −iθ t − t where θ(x) = 1 for x > 0 and θ(x) = 0 for x < 0. The corresponding causal outgoing wave solution of (47.150) is t   (+) Ψk (r, t) = χk (r, t) + dt dr G (+) r, t; r , t 0 −∞

    × V r Ψk(+) r , t ,

(47.155)

and the S-matrix element for a transition ki → k f in the presence of the laser field is given by       , (47.156) Sk f ,ki = −i χk f V Ψk(+) i where an integration is carried out over all space and time in this matrix element. The time integration in (47.156) can be performed using the relation exp (ix sin u) =

∞
n=−∞

Jn (x) exp (inu) ,

(47.157)

Electron–Atom, Electron–Ion, and Electron–Molecule Collisions

where Jn (x) is an ordinary Bessel function of order n. Then ∞
  Sk f ,ki = −2πi δ E k f − E ki − nω Tknf ,ki n=−∞

(47.158)

where the delta function ensures energy conservation, and n is the number of photons absorbed or emitted. The differential cross section for the scattering process ki → k f with exchange of n photons can then be defined in terms of the T -matrix elements Tknf ,ki by 2 kf  dσn = (2π)4 Tknf ,ki  . (47.159) dΩ ki There are two limiting cases in which considerable simplification in this expression occurs. First, at high energies or for weak potentials, the first Born approximation can be used to describe scattering by the potential V(r). In this case (47.159) reduces to kf 2 dσn1,B = (2π)4 J (∆ · α0 ) |V(∆)|2 , (47.160) dΩ ki n where ∆ = ki − k f is the momentum transfer vector and (47.161) V(∆) = (2π)−3 exp(i∆ · r)V(r) dr . It follows immediately that dσn1,B

dσ 1,B , (47.162) dΩ dΩ where dσ 1,B / dΩ is the field-free first Born differential cross section. Using the sum rule ∞


= Jn2 (∆ · α0 )

Jn2 (x) = 1 ,

(47.163)

n=−∞

(47.162) immediately yields (47.164)

The second limiting case is the low frequency (soft photon) limit, where the laser photon energy ω is small compared with the electron energy E ki . In this limit, the T -matrix element is given by [47.128]        Tknf ,ki = Jn (∆ · α0 ) χk f T E ki χki + O w2 , (47.165)

where

ki

and

ki = ki +

k f

are the shifted wave vectors

nω nω α0 , k f = k f + α0 (47.166) ∆ · α0 ∆ · α0

725

and T (E ki ) is the T -operator in the absence of the 2 laser corresponding to the energy E ki = ki /2. The differential cross section for the transfer of n photons is then   kf 2 dσ   dσn = k , k + O ω2 , J (∆ · α0 ) dΩ ki n dΩ f i (47.167)

where dσ(k f , ki )/ dΩ refers to the transition ki → k f in the absence of the laser. If the frequency is small enough to neglect the n dependence of ki and k f , then using the sum rule (47.163), (47.167) becomes ∞
dσ dσn = , (47.168) dΩ dΩ n=−∞ where dσ/ dΩ is the field-free differential cross section. The Kroll–Watson result (47.167) has been found to be surprisingly accurate, even for cases where ω/E ki ≈ 0.5 [47.129, 130]. A nonrigorous extension of the Kroll–Watson result to inelastic processes has been considered by a number of authors, and has been found to give qualitative agreement with experiments on helium [47.131].

47.3.2 Scattering by Complex Atoms and Ions The time-dependent Schrödinger equation describing an electron scattered by an N-electron atom or ion in the presence of a laser field can be written in analogy with (47.147) as ∂ i Ψ (X N+1 , t) ∂t $ N+1 i
= HN+1 − A (ri , t) · ∇i c i=1 % N+1 1
2 + 2 A (ri , t) Ψ (X N+1 , t) , 2c i=1

(47.169)

where HN+1 is the (N + 1)-electron Hamiltonian defined by (47.3) and X N+1 represents the space and spin coordinates of all N + 1 electrons defined as in (47.8). With the same assumptions made in the reduction of (47.147) to (47.150), (47.163) can be rewritten in the velocity gauge form ∂ i ΨV (X N+1 , t) ∂t   1 = HN+1 + A (t) · PN+1 ΨV (X N+1 , t) , c (47.170)

Part D 47.3

∞
dσ 1,B dσn1,B = . dΩ dΩ n=−∞

47.3 Electron–Atom Collisions in a Laser Field

726

Part D

Scattering Theory

where PN+1 =

N+1


−i∇i

(47.171)

i=1

Part D 47.3

is the total momentum operator. The solution of (47.170) for e− –H scattering in a strong laser field at high electron impact energies has been considered by Francken and Joachain [47.132]. They discussed the use of the Born series and the eikonal Born series (EBS) approximations, given by (47.104), to describe the electron–atom collision, and included the dressed wave function of the atomic hydrogen target to first-order in the field strength E 0 . Important effects related to the dressing of the target are the appearance in the cross sections of asymmetries between the absorption and the emission of a given number of laser photons, as well as the appearance of new resonance structures in the cross sections. So far, detailed studies of low-energy electron–atom and electron–ion collisions in laser fields have been very limited. Pioneering work on e− –H+ collisions by Dimou and Faisal [47.133], and by Collins and Csanak [47.134] have shown important resonance effects caused by the field coupling bound states to the continuum. In addition, multichannel quantum defect theory has been applied by Zoller and co-workers [47.135, 136] to study the behavior of Rydberg states in laser fields. We conclude this section by briefly describing the R-matrix–Floquet method [47.137] for treating electron collisions with complex atoms and ions in a laser field, based on the R-matrix method discussed in Sect. 47.1.5. In this approach, configuration space is divided into internal and external regions, as in the field-free case. In the internal region, a further gauge transformation of the field is made to the length gauge, so that the timedependent Schrödinger equation (47.170) now becomes ∂ i ΨL (X N+1 , t) ∂t = [HN+1 + E (t) · RN+1 ]ΨL (X N+1 , t) , (47.172)  N+1 where RN+1 = i=1 ri , and we assume that the electric field E (t) is given by 1 d E(t) = − A(t) = ˆ E0 cos ωt . (47.173) c dt In order to solve (47.172), we introduce the Floquet– Fourier expansion [47.138, 139] ΨL (X N+1 , t) = e−iEt

∞


e−inωt ΨnL (X N+1 ) .

n=−∞

(47.174)

Substituting this equation into (47.172), using (47.173) and equating the coefficients of exp[−i(E + nω)t] to zero gives  L  L =0, (HN+1 − E − nω)ΨnL + D N+1 Ψn−1 + Ψn+1 (47.175)

where we have introduced the operator 1 D N+1 = E0 ˆ · RN+1 . 2

(47.176)

The functions ΨnL can be regarded as the components of a vector Ψ L in photon space. Equation (47.175) can then be written in this space as (HF − E I)Ψ L = 0 ,

(47.177)

where the Floquet Hamiltonian HF is an infinite tridiagonal matrix. In order to solve (47.177) in the internal region, the components ΨnL are expanded in a basis which, in analogy with (47.69), has the form L Ψkn (X N+1 )
Γ  Φ i x1 , . . . , xN ; rˆ N+1 σ N+1 r −1 =A N+1 Γ ij Γ × u j (r N+1 ) aijkn
Γ + χiΓ (x1 , . . . , xN+1 ) bikn ,

(47.178)

Γi

where the summation over Γ is required since the total orbital angular momentum L and the total parity π in (47.9) are no longer conserved. The coeffiΓ and bΓ are then determined by diagonalizcients aijkn ikn ing HF + Lb , where Lb is an appropriate Bloch operator. In the external region, the wave function describing the scattered electron is transformed to the velocity gauge, and the corresponding coupled equations integrated outwards from the internal region boundary for each energy of interest [47.140]. After a further transformation to the acceleration frame (or Kramers–Henneberger frame [47.141]) the wave function can be fitted to an asymptotic form to yield the K -matrix, S-matrix and collision cross sections. This approach has been used to calculate laser-assisted electron scattering by H and He atoms [47.142, 143] and a detailed discussion of the theory has been given [47.144].

Electron–Atom, Electron–Ion, and Electron–Molecule Collisions

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47.28 47.29 47.30 47.31

47.35 47.36 47.37 47.38 47.39 47.40 47.41

47.42 47.43 47.44 47.45 47.46 47.47 47.48 47.49 47.50 47.51 47.52 47.53 47.54 47.55 47.56 47.57 47.58 47.59 47.60 47.61 47.62 47.63 47.64

47.65 47.66

47.67 47.68 47.69 47.70 47.71

47.72 47.73

47.74

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47.75 47.76 47.77 47.78 47.79 47.80 47.81 47.82 47.83 47.84 47.85 47.86 47.87 47.88 47.89

47.90 47.91 47.92 47.93 47.94 47.95

Part D 47

47.96 47.97 47.98 47.99 47.100 47.101 47.102 47.103 47.104 47.105 47.106 47.107

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Electron–Atom, Electron–Ion, and Electron–Molecule Collisions

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Part D 47

731

Positron Collis 48. Positron Collisions

The positron is the antiparticle of the electron, having the same mass but opposite charge. Positrons undergo collisions with atomic and molecular systems in much the same way as electrons do. Thus, the standard scattering theory for electrons (see Chapt. 47) can also be applied to positron scattering. However, there are a number of important differences from electron scattering which we outline below. Since the positron is a distinct particle from the atomic electrons, it cannot undergo an exchange process with the bound electrons during a collision, as is possible with electrons. Thus, the nonlocal exchange terms which arise in the description of electron scattering are not present for positrons. This leads to a simplification of the scattering equations from those for electrons. However, there are scattering channels available with positron scattering which do not exist with electrons. These are dealt with in Sect. 48.1. Historically beams of low-energy positrons were difficult to obtain and consequently there is considerably less experimental data available for positrons than for electrons. This was particularly true for quantities which required large incident

48.1 Scattering Channels ............................. 731 48.1.1 Postronium Formation ............... 731 48.1.2 Annihilation ............................. 732 48.2 Theoretical Methods ............................ 733 48.3 Particular Applications ......................... 48.3.1 Atomic Hydrogen....................... 48.3.2 Noble Gases.............................. 48.3.3 Other Atoms ............................. 48.3.4 Molecular Hydrogen .................. 48.3.5 Other Molecules ........................

735 735 735 736 737 737

48.4 Binding of Positrons to Atoms .............. 737 48.5 Reviews .............................................. 738 References .................................................. 738 positron fluxes, such as differential scattering cross sections and coincidence parameters. However, the recent development of cold trap-based positron beams with high resolution and high brightness by the San Diego group [48.1] has the potential to revolutionize this field and put it on a par with electron scattering. Throughout this chapter we will employ atomic units unless otherwise noted.

48.1 Scattering Channels Positrons colliding with atomic and molecular systems have the same scattering channels available as for electrons, viz., elastic, inelastic, ionization, and for molecules, dissociation. However, two channels exist for positrons which do not exist for electrons, viz., positronium formation and annihilation.

Positronium, a bound state of an electron–positron pair (Chapt. 27), can be formed during the collision of a positron with an atomic or molecular target. The positronium ‘atom’ can escape to infinity leaving the target in a ionized state with a positive charge of one.

Part D 48

48.1.1 Postronium Formation

Thus, this process can be difficult to distinguish experimentally from true ionization where both the incident positron and the ionized electron are asymptotically free particles. The positronium atom can exist in its ground state or in any one of an infinite number of excited states after the collision. The level structure of positronium is, to order α2 , where α is the fine-structure constant, identical to that of hydrogen but with each level having half the energy of the corresponding hydrogenic state. Positronium formation is a rearrangement channel, and thus, is a two-centre problem. Because positronium is a light particle, having a reduced mass one-half of that of an electron, the semi-classical type of approximations used in ion–atom collisions (Chapt. 50) are not

732

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applicable here. We will discuss various theoretical approaches to this process in Sect. 48.2 and give references to experimental results in Sect. 48.3. Positronium formation in the ground state has a threshold which is 1/4 of a Hartree (6.802 85 eV) below the ionization threshold of the target. This means that it is normally the lowest inelastic channel in positron scattering from neutral atoms. For atoms with a small ionization potential, such as the alkalis, this channel is always open. The energy range between the positronium threshold and the first excited state of the atom is known as the Ore gap. In this range, positronium formation is the only possible inelastic process.

48.1.2 Annihilation Annihilation is a process in which an electron–positron pair is converted into two or more photons. It can occur either directly with a bound atomic electron or after positronium formation has taken place. The direct annihilation cross section for a positron of momentum k colliding with an atomic or molecular target can be written as [48.2] α3 Z eff
2  πa0 , (48.1) k where Z eff can be thought of as the effective number of electrons in the target with which the positron can annihilate. If Ψ(r1 , r2 , . . . , r N , x) is the wave function for the system of a positron, with coordinate x, colliding with an N-electron target, then σa =

N    2 dr1 dr2 . . . dr N Ψ(r1 , r2 , . . . , r N ; ri ) . Z eff = i=1

(48.2)

Part D 48.1

While this formula can be naively derived by assuming that the positron can only annihilate with an electron if it is at the identical location, it actually follows from a quantum electrodynamical treatment of the process [48.3]. If the wave function Ψ is approximated by the product of the undistorted target wave function times a positron scattering function F(x), then   2 Z eff = drρ(r) F(r) , (48.3) where ρ is the electron number density of the target. Thus, in the Born approximation, where F is taken as a plane wave, Z eff simply becomes the total number of electrons Z in the target. However, a pronounced enhancement of the annihilation rate in the vicinity of the

Ps formation threshold due to virtual Ps formation was predicted [48.4, 5]. Subsequently, the Born approximation was shown to be grossly inadequate by the San Diego group, who found annihilation rates Z eff at room temperature which are an order of magnitude larger for some atoms and even up to five orders of magnitude larger in large hydrocarbon molecules. Furthermore, there is evidence that only the outer shell of electrons takes part in the annihilation process. Two mechanisms have been proposed in order to explain these large values for Z eff . One involves the enhancement of the direct annihilation process below the Ps formation threshold due to the attractive nature of the positron–electron interaction, which increases the overlap of positron and electron densities on the atom or molecule. The second mechanism is referred to as resonant annihilation, which occurs after the positron has been captured into a Feshbach resonance, where the positron is bound to a vibrationally excited molecule. A summary of the above results can be found in the recent article by Barnes et al. [48.6]. When a positron annihilates with an atomic electron, two 511 keV photons is the most likely result, if the positron–electron pair are in a singlet spin state (parapositronium). In the centre-of-mass frame of the pair, the photons are emitted in opposite directions to conserve momentum. However, in the laboratory frame the bound electron has a momentum distribution which is reflected in the photon directions not being exactly 180 degrees apart. This slight angular deviation, called the angular correlation, can be measured, and gives information about the momentum distribution of the bound electrons. This quantity is given by [48.3] N   S(q) = dr1 . . . dri−1 dri+1 . . . dr N (48.4)  ×

i=1

 2 dri dx eiq·x Ψ(r1 , r2 , . . . , r N ; x) δ(ri − x) ,

where q is the resultant momentum of the annihilating pair. In evaluating this quantity, the positron is assumed to be thermalized in the gas before undergoing annihilation. Experimentally, only one component of q is measured, so that S(q) is integrated over the other two components of the momentum to obtain the measured quantity. The spin triplet component of an electron– positron pair (orthopositronium) can only decay with the emission of three or more photons which do not have well defined energies. This is a much less probable process than the two photon decay from the singlet component.

Positron Collisions

48.2 Theoretical Methods

733

48.2 Theoretical Methods The basic theoretical approaches to the calculation of positron scattering from atoms and molecules were originally developed for electron scattering and later applied to the positron case. Thus, we emphasize here only the differences that arise between the electron and positron cases, both in the theoretical formulations, and in later sections, in the nature of the results. The lowest-order interaction between a free positron and an atomic or molecular target is the repulsive static potential of the target     Vs = ψ0 V ψ0 , (48.5) where ψ0 is the unperturbed target wave function and V is the electrostatic interaction potential between the positron and the target. Since this interaction has the opposite sign from that for electron scattering, the static potential also has the opposite sign in these two cases. On the other hand, the next higher-order of interaction is polarization, which arises from the distortion of the atom by the incident particle. If we represent this distortion of the target to first-order by the wave function ψ1 , as in the polarized-orbital approximation, for example [48.9], then the polarization interaction can be represented by the potential     Vp = ψ0 V ψ1 . (48.6)

–0.1

–0.5

–0.9

–1.3 0.0

0.4

0.8

1.2 Momentum (a.u.)

Fig. 48.1 Variational s-wave phase shifts for electron [48.7]

(dashed line) and positron [48.8] (solid line) scattering from helium atoms

p-wave phase shifts leads to the large difference in the total elastic cross sections as shown. Higher-order terms in the interaction potential may also give important contributions to scattering cross sections. For a detailed discussion, see the article by Drachman and Temkin [48.10]. A simple potential scattering calculation using the sum of the static and polarization potentials, but without the exchange terms that are present for the electron case, can be applied to elastic scattering calculations for closed shell systems (see Sect. 48.3.3). The potentials defined above can also be used in a distorted-wave approximation (Chapt. 47) which can be applied to excitation and ionization by positron impact. Once again, the complicated exchange terms which arise in electron scattering are absent here. Sienkiwicz and Baylis [48.11] have included such potentials in a Dirac–Fock formulation of positron scattering which treats the positron as an electron with negative energy. For positrons with high enough incident energies (≈ 1 keV), the first Born approximation will become valid (Chapt. 47). Since the first Born approximation is independent of the sign of the charge of the incident particles, this indicates that as the incident energy increases, the corresponding cross sections for electron

Part D 48.2

This potential is attractive for both positron and electron scattering and has an asymptotic form with leading term −αd /2r 4 , where αd is the static dipole polarizability (Sect. 23.2.3) of the target. Thus, the static and polarization potentials for positron scattering from ground state systems are of opposite sign and tend to cancel one another. This leads to very different behaviour from the electron case where they are of the same sign. In particular, the elastic scattering cross sections for positron scattering from an atom are much smaller than for electron scattering, and the phase shifts (Sect. 47.1.1) have very different magnitudes and dependences on energy. This is illustrated for the case of scattering from helium in Figs. 48.1–48.3, where the results of the highly accurate variational calculations for scattering by electrons and positrons are shown. Note the difference in sign between the electron and positron s-wave phase shifts for very small values of the incident momentum. The fact that the positron phase shift goes through zero leads to the Ramsauer minimum in the positron total cross section, as shown in Fig. 48.3. The large difference in magnitudes between the electron and positron s- and

s-wave phase shift

734

Part D

Scattering Theory

2

p-wave phase shift

Total cross section (πa 0)

0.3

0.2

1

0.1 0.1

0.0 0.00

0.00 0.40

0.80 1.20 Momentum (a.u.)

Fig. 48.2 Variational

p-wave phase shifts for electron [48.7] (dashed line) and positron [48.15] (solid line) from helium atoms

Part D 48.2

and positron scattering will eventually merge. From flux conservation arguments, this means that the positronium formation cross section will rapidly decrease as the incident energy increases. In fact, from experimental measurements, the total cross sections (summed over all possible channels) for electron and positron scattering appear to merge at a much lower energy than the cross sections for individual channels [48.12]. More elaborate calculations for high energy scattering have been carried out in the eikonal-Born series [48.13] (Chapt. 47). These approximations allow for both polarization and absorption (i. e., inelastic processes) and yield good agreement with elastic experimental measurements of differential cross sections at energies above 100 eV. A detailed analysis [48.13, 14] of the various contributions to the scattering indicates that absorption effects due to the various open inelastic channels plays a much more important role here than for electron scattering. A more elaborate treatment of positron scattering is based on the close-coupling approximation (Chapt. 47), where the wave function for the total system of positron plus target is expanded using a basis set comprised of the wave functions of the target. Once again, there are no exchange terms involving the positron and, in principle, a complete expansion including the continuum states of the target would include the possibility of positronium formation. However, such an expansion is not practi-

0.40

0.80 Momentum (a.u.)

Fig. 48.3 Total elastic cross sections for electron (dashed line) and positron (solid line) from helium atoms calculated from the phase shifts shown in Figs. 48.1 and 48.2. The higher-order phase shifts were calculated from effective range theory (Sect. 47.1.1)

cable if one wants to calculate explicit cross sections for positronium formation. Even in cases where such cross sections are not required, the considerable effect that the positronium formation channels can have on the other scattering cross sections is best included by a close-coupling expansion that includes terms representing positronium states plus the residual target ion. There is a problem of double counting of states in such an expansion but, in practice, this does not appear to be a problem if the number of states in the expansion is not large. Also, in many cases, additional pseudostates have to be included in the expansion in order to correctly represent the long-range polarization interaction. A close-coupling expansion including positronium states is a two-centre problem, i. e., it includes the centres of mass of both the target and the positronium states. Since positronium is a light system, the semi-classical approach often used to treat rearrangement collisions between heavy systems (Chapt. 50) is not applicable here. This means that one is faced with a problem of considerable computational complexity [48.16–21]. Another way to take into account the effects of open inelastic channels without the complications of a full close-coupling approach is to use optical potentials. These are often based on a close-coupling formal-

Positron Collisions

ism [48.14, 22] and lead to a complex potential, the real part of which represents distortions of the target (such as polarization) while the complex part allows for absorption (i. e., flux into open channels not explicitly represented). Bray and Stelbovics [48.23] have applied the convergent close-coupling method to the scattering of positrons from atoms. This method includes contributions from the continuum states of the target and sufficient terms in the expansion are included to ensure numerical convergence. At the present time, however, positronium states have not been explicitly included. Finally, there is the variational method (Chapt. 47), which uses an analytic form of trial wave function to represent the total system. The parameters of this analytic function are determined as part of the method. Given a trial wave function with sufficient flexibility and a large enough number of parameters, essentially exact results can be obtained in the elastic energy range and

48.3 Particular Applications

735

the Ore gap. Because the complexity of the trial function increases as the square of the number of electrons in the target, only positron scattering from hydrogen, helium and lithium and the hydrogen molecule have been treated by this method, to date [48.8, 24]. In the case of ionization there appears to be quite distinct threshold behaviour of the cross sections for electron and positron collisions. For electrons, the Wannier threshold law (Sect. 52.2.1) has exponent 1.127, while a similar analysis for positrons [48.25] yields an exponent of 2.651. However, the existence of the positronium formation channel leaves in question whether this analysis will give the dominant term at threshold. For a fuller discussion see [48.26], and references therein. There has been an investigation [48.27] of the behaviour of the elastic cross sections at the positronium formation threshold which predicts the occurrence of a Wigner cusp for the lighter noble gases.

48.3 Particular Applications 48.3.1 Atomic Hydrogen

48.3.2 Noble Gases Because the noble gases are convenient experimental targets, a good deal of effort has gone into calculations for these targets, particularly for elastic scattering, ionization, and Ps formation. In the purely elastic energy range, i. e., for energies below the positronium formation threshold, the simple potential scattering approach using the static and polarization potentials defined above yields quite good results. Since the long-range behaviour of the sum of the potentials is attractive, the scattering phase shifts for positrons must be positive for suffi-

Part D 48.3

Because of the difficulty of making measurements in atomic hydrogen, the available experimental data is restricted so far to total cross sections, as well as to total ionization and positronium formation cross sections. Essentially exact variational calculations have been carried out in the elastic energy regime and the Ore gap [48.8]. Ionization cross sections have been measured by both the Bielefeld and London groups [48.28, 29] (and references therein) and have been calculated in a number of approximations [48.23, 30–32]. However, disagreements between the experimental measurements mean that there is at present no reliable way of assessing the various approximations used. More elaborate calculations with asymptotically correct wave functions have been used to determine triple differential cross sections for ionization [48.33, 34]. However, the task of integrating these to produce total cross sections is a formidable one. The total positron–hydrogen cross section has also been measured by the Detroit group [48.35], and is in quite good agreement with calculations based upon the coupled-pseudostate method [48.32] except at very low energies where the experimental uncertainties are the greatest.

In order to determine reliable positronium formation cross sections, the explicit positronium states have to be included. Several such calculations have been carried out [48.17–20]. These indicate the necessity of explicitly including positronium formation channels in the expansion of the total wave function in order to obtain accurate results, even for elastic scattering. The most recent calculations [48.32, 36] are in quite good agreement with various experiments [48.28, 37] over the majority of the energy range. As is the case for electron scattering, positron cross sections exhibit resonances (Sect. 47.1.3). These have been extensively studied by Ho [48.38] using variational and complex rotation methods.

736

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Part D 48.3

ciently low energies. However, as its incident energy increases, the positron probes the repulsive inner part of the potential and the phase shifts become negative. This behaviour leads to the well-known Ramsauer minimum in the integral elastic cross sections (Chapt. 47) for the lighter noble gases (helium, neon and possibly argon), but not for krypton and xenon [48.15, 39] (and references therein). This differs from electron scattering, where some low-energy phase shifts can be negative (modulo π) because of the existence of bound orbitals of the same symmetry. Another difference between positron and electron scattering is exhibited by the differential cross sections (Chapt. 47). As a result of the low intensity and width of positron beams, most measurements of differential cross sections are relative, with the normalization often being made to theoretical calculations at specific angles. For electrons, the shape of the cross section is determined by a few dominant phase shifts, whereas for positrons, many phase shifts contribute to the final shape [48.40]. Because of this behaviour, the differential cross sections for positron scattering have much less overall structure than for electron scattering. However, the differential cross sections for positrons for many of the noble gases have a single minimum at relatively small angles, both below and above the first inelastic channel. These have been reviewed by Kauppila et al. [48.41]. At intermediate energies, the simple potential scattering approximation is no longer sufficient and the inelastic channels have to be taken into account via, for example, the use of an optical potential [48.13, 14, 42]. Furthermore, in the inelastic scattering regime, the existence of open channels has a much more marked effect on the shape of the differential cross sections for positron scattering than for electrons [48.43]. The first absolute differential cross sections were measured for argon and krypton at very low energies using a magnetized beam of cold positrons [48.44, 45]. These results are in excellent agreement with a variety of different theoretical predictions [48.46–48]. There is relatively very little experimental data for the excitation of the noble gases. Some experimental work has been carried out for the lighter noble gases, helium, neon and argon [48.49] (and references therein), and there is satisfactory agreement between these measurements and close-coupling [48.50], as well as distorted-wave [48.51] calculations. The first state-resolved absolute excitation cross sections for the 4s [1/2]o1 and 4s [3/2]o1 states of argon and the 6s [3/2]o1 state of xenon have been measured by the

San Diego group [48.6, 52]. Relativistic distorted-wave calculations are in satisfactory agreement with the experiment for argon [48.53], but less so for xenon [48.54]. The total ionization and positronium cross sections have been measured extensively for all of the noble gases. In general, there is good agreement amongst the various experiments for the ionization cross section, but much less so for the positronium formation cross section. A summary of the experimental work on the ionization and positronium cross sections for neon, argon, krypton and xenon can be found in the article by Laricchia et al. [48.55] (and references therein), while a more detailed analysis of these cross sections in argon can be found in recent articles by the San Diego group [48.6, 56]. There has also been an extensive investigation of the energy dependence of the elastic and positronium formation cross section near the Ps formation threshold for all the inert gases [48.57] (and references therein). There exist some measurements of Z eff and angular correlation parameters for these gases [48.58], mainly at room temperature, and calculations for them have been made in the polarized-orbital approximation [48.59].

48.3.3 Other Atoms In the case of positron scattering from the alkali atoms, the positronium formation channel is always open and the simple potential scattering approach does not yield reliable results. The Detroit group has measured the total cross section, as well as upper and lower bounds to the positronium formation cross section for sodium, potassium, and rubidium. Although early close-coupling calculations of the elastic and excitation cross sections [48.60–62] were in surprisingly good agreement with the experimental total cross section, these calculations did not include the positronium formation channel. Subsequently, much more sophisticated calculations were carried out by the Belfast group using the coupled-pseudostate method, which included both eigenstates of the target as well as positronium [48.32, 63–65]. These calculations also showed the increasing importance of positronium formation in excited states for the alkalis potassium, rubidium, and caesium. The overall agreement between experimental results and those from the coupled-pseudostate method are quite good for both potassium and rubidium, however, for sodium, the experimental positronium formation cross section is significantly above these theoretical calculations. A summary of the experimental work on the alkalis can be found in [48.35], while

Positron Collisions

48.4 Binding of Positrons to Atoms

a corresponding summary of theoretical work is given in [48.32]. Substantial resonance features have been found in these positron–alkali atom cross sections [48.66]. The positronium formation threshold for magnesium is very low, only 0.844 eV, and hence, the elastic and positronium formation cross sections will dominate in the low energy region. Upper and lower bounds to the Ps formation cross section in magnesium have been determined [48.35, 67], and are in agreement with both close-coupling calculations [48.68] and the results of many-body theory [48.69].

there are some differences between these experiments, near threshold the positron cross section increases less rapidly, in general, than the corresponding electron cross section, in accordance with the Wannier law. Theoretical calculations are in satisfactory agreement with the measurements [48.82] (and references therein). The positronium formation cross section has also been measured by a number of different groups with coupled-channels calculations being carried out for this process [48.83].

48.3.4 Molecular Hydrogen

For diatomic and triatomic molecules, most of the experimental and theoretical work has been carried out for CO, CO2 , O2 , and N2 . Total cross sections for O2 , N2 , and CO2 have been measured from threshold to several hundred eV [48.70] (and references therein). Relative differential cross sections have been measured for CO, CO2 , O2 , N2 , as well as N2 O, on both sides of the positronium formation threshold [48.84]. Absolute differential cross sections have been measured for CO at 6.75 eV [48.45]. At low energies the gases N2 , O2 , and CO exhibit a minimum in the DCS at small angles, as per the heavier noble gases. This minimum gradually disappears as the energy increases. Vibrational excitation cross sections for CO and CO2 have been measured [48.75] and are in excellent agreement with the theoretical calculations of [48.85] for CO, and in satisfactory agreement with theory [48.86] for CO2 . Electron excitation of the a1 Π and a1 Σ states of N2 have been measured from threshold to 20 eV [48.52]. Interestingly, the positron cross section near threshold is approximately double that for electrons. For polyatomic molecules, the majority of experimental and theoretical work has been carried out for CH4 . This includes the total cross section and quasi-elastic (summed over vibration–rotational levels) differential cross sections. At low energies there is a minimum in these DCS at small angles, as per the heavier noble gases which, in turn, also disappears at higher energies. The positronium formation cross section has also been measured.

By its fundamental nature, molecular hydrogen has attracted considerable attention both experimentally and theoretically. The total elastic cross section has been measured by both the Detroit group [48.37] and the London group [48.70], with good agreement between both sets of data. There have been several theoretical calculations of this cross section by a variety of methods: Kohn variational [48.71], R-matrix [48.72], distributed positron model [48.73], and recently, a Schwinger multichannel method [48.74], with [48.72, 73] being in satisfactory agreement with experiment. Once again, the elastic cross section is strongly influenced by the positronium formation channel near threshold. Quite recently the vibrational (0 → 1) excitation cross section of molecular hydrogen has been measured between 0.55 and 4 eV by Sullivan et al. [48.75]. Their data are in quite good agreement with theoretical calculations. The San Diego group has also measured the electronic excitation of the B1 Σ state from threshold to 30 eV [48.52]. Their data are in reasonable agreement with the Schwinger multichannel calculation of Lino et al. [48.76]. Interestingly, the measured positron excitation cross section appears to be larger than that determined for electron excitation. The ionization cross section has been determined over a wide range of energies by a number of different groups [48.77–81] (and references therein). Since all of the above measurements are relative, they must be normalized to one another at particular energies. Although

48.3.5 Other Molecules

greatly enhance the annihilation cross section and help to explain the large measured values of Z eff for both atoms and molecules. It has been shown theoretically

Part D 48.4

48.4 Binding of Positrons to Atoms There have been many recent investigations of the possible binding of positrons to a variety atoms. As was mentioned in Sect. 46.2.2, such binding could

737

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Scattering Theory

that a positron will bind to a large number of oneand two-electron atomic systems [48.87] (and references therein). For one-electron systems, where the ionization potential is less than 6.802 85 eV, the dominant configuration is a polarized positronium (Ps) cluster moving in the field of the residual positive ion, while for two-electron systems, with an ioniza-

tion potential greater than 6.802 85 eV, the dominant configuration involves a positron orbiting a polarized neutral atom [48.88]. So far, there is no experimental evidence for these positronic atoms. However, there is considerable evidence that positrons will bind to large hydrocarbon molecules [48.6] (and references therein).

48.5 Reviews For a number of years a Positron Workshop has been held as a satellite of the International Conference on the Physics of Electronic and Atomic Collisions. Their proceedings [48.89–100] give an excellent summary of the state of positron scattering research, both experimental and theoretical, including such additional topics as positronium scattering from atoms, the formation of antihydrogen, inner shell ionization, and applications to astrophysics.

There are several review articles on positron scattering in gases, including the early historical development [48.101], more comprehensive articles [48.2, 102–104], as well as a more recent review [48.105], which also discusses the future for positron physics. A recent book [48.106] discusses various aspects of both experimental and theoretical positron physics.

References 48.1 48.2 48.3 48.4 48.5 48.6 48.7 48.8 48.9 48.10

48.11 48.12

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48.13 48.14 48.15

C. Kurz, S. J. Gilbert, R. G. Greaves, C. M. Surko: Nucl. Instrum. Methods B 143, 188 (1998) P. A. Fraser: Adv. At. Mol. Phys. 4, 63 (1968) R. Ferrell: Rev. Mod. Phys. 28, 308 (1956) J. W. Humberston, P. Van Reeth: Nucl. Instrum. Methods B 143, 127 (1998) G. Laricchia, C. Wilkin: Nucl. Instrum. Methods B 143, 135 (1998) L. D. Barnes, J. P. Marler, J. P. Sullivan, C. M. Surko: Phys. Scr. T110, 280 (2004) R. K. Nesbet: Phys. Rev. A 20, 58 (1979) E. A. G. Armour, J. W. Humberston: Phys. Rep. 204, 165 (1991) R. P. McEachran, A. D. Stauffer: J. Phys. B 10, 663 (1977) R. J. Drachman, A. Temkin: Case Studies in Atomic Collision Physics II, ed. by E. W. McDaniel, M. R. C. McDowell (North-Holland, Amsterdam 1972) Chap. 6 J. E. Sienkiewicz, W. E. Baylis: Phys. Rev. A 43, 1331 (1991) W. E. Kauppila, T. S. Stein: Can. J. Phys. 60, 471 (1982) C. J. Joachain, R. M. Potvliege: Phys. Rev. A 35, 4873 (1987) K. Bartschat, R. P. McEachran, A. D. Stauffer: J. Phys. B. 21, 2789 (1988) J. W. Humberston, R. I. Campeanu: J. Phys. B 13, 4907 (1980)

48.16 48.17 48.18 48.19 48.20 48.21

48.22 48.23 48.24 48.25 48.26 48.27 48.28 48.29

48.30 48.31

R. N. Hewitt, C. J. Noble, B. H. Bransden: J. Phys. B 26, 3661 (1993) K. Higgins, P. G. Burke: J. Phys. B 26, 4269 (1993) J. Mitroy: J. Phys. B 26, 4861 (1993) N. K. Sarkar, A. S. Ghosh: J. Phys. B 27, 759 (1994) G. Liu, T. T. Gien: Phys. Rev. A 46, 3918 (1992) C. P. Campbell, M. T. McAlinden, A. A. Kernoghan, H. R. J. Walters: Nucl. Instrum. Methods B 143, 41 (1998) I. E. McCarthy, K. Ratnavelu, Y. Zhou: J. Phys. B 26, 2733 (1993) I. Bray, A. Stelbovics: Phys. Rev. A 49, R2224 (1994) J. W. Humberston: J. Phys. B 25, L491 (1992) H. Klar: J. Phys. B 14, 4165 (1981) T.-Y. Kuo, H.-L. Sun, K.-N. Huang: Phys. Rev. A 67, 012705 (2003) G. Laricchia, J. Moxom, M. Charlton, Á. Kövér, W. E. Meyerhoff: Hyperfine Interact. 89, 209 (1994) M. Weber, A. Hofmann, W. Raith, W. Sperber: Hyperfine Interact. 89, 221 (1994) G. O. Jones, M. Charlton, J. Slevin, G. Laricchia, Á. Kövér, M. R. Poulsen, S. N. Chormaic: J. Phys. B 26, L483 (1993) P. Acacia, R. I. Campeanu, M. Horbatsch, R. P. McEachran, A. D. Stauffer: Phys. Lett. A 179, 205 (1993) S.-W. Hsu, T.-Y. Kuo, C. J. Chen, K.-N. Huan: Phys. Lett. 167A, 277 (1992)

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48.32 48.33 48.34 48.35

48.36 48.37 48.38 48.39 48.40

48.41 48.42 48.43 48.44 48.45

48.46 48.47 48.48 48.49 48.50 48.51 48.52 48.53 48.54 48.55 48.56

48.58 48.59

48.60 48.61 48.62 48.63 48.64 48.65 48.66 48.67 48.68 48.69 48.70 48.71 48.72 48.73 48.74 48.75 48.76 48.77 48.78 48.79 48.80 48.81

48.82 48.83 48.84

48.85 48.86 48.87 48.88

S. J. Ward, M. Horbatsch, R. P. McEachran, A. D. Stauffer: J. Phys. B 22, 1845 (1989) R. P. McEachran, M. Horbatsch, A. D. Stauffer: J. Phys. B 24, 2853 (1991) M. T. McAlinden, A. A. Kernoghan, H. R. J. Walters: Hyperfine Interact. 89, 161 (1994) M. T. McAlinden, A. A. Kernoghan, H. R. J. Walters: J. Phys. B 29, 555 (1996) A. A. Kernoghan, M. T. McAlinden, H. R. J. Walters: J. Phys. B 29, 3971 (1996) M. T. McAlinden, A. A. Kernoghan, H. R. J. Walters: J. Phys. B 30, 1543 (1997) S. J. Ward, M. Horbatsch, R. P. McEachran, A. D. Stauffer: J. Phys. B 22, 3763 (1989) E. Surdutovich, M. Harte, W. E. Kauppila, C. K. Kwan, S. Zhou: Phys. Rev. A 68, 022709 (2003) R. N. Hewitt, C. J. Noble, B. H. Bransden: Can. J. Phys. 74, 559 (1996) G. F. Gribakin, W. A. King: Can. J. Phys. 74, 449 (1996) M. Charlton, T. C. Griffith, G. R. Heyland, G. L. Wright: J. Phys. B 16, 323 (1983) E. A. G. Armour, D. J. Baker, M. Plummer: J. Phys. B 23, 3057 (1990) G. Danby, J. Tennyson: J. Phys. B 23, 2471S (1990) T. L. Gibson: J. Phys. B 25, 1321 (1992) J. S. E. Germano, M. A. P. Lima: Phys. Rev. A 47, 3976 (1993) J. P. Sullivan, S. J. Gilbert, C. M. Surko: Phys. Rev. Lett. 86, 1494 (2001) J. L. S. Lino, J. S. E. Germano, M. A. P. Lima: J. Phys. B 27, 1881 (1994) D. Fromme, G. Kruse, W. Raith, G. Sinapius: J. Phys. B 21, L261 (1988) H. Knudsen, L. Brun-Nielsen, M. Charlton, P. M. Poulsen: J. Phys. B 23, 3955 (1990) J. Moxom, P. Ashley, G. Laricchia: Can. J. Phys. 74, 367 (1996) P. Ashley, J. Moxom, G. Laricchia: Phys. Rev. Lett. 77, 1250 (1996) F. M. Jacobsen, N. P. Frandsen, H. Knudsen, U. Mikkelson, D. M. Schrader: J. Phys. B 28, 4675 (1995) R. I. Campeanu, V. Chis, L. Nagy, A. D. Stauffer: Phys. Lett. A 310, 445 (2003) P. K. Biswas, J. S. E. Germano, T. Frederico: J. Phys. B 35, L409 (2002) D. A. Przybyla, W. Addo-Asah, W. E. Kauppila, C. K. Kwan, T. S. Stein: Nucl. Instrum. Methods B 143, 57 (1998) F. A. Gianturco, T. Mukherjee, P. Paioletti: Phys. Rev. A 56, 3638 (1997) M. Kimura, M. Takekawa, Y. Itikawa, H. Takaki, O. Sueoka: Phys. Rev. Lett. 80, 3936 (1998) J. Mitroy, M. W. J. Bromley, G. G. Ryzhikh: J. Phys. B 35, R81 (2002) J. Mitroy: Phys. Rev. A 66, 010501R (2002)

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A. A. Kernoghan, D. J. R. Robinson, M. T. McAlinden, H. R. J. Walters: J. Phys. B 29, 2089 (1996) M. Brauner, J. S. Briggs, H. Klar: J. Phys. B 22, 2265 (1989) S. Jetzke, F. H. M. Faisal: J. Phys. B 25, 1543 (1992) T. S. Stein, J. Jiang, W. E. Kauppila, C. K. Kwan, H. Li, A. Surdutovich, S. Zhou: Can. J. Phys. 74, 313 (1996) J. Mitroy: J. Phys. B 29, L263 (1996) S. Zhou, H. Li, W. E. Kauppila, C. K. Kwan, T. S. Stein: Phys. Rev. A 55, 361 (1997) Y. K. Ho: Hyperfine Interact. 73, 109 (1992) G. Sinapius, W. Raith, W. G. Wilson: J. Phys. B 13, 4079 (1980) R. P. McEachran, A. D. Stauffer: Positron (Electron)Gas Scattering, ed. by W. E. Kauppila, T. S. Stein, J. M. Wadehra (World Scientific, Singapore 1986) p. 122 W. E. Kauppila, C. K. Kwan, D. Przybyla, S. J. Smith, T. S. Stein: Can. J. Phys. 74, 474 (1996) A. Jain: Phys. Rev. A 41, 2437 (1990) W. E. Kauppila, T. S. Stein: Hyperfine Interact. 40, 87 (1990) S. J. Gilbert, R. G. Greaves, C. M. Surko: Phys. Rev. Lett. 82, 5032 (1999) J. P. Sullivan, S. J. Gilbert, J. P. Marler, R. G. Greaves, S. J. Buckman, C. M. Surko: Phys. Rev. A 66, 042708 (2002) R. P. McEachran, A. G. Ryman, A. D. Stauffer: J. Phys. B 12, 1031 (1979) R. P. McEachran, A. D. Stauffer, L. E. M. Campbell: J. Phys. B 13, 1281 (1980) V. A. Dzuba, V. V. Flambaum, G. F. Gribakin, W. A. King: J. Phys. B 29, 3151 (1996) S. Mori, O. Sueoka: J. Phys. B 27, 4349 (1994) R. N. Hewitt, C. J. Noble, B. H. Bransden: J. Phys. B 25, 2683 (1992) L. A. Parcell, R. P. McEachran, A. D. Stauffer: Nucl. Instrum. Methods B 177, 113 (2000) J. P. Sullivan, J. P. Marler, S. J. Gilbert, S. J. Buckman, C. M. Surko: Phys. Rev. Lett. 87, 073201 (2001) R. P. McEachran, A. D. Stauffer: Phys. Rev. A 65, 034703 (2002) L. A. Parcell, R. P. McEachran, A. D. Stauffer: Nucl. Instrum. Methods B 221, 93 (2004) G. Laricchia, P. Van Reeth, J. Moxom: J. Phys. B 35, 2525 (2002) J. P. Marler, L. D. Barnes, S. J. Gilbert, J. A. Young, J. P. Sullivan, C. M. Surko: Nucl. Instrum. Methods B 221, 84 (2004) W. E. Meyerhof, G. Laricchia: J. Phys. B 30, 2221 (1997) K. Iwata, G. F. Gribakin, R. G. Greaves, C. Kurz, C. M. Surko: Phys. Rev. A 61, 022719 (2000) D. M. Schrader, R. E. Svetic: Can. J. Phys. 60, 517 (1982)

References

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Scattering Theory

48.89

48.90

48.91

48.92 48.93

48.94 48.95

48.96

48.97

J. W. Darewych, J. W. Humberston, R. P. McEachran, D. A. L. Paul, A. D. Stauffer (Eds.): Can. J. Phys. 60, 461–617 (1981) J. W. Humberston, M. R. C. McDowell (Eds.): Positron Scattering in Gases (Plenum, New York 1984) W. E. Kauppila, T. S. Stein, J. M. Wadehra (Eds.): Positron (Electron)-Gas Scattering (World Scientific, Singapore 1986) J. W. Humberston, E. A. G. Armour (Eds.): Atomic Physics with Positrons (Plenum, New York 1987) R. J. Drachman (Ed.): Annihilation in Gases and Galaxies (NASA Conference Publication, Washington 1990) p. 3058 L. A. Parcell (Ed.): Positron interaction with gases, Hyperfine Interact. 73, 1–232 (1992) W. Raith, R. P. McEachran (Eds.): Positron interactions with atoms, molecules and clusters, Hyperfine Interact. 89, 1–496 (1994) R. P. McEachran, A. D. Stauffer (Eds.): Proceedings of the 1995 positron workshop, Can. J. Phys. 74, 313–563 (1996) H. H. Andersen, E. A. G. Armour, J. W. Humberston, G. A. Laricchia (Eds.): Proceedings of the 1997

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48.104 48.105

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positron workshop, Nucl. Instrum. Methods B 143, 1–232 (1998) S. Hara, T. Hyodo, Y. Nagashima, L. Rehn (Ed.): Proceedings of the 1998 positron workshop, Nucl. Instrum. Methods B 171, 1–250 (2000) M. H. Holzscheiter (Ed.): Proceedings of the 2001 positron workshop, Nucl. Instrum. Methods B 192, 1–237 (2002) U. Uggerhøj, T. Ichioka, H. Knudsen (Eds.): Proceedings of the 2003 positron workshop, Nucl. Instrum. Methods B 221, 1–242 (2004) H. S. W. Massey: Can. J. Phys. 60, 461 (1982) W. E. Kauppila, T. S. Stein: Adv. At. Mol. Opt. Phys. 26, 1 (1990) B. H. Bransden: Case Studies in Atomic Collision Physics I, ed. by E. W. McDaniel, M. R. C. McDowell (Wiley, New York 1969) Chap. 4 T. S. Stein, W. E. Kauppila: Adv. At. Mol. Opt. Phys. 18, 53 (1982) S. J. Buckman: New Directions in Antimatter in Chemistry and Physics, ed. by C. M. Surko, F. A. Gianturco (Kluver, Amsterdam 2001) p. 391 M. Charlton, J. W. Humberston: Positron Physics (Cambridge Univ. Press, Cambridge 2000)

Part D 48

741

Adiabatic and diabatic electronic states of a system of atoms are defined and their properties are described. Nonadiabatic interaction for slow quasiclassical motion of atoms is discussed within two-state common-trajectory approximation. Analytical formulae for nonadiabatic transition probabilities are presented for particular modles with reference to single and double passage of coupling regions (Landau– Zener–Stückelberg, Rosen–Zener–Demkov, Nikitin models). Generalization for multiple passage is described.

49.1 Basic Definitions.................................. 49.1.1 Slow Quasiclassical Collisions ...... 49.1.2 Adiabatic and Diabatic Electronic States ..... 49.1.3 Nonadiabatic Transitions: The Massey Parameter ...............

741 741

49.2 Two-State Approximation .................... 49.2.1 Relation Between Adiabatic and Diabatic Basis Functions ...... 49.2.2 Coupled Equations and Transition Probabilities in the Common Trajectory Approximation.......................... 49.2.3 Selection Rules for Nonadiabatic Coupling ..........

743

742 742

743

744 745

49.3 Single-Passage Transition Probabilities: Analytical Models ................................ 49.3.1 Crossing and Narrow Avoided Crossing of Potential Energy Curves: The Landau–Zener Model in the Common Trajectory Approximation.......................... 49.3.2 Arbitrary Avoided Crossing and Diverging Potential Energy Curves: The Nikitin Model in the Common Trajectory Approximation.......................... 49.3.3 Beyond the Common Trajectory Approximation.......................... 49.4 Double-Passage Transition Probabilities and Cross Sections ............................... 49.4.1 Mean Transition Probability and the Stückelberg Phase ......... 49.4.2 Approximate Formulae for the Transition Probabilities.... 49.4.3 Integral Cross Sections for a Double-Passage Transition Probability ................ 49.5 Multiple-Passage Transition Probabilities ........................ 49.5.1 Multiple Passage in Atomic Collisions ................... 49.5.2 Multiple Passage in Molecular Collisions ............... References ..................................................

746

746

747 748 749 749 750

751 751 751 751 752

49.1 Basic Definitions 49.1.1 Slow Quasiclassical Collisions Slow collisions of atoms or molecules (neutral or charged) are defined as collisions for which the velocity of the relative motion of colliding particles v is substantially lower than the velocity of valence electrons ve : v/ve
1 .

(49.1)

If ve is estimated as ve ≈ 1 a.u. ≈ 108 cm/s, then (49.1) is fulfilled for medium mass nuclei (∼ 10 amu) up to several keV.

Quasiclassical collisions are those for which the de Broglie wavelength λdB for the relative motion is substantially smaller than the range parameter a of the interaction potential λdB
a .

(49.2)

The two conditions (49.1) and (49.2) define the energy range within which collisions are slow and quasiclassical. For medium mass nuclei, this energy range covers collision energies above room temperature and below hundreds of eV. The paramater a should not be confused with another important parameter L 0 which character-

Part D 49

Adiabatic and

49. Adiabatic and Diabatic Collision Processes at Low Energies

742

Part D

Scattering Theory

Part D 49.1

izes the extent of the interaction region. For instance, for the exchange interaction between two atoms, L 0 corresponds to the distance of closest approach of the colliding particles, while a is the range of the exponential decrease of the interaction. Typically, L 0 noticeably exceeds a.

49.1.2 Adiabatic and Diabatic Electronic States Let r refer to a set of electronic coordinates in a body-fixed frame related to the nuclear framework of a colliding system, and let R refer to a set of nuclear coordinates determining the relative position of nuclei in this system. A configuration of electrons and nuclei in a frame fixed in space is completely determined by r, R, and the set of Euler angles Ω, which relate the body-fixed frame to the space-fixed frame. If the total Hamiltonian of the system is H(r, R, Ω), the stationary state wave function satisfies the equation H(r, R, Ω)ΨE (r, R, Ω) = EΨE (r, R, Ω) . (49.3) The electronic adiabatic Hamiltonian H(r; R) is defined to be the part of H(r, R, Ω) in which the kinetic energy of the nuclei is ignored. The adiabatic electronic functions ψn (r; R) are defined as eigenfunctions of H(r; R) at a fixed nuclear configuration R: H(r; R)ψn (r; R) = Un (R)ψn (r; R) .

(49.4)

The eigenvalues Un (R) are called adiabatic potential energy surfaces (adiabatic PES). In the case of a diatom, the set R collapses into a single coordinate, the internuclear distance R, and the PES become potential energy curves, Un (R). The functions ψn (r; R) depend explicitly on R and implicitly on the Euler angles Ω. The significance of the adiabatic PES is related to the fact that in the limit of very low velocities, a system of nuclei will move across a single PES. In this approximation, called the adiabatic approximation, the function Un (R) plays the part of the potential energy which drives the motion of the nuclei. An electronic diabatic Hamiltonian is defined formally as a part of H, i. e., H0 = H + ∆H. The partitioning of H into H0 and ∆H is dictated by the requirement that the eigenfunctions of H0 , called diabatic electronic functions φn , depend weakly on the configuration R. The physical meaning of this weak dependence is different for different problems. A perfect diabatic basis set φn (r) is R-independent; for practical

purposes one can use a diabatic set which is considered as R-independent within a certain region of the configuration space R. Two basis sets ψn and φn generate the matrices φm |H|φn  = Hmn , φm |H|φn  = Hmn + Dmn , ψm |H|ψn  = Un (R)δmn + Dmn .

(49.5)

The eigenvalues of the matrix Hmn are Un . Dmn is the matrix of dynamic coupling in the diabatic basis, and Dmn is the matrix of dynamic coupling in the adiabatic basis; the former matrix vanishes for a perfect diabatic basis. All the above matrices are, in principle, of infinite order. For low-energy collisions, the use of finite matrices of moderate dimension, will usually suffice. Diabatic PES are defined as the diagonal elements Hnn . The significance of the diabatic PES is that for velocities which are high [but still satisfy (49.1)] the system moves preferentially across diabatic PES, provided that the additional conditions discussed in Sect. 49.3 are fulfilled. For a given finite adiabatic basis ψn (r; R), a perfect diabatic basis φn  (r) can be constructed by diagonalizing the matrix Dnm (R). The two basis sets are related by a unitary transformation
Cnn  (R) φn  (r) . (49.6) ψn (r; R) = n

49.1.3 Nonadiabatic Transitions: The Massey Parameter Deviations from the adiabatic approximation manifest themselves in transitions between different PES which are induced by the dynamic coupling matrix D. At low energies, the transitions usually occur in localized regions of nonadiabatic coupling (NAR). In these regions, the motion of nuclei in different electronic states is coupled, and in general it cannot be interpreted as being driven by a single potential. An important simplifying feature of slow adiabatic collisions is that typically the distance between different NAR is substantially larger than the extents of each NAR. This makes it possible to formulate simple models for the coupling in isolated NAR, and subsequently to incorporate the solution for nonadiabatic coupling into the overall dynamics of the system. For a system of s nuclear degrees of freedom, there are the following possibilities for the behavior of PES within NAR:

Adiabatic and Diabatic Collision Processes at Low Energies

ζ(R) = ωel τnuc = ∆U(R)∆L/~v(R) .

(49.7)

The nonadiabatic coupling is inefficient at those configurations R where ζ(R)  1. If ζ(R) is less than or of the order of unity, the nonadiabatic coupling is efficient, and a change in adiabatic dynamics of nuclear motion is very substantial.

The following relations usually hold for the parameters ∆L, a, L 0 for slow collisions: ∆L
a
L 0 .

(49.8)

When the nonadiabatic coupling is taken into account, the total (electronic and nuclear) wave function ΨE can be represented as a series expansion in ψn or φn (the Euler angles Ω are suppressed for brevity):
ψn (r; R)χnE (R) ΨE (r, R) = n

=




φn (r)κnE (R) .

(49.9)

n

Here χnE (R) and κnE (R) are the functions which have to be found as solutions to the coupled equations formulated in the adiabatic or diabatic electronic basis, repectively [49.1, 2]. In general, different contributions to the first sum in (49.9) can be associated with nonadiabatic transition probabilities between different electronic states. A practical means of calculating functions χnE (R) [or κnE (R)] consists of expanding them over certain basis functions Ξnν (R ), where R denotes all coordinates R except for the interparticle distance R. Writing
Ξnν (R )ξnνE (R) , (49.10) χnE (R) = ν

one arrives at a set of coupled second-order equations for the unknown functions ξnνE (R) (the scattering equations) [49.1]. In the semiclassical approximation, these equations become a set of first-order equations for the amplitudes of the WKB counterparts of ξnνE (R). At the next step of simplification, in the common trajectory approximation, the variable R is changed into the time variable t, the latter being related to R via the classical trajectory R = R(t) [49.2]. In the adiabatic approximation, the total wave function is represented by a single term in the first sum of (49.9): ΨE (r, R) = ψn (r; R)χnE (R) .

(49.11)

49.2 Two-State Approximation 49.2.1 Relation Between Adiabatic and Diabatic Basis Functions In the two-state approximation, the basis of electronic functions consists of two states. In this case, the elements of the matrix C in (49.6) are expressed through a single parameter only, a mixing or rotation angle θ: ψ1 (r; R) = cos θ(R)φ1 (r) + sin θ(R)φ2 (r) ,

ψ2 (r; R) = − sin θ(R)φ1 (r) + cos θ(R)φ2 (r) . (49.12)

The rotation angle, θ(R), is expressed via the diagonal and off-diagonal matrix elements of the adiabatic Hamiltonian H in the diabatic basis φ1 , φ2 : tan 2θ(R) =

2H12 (R) . H11 (R) − H22 (R)

(49.13)

743

Part D 49.2

(i) If two s-dimensional PES correspond to electronic states of different symmetry, they can cross along an (s − 1)-dimensional line. For a system of two atoms, s = 1, and so two potential curves of different symmetry can cross at a point. (ii) If two s-dimensional PES correspond to electronic states of the same symmetry, they can cross along an (s − 2)-dimensional line. For a system of two atoms, s = 1, and so two potential curves of different symmetry cannot cross. If they have a tendency to cross, they will exhibit a pattern which is called an avoided crossing or a pseudocrossing. (iii) If two s-dimensional PES correspond to electronic states of the same symmetry in the presence of spin–orbit coupling, they can cross along an (s − 3)-dimensional line. Statement (ii) applied to a two-atom system is known as the Wigner–Witmer noncrossing rule. The efficiency of the nonadiabatic coupling between two adiabatic electronic states is determined, according to the adiabatic principle of mechanics (both classical and quantum), by the value of the Massey parameter ζ , which represents the product of the electronic transition frequency ωel and the time τnuc that characterizes the rate of change of electronic function due to nuclear motion. Putting ωel ≈ ∆U(R)/~, (∆U is the spacing between any two adiabatic PES), and τnuc = ∆L/v(R), (∆L is a certain range which depends on the type of coupling), we get

49.2 Two-State Approximation

744

Part D

Scattering Theory

Part D 49.2

The eigenvalues of H in terms of Hik are U1,2 (R) = [H11 (R) + H22 (R)] /2 ± ∆U(R)/2 , (49.14)

db1 = H11 (Q)b1 + H12 (Q)b2 , dt db2 = H21 (Q)b1 + H22 (Q)b2 . (49.20) i~ dt Clearly, for a system of two atoms, Q ≡ R. Solutions to (49.19) and (49.20) are equivalent, provided that the initial conditions are matched, and the transition probability is properly defined. For a given trajectory, it is customary to identify the center of the NAR with a value of Q = Qp which corresponds to the real part of the complex-valued coordinate Qc at which two adiabatic PES cross. The crossing conditions in the adiabatic and diabatic representations are i~

where 1/2  2 (R) . ∆U(R) = [H11 (R) − H22 (R)]2 + 4H12 (49.15)

The matrix elements Hik are expressed via the adiabatic potentials and the rotation angle by H11 (R) + H22 (R) = U1 (R) + U2 (R) , H11 (R) − H22 (R) = ∆U(R) cos 2θ(R) , H12 (R) = (1/2)∆U(R) sin 2θ(R) . (49.16)

49.2.2 Coupled Equations and Transition Probabilities in the Common Trajectory Approximation

U1 (Qc ) − U2 (Qc ) = 0 , or

A two-state nonadiabatic wave function Ψ(r, R) can be written as an expansion into either adiabatic or diabatic electronic wave functions: Ψ(r; R) = ψ1 (r; R) α1 (R) + ψ2 (r; R) α2 (R) , (49.17) Ψ(r; R) = φ1 (r)β1 (R) + φ2 (r)β2 (R) , in which the nuclear wave functions satisfy two coupled s-dimensional Schrödinger equations [49.1]. In the common trajectory approximation, the motion of the nuclei is described by the classical trajectory, i. e., by a one-dimensional manifold Q(t) embedded in the s-dimensional manifold R. A section of PES along this one-dimensional manifold determines a set of effective potential energy curves (PEC). In the case of atomic collisions, Q coincides with the interatomic distance R, and the effective PEC are just ordinary PEC. A common trajectory counterpart of (49.17) is Ψ(r, t) = ψ1 [r; Q(t)] a1 (t) + ψ2 [r; Q(t)] a2 (t) , (49.18) Ψ(r; t) = φ1 (r)b1 (t) + φ2 (r)b2 (t) . The adiabatic expansion coefficients ak (t) satisfy the set of equations da1 ˙ = U1 (Q)a1 + iQg(Q)a i~ 2, dt da2 ˙ = − iQg(Q)a i~ 1 + U2 (Q)a2 , dt

where g(Q) = ψ1 |∂/∂Q|ψ2  = dθ/ dQ, and Q = Q(t). The diabatic expansion coefficients bk (t) satisfy the set of equations



2 2 (Qc ) = 0 . H11 (Qc ) − H22 (Qc ) + 4H12

(49.21)

(49.22)

Since Q represents a one-dimensional manifold, the crossing condition (49.22) is satisfied for a complex value of Q = Qs unless H12 = 0. Then, by definition, the location of the NAR centeris identified with Qp through Qp = Re(Qc ) ,

(49.23)

where Qc is that value of Qs which possesses the smallest imaginary part, and Re denotes the real part. For the case when the regions of nonadiabatic coupling are well localized, the function g(Q) possesses a pronounced maximum at (or close to) Qp , the width ∆Qp of which determines the range of the NAR; normally ∆Qp is about Im (Qc ), with Im denoting the imaginary part. The two Eqs. (49.19) decouple on both sides of this maximum. A solution of the equations for the nonadiabatic coupling across an isolated maximum of g(Q) yields the so-called single-passage (or one-way) transition amplitude and transition probability. For this problem, the time t = 0 can be assigned to the maximum point of g[Q(t)]. Assuming that away from t = 0 the decoupling occurs rapidly enough, the nonadiabatic transition probability P12 = |a2 (∞)|2 ,

(49.24)

provided that a solution to (49.19) corresponds to the initial conditions, (49.19)

a1 (−∞) = 1 ,

a2 (−∞) = 0 .

(49.25)

Adiabatic and Diabatic Collision Processes at Low Energies

  tc

    2  U1 [Q(t)]−U2 [Q(t)] dt  , P12 = exp −  Im ~ 

tr

(49.26)

where tc is a root of Q(tc ) = Qc .

(49.27)

Here tr is any real-valued time. Equation (49.26) is valid when the exponent is large, so that P12 is exponentially small. The property of the function g(Q) to pass through a single narrow maximum ensures that the rotation angle away from the maximum tends to constant values, and adiabatic functions in these regions are expressed by certain linear combinations of diabatic functions with constant mixing coefficients. These linear combinations should serve as the initial condition on one side of the coupling region, and as the proper final state on the other, when the problem of a nonadiabatic transition between adiabatic states is treated in the diabatic representation. The same property of the function g(Q) implies that a common trajectory needs to be defined only locally, within a given NAR, and not globally, in the full configuration space.

49.2.3 Selection Rules for Nonadiabatic Coupling In the general case, the coupling between adiabatic states or diabatic states is controlled by certain selection rules. The most detailed selection rules exist for a system of two colliding atoms, since this system possesses a high

symmetry (C∞v or, for identical atoms, D∞h point symmetry in the adiabatic approximation). In the adiabatic representation, the coupling is due to the elements of the matrix D. They fall into two different categories: those proportional to the radial nuclear velocity (coupling by radial motion or radial coupling), and those proportional to the angular velocity of rotation of the molecular axis (coupling by rotational motion or Coriolis coupling). In a diabatic representation, provided that the effect of the D matrix is neglected, the coupling is due to the parts of the interaction potential neglected in the definition of the diabatic Hamiltonian H0 . In typical cases, these parts are the electrostatic interaction between different electronic states constructed as certain electronic configurations (H0 corresponds to a self-consistent field Hamiltonian); spin–orbit interaction (H0 corresponds to a nonrelativistic Hamiltonian); hyperfine interaction (H0 ignores the magnetic interaction of electronic and nuclear spins as well as the electrostatic interaction between electrons and nuclear quadrupole moments). The selection rules for the above interactions in the case of two atoms are listed in Table 49.1 for two conventional nomenclatures for molecular states: Hund’s case (σ) (a), 2S+1 Λ(σ) w and Hund’s case (c), Ωw [49.3]. For molecular systems with more than two nuclei, the selection rules cannot be put in a detailed form since, in general, the symmetry of the system is quite low. For the important case of three atoms, a general configuration is planar (Cs symmetry); particular configurations correspond to an isosceles triangle if two atoms are identical (C2v symmetry), to an equilateral triangle for three identical atoms (D3h symmetry) or to a linear configuration. For the last case, the selection rules are the same as for a system of two atoms.

Table 49.1 Selection rules for the coupling between diabatic and adiabatic states of a diatomic quasimolecule (w = g, u;

σ = +, −)

Interaction

2S+1 Λ (σ) w

Configuration interaction (electrostatic)

∆Λ = 0, ∆S = 0 g u, + −

∆Ω = 0 g u, +

−

Spin–orbit interaction

∆Λ = 0, ±1, ∆S = 0, ±1 g u, + −

∆Ω = 0 g u, +

−

Radial motion

∆Λ = 0, ∆S = 0 g u, + −

∆Ω = 0 g u, +

−

Rotational motion

∆Λ = ±1, ∆S = 0 g u, + −

∆Ω = ±1 g u, +

−

Hyperfine interaction

∆Λ = 0, ±1, ∆S = 0, ±1 g u, + −

∆Ω = 0, ±1 g u, + −

nomenclature

Ωw(σ) nomenclature

745

Part D 49.2

In the limit of almost adiabatic conditions where P12 is very small, the following equation holds [49.3]:

49.2 Two-State Approximation

746

Part D

Scattering Theory

Part D 49.3

The selection rules for the dynamic coupling between adiabatic states classified according to the irreducible representations of the Cs and C2v groups are listed in Table 49.2. In this table, z and y refer to two modes of the relative nuclear motion in the system plane, Rz and R y refer to two rotations about principal axes of inertia lying in the system plane, and Rx refers to a rotation about the principal axis of inertia perpendicular to the system plane.

Table 49.2 Selection rules for dynamic coupling between

adiabatic states of a system of three atoms A


Cs A A

C2v A1 B1 A2 B2

A1 z y, Rx Rz Ry

A

B1 y, Rx z Ry Rz

A2 Rz Ry z y, Rx

B2 Ry Rz y, Rx z

49.3 Single-Passage Transition Probabilities: Analytical Models with ∆F = |F1 − F2 |. The common trajectory is assumed to be a linear function of t,

49.3.1 Crossing and Narrow Avoided Crossing of Potential Energy Curves: The Landau–Zener Model in the Common Trajectory Approximation

Q = Qp + vp t ,

The Landau–Zener model applies to a situation when the effective adiabatic PEC cross or show a narrow avoided crossing. The latter is defined by the condition that the spacing between adiabatic PEC within a NAR is much smaller than the spacing between adiabatic PEC away from the NAR. The cases of crossing and narrow avoided crossing of adiabatic PEC can be considered within a unified model since a narrow avoided crossing of adiabatic PEC corresponds to a crossing of diabatic PEC. Therefore, in both cases, one considers the crossing of zero-order PES (adiabatic or diabatic) and the interaction between them (dynamic or static). However, the definition of transition probability is different for crossing and avoided crossing. In the framework of the Landau–Zener model [49.4– 6], the two zero-order PEC which cross at a point Qp along a trajectory Q(t) are approximated by functions linear in ∆Q = Q − Qp , and the off-diagonal matrix element is assumed to be a constant. For the avoided crossing adiabatic PEC (crossing diabatic PEC), the matrix H jk within a NAR is approximated as H11 (Q) = E 0 − F1 (Q − Qp ) , H22 (Q) = E 0 − F2 (Q − Qp ) , H12 (Q) = V = constant ,

(49.28)

from which the spacing between adiabatic PEC is

1/2 ∆U = (∆F)2 (Q − Qp )2 + 4|V |2 , (49.29)

(49.30)

where vp is the velocity of Q-motion at point Qp . For this model, adiabatic wave functions on both sides of the nonadiabaticity region (in the limits −∞ < t < +∞) coincide with diabatic functions, but their ordering is reversed. Explicitly, ψ1 = φ1 ; ψ1 = φ2 ;

ψ2 = φ2 for t → −∞ , ψ2 = −φ1 for t → +∞ .

(49.31)

The transition probability between pseudocrossing adiabatic curves for the Hamiltonian (49.28) and the trajectory (49.30) is given by the Landau–Zener formula   psc LZ = exp − 2πζ LZ ; P12 = P12 ζ LZ = V 2 /(~∆Fvp ) ,

(49.32)

where ζ LZ is the appropriate Massey parameter at the pseudocrossing point. Note that for the LZ model, the single-passage transition probability depends on one dimensionless parameter ζ LZ . A remarkable property of LZ is the Landau–Zener model is that the probability P12 given by (49.26) for an arbitrary value of the exponent, and not only for large ones when the probability is very low. With a change of the velocity from very low to very high values, the transition probability varies from zero to unity. In the near-adiabatic limit, ζ LZ  1, the nuclei preferentially move across single adiabatic PEC, while in the sudden limit, ζ LZ
1, they move across single diabatic PEC. It is the latter property of the LZ transition probability that allows one to interpret diabatic energies H11 and H22 as the potentials which drive the nuclear motion at high velocities.

Adiabatic and Diabatic Collision Processes at Low Energies

and

2V/∆F
a ,

(49.33)

1/2 
a. 2 ~vp /∆F

(49.34)

Clearly, the range parameter a does not enter the LZ formula since it controls the behavior of adiabatic curves away from the crossing point. The constant velocity approximation (49.30) imposes yet another condition: V
µvp2 /2 .

(49.35)

The actual application of (49.32) requires the specification of V and vp for each particular trajectory Q(t). For the case of avoided crossing between two potential curves of a diatom, V does not depend on the trajectory and represents, according to Table 49.2, the matrix element of the electrostatic interaction, spin–orbit interaction or hyperfine interaction. For crossing adiabatic PEC, the Landau–Zener model assumes the following approximation for adiabatic potentials and the dynamic coupling: U1 (Q) = E 0 − F1 (Q − Qc ) , U1 (Q) = E 0 − F2 (Q − Qc ) , D12 (Q) = D = constant ,

psc

(49.36)

(49.37)

provided that D and vc in the crossing situation are replaced by V and vp in the pseudocrossing situation. Conditions (49.33) and (49.34) applied to the case of the dynamic coupling often imply that this coupling is weak [49.2]. Therefore, (49.37) yields c P12 =

2πD2

~∆Fvc

.

49.3.2 Arbitrary Avoided Crossing and Diverging Potential Energy Curves: The Nikitin Model in the Common Trajectory Approximation The restrictions of narrow avoided crossing [(49.33) and (49.34) for the LZ model] are relaxed in a more general model suggested by Nikitin [49.7]. This model uses a more flexible exponential parametrization, instead of the linear parametrization for diabatic matrix elements (49.36). In a diabatic basis, the model is formulated with the Hamiltonian H11 (Q) =U0 (Q)−∆E/2+(A/2)cos 2ϑ exp(−αQ) , H22 (Q) =U0 (Q)+∆E/2−(A/2)cos 2ϑ exp(−αQ) , H12 (Q) = (A/2) sin 2ϑ exp(−αQ) . (49.39) The spacing between adiabatic PEC is     ∆U = ∆E 1 − 2 cos 2ϑ exp − α Q − Qp   1/2 + exp − 2α Q − Qp , (49.40) where Qp is introduced instead of A via (49.21) and (49.23). At the center of an NAR, where Q = Qp , the spacing between adiabatic PEC, ∆Up =∆U(Qp ), is ∆Up = 2∆E sin ϑ .

with the trajectory parametrization given by (49.30) where vp is replaced by vc . Since the ordering of adiabatic PEC for crossing and pseudocrossing is reversed on one side of a NAR, the following relation exists bec , tween transition probabilities for the crossing case P12 and the survival probability for the pseudocrossing case psc 1 − P12 : c P12 = 1 − P12 ,

Usually, the matrix element D is related to the Coriolis coupling, and it is proportional to the angular velocity of rotation of the molecular frame at the crossing point Qc .

(49.38)

(49.41)

The common trajectory within the NAR is taken in a form identical to (49.30) in which vp is now the velocity of Q motion at the center of the coupling region Qp . For this model, adiabatic wave functions coincide with diabatic functions before entering the coupling region (in the limit α(Q − Qp )  1), but after exiting the coupling region [in the limit α(Q − Qp )
−1] they are linear combinations of the diabatic functions ψ1 (r; Q) = φ1 (r) , ψ2 (r; Q) = φ2 (r) , for α(Q − Qp )  1 ; ψ1 = φ1 cos ϑ + φ2 sin ϑ , ψ2 = − φ1 sin ϑ + φ2 cos ϑ , for α(Q − Qp )
−1 .

(49.42)

The latter equation identifies the parameter ϑ that enters into the definition of the diabatic Hamiltonian in (49.39)

747

Part D 49.3

The region of applicability of the Landau–Zener formula is determined by the condition that the extension of the region of nonadiabatic interaction ∆Q should be small compared with the range a over which potential curves deviate substantially from linear functions. The condition ∆Q
a actually implies the two conditions [49.2]

49.3 Single-Passage Transition Probabilities: Analytical Models

748

Part D

Scattering Theory

Part D 49.3

with the asymptotic value of the mixing angle θ (49.12) for α(Q − Qp )
−1. The transition probability P12 between adiabatic PEC for the Hamiltonian (49.39) and the trajectory (49.30) is N P12 = exp(−πζp )

sinh(πζ − πζp ) , sinh(πζ)

where ζ = ∆E/(~αvp ) and ζp = ζ With the change in velocity from very low to very high values, the transition probability varies from zero to cos2 ϑ. As ϑ changes from very small values to π, the pattern of adiabatic potential curves changes from narrow to wide pseudocrossing and ultimately to strong divergence. Since the single-passage transition probability (49.43) depends on two parameters, the Nikitin model is more versatile than the Landau–Zener one. In three limiting cases, ϑ
1, ζ  1, ϑ = π/4, ζ arbitrary, and ϑ arbitrary, ζ = 0, (49.43) may be simplified. In the first case, the diabatic Hamiltonian (49.39) becomes the Landau–Zener Hamiltonian (49.28). Also, (49.43) reduces to a single exponential which gives the LZ transition probability, (49.44)

with ζp identical to ζ LZ . The two conditions ϑ
1 and ζ  1 are equivalent to the two conditions (49.33) and (49.34). In the second case (ϑ = π/4), the diabatic Hamitonian reads H11 (R) = E 0 − ∆E/2 , H22 (R) = E 0 + ∆E/2 , H12 (R) = (A/2) exp(−αQ) .

(49.45)

The transition probability in this case is given by the Rosen–Zener–Demkov formula [49.8, 9] RZD P12 =

exp(−πζ) , ζ = ∆E/(~vp α) . (49.46) 1 + exp(−πζ)

In the third case (ζ = 0), also called the resonance case since ∆E = 0, the transition probability reads Res = cos2 ϑ . P12

49.3.3 Beyond the Common Trajectory Approximation

(49.43)

sin2 ϑ.

N LZ P12 = P12 = exp(−2πζp ) ,

ζ  1, while in the sudden limit, ζ
1, they move across both diabatic PEC, unless the condition of narrow avoided crossing, ϑ
1, is fulfilled.

(49.47)

Equation (49.47) is a particular example of transitions between initially degenerate states. This kind of transition occurs in the recoupling of angular momenta in collisions of atoms possessing nonzero electronic angular momentum [49.10]. For the general case, the nuclei preferentially move across single adiabatic PEC in the near-adiabatic limit,

The common trajectory approximation is valid when the spacing between adiabatic PEC within an NAR is small compared to the local kinetic energy of the nuclei. The relaxation of this restriction is not unambiguous since one should pass from a one-dimensional manifold (time as a progress variable) to a multi-dimensional coordinate (configuration space manifold). Only if the latter is one-dimensional (a single coordinate as a progress variable, as is the case for atom-atom collisions), one can suggest a generalization of the common trajectory transition probability. We consider this case, taking R to be such a single coordinate, and assume that the quantum motion across the adiabatic PEC satisfies standard quasiclassical conditions [49.3]. For a two-state problem with adiabatic potentials U1 (R) and U2 (R), the general condition of the common trajectory apporximation reads    1 E − U1 (Rp ) + U2 (Rp )  U1 (Rp ) − U2 (Rp ) , 2 (49.48)

where E is the total (conserved) energy and Rp is the coordinate of the NAR center. The quantum generalization of the expression for the transition probability in the near-adiabatic condition, (49.26), is given by the original Landau formula [49.4, 5]   R c  2      2µ E − U1 (R) dR P12 = exp − Im  ~ R   Rc      2µ E − U2 (R) dR  , (49.49) −  R

where µ is the reduced mass of the colliding atoms, Rc is the complex-valued coordinate of the crossing of U1 (R) and U2 (R) and R is any value of the coordinate in the classically accessible region of motion of the nuclei. The nonadiabatic transition is localized in the region of width ∆R = Im(Rc ) centered at Rp = Re(Rc ). Equation (49.49) becomes the common trajectory equation (49.26) under the condition (49.48). The quantum generalization of the LZ transition probability can not be represented by an exact analytical expression though it is known that it depends on two

Adiabatic and Diabatic Collision Processes at Low Energies

LZ P12

1/2     = exp−2πζ LZ  ,  −2  1 LZ 2 1+ 1+p 160 p (ζ ) +0.7 



2

(49.50)

where p =

µvp2

∆F . √ 2V 2 |F1 F2 |

Equation (49.50) becomes the common trajectory (49.32) under the condition p  1; it turns out that the latter condition may be less restrictive than the general condition (49.48).

The quantum generalization of the Nikitin transition probability is possible provided U0 (R) in (49.39) is given by an exponential function, U0 (R) ≈ exp(−αR). The transition probability reads [49.11]: N P12 = exp(−πδp )

sinh(πδ − πδp ) sinh(πδ)

(49.51)

and depends on three parameters of the model (and not two as is the case for the common trajectory approximation). These parameters enter into δp and δ through complicated contour integrals. If the general condition of the common trajectory approximation, (49.48) , is fulfilled, δp and δ reduce to ζp and ζ so that (49.51) becomes (49.43). More discussions of two-state models within and beyond the common trajectory approximation can be found elsewhere [49.2, 11–14].

49.4 Double-Passage Transition Probabilities and Cross Sections 49.4.1 Mean Transition Probability and the Stückelberg Phase

colliding atoms is conserved, and we have (0) ∆Φ12 /2 =

In the case of an atomic collision, the set R shrinks into a single coordinate R. If there is only one NAR over the whole range of R, the colliding system traverses it twice, as the atoms approach and then recede. In this case, there are two different paths between the center of the NAR, Rp , and the turning points Rt1 and Rt2 on the adiabatic potential curves U1 (R) and U2 (R). The double-passage transition probability P12 is expressed via the single-passage transition probability P12 , the single-passage survival probability 1 − P12 , and the Stückelberg interference term cos ∆Φ12 [49.15], P12 = 2P12 (1 − P12 )(1 − cos ∆Φ12 ) = 4P12 (1 − P12 ) sin2 (∆Φ12 /2) .

(49.52)

The Stückelberg phase ∆Φ12 /2 is expressed as the (0) phase difference ∆Φ12 /2 which is accumulated during the motion of a diatom from the center of the NAR to the turning points, together with an additional phase φ12 by (0) /2 + φ12 . ∆Φ12 /2 = ∆Φ12

Rt1  1/2 2µE − ~2 ( + 1)/R2 − 2µU1 (r) dR/~ Rp

Rt2  1/2 2µE − ~2 ( + 1)/R2 − 2µU2 (r) − dR/~ , Rp

(49.54)

where Rt1 and Rt2 are the turning points for adiabatic motion on potential curves U1 and U2 , E is the total (0) energy. Once Rp is chosen, ∆Φ12 /2 is well-defined and is independent of the dynamic details of a nonadiabatic transition. On the other hand, P12 and φ12 do depend on these details. In particular, for the models discussed in Sect. 49.3, the velocity vp that enters into the Massey parameter, is "1/2  1/2 ! 2 ~2 ( + 1) , E − Up − vp = µ 2µRp2

(49.53)

(49.55)

Generally, (49.53) is valid provided ∆Φ12 /2  1. For transitions between electronic states of the same axial symmetry, the relative angular momentum  of the

where Up ≈ U1 (Rp ) ≈ U2 (Rp ). As a function of E and , the double-passage transition probability is symmetric with respect to the initial and final states.

749

Part D 49.4

parameters (and not on one, as in the case for the common trajectory approximation) [49.2]. A recommended approximate expression for E > E 0 reads [49.11]

49.4 Double-Passage Transition Probabilities and Cross Sections

750

Part D

Scattering Theory

Part D 49.4

In many applications, one can use the mean transition probability P12 , which is obtained from P12 by averaging over several oscillations: P12  = 2P12 (1 − P12 ) .

(49.56)

The important limiting cases of the double-passage Nikitin model are: (i) Double-passage Landau–Zener–Stückelberg equation,     LZS = 4 exp −2πζ LZ 1 − exp −2πζ LZ P12 # $ (0) LZS × sin2 ∆Φ12 /2 + φ12 (49.57) ,

49.4.2 Approximate Formulae for the Transition Probabilities Several approximate formulae are available for P12  in the case where H12 depends on time in a bell-shaped manner, and ∆H = H11 − H22 can be represented as ~ω + ∆V , with ∆V also having a bell-shaped form. Define  +∞ 1 v= H12 (t) dt ; ~ −∞

 +∞ 1 H12 (t) exp(iωt) dt , w0 = ~

(0) where ζ LZ is given by (49.32), ∆Φ12 /2 by (49.54) LZS and the expression for φ12 is available [49.2]. When & % LZS passes through ζ LZ changes from zero to infinity, P12 LZS decreases a maximum, P12 max = 1/2, and φ12 from π/4 to zero. (ii) Double-passage Rosen–Zener–Demkov equation:   (0) RZD sin2 ∆Φ12 /2 + φ12 RZD P12 = (49.58) . cosh2 (πζ/2) RZD is given by (49.46) and the expression for φ12

where ζ is available [49.2]. Under certain conditions [49.2, 9], the Stückelberg phase in (49.58) can be identified with the phase accumulated during the motion of a diatom from infinitely large distance to the turning points. (iii) Double-passage equation for a resonance process (∆E =0):     res P12 = sin2 2ϑ sin2 ∆Φ12 /2 . (49.59) The general resonance case (zero energy change, ∆E = 0) is also called an accidental resonance. For the accidental resonance, the diagonal diabatic matrix elements are not equal to each other. A particular case of an accidental resonance is a symmetric resonance, for which the diabatic matrix elements are the same. For the Nikitin model, symmetric resonance corresponds to ϑ = π/4, and (49.59) reads   symm P12 = sin2 ∆Φ12 /2 . (49.60) Equation (49.60) also follows from (49.58) in the limit ζ → 0. Actually, (49.60) is valid for any symmetric resonance case [not necessarily for the model Hamiltonians (49.39) and (49.45)] and for the arbitrary values of the phase ∆Φ12 /2. This phase can be identified (0) with ∆Φ12 /2 from (49.54) provided Rp is taken to be infinitely large.

−∞

w=

 +∞   t 1 H12 (t) exp (i/~) ∆H(t) dt dt , ~ −∞

 u=

1 2~

0

+∞ ∆V12 (t) dt ,

(49.61)

−∞

and t S(t) = (1/~)

 1/2 2 ∆H 2 (t) + 4H12 (t) dt . (49.62)

0

Then the various approximate formulae, as suggested by different authors [49.2], read P12 ∼ = (w0 /v)2 sin2 v ,

(49.63)

P12 ∼ = sin2 w ,

(49.64)

P12 ∼ =

w20

sin2

 (u 20 + w20 ) ,

u 20 + w20  +∞ 2     ∼  P12 =  H12 (t) exp[iS(t)] dt/~ ,  

(49.65)

(49.66)

−∞

sin2 Sc P12 ∼ . = cosh2 Sc

(49.67)

In (49.67), Sc and Sc are the real and imaginary parts of the complex quantity Sc = S(tc ) from (49.62). The complex-valued time tc is found from 2 (tc ) = 0 , ∆H 2 (tc ) + 4H12

(49.68)

under the condition that tc possesses the smallest imaginary part of all roots of this equation.

Adiabatic and Diabatic Collision Processes at Low Energies

49.5 Multiple-Passage Transition Probabilities

constant velocity v and impact parameter b = ~/µv: 1/2  . (49.70) R(t) = b2 + v2 t 2

The quasiclassical inelastic integral cross section σi f for the transition i → f is related to Pi f by

For instance, for the Landau–Zener model m = µvRp /~, and the cross section depends on one dimensionless parameter γ = 2πV 2 /(∆F ~v) according to

σi f

π = µE i

∞ Pi f  d ,

(49.69)

0

where E i is the initial collision energy, E i = E − Ui (∞). The cross section defined by (49.69) typically shows the following qualitative dependence on the collision velocity: σi f increases rapidly with E i at low energies, reaches a maximum and then slowly falls off at high energies. The position of the maximum roughly corresponds to the energy E i = E i∗ at which the relevant Massey parameter at the NAR center, ζ(Rp ), is of the order of unity. The conditions E i < E i∗ and E i > E i∗ correspond to the near-adiabatic and strongly nonadiabatic (also called diabatic) regimes, respectively. In calculating σi f , one usually neglects the Stückelberg oscillating term and sets the upper limit in the integral in (49.69) to a value  = m beyond which the integrand begins to fall off quickly. Yet another simplification is possible, in the framework of the impact parameter approximation, when the relative motion of atoms is described by a rectilinear trajectory R(t) with

1 √  √  exp(−γ/ x ) 1− exp(−γ/ x ) dx

σ12 (γ) = 2πRp2

0

  = 4πRp2 E 3 (γ) − E 3 (2γ) ,

(49.71)

where E 3 (z) is the exponential integral. For the symmetric resonance, the cross section reads " ! ∞ ∆ΦiRes π f (b, v) 2 b db ≈ b2m (v) , σi f (v) = 2π sin 2 2 0

(49.72)

where bm is found from the Firsov criterion [49.2]:   ∞   ∆ΦiRes Ui (R) − U f (R) 2 f bm , v ' = dR = . 2 2 2 π ~v R − bm b∗

(49.73)

The cross section in (49.71) first increases and then decreases with the collision velocity v, while that in (49.72) slowly decreases with v.

49.5 Multiple-Passage Transition Probabilities 49.5.1 Multiple Passage in Atomic Collisions In the case of atomic collisions, there is only one nuclear coordinate R. If there exist several NAR on the R-axis, those which are classically accessible (for given total energy E and total angular momentum J) can be traversed several times. In the semiclassical approximation [49.16], the multiple-passage transition amplitude Ai f for a given transition between inital state i and final state f can be calculated as a sum of transition amplitudes AiLf , over all possible classical ways L which connect these states, and which run along a one-dimensional manifold R:
AiLf , (49.74) Ai f = L

where each AiLf can be expressed through the probability PiLf and the phase ΦiLf by [49.13] 1/2    AiLf = PiLf exp iΦiLf . (49.75)

The net transition probability is then Pi f = |Ai f |2 (49.76)

     1/2 PiLf PiLf = PiLf + cos ΦiLf − ΦiLf . L,L

L

The first sum runs over all different paths, and the second (primed) over all different pairs of paths. The primed sum usually yields a contribution to the transition probability which oscillates rapidly with a change of the parameters entering into Pi f (i. e., E and J) and represents a multiple-passage counterpart to the Stückelberg oscillations. If the Stückelberg oscillations are neglected, Pi f is equivalent to a mean transition probability Pi f :
PiLf . (49.77) Pi f  = L

For one NAR, there are two equivalent paths, and Pi(1) f = P (2) i f = Pi f (1 − Pi f ). Then (49.77) yields (49.56).

Part D 49.5

49.4.3 Integral Cross Sections for a Double-Passage Transition Probability

751

752

Part D

Scattering Theory

Part D 49

49.5.2 Multiple Passage in Molecular Collisions For molecular collsions, (49.74) and (49.75) apply as well. However, the manifold of R to which a trajectory Q(t) belongs now comprises 3N − 5 (for a linear arrangement of nuclei) or 3N − 6 (for a nonlinear arrangement) degrees of freedom, where N is the number of atoms in the system. The approximation (49.77) is called, in the context of inelastic molecular collisions, the surface-hopping approximation [49.17, 18]. Each time a trajectory reaches an NAR, it bifurcates, and the system makes a hop from one PES to another with a certain probability. Keeping track of all the bifurcations and associated probabilities, one calculates PiLf along a path L made up of different portions of trajectories running across different PES. Because of the complicated sequence of nonadiabatic events leading from the initial state to the final state, each PiLf is a complicated function of different single passage transition probabilities Pnm , and survival probabilities 1 − Pnm . Even if

all Pnm are known in analytical form, the calculation of Pi f  requires numerical computations to keep track of individual nonadiabatic events [49.17]. The manifold R can be reduced in size if one treats other degrees of freedom, besides electronic ones, on the same footing. In this way one introduces adiabatic vibronic (vibrational + electronic) states and adiabatic vibronic PES, and considers nonadiabatic transitions between them [49.19]. In the vibronic representation, the formal theory remains the same; however its implementation is more difficult since there are many more possibilities for trajectory branching. Finally, under certain conditions, one can use a fully adiabatic description of all degrees of freedom save one – the intermolecular distance R. This approach provides a basis for the statistical adiabatic channel model (SACM) of unimolecular reactions [49.20] where the receding fragments are scattered adiabatically in the exit channels after leaving the region of a statistical complex. For a latest review of the theory of molecular nonadiabatic dynamics, see [49.21] and papers in [49.22].

References 49.1

49.2 49.3 49.4 49.5 49.6 49.7 49.8 49.9 49.10

49.11

49.12 49.13

R. B. Bernstein (Ed.): Atom-Molecule Collision Theory: A Guide for the Experimentalist (Plenum, New York 1979) E. E. Nikitin, S. Ya. Unamskii: Theory of Slow Atomic Collisions (Springer, Berlin, Heidelberg 1984) L. D. Landau, E. M. Lifshitz: Quantum Mechanics (Pergamon, Oxford 1977) L. D. Landau: Phys. Z. Sowjetunion 1, 88 (1932) L. D. Landau: Phys. Z. Sowjetunion 2, 46 (1932) C. Zener: Proc. Roy. Soc. 137, 396 (1932) E. E. Nikitin: Discuss. Faraday Soc. 33, 14 (1962) N. Rosen, C. Zener: Phys. Rev. 40, 502 (1932) Yu. N. Demkov: Sov. Phys. JETP 18, 138 (1964) E. I. Dashevskaya, E. E. Nikitin: Quasiclassical approximation in the theory of scattering of polarized atoms. In: Atomic Physics Methods in Modern Research, Lecture Notes in Physics, Vol. 499, ed. by K. Jungmann, J. Kowalski, I. Reinhard, F. Träger (Springer, Berlin, Heidelberg 1997) p. 185 H. Nakamura: Nonadiabatic Transition: Concepts, Basic Theories and Applications (World Scientific, Singapore 2002) M. S. Child: Semiclassical Mechanics with Molecular Applications (Clarendon, Oxford 1994) S. F. C. O’Rourke, B. S. Nesbitt, D. S. F. Crothers: Adv. Chem. Phys. 103, 217 (1998)

49.14

49.15 49.16 49.17 49.18

49.19 49.20

49.21

49.22

E. S. Medvedev, V. I. Osherov: Radiationless Transitions in Polyatomic Molecules (Springer, Berlin, Heidelberg 1994) E. C. G. Stückelberg: Helv. Phys. Acta 5, 369 (1932) W. H. Miller: Adv. Chem. Phys. 30, 77 (1975) S. Chapman: Adv. Chem. Phys. 82, 423 (1992) J. C. Tully: Nonadiabatic dynamics. In: Modern Methods for Multidimensional Dynamics Computations in Chemistry, ed. by D. L. Thompson (World Scientific, Singapore 1998) p. 34 V. Sidis: Adv. At. Opt. Phys. 26, 161 (1990) M. Quack, J. Troe: Statistical adiabatic channel models. In: Encyclopedia of Computational Chemistry, Vol. 4, ed. by P. v. R. Schleyer, N. L. Allinger, T. Clark, J. Gasteiger, P. A. Kollman, H. F. Schaefer III, P. R. Schreiner (Wiley, Chichester 1998) p. 2708 A. W. Jasper, B. K. Kendrick, C. A. Mead, D. G. Truhlar: Non-Born–Oppenheimer chemistry: Potential surfaces, couplings, and dynamics. In: Modern Trends in Chemical Reaction Dynamics: Experiment and Theory (Part I), ed. by X. Yang, K. Lui (World Scientific, Singapore 2004) p. 329 A. Lagana, G. Lendvay (Eds.): Theory of Chemical Reaction Dynamics (Kluwer, Dordrecht 2004)

753

Ion–Atom and 50. Ion–Atom and Atom–Atom Collisions

Low energy collisions are treated in Chapt. 49. Charge transfer reactions are treated in Chapt. 51. We will emphasize therefore excitation and ionization transitions, and discuss charge transfer only as it is interrelated with these transitions. The description of these heavy particle collisions has many features in common with electron and positron collisions with atoms and ions (Chapts. 47 and 48) and this chapter should be studied in parallel with those. There are also chapters dealing with special phenomena in ion– atom collisions: excitation at high collision energies and the Thomas peak (Chapt. 57), electron emission in high energy ion–atom collisions (Chapt. 53), and alignment and orientation (Chapt. 46). Other chapters deal with certain specific theoretical methods: continuum distorted wave (CDW) approximations (Chapt. 52), the binary encounter approximation (Chapt. 56), and classical trajectory Monte Carlo (CTMC) techniques (Chapt. 58). The emphasis of the present chapter is coupled-states calculations of excitation and ionization. There are several review articles and monographs [50.1, 2]. The collisions considered here involve a projectile ion or atom and a target atom. The collision kinematics can be described in the lab frame, where the target atom is assumed to be initially at rest and the collision energy is the kinetic energy of the projectile when it is far from the target prior to the collision, or in the center of mass frame. The primary quantities of interest are the cross sections for producing various final states of

50.1 Treatment of Heavy Particle Motion ...... 754 50.2 Independent-Particle Models Versus Many-Electron Treatments ......... 755 50.3 Analytical Approximations Versus Numerical Calculations ............... 50.3.1 Single-Centered Expansion ........ 50.3.2 Two-Centered Expansion ........... 50.3.3 One-and-a-Half Centered Expansion ................................

756 757 758 758

50.4 Description of the Ionization Continuum 758 References .................................................. 759

the system for given initial states of the target and projectile. The total cross sections depend on the initial and final quantum state of the target and projectile and on the collision energy. Let A denote the projectile atom or ion with ionic charge q, and B denote the target atom. Let A∗ and B ∗ denote excited states. Some examples of processes for which the cross section is of interest are Aq+ + B →    (Aq+ )∗ + B ,     q+ ∗   A + B , Aq+ + B + + e− ,      A(q+1)+ + B + e− ,     A(q−1)+ + B + ,

projectile excitation target excitation target single ionization projectile single ionization single e− charge transfer .

For a multi-electron collision system, combinations of the above quantities are possible. A few representative examples are Aq+ + B →    Aq+ + B ++ + 2 e− ,     q+ + ∗ −    A + (B ) + e ,

target double ionization target excitation-

ionization    (q−1)+ + ∗   + (B ) , transfer-excitation A     A(q−1)+ + B ++ + e− , transfer-ionization .

Part D 50

This chapter summarizes the principal features of theoretical treatments of ion–atom and atom–atom collisions. This is a broad topic and the goal here is a general overview that introduces the main concepts, terminology, and methods in the field. Attention will focus on intermediate and high collision velocities, for which the relative velocity between the projectile and target is on the order of, or larger than, the orbital speed of the electrons active in the transition.

754

Part D

Scattering Theory

Part D 50.1

For each of these processes, the projectile and/or target can be initially in excited states. A class of ion–atom collisions that has received much theoretical attention because of the relative simplicity is the one where the projectile ion is initially bare, with charge q = Z p , the projectile nuclear charge. One can also consider various differential cross sections: differential in the projectile scattering angle, in the energy and angle of emission of ionized electrons, in recoil momentum of the target, etc.

Theoretical calculations of these cross sections can be classified according to the approximations and/or methods used. We will discuss four such classifications: treatment of the heavy particle motion, independent particle model (IPM) versus inclusion of correlation in multi-electron systems, analytical approximations (PWBA, SCA, Glauber, etc.) versus numerical methods (such as coupled-states) for obtaining cross sections, and treatments of the ionization continuum.

50.1 Treatment of Heavy Particle Motion Only at very low energies must the motion of the nuclei be described quantum mechanically. At the intermediate and high collision velocities considered here the semiclassical approximation, in which the motion of the nuclei can be described classically and only the electrons need be described by quantum mechanical wave functions, is accurate [50.3]. The projectile nucleus then moves on a predetermined classical path. At high collision energies this path can often be taken to be a straight line path with constant speed. At somewhat lower energies the deflection and change in speed of the projectile due to the projectile–target interaction is often incorporated [50.4–6]. The Coulomb trajectory due to the nucleus–nucleus interaction can be used, and the screening effects of the projectile and target electrons can be included. For a bare positive ion projectile, the Coulomb trajectory effects increase the distance of closest approach for a given impact parameter and reduce the projectile speed in the interaction region. Projectile trajectory effects are particularly strong for projectiles less massive than protons (positive or negative muons [50.7], for example). The semiclassical approach with trajectory effects has even been used for electron impact excitation and ionization [50.7]. Coulomb trajectories are also important when small impact parameter collisions are considered. The recoil of the target nucleus can also be treated classically and this recoil can affect the cross sections [50.8–10]. In the semiclassical approximation, the vector R(b, t) that locates the projectile relative to the target nucleus is a function of the impact parameter b and time t. The specific functional dependence is determined by the trajectory being used. For a straight line, constant velocity v path, R = b + vt. For the case of a bare projectile ion and a one-electron target, the time-dependent

projectile–target interaction is given by V(b, t) =

−Z P e2 Z P Z T e2 + , |r − R(b, t)| R(b, t)

(50.1)

where Z P and Z T are the projectile and target nuclear charges and r is the position vector of the electron relative to the target nucleus. If the collision calculation starting from (50.1) is done exactly the nuclear repulsion term Z P Z T e2 /R(b, t) does not involve the electronic coordinates and makes no contribution to total cross sections for anything other than elastic scattering. This term makes a nonzero contribution in an approximate calculation, such as a first-order perturbation theory calculation with nonorthogonal initial and final states, and this is a defect of such calculations. A deficiency of the semiclassical approximation as described above is that, since a predetermined classical path is used, there is no coupling between the energy and momentum given to the target electron and that lost by the projectile. A simple improvement, in cases where the energy lost by the projectile when the target transition occurs is an appreciable fraction of its total energy, is to use some average of the projectile’s initial and final speeds as the asymptotic projectile speed. This results in a projectile trajectory that depends on the cross section being calculated. This lack of coupling between the projectile motion and the states of the electrons can be a particular deficiency when cross sections differential in the scattering angle Θ of the projectile are computed. If the collision energy is large and the projectile is scattered primarily by the static potential of the target, a classical treatment of the scattering can be used to relate b to Θ. Even when straight line, constant speed projectile paths are used to calculate the transition probabilities, differential cross sections can be extracted by relating b to Θ. At lower energies, where the de Broglie wavelength of

Ion–Atom and Atom–Atom Collisions

tered by up to 180◦ as b is decreased to zero. But the projectile can also be scattered by a close interaction with a target electron, and the kinematics then can be quite different. Such an interaction has a low probability for scattering the projectile to a large angle. Classically, from energy and momentum considerations, the maximum angle through which a proton can be scattered by an electron initially at rest is about 0.5 mrads; and if the proton is scattered through 0.5 mrads, the electron acquires large energy and momentum. The combined effects of both projectile scattering mechanisms are difficult to treat in a semiclassical model [50.16].

50.2 Independent-Particle Models Versus Many-Electron Treatments In the semiclassical impact parameter method, as described in Sect. 50.1, for a single electron collision system one has to calculate a single particle wave function and from it single-electron transition amplitudes. A multi-electron collision problem can be treated as an effective single-electron problem if only one projectile– electron interaction is considered and the other electrons merely provide an effective single-particle potential in which the active electron moves. In an independent particle model (IPM), the electron–electron interactions are replaced by effective single-electron potentials. Since the projectile–target interaction is a sum of single-electron interactions, the many-electron collision problem reduces to an uncoupled set of single electron problems. Their solution at each impact parameter gives single electron transition amplitudes aij and transition probabilities ρij = |aij |2 . Cross sections for multi-electron transitions can still be calculated by combining the single-electron amplitudes in an appropriate way [50.17, 18]. In doing this, it is important to distinguish between inclusive and exclusive processes. For an inclusive cross section, one final orbital occupancy, or a few final orbital occupancies, is specified, but the final states of the remaining electrons are not specified and all possibilities are summed over. For a totally exclusive process, all final orbital occupancies are specified. For example, consider a p + Ne collision. For the inclusive cross section for K -shell vacancy production, at least one K -shell vacancy in the final state is specified. The other electrons could remain in their original orbitals, there could be two K -shell vacancies, the K -shell vacancy could be accompanied by any number of L-shell vacancies, etc., and all these possible final states are summed over. An example of an

exclusive cross section is the cross section for producing a single K -shell vacancy with the specification that the final state of the target have no additional vacancies. In calculating inclusive cross sections from singleelectron transition amplitudes in an IPM it is important to take proper account of time-ordering and Pauli exclusion effects, as well as all multi-electron processes that lead to the specified final state. Again using the p + Ne collision example, a two-electron process that leads to a K -shell vacancy is for the projectile to first ionize an L-shell electron, and then in the same collision to excite a K shell electron into the L-shell hole just produced. But the K -shell electron must have the same spin component as the L-shell electron that was removed, and the K to L excitation can occur only after the L-hole has been made. This can lead to correlations among the single-electron amplitudes that have been called Pauli correlations or Pauli blocking effects. For a totally inclusive process where only one final occupancy is specified, all the multi-electron contributions cancel in any IPM and the probability for producing, for example, a hole in the initially occupied orbital labeled 1, without any other specification of final state orbital occupancies, is given by  ρ1 = |ak1 |2 , (50.2) k 0.4◦ ). We have also plotted on Fig. 51.16 the experimental data of Roncin et al. [51.35]. The agreement with experiment is very satisfactory.

1.5 × 104

RHC LHC (arb. units) B+3/ He Elab = 1.5 keV

104

5000

LHC

RHC

0 0

0.1

0.2

0.3

0.4 Angle (deg)

Fig. 51.15 Right-hand circular polarization (full curve) and

left-hand circular polarization (dashed curve) for 2 P electron capture in B3+ /He collisions as a function of scattering angle for E = 1.5 keV

(51.33)

and the circular polarization as  2 Im f Σ Z f Π∗ + RHC − LHC Z L= =    . RHC + LHC  f Σ 2 +  f + 2 Z Π

Circular polarization 1

(51.34)

0.5

Z

Figures 51.15 and 51.16 show the RHC, LHC and L quantities for an incident ion energy of 1.5 keV, where comparison with experiments [51.35, 36] can

0

–0.5 X, z v –1 b

–0.5 Z, y

0

0.5 Angle (deg)

Fig. 51.16 Circular polarization for 2 P electron capture in Y, x

Fig. 51.14 coordinate systems for scattering

771

Part D 51.7

ments, it is more convenient to define the quantization axis with respect to an axis perpendicular to the collison plane. But it is straightforward to express the orientation and alignment parameters in terms of the scattering amplitudes obtained with respect to the molecular frame. In accordance with customary conventions, in Fig. 51.14, the scattering plane contains the X and Z axes. The Y axis, perpendicular to the scattering plane, is taken to be the quantization axis. Let XYZ be the laboratory frame and xyz the body-fixed frame defined as above. The scattering amplitudes calculated in Sect. 51.3 are expressed with respect to the body-fixed frame xyz. The scattering amplitudes f MY =±1 in the laboratory frame are related to the amplitudes f Σ Z , f Π + in Z the body-fixed frame by  1  fΣZ ∓ fΠ + . f MY =±1 = √ (51.31) Z 2

51.7 Orientation Effects

B3+ /He collisions as a function of scattering angle for E = 1.5 keV. The solid circles (with the error bars) are taken from [51.35]

772

Part D

Scattering Theory

51.8 New Developments

Part D 51

During the last eight years there have been some interesting new developments [51.38–40] using hyperspherical coordinates to describe the dynamics of ion–atom collisions. Of particular interest for this chapter are the calculations of Le et al. [51.40] for charge transfer in the Si4+ /H(D) and Be4+ /H systems. Their calculated cross sections are almost identical to those obtained by Pieksma et al. [51.41] using the Thorson–Delos-type [51.8] approximate Jacobi coordinates introduced in Sect. 49.3. This confirms the close connection between the hyerspherical and the Thorson–Delos-type coordinates which had already been observed by Gargaud et al. [51.10] for two-state systems. Another aspect of the theoretical formulation which has been clarified recently concerns the adiabatic– diabatic transformation (51.20). The radial differential

equations (51.21) only take this simple form if it can be assumed that B=

d A − A2 . dξ

(51.35)

However, (51.35) is only strictly satisfied if the basis set is complete. And indeed, it has been found [51.10] from direct calculations of the matrix elements Bmn that it is not well satisfied for any choice of Jacobi coordinates. On the other hand, the calculations show that (51.35) is well satisfied for a minimal basis set using the Thorson–Delos reaction coordinates. This result confirms that convergence of an adiabatic basis set can indeed be achieved using appropriate reaction coordinates and also explains why the calculations of Le et al. [51.40] and Pieksma et al. [51.41, 42] agree so well.

References 51.1 51.2 51.3 51.4 51.5 51.6 51.7 51.8 51.9 51.10 51.11 51.12 51.13 51.14 51.15 51.16

A. Dalgarno, S. E. Butler: Comments At. Mol. Phys. 7, 129 (1978) D. Pequignot, S. M. V. Aldrovandi, G. Stasinska: Astron. Astrophys. 63, 313 (1978) M. Gargaud, J. Hanssen, R. McCarroll, P. Valiron: J. Phys. B 14, 2259 (1981) V. H. S. Kwong, Z. Fang: Phys. Rev. Lett. 71, 4127 (1993) T. K. McLaughlin, S. M. Wilson, R. W. McCullough, H. B. Gilbody: J. Phys. B 23, 737 (1990) D. R. Bates, R. McCarroll: Proc. R. Soc. London A 245, 175 (1958) S. B. Schneiderman, A. Russek: Phys. Rev. A 181, 311 (1969) W. R. Thorson, J. B. Delos: Phys. Rev. A 18, 117 (1978) R. McCarroll, D. S. F. Crothers: Adv. At. Mol. Opt. Phys. 32, 253 (1994) M. Gargaud, R. McCarroll, P. Valiron: J. Phys. B 20, 1555 (1987) L. F. Errea, L. Mendez, A. Riera: J. Phys. B 15, 2255 (1982) L. F. Errea, C. Harel, H. Jouin, L. Mendez, B. Pons, A. Riera: J. Phys. B 27, 3603 (1994) M. Gargaud, R. McCarroll, L. Opradolce: J. Phys. B 21, 521 (1988) M. Gargaud, F. Fraija, M. C. Bacchus-Montabonel, R. McCarroll: J. Phys. B 29, 179 (1994) P. Honvault, M. C. Bacchus-Montabonel, R. McCarroll: J. Phys. B 27, 3115 (1994) P. Honvault, M. Gargaud, M. C. Bacchusmontabonel, R. McCarroll: Astronom. Astrophys. 302, 931 (1995)

51.17 51.18 51.19 51.20 51.21 51.22 51.23 51.24 51.25 51.26

51.27 51.28

51.29 51.30 51.31 51.32

51.33

D. L. Cooper, M. J. Ford, J. Gerratt, M. Raimondi: Phys. Rev. A 34, 1752 (1986) S. Bienstock, T. G. Heil, A. Dalgarno: Phys. Rev. A 25, 2850 (1982) C. Bottcher, A. Dalgarno: Proc. Soc. A 340, 187 (1974) P. Valiron, R. Gayet, R. McCarroll, F. MasnouSeeuws, M. Philippe: J. Phys. B 12, 53 (1979) R. Grice, D. R. Herschbach: Mol. Phys. 27, 159 (1974) M. Gargaud, R. McCarroll: Phys. Scr. 51, 752 (1995) M. Gargaud, R. McCarroll, L. Opradolce: Astron. Astrophys. 208, 251 (1989) L. F. Errea, B. Herrero, L. Méndez, O. Mó, A. Riera: J. Phys. B 24, 4049 (1991) R. E. Olson, A. Salop: Phys. Rev. A 14, 579 (1976) M. Kimura, T. Iwai, Y. Kaneko, N. Kobayashi, A. Matumoto, S. Ohtani, K. Okuno, S. Takagi, H. Tawara, S. Tsurubuchi: J. Phys. Soc. (Japan) 53, 2224 (1984) K. Taulbjerg: J. Phys. B 19, L367 (1986) M. Gargaud: Transfert de charge entre ions multichargés et hydrogène atomique (et moléculaire) aux basses énergies. Ph.D. Thesis (Université de Bordeaux, France 1987) S. E. Butler, A. Dalgarno: Astrophys. J. 241, 838 (1980) F. T. Smith: Phys. Rev. 179, 111 (1969) B. R. Johnson: J. Comput. Phys. 13, 445 (1973) P. Valiron: Echange de charge des ions C2+ et Si2+ avec l’hydrogène atomique dans le milieu interstellaire. Ph.D. Thesis (Université de Bordeaux, France 1976) F. Fraija, M. C. Bacchus-Montabonel, M. Gargaud: Z. Phys. D 29, 179 (1994)

Ion–Atom Charge Transfer Reactions at Low Energies

51.34 51.35

51.36

51.37 51.38

M. Gargaud, R. McCarroll, P. Valiron: Astron. Astrophys. 106, 197 (1982) P. Roncin, C. Adjouri, N. Andersen, M. Barat, A. Dubois, M. N. Gaboriaud, J. P. Hansen, S. E. Nielsen, S. Z. Szilagyi: J. Phys. B 27, 3079 (1994) P. Roncin, C. Adjouri, M. N. Gaboriaud, L. Guillemot, M. Barat, N. Andersen: Phys. Rev. Lett. 65, 3261 (1990) M. Gargaud, M. C. Bacchus-Montabonel, T. Grozdanov, R. McCarroll: J. Phys. B 27, 4675 (1994) A. Igarashi, C. D. Lin: Phys. Rev. Lett. 83, 4041 (1999)

51.39 51.40 51.41 51.42

References

773

C.-N. Liu, A.-T. Le, T. Morishita, B. D. Esry, C. D. Lin: Phys. Rev. A 67, 52705 (2003) A.-T. Le, M. Hesse, T. G. Lee, C. D. Lin: Phys. Rev. A 67, 52705 (2003) M. Pieksma, M. Gargaud, R. McCarroll, C. Havener: Phys. Rev. A 54, 13 (1996) D. Rabli: Extension de la méthode du potentiel modèle pour traiter la dynamique des systèmes diatomiques. Application au transfert de charge dans les collisions Si3+ + He et He2+ + He métastable. Ph.D. Thesis (Université Pierre et Marie Curie, Paris 2001)

Part D 51

775

Continuum Dis

52. Continuum Distorted Wave and Wannier Methods

The continuum distorted wave model has been extensively applied to charge transfer and ionization processes. We present both the perturbative and variational capture theories as well as highlighting the suitability of this model in describing the continuum final states in both heavy and light particle ionization. We then develop the Wannier theory for threshold ionization, and further theoretical work which led to the modern quantal semiclassical approximation. This very successful theory has provided the first absolute cross sections which are in good agreement with experiment.

Continuum Distorted Wave Method ....... 52.1.1 Perturbation Theory................... 52.1.2 Relativistic Continuum-Distorted Waves ...................................... 52.1.3 Variational CDW ........................ 52.1.4 Ionization ................................

775 775

52.2 Wannier Method ................................. 52.2.1 The Wannier Threshold Law ........ 52.2.2 Peterkop’s Semiclassical Theory ..................................... 52.2.3 The Quantal Semiclassical Approximation..........................

781 781

778 778 779

782 783

References .................................................. 786

fects. In the final channel the usual product of two continuum distorted wave functions, each associated with a distinct electron–nucleus interaction, is used [52.3]. In addition, Sect. 52.2 on the Wannier Method reports the major progress made over the last eight years. These major advances include (a) the development of below-threshold semiclassical theory for the study of doubly excited states [52.4–7] (b) a more accurate variant of the semi-classical quantum-mechanical treatment of Crothers [52.8].

52.1 Continuum Distorted Wave Method 52.1.1 Perturbation Theory Continuum distorted wave theory (CDW) is one of the most advanced and complete perturbative theories of heavy particle collisions which has been formulated to date. It was originally introduced by Cheshire [52.9] to model the process of charge transfer during the collision of an atom/ion with an ion (specifically the resonant process of p + H(1s) → H(1s) + p). These types of three-body collisions are made amenable to the perturbative approach when the ratio of the projectile impact velocity v to the electron initial bound state mean velocity vb

satisfies v
3. (52.1) vb The criterion for nonrelativistic collisions, in which electron capture is a dominant process, is that both v and vb are small compared with the speed of light. The theoretical description of collisions which involve the disturbance of a bound electron of mass m e by the field of a fast moving heavy particle of mass M can be greatly simplified by exploiting the fact that since the ratio m e /M is so small, the heavy particle follows a straight-line trajectory throughout the collision.

Part D 52

The most recent developments include third-order continuum distorted wave double-scattering 1s–1s transitions (Sect. 52.1.1), relativistic continuum distorted waves (Sect. 52.1.2), and new theory on magnetically quantized continuum distorted waves [52.1, 2] (Sect. 52.1.4). A novel ionization theory for low energies (below 80 keV) is also reported in which the target is considered as a one electron atom and the interactions between this active electron and the remaining target electrons are treated by a model potential including both short and long range ef-

52.1

Scattering Theory

This allows the parametrization of the internuclear vector R in terms of an impact parameter b, such that R = b + vt .

with N(ζ) = exp(πζ/2)Γ(1 − iζ) ,

(52.2)

This impact parameter picture (IPP) of the collision is equivalent to the full quantal or wave treatment when the eikonal criterion for small angle scattering is satisfied [52.10]. It has become standard to work in a generalized nonorthogonal coordinate system in which the vectors rT (rP ) from the target (projectile) to the electron are treated, along with R, as independent variables [52.11]. Working in the frame centered on the target nucleus and using atomic units, the Lagrangian is given by d 1 ∂ 1 H−i = − ∇r2T + VT (rT ) − i − ∇r2P dt rT 2 ∂t 2 + VP (rP ) + iv · rP + VTP (R) − rP · rT , (52.3) ∆

∆

∆

(52.7)

and for the internuclear function C,   ZP ZT 2 ln(vR − v t) . C(R, t) = exp i v

∆

∆

± ξi± = D−v (rP )Φi (rT , t)C(R, t) ,

(52.5)

± where D−v is the distortion from the projectile, and C is due to the internuclear potential, VTP . The bound state Φi = φi (rT ) exp (−ii t), where φi (rT ) is the initial eigenstate, and i is the initial eigenenergy. In this form, the action of the Lagrangian can be split into three separate differential equations plus a residual interaction. This gives the following solutions: for the distortion D, + D−v = N(ζP )1 F1 (iζP ; 1; iv · rP + ivrP ) ,
+ ∗ Dv− = D−v ,

± ∗ ξ± f = Dv (rT )Φ f (rP , t)C (R, −t)   v2 × exp iv · rT − i t , 2

(52.9)

where iv · rT − i v2 t results from the Galilean transformation to the target frame. The superscripts plus and minus refer to outgoing and incoming Coulomb boundary conditions respectively. These are determined by the asymptotic form of the wave functions 2

lim ξ + t→−∞ i

∼ Φi (rT , t)   Z P (Z T − 1) ln(vR − v2 t) , × exp i v (52.10)

and

  v2 t ∼ Φ (r , t) exp iv · r − i lim ξ − f P T t→+∞ f 2   Z T (Z P − 1) 2 ln(vR + v t) . × exp −i v (52.11)

ξi+

ξ− f

Of course, and are not exact solutions of the three-body Schrödinger equation; in fact,   d + H −i ξ + = Wi ξi+ = − rP D−v · rT Φi , dt rT i (52.12)

and

 H −i

d dt rT



− ξ− f = Wfξf v2

= − eiv·rT −i 2 t ∇rT Dv− · ∇rP φ f . (52.13)

The CDW transition amplitude is written as Ai f = −i

(52.6)

(52.8)

Similarly it can be shown that

∆

Part D 52.1

where d/dtrT refers to differentiation with respect to t, keeping rT fixed, and where −Z T −Z P ZT ZP . (52.4) , VP = , VTP = VT = rT rP R Since these potentials are pure Coulomb potentials, they continue to affect the relevant wave functions even at infinity. These long range Coulomb boundary conditions are defined in (52.10) and (52.11). The + iv · rP term gives rise to the Bates–McCarroll electron translation factors which are required to satisfy Galilean invariance. The Lagrangian above has been written in such a way as to highlight the three two-body decompositions exploited in CDW, with the − rP · rT term, the so-called nonorthogonal kinetic energy, coupling the systems. The essence of CDW is to treat the bound electron as simultaneously being in the continuum of the other heavy particle. The initial wave function can be written as

ζT,P = Z T,P /v ;

∆

Part D

∆

776

+∞ 

+ dt ξ − f TCDW ξi .

−∞

(52.14)

Continuum Distorted Wave and Wannier Methods

A perturbative expansion via the distorted wave Lippmann–Schwinger equation can be made for TCDW , either in the post form †

+ + TCDW = W f (1 + G V Wi ) + TCDW G i VG V Wi ,

(52.15)

or in the prior form  † † − − TCDW = 1 + W f G V Wi + W f G V VG f TCDW , (52.16)

encounter peak, small angles, and the interference minimum, the CDW series has converged at second order. Moreover, it is proven that the third-order correction makes no contribution to the velocity dependent v−11 and v−12 behavior of the Thomas double-scattering total cross section at the leading angles. In contrast, it may be seen in [52.15] that the Oppenheimer–Brinkman– Kramers (OBK) travelling atomic orbital theory (which in general suffers from a common phase factor which embraces intermediate elastic divergences [52.16] in the first and higher-order terms) has not converged at second order. It remains an open question as to whether fourth-order terms or higher in the OBK approximation contribute to various differential cross-sections. It is concluded that the CDW model gives a superior description of the Thomas double-scattering mechanism when compared with the OBK model. Anomalously large cross sections are obtained at low energies if the CDW wave function is not normalized at all times throughout the collision [52.11]. This is best demonstrated by the presence of the N(ζ) terms in lim ξ + t→+∞ i

= N(ζP )Φi (rT , t)   ZP ZT ln(vR − v2 t) (52.19) × exp i v

(dσ/dΩ )lab (cm2/sr) 10–18

10–19

10–20

10–21

0.0

0.2

0.4

0.6

0.8 θlab (mrad)

Fig. 52.1 Differential cross sections for electron capture in

the collision H+ + H(1s) → H(1s) + H+ as a function of laboratory scattering angle (θlab ) for impact energy of 5 MeV: solid line TCDW, folded over the experimental resolution of Vogt et al. [52.12]. Unfolded theoretical results: dashed line TCDW; dotted line CDW1. Experimental data; circles, Vogt et al. [52.12]

777

Part D 52.1

where the Green functions are given by  −1 d − H + Wi, f + i , (52.17) G i, f = i dtrT  −1 d GV = i − H + V + i , (52.18) dt rT and V is any potential which ensures that the kernels of the integral equations for TCDW are continuous [52.13]. By taking the first term in the expansions (52.15) and (52.16), we get the post and prior forms of the CDW1 amplitude as used by Cheshire. When calculating these amplitudes, the separable nature of the CDW wave function is best exploited by using Fourier transforms to move to the time-independent wave picture. A similar transformation is not suitable in the coupled channel approach discussed in Sect. 52.1.3. Crothers [52.13], working in the wave treatment, has calculated the second order CDW2 amplitude using various approximations for the Green functions, and has shown that the CDW perturbation series has converged very well to first-order in most parts of the differential cross section. This is in contrast to the standard Born or Brinkman–Kramers approximations which do not start to converge until expanded to second-order. CDW1 is the only first-order perturbation theory, apart from asymmetric hybrid models derived from it, which produces a Thomas peak. Unfortunately, due to the accidental cancellation of the leading order terms, CDW1 has an extreme dip at the Thomas angle, a defect removed in CDW2 [52.14]. This is illustrated in Fig. 52.1 which also includes both folded and unfolded versions of the asymmetric target CDW (TCDW) theory discussed below, as well as experimental data [52.12]. Further work in this area has included the development of the Thomas double-scattering electron capture at asymptotically high velocity within the thirdorder continuum distorted-wave perturbation theory for 1s–1s transitions in proton hydrogen collisions. It has been shown [52.15] that at the critical proton scattering angles, namely the forward peak, Thomas double

52.1 Continuum Distorted Wave Method

778

Part D

Scattering Theory

and

  v2 t ∗ lim ξ − = N (ζ )Φ (r , t) exp iv · r − i T f P T t→−∞ f 2   ZT ZP ln(vR + v2 t) . × exp −i v (52.20)

Using 2πZ (52.21) v it is clear that the problem gets worse as v decreases. It can be corrected by defining  −1/2 ξˆi,±f = ξi,±f ξi,±f ξi,±f . (52.22) lim | N(Z/v)|2 ∼

v→0

Part D 52.1

Simpler distorted wave models can be generated through further approximations. Two of the most popular are Target CDW [D−v (rP ) → 1] and Projectile CDW [Dv (rT ) → 1]. These approximations are justified when Z T > Z P and Z P > Z T , respectively, and are particularly simple to calculate when the simple Born-like residual interaction is used rather than the full CDW form. The asymptotic forms of the CDW wave functions can be used throughout the collision, ensuring normalization, and this leads to the eikonal or symmetric eikonal models.

52.1.2 Relativistic Continuum-Distorted Waves The CDW model can be naturally extended to the twocenter time-dependent Dirac equation, so that a Lorentz invariant theory is obtained. When the electron orbital velocity αZc, or the collision velocity v, approaches the speed of light c, the kinematics are modified by time dilation. In addition, the particle interactions change because of retardation and the fact that spin-orbit effects are now important. Moreover, vacuum interactions such as radiative emissions and electron–positron pair production begin to play a role. A comprehensive account of atomic processes in relativistic heavy-particle collisions can be found in two recent books [52.17, 18]. At high
collision energies, γ ≡ 1 − v2 /c2 −1/2  1, and highcharge states of the ions, the Dirac sea of negative energy states becomes energetically accessible and strongly coupled. The process of electron capture, for example, may be mediated by spin-flip transitions [52.19], or spontaneous X-ray emission (radiative electron capture) [52.20] and even electron capture via pair production [52.21, 22]. Although the importance of vacuum processes diminishes with energy, these mechanisms

dominate in the extreme relativistic regime. Indeed the last of these processes was used to produce antihydrogen in the laboratory [52.23] at GeV u−1 energies. At relativistic energies, the principal inelastic process is collisional ionization [52.17, 18]. The extensions of the distorted-wave theory to accommodate Lorentz invariance has been developed by Rivarola and Deco [52.24, 25] and Crothers and coworkers [52.19] following work on the Born series [52.26] and impulse approximation [52.27]. In practical applications to electron capture cross sections, the symmetric semirelativistic CDW theory of Glass et al. [52.19] was found to be in very good agreement with experiments in the GeV u−1 energy range with charges Z P,T ∼ 6–80. For non-radiative electron capture, it was found that second-order retardation dominates at extreme relativistic velocities so that σ ∼ γ −1 (ln γ)2 [52.28]. However, the momentum transfer kinematics for this process are unfavorable, and a more efficient mechanism based on electron–positron pair production with capture of the created electron is more strongly coupled. Theoretical estimates of this process using relativistic CDW [52.29] compared with experiments [52.30] are in very good agreement. The same model has been applied to estimate yields of antihydrogen following antiproton impact with neutral high-Z atoms [52.23] following experiments at CERN and Fermilab. The virtual photon model of Baur [52.30] gives cross sections that agree well with the limited data [52.23]. However, these estimates are roughly ten times larger than the relativistic CDW results [52.17] and one hundred times the first-Born estimate [52.31]. It appears that additional studies, both experimental and theoretical, would be worthwhile in order to understand this process more fully.

52.1.3 Variational CDW As the ratio of v/vb decreases, perturbation theory starts to fail. This is due to the effective interaction time between the projectile and target atoms being long enough for strong three-body coupling. In this environment variational methods have proved successful. This procedure ensures both gauge invariance and unitarity – two fundamental attributes perturbation theory usually cannot guarantee. Continuing in the IPP, we use the Sil variational principle, which gives δ

+∞ d dtΨ |H − i |Ψ = 0 . dtrT

−∞

(52.23)

Continuum Distorted Wave and Wannier Methods

In the two-state CDW approximation we may assume ΨCDW = c0 (t)ξi+ + c1 (t)ξ − f

(52.24)

subject to the boundary conditions c0 (−∞) = 1 and c1 (−∞) = 0. Variation of c∗0 and c∗1 gives the standard coupled equations iN00 c˙ 0 + iN01 c˙ 1 = H00 c0 + H01 c1 , iN10 c˙ 0 + iN11 c˙ 1 = H10 c0 + H11 c1 ,

(52.25) (52.26)

where



N00 = ξi+ ξi+ , 

− N11 = ξ − f ξ f , 

∗ N01 = ξi+ ξ − f = N10 ,

†

S = Ω− Ω+ ,

(52.28)

†

where Ω− represents the propagation of the initial state from t = −∞ to t = 0− , while Ω+ represents the propagation of the final state from t = +∞ to t = 0+ . The total wave function is similarly split into two expansions over an orthogonal basis ψ, with Ψ

+

+

−

=c ψ ,

t0,

(52.30)

i˙c− = H++ c− ,

t0,

(52.32)

where H±± = Ψ ± | H − i

Equations (52.25) and (52.26) can clearly be written as a matrix equation, (52.27)

which is then easily generalized for larger expansions of Ψ . By using an orthogonalized basis set of normalized functions in the manner of Löwdin [52.32,33], the N matrix reduces to the unit matrix. This will be understood when considering expansions for Ψ from now on. Another interesting, but potentially ruinous, result of the asymptotic forms (52.19) and (52.20) is their failure to obey the correct long-range Coulomb boundary conditions; compare this with their expressions at the opposite time extreme in (52.10) and (52.11). This has no consequence until second-order VCDW is calculated in which divergent integrals arise as a direct result of this feature of the wave functions. These terms are analogous to the well-known intermediate elastic divergences which occur in Born-type expansions which do not have the correct Coulomb phases. A novel way to avoid this problem [52.34, 35] is to split the time plane into two parts, allowing the well-

d | Ψ ± . dt rT

(52.33)

The coefficients c± then have to be matched over a local discontinuity in the total wave function at t = 0, such that c+ (0) = c− (0). Halfway house VCDW has all the appealing attributes of a variational theory but, by explicitly satisfying the long-range Coulomb boundary conditions, it is divergence free.

52.1.4 Ionization CDW, by treating the Coulomb interactions to such a high degree, has obvious attractions for modelling the ionization process. Single ionization of an electron from an atom by a high-energy projectile is a perturbative process and the Born approximation will match experimental total cross sections rather well. However CDW-like representations of the initial and final states generate better results at lower energies, as well as producing features in the differential cross sections which are beyond the reach of the first Born approximation. In full CDW ionization theory, the initial state is given by the usual charge transfer wave function ξi

Part D 52.1

where the superscripts on the Ψ correspond to the respective heavy-particle motion. This in turn divides the coupled equations into two sets

d +  H00 = ξi+ H − i ξ , dtrT i

d −  H01 = ξi+ H − i ξ , dtrT f

d +  H10 = ξ − ξ , f H −i dtrT i

d −  H11 = ξ − ξ . f H −i dtrT f

779

behaved set {ξ + } to be used exclusively for t ≤ 0, while the set {ξ − } forms the basis for t ≥ 0. This phase integral halfway house VCDW is based on the factorization of the scattering matrix S into a product of two Møller matrices,

Ψ − = c− ψ + ,

and

iN˙c = H c ,

52.1 Continuum Distorted Wave Method

780

Part D

Scattering Theory

(52.5), while the final state takes the form   k2 − −2/3 exp ik · rT − i t ξ f = (2π) 2   ZT ZP ln(vR + v · R) × exp −i v × N ∗ (Z T /k)1 F1 × (−iZ T /k; 1; −ik · rT − ikrT ) × N ∗ (Z P / p)1 F1 × (−iZ P / p; 1; −i p · rP − i prP ) ,

(52.34)

Part D 52.1

where k ( p = k − v) is the momentum of the electron relative to the target (projectile) nucleus. CDW ionization theories presented prior to 1982 produced spuriously large results due to the unnormalized initial state. Since the matrix element ξi+ | ξi+ is computationally expensive to calculate as a function of b and t, a very successful alternative is to take the initial state as an eikonal distorted state [52.36], thus ensuring normalization. The initial state in this CDW-EIS theory is taken to be + (rP )Φi (rT , t)C(R, t) , ξiEIS = Dˆ −v

where now

(52.35)

  ZP + ˆ ln(vrP + v · rP ) . (52.36) D−v (rP ) = exp −i v The final state remains as in (52.34). The CDW final state is most effective when differential cross sections are studied. The most interesting features are in the forward θ = 0 ejection angle; i. e., the soft collision peak (k 0), the electron capture to the continuum peak ECC (k v) and the binary encounter peak (k 2v). CDW theories are especially suited to the description of the ECC peak which results theoretically from the presence of the N(Z P / p) factor in the wave function. This peak can be analyzed in detail via a multipole expansion and much theoretical work has centered on the dipole parameter β. A negative β is strongly suggested by both experiment and physical intuition. CDW, CDW-EIS and halfway house VCDW all predict different values for β with the last theory being the only one which gives a high energy limit which remains negative [52.37]. Magnetically quantized continuum distorted wave theory also has been considered [52.1] in the description of ionization in ion–atom collisions. This generalizes the CDW-EIS theory of Crothers and McCann [52.2] to incorporate the azimuthal angle dependence of each CDW in the final state wave function. This is accomplished by the analytic continuation of hydrogenic-like

wave functions from below to above threshold, using parabolic coordinates and quantum numbers, including magnetic quantum numbers, thus providing a more complete set of states. The continuation applies to excitation, charge transfer, ionization, and double and hybrid events for both light- and heavy-particle collisions. It has successfully been applied to the calculation of double differential cross sections for the single ionization of the hydrogen atom and for a hydrogen molecule by a proton for electrons ejected in the forward direction at a collision energy of 50 keV and 100 keV, respectively. It is well known that the CDW-EIS models are the best suited to the intermediate and high energy regions. Recent results for proton-argon total ionization cross sections [52.38] highlight large discrepancies between CDW-EIS theory and experiment for energies below 80 keV. This problem has recently been addressed [52.3]. Here, following the theory of [52.39] the authors in [52.3] use a Born initial state wave function. In the final channel, the usual product of two continuum distorted wave functions each associated with a distinct electron–nucleus interaction is used. In their treatment the target is considered as a one-electron atom and the interactions between this active electron and remaining target electrons are treated by a model potential including both short- and long-range effects. The success of this new theory for low energies is shown in Fig. 52.2. Here it is clear that the calculation in [52.3] gives good agreement for the total cross sections in the energy range 10–300 keV with the measurements of Rudd et al. [52.40]. Double ionization in general remains an extremely difficult area for theoretical models based on perturbative expansions – even those with explicit distortions built in. In this process, the explicit correlation between the electrons in the target atom is vitally important. However, one example where CDW theory can overcome these problems is in the bound state wave function of Pluvinage [52.41]. This very successful treatment, which also includes a variationally determined parameter, is just the CDW analogy in bound state theory, although this appeared well before Cheshire’s paper on scattering. The Pluvinage wave function for the ground state of a two-electron atom is given by   Z T3 φ = c(κ) exp [−Z T (r1 + r2 ) + iκr12 ] π   1 × 1 F1 1 + ; 2; 2iκr12 , (52.37) 2iκ

Continuum Distorted Wave and Wannier Methods

10

Total cross sections (10–16 cm2)

1

0.1 10

100 1000 Incident proton energy (keV)

Fig. 52.2 Total ionization cross section for the protonimpact single ionization of Ar: solid line theoretical results of [52.3], dashed line theoretical results of OPM [52.38] and dotted line HFS [52.38]. Experimental data; circles, Rudd et al. [52.40]

The constant κ is variationally determined to be 0.41, giving the normalization constant c = (0.364 05)1/2 . An analogous wave function for two electrons in the target continuum was derived [52.11] and implemented later in a successful CDW treatment of ionization by electrons and positrons [52.42]. This BBK theory, so named in deference to the authors, demonstrates the high resolution CDW final states obtained, right down to triply differential cross sections. However, this model suffers from the low-energy normalization problems associated with CDW, and is also unable to describe the threshold effects which are in the domain of Wannier theory.

52.2 Wannier Method 52.2.1 The Wannier Threshold Law In 1953, Wannier [52.43] deduced the relationship between the cross section at the threshold of a reaction, and the excess-of-threshold energy of the incident particle for a three-body ionization problem, where the final state consists of a residual unit positive charge and two electrons, with each body moving in the continuum of the other two. This extended to three bodies the earlier two-body threshold law derived by Wigner [52.44] (see Sect. 60.2.1). For final states with L = 0, Wannier employed hyperspherical coordinates (ρ, α, θ12 ), where   2 2 2 −1 r2 , ρ = r1 + r2 , α = tan r1 θ12 = cos−1 (ˆr1 · rˆ2 ) . (52.38) Here we assume that the residual ion is infinitely massive and at rest with respect to the two electrons. By converting the two-electron problem to the case of motion of a single point in six-dimensional space, we can take the hyperradius ρ as the ‘size’ of the hypersphere, α as the radial correlation of the electrons and θ12 as their angular correlation. This allows the Schrödinger equation for the final state to be written

781

(in a.u.) as

  l 2 rˆ1 l 2 rˆ2 h0 − 2 − + 2E ρ cos2 α ρ2 sin2 α  2Z(α, θ12 ) Ψ(r1 , r2 ) = 0 , + ρ where h0 =

(52.39)

  1 ∂ 1 ∂ 5 ∂ ρ + 2 2 5 2 ∂ρ ρ ∂ρ ρ sin α cos α ∂α   ∂ 2 2 × sin α cos α (52.40) ∂α

and Z(α, θ12 ) =

1 1 1 + − cos α sin α (1 − cos θ12 sin 2α) 21 (52.41)

is the potential surface on which the particle is moving [52.45]. The most likely configuration of the electrons leading to double escape at threshold corresponds to the region r1 = −r2 , i. e., the two electrons escape in opposite directions from the reaction zone, corresponding to the saddle point of the potential surface Z(α, θ12 ), and defined as the Wannier ridge. Also, dynamic screening

Part D 52.2

where r1,2 are the distances of the two electrons from the target nucleus, and r12 is the inter-electron separation.

52.2 Wannier Method

782

Part D

Scattering Theory

between the two electrons means there would be equal partitioning of the available energy for the two particles, and so they would have equal but opposite velocities on escape. In hyperspherical coordinates, the most important region for double escape is, therefore, α = π/4, and θ12 = π. The main conclusion of Wannier’s theory is that the total cross section for electron impact single ionization scales as σ = kE

m 12

,

(52.42)

where E is the excess-of-threshold energy, 1 1 m 12 = − + 4 4



100Z − 9 4Z − 1



1 2

,

where 1 ∂ 5 (ρ f ) , ρ5 ∂ρ  ∂
1 f sin2 2α , D1 f = sin2 2α ∂α 1 4 ∂ D2 f = ( f sin θ12 ) . sin2 2α sin θ12 ∂θ12 D0 f =

(52.49) (52.50) (52.51)

Following Wannier’s hypothesis, solutions of these equations are found in the region α = π/4, θ12 = π. Taking the Taylor expansion for Z(α, θ12 ) as 1 1 Z(α, θ12 ) = Z 0 + Z 1 (∆α)2 + Z 2 (∆θ12 )2 +· · · , 2 8 (52.52)

(52.43)

and Z the residual charge; m 12 is 1.127 for unit residual charge, with m 12 → 1 as Z → ∞.

Part D 52.2

52.2.2 Peterkop’s Semiclassical Theory

where ∆α = α − π/4 and ∆θ12 = θ12 − π, it follows from (52.41) that 3 11 1 Z0 = √ , Z1 = √ , Z2 = − √ . 2 2 2

(52.53)

Similarly, taking the solution of (52.47) in the form The Wannier threshold law has been verified both semiclassically [52.46] and quantum mechanically [52.47]. Peterkop [52.46] adopted a semiclassical JWKB approach to the problem, by using the three-dimensional WKB ansatz   1 iS , (52.44) Ψ0 = P 2 exp ~ for the final-state wave function, where S and P are, respectively, the solutions of the Hamilton–Jacobi equation, (∇1 S)2 + (∇2 S)2 = 2(E − V ) ,

(52.45)

(52.46)

In hyperspherical coordinates, these equations become  2     ∂S ∂S 2 1 ∂S 2 4 + 2 + 2 2 ∂ρ ∂α ρ ρ sin 2α ∂θ12 2Z = 2E + (52.47) ρ and



 ∂S 1 + 2 D0 P ∂ρ ρ      ∂S ∂S =0, + D2 × D1 P ∂α ∂θ12

(52.54)

gives dS0 =ω, dρ dSi Si2 Zi + , ω = dρ ρ2 ρ

(52.55)

i = 1, 2 ,

(52.48)

(52.56)

1

where ω = (2E + 2Z 0 /ρ) 2 . The solutions are Z 0 ρ(χ + ω)2 ln , χ 2Z 0 1 du i , i = 1, 2 , Si = ρ2 ω u i dρ

S0 = ρω +

and the continuity equation, ∇1 (P∇1 S) + ∇2 (P∇2 S) = 0 .

1 1 S = S0 (ρ) + S1 (ρ)(∆α)2 + S2 (ρ)(∆θ12 )2 + · · · 2 8

(52.57) (52.58)

1

where χ = (2E ) 2 and u i = Ci1 u i1 + Ci2 u i2 , (52.59)   3 −Eρ , u ij = ρm ij 2 F1 m ij , m ij + 1; 2m ij + ; 2 Z0 (52.60)

1 1 1 1 m i1 = − − µi , m i2 = − + µi , 4 2 4 2 1  8Z i 2 1 1+ µi = . 2 Z0

(52.61) (52.62)

where, for i = 2, the principal branch is understood.

Continuum Distorted Wave and Wannier Methods

Expanding P in the same form as S, and restricting the solution to finding P0 , gives P0 =

C , ρ5 ωu 1 u 22

(52.63)

where C ∼ C12 E 1−m 12 . By solving these equations, Peterkop extracted the Wannier cross section behavior by matching the exact wave function with an approximate one for which the energy dependence is known at some arbitrarily finite value r0 of ρ, giving the total cross section as σtot ∼ E 1.127 ,

(52.64)

as required. However, neither this method of Peterkop nor the quantum mechanical approach of [52.47] were able to deduce the constant of proportionality.

S = s0 ln |∆α| + s1 ln(∆θ12 ) + S0 (ρ) 1 1 + S1 (ρ)(∆α)2 + S2 (ρ)(∆θ12 )2 + . . . , 2 8 (52.67)

where the extra logarithmic phases indicate longrange Coulomb potentials. By applying the Kohn variational principle perturbatively, and invoking the Jeffreys’ [52.48] connection formula on the Wannier ridge with ρ = 0 as the classical turning point, the final state wave function is [52.8] Ψ −∗ f =

x| sin(α − π/4)|1/2 , ρ5/2 sin α cos α(sin θ12 )1/2

(52.65)

so that 

  ∂ ∂ 1 ∂2 sin |∆α| + ∂α ∂ρ2 ρ2 sin |∆α| ∂α ∂2 2Z + 2E + 2 ρ ∂θ12  1 1 2 2 ( 4 + 4 csc ∆θ12 ) csc |∆α| + 2 2 − x=0, ρ sin α cos2 α 4ρ2

+

1

ρ2 sin2 α cos2 α

(52.66)

where |∆α| and θ12 are, respectively, the polar and azimuthal angles. Near θ12 = π and α = π/4, (i. e., at ∆θ12 = ∆α = 0), the term (4ρ2 )−1 is negligible compared with csc2 θ12 /(4ρ2 ). Also the θ12 pseudopotential is clearly attractive while the α potential is repulsive, and both potentials are large just at the region of importance, i. e., at ∆α = 0 = ∆θ12 . Again, following the method of Peterkop, the final state wave function is written in the form (52.44),

c1/2 E m 12 /2 ρm 12 /2+1/4r(2Z 0 )1/4 (−χ/2π)1/2 (2Z 0 /ρ)1/4 ρ5/2 sin α cos α

 × δ kˆ 1 − rˆ1 δ kˆ 2 − rˆ2   −2 × exp 4i(8Z 0 ρ)−1/2(∆θ12 )   1 × exp −i(8Z 0 ρ)1/2 − i(∆α)2 2 1 × (2Z 0 ρ)1/2 m 12 − i(∆θ12 )2 8   1 1/2 ×(2Z 0 ρ) m 21 − iπ − c.c , (52.68) 4

where χ = 2πIm(m 21 ). Taking the total cross section for distinguishable particles as    2 π 2 a02 π Z 2 tanh χ
σ= dkˆ 1 dkˆ 2 f kˆ 1 , kˆ 2 1/2 k0 (2E )   −Z 2 2 × exp (∆Θ12 ) π tanh χ , (52.69) 4(2E )1/2 where f is the scattering amplitude, then the corresponding triple-differential cross section is d3 σ  dkˆ 1 dkˆ 2 d 12 k12 =

2π 2 a02 d πZ 2 tanh χ k0 dE (2E )1/2   −Z 2 2 × exp (Θ − Rπ) π tanh χ 12 4(2E )1/2  2 (52.70) × f kˆ 1 , kˆ 2 .

Assuming the contribution from triplet states is negligible, | f |2 can be written as 
 2 1
(52.71) f kˆ 1 , kˆ 2 + f kˆ 2 , kˆ 1 , 4

Part D 52.2

Ψ −∗ =

783

(52.47) and (52.48), where now the action perturbation expression S must be generalized to

52.2.3 The Quantal Semiclassical Approximation As can be seen from the form of (52.63) for P0 , u 2 vanishes in the double limit ρ → +∞, E → 0, and so, the semiclassical theory breaks down at the very configuration of importance. To avoid this problem, Crothers [52.8] adopted a change of the dependent variable to obtain a uniform JWKB approximation. Taking α = π/4, (i. e., ∆α = 0) as the natural barrier, he set the final-state wave function as

52.2 Wannier Method

784

Part D

Scattering Theory



 where f kˆ 1 , kˆ 2 (and by permutation f kˆ 2 , kˆ 1 , hereafter referred to as f and g respectively) is given by  2i dr1 dr2 dr3 Ψ −∗ f f φ f (2, r3 ) π (52.72) × (H − E ) exp(ik0 · r1 )ψi (r2 , r3 ) .

and where f (and similarly g for Θ2 ) is given by ∞ f= 0

× PL (cos Θ1 ) exp

φ(r2 , z 0 )φ(r3 , β) + φ(r3 , z 0 )φ(r2 , β) , [2(1 + S)]1/2 (52.73)

where

Part D 52.2

φ(r, z 0 ) = z 0 π −1/2 exp(−z 0r) ,  2 φ(r, z 0 )φ(r, β)dr S= 3/2

 =

4z 0 β (z 0 + β)2

(52.74)

3 ,

(52.75)

and z 0 and β take the physical values z 0 = 1.80721/2 and β = 2. The total singlet cross section was found to be (in atomic units) σ = 2.37E m 12 a02 ,

(52.76)

in line with Wannier’s threshold law, and with experiment [52.49], while the corresponding absolute triple differential cross sections (TDCS) were expressed as 70cz 20 21/2 χ tanh χ d3 σ =  | f + g|2 1/2 1 2 ˆ ˆ πR Z m dk1 dk2 d 2 k1 ∞ 0 12    d m 12 −1/2 −Z 2 (Θ12 − π)2 π tanh χ E × exp dE 4(2E )1/2 (52.77)

in units of 10−19 cm2 sr−2 eV−1 , where c=

Γ(m 12 + 3/2)Γ(m 12 + 1) , 2πZ 0m 12 Γ(2m 12 + 3/2)

(52.78)



L max

i L (2L + 1) jL

L=0

 ρz 0

21/2

1 8

  × Im m 21 (Θ12 − π)2 (2Z 0 ρ)1/2  1 × 2 cos (8Z 0 ρ)1/2 + 8   2 × Re m 21 (Θ12 − π) (2Z 0 ρ)1/2

As a test of the above formulation for the process e− + He → He+ + 2 e− near the ionization threshold, Crothers [52.8] used an independent-electron open-shell wave function for the initial bound state helium target, written as ψi (r2 , r3 ) =

dρρ3/2+m 21 /2+1/4

× r(ρ, Θ12 )

(52.79)

with (1 + S)1/2r(ρ, Θ12 )   ρz  1 0 1/2 = exp − 1/2 2 z0 − 2 (1 − cos Θ12 )1/2  1/2 64(2) (z 0 − 1) + exp(−21/2 ρ) (z 0 + 2)3 32 + [2(2)1/2 (z 0 + 2)3   −ρ(z 0 + 2) . + (z 0 + 2)ρ] exp (52.80) 21/2 These results have been found to compare favorably with both the relative experimental results of [52.50] and the absolute experimental results of [52.51]. The Crothers quantal semiclassical approximation has been successfully applied to other threshold (e, 2e) and (photon, 2e) collisions, namely two-electron photode
tachment from H− [52.52], He 4P 05/2 [52.53] and K− [52.54]. Further investigations of the TDCS for helium at threshold have since been carried out. The 3P 0 triplet contribution to the TDCS was studied in [52.55], where small but notable improvements in the comparison with experimental results [52.50] and [52.51] were achieved for most configurations of the angles θ1 and θ2 . The inclusion of contributions from 3D e,0 or 3F 0 to the absolute TDCS were found to be negligible in comparison with the effect of the 3P 0 [52.56], although the admittedly non-Wannier effective-charge investigation of θ12 = π by Pan and Starace [52.57] suggests that 3F 0 may be important at θ1 = π/6 or 5π/6, in line with the experiment of Rösel et al. [52.51]. Another aspect which is thought to contribute to the TDCS is explicit correlation in the initial bound state wave

Continuum Distorted Wave and Wannier Methods

a)

give the explicit widths of the resonances from which the intensities have been estimated. The theory in [52.4] was considered initially for the inaugural case of L = 0. Further investigation [52.5–7] has extended the theory to include resonant states for L = 1 and L = 2. In the case of L = 1, an irrational quantum number was obtained and attosecond lifetimes were obtained. Excellent results were obtained for the resonance positions, lifetimes, intensities, and scaling rules in comparison with the experimental data [52.60]. This success persuaded Deb and Crothers [52.62] to re-visit the problem of quantal near-threshold ionization of He by electron impact. In particular they re-examined the problem of above threshold ionization of He by electron impact by retaining the term 2L(L + 1)/ρ2 in the hyperspherical equation  1 ∂ ∂ 1 ∂ 5 ∂ ρ + sin2 2α ∂α ρ5 ∂ρ ∂ρ ρ2 sin2 2α ∂α 4 ∂ ∂ sin θ12 + 2E ∂θ12 ρ2 sin θ12 ∂θ12  2ζ(α, θ12 ) 2L(L + 1) Ψ =0. + ρ ρ2 +

(52.81)

b)

(a) θ1 = 60°

(b) θ1 = 90°

Fig. 52.3a,b Helium triply differential ionization cross section for coplanar geometry, E 1 = E 2 = 1 eV and E = 2 eV, calculated (full curve) TDCS of Copeland and Crothers [52.58] in polar coordinates, with polar coordinates as θ2 , for scattering angles θ1 = 60◦ (a) and θ1 = 90◦ (b), in comparison with absolute experimental (circles) data of Rösel et al. [52.44], and theoretical results (broken curve) of Crothers [52.40]. The radius of each circle is 1.0 × 10−19 cm2 sr−2 eV−1

785

Part D 52.2

function for the helium target, in which the interelectronic distance r23 is explicitly contained. Absolute singlet triple-differential cross sections have been obtained [52.58], using a helium ground state wave function developed by Le Sech [52.59]. Again, excellent agreement with the singlet results of [52.8] has been achieved, and, in most configurations, notable improvements with the corresponding experimental data are obtained, as shown in Fig. 52.3 for scattering angle θ1 = 60◦ (a) and 90◦ (b), indicating that electron correlation should also be considered if a full picture of threshold ionization is to be achieved. The last eight years has seen significant new developments and contributions to the Wannier theory. One notable achievement has been the analytical continuation of the uniform semi-classical wave function [52.8] to below the energy threshold to calculate the complex eigenenergies for doubly excited states of helium using a complex Bohr–Sommerfeld quantization rule with at least one complex transition point [52.4, 5]. The real parts of the eigenvalues were found to be in good agreement with the experimental results of Buckman et al. [52.60, 61] for the resonance positions while the imaginary parts

52.2 Wannier Method

786

Part D

Scattering Theory

Following the procedure of Crothers [52.8], they obtained Ψ −∗ f =

1/2
 c1/2 E m 12 /2 u 1 δ kˆ 1 − rˆ1 ω˜ 1/2 ρ5/2 sin α cos α  
 × δ kˆ 2 − rˆ 2 exp 4i(8Z 0 ρ)−1/2 (∆θ12 )−2   1 × exp −i S0 + S1 (∆α)2 2   1 π − conjugate , + S2 (∆θ12 )2 + 8 4

(52.82)

where the classical action variables are given by ρ S0 =

dρ˜ ω˜ ,

(52.83)

0

Part D 52

Si = ρ2 ω(ln u i ) ,

i = 1, 2

(52.84)

with  1/2 ω˜ = ω2 − ω (lnu2 −i ln u 1 ) ,

(52.85)

ω = 2E + 2Z 0 /ρ − 2L(L + 1)/ρ .

(52.86)

2

2

The primes in (52.84) and (52.85) denote derivatives with respect to ρ and ρ˜ respectively. It is to be noted here that the original work used the approximated form of ω by dropping the L-dependent term in (52.86). The inclusion of this angular momentum term moves the classical turning point from the origin to ρ+ , where  −Z 0 + Z 02 + 4EL(L + 1) . (52.87) ρ+ = 2E

As a result, the lower limit of ρ integration will be replaced by ρ+ . The classical action variables S1 and S2 are now evaluated without introducing the limit Eρ → 0. Using the final state wave function in (52.82) we have calculated first the direct ionization amplitude. The exchange ionization amplitudes for the two indistinguishable atomic electrons were then obtained by interchanging the angles θ1 and θ2 in the direct amplitude. Singlet and triplet contributions are then accounted for in the usual ratio of 1:3. Equation (52.82) is a more accurate variant of (52.68) used in the original theory of Crothers [52.8]. Using this refinement of the wave function, all partial wave contributions for singlets and triplets are accounted for up to L = 6 for the case of He by electron impact at an excess of 2 eV above threshold. It has been found that within the co-planar geometry, both the symmetric and asymmetric triple differential cross sections, peaking at and near the Wannier ridge, are greatly improved when compared with experiment [52.63]. However, far away from the Wannier ridge the triple differential cross sections tend to show qualitative differences from measurement [52.63]. The improved theory [52.62] has also successfully been applied to the calculation of total cross sections of positron impact ionization of helium for energies 0.5–10 eV above threshold [52.64]. Excellent agreement with available experimental data [52.65] was obtained for the absolute theoretical calculation of positron impact ionization [52.64]. Finally this recent work has answered some questions concerning near-threshold processes, a large and interesting area of study and we firmly believe that further development of this powerful theory will answer many more.

References 52.1

52.2 52.3 52.4 52.5

D. S. F. Crothers, D. M. McSherry, S. F. C. O’Rourke, M. B. Shah, C. McGrath, H. B. Gilbody: Phys. Rev. Lett 88, 053201 (2002) S. F. C. O’Rourke, D. M. McSherry, D. S. F. Crothers: J. Phys. B 36, 314 (2003) S. Bhattacharya, R. Das, N. C. Deb, K. Roy, D. S. F. Crothers: Phys. Rev. A 68, 052702 (2003) A. M. Loughan, D. S. F. Crothers: Phys. Rev. Lett. 79, 4966 (1997) A. M. Loughan: Adv. Chem. Phys. 114, 312 (2000)

52.6 52.7 52.8 52.9 52.10

52.11

A. M. Loughan, D. S. F. Crothers: J. Phys. B 31, 2153 (1998) D. S. F. Crothers, A. M. Loughan: Philos. Trans. R. Soc. London A 357, 1391 (1999) D. S. F. Crothers: J. Phys. B 19, 463 (1986) I. M. Cheshire: Proc. Phys. Soc. 84, 89 (1964) B. H. Bransden, M. R. C. McDowell: Charge Exchange and the Theory of Ion-Atom Collisions (Clarendon, Oxford 1992) D. S. F. Crothers: J. Phys. B 15, 2061 (1982)

Continuum Distorted Wave and Wannier Methods

52.12 52.13 52.14 52.15 52.16 52.17

52.18 52.19 52.20 52.21 52.22 52.23

52.26 52.27 52.28 52.29 52.30 52.31 52.32 52.33 52.34 52.35 52.36

52.37 52.38 52.39 52.40 52.41 52.42 52.43 52.44 52.45 52.46 52.47 52.48 52.49 52.50 52.51 52.52 52.53 52.54 52.55 52.56 52.57 52.58 52.59 52.60 52.61 52.62 52.63 52.64 52.65

D. S. F. Crothers, S. F. C. O’Rourke: J. Phys. B 25, 2351 (1992) T. Kirchner, L. Gulyas, H. J. Ludde, A. Henne, E. Engel, R. M. Dreizler: Phys. Rev. Lett 79, 1658 (1997) S. Sahoo, R. Das, N. C. Sil, S. C. Mukherjee, K. Roy: Phys. Rev. A 62, 022716 (2000) M. E. Rudd, Y. K. Kim, D. H. Madison, J. W. Gallagher: Rev. Mod. Phys. 57, 965 (1985) P. Pluvinage: Ann. Phys. N. Y. 5, 145 (1950) M. Brauner, J. S. Briggs, H. Klar: J. Phys. B 22, 2265 (1989) G. H. Wannier: Phys. Rev. 90, 817 (1953) E. P. Wigner: Phys. Rev. 73, 1002 (1948) M. R. H. Rudge, M. J. Seaton: Proc. R. Soc. A 283, 262 (1965) R. K. Peterkop: J. Phys. B 4, 513 (1971) A. R. P. Rau: Phys. Rev. A 4, 207 (1971) H. Jeffreys: Proc. Lond. Math. Soc. 23, 428 (1923) S. Cvejanovi´c, F. H. Read: J. Phys. B 7, 1841 (1974) P. Selles, A. Huetz, J. Mazeau: J. Phys. B 20, 5195 (1987) T. Rösel, J. Röder, L. Frost, K. Jung, H. Ehrhardt: J. Phys. B 25, 3859 (1992) J. F. McCann, D. S. F. Crothers: J. Phys. B 19, L399 (1986) D. S. F. Crothers, D. J. Lennon: J. Phys. B 21, L409 (1988) D. R. J. Carruthers, D. S. F. Crothers: J. Phys. B 24, L199 (1991) D. R. J. Carruthers, D. S. F. Crothers: Z. Phys. D 23, 365 (1992) D. R. J. Carruthers: Ph. D. Thesis (Queen’s University Belfast, Belfast 1993) C. Pan, A. F. Starace: Phys. Rev. A 45, 4588 (1992) F. B. M. Copeland, D. S. F. Crothers: J. Phys. B 27, 2039 (1994) L. D. A. Siebbeles, D. P. Marshall, C. Le Sech: J. Phys. B 26, L321 (1993) S. J. Buckman, P. Hammond, F. H. Read, G. C. King: J. Phys B 16, 4039 (1983) S. J. Buckman, D. S. Newman: J. Phys. B 20, L711 (1987) N. C. Deb, D. S. F. Crothers: J. Phys. B 33, L623 (2000) T. Rösel, J. Röder, L. Frost, K. Jung, H. Ehrhardt, S. Jones, D. H. Madison: Phys. Rev. A 46, 2539 (1992) N. C. Deb, D. S. F. Crothers: J. Phys. B 35, L85 (2002) P. Ashley, J. Moxom, G. Laricchia: Phys. Rev. Lett. 77, 1250 (1996)

787

Part D 52

52.24 52.25

H. Vogt, R. Schuch, E. Justiniano, M. Schultz, W. Schwab: Phys. Rev. Lett. 57, 2256 (1986) D. S. F. Crothers: J. Phys. B 18, 2893 (1985) D. S. F. Crothers, J. F. McCann: J. Phys. B 17, L177 (1984) S. F. C. O’Rourke, D. S. F. Crothers: J. Phys. B 29, 1969 (1996) D. P. Dewangan, J. Eichler: J. Phys. B 18, L65 (1985) D. S. F. Crothers: Relativistic Heavy-Particle Collision Theory (Kluwer Academic Plenum, New York 2000) J. Eichler, W. E. Meyerhof: Relativistic Atomic Collisions (Academic, San Diego 1995) J. T. Glass, J. F. McCann, D. S. F. Crothers: J. Phys. B 27, 3445 (1994) J. T. Glass, J. F. McCann, D. S. F. Crothers: Proc. R. Soc. Lond. A 453, 387 (1997) U. Becker: J. Phys. B 20, 6563 (1987) C. A. Bertulani, G. Baur: Phys. Rev. D 58, 4005 (1998) G. Baur, G. Boero, A. Brauksiepe, A. Buzzo, W. Eyrich, R. Geyer, D. Grzonka, J. Hauffe, K. Kilian, M. LoVetere, M. Macri, M. Moosburger, R. Nellen, W. Oelert, S. Passaggio, A. Pozzo, K. Röhrich, K. Sachs, G. Schepers, T. Sefzick, R. S. Simon, R. Stratmann, F. Stinzing, M. Wolke: Phys. Lett. B 368, 251 (1996) G. R. Deco, R. D. Rivarola: J. Phys. B 19, 1759 (1986) G. R. Deco, O. Fojon, J. Maidagan, R. D. Rivarola: Phys. Rev. A 47, 3769 (1993) W. J. Humphries, B. L. Moiseiwitsch: J. Phys. B 17, 2655 (1984) D. H. Jakubassa-Amundsen, P. A. Amundsen: Z. Phys. A 298, 13 (1980) J. F. McCann, J. T. Glass, D. S. F. Crothers: J. Phys. B 29, 6155 (1996) R. J. S. Lee, J. V. Mullan, J. F. McCann, D. S. F. Crothers: Phys. Rev. A 63, 062712 (2001) G. Baur: Phys. Lett. B 311, 343 (1993) J. Eichler: Phys. Rev. Lett. 75, 3653 (1995) P. O. Löwdin: Ark. Nat. Astron. Phys. 35A, 918 (1947) P. O. Löwdin: J. Chem. Phys. 18, 365 (1950) D. S. F. Crothers: Nucl. Instrum. Methods B 27, 555 (1987) D. S. F. Crothers, L. J. Dubé: Adv. At. Mol. Phys. 30, 287 (1993) D. S. F. Crothers, J. F. McCann: J. Phys. B 16, 3229 (1983)

References

789

Ionization in H

53. Ionization in High Energy Ion–Atom Collisions

Atomic species moving at high velocities form an important component of ionizing radiation. When these species interact with matter, they effect chemical and biological changes which originate with primary collision processes, usually the ejection of electrons. This chapter gives an overview of this primary process that has emerged from studies of the energy and angular distribution of electrons ejected by the impact of atomic or ionic projectiles on atomic targets. It seeks to highlight those features which are most

789 792 792 796 796 796

ubiquitous [53.1, 2]. Atomic units [53.3] are used, although e, m, and ~ are sometimes exhibited explicitly to show the connection with standard treatments of the first Born approximation [53.1].

(DDCS). Two somewhat different cross sections are used. The Galilean invariant cross sections are differential in the wave vectors k of the Schrödinger waves for the ejected electrons and are denoted by d3 σ/ dk3 . The wave vectors k refer to the laboratory or target frame. The Galilean invariant cross sections take the same form in any reference frame, including the projectile frame. In this frame the electron wave vectors are denoted by primes, k
. The DDCS is given in terms of an alternative expression in Sect. 53.1, (53.9).

53.1 Born Approximation The first Born approximation, often referred to as FBA or B1, provides an excellent framework to understand qualitatively most of the prominent features observed in fast ion–atom collisions. For a bare ion projectile of charge Z P impinging upon a neutral atom target of nuclear charge Z T , the Hamiltonian of the system is written as H = H0 + H1 ,

(53.1)

where

  ZT ZT 2 2
p2j PP2
Z e e T ,  + − + H0 = 2µ 2m rj |ri − r j | i> j

j=1

H1 =

ZP ZT R

e2

−

ZT


Z P2

j=1

|R− r j |

,

(53.2)

and where µ is the projectile reduced mass, m the electron mass, R the position vector of the projectile with respect to the target nucleus, PP the corresponding momentum operator, r j the position vector of the ith target electron and pj the corresponding momentum operator. The first Born approximation consists of breaking H as in (53.1): the wave functions employed are antisymmetric products of the target eigenfunctions of H0 and a plane wave of relative motion for the target and projectile. Transitions are induced by the interaction term H1 . For an inelastic collision, the target wave function is orthogonal to the initial state wave function and the matrix element of the first term in H1 , Z P Z T e2 /R, vanishes. Using Bethe’s

Part D 53

When high-velocity projectiles strike atomic targets, electrons are ejected from the target atoms, and, if the projectiles have some electrons, from the projectile ions also. For partially ionized projectiles, there are two groups of electrons, one from the target and one from the projectile. In principle, these two groups of electrons cannot be separately identified, but in practice each group often predominates in separate energy and angular regions, and can be identified with the help of computed distributions. These disstributions are expressed in terms of doubly differential cross sections

53.1 Born Approximation ............................ 53.2 Prominent Features ............................. 53.2.1 Target Electrons ........................ 53.2.2 Projectile Electrons .................... 53.3 Recent Developments........................... References ..................................................

790

Part D

Scattering Theory

integral 

The DDCS, as defined, is related to the Galilean invariant cross section d3 σ/ dk3 by [53.6]

eiq·R 3 4π d r = 2 eiq·R , |R− r| q

(53.3)

the inelastic cross section becomes [53.4]  Z2 4µ2 e4 K f dΩ K f 4P σi f = 4 Ki ~ q
× |Φ f | exp(iq · r j )|Φi |2 ,

(53.4)

j

where ~ K i and ~ K f are the initial and final momenta of the projectile, ~q = ~(K f − K i ) is the momentum transferred from the projectile to the target, Φ( f,i) represents the wave functions for the final ( f ) and initial (i) states of the target electrons, Ω K f is the solid angle of the scattered projectile, j sums over all target electron coordinates r j , and an average over initial and sum over final states is assumed [53.4]. The integral over dΩ K f can be performed by making the change of variables dΩ K f ≡ sin θ K f dθ K f dφ K f =

1 q dq dφ K f Ki K f (53.5)

and substituting ~ K i = µvi to obtain

Part D 53.1

σi f =

8πa02 Z P2 (vi /vB )2 ZT K + 2 dq |Φ f | j=1 exp(iq · r j )|Φi | × , q (qa0 )2 K−

(53.6)

where vi is the initial relative velocity and vB = αc is the atomic unit of velocity. (vi /vB )2 = T/R in the notation of [53.4]. From the definition of q, the limits of integration are K ± = |K i ± K f |. The cross section is independent of the mass of the incident projectile. The dimensionless generalized oscillator strength (GOS) is defined by [53.5]  2 ∆E |Φ f | j exp(iq · r j )|Φi | f i f (q) ≡ , (53.7) R∞ (qa0 )2 where ∆E is the energy lost by the projectile in the collsion; again, an average over initial and sum over final magnetic substates is assumed. In terms of the GOS, the cross section becomes σi f

8πa02 Z P2 = (vi /vB )2

K + K−

f i f (q) dq . ∆E/R∞ q

(53.8)

 m d3 σ d3 σi f if = 2 k 3 , dE k dΩk ~ d k

(53.9)

but it is more useful for describing the low energy ejected electrons since it connects smoothly with cross sections for discrete excitations of the target. The first Born approximation for both excitation and ionization by a bare projectile scale as Z P2 . The advantage of the GOS is that, in the limit q → 0, it approaches the optical oscillator strength – a relationship that connects ionization in fast ion–atom collisions with photoionization [53.7]. For an ionization process where an electron of momentum ~k is ejected from the target, the DDCS, differential in ejected electron energy E k and angle Ωk , denoted by d3 σi f / dE k dΩk , is also given by (53.4) with the proviso that the Φ f becomes Φ − f , representing a target ion plus an ejected (continuum) electron with incoming wave boundary conditions [53.5], normalized on the energy scale [53.3], i. e., 


−

Φ f (E)Φ − = δ E − E
. (53.10) f E If the energy is in Rydbergs, the asymptotic form of the radial part, r → ∞, of the continuum wave function of angular momentum is given by [53.7]

1  1 12 sin kr − π/2 + k−1 ln 2kr + σ + δ , r πk

(53.11)

where σ ≡ arg Γ( + 1 − i/k) is the Coulomb phase shift, and δ is the non-Coulomb phase shift due to the short range part of the potential. For normalization on the energy scale, in atomic units (1 a.u. √ = 2R∞ ), this asymptotic form is multiplied by 2, and for normalization on the k-scale, (53.11) is multiplied by √ 2k [53.3]. One can also obtain the single differential cross section (SDS), dσi f / dE k , by integrating over Ωk . This cross section can be expressed in terms of the differential GOS density in the continuum [53.5], d f i f (q) ∆E 1 = dE k R∞ (qa0 )2  ZT
× |Φ − exp(iq · r j )|Φi |2 dΩk , f | j=1

(53.12)

Ionization in High Energy Ion–Atom Collisions

as

53.1 Born Approximation

791

given by 8πa02 dσi f = dE k (vi /vB )2



d f i f (q)/ dE k dq , ∆E/R q

(53.13)

with ∆E = E k + IT , and IT is the ionization energy of the target. Finally, the total ionization cross section is obtained by integration of dσi f / dE k over E k , σi f =

8πa02 Z P2 (vi /vB )2

 Z P2

d f i f (q)/ dE k dq dE k . ∆E/R q (53.14)

Let us now consider a collision in which the projectile brings in its own NP electrons. The Hamiltonian for the system becomes

NP  


exp(iq · rk )ΦiP , Fii (q) = ΦiP 

(53.16)

k=1

and Φ(Pf,i) represents the wave function for the final and initial electron state of the projectile. Thus, the effect of the projectile electrons on process (a) is to screen the projectile nucleus, and the dynamical screening depends upon the momentum transfer q. Clearly from (53.16), Fii (0) = NP and Fii (∞) = 0. Thus, for small energy transfer, which implies small q (large impact parameter), Z P → Z P − NP , i. e., full screening of the projectile by its electrons. For large energy transfer, which implies q (small impact parameter), Z P remains unmodified, i. e., no screening.

H = H0 + H1

  ZT ZT 2 2
p2j p2P
e e Z T   + − H0 = + 2µ 2m rj |ri − r j | i> j

j=1

+

NP




k=1

ZP ZT R +

2m e2

−

e2

ZP + rk

NP


e2

i>k

|ri − rk |

ZT
NP
j=1 k=1

e2 |R+ rk − r j |

10–18 υ = 60°

 He++

ZT NP

Z P e2 Z T e2 − − |R− r j | |R+ rk | j=1

cm2 eV sr

10–19

Part D 53.1

H1 =

p2k

δ2σ δεδ⍀

k=1

(53.15)

where pk and rk refer to projectile electrons. Under these conditions, the solutions of H0 include the wave function of the projectile electrons in the antisymmetric product. For an inelastic collision, the matrix element of the first term of H1 vanishes as before. However, there are two extra terms in H1 : the interaction of the projectile electrons with the target nucleus, and the interaction of the target electrons with projectile electrons. These projectile electrons open physically distinct, alternative possibilities for the ionization process [53.2]: (a) target ionization, projectile remains in initial state; (b) target ionization, projectile excited (including ionization); (c) projectile ionization, target remains in initial state; (d) projectile ionization, target excited (including ionization). For process (a), evaluation of the first Born cross section gives (53.4) with the projectile charge Z P2 replaced by |Z P − Fii (q)|2 , where the elastic form factor Fii (q) is

10–20

He+

10–21

10–22

H+

0

5

10

15

20

25

30

35 ε (Ry)

Fig. 53.1 Theoretical double differential cross sections

(DDCS) for the ionization of He by equal velocity H+ , He++ and He+ (target ionization with no projectile excitation only) as a function of ejected electron energy in Ry for an ejection angle of 60◦ . The incident velocity corresponds to 0.5 MeV H+ and 2.0 MeV He+ and He++ which all have the same velocity as a 20 Ry (272 eV) electron

792

Part D

Scattering Theory

To illustrate this effect, a calculation of process (a) is presented in Fig. 53.1 for the DDCS for 2 MeV He+ + He collisions, along with a DDCS calculation for equal velocity H+ and He++ projectiles [53.8]. From these results it is seen that for small energy transfer, the He+ projectile behaves almost like a heavy proton. With increasing ejected electron energy, however, it is seen to approach He++ -like behavior. Note that this result is process (a) alone. Looking at process (b), the situation is rather different. Now two electrons change their state so that transition matrix elements of all terms except the electron–electron term in H1 vanish. In this case, Z P2 in (53.4) is replaced by the square of the inelastic projectile form factor |Ffi |2 , where NP  


exp(iq · rk )ΦiP . F fi (q) = Φ Pf 

(53.17)

k=1

Part D 53.2

Owing to the orthogonality of initial and final projectile states, this form factor vanishes in both the limits, q → 0 and q → ∞. Thus, the total target DDCS is process (a) plus the sum over all of the possible projectile excitations (including projectile ionization) of process (b). Since there is an infinite number of projectile excitations, it is useful to have a method for summing over all of them. Using the closure relation,

the sum over all projectile excitations at fixed q is given by [53.2]


 |Ffi (q)|2 = NP − |Fii (q)|2 + Φ P  i

f =i

×

NP


  exp[iq · (r j − r j
)]ΦiP ,

j, j
, j = j


(53.18)

where only the initial state wave function of the projectile electrons appears. While this sum rule is exact, it cannot be substituted exactly into (53.4) because of the transformation between cos θ K f and q, and hence the limits of integration over q vary with the excitation energy of the final state of the projectile. Various ways of choosing approximate integration limits have been suggested [53.9]. Projectile ionization alone, and with target excitation, process (c) and (d) above, are handled exactly as (a) and (b), but in the projectile reference frame. Thus, ionization of the projectile by the target is calculated using the methodology detailed above, after which the results are transformed into the laboratory frame using the invariance of d3 σi f / d3 k and kL = kP + K i /µ. The subscripts L and P refer to the laboratory and projectile frame, respectively.

53.2 Prominent Features A plot of d3 σ/ dk3 superimposes electrons from the target and electrons from the projectile. To sort out the main features of the two groups of electrons, first consider impact of bare ions on neutral targets, where the DDCS exhibits only electrons ejected from the target. Figure 53.2 shows a computed cross section for protons on H. The wave vectors are resolved into two components: a component k , which is parallel to the wave vector K i of the projectile in the laboratory frame, and a component k⊥ perpendicular to K i . In most cases, integration is over the direction of K f so that K i is an axis of symmetry. Then the DDCS is independent of the direction of k⊥ . Because Fig. 53.2 was computed using an approximate theory described in Sect. 53.2.1, the details are not necessarily accurate, but these computations are qualitatively reliable over an electron energy and angular range which excludes very low electron energies, i. e., the region around k = 0.

53.2.1 Target Electrons The Bethe Ridge Figure 53.2 provides a useful starting point to examine electron energy and angular distributions. The most prominent feature is the semicircular ridge, called the Bethe ridge, surrounding a valley. The center of the semicircle is at k = v, where v = K i /MP is the velocity of the projectile in the lab frame. In the projectile frame, where k
= k − v, the center of the circle is at k
= 0 and the ridge is at k
= v. This region corresponds to ionization events where the momementum J = q − k of the recoiling target ion vanishes. Momentum conservation implies that Ki = K f + k + J , (53.19) while energy conservation is expressed as K 2f K i2 J2 k2 + εi = + + , (53.20) 2MP 2MP 2 2MT

Ionization in High Energy Ion–Atom Collisions

where MT is the mass of the ionized target. At the Bethe ridge, J ≈ 0 so that (53.19) and (53.20) combine to determine the value of k at this point, called kB ;

0 (53.21)

–2

where εi is the initial electron eigenenergy. In the approximation that m/MP is set equal to zero and the inital binding energy is small compared with 1/2v2 , (53.21) shows that the prominent ridge seen in Fig. 53.2 at k
= v corresponds to collisions where all of the momentum lost by the projectile is transferred to the ejected electron. This ridge extends from E k = 0 up to electron energies of the order of 2v2 . Generally, ejected electrons move in the combined field of both the target and the projectile. Electrons with momenta k in the region 0 < k < vZ P /(Z P + Z P ) will be referred to as low energy electrons since they move in regions where the target potential is stronger than that of the projectile. For very fast projectiles, many more electrons are in the low energy region than at higher energies, thus this region is of special interest. Since the target potential influences the motion significantly, the ionization process in this region represents a continuation of excitation across the ionization threshold. To analyze this region, cross sections differential in E k and angle Ωk in the target frame are preferable since they connect smoothly with cross sections for exciting target states (53.4). Because the DDCS is independent of the azimuthal angle ϕk , the DDCS integrated over ϕk is often employed. As the variables E k , θk suggest, the main features of the low energy electrons emerge in plots of energy and angular distributions. For fixed energy, the Bethe ridge appears at an angle θB given by (53.21) and can be written as a relation between cos θB and E k as   E k + εi 1 . (53.22) cos θB = √ 2E k v

–4

(53.23)

–6 –8 –10 –12 –14 0 2 4 6 8 10 12

kP perpendicular 14

15

10

5

0

–5

–15 –10 kP parallel

Fig. 53.2 Plot of the Galilean invariant DDCS for electrons ejected from atomic hydrogen by 1.5 keV proton impact computed in the distorted wave strong potential Born approximation (DSPB)

For electron energies E k of the order of the initial binding energy |εi | and large v, the parallel component of q is much smaller than the magnitude given by (53.19), so that q is predominantly perpendicular to the beam direction. From (53.4), electrons ejected

2

DDCS(10–24m2/ sr)

1.5

1

0.5

0

0

40

80

120

160 Angle (deg)

Fig. 53.3 Plot of the angular distribution of 13.6 eV electrons ejected from He by 5 keV proton impact [53.10]. The solid curve is the Born approximation and the points are the measured values

Part D 53.2

For fixed E k = 0, the DDCS maximizes at an angle that approaches 90◦ as v → ∞; in particular this angle is obtained when E k = −εi . Figure 53.3 shows the angular distribution of 13.6 eV electrons ejected from He by 5 MeV proton impact [53.10]. The cross section maximizes at, and is nearly symmetric about, 90◦ . This feature is understandable from the Born approximation. For small deflection angles, (53.20) gives E k + εi . v

793

log (DDCS (arb. units)) 2

[(1 + m/MP )kB − v]2 = v2 + (1 + m/MP )εi

q ≈ K i − K f ≈

53.2 Prominent Features

794

Part D

Scattering Theory

Part D 53.2

from isotropic inital states are distributed symmetrically about this axis, i. e., about 90◦ , in first approximation. Because q is averaged over in forming the cross section, all directions contribute to some extent, and the distribution is not completely symmetric. The solid curve in Fig. 53.3 represents a distribution computed in the Bethe–Born approximation, and it is seen to peak at 90◦ but is somewhat asymmeteric about this angle, reflecting effects of averaging over the direction of q. Such distributions are well described by only a few partial waves for the outgoing electron. Indeed, the theoretical distribution is well described by a combination of s and p waves. The calculations fit the experimental data except in the forward direction where the experimental distribution increases sharply. Such sharp increases must reflect the presence of many partial waves not described by the Born approximation. Since the main disagreement between theory and experiment is confined to a small angular region near 0◦ which contributes only a small part to the total cross section, the Bethe–Born approximation represents a well-founded theoretical framework for interpreting features that appear in the low-electron-energy portion of the DDCS. While the main contribution to total ionization cross sections comes from the low energy region, electrons in the high energy region (v/2 < k) carry more energy per electron, and therefore play a prominent role in processes initiated by fast electrons. The fast electrons on the Bethe ridge correspond to collisions where the momentum q lost by the incident projectile is transferred mainly to the ejected electron. Because the momentum k of the ejected electrons is much larger than the mean value of √ the electron momentum in the initial state, given by 2εi , this portion of the spectrum can be interpreted in terms of a binary collision between the incident projectile and the target electron treated as quasifree. By quasifree we understand that the electron has a nonzero binding energy εi , but is otherwise regarded as a free electron with an initial momentum s, distributed over a range of values centered around the mean value. In the projectile reference frame, the electron has the momentum s − v. The electron scatters elastically from the projectile and emerges with a final momentum k
= k − v. This simple picture is known as the elastic scattering model, and is often employed for processes involving weakly bound electrons [53.11]. There are several quantal versions of this picture, indeed any theory of ionization must reduce to the electron scattering model in the limit that the initial binding energy εi vanishes.

When the projectile P is a bare ion so that the binary scattering process is just Rutherford scattering, the Bethe–Born approximation is in accord with this picture. It works because the electrons are fast so that effects of the target potential in the final state are unimportant, and because a first-order computation of Rutherford scattering gives the same scattering cross section as the exact Rutherford amplitude. For that reason the domain of applicability of the first Born approximation includes the Bethe ridge, even in the high energy region. The binary encounter peak shows new features when the projectile carries electrons. The target electrons scatter from a partially screened ion and the Bethe ridge reflects properties of the elastic scattering cross section for the complex projectile species. In contrast to the Rutherford cross section, elastic scattering cross sections for complex species may have maxima and minima as functions of the electron energy and angle. Such features have been identified in collisions of highly charged ions with neutral atomic targets [53.12]. The Continuum Electron Capture Cusp Figure 53.2 shows a sharp cusp-like structure when the momentum of the ejected electron in the projectile frame k
vanishes. Then the ejected electron moves with a velocity exactly equal to the projectile velocity. For charged projectiles, it is virtually impossible to determine whether such electron states are Rydberg states of high principal quantum number n
, or continuum states with E k
= 12 k
2 much less than εi . Indeed, the physical similarity of such atomic states implies that the cross section for electron capture to states of high n
differs from the ionization cross section near k
= 0 only by a density of states factor. This factor is just dE k
for ionization processes and dE n
= Z P2 n
−3 for capture. It follows that the cross section d3 σ/ dE k
dΩk
is a smooth function of E k
which is nonzero and finite at E k
= 0. The Galilean invariant cross section of Fig. 53.2, however, is not smooth; rather it is given by

d3 σ 1 d3 σ d3 σ = =
. 3 3 k dE k
dΩk
dk dk


(53.24)

Since d3 σ/ dE k
dΩk
is nonzero at k
= 0, it follows that the DDCS has a k
−1 singularity at k
= 0. Experiments measure cross sections averaged over this singular factor. The averaging results in the cusp-shaped feature at k
= 0 seen in Fig. 53.2 [53.13–15].

Ionization in High Energy Ion–Atom Collisions

The theoretical description of the cusp feature requires that the electron in the final state move mainly in the field of the projectile rather than that of the target. While the appropriate final state wave function can be introduced into (53.4), the result is quantitatively inaccurate. A more accurate amplitude emerges when the exact amplitude is expanded in powers the interaction potential VT of the electron with the final state target ion [53.16], since VT has a much smaller effect than the interaction VP . Approximate evaluation of the ionization amplitude in an independent particle approximation gives [53.17]  Tfi = d3 s V˜ T (J + s)ϕ˜ i (s) × ψk−
(r)| exp[i(J + s)]|ψs−v (r) , and d3 σ = (2π)4 µ2 dE k dΩk

(53.26)

where µ is the reduced mass given by µ = MP (MP + 1)/(MP + MT + 1) ,

s-v

a)

795

e–

rP rT

P

b)

1– 2

(s-v)2

T

~ V T (J + s) e–i(J + S)r

1– 2

k' 2

(53.25)

 |Tfi |2 dΩ K f

53.2 Prominent Features

Fig. 53.4a,b Free–free transition picture of ionization. (a) Schematic representation of a target electron with momentum s − v in the projectile frame. (b) A free–free − to the fitransition from the projectile eigenstate ψs−v nal eigenstate ψk+
occurs with the target interaction as the transition operator

(53.27)

from the projectile into a final state with projectile frame momentum k
. This quasi-elastic cross section is averaged over the initial momentum distribution of the electron |ϕ˜ i (−J)|2 d3 J so that   k
d5 σ ESM = (2π)4 |T elas (k
, −J − v)|2 | dΩk
v × |ϕ˜ i (−J)|2 d3 J . (53.28) The projectile frame cross section is recovered upon using the relation d3 J = µ f µi v dΩ K f k
2 dk
in (53.25). The two approximations (53.4) and (53.25) describe all of the major features that emerge in the ionization of one-electron targets by fully ionized projectiles. Both theories also incorporate some multi-electron effects that appear when either the target or projectile (or both) have several electrons. Equation (53.28) is useful for describing effects of target ionization by multi-electron projectiles when the projectile electrons remain in their ground state since T elas incorporates, in principle, the exact amplitude for electrons to scatter from the projectile in its ground state. Of course, the multi-electron projectile can also become excited or ionized owing to its interaction with the target electrons. These processes are not incorporated in the ESM cross section (53.28); rather the Born amplitude of (53.4) represents a more tractable theory to analyze these features.

Part D 53.2

and ϕ˜ i (s) is the momentum space independent-particle orbital wave function for the active electron in the intial state, V˜ T is the corresponding effective interaction potential, and the ψk±
are normalized on the momentum scale. Figure 53.4 gives a pictorial interpretation of this amplitude. The initial momentum distribtution of the electron is given by ϕ˜ i (s). In the projectile frame, this electron has momentum s − v and moves in + a projectile continuum state represented by ψs−v (r). Owing to the interaction with the target represented by exp[i(J +s)] V˜T (J + s), a transition to the final projectile state ψk−
occurs as√illustrated in Fig. 53.4 (b). The final state ψk−
has a 1/ k
normalization that gives rise to the cusp at k = v seen in Fig. 53.2. This picture describes ionization in terms of the free–free transition, − ψs−v → ψk+
. The amplitude in (53.25), known as the distorted wave strong potential Born amplitude (DSPB) [53.18], or the projectile impulse approximation amplitude [53.19], also describes binary encounter electrons. Thus the regions of applicability of (53.4) and (53.25) overlap considerably. In the binary encounter region, where k > v/2 and J ≈ 0, the cross section given by (53.9) and (53.25) reduces to the elastic scattering model (ESM) [53.20], where the incident electron with projectile frame momentum −J − v quasi-elastically scatters

796

Part D

Scattering Theory

53.2.2 Projectile Electrons When the incident atomic species carries some attached electrons, these electrons may be removed in the collision with a target. Essentially all of the features that appear for the target electrons also appear for the projectile electrons; however the DDCS is shifted by the Galilean transformation to the projectile frame. The transformation from the laboratory frame to the projectile frame is an essential step for interpreting features of the projectile electrons. This transformation has the effect of spreading out the electron energy distribution since at θk = 0◦ a small energy interval ∆E k in the

lab frame relates to a small energy interval ∆E k
in the projectile frame according to   v (53.29) ∆E k = ∆E k
1 + √ . 2vE k
For example, when the projectile energy is 1.5 MeV/au, a projectile frame electron energy interval 0 < E k
< 0.1 eV maps into an interval of ±8.3 eV centered at E k = 817 eV. This amplification of both the electron energy and the electron energy interval proves useful for measuring features of projectile species with high resolution [53.21].

53.3 Recent Developments Added by Mark M. Cassar. The theoretical study of ion-

Part D 53

ization processes in atomic collisions remains an active research field. At present, there are three main quantum mechanical approaches to the simplest colliding systems with one active electron. The first approach is to solve directly the time-dependent Schrödinger equation by taking advantage of modern-day computing power. The accuracy of these calculations, however, is still insufficient, and further computational and technological advancements are necessary. The second approach expands the total wave function in atomic or molecular bases; satisfactory agreement with experimental results is obtained, but at the expense of having to include a large

number of basis set members in the expansions. The third approach involves the use of Sturmian expansions, along with a specific scaling and transformation of the wave functions. This technique overcomes many of the difficulties that others suffer from, and at the same time provides the best agreement with experimental data over a broad range of energies. Problems in our understanding of the ionization process remain, even for the simplest colliding system H+ -H, and extensions and refinements of the present theoretical and experimental methods are needed. The reader is referred to the in-depth review by Macek et al. [53.22].

References 53.1 53.2

53.3

53.4 53.5 53.6 53.7

U. Fano: Ann. Revs. Nucl. Sci. 13, 1 (1963) J. S. Briggs, K. Taulbjerg: Theory of inelastic atom– atom collisions. In: Structure and Collisions in Ions and Atoms, ed. by I. A. Sellin (Springer, Berlin, Heidelberg 1978) pp. 105–153 H. A. Bethe, E. E. Salpeter: Quantum Mechanics of One- and Two-Electron Atoms (Plenum, New York 1977) M. Inokuti: Rev. Mod. Phys. 43, 297 (1971) S. Geltman: Topics in Atomic Collision Theory (Academic, New York 1969) pp. 41–44 F. Drepper, J. S. Briggs: J. Phys. B 9, 2063 (1976) A. F. Starace: Theory of atomic photoionization. In: Handbuch der Physik, Vol. 31, ed. by W. Mehlhorn (Springer, Berlin, Heidelberg 1982)

53.8 53.9 53.10 53.11 53.12

53.13 53.14

S. T. Manson, L. H. Toburen: Phys. Rev. Lett. 46, 529 (1981) H. M. Hartley, H. R. J. Walters: J. Phys. B 20, 1983 (1987) S. T. Manson, L. Toburen, D. Madison, N. Stolterfoht: Phys. Rev. A 12, 60 (1975) M. M. Duncan, M. G. Menendez: Phys. Rev. A 16, 1799 (1977) D. H. Lee, P. Richard, T. J. M. Zouros, J. M. Sanders, J. L. Shinpaugh, H. Hidmi: Phys. Rev. A 41, 4816 (1990) M. E. Rudd, J. H. Macek: Case Studies in Atomic Physics, 3, 47 (1972) G. G. Crooks, M. E. Rudd: Phys. Rev. Lett. 25, 1599 (1970)

Ionization in High Energy Ion–Atom Collisions

53.15 53.16 53.17 53.18

K. G. Harrison, M. W. Lucas: Phys. Lett. 33 A, 142 (1970) J. S. Briggs: J. Phys. B 10, 3075 (1977) M. Brauner, J. H. Macek: Phys. Rev. A 46, 2519 (1992) K. Taulbjerg, R. Barrachina, J. H. Macek: Phys. Rev. A 41, 207 (1990)

53.19 53.20 53.21 53.22

References

797

D. Jakubassa-Amundsen: J. Phys. B 16, 1767 (1983) J. Macek, K. Taulbjerg: J. Phys. B 26, 1353 (1993) N. Stolterfoht: Phys. Rep. 146, 315 (1987) S. Yu. Ovchinnikov, G. N. Ogurtsov, J. H. Macek, Yu. S. Gordeev: Phys. Rep. 389, 119 (2004)

Part D 53

799

Electron–Ion a

54. Electron–Ion and Ion–Ion Recombination

54.4.2 Quantal Cross Section................. 808 54.4.3 Noncrossing Mechanism............. 810

Electron–ion and ion–ion recombination processes are of key importance in understanding the properties of plasmas, whether they are in the upper atmosphere, the solar corona, or industrial reactors on earth. This is a collection of formulae, expressions, and specific equations that cover the various aspects, approximations, and approaches to electron-ion and ion-ion recombination processes.

54.1 Recombination Processes ..................... 54.1.1 Electron–Ion Recombination ...... 54.1.2 Positive–Ion Negative–Ion Recombination ......................... 54.1.3 Balances .................................. 54.2 Collisional-Radiative Recombination ..... 54.2.1 Saha and Boltzmann Distributions............................. 54.2.2 Quasi-Steady State Distributions..................... 54.2.3 Ionization and Recombination Coefficients .............................. 54.2.4 Working Rate Formulae ..............

54.6 One-Way Microscopic Equilibrium Current, Flux, and Pair-Distributions..... 811

800 800 800 800 801 801 802 802 802 803

803

804

805

806

54.4 Dissociative Recombination .................. 807 54.4.1 Curve-Crossing Mechanisms........ 807

54.7 Microscopic Methods for Termolecular Ion–Ion Recombination ....................... 54.7.1 Time Dependent Method: Low Gas Density ........................ 54.7.2 Time Independent Methods: Low Gas Density ........................ 54.7.3 Recombination at Higher Gas Densities .................................. 54.7.4 Master Equations ...................... 54.7.5 Recombination Rate .................. 54.8 Radiative Recombination ..................... 54.8.1 Detailed Balance and Recombination-Ionization Cross Sections ........................... 54.8.2 Kramers Cross Sections, Rates, Electron Energy-Loss Rates and Radiated Power for Hydrogenic Systems .............. 54.8.3 Basic Formulae for Quantal Cross Sections ................................... 54.8.4 Bound-Free Oscillator Strengths.. 54.8.5 Radiative Recombination Rate .... 54.8.6 Gaunt Factor, Cross Sections and Rates for Hydrogenic Systems ................................... 54.8.7 Exact Universal Rate Scaling Law and Results for Hydrogenic Systems ...................................

812 813 814 815 816 816 817

817

818 819 822 822

823

823

54.9 Useful Quantities ................................. 824 References .................................................. 824

Part D 54

54.3 Macroscopic Methods ........................... 54.3.1 Resonant Capture-Stabilization Model: Dissociative and Dielectronic Recombination . 54.3.2 Reactive Sphere Model: Three-Body Electron–Ion and Ion–Ion Recombination....... 54.3.3 Working Formulae for Three-Body Collisional Recombination at Low Density .... 54.3.4 Recombination Influenced by Diffusional Drift at High Gas Densities ..................................

54.5 Mutual Neutralization .......................... 810 54.5.1 Landau–Zener Probability for Single Crossing at R X ............. 811 54.5.2 Cross Section and Rate Coefficient for Mutual Neutralization ........... 811

800

Part D

Scattering Theory

54.1 Recombination Processes 54.1.1 Electron–Ion Recombination This proceeds via the following four processes: (a) radiative recombination (RR) e− + A+ (i) → A(n
) + hν ,

(54.1)

(b) three-body collisional-radiative recombination e− + A+ + e− → A + e− , −

(54.2a)

+

e + A +M → A+M ,

(54.2b)

where the third body can be an electron or a neutral gas. (c) dielectronic recombination (DLR)
 e− + A Z+ (i)
A Z+ (k) − e− →

A(Z−1)+ (f n 


n


) + hν ,

(54.3)

(d) dissociative recombination (DR) e− + AB + → A + B ∗ .

(54.4)

Processes (a), (c), (d) and (e) are elementary processes in that microscopic detailed balance (proper balance) exists with their true inverses, i. e., with photoionization (both with and without autoionization) as in (c) and (a), associative ionization and ion-pair formation as in (d) and (e), respectively. Processes (b), (f) and (g) in general involve a complex sequence of elementary energy-changing mechanisms as collisional and radiative cascade and their overall rates are determined by an input-output continuity equation involving microscopic continuum-bound and bound–bound collisional and radiative rates.

54.1.3 Balances Proper Balances Proper balances are detailed microscopic balances between forward and reverse mechanisms that are direct inverses of one another, as in

(a) Maxwellian: e− (v1 ) + e− (v2 )
e− (v1 ) + e− (v2 ) , (54.8)

Electron recombination with bare ions can proceed only via (a) and (b), while (c) and (d) provide additional pathways for ions with at least one electron initially or for molecular ions AB + . Electron radiative capture denotes the combined effect of RR and DLR.

54.1.2 Positive–Ion Negative–Ion Recombination Part D 54.1

(54.5)

between direct ionization from and direct recombination into a given level n
; (54.10)

(d) Planck: e− + H+
H(n
) + hν ,

(54.11)

which involves interaction between radiation and atoms in photoionization/recombination to a given level n
. (54.6)

(g) tidal recombination AB + + C − + M → AC + B + M → BC + A + M ,

(54.9)

between excitation and de-excitation among bound levels;

(f) three-body (termolecular) recombination A+ + B − + M → AB + M ,

(b) Saha: e− + H(n
)
e− + H+ + e−

(c) Boltzmann: e− + H(n
)
e− + H(n  ,
 )

This proceeds via the following three processes: (e) mutual neutralization A+ + B − → A + B ∗ ,

where the kinetic energy of the particles is redistributed;

(54.7a) (54.7b)

where M is some third species (atomic, molecular or ionic). Although (e) always occurs when no gas M is present, it is greatly enhanced by coupling to (f). The dependence of the rate αˆ on density N of background gas M is different for all three cases, (e)–(g).

Improper Balances Improper balances maintain constant densities via production and destruction mechanisms that are not pure inverses of each other. They are associated with flux activity on a macroscopic level as in the transport of particles into the system for recombination and net production and transport of particles (i. e. e− , A+ ) for ionization. Improper balances can then exist between dissimilar elementary production–depletion processes as in (a) coronal balance between electron-excitation into and radiative decay out of level n. (b) radiative

Electron–Ion and Ion–Ion Recombination

balance between radiative capture into and radiative cascade out of level n. (c) excitation saturation balance between upward collisional excitations n−1 → n → n+

54.2 Collisional-Radiative Recombination

801

1 between adjacent levels. (d) de-excitation saturation balance between downward collisional de-excitations n+1 → n → n−1 into and out of level n.

54.2 Collisional-Radiative Recombination Radiative Recombination Process (54.1) involves a free-bound electronic transition with radiation spread over the recombination continuum. It is the inverse of photoionization without autoionization and favors high energy gaps with transitions to low n ≈ 1, 2, 3 and low angular momentum states
≈ 0, 1, 2 at higher electron energies.

Collisional-Radiative Recombination Here the cascade collisions and radiation are coupled via the continuity equation. The population n i of an individual excited level i of energy E i is determined by the rate equations

Production Rates and Processes The production rate for a level i is  R Pi = n e n f K cfi + n 2e N + kci f =i

+



  n f A fi + B fi ρν

f>i

  + n e N + αˆ iRR + βi ρν ,

(54.14)

where the terms in the above order represent (1) collisional excitation and de-excitation by e− –A( f ) collisions, (2) three-body e− –A+ collisional recombination into level i, (3) spontaneous and stimulated radiative cascade, and (4) spontaneous and stimulated radiative recombination. Destruction Rates and Processes The destruction rate for a level i is  K icf + n e n i Si n i Di = n e n i f =i

+ ni



Ai f + Bi f ρν



f

dn i ∂n i = + ∇ · (n i vi ) dt ∂t    n f ν fi − n i νi f = Pi − n i Di , = i= f

(54.12)

+ ni



Bi f ρν + n i Bic ρν ,

(54.15)

f>i

(54.13)

where the terms in the above order represent (1) collisional destruction, (2) collisional ionization,

Part D 54.2

Three-Body Electron–Ion Recombination Processes (54.2a,b) favor free-bound collisional transitions to high levels n, within a few kB T of the ionization limit of A(n) and collisional transitions across small energy gaps. Recombination becomes stabilized by collisional-radiative cascade through the lower bound levels of A. Collisions of the e−−A+ pair with third bodies becomes more important for higher levels n and radiative emission is important down to and among the lower levels n. In optically thin plasmas this radiation is lost, while in optically thick plasmas it may be re-absorbed. At low electron densities radiative recombination dominates with predominant transitions taking place to the ground level. For process (54.2a) at high electron densities, threebody collisions into high Rydberg levels dominate, followed by cascade which is collision dominated at low electron temperatures T e and radiation dominated at high T e . For process (54.2b) at low gas densities N, the recombination is collisionally-radiatively controlled while, at high N, it eventually becomes controlled by the rate of diffusional drift (54.61) through the gas M.

which involve temporal and spatial relaxation in (54.12) and collisional-radiative production rates Pi and destruction frequencies Di of the elementary processes included in (54.13). The total collisional and radiative transition frequency between levels i and f is νi f and the f -sum is taken over all discrete and continuous (c) states of the recombining species. The transition frequency νi f includes all contributing elementary processes that directly link states i and f , e.g., collisional excitation and deexcitation, ionization (i → c) and recombination (c → i) by electrons and heavy particles, radiative recombination (c → i), radiative decay (i → f ), possibly radiative absorption for optically thick plasmas, autoionization and dielectronic recombination.

802

Part D

Scattering Theory

(3) spontantaneous and stimulated emission, (4) photoexcitation, and (5) photoionization.

QSS-condition, dn i / dt = 0 for the bound levels i = 1, therefore holds. The QSS distributions n i therefore satisfy Pi = n i Di .

54.2.1 Saha and Boltzmann Distributions Collisions of A(n) with third bodies such as e− and M are more rapid than radiative decay above a certain excited level n ∗ . Since each collision process is accompanied by its exact inverse the principle of detailed balance determines the population of levels i > n ∗ . Saha Distribution This connects equilibrium densities n˜ i , n˜ e and N˜ + of bound levels i, of free electrons at temperature T e and of ions by  n˜ i g(i) h3 = exp(Ii /kB T e ) , (2πm e kT e )3/2 g e g+ n˜ e N˜ + A (54.16)

where the electronic statistical weights of the free electron, the ion of charge Z + 1 and the recombined e−−A+ species of net charge Z and ionization potential Ii are g e = 2, g+ ˜ i for A and g(i), respectively. Since n i ≤ n all i, then the Saha–Boltzmann distributions imply that n 1  n i and n e  n i for i = 1, 2, where i = 1 is the ground state.

Part D 54.2

Boltzmann Distribution This connects the equilibrium populations of bound levels i of energy E i by   n˜ i /n˜ j = [g(i)/g( j )] exp −(E i − E j )/kB T e . (54.17)

54.2.2 Quasi-Steady State Distributions The reciprocal lifetime of level i is the sum of radiative and collisional components and is therefore shorter than the pure radiative lifetime τR ≈ 10−7 Z −4 s. The lifetime τ1 for the ground level is collisionally controlled, is dependent upon n e , and generally is within the range of 102 and 104 s for most laboratory plasmas and the solar atmosphere. The excited level lifetimes τi are then much shorter than τ1 . The (spatial) diffusion or plasma decay (recombination) time is then much longer than τi and the total number of recombined species is much smaller than the ground-state population n 1 . The recombination proceeds on a timescale much longer than the time for population/destruction of the excited levels. The condition for quasisteady state, or

54.2.3 Ionization and Recombination Coefficients Under QSS, the continuity equation (54.13) then reduces to a finite set of simultaneous equations Pi = n i Di . This gives a matrix equation which is solved numerically for n i (i = 1) ≤ n˜ i in terms of n 1 and n e . The net groundstate population frequency per unit volume (cm−3 s−1 ) can then be expressed as dn 1 = n e N + αˆ CR − n e n 1 SCR , dt

(54.18)

where αˆ CR and SCR , respectively, are the overall rate coefficients for recombination and ionization via the collisional-radiative sequence. The determined αˆ CR equals the direct (c → 1) recombination to the ground level supplemented by the net collisional-radiative cascade from that portion of bound-state population which originated from the continuum. The determined SCR equals direct depletion (excitation and ionization) of the ground state reduced by the de-excitation collisional radiative cascade from that portion of the bound levels accessed originally from the ground level. At low n e , αˆ CR and SCR reduce, respectively, to the radiative recombination coefficient summed over all levels and to the collisional ionization coefficient for the ground level. C, E and
Blocks of Energy Levels For the recombination processes (54.2a), (54.2b) and (54.6) which involve a sequence of elementary reactions, the e− − A+ or A+ − B − continuum levels and the ground A(n = 1) or the lowest vibrational levels of AB are therefore treated as two large particle reservoirs of reactants and products. These two reservoirs act as reactant and as sink blocks C and
which are, respectively, drained and filled at the same rate via a conduit of highly excited levels which comprise an intermediate block of levels E. This C draining and
filling proceeds, via block E, on a timescale large compared with the short time for a small amount from the reservoirs to be re-distributed within block E. This forms the basis of QSS.

54.2.4 Working Rate Formulae For electron–atomic–ion collisional-radiative recombination (54.2a), detailed QSS calculations can be fitted by

Electron–Ion and Ion–Ion Recombination

the rate [54.1]

αˆ CR = 3.8 × 10−9 T e−4.5 n e + 1.55 × 10−10 T e−0.63  +6 × 10−9 T e−2.18 n 0.37 (54.19) cm3 s−1 e agrees with experiment for a Lyman α optically thick plasma with n e and T e in the range 109 cm−3 ≤ n e ≤ 1013 cm−3 and 2.5 K ≤ T e ≤ 4000 K. The first term is the pure collisional rate (54.49), the second term is the radiative cascade contribution, and the third term arises from collisional-radiative coupling.

54.3 Macroscopic Methods

803

  For e− − He+ recombination in a high (5– 2 100 Torr) pressure helium afterglow the rate for (54.2b) is [54.2]
 αˆ CR = (4 ± 0.5) × 10−20 n e (T e /293)−(4±0.5)
+ (5±1) × 10−27 n(He)  + (2.5 ± 2.5) × 10−10 × (T e /293)−(1±1) cm3 /s.

(54.20)

The first two terms are in accord with the purely collisional rates (54.49) and (54.52b), respectively.

54.3 Macroscopic Methods 54.3.1 Resonant Capture-Stabilization Model: Dissociative and Dielectronic Recombination The electron is captured dielectronically (54.41) into an energy-resonant long-lived intermediate collision complex of super-excited states d which can autoionize or be stabilized irreversibly into the final product channel f either by molecular fragmentation kc

νs

e− + AB + (i)
AB ∗∗ → A + B ∗ , νa

(54.21)

νa νs

→

n
(Z−1)+ An 
 ( f ) + hν

.

(54.22)

Production Rate of Super-Excited States d

dn ∗d = n e N + kc (d) − n ∗d [ν A (d) + νS (d)] ; (54.23) dt  ν A (d) = νa (d → i  ) , (54.24a)

and the rate to all product channels is  kc (d)νS (d) . αˆ = ν A (d) + νS (d)

νS (d) =

νs (d → f  ) .

(54.24b)

f

Steady-State Distribution For a steady-state distribution, the capture volume is n ∗d kc (d) . = (54.25) n e N+ ν A (d) + νS (d)

(54.26b)

d

In the above, the quantities P Sf (d) = νs (d → f )/ [ν A (d) + ν S (d)] ,

(54.27)

P (d) = νS (d)/ [ν A (d) + νS (d)] ,

(54.28)

S

represent the corresponding stabilization probabilities. Macroscopic Detailed Balance and Saha Distribution n˜ ∗d kc (d) = kc (d)τa (d → i) K di (T ) = = νa (d → i) n˜ e N˜ +

=

i



(54.26a)

d



ω(d) (2πm e kB T )3/2 2ω+   ∗ × exp −E di /kB T , h3

(54.29a)

(54.29b)

∗ is the energy of super-excited neutral levels where E di AB ∗∗ above that for ion level AB + (i), and ω are the corresponding statistical weights.

Alternative Rate Formula

 νa (d → i)νs (d → f ) αˆ f = . K di ν A (d) + νS (d) d

(54.30)

Part D 54.3

as in direct dissociative recombination (DR), or by emission of radiation as in dielectronic recombination (DLR)  kc
e− + A Z+ (i)
A Z+ (k) − e−

Recombination Rate and Stabilization Probability The recombination rate to channel f is  kc (d)νs ( d → f ) αˆ f = , ν A (d) + νS (d)

804

Part D

Scattering Theory

Normalized Excited State Distributions

νa (d → i) ρd = n ∗d /n˜ ∗d = , [ν A (d) + νS (d)]   kc (d)P S (d) = K di ρd νS (d) αˆ = d

=



(54.31) (54.32a)

d

kc (d) [ρd νS (d)τa (d → i)] .

(54.32b)

Normalized Distributions n∗ ρ∗ = ∗ = P D γc (t) + P S γs (t) , n˜ n A (t)n B (t) n s (t) γc (t) = , γs (t) = . n˜ A n˜ B n˜ s

(54.36a) (54.36b)

Stabilization and Dissociation Probabilities

d

Although equivalent, (54.26a) and (54.30) are normally invoked for (54.21) and (54.22), respectively, since P S ≤ 1 for DR so that αˆ DR → kc ; and ν A  ν S for DLR with n 50 so that αˆ → K di νs . For n  50, νS  ν A and αˆ → kc . The above results (54.26a) and (54.30) can also be derived from microscopic Breit– Wigner scattering theory for isolated (nonoverlapping) resonances.

54.3.2 Reactive Sphere Model: Three-Body Electron–Ion and Ion–Ion Recombination Since the Coulomb attraction cannot support quasibound levels, three body electron–ion and ion–ion recombination do not in general proceed via time-delayed resonances, but rather by reactive (energy-reducing) collisions with the third body M. This is particularly effective for A–B separations R ≤ R0 , as in the sequence kc

A + B
AB ∗ (R ≤ R0 ) , νd νs

Part D 54.3

AB ∗ (R ≤ R0 ) + M
AB + M . ν−s

(54.33a) (54.33b)

In contrast to (54.21) and (54.22) where the stabilization is irreversible, the forward step in (54.33b) is reversible. The sequence (54.33a) and (54.33b) represents a closed system where thermodynamic equilibrium is eventually established. Steady State Distribution of AB ∗ Complex

kc ν−s n∗ = n A (t)n B (t) + n s (t) . νs + νd νs + νd (54.34)

Saha and Boltzmann balances: Saha: n˜ A n˜ B kc = n˜ ∗ νd , Boltzmann: n˜ s ν−s = n˜ ∗ νs .

(54.35)

n˜ ∗ is in Saha balance with reactant block C and in Boltzmann balance with product block
.

PS =

νs , (νs + νd )

PD =

νd . (νs + νd )

(54.37)

Time Dependent Equations dn c = −kc P S n˜ A n˜ B [γc (t) − γs (t)] , (54.38a) dt dn s = −ν−s P D n˜ s [γs (t) − γc (t)] , (54.38b) dt dn c = −αˆ 3 n A (t)n B (t) + kd n s (t) . (54.39) dt  3  where the recombination rate coefficient cm /s and dissociation frequency are, respectively,

kc νs , (54.40) (νs + νd ) ν ν −s d , kd = ν−s P D = (54.41) (νs + νd ) which also satisfy the macroscopic detailed balance relation αˆ 3 = kc P S =

αˆ 3 n˜ A n˜ B = kd n˜ s .

(54.42)

Time Independent Treatment The rate αˆ 3 given by the time dependent treatment can also be deduced by viewing the recombination process as a source block C kept fully filled with dissociated species A and B maintained at equilibrium concentrations n˜ A , n˜ B (i. e. γc = 1) and draining at the rate αˆ 3 n˜ A n˜ B through a steady-state intermediate block E of excited levels into a fully absorbing sink block
of fully associated species AB kept fully depleted with γs = 0 so that there is no backward re-dissociation from block
. The frequency kd is deduced as if the reverse scenario, γs = 1 and γc = 0, holds. This picture uncouples αˆ and kd , and allows each coefficient to be calculated independently. Both dissociation (or ionization) and association (recombination) occur within block E. If γc = 1 and γs = 0, then

ρ∗ = n ∗ /n˜ ∗ = νd /(νs + νd ) , K = n˜ ∗ /n˜ A n˜ B = kc /νd = kc τd ,

(54.43b)

P S = νs /(νs + νd ) = ρ∗ νs τs ,

(54.43c)

(54.43a)

Electron–Ion and Ion–Ion Recombination

and the recombination coefficient is   αˆ = kc P S = kc ρ∗ νs τd = Kρ∗ νs .

(54.44)

Microscopic Generalization From (54.167), the microscopic generalizations of rate (54.40) and probability (54.43c) are, respectively,

∞ αˆ = v

−ε

εe

b0 dε

0

(a) e− + A+ + e− Here, σ0 = 19 πR2e for ( e− − e− ) collisions for scattering angles θ ≥ π/2 so that c αˆ ee (T ) = 2.7 × 10−20



300 T

4.5

n e cm3 s−1 (54.49)

ρi (R)νib (R) dt ≡ ρνs τd ,

P S (ε, b; R0 ) = Ri

(54.45b)

νi(b)

where ρi (R) = n(ε, b; R)/n(ε, is the fre˜ b; R); quency (54.164a) of (A–B)–M continuum-bound collisional transitions at fixed A–B separation R, Ri is the pericenter of the orbit, |i ≡ |ε, b , and b20 = R02 [1 − V(R)/E] , ε = E/kB T , (54.45c)  S 1/2 αˆ ≡ kc P ε,b , v = (8kB T/πM AB ) , (54.45d)   kc = πR02 [1 − V(R0 )/kB T ] v . (54.45e) where M AB is the reduced mass of A and B. Low Gas Densities Here ρi (R) = 1 for E > 0, R0 R0 S P (ε, b; R0 ) = ν(t) dt = ds/λi . Ri

in agreement with Mansbach and Keck [54.3]. (b) A+ + B− + M Here, σ0 v ≈ 10−9 cm3 s−1 , which is independent of T for polarization attraction. Then

αˆ 3 (T ) = 2 × 10−25

Ri

0

which is linear in the gas density N.

300 T

2.5

N cm3 s−1 .

(54.50)

αˆ eM = σ0 N

Em E)v dE 4πR dR n(R, ˜ 2

0

(54.51a)

0

R0 = ve σ0 N

4πR2 dR 0

εm V(R) ε e−ε dε 1− E 0

(54.51b)

where ε = E/kB T , and E m = δe2 /R = εm kB T is the maximum energy for collisional trapping. Hence,

8kB T e 1/2 2 R e R0 [σ0 N] (54.52a) πm e

10−26 300 2.5 ≈ Ncm3 s−1 , (54.52b) M T

54.3.3 Working Formulae for Three-Body Collisional Recombination at Low Density

αˆ eM (T e ) = 4πδ

For three-body ion–ion collisional recombination of the form A+ + B − + M in a gas at low density N, set V(R) = − e2 /R. Then (54.47) yields



3 Re 8kB T 1/2 4 3 αˆ c (T ) = (σ0 N ) , πR0 1 + πM AB 3 2 R0 (54.48)

where the mass M of the gas atom is now in u. This result agrees with the energy diffusion result of Pitaevski˘ı [54.4] when R0 is taken as the Thomson radius RT = 23 R e .

Part D 54.3

λi = (Nσ)−1 is the microscopic path length towards the (A–B)–M reactive collision with frequency ν = Nvσ. For λi constant, the rate (54.45a) reduces at low N to

R0 V(R) 1− 4πR2 dR (54.47) αˆ = (vσ0 N ) kB T



(c) e− + A+ + M Only a small fraction δ = 2m/M of the electron’s energy is lost upon ( e− − M ) collision so that (54.45a) for constant λ is modified to

R0 (54.46)

805

where R e = e2 /kB T , and the trapping radius R0 , determined by the classical variational method, is 0.41R e , in agreement with detailed calculation. The special cases are:

2πb dbP S (ε, b; R0 ) , (54.45a)

0 R0

54.3 Macroscopic Methods

806

Part D

Scattering Theory

54.3.4 Recombination Influenced by Diffusional Drift at High Gas Densities

Recombination Rate

αˆ RN (R0 )αˆ TR (R0 ) αˆ RN (R0 ) + αˆ TR (R0 )  αˆ RN , N → 0 → αˆ TR , N → ∞ .

αˆ =

Diffusional-Drift Current The drift current of A+ towards B − in a gas under an A+ –B − attractive potential V(R) is   K J(R) = −D∇n(R) − ∇V(R) n(R) (54.53a) e

∂ρ ˆ . (54.53b) R = − D N˜ A N˜ B e−V(R)/kB T ∂R

De = K(kB T ) ,

(54.54)

R0

 −1 , (54.61) αˆ TR (R0 ) = 4πKe 1 − exp(−R e /R0 )

where R e = e2 /kB T provides a natural unit of length. Langevin Rate For R0 R e , the transport rate

Normalized Ion-Pair R-Distribution

n(R) . ˜ ˜ N A N B exp [−V(R)/kB T ]

(54.55)

αˆ TR → αˆ L = 4πKe ,

∂n + ∇ · J = 0 , R ≥ R0 ∂t αˆ RN (R0 )ρ(R0 ) = αρ(∞) ˆ

(54.56a)

Reaction Rate   When R0 is large enough that R0 -pairs are in E, L 2 equilibrium (54.167),

∞

(54.56b)

Part D 54.3

The rate of reaction for ion-pairs with separations R ≤ R0 is αRN (R0 ). This is the recombination rate that would be obtained for a thermodynamic equilibrium distribution of ion pairs with R ≥ R0 , i. e. for ρ(R ≥ R0 ) = 1.

αˆ RN (R0 ) = v

R0

∂n ∂t

ε e−ε dε

0

0

≡v

ε e−ε


 dε πb20 P S (ε; R0 ) (54.63b)

0

dR = −4πR02 J(R0 ) . (54.57)

R ≥ R0 (54.58a) (54.58b)

(54.63c)

where b20 = R02 [1 − V(R0 )/E] , ε = E/kB T , v = (8kT/πM AB ) ,

V(R0 ) . b2max = R02 1 − kB T 1/2

  ρ(R0 ) = ρ(∞) α/ ˆ αˆ RN (R0 ) .

2πb dbP S (ε, b; R0 )

≡ vπb2max P S (R0 ) ,

Steady-State Solution

αˆ , ρ(R) = ρ(∞) 1 − αˆ TR (R)

b0

(54.63a)

∞

Steady-State Rate of Recombination

αˆ N˜ A N˜ B =

(54.62)

tends to the Langevin rate which varies as N −1 .

Continuity Equations for Currents and Rates

∞

(54.60)

With V(R) = −e2 /R,

where the Di and K i are, respectively, the diffusion and mobility coefficients of species i in gas M.

ρ(R) =

(54.59b)

Diffusional-Drift Transport Rate  −1 ∞ V(R)/kB T  e  αˆ TR (R0 ) = 4πD  dR . R2

Relative Diffusion and Mobility Coefficients

D = D A + DB , K = KA + KB ,

(54.59a)

(54.64a) (54.64b) (54.64c)

The probability P S and its averages over b and (b, E) for reaction between pairs with R ≤ R0 is determined in (54.63a–c) from solutions of coupled master equations. P S increases linearly with N initially and tends to unity

Electron–Ion and Ion–Ion Recombination

54.4 Dissociative Recombination

at high N. The recombination rate (54.59a) with (54.63a) and (54.61) therefore increases linearly with N initially, reaches a maximum when αˆ TR ≈ αˆ RN and then decreases eventually as N −1 , in accord with (i).

then bound A − B pairs are formed with R ≤ RT . Since E = 32 kB T − e2 /R, then 2 e 2 2 = Re . RT = (54.69) 3 kB T 3

Reaction Probability The classical absorption solution of (54.157) is   R0 dsi   P S (E, b; R0 ) = 1 − exp − (54.65) . λi

Thomson Straight-Line Probability The E → ∞ limit of (54.65) is 
  T PA,B (b; RT ) = 1 − exp −2 RT2 − b2 /λ A,B .

Ri

(54.70)

−1 With the binary decomposition λi−1 = λi−1 A + λi B ,

P = PA + PB − PA PB . S

(54.66)

Exact b2 –Averaged Probability With Vc = −e2 /R for the A+ –B − interaction in (54.63b), and at low gas densities N,

4R0 3Vc (R0 ) 1− 3λ A,B 2E i PA,B (E, R0 ) = (54.67) [1 − Vc (R0 )/E i ]

appropriate for constant mean free path λi . (E,b2 )–Averaged Probability 0 ) in (54.63c) at low gas density is

P S (R

PA,B (R0 ) = PA,B (E = kB T, R0 ) .

(54.68)

Thomson Trapping Distance When the kinetic energy gained from Coulomb attraction is assumed lost upon collision with third bodies,

The

b2 -average

is the Thomson probability  1
T −2X 1 − e PA,B (RT ) = 1 − (1 + 2X) 2X 2 (54.71a)

for reaction of (A − B) pairs with R ≤ RT . As N → 0  3 2 4

1 T (RT ) → X 1 − X + X 2 − X 3 + · · · PA,B 3 4 5 6 (54.71b)

and tends to unity at high N. X = RT /λ A,B = N(σ0 RT ). These probabilites have been generalized [54.5] to include hyperbolic and general trajectories. Thomson Reaction Rate   αˆ T = πRT2 v PAT + PBT − PAT PBT   4 πR3 λ−1 + λ−1 , N → 0 B → 3 T A πR2 v, N →∞. T

(54.72)

54.4.1 Curve-Crossing Mechanisms Direct Process. Dissociative recombination (DR) for diatomic ions can occur via a crossing at R X between the bound and repulsive potential energy curves V + (R) and Vd (R) for AB + and AB ∗∗ , respectively. Here, DR involves the two-stage sequence νd

e− + AB + (vi )
(AB ∗∗ )R −→ A + B ∗ . (54.73) νa

The first stage is dielectronic capture whereby the free electron of energy ε = Vd (R) − V + (R) excites an electron of the diatomic ion AB + with internal separation R and is then resonantly captured by the ion, at rate kc , to

form a repulsive state d of the doubly excited molecule AB ∗∗ , which in turn can either autoionize at probability frequency νa , or else in the second stage predissociate into various channels at probability frequency νd . This competition continues until the (electronically excited) neutral fragments accelerate past the crossing at R X . Beyond R X the increasing energy of relative separation reduces the total electronic energy to such an extent that autoionization is essentially precluded and the neutralization is then rendered permanent past the stabilization point R X . This interpretation [54.6] has remained intact and robust in the current light of ab initio quantum chemistry and quantal scattering calculations  + + for the simple diatomics O+ 2 , N2 , Ne2 , etc. . Mechanism (54.73) is termed the direct process which, in terms

Part D 54.4

54.4 Dissociative Recombination

kc

807

808

Part D

Scattering Theory

of the macroscopic frequencies in (54.73), proceeds at the rate αˆ = kc PS = kc [νd /(νa + νd )] ,

(54.74)

where PS is probability for A − B ∗ survival against autoionization from the initial capture at Rc to the crossing point R X . Configuration mixing theories of this direct process are available in the quantal [54.7] and semiclassical-classical path formulations [54.8]. Indirect Process In the three-stage sequence   e− + AB + (vi+ ) → AB + (v f ) − e− n → (AB ∗∗ )d

→ A + B∗

(54.75)

Interrupted Recombination The process −

kc

+

νd

∗∗

e + AB (vi )
(AB )d → A + B 

νnd νdn +

−

AB (v) − e

 n

54.4.2 Quantal Cross Section The cross section for direct dissociative recombination     e− + AB + vi+
AB ∗∗ r −→ A + B ∗ (54.77) of electrons of energy ε, wavenumber k e and spin statistical weight 2, for a molecular ion AB + (vi+ ) of + electronic statistical weight ω+ AB in vibrational level vi is

π ω∗AB 2 aQ σDR (ε) = 2 k e 2ω+

∗ ω AB h2 2 aQ . = (54.78) 8πm e ε 2ω+ Here ω∗AB is the electronic statistical weight of the dissociative neutral state of AB ∗ whose potential energy curve Vd crosses the corresponding potential energy curve V + of the ionic state. The transition T-matrix element for autoionization of AB ∗ embedded in the (moving) electronic continuum of AB + + e− is the quantal probability amplitude ∞ a Q (v) = 2π

  ∗ Vdε (R) ψv+∗ (R)ψd (R) dR

0

(54.79)

for autoionization. Here ψv+ and ψd are the nuclear bound and continuum vibrational wave functions for AB + and AB ∗ , respectively, while V dε (R) = φd |Hel (r, R(t))|φε (r, R) r,ˆε ∗ = Vεd (R)

∗

νa




Part D 54.4

the so-called indirect process [54.7] might contribute. Here the accelerating electron   loses energy by vibrational excitation vi+ → v f of the ion and is then resonantly captured into a Rydberg orbital of the bound molecule AB ∗ in vibrational level v f , which then interacts one way (via configuration mixing) with the doubly excited repulsive molecule AB ∗∗ . The capture initially proceeds via a small effect – vibronic coupling (the matrix element of the nuclear kinetic energy) induced by the breakdown of the Born-Oppenheimer approximation   – at certain resonance energies εn = E(v f ) − E vi+ and, in the absence of the direct channel (54.73), would therefore be manifest by a series of characteristic very narrow Lorentz profiles in the cross section. Uncoupled from (54.73) the indirect process would augment the rate. Vibronic capture proceeds more easily when v f = vi+ + 1 so states with n ≈ 7 − 9 would be involved   that+Rydberg  for H2 vi+ = 0 so that the resulting longer periods of the Rydberg electron would permit changes in nuclear motion to compete with the electronic dissociation. Recombination then proceeds as in the second stage of (54.73), i. e., by electronic coupling to the dissociative state d at the crossing point. A multichannel quantum defect theory [54.9] has combined the direct and indirect mechanisms

with frequency νdn and νnd between the d and n states. All (n, v) Rydberg states can be populated, particularly those in low n and high v since the electronic d − n interaction varies as n −1.5 with broad structure. Although the dissociation process proceeds here via a second order effect (νdn and νnd ), the electronic coupling may dominate the indirect vibronic capture and interrupt the recombination, in contrast to (54.75) which, as written in the one-way direction, feeds the recombination. Both dip and spike structure has been observed [54.10].

(54.76)

proceeds via the first (dielectronic capture) stage of (54.73) followed by a two-way electronic transition

(54.80)

are the bound-continuum electronic matrix elements coupling the diabatic electronic bound state wave functions ψd (r, R) for AB ∗ with the electronic continuum state wave functions φε (r, R) for AB + + e− . The matrix element is an average over electronic coordinates r and

Electron–Ion and Ion–Ion Recombination

all directions ˆ of the continuum electron. Both continuum electronic and vibrational wave functions are energy normalized (Sect. 54.8.3), and ∗ Γ (R) = 2π Vdε (R)

2

(54.81)

is the energy width for autoionization at a given nuclear separation R. Given Γ(R) from quantum chemistry codes, the problem reduces to evaluation of continuum vibrational wave functions in the presence of autoionization. The rate associated with a Maxwellian distribution of electrons at temperature T is  αˆ = ve ε σDR (ε) e−ε/kB T dε/(kB T )2 (54.82) where ve is the mean speed (Sect. 54.9). Maximum Cross Section and Rate Since the probability for recombination must remain less 2 than unity, a Q ≤ 1 so that the maximum cross section and rates are



π ω∗AB h2 max σDR (ε) = 2 = (2
+ 1) , 8πm e ε k e 2ω+ (54.83)

where ω∗AB has been replaced by 2(2
+ 1)ω+ under the assumption that the captured electron is bound in a high level Rydberg state of angular momentum
, and max αˆ max (T ) = v e σDR (ε = kB T ) (54.84a)

1/2 300 ≈ 5 × 10−7 (2
+ 1) cm3 /s . T

54.4 Dissociative Recombination

which is then the cross section for initial electron capture since autoionization has been precluded. Although the Born T -matrix (54.85) violates unitarity, the capture cross section (54.86) must remain less then the maximum value



∗ ω AB h2 π ω∗AB σcmax = 2 = , + 8πm e ε 2ω+ k e 2ω (54.87) 2

since a Q ≤ 1. So as to acknowledge after the fact the effect of autoionization, assumed small, and neglected by (54.85), the DR cross section can be approximated as     (54.88) σDR ε, v+ = σc ε, v+ PS , where PS is the probability of survival against autoionization on the Vd curve until stabilization takes place at some crossing point R X . Approximate Capture Cross Section With the energy-normalized Winans–Stückelberg vibrational wave function

ψd(0) (R) = Vd (R)

−1/2

δ(R − Rc ) ,

(54.89)

where Rc is the classical turning point for (A − B ∗ ) relative motion, (54.86) reduces to ! 

2 ψv+ (Rc ) π ω∗AB + σc (ε, v ) = 2 [2πΓ(Rc )] |Vd (Rc )| k e 2ω+ (54.90)

Cross section maxima of 5(2
+ 1)(300/T ) × 10−14 cm2 are therefore possible, being consistent with the rate (54.84b).

where the term inside the braces in (54.90) is the effective Franck–Condon factor.

0

(54.85)

ψd(0)

where is ψd in the absence of the back reaction of autoionization. Under this assumption, (54.78) reduces to

 2 π ω∗AB T B v+ , (54.86) σc (ε, v+ ) = 2 + k e 2ω

Six Approximate Stabilization Probabilities (1) A unitarized T -matrix is

T=

TB 1+

1 2 TB

2

,

(54.91)

so that PS = |T |2 / |TB |2 to give PS (low ε)

−2 1 = 1 + |TB |2 4   2 −2  ∞  
 ∗ = 1 + π2 Vdε (R) ψv+∗ (R)ψd(0) (R) dr     0

(54.92a)

Part D 54.4

(54.84b)

First-Order Quantal Approximation When the effect of autoionization on the continuum vibrational wave function ψd (R) for AB ∗ is ignored, then a first-order undistorted approximation to the quantal amplitude (54.79) is ∞
 + ∗ TB (v ) = 2π Vdε (R) ψv+∗ (R)ψd(0) (R) dR

809

810

Part D

Scattering Theory

which is valid at low ε when only one vibrational level v+ , i. e., the initial level of the ion is repopulated by autoionization. (2) At higher ε, when population of many other ionic levels v+f occurs, then −2    + 2 1 TB v f  , PS (ε) = 1 + (54.92b) 4 f

where the summation is over all the open vibrational levels v+f of the ion. When no intermediate Rydberg AB ∗ (v) states   are energy resonant with the initial e− + AB + v+ state, i. e., coupling with the indirect mechanism is neglected, then (54.88) with (54.92b) is the direct DR cross section normally calculated. (3) In the high-ε limit when an infinite number of v+f levels are populated following autoionization, the survival probability, with the aid of closure, is then  −2 R X 2 2   ∗ PS = 1 + π 2 Vdε (R) ψd(0) (R) dR . Rc

(54.93)

(4) On adopting in (54.93) the JWKB semiclassical wave function for ψd(0) , then −2  R X Γ(R) 1   PS (high ε) = 1 + dR 2~ v(R) 

Part D 54.5

1 = 1 + 2

Rc

t X

−2

νa (t) dt 

,

(54.94)

tc

where v(R) is the local radial speed of A − B relative motion, and where the frequency νa (t) of autoionization is Γ/~.

(5) A classical path local approximation for PS yields  t  X PS = exp − νa (t) dt  , (54.95) tc

which agrees to first-order for small ν with the expansion of (54.94). (6) A partitioning of (54.73) yields PS = νd /(νa + νd ) = (1 + νa τd )−1 ,

(54.96)

on adopting macroscopic averaged frequencies νi and associated lifetimes τi = νi−1 . The six surivival probabilities in (54.92a,b), (54.93–54.96) are all suitable for use in the DR cross section (54.88).

54.4.3 Noncrossing Mechanism The dissociative recombination (DR) processes e− + H+ 3 → H2 + H → H+H+H

(54.97)

at low electron energy ε, and e− + HeH+ → He + H(n =2)

(54.98)

have spurred renewed theoretical interest because they both proceed at respective rates of 2 × 10−7 to 2 × 10−8 cm3 s−1 and 10−8 cm3 s−1 at 300 K. Such rates are generally associated with the direct DR, which involves favorable curve crossings between the potential energy surfaces, V + (R) and Vd (R) for the ion AB + and neutral dissociative AB ∗∗ states. The difficulty with (54.97) and (54.98) is that there are no such curve crossings, except at ε ≥ 8 eV for (54.97). In this instance, the previous standard theories would support only extremely small rates when electronic resonant conditions do not prevail at thermal energies. Theories [54.11,12] are currently being developed for application to processes such as (54.97).

54.5 Mutual Neutralization A+ + B − → A + B .

(54.99)

Diabatic Potentials and V (0) f (R) for initial (ionic) and final (covalent) states are diagonal elements of

Vi(0) (R)

Vi f (R)= Ψi (r, R)|Hel (r, R)|Ψ f (r, R) r , (54.100) where Ψi, f are diabatic states and Hel is the electronic Hamiltonian at fixed internuclear distance R.

Adiabatic Potentials for a Two-State System


2 1/2 V ± (R) = V0 (R) ± ∆2 (R) + Vi f (R) ,  1
(0) Vi (R) + V (0) V0 (R) = f (R) ,
2  ∆(R) = Vi(0) (R) − V (0) f (R) .

(54.101a) (54.101b) (54.101c)

Electron–Ion and Ion–Ion Recombination

54.6 One-Way Microscopic Equilibrium Current, Flux, and Pair-Distributions

For a single crossing of diabatic potentials at R X then Vi(0) (R X ) = V (0) f (R X ) and the adiabatic potentials at R X are, V ± (R X ) = Vi(0) (R X ) ± Vi f (R X )

54.5.2 Cross Section and Rate Coefficient for Mutual Neutralization b X

(54.102)

σ M (E) = 4π

with energy separation 2Vi f (R X ).

Pi f (1 − Pi f )b db 0

= πb2X PM ,  Vi(0) (R X ) 2 R2X πb X = π 1 − E  14.4 = π 1+   R2X . R X Å E(eV)

54.5.1 Landau–Zener Probability for Single Crossing at R X On assuming ∆(R) = (R − R X )∆ (R X ), where ∆ (R) = d∆(R)/ dR, the probability for single crossing is Pi f (R X ) = exp [η(R X )/v X (b)] (54.103a) 2 Vi f (R X ) 2π η(R X ) = (54.103b) ~ ∆ (R X )
1/2 v X (b) = 1 − Vi(0) (R X )/E − b2 /R2X .

∞ αˆ M = (8kB T/πM AB )1/2

Overall Charge-Transfer Probability From the incoming and outgoing legs of the trajectory,

P (E) = 2Pi f (1 − Pi f ) .

(54.104)

(54.105a)

(54.105b)

PM is the b2 -averaged probability (54.104) for chargetransfer reaction within a sphere of radius R X . The rate is

(54.103c)

X

811

σ M () e− d (54.106)

0

where  = E/kB T .

54.6 One-Way Microscopic Equilibrium Current, Flux, and Pair-Distributions

v ε ni n i±

mean relative speed (8kT/πM AB )1/2 normalized energy E/kB T pair distribution function n i+ + n i− component of n i with R˙ > 0 (+) and R˙ < 0 (−).

All quantities on the RHS in the expressions (a)–(e) below are to be multiplied by N˜ A N˜ B [ω AB /ω A ω B ] where the ωi denote the statistical weights of species i which are not included by the density of states associated with the E, L 2 orbital degrees of freedom.



Case (a). |i ≡ R, E, L 2 .

Current: ji± (R) = n ± (R, E, L 2 )vR ≡ n i± vR Flux: 4πR2 ji± (R) dE dL 2 =

4π 2 e−E/kB T dE dL 2 . (2πMkB T )3/2 (54.107)

This flux is independent of R. For dissociated pairs E > 0,   4πR2 ji± (R) dE dL 2 = vε e−ε dε [2πb db] . (54.108)



 R, E, L 2 -Distribution:   n R, E, L 2 dRdE dL 2  2 

8π /vR e−E/kB T dR dE dL 2 . (54.109) = (2πMkB T )3/2 4πR2

Part D 54.6

Notation: M reduced mass M A M B /(M A + M B ) R internal separation of A − B E orbital energy 12 Mv2 + V(R) L orbital angular momentum L 2 2MEb2 for E > 0 ˙ vR radial speed | R|

812

Part D

Scattering Theory

Case (b). |i ≡ |R, E ; L 2 -integrated quantities.

1 1 ji± (R) = vn ± (R, E) ≡ vn i± , (54.110) 2 2   4πR2 ji± (R) dE = vε e−ε dε πb20 ,

Current: Flux:

(54.111a)

= πR [1 − V(R)/E] , (54.111b) (R, E)-Distribution: n(R, E) dRdE   2 E − V(R) 1/2 −ε =√ e dε dR kB T π (54.112) ≡ G MB (E, R) dR , πb20

2

which defines the Maxwell–Boltzmann velocity vistribution G MB in the presence of the field V(R). 



Case (c). E, L 2 -integrated quantities.

1 j ± (R) = v e−V(R)/kB T , (54.113) 4 2 ± 2 −V(R)/kB T 4πR j (R) = πR v e , (54.114)

Current: Flux:

n(R) = e−V(R)/kB T .

Distribution:

(54.115)

When E-integration is only over dissociated states (E > 0), the above quantities are 1 (54.116) jd± (R) = v [1 − V(R)/kB T ] , 4

V(R) 4πR2 jd± (R) = πR2 1 − v ≡ πb2max v , kB T (54.117)

Part D 54.7

n(R) = [1 − V(R)/kB T ] . 

(54.118)



Case (d). E, L 2 -distribution. For Bound Levels

4π 2 τR (E, L) −E/kB T n(E, L 2 ) dE dL 2 = e dE dL 2 , (2πMkB T )3/2 ,

(54.119)

where τR = dt = (∂JR /∂E) is the period for bounded radial motion, of energy E and radial action JR (E, L) = M vR dR.

Case (e). E-distribution. For bound levels

2 e−ε n(E) dE = √ dε π

R A 0

E−V kB T

1/2 dR , (54.120)

where R A is the turning point E = V(R A ). Example. For electron–ion bounded motion, V(R) =

−Ze2 /R, R A = Ze2 /|E|, R e = Ze2 /kB T , ε = E/kB T . Then τR = 2π(m/Ze2 )1/2 (R A /2)3/2 , R A

1/2 Re π 2 5/2 1/2 − |ε| R R , dR = R 4 A e

(54.121)

0

and

2 π 5/2 1/2 2 e−ε R R (54.122) √ dε 4 A e π −ε 2 3 π Re 2e . = √ dε (54.123) π 4 |ε|5/2 For closely spaced levels in a hydrogenic e− − A Z+ system,

2  dE  dL n s ( p,
) = n E, L 2 (54.124a) dp d


dE . n s ( p) = n(E) (54.124b) dp −1  2 2   Z e /a0 and L 2 = (
+ Using E = − 2 p2 2 2 1/2) ~ for level ( p,
) then



2 dJR dL dE dL 2 = (54.125) τR (E, L) d p d
dp d
  = h (2
+ 1)~2 (54.126)

n s (E) dE =

n s ( p,
) 2(2
+ 1) h3 = e I p /kB T , + n e N+ 2ω A (2πm e kB T )3/2 (54.127a)

n s ( p)

2 p2

h3

e I p /kB T , (54.127b) (2πm e kB T )3/2 2ω+ A in agreement with the Saha ionization formula (54.16) where N + is the equilibrium concentration of A Z+ ions in their ground electronic states. The spin statistical weights are ωeA = ω e = 2. n e N+

=

54.7 Microscopic Methods for Termolecular Ion–Ion Recombination At low gas density, the basic process +

−

A + B + M → AB + M

(54.128)

is characterized by nonequilibrium with respect to E. Dissociated and bound A+ –B − ion pairs are in equilibrium with respect to their separation R, but bound pairs

Electron–Ion and Ion–Ion Recombination

are not in E-equilibrium with each other. L 2 -equilibrium can be assumed for ion–ion recombination but not for ion–atom association reactions. At higher gas densities N, there is nonequilibrium in the ion-pair distributions with respect to R, E and L 2 . In the limit of high N, there is only nonequilibrium with respect to R. See [54.13] and the appropriate reference list for full details of theory.

54.7 Microscopic Methods for Termolecular Ion–Ion Recombination

Master Equation for γi (t) ∞   dγi (t) =− γi (t) − γ f (t) νi f dE f . dt

Quasi-Steady State (QSS) Reduction

Set t→∞

−D

where n i dE i is the number density of pairs in the interval dE i about E i , and νi f dE f is the frequency of i-pair collisions with M that change the i-pair orbital energy from E i to between E f and E f + dE f . The greatest binding energy of the A+ –B − pair is D. Association Rate

∞ A S dn i dE i R (t) = Pi dt

(54.130a)

−D

= αN ˆ A (t)N B (t) − kn s (t) ,

dγi (t) = − [γc (t) − γs (t)] dt

One-Way Equilibrium Collisional Rate and Detailed Balance (54.131)

∞

 PiD − P D f Ci f dE f .

−D

(54.136)

Recombination and Dissociation Coefficients Equation (54.135) in (54.130a) enables the recombination rate in (54.130b) to be written as ∞ ∞

 D ˜ ˜ αˆ N A N B = PiD − P D Pi dE i f Ci f dE f . −D

−D

(54.137)

The QSS condition ( dn i / dt = 0 in block E) is then ∞

E νi f dE f =

PiD −D

νi f P D f dE f ,

which involves only time independent quantities. Under QSS, (54.137) reduces to the net downward current across bound level −E, ∞ −E

 ˜ ˜ αˆ N A N B = PiD − P D dE i f Ci f dE f , −E

−D

where the tilde denotes equilibrium (Saha) distributions. Normalized Distribution Functions

γi (t) = n i (t)/n˜ iS ,

γs (t) = n s (t)/n˜ sB (t) , (54.132) γc (t) = N A (t)N B (t)/ N˜ A N˜ B , (54.133) where tions.

n˜ iS

and ˜ are the Saha and Boltzmann distribunB

(54.138)

−D

(54.139)

which is independent of the energy level (−E) in the range 0 ≥ −E ≥ −S of block E. The dissociation frequency k in (54.130b) is −E kn˜ s = −D

∞

dE i −E

 PiS − P Sf Ci f dE f ,

(54.140)

Part D 54.7

where is the probability for collisional stabilization (recombination) of i-pairs via a sequence of energy changing collisions with M. The coefficients for C →
recombination out of the C-block with ion concentrations N A (t), N B (t) (in cm−3 ) into the
block of total ion-pair   n s (t)  and for
→ C dissociation  concentrations are αˆ cm3 s−1 and k s−1 , respectively.

Ci f = n˜ i νi f = n˜ f ν fi = C fi ,

where and are the respective time-independent portions of the normalized distribution γi which originate, respectively, from blocks C and
. The energy separation between the C and
blocks is so large that PiS = 0 (E i ≥ 0, C block), PiS ≤ 1 (0 > E i ≥ −S, E block), PiS = 1 (−S ≥ E i ≥ −D,
block). Since PiS + PiD = 1, then

(54.130b)

PiS

(54.135)

PiS

PiD

Energy levels E i of A+ –B − pairs are so close that they form a quasicontinuum with a nonequilibrium distribution over E i determined by the master equation ∞   dn i (t) = (54.129) n i νi f − n f ν fi dE f , dt

(54.134)

−D

γi (t) = PiD γc (t) + PiS γs (t) −→1

54.7.1 Time Dependent Method: Low Gas Density

813

814

Part D

Scattering Theory

and macroscopic detailed balance αˆ N˜ A N˜ B = kn˜ s is automatically satisfied. αˆ is the direct (C →
) collisional contribution (small) plus the (much larger) net collisional cascade downward contribution from that fraction of bound levels which originated in the continuum C. kd is the direct dissociation frequency (small) plus the net collisional cascade upward contribution from that fraction of bound levels which originated in block
.

54.7.2 Time Independent Methods: Low Gas Density QSS-Rate. Since recombination and dissociation (ionization) involve only that fraction of the bound state population which originated from the C and
blocks, respectivel,y recombination can be viewed as time independent with N A N B = N˜ A N˜ B , n s (t) = 0 , (54.141a)

ρi = n i /n˜ i ≡ PiD (54.141b) ∞ −E   αˆ N˜ A N˜ B = dE i ρi − ρ f Ci f dE f . (54.141c) −E

−D

ρi

∞ νi f dE f =

−D

ρ f νi f dE f

picture corresponds to n s (t) = n˜ s ,

γc (t) = 0,

ρi = n i /n˜ i ≡ PiS , (54.146)

in analogy to the macroscopic reduction of (54.38a,b). Variational Principle The QSS-condition (54.135) implies that the fraction PiD of bound levels i with precursor C are so distributed over i that (54.137) for αˆ is a minimum. Hence PiD or ρi are obtained either from the solution of (54.142) or from minimizing the variational functional

αˆ N˜ A N˜ B =

∞

∞ n i dE i

−D

  ρi − ρ f νi f dE f

−D

(54.147a)

=

1 2

∞

∞ dE i

−D

 2 ρi − ρ f Ci f dE f

−D

(54.147b)

QSS Integral Equation.

∞

Time Independent Dissociation. The time independent

(54.142)

−S

with respect to variational parameters contained in a trial analytic expression for ρi . Minimization of the quadratic functional (54.147b) has an analogy with the principle of least dissipation in the theory of electrical networks.

is solved subject to the boundary condition ρi = 1(E i ≥ 0) ,

ρi = 0(−S ≥ E i ≥ −D) .

Part D 54.7

(54.143)

Collisional Energy-Change Moments.

D

(m)

1 (E i ) = m!

Di(m)

∞ (E f − E i )m Ci f dE f , (54.144) −D

 1 d  (∆E)m . = m! dt

with the QSS analytical solution (54.145)

Averaged Energy-Change Frequency. For an equilib-

rium distribution n˜ i of E i -pairs per unit interval dE i per second, d ∆E . Di(1) = dt Averaged Energy-Change per Collision.

∆E = Di(1) /Di(0) .

Diffusion-in-Energy-Space Method Integral Equation (54.142) can be expanded in terms of energy-change moments, via a Fokker–Planck analysis to yield the differential equation

∂ (2) ∂ρi Di =0, (54.148) ∂E i ∂E i

 0  ρi (E i ) =  Ei

 dE    D(2) (E)

0

−S

−1 dE  (54.149) D(2) (E)

of Pitaevski˘ı [54.4] for ( e− + A+ + M) recombination where collisional energy changes are small. This distribution does not satisfy the exact QSS condition (54.142). When inserted in the exact non-QSS rate (54.147b), highly accurate αˆ for heavy-particle recombination are obtained.

Electron–Ion and Ion–Ion Recombination

Bottleneck Method   The one-way equilibrium rate cm−3 s−1 across −E, i. e., (54.141c) with ρi = 1 and ρ f = 0, is

α(−E) N˜ A N˜ B = ˆ

∞

−E Ci f dE f .

dE i

−E

(54.150)

−D

This is an upper limit to (54.141c) and exhibits a minimum at −E ∗ , the bottleneck location. The least upper ∗ ). limit to αˆ is then α(−E ˆ Trapping Radius Method Assume that pairs with internal separation R ≤ RT recombine with unit probability so that the one-way equilibrium rate across the dissociation limit at E = 0 for these pairs is

α(R ˆ T ) N˜ A N˜ B =

RT

Ci f (R) dE f ,

Low N

Equilibrium in R, but not in E, L 2   → master equation for n E, L 2 .

Pure Coulomb attraction

Equilibrium in L 2 → master equation for n(E).

High N

Nonequilibrium in R, E, L 2 → master equation for n i± (R).   Equilibrium in E, L 2 but not in R → macroscopic transport equation (54.56a) in n(R).

Highest N

(54.151)

V(R)

where V(R) = −e2 /R, and Ci f (R) = n˜ i (R)νi f (R) is the rate per unit interval ( dRdE i ) dE f for the E i → E f collisional transitions at fixed R in  +    A − B − E ,R + M → A+ − B − E ,R + M . i

815

and n ± (R, E) of expanding (+) and contracting (−) pairs with respect to A–B separation R, orbital energy E and orbital angular momentum L 2 . With  n R, Ei , L i2 ≡ n i (R), and using the notation defined at the beginning of Sect. 54.6, the distinct regimes for the master equations discussed in Sect. 54.7.4 are:

0 dR

0

54.7 Microscopic Methods for Termolecular Ion–Ion Recombination

f

(54.152)

54.7.3 Recombination at Higher Gas Densities As the density N of the gas M is raised, the recombination rate αˆ increases initially as N to such an extent that there are increasingly more pairs n i− (R, E) in a state of contraction in R than there are those n i+ (R, E) in a state of expansion; i. e., the ion-pair distribution densities n i± (R, E) per unit interval dE dR are not in equilibrium with respect to R in blocks C and E . Those in the highly excited block E in addition are not in equilibrium with respect to energy E. Basic sets of coupled master equations have been developed [54.13]  for the microscopic nonequilibrium distributions n ± R, E, L 2

n ± (R) n i (R) , ρi± (R) = i± , n˜ i (R) n˜ i (R)  1 ρi (R) = ρi+ + ρi− . 2 ρi (R) =

(54.153)

Orbital Energy and Angular Momentum

1 M AB v2 + V(R) , 2 1 E i = M AB vR2 + Vi (R) , 2 L i2 Vi (R) = V(R) + , 2M AB R2 L i = |R× M AB v| , Ei =

L i2 = (2M AB E i )b2 , E i > 0 .

(54.154a) (54.154b) (54.154c)

(54.154d)

Maximum Orbital Angular Momenta (1) A specified separation R can be accessed by all orbits of energy Ei with L i2 between 0 and 2 L im (E i , R) = 2M AB R2 [E i − V(R)] .

(54.155a)

(2) Bounded orbits of energy E i < 0 can have L i2 between 0 and 2 (E i ) = 2M AB Rc2 [E i − V(Rc )] , L ic

(54.155b)

where Rc is the radius of the circular orbit determined by ∂Vi /∂R = 0, i. e., by E i = V(Rc ) + 12 Rc (∂V/∂R) Rc .

Part D 54.7

  The concentration cm−3 of pairs with internal separation R and orbital energy E i in the interval dRdE i about (R, E i ) is n˜ i (R) dRdE i . Agreement with the exact treatment  [54.13]  is found by assigning RT = (0.48 - 0.55) e2 /kB T for the recombination of equal mass ions in an equal mass gas for various ion– neutral interactions. For further details on the above methods, see the appropriate references on termolecular recombination in the general references on page 825.

Normalized Distributions  For a state |i ≡ E, L 2 ,

816

Part D

Scattering Theory

54.7.4 Master Equations

54.7.5 Recombination Rate

  Master Equation for n i± (R)± R, E i , L i2 [54.13]

±

 1 ∂
2 ± | |v n (R) R R i E i ,L i2 R2 ∂R 2

L f m

∞ =−

dE f 0

V(R)

αˆ N˜ A N˜ B = −4πR02 J(R0 )



(54.156)

∞

dL 2f

(54.157)

ρi− (R0 ) − ρi+ (R0 ) =

(54.163)

2

L f m

ρi (R)νib (R) =

dE f

  dL 2f ρi (R) − ρ f (R)

0

ρi (R)νic (R) =

(54.164a) L 2f m

∞

 dE f

V(R0 )

  dL 2f ρi (R) − ρ f (R)

0

× νi f (R) .

(54.164b)

Part D 54.7

Collisional Representation

(54.158)

dL 2f

 × n i (R)νi f (R) − n f (R)ν fi (R) ,  ρi+ (R) − ρi− (R)

R0 n˜ i (R) dR

dL i2

dE i 0

R


 i b × ρi (R)νi (R) ,

0



L ic

V(R0 )

2

dE f

2

∞

αˆ N˜ A N˜ B =

L f m

∞

V(R)

V(R)


 ρi (R) νib (R) + νic (R) dt ,

V(R)

    Ji = n i+ (R) − n i− (R) |vR | = ρi+ − ρi− j˜i±

(54.159)

∂R

(54.165)

which is the microscopic generalization of the macroscopic result αˆ = Kρ∗ νs = α RN (R0 )ρ(R0 ). The flux for dissociated pairs E i > 0 is 4πR2 |vR | n˜ i± (R) dE dL 2   = vε e−ε dε [2πb db] N˜ A N˜ B ,

(54.166)

so the rate (54.165) as R0 → ∞ is

2

L f m dL 2f

dE f

R0

× νi f (R) ,

Continuity Equations

=−

(54.162)

where is given by (54.155b) with Rc = R0 for bound states and is infinite for dissociated states, and where

0

Corresponding Master equations for the integrated distributions n ± (R, E) and ρ± (R, E) have been derived [54.13].

∞

,

2 L ic

V(R  0)

L2

1 |vR | 2

0  − ρi (R0 ) − ρi+ (R0 )

with

 dE f

×




 × ρi± (R) − ρ±f (R) νi f (R) .

∂


dL i2 4πR02 j˜i± (R0 )

dE i

Ri

L 2f m

V(R)



L ic

V(R0 )

.

Master Equations for Normalized Distributions [54.13]

1 ∂ (R2 Ji ) = − R2 ∂R

2

∞

αˆ N˜ A N˜ B =

The set of master equations [54.13] for n i+ is coupled  to the ni− set by the boundary conditions n i− Ri∓ = n i+ Ri∓ at the pericenter Ri− for all E i and apocenter Ri+ for E i < 0 of the E i , L i2 -orbit.

∂ρ± ± |vR | i = − ∂R

(54.161)

has the microscopic generalization


dL 2f n i± (R)νi f (R)

− n ±f (R)ν fi (R)

Flux Representation The R0 → ∞ limit of



 ρi (R) − ρ f (R) νi f (R) .

0

(54.160)

∞ αˆ = v

−ε

εe 0

b0 dε

R0 ρi (R)νib (R) dt ,

2πb db 0

Ri

(54.167)

Electron–Ion and Ion–Ion Recombination

which is the microscopic generalization (54.45) of the macroscopic result αˆ = kc P S of (54.44). Reaction Rate α RN (R0 ) On solving (54.157) subject to ρ(R0 ) = 1, then according to (54.56b), αˆ determined by (54.162) is the rate αˆ RN of recombination within the (A − B) sphere of radius R0 . The overall rate

54.8 Radiative Recombination

817

of recombination αˆ is then given by the full diffusional-drift reaction rate (54.59b) where the rate of transport to R0 is determined uniquely by (54.60). For development of theory [54.13] and computer simulations, see the reference list on Termolecular Ion–Ion Recombination: Theory, and Simulations, respectively.

54.8 Radiative Recombination In the radiative recombination (RR) process −



e (E,
) + A

Z+

(c) → A

(Z−1)+

(c, n
) + hν , (54.168)

e−

-0 . = ve σRn
(T e ) ,

(54.169)

where ε = E/kB T e , or from the Milne DB relation (54.243) between the forward and reverse macroscopic rates of (54.168). Using the hydrogenic semiclassical σ In of Kramers [54.5], together with an asymptotic expansion [54.14] for the g-factor of Gaunt [54.15], the quantal/semiclassical cross section ratio in (54.249), Seaton [54.16] calculated αˆ Rn
. The rate of electron energy loss in RR is 0 / ∞ dE = n e ve (kB T e ) ε2 σRn
(ε) e−ε dε , dt n
0

(54.170)

(54.171)

0

Standard Conversions

  E = p2e /2m e = ~2 k2e /2m e = k2e a02 e2 /2a0

  = κ Z e /2a0 = ε Z 2 e2 /2a0 , E ν = hν = ~ω = ~kν c = (In + E)    ≡ 1 + n 2 ε Z 2 e2 /2n 2 a0 ,   hν/In = 1 + n 2 ε, k2e a02 = 2E/ e2 /a0 ,   kν a0 = (hν)α/ e2 /a0 ,   kν2 /k2e = (hν)2 / 2Em e c2
  = α2 (hν)2 / 2E e2 /a0 , 2

α

IH −2



2 2



(54.172a) (54.172b) (54.172c) (54.172d) (54.172e) (54.172f) (54.172g) (54.172h)

= e /2a0 , α = e /~c = 1/137.035 9895 ,     = m e c2 / e2 /a0 , In = Z 2 /n 2 IH . 2

2

(54.172i)

The electron and photon wavenumbers are k e and kν , respectively.

54.8.1 Detailed Balance and Recombination-Ionization Cross Sections Cross sections σRn
(E) and σIn
(hν) for radiative recombination (RR) into and photoionization (PI) out of level n
of atom A are interrelated by the detailed balance relation 2 n
2 n
g e g+ A k e σR (E) = gν g A kν σI (hν) ,

(54.173)

where g e = gν = 2. Electronic statistical weights of A and A+ are g A and g+ A , respectively. Thus, using

Part D 54.8

the accelerating electron with energy and angular momentum (E,
 ) is captured, via coupling with the weak quantum electrodynamical interaction (e/m e c)A· p associated with the electromagnetic field of the moving ion, into an excited state n
with binding energy In
about the parent ion A Z+ (initially in an electronic state c). The simultaneously emitted photon carries away the excess energy hν = E + In
and angular momentum difference between the initial and final electronic states. The cross section σRn
(E) for RR is calculated (a) from the Einstein A coefficient for free–bound transitions or (b) from the cross section σIn
(hν) for photoionization (PI) via the detailed balance (DB) relationship appropriate to (54.168). The rates v e σR and averaged cross sections σR for a Maxwellian distribution of electron speeds v e are then determined from either ∞ αˆ Rn
(T e ) = ve εσRn
(ε) exp(−ε) dε

and the radiated power produced in RR is / 0 ∞ d(hν) = n e ve εhνσRn
(ε) e−ε dε . dt n


818

Part D

Scattering Theory

(54.172g) for kν2 /k2e , 

(hν)2 gA n
σIn
(hν) . σR (E) = Em e c2 2g+ A

(54.174)

The statistical factors  are:   (a) For A+ + e− state c Sc , L c ; ε,
 , m  : g+ A = (2Sc + 1)(2L c + 1).

PI and RR Cross Sections for Level n. In the Kramer (K) semiclassical approximation, 3 In n n σ (hν) = σI0 = K σIn
(hν) , (54.181) K I hν

In 2 n σ (E) = σ (E) (54.182) K R R0 n In + E

= 3.897 × 10−20
 −1 2 × nε 13.606 + n 2 ε2 cm ,

(b) For A(n
) state b[Sc , L c ; n,
]SL: g A = (2S + 1)(2L + 1). (c) For n
electron outside a closed shell: g+ A = 1, g A = 2(2
+ 1). Cross sections are averaged over initial and summed over final degenerate states. For case (c), σIn =

n−1 1  (2
+ 1)σIn
; n2

(54.175a)

where ε is in units of eV and is given by    ε = E/Z 2 ≡ 2.585 × 10−2 /Z 2 T e /300 . (54.183)

Equation (54.182) illustrates that RR into low n at low E is favored.


=0

σRn =

n−1 

2(2
+ 1)σRn
.

Cross Section for RR into Level n
. (54.175b)


=0

54.8.2 Kramers Cross Sections, Rates, Electron Energy-Loss Rates and Radiated Power for Hydrogenic Systems These are all calculated from application of detailed balance (54.173) to the original σIn (hν) of Kramers [54.5].

n
K σR


= (2
+ 1)/n 2 K σRn .

Rate for RR into Level n.

αˆ Rn (T e ) = αˆ 0 (T e ) (2/n) bn ebn E 1 (bn ) ,

Part D 54.8

Z 2 e2 , 2n 2 a0

hν = In + E .

αˆ Rn (T e → 0) = αˆ 0 (T e ) (2/n) 

−2 −3 × 1 − b−1 n + 2bn − 6bn + · · · . (54.185b)

(54.176)

The results below are expressed in terms of the quantities In , (54.177) kB T e 64πa2 α n  n σI0 = √0 Z2 3 3   = 7.907 071 × 10−18 n/Z 2 cm2 , bn =





 8πa02 α3 Z 2 e2 /a0 , √ E 3 3    8πa02 α3 Z 2 e2 /a0 αˆ 0 (T e ) = ve √ kB T e 3 3

σR0 (E) =

(54.185a)

which tends for large bn (i. e., kB T e In ) to

Semiclassical (Kramers) Cross Sections For hydrogenic systems,

In =

(54.184)

(54.178)

The Kramers cross section for photoionization at n and threshold is σI0 n = 2σR0 /n; σR0

αˆ 0n = 2αˆ 0 /n

(54.186)

provide the corresponding Kramers cross section and rate for recombination as E → 0 and T e → 0, respectively. RR Cross Sections and Rates into All Levels n ≥ n f .

∞ σRT (E) =

σRn (E) dn nf

(54.179)

(54.180)

= σR0 (E) ln(1 + I f /E) , (54.187a)
 αˆ RT (T e ) = αˆ 0 (T e ) γ + ln b f + eb f E 1 (b f ) (54.187b)

Electron–Ion and Ion–Ion Recombination

e−x ln x dx = γ ,

/

(54.188a)

0

∞

x −1 e−x dx = E 1 (b) ,

(54.188b)

ex E 1 (x) dx = γ + ln b + eb E 1 (b) ,

(54.188c)

b

0


−1 = n e αˆ Rn (T e )In bn ebn E 1 (bn ) ,

n

which for large bn (i. e. (kB T e ) In ) tends to

 −2 −3 n e αˆ Rn (T e )In 1 + b−1 n − bn + 3bn + · · · . (54.191b)

Radiated Power for RR into All Levels n ≥ nf .

0

/

  1 − x ex E 1 (x) dx

0

= γ + ln b + eb (1 − b)E 1 (b) ,

(54.188d)

where γ = 0.577 2157 is Euler’s constant, and E 1 (b) is the first exponential integral such that b eb E 1 (b) b1

−→1 − b−1 + 2b−2 − 6b−3 + 24b−4 + · · · . Electron Energy Loss Rate Energy Loss Rate for RR into Level n. / 0

dE 1 − bn ebn E 1 (bn ) , = n e αˆ Rn (T e )kB T e dt n ebn E 1 (bn ) (54.189a)

(54.189b)

with (54.185a) for αˆ Rn . 0


 = n e kB T e αˆ 0 (T e ) γ + ln b f + eb f E 1 (b f )(1 − b f ) (54.190a)


 = n e (kB T e ) αˆ RT (T e ) − αˆ 0 (T e )b f eb f E 1 (b f )

(54.190b)

with (54.187b) and (54.180) for αˆ RT and αˆ 0 .

(54.192)

To allow n-summation, rather than integration as in (54.187a), to each of the above expressions is added n n 1/2σR f , 1/2αˆ R f , 1/2 dE/ dt n f and 1/2 d(hν)/ dt n f , respectively. The expressions valid for bare nuclei of charge Z are also fairly accurate for recombination to a core of charge Z c and atomic number Z A , provided that Z is identified as 1/2(Z A + Z c ). Differential Cross Sections for Coulomb Elastic Scattering.

σc (E, θ) =

b20 4 sin4 12 θ

,

b20 = (Ze2 /2E)2 . (54.193)

The integral cross section for Coulomb scattering by θ ≥ π/2 at energy E = (3/2)kB T is 1 σc (E) = πb20 = πR2e , 9

R e = e2 /kB T .

(54.194)

Photon Emission Probability.

Pν = σRn (E)/σc (E) .

Energy Loss Rate for RR into All Levels n ≥ nf .

dE dt

0 d(hν) = n e αˆ 0 (T e )I f . dt

(54.195a)

This is small and increases with decreasing n as 3 In 8α 8 E . (54.195b) Pν (E) = √ 3 3 n (e2 /a0 ) hν

54.8.3 Basic Formulae for Quantal Cross Sections Radiative Recombination and Photoionization Cross Sections The cross section σRn
for recombination follows from the continuum-bound transition probability Pi f per unit

Part D 54.8

which for large bn (i. e. (kB T e ) In ) tends to

 −2 −3 n e αˆ Rn (T e )kB T e 1 − b−1 n + 3bn − 13bn + · · ·

/

d(hν) dt

(54.191a)

b

b

819

Radiated Power Radiated Power for RR into Level n.

Useful Integrals.

∞

54.8 Radiative Recombination

820

Part D

Scattering Theory

time. It is also provided by the detailed balance relation (54.173) in terms of σIn
which follows from P fi . The number of radiative transitions per second is 

g e g+ ρ(E) dE dkˆ e Pi f ρ(E ν ) dE ν dkˆ ν

RR Cross Section into Level (n
m).

 1 σRn
m (k e ) dkˆ e 4π    h 3 ρ(E) = An
m E, kˆ e dkˆ e . (54.205) 8πm e E

σRn
m (E) =

A

= g e g+ Ave

dkν d pe σR (k e ) = gν g A c σI (kν ) , (2π ~)3 (2π)3

 where the electron cm−2 s−1 is current

ve d pe 2m E = dE dkˆ e , 3 (2π ~) h3   and the photon current cm−2 s−1 is dkν (hν)2 c = c dE ν dkˆ ν . (2π)3 (2π ~c)3 

(54.196)



(αhν)3 8π 2 ρ(E)RIn
(E) 3 2(e2 /a0 )E   n
| Ψn
m |r|Ψi (ke ) |2 . (54.206) RI (E) = dkˆ e

σRn
(E) = (54.197)

m

(54.198)

Time Dependent Quantum Electrodynamical Interaction.



2πhν 1/2 e A · p = ie V(r, t) = (
ˆ · r) e−i(kν ·r−ωt) mc V ≡ V(r) eiωt . (54.199) In the dipole approximation, e−ikν ·r ≈ 1. Continuum-Bound State-to-State Probability.

2π 2 V fi δ [E ν − (E + In )] ~ V fi = Ψn
m (r)| V(r) |Ψi (r, k e ) .

RR Cross Section into Level (n
).

Pi f =

Transition T-Matrix for RR.

πa02 |TR |2 ρ(E) , (ka0)2  2 |TR |2 = 4π 2 D fi dkˆ e .

σRn
(E) =

(54.207) (54.208)

m

Photoionization Cross Section. From detailed balance in (54.196), σIn
is  2 g+ 8π n
A αhν σI (hν) = ρ(E)RIn
(E) . 3 gA (54.209)

(54.200)

Number of Photon States in Volume V.

  ρ E ν , kˆ ν dE ν dkˆ ν = V(hν)2 /(2π ~c)3 dE ν dkˆ ν

Continuum Wave Function Expansion.

Ψi (k e , r) =



   i
eiη
 R E
 (r)Y
 m  kˆ e Y
 m  (ˆr ) .


 m 

(54.210)

(54.201a)

Part D 54.8


= V ω2 /(2πc)3 dω dkˆ ν .

(54.201b)

Continuum-Bound Transition Rate. On summing over the two directions (gν = 2) of polarization, the rate for transitions into all final photon states is    ˆ An
m E, k e = Pi f ρ(E ν ) dE ν dkˆ ν

=

4e2 (hν)3 | Ψn
m |r|Ψi (ke ) |2 . 3~ (3~c)3 (54.202)



An
m E, kˆ e = (2π/~) D fi

2

,



      Ψi (k e ; r)Ψi∗ ke ; r dr = δ E − E  δ kˆ e − kˆ e . (54.211)

Plane Wave Expansion.

eik·r = 4π

∞ 
=0

  ∗ ˆ k Y
m (ˆr ) i
j
(kr)Y
m

1 j
(kr) ∼ sin kr −
π /(kr) . 2 For bound states, Ψn
m (r) = Rn
(r)Y
m (ˆr ) .

Transition Frequency: Alternative Formula.



Energy Normalization. With ρ(E) = 1,

(54.203)

where the dipole atom-radiation interaction coupling is 3 1/2 2ω Ψn
m |e r|Ψi (k e ) . (54.204) D fi (k e ) = 3πc3

(54.212)

(54.213)

(54.214)

RR and PI Cross Sections and Radial Integrals.

σRn
(E) =

8π 2 3



(αhν)3 2(e2 /a0 )E

ρ(E)RI (E; n
) . (54.215)

Electron–Ion and Ion–Ion Recombination

For an electron outside a closed core, g+ A

= 1,

σIn
(hν) = 

ε,
Rn
=

g A = 2(2
+ 1) 2 4π αhνρ(E) 3(2
+ 1)

RI (E; n
) ,

(54.216a)

∞

(Rε
 r Rn
) r 2 dr ,

(54.216b)

0 2

ε,
−1 ε,
+1 RI (E; n
) =
Rn
+ (
+ 1) Rn


2

.

(54.216c)

For an electron  outside an unfilled core (c) in the process A+ + e− → A(n
), the weights are State i: [Sc , L c ; ε] , g+ A = (2Sc + 1)(2L c + 1) State f : [(Sc , L c ; n
)S, L] , g A = (2S + 1)(2L + 1). (2L + 1) RI (E; n
) = (2L c + 1) !2    
L Lc  2L + 1 × L 
 1
 =
±1 L  ×
max

2 ∞ 2 (Rε
 r Rn
) r dr . 0

(54.217)

This reduces to (54.216c) when the radial functions Ri, f do not depend on (Sc , L c , S, L). Cross Section for Dielectronic Recombination

πa02 |TDLR (E)|2 ρ(E) , (ka0)2

|TDLR (E)| = 4π 2

2

(54.218)

Continuum Wave Normalization and Density of States The basic formulae (54.206) for σRn
depends on the density of states ρ(E) which in turn varies according to the particular normalization constant N adopted for the continuum radial wave,

2 1 r, (54.220) R E
(r) ∼ N sin kr −
π + η
2

in (54.210) where the phase is η
= arg Γ(
+ 1 + iβ) − β ln 2kr + δ
.

(54.221)

The phase corresponding to the Hartree–Fock shortrange interaction is δ
. The phase shift for  Coulomb  electron motion under − Ze2 /r is (η
− δ
) with β = Z/(ka0 ). For a plane wave φk (r) = N  exp(ik · r),     2 φk (r)|φk (r) dk = (2π)3 N  ρ(k) dk δ k − k 3     h 2 ≡ N  ρ(E, kˆ ) dE dkˆ δ E − E  δ kˆ − kˆ  . mp (54.222)

On integrating (54.222) over all E and kˆ for a single particle distributed over all |E, kˆ states, N  and ρ are then interrelated by  2  (54.223) N  ρ E, kˆ = m p/h 3 . The incident current is (54.224a) j dE dkˆ e = v N  ρ(E, kˆ ) dE dkˆ e 

= 2m E/h 3 dE dkˆ e = v d p e /h 3 . 2

(54.224b)

dkˆ e

  2  Ψ f D Ψ j Ψ j V |Ψi (k e )   × , E − ε j + iΓ j /2 j (54.219)

which is the generalization of the T -matrix (54.208) to include the effect of  intermediate doubly-excited autoionizing states Ψ j in energy resonance to within width Γ j of the initial continuum state Ψi . The 1N electrostatic interaction V = e2 i=1 (ri − r N+1 )−1 initially produces dielectronic capture by coupling the initial state i with the resonant states j which become stabilized by coupling via the  1/2dipole 1 N+1radiation field interaction D = 2ω3 /3πc3 i=1 (eri ) to the final stabilized state f . The above cross section for (54.3) is valid for isolated, nonoverlapping resonances.

821

Radial Wave Connection. From (54.210) and (54.212),

N = (4π N  /k), so that the connection between N of (54.220) and ρ(E) is   2m/~2 (2/π) |N|2 ρ(E, kˆ ) = = . (54.225) πk ka0 e2 RR Cross Sections for Common Normalization Factors of Continuum Radial  Functions  2m/~2 (2/π) (a) N = 1; ρ(E) = = , πk (ka0 )e2   8π 2 a02 2 D fi dkˆ e , σRn
(E) = (ka0 )3 m

where D fi of (54.204) is dimensionless.

(54.226) (54.227)

Part D 54.8

n
σDLR (E) =

54.8 Radiative Recombination

822

Part D

Scattering Theory

  ρ(E) = 2m/~2 (k/π) , (54.228) 3  

3 e2 /a0 16πa2 αhν RI σRn
(E) = √ 0 , 2 E a05 3 2 e /a0

(b) N = k−1 ;

Bound–Bound Absorption Oscillator Strength. For

a transition n → n  ,     n
m f n
m Fnn  = 2
m
 m  4 26

(54.229)

where and (54.216c) for RI has dimen  (54.216b) sion L 5 .   2m/~2 (c) N = k−1/2 ; ρ(E) = , (54.230) π   8πa02 RI α3 (hν)3 σRn
(E) = , (54.231)  2 4 2 3 a e /a0 E 0   where RI has dimensions of L 4 .   (d) N = 2m/~2 π 2 E 1/4 ; ρ(E) = 1 , (54.232) 

2 3 3 RI 4(πa0 ) α (hν) , σRn
(E) =  2 2a 2 3 e 0 e /a0 E (54.233)

  where RI has dimensions of L 2 E −1 .

54.8.4 Bound-Free Oscillator Strengths For a transition n
→ E to E + dE,  d f n
2 (hν) 1   r ε
m = 2 dE 3 (e /a0 ) (2
+ 1) m   n
m 

dkˆ e 

Part D 54.8

=



,

| Ψn
m |r|Ψi (ke ) E|2

m 

m,
 ,m 





σRn
(E) = 2π 2 αa02 g A

σIn
(hν) = 2π 2 αa02 g+ A



2

,

kν2 k2e

(54.235)



e2 a0

d f n
, dE

e2 d f n
. a0 dE

σRn (E) =

n−1 
=0

(54.236b)

g+ A

=1, 2 k dFn , σRn
(E) = 2π 2 αa02 2ν k e dE (54.237)

dFn = dE

n−1 
=0

 d f n
m d f n
=2 . gn
dE dE
,m

(54.239b) (54.239c) (54.239d)

(54.239e)

(54.239f)

This semiclassical analysis yields exactly Kramers PI and associated RR cross sections in Sect. 54.8.2.

αˆ Rn
(T e ) = ve

ε σRn
(ε) e−ε dε

(54.240a)

0

. ≡ ve σRn
(T e ) , (54.240b)  n
 where ε = E/kB T and σR (T e ) is the Maxwellianaveraged cross section for radiative recombination. In terms of the continuum-bound An
(E),

∞ dAn
h3 αˆ Rn
(T e ) = e−ε dε , dε (2πm e kB T )3/2 0

(54.236a)

Semiclassical Hydrogenic Systems

g A = gn
= 2(2
+ 1) ,

1 1 , n 3 n 3

25 dFn d f n
I2 = √ n n 3 = 2n 2 , dE dE 3 3π (hν)

25 α3 n In2 πa02 , σRn (E) = √ 3 3 E(hν) 3 26 α n In n
σI (hν) = √ 2 πa02 , hν Z 3 3

n  I 3 n = 7.907 071 Mb . hν Z2

∞ 2

(54.234)

ε
m rn
m

−3 5

54.8.5 Radiative Recombination Rate


m

RI (ε; n
) =

= √ 3 3π

1 1 − n 2 n 2

(54.239a)

dAn
= ρ(E) dE



(54.241)

An
m (E, kˆ e ) dkˆ e . (54.242)

m

Milne Detailed Balance Relation In terms of σIn
(hν),

αˆ Rn
(T e )  = ve

gA 2g+ A



kB T e mc2



In kB T e

2 . σIn
(T e ) , (54.243)

(54.238)

where, in reduced units ω = hν/In , T = kB T e /In = b−1 n , the averaged PI cross section corresponding to (54.174)

Electron–Ion and Ion–Ion Recombination

is . e1/T σIn
(T ) = T

∞

ω2 σIn
(ω) e−ω/T dω . (54.244)

1   When σIn
(ω) is expressed in Mb 10−18 cm2 ,

1/2 2  In gA n
−13 300 αˆ R (T e ) = 1.508 × 10 Te IH 2g+ A . n
3 −1 (54.245) × σI (T) cm s .

When σI can be expressed in terms of the threshold cross section σ0n (54.178) as σIn
(hν) = (In /hν) p σ0 (n); ( p = 0, 1, 2, 3) , then





σIn
(T)

(54.246)

= S p (T )σ0 (n), where

S0 (T ) = 1 + 2T + 2T 2 ,

S1 (T ) = 1 + T , (54.247a)

S2 (T ) = 1 ,

 S3 (T ) = e1/T /T E 1 (1/T ) T 1

∼ 1 − T + 2T 2 − 6T 3 .

(54.247b) (54.247c) (54.247d)

The case p = 3 corresponds to Kramers PI cross section (54.181) so that (2
+ 1) 2 αˆ 0 (T e )S3 (T ) (54.248a) ˆ Rn
(T e ) = Kα n2 n ≡ K αˆ Rn
(T e → 0)S3 (T ) , (54.248b)  3 1/2  n
2 such that K αˆ R ∼ Z / n T e as T = (kB T e /In ) → 0.

The Gaunt factor G n
is the ratio of the quantal to Kramers (K) semiclassical PI cross section such that σIn
(hν) = K σIn (hν)G n
(ω) ; ω = hν/In = 1 + E/In .

(54.249)

(a) Radiative Recombination Cross Section 

gA α2 (hν)2 n
G n
(ω)K σIn (hν) σR (E) = 2E(e2 /a0 ) g+ A (54.250a)

= G n
(ω)K σRn
(E)   (2
+ 1) = G (ω) σRn (E) , n
n2 K σRn (E) = G n (ω)K σRn (E)

(54.250b) (54.250c) (54.250d)

823

where the quantum mechanical correction, or Gaunt factor, to the semiclassical cross sections  1, ω→1 G n
(ω) → (54.251) ω−(
+1/2) , ω → ∞ favors low n
states. The
-averaged Gaunt factor is n−1   (2
+ 1)G n
(ω) . G n (ω) = 1/n 2

(54.252)


=0

Approximations for G n : as ε increases from zero,   4 28 2 −3/4 G n (ε) = 1 + (an + bn ) + an (54.253a) 3 18 7 7  1 − (an + bn ) + an bn + b2n (54.253b) 3 6   2 2 2 where E = ε Z e /2a0 , ω = 1 + n ε, and   an (ε) = 0.172 825 1 − n 2 ε cn (ε) , (54.254a)

4 2 bn (ε) = 0.049 59 1 + n ε + n 4 ε2 c2n (ε) , 3 cn (ε) = n

−2/3

 −2/3 1 + n2ε .

(54.254b) (54.254c)

Radiative Recombination Rate

αˆ Rn
(T e ) = K αˆ Rn
(T e → 0)Fn
(T ) , (54.255)

(2
+ 1) αˆ Rn
(T e → 0) = n2

2 αˆ 0 (T e ) , (54.256) n

in accordance with (54.185b). e1/T Fn
(T ) = T

∞ 1

G n
(ω) −ω/T e dω . ω

(54.257)

The multiplicative factors F and G convert the semiclassical (Kramers) T e → 0 rate and cross section to their values. Departures from the scaling rule  2 quantal 1/2  Z /n 3 T e for RR rates is measured by Fn
(T ).

54.8.7 Exact Universal Rate Scaling Law and Results for Hydrogenic Systems   αˆ Rn
(Z, T e ) = Z αˆ Rn
1, T e /Z 2

(54.258)

as exhibited by (54.243) with (54.239e) and (54.244). Recombination rates are greatest into low n levels and the ω−
−1/2 variation of G n
preferentially populates states with low
≈ 2–5. Highly accurate analytical

Part D 54.8

54.8.6 Gaunt Factor, Cross Sections and Rates for Hydrogenic Systems

54.8 Radiative Recombination

824

Part D

Scattering Theory

fits for G n
(ω) have been obtained for n ≤ 20 so that (54.249) can be expressed in terms of known functions of fit parameters [54.17]. This procedure (which does not violate the S2 sum rule) has been extended to nonhydrogenic systems of neon-like Fe XVII, where σIn
(ω) is a monotonically decreasing function of ω. The variation of the
-averaged values n −2

n−1  (2
+ 1)Fn
(T )
=0

system is α(Z, T ) = 5.2 × 10−14 Zλ1/2 ˆ

1 1/2 , × 0.43 + ln λ + 0.47/λ 2 (54.259)

λ = 1.58 × 105 Z 2 /T

[α] = cm3 /s.

where and ˆ [54.19] exist for the effective rate

Tables



αˆ n
E (T ) =

∞ n −1 

 

αˆ Rn
Cn 
 ,n


(54.260)

n  =n
 =0

is close in both shape and magnitude to the corresponding semiclassical function S3 (T ), given by (54.257) with G n
(ω) = 1. Hence the
-averaged recombination rate is   αˆ Rn (Z, T ) = (300/T )1/2 Z 2 /n Fn (T ) × 1.1932 × 10−12 cm3 s−1 , where Fn can be calculated directly from (54.257) or be approximated as G n (1)S(T ). A computer program based on a three-term expansion of G n is also available [54.18]. From a three-term expansion for G, the rate of radiative recombination into all levels of a hydrogenic

of populating a given level n
of H via radiative recombination into all levels n  ≥ n with subsequent radiative cascade (i → f ) with probability Ci, f via all possible intermediate paths. Tables [54.19] also exist for the full rate αˆ FN (T ) =

n−1 ∞  

αˆ Rn


(54.261)

n=N
=0

of recombination, into all levels above N = 1, 2, 3, 4, of hydrogen. They are useful in deducing time scales of radiative recombination and rates for complex ions.

54.9 Useful Quantities (a) Mean Speed



T 1/2 8kB T 1/2 7 ve = = 1.076 042 × 10 cm/s πm e 300

(c) Boltzmann Average Momentum

∞ p =

Part D 54

1/2

= 6.692 38 × 107 TeV cm/s

1/2 T 1/2  5 m p /m i vi = 2.511 16 × 10 cm/s 300 where (m p /m e )1/2 = 42.850 352, and T = 11 604.45 TeV relates the temperature in K and in eV.

e− p

2 /2mk

BT

d p = (2πm e kB T )1/2 .

−∞

(d) De Broglie Wavelength

λdB = =

h h = p (2πm e kB T )1/2 7.453 818 × 10−6

cm 1/2 Te

6.9194 300 1/2 Å = 1/2 Å . = 43.035 Te TeV

(b) Natural Radius |V(R e )| = e2 /R e = kB T .



e2 300 14.4 Å. = 557 Å= Re = kB T T TeV

References 54.1 54.2

J. Stevefelt, J. Boulmer, J-F. Delpech: Phys. Rev. A 12, 1246 (1975) R. Deloche, P. Monchicourt, M. Cheret, F. Lambert: Phys. Rev. A 13, 1140 (1976)

54.3 54.4 54.5 54.6

P. Mansbach, J. Keck: Phys. Rev. 181, 275 (1965) L. P. Pitaevski˘ı: Sov. Phys. JETP 15, 919 (1962) H. A. Kramers: Philos. Mag. 46, 836 (1923) D. R. Bates: Phys. Rev. 78, 492 (1950)

Electron–Ion and Ion–Ion Recombination

54.7 54.8

54.9 54.10 54.11

J. N. Bardsley: J. Phys. A Proc. Phys. Soc. 1, 365 (1968) M. R. Flannery: Atomic Collisions: A Symposium in Honor of Christopher Bottcher, ed. by D. R. Schultz, M. R. Strayer, J. H. Macek (American Institute of Physics, New York 1995) p. 53 A. Giusti: J. Phys. B 13, 3867 (1980) P. van der Donk, F. B. Yousif, J. B. A. Mitchell, A. P. Hickman: Phys. Rev. Lett. 68, 2252 (1992) S. L. Guberman: Phys. Rev. A 49, R4277 (1994)

54.12 54.13 54.14 54.15 54.16 54.17 54.18 54.19

General References

825

M. R. Flannery: Int. J. Mass Spectrom. Ion Process 149/150, 597 (1995) M. R. Flannery: J. Chem. Phys. 95, 8205 (1991) A. Burgess: Mon. Not. R. Astron. Soc. 118, 477 (1958) J. A. Gaunt: Philos. Trans. Roy. Soc. A 229, 163 (1930) M. J. Seaton: Mon. Not. R. Astron. Soc. 119, 81 (1959) B. F. Rozsnyai, V. L. Jacobs: Astrophys. J. 327, 485 (1988) D. R. Flower, M. J. Seaton: Comp. Phys. Commun. 1, 31 (1969) P. G. Martin: Astrophys. J. Supp. Ser. 66, 125 (1988)

General References (General Recombination) G.1 G.2

G.3

G.4

G.5

M. R. Flannery: Adv. At. Mol. Phys. 32, 117 (1994) M. R. Flannery: Recombination Processes. In: Molecular Processes in Space, ed. by T. Watanabe, I. Shimamura, M. Shimiza, Y. Itikawa (Plenum, New York 1990), Chapt. 7, p. 145 D. R. Bates: Recombination. In: Electronic and Atomic Collisions, ed. by H. B. Gilbody, W. R. Newell, F. H. Read, A. C. H. Smith (North-Holland, Amsterdam 1988), p. 3 D. R. Bates: Recombination. In: Case Studies in Atomic Physics, ed. by E. W. McDaniel, M. R. C. McDowell (North-Holland, Amsterdam 1974), Vol. 4, p. 57 D. R. Bates, A. Dalgarno: Electronic Recombination. In: Atomic and Molecular Processes,

G.6 G.7 G.8

G.9 G.10 G.11 G.12

ed. by D. R. Bates (Academic Press, New York 1962) D. R. Bates: Adv. At. Mol. Phys. 15, 235 (1979) W. G. Graham (Ed.): Recombination of Atomic Ions (Plenum Press, New York 1992) J. N. Bardsley: Recombination Processes in Atomic and Molecular Physics. In: Atomic and Molecular Collision Theory, ed. by F. A. Gianturco (Plenum Press, New York 1980), p. 123 J. Dubau, S. Volonte: Rep. Prog. Phys. 43, 199 (1980) Y. Hahn: Adv. At. Mol. Phys. 21, 123 (1985) Y. Hahn, K. J. LaGattuta: Phys. Rep. 116, 195 (1988) E. W. McDaniel, E. J. Mansky: Adv. At. Mol. Opt. Phys. 33, 389 (1994)

(Three-Body Electron–Ion Collisional-Radiative Recombination: Theory)

G.14 G.15

D. R. Bates, A. E. Kingston, R. W. P. McWhirter: Recombination between Electrons and Atomic Ions: I. Optically Thin Plasmas; II. Optically Thick Plasmas; Proc. R. Soc. (London) Ser. A 267, 297 (1962); 270, 155 (1962) D. R. Bates, S. P. Khare: Proc. Phys. Soc. 85, 231 (1965) D. R. Bates, V. Malaviya, N. A. Young: Proc. R. Soc. (London) Ser. A 320, 437 (1971)

G.16 G.17 G.18

A. Burgess, H. P. Summers: Mon. Not. R. Astron. Soc. 174, 345 (1976) H. P. Summers: Mon. Not. R. Astron. Soc. 178, 101 (1977) N. N. Ljepojevic, R. J. Hutcheon, J. Payne: Comp. Phys. Commun. 44, 157 (1987)

(Electron–Ion Recombination: Molecular Dynamics Simulations) G.19 G.20 G.21

W. L. Morgan: J. Chem. Phys. 80, 4564 (1984) W. L. Morgan: Phys. Rev. A 30, 979 (1984) W. L. Morgan, J. N. Bardsley: Chem. Phys. Lett. 96, 93 (1983).

G.22

W. L. Morgan: Recent Studies in Atomic and Molecular Processes, ed. by A. E. Kingston (Plenum Press, New York 1987), p. 149

Part D G

G.13

826

Part D

Scattering Theory

(Ion–Ion Recombination: Review Articles) G.23

G.24

G.25

G.26

M. R. Flannery: Three-Body Recombination between Positive and Negative Ions. In: Case Studies in Atomic Collision Physics, ed. by E. W. McDaniel, M. R. C. McDowell (North-Holland, Amsterdam 1972), Vol. 2, 1, p. 1 M. R. Flannery: Ionic Recombination. In: Atomic Processes and Applications, ed. by P. G. Burke, B. L. Moiseiwitsch (North-Holland, Amsterdam 1976), 12, p. 407 M. R. Flannery: Ion–Ion Recombination in High Pressure Plasmas. In: Applied Atomic Collision Physics, ed. by E. W. McDaniel and W. L. Nighan (Academic Press, New York 1983), Vol. 3 Gas Lasers, 5, p. 393 M. R. Flannery: Microscopic and Macroscopic Perspectives of Termolecular Association of Atomic

G.27 G.28 G.29

G.30

Reactants in a Gas. In: Recent Studies in Atomic and Molecular Processes, ed. by A. E. Kingston (Plenum Press, London 1987), p. 167 D. R. Bates: Adv. At. Mol. Phys. 20, 1 (1985) B. H. Mahan: Adv. Chem. Phys. 23, 1 (1973) J. T. Moseley, R. E. Olson, J. R. Peterson: Case Studies in Atomic Collision Physics, ed. by M. R. C. McDowell, E. W. McDaniel (North-Holland, Amsterdam 1972), p. 1 D. Smith, N. G. Adams: Studies of Ion–Ion Recombination using Flowing Afterglow Plasmas. In: Physics of Ion–Ion and Electron–Ion Collisions, ed. by F. Brouillard, J. W. McGowan (Plenum Press, New York 1982), p. 501

(Termolecular Ion–Ion Recombination: Theory)(A) Low Gas Densities: Linear Region G.31 G.32 G.33 G.34 G.35 G.36

M. R. Flannery, E. J. Mansky: J. Chem. Phys. 88, 4228 (1988) M. R. Flannery, E. J. Mansky: J. Chem. Phys. 89, 4086 (1988) M. R. Flannery: J. Chem. Phys. 89, 214 (1988) M. R. Flannery: J. Chem. Phys. 87, 6947 (1987) M. R. Flannery: J. Phys. B 13, 3649 (1980) M. R. Flannery: J. Phys. B 14, 915 (1981)

G.37 G.38 G.39 G.40

D. R. Bates, I. Mendaˇs: J. Phys. B 15, 1949 (1982) D. R. Bates, P. B. Hays, D. Sprevak: J. Phys. B 4, 962 (1971) D. R. Bates, M. R. Flannery: Proc. R. Soc. (London) Ser. A 302, 367 (1968) D. R. Bates, R. J. Moffett: Proc. R. Soc. (London) Ser. A 291, 1 (1966)

(B) All Gas Densities: Non-Linear Region

Part D G

G.41

M. R. Flannery: Microscopic and Macroscopic Theories of Termolecular Recombination between Atomic Ions. In: Dissociative Recombination, ed. by B. R. Rowe, J. B. A. Mitchell, A. Canosa (Plenum Press, New York 1993), p. 205

G.42 G.43 G.44 G.45

M. R. Flannery: J. Chem. Phys. 95, 8205 (1991) M. R. Flannery: Phil. Trans. R. Soc. (London) Ser. A 304, 447 (1982) D. R. Bates, I. Mendaˇs: Proc. R. Soc. (London) Ser. A 359, 275 (1978) J. J. Thomson: Phil. Mag. 47, 337 (1924)

Ion–Ion Recombination: Monte-Carlo Simulations G.46 G.47 G.48 G.49 G.50

P. J. Feibelman: J. Chem. Phys. 42, 2462 (1965) A. Jones, J. L. J. Rosenfeld: Proc. R. Soc. (London) Ser. A 333, 419 (1973) D. R. Bates, I. Mendaˇs: Proc. R. Soc. (London) Ser. A 359, 287 (1978) D. R. Bates: Chem. Phys. Lett. 75, 409 (1980) J. N. Bardsley, J. M. Wadehra: Chem. Phys. Lett. 72, 477 (1980)

G.51 G.52 G.53 G.54 G.55

D. R. Bates: J. Phys. B 14, 4207 (1981) D. R. Bates: J. Phys. B 14, 2853 (1981) D. R. Bates, I. Mendaˇs: Chem. Phys. Lett. 88, 528 (1982) W. L. Morgan, J. N. Bardsley, J. Lin, B. L. Whitten: Phys. Rev. A 26, 1696 (1982) B. L. Whitten, W. L. Morgan, J. N. Bardsley: J. Phys. B 15, 319 (1982)

Electron–Ion and Ion–Ion Recombination

General References

827

Ion–Ion Tidal Recombination: Molecular Dynamics Simulations G.56

D. R. Bates, W. L. Morgan: Phys. Rev. Lett. 64, 2258 (1990)

G.57

W. L. Morgan, D. R. Bates: J. Phys. B 25, 5421 (1992)

G.61 G.62

Y. S. Kim, R. H. Pratt: Phys. Rev. A 27, 2913 (1983) D. J. McLaughlin, Y. Hahn: Phys. Rev. A 43, 1313 (1991) F. D. Aaron, A. Costescu, C. Dinu, J. Phys. II (Paris) 3, 1227 (1993)

Radiative Recombination: Theory G.58 G.59 G.60

M. J. Seaton: Mon. Not. R. Astron. Soc. 119, 81 (1959) D. R. Flower, M. J. Seaton: Comp. Phys. Commun. 1, 31 (1969) A. Burgess, H. P. Summers: Mon. Not. R. Astron. Soc. 226, 257 (1987)

G.63

Dissociative Recombination: Theory and Experiment G.64

G.65 G.66

G.67

D. Zajfman, J. B. A. Mitchell, B. R. Rowe, D. Schwalin (Eds.): Dissociative Recombination: Theory, Experiment and Applications III (World Scientific, Singapore 1996) D. R. Bates: Adv. At. Mol. Phys. 34, 427 (1994) B. R. Rowe, J. B. A. Mitchell, A. Canosa (Eds.): Dissociative Recombination: Theory, Experiment and Applications II (Plenum, New York 1993) J. B. A. Mitchell, S. L. Guberman (Eds.): Dissociative Recombination: Theory, Experiment and Applications I (World Scientific, Singapore 1989)

G.68 G.69

G.70

J. B. A. Mitchell: Phys. Rep. 186, 215 (1990) A. Giusti-Suzor: Recent Developments in the Theory of Dissociative Recombination and Related Processes. In: Atomic Processes in Electron-Ion and Ion-Ion Collisions, ed. by F. Brouillard (Plenum Press, New York 1986) J. N. Bardsley, M. A. Biondi: Adv. At. Mol. Phys. 6, 1 (1970)

Part D G

829

Dielectronic Re 55. Dielectronic Recombination

Dielectronic recombination (DR) is a two-step process that greatly increases the effciency for electrons and ions to recombine in a plasma. The process therefore plays an important role in the theoretical modeling of plasmas, whether in the laboratory or in astrophysical sources such as the solar corona. The purpose of this chapter is to present the theoretical formulation for DR, and the principal methods for calculating rate coefficients. The results are compared with experiment over a broad range of low-Z ions and high-Z ions where relativistic effects become important.

Electron–ion recombination into a particular final recombined state may be schematically represented as (q−1)+

e− + Ai

→ Af

e− + Ai


(q−1)+ (q−1)+ → Af → Aj + ~ω , (55.2)

q+

and q+

+ ~ω ,

(55.1)

55.2 Comparisons with Experiment .............. 831 55.2.1 Low-Z Ions............................... 831 55.2.2 High-Z Ions and Relativistic Effects ............... 831 55.3 Radiative-Dielectronic Recombination Interference ........................................ 832 55.4 Dielectronic Recombination in Plasmas .......................................... 833 References .................................................. 833

In 1961, Unsold, in a letter to Seaton, suggested that DR might account for a well-known temperature discrepancy in the solar corona. Seaton initially concluded that DR would not significantly increase recombination in the solar corona. However, he had only included the lower energy resonance states in his analysis; Burgess [55.3] showed that when one includes the higher members of the Rydberg series of resonance states that are populated at coronal temperatures, DR can indeed explain this discrepancy. Dielectronic recombination has since received much theoretical attention due, in part, to its importance in modeling high temperature plasmas. Various approaches to the theory are discussed in a review by Hahn [55.4] and in [55.5]. Recently, there have been various projects aimed at the generation of large quantities of DR data for use in astrophysical and fusion plasma modeling. One such project is based on the results of the AUTOSTRUCTURE code, with both the total and partial (i. e., resolved by recombined level) DR rate coefficients being archived. The methodology is outlined in Badnell et al. [55.6]. Data are calculated for all members of an isoelectronic sequence from H to Ar, along with various ions relevant to astrophysics or fusion, namely Ca, Ti, Cr, Fe, Ni, Zn, Kr, Mo, and Xe. Work has been completed for the oxygen [55.7], beryllium [55.8], carbon [55.9], lithium [55.10], boron [55.11], neon [55.12] and nitrogen [55.13] isoelectronic sequences. There has also been

Part D 55

where q is the charge on the atomic ion A, ω is the frequency of the emitted light, and the brackets in (55.2) indicate a doubly-excited resonance state. The first process is called radiative recombination (RR), while the second is called dielectronic recombination (DR). Both recombination mechanisms are the inverse of photoionization. At sufficiently high electron density, three-body recombination becomes possible. The three-body mechanism is the inverse of electron impact ionization. The review article by Seaton and Storey [55.1] includes an interesting history of the theoretical work on dielectronic recombination. The process was first referred to as dielectronic recombination by Massey and Bates [55.2], after a suggestion of its possible importance in the ionosphere by Sayers in 1939. However, estimates of the rate coefficient for this process indicated that DR is not an important process in the ionosphere, where the temperatures are too low to excite anything but the lower energy resonance states.

55.1 Theoretical Formulation ....................... 830

830

Part D

Scattering Theory

a large quantity of data generated using a fully relativistic Dirac–Fock code [55.14]. This data includes calculations of Na-like ions [55.15] and H-like through to Ne-like [55.16] ions of certain astrophysically important elements. Interest in dielectronic recombination has increased dramatically in the last twenty years. Mitchell et al. [55.17] published the DR cross section for C+ using a merged electron–ion beams apparatus, and Belic et al. [55.18] reported on a crossed beams measurement of the DR cross section for Mg+ . Also, Dittner et al. [55.19] published merged beams measurements of the DR cross section for the multiply charged ions

B2+ and C3+ . Since that time, atomic physics experiments carried out using heavy-ion traps, accelerators, and storage-cooler rings have produced high-resolution mappings of the resonance structures associated with electron-ion recombination. The experiments have been carried out using a wide range of facilities and technologies, such as the test storage ring (TSR) at Heidelberg, the experimental storage ring (ESR) at Darmstadt, the accelerator-cooler ring facility at Aarhus, the electron beam ion trap (EBIT) at Livermore, and the electron beam ion source (EBIS) at Kansas State. A good review of the dramatic experimental progress in DR measurements is again found in the NATO proceedings [55.5].

55.1 Theoretical Formulation In the independent-processes approximation, the two paths for recombination are summed incoherently. The radiative recombination cross section for (55.1), in lowest-order of perturbation theory, is given by 8π 2 σRR = 3 k    1    2 ×  α0 J0 M0  Dα1 J1  jJM  . 2g1 j

JM M0

(55.3)

The set (α1 J1 ) represents the quantum numbers for the N-electron target ion state, (α0 J0 M0 ) represents the 具σν典 (10–10cm3/s) 12.0

quantum numbers for the (N + 1)-electron recombined ion state, (k j ) represents the quantum numbers for the continuum electron state, (JM) represents the quantum numbers for the (N + 1)-electron system of target plus free electron state, and g1 is the statistical weight of a J1 level. The dipole radiation field operator is given by  N+1 2ω3  D= rs . (55.4) 3πc3 s=1

Continuum normalization is chosen as one times a sine function, and atomic units (e = ~ = m = 1) are used. In the isolated-resonance approximation, the dielectronic recombination cross section for (55.2), in lowest-order perturbation theory, is given by 8π 2    1 2g1 k3  j JM M0    α0 J0 M0 |D|αi Ji Mi αi Ji Mi |V |α1 J1  jJM 2   , ×   E 0 − E i + iΓi /2

10.0

σDR =

8.0

Part D 55.1

6.0

αi Ji Mi

(55.5)

4.0 2.0 0.0 0.0

2.5

5.0

7.5

Fig. 55.1 Dielectronic Recombination for

10.0 12.5 Energy (eV)

O5+ .

Calculations were performed for fields of 0 V/cm (dotted curve), 3 V/cm (dashed curve), 5 V/cm (chain curve), and 7 V/cm (solid curve)

where the set (αi Ji Mi ) represents the quantum numbers for a resonance state with energy E i and total width Γi , and the electrostatic interaction between electrons is given by V=

N 

|rs − r N+1 |−1 .

(55.6)

s=1

By the principle of detailed balance, σRR of (55.3) is proportional to the photoionization cross section from

Dielectronic Recombination

55.2 Comparisons with Experiment

the bound state, while the energy-averaged σDR may be written as 2π 2  1 Aa Ar σDR  = , (55.7) 2g1 Γi ∆k2

the radiative decay rate Ar is given by  |α0 J0 M0 |D|αi Ji Mi |2 , Ar = 2π

where the autoionization decay rate Aa is given by 4  |α1 J1  jJM|V |αi Ji Mi |2 , (55.8) Aa = k

and ∆ is the energy bin width. Each resonance level in (55.7) makes a contribution at a fixed continuum energy k2 /2; thus σDR  plotted as a function of energy is a histogram.

αi Ji Mi

j

JM

831

(55.9)

M0

55.2 Comparisons with Experiment 55.2.1 Low-Z Ions For the most part, the agreement between the recent high-resolution measurements and theoretical calculations based on the independent-processes and isolated-resonance (IPIR) approximations is quite good [55.20–38]. We illustrate the agreement with experiment obtained by calculations employing the IPIR approximations with three examples from the Li isoelectronic sequence. The pathways for dielectronic capture, in terms of specific levels, are given by
 e− + Aq+ (2s1/2 ) → A(q−1)+ (2p1/2 nl j ) 


 A(q−1)+ (2p3/2 nl j ) , (55.10)

55.2.2 High-Z Ions and Relativistic Effects Dielectronic recombination cross section calculations [55.31] in the IPIR approximation for Cu26+ are compared with experiment in Fig. 55.2. The two finestructure Rydberg series are now clearly resolved; the fine-structure splitting is about 27 eV for Cu26+ . The 2p 1/2 13 resonances are just above threshold, while the 2p 3/2 11 resonances are found around 5.0 eV. Electric fields in the range 0–50 V/cm have little effect on the Cu26+ spectrum. Overall, the agreement between theory and experiment is excellent. Electron-ion recombination cross section calculations [55.32] in the IPIR approximation for low-lying resonances in Au76+ are compared with experiment in Fig. 55.3. The 2p 1/2 nl j series limit is at 217 eV, while the 2p 3/2 nl j series limit is at 2.24 keV; yielding a fine

Part D 55.2

where a 1s 2 core is assumed to be present. Both Rydberg series autoionize by the reverse of the paths in (55.10), while for sufficiently high n, the 2p 3/2 nl j levels may autoionize to the 2p 1/2 continuum. Both series radiatively stabilize by either a 2p → 2s core orbital transition, or by a nl j → n l  j  valence orbital transition, where 2p j n  l  j  with j = 12 , 32 is a bound level. Dielectronic recombination cross section calculations [55.28] in the IPIR approximation for O5+ are compared with experiment in Fig. 55.1. Fine-structure splitting of the two series of (55.10) is minimal for this light ion, so there appears only one Rydberg series. The 2p 6 resonances are located at 2.5 eV, the 2p 7 at 5.0 eV, and so on; accumulating at the series limit around 11.3 eV. Electric field effects on the high-n resonances are strong in O5+ , so that calculations were done for fields of 0, 3, 5, and 7 V/cm. Since the precise electric field strength in the experiment is not known, the accuracy of an electric field dependent theory, in this case, has yet to be determined. However, the effects of state mixing by extrinsic fields in the collision region for the dielectronic recombination of Mg+

have been investigated both experimentally [55.39] and theoretically [55.40]. There have been a significant number of recent experiments on low-Z ions. These experiments, in general, show good agreement with theory, see Fogle et al. [55.37] and Schnell et al. [55.38]. However, it is also clear that discrepancies remain between theory and experiment for certain low energy resonances, due to the difficulty in calculating the energy positions of such resonances. As has been pointed out in Savin et al. [55.41] and Schippers et al. [55.42], this can lead to significant uncertainties in low temperature DR rate coefficients. Calculating such low energy resonances to sufficient accuracy for low temperature DR rate coefficients remains a significant challenge for theory. Some success in this area has been achieved using relativistic many-body perturbation theory, obtaining very good agreement with low energy resonance positions for a range of systems, see, for example, Lindroth et al. [55.43], Fogle et al. [55.44], and Tokman et al. [55.45].

832

Part D

Scattering Theory

150

具σ ν典(10–10cm3/s)

a) Recombination rate (10–9cm3/s)

11

120

19 2p1 / 2 ⬁

14 90

⬁

2p 3 /2

2p1/2 nlj

5.6 n = 20

5.2 Experiment

n = 21 2p3/2 6lj

j = 3/2

4.8

3/2

5/2

7/2

4.4

60

4.0

30

3.6 0

150

0

20

40

60

具σ ν典(10–10cm3/s)

80 Energy (eV)

0

10

20

30

40 Energy (eV)

30

40 Energy (eV)

30

40 Energy (eV)

b) Recombination rate (10–9cm3/s) 4.8

× 10

Theory 4.4

120

4.0 90 3.6 0

60

10

20

c) Recombination rate (10–9cm3/s) 30 0

4.8 0

20

40

60

80 Energy (eV)

Fig. 55.2 Dielectronic Recombination for Cu26+

4.4 4.0 3.6

Part D 55.3

structure splitting of 2.03 keV. QED effects alone shift the 2p 3/2 nl j series limit by 22.0 eV. Thus accurate atomic structure calculations must be made to locate the 2p 3/2 6l j resonances in the 0–50 eV energy range of the experiment. The figure shows that the perturbative relativistic, semirelativistic, and fully relativistic calculations for the dielectronic recombination cross section ride on top of a strong radiative recombination background. In principle, the fully relativistic theory contains the most physics, and thus it is comforting that on the whole it is in good agreement with the experiment. It is instructive, however, to see how well the computationally simpler perturbative relativistic and semirelativistic theories do for such a highly charged ion.

0

10

20

Fig. 55.3a–c Dielectronic recombination for Au76+ . The curves show (a) perturbative relativistic, (b) semirelativistic, and (c) fully relativistic calculations for the dielectronic

recombination cross section

There have been several recent experimental measurements on high-Z ions, in particular, for astrophysically abundant species. In general, there is good agreement between theory and experiment. Examples of high-Z element DR studies include those done on Fe XXI and Fe XXII by Savin et al. [55.41], and on Fe XX by Savin et al. [55.46].

55.3 Radiative-Dielectronic Recombination Interference There has been a great deal of effort in recent years to develop a more general theory of electron-ion recombi-

nation which would go beyond the IPIR approximation to include radiative–dielectronic recombination inter-

Dielectronic Recombination

ference and overlapping (and interacting) resonance structures [55.47–54]. In almost all cases, the interference between a dielectronic recombination resonance and the radiative recombination background is quite small and difficult to observe. The best possibility for observation of RR-DR interference appears to be in highly charged atomic ions. In the cases studied to date, the combination of electron and photon continuum coupling selection rules and the requirement of near energy degeneracy make the overlapping (and interacting) resonance effects small and difficult to observe. Heavy ions in relatively low stages of ionization are the best place to look, since there are resonance series attached to the large numbers of L S terms or fine structure levels.

References

833

The distorted wave approximation (Chapt. 52) has been so successful in describing dielectronic recombination cross sections for most atomic ions because, for low charged ions, the DR cross section is proportional to the radiative rate, while for highly charged ions the DR cross section is proportional to the autoionization rate. Thus the weakness of the distorted wave method in calculating accurate autoionization rates for low charged ions is masked by a DR cross section that is highly dependent on radiative atomic structure. As one moves to more highly charged ions, the DR cross section becomes more sensitive to the autoionization rates, but at the same time, the distorted-wave method becomes increasingly more accurate.

55.4 Dielectronic Recombination Dielectronic recombination is an important atomic process that is included in the theoretical modeling of the ionization state and emission of radiating ions, which is fundamental to the interpretation of spectral emission from both fusion and astrophysical plasmas (Chapts. 82, 86, and 87). The dielectronic recombination rate coefficient, into a particular final recombined state, is given by  2

2 2 −k αDR = , k ∆ σDR (i → f ) exp 2T πT 3

where T is the electron temperature. Dielectronic recombination rate coefficients, from the ground and metastable states of a target ion into fully resolved low-lying states and bundled high-lying states of a recombined ion, are required for a generalized collisional radiative treatment [55.55, 56] of highly populated metastable states, the influence of finite plasma density on excited state populations, and of ionization in dynamic plasmas.

(55.11)

References 55.1

55.2

55.6

55.7

55.8

55.9

55.10 55.11

55.12

55.13 55.14 55.15 55.16 55.17

O. Zatzarinny, T. W. Gorczyca, K. T. Korista, N. R. Badnell, D. W. Savin: Astron., Astrophys. 417, 1173 (2003) J. Colgan, M. S. Pindzola, N. R. Badnell: Astron., Astrophys. 417, 1188 (2004) Z. Altun, A. Yumak, N. R. Badnell, J. Colgan, M. S. Pindzola: Astron., Astrophys. 420, 775 (2004) O. Zatzarinny, T. W. Gorczyca, K. T. Korista, N. R. Badnell, D. W. Savin: Astron., Astrophys. 426, 699 (2004) D. M. Mitnik, N. R. Badnell: Astron., Astrophys. 425, 1153 (2004) M. F. Gu: Astrophys. J. 582, 1241 (2003) M. F. Gu: Astrophys. J. 153, 389 (2004) M. F. Gu: Astrophys. J. 590, 1131 (2003) J. B. A. Mitchell, C. T. Ng, J. L. Forand, D. P. Levac, R. E. Mitchell, A. Sen, D. B. Miko, J. Wm. McGowan: Phys. Rev. Lett. 50, 335 (1983)

Part D 55

55.3 55.4 55.5

M. J. Seaton, P. J. Storey: Atomic Processes and Applications, ed. by P. G. Burke, B. L. Moiseiwitsch (North-Holland, Amsterdam 1976) p. 133 H. S. W. Massey, D. R. Bates: Rep. Prog. Phys. 9, 62 (1942) A. Burgess: Astrophys. J. 139, 776 (1964) Y. Hahn: Adv. At. Mol. Phys. 21, 123 (1985) W. G. Graham, W. Fritsch, Y. Hahn, J. A. Tanis (Eds.): Recombination of Atomic Ions, NATO ASI Ser. B, Vol. 296 (Plenum, New York 1992) N. R. Badnell, M. G. O’Mullane, H. P. Summers, Z. Altun, M. A. Bautista, J. Colgan, T. W. Gorczyca, D. M. Mitnik, M. S. Pindzola, O. Zatzarinny: Astron., Astrophys. 406, 1151 (2003) O. Zatzarinny, T. W. Gorczyca, K. T. Korista, N. R. Badnell, D. W. Savin: Astron., Astrophys. 412, 587 (2003) J. Colgan, M. S. Pindzola, A. D. Whiteford, N. R. Badnell: Astron., Astrophys. 412, 597 (2003)

834

Part D

Scattering Theory

55.18 55.19

55.20

55.21 55.22 55.23 55.24 55.25 55.26 55.27 55.28 55.29 55.30 55.31 55.32

55.33 55.34

Part D 55

55.35

55.36

D. S. Belic, G. H. Dunn, T. J. Morgan, D. W. Mueller, C. Timmer: Phys. Rev. Lett. 50, 339 (1983) P. F. Dittner, S. Datz, P. D. Miller, C. D. Moak, P. H. Stelson, C. Bottcher, W. B. Dress, G. D. Alton, N. Neskovic: Phys. Rev. Lett. 51, 31 (1983) G. Kilgus, J. Berger, P. Blatt, M. Grieser, D. Habs, B. Hochadel, E. Jaeschke, D. Kramer, R. Neumann, G. Neureither, W. Ott, D. Schwalm, M. Steck, R. Stokstad, E. Szmola, A. Wolf, R. Schuch, A. Müller, M. Wagner: Phys. Rev. Lett. 64, 737 (1990) M. S. Pindzola, N. R. Badnell, D. C. Griffin: Phys. Rev. A 42, 282 (1990) D. R. DeWitt, D. Schneider, M. W. Clark, M. H. Chen, D. Church: Phys. Rev. A 44, 7185 (1991) L. H. Andersen, P. Hvelplund, H. Knudsen, P. Kvistgaard: Phys. Rev. Lett. 62, 2656 (1989) N. R. Badnell, M. S. Pindzola, D. C. Griffin: Phys. Rev. A 41, 2422 (1990) L. H. Andersen, G. Y. Pan, H. T. Schmidt, N. R. Badnell, M. S. Pindzola: Phys. Rev. A 45, 7868 (1992) D. A. Knapp, R. E. Marrs, M. B. Schneider, M. H. Chen, M. A. Levine, P. Lee: Phys. Rev. A 47, 2039 (1993) R. Ali, C. P. Bhalla, C. L. Cocke, M. Stockli: Phys. Rev. Lett. 64, 633 (1990) D. C. Griffin, M. S. Pindzola, P. Krylstedt: Phys. Rev. A 40, 6699 (1989) L. H. Andersen, J. Bolko, P. Kvistgaard: Phys. Rev. A 41, 1293 (1990) L. H. Andersen, G. Y. Pan, H. T. Schmidt, M. S. Pindzola, N. R. Badnell: Phys. Rev. A 45, 6332 (1992) G. Kilgus, D. Habs, D. Schwalm, A. Wolf, N. R. Badnell, A. Müller: Phys. Rev. A 46, 5730 (1992) W. Spies, A. Müller, J. Linkemann, A. Frank, M. Wagner, C. Kozhuharov, B. Franzke, K. Beckert, F. Bosch, H. Eickhoff, M. Jung, O. Klepper, W. König, P. H. Mokler, R. Moshammer, F. Nolden, U. Schaaf, P. Spädtke, M. Steck, P. Zimmerer, N. Grün, W. Scheid, M. S. Pindzola, N. R. Badnell: Phys. Rev. Lett. 69, 2768 (1992) N. R. Badnell, M. S. Pindzola, L. H. Andersen, J. Bolko, H. T. Schmidt: J. Phys. B 24, 4441 (1991) A. Lampert, D. Habs, G. Kilgus, D. Schwalm, A. Wolf, N. R. Badnell, M. S. Pindzola: AIP Conference Proceedings # 274, The VIth International Conference on the Physics of Highly-Charged Ions, 1992, ed. by M. Stockli, P. Richard (American Institute of Physics, New York 1992) p. 537 D. R. DeWitt, D. Schneider, M. H. Chen, M. W. Clark, J. W. McDonald, M. B. Schneider: Phys. Rev. Lett. 68, 1694 (1992) M. B. Schneider, D. A. Knapp, M. H. Chen, J. H. Scofield, P. Beiersdorfer, C. L. Bennett,

55.37 55.38

55.39

55.40 55.41

55.42

55.43

55.44 55.45

55.46

55.47 55.48 55.49 55.50 55.51 55.52 55.53 55.54 55.55 55.56

J. R. Henderson, R. E. Marrs, M. A. Levine: Phys. Rev. A 45, R1291 (1992) M. Fogle, N. R. Badnell, N. Eklöw, T. Mohamed, R. Schuch: Astron., Astrophys. 409, 781 (2003) M. Schnell, G. Gwinner, N. R. Badnell, M. E. Bannister, R. Böhm, J. Colgan, S. Klieslich, S. D. Loch, D. Mitnik, A. Müller, M. S. Pindzola, S. Schippers, D. Schwalm, W. Shi, A. Wolf, S.-G. Zhou: Phys. Rev. Lett. 91, 043001 (2003) A. Müller, D. S. Belic, B. D. DePaola, N. Djuric, G. H. Dunn, D. W. Mueller, C. Timmer: Phys. Rev. Lett. 56, 127 (1986) C. Bottcher, D. C. Griffin, M. S. Pindzola: Phys. Rev. A 34, 860 (1986) D. W. Savin, S. M. Kahn, G. Gwinner, M. Grieser, R. Repnow, G. Saathoff, D. Schwalm, A. Wolf, A. Müller, S. Schippers, P. A. Zavodszky, M. H. Chen, T. W. Gorczyca, O. Zatzarinny, M. F. Gu: Astrophys. J. Suppl. Ser. 147, 421 (2003) S. Schippers, M. Schnell, C. Brandau, S. Klieslich, A. Müller, A. Wolf: Astron., Astrophys. 421, 1185 (2004) E. Lindroth, H. Danared, P. Glans, Z. Pesic, M. Tokman, G. Vikor, R. Schuch: Phys. Rev. Lett. 86, 5027 (2001) M. Fogle, N. Eklöw, E. Lindroth, T. Mohamed, R. Schuch, M. Tokman: J.Phys.B 36, 2563 (2003) M. Tokman, N. Eklöw, P. Glans, E. Lindroth, R. Schuch, G. Gwinner, D. Schwalm, A. Wolf, A. Hoffknecht, A. Müller, S. Schippers: Phys. Rev. A 66, 012703 (2002) D. W. Savin, E. Behar, S. M. Kahn, G. Gwinner, A. A. Saghiri, M. Schmitt, M. Grieser, R. Repnov, D. Schwalm, A. Wolf, T. Bartsch, A. Müller, S. Schippers, N. R. Badnell, M. H. Chen, T. W. Gorczyca: Astrophys. J. Suppl. Ser. 138, 337 (2003) P. C. W. Davies, M. J. Seaton: J. Phys. B 2, 757 (1969) E. Trefftz: J. Phys. B 3, 763 (1970) R. H. Bell, M. J. Seaton: J. Phys. B 18, 1589 (1985) G. Alber, J. Cooper, A. R. P. Rau: Phys. Rev. A 30, 2845 (1984) K. J. LaGattuta: Phys. Rev. A 40, 558 (1989) S. L. Haan, V. L. Jacobs: Phys. Rev. A 40, 80 (1989) N. R. Badnell, M. S. Pindzola: Phys. Rev. A 45, 2820 (1992) M. S. Pindzola, N. R. Badnell, D. C. Griffin: Phys. Rev. A 46, 5725 (1992) H. P. Summers, M. B. Hooper: Plasma Phys. 25, 1311 (1983) N. R. Badnell, M. S. Pindzola, W. J. Dickson, H. P. Summers, D. C. Griffin: Astrophys. J. 407, L91 (1993)

835

Rydberg Collis 56. Rydberg Collisions: Binary Encounter, Born and Impulse Approximations

Rydberg collisions are collisions of electrons, ions and neutral particles with atomic or molecular targets which are in highly excited Rydberg states characterized by large principal quantum numbers (n
1). Rydberg collisions of atoms and molecules with neutral and charged particles include the study of collision-induced transitions both to and from Rydberg states and transitions among Rydberg levels. The basic quantum mechanical structural properties of Rydberg states are given in Chapt. 14. This Chapter collects together many of the equations used to study theoretically the collisional properties of both charged and neutral particles with atoms and molecules in Rydberg states or orbitals. The primary theoretical scattering approximations enumerated in this Chapter are the impulse approximation, binary encounter approximation and the Born approximation. The theoretical techniques used to study Rydberg collisions complement and supplement the eigenfunction expansion approximations used for collisions with target atoms and molecules in their ground (n = 1) or first few excited states (n > 1), as discussed in Chapt. 47. Direct application of eigenfunction expansion techniques to Rydberg collisions, wherein the target particle can be in a Rydberg orbital with principal quantum number in the range n ≥ 100, is prohibitively difficult due to the need to compute numerically and store wave functions with n3 nodes. For n = 100 this amounts to ∼ 106 nodes for each of the wave functions represented in the eigenfunction expansion. Therefore, a variety of approximate scattering theories have been developed to deal specifically with the pecularities of Rydberg collisions.

56.1 Rydberg Collision Processes .................. 836 836 836 836 837

56.3 Correspondence Principles .................... 56.3.1 Bohr–Sommerfeld Quantization .. 56.3.2 Bohr Correspondence Principle ... 56.3.3 Heisenberg Correspondence Principle .................................. 56.3.4 Strong Coupling Correspondence Principle .................................. 56.3.5 Equivalent Oscillator Theorem.....

839 839 839 839 840 840

56.4 Distribution Functions ......................... 840 56.4.1 Spatial Distributions .................. 840 56.4.2 Momentum Distributions ........... 840 56.5 Classical Theory ................................... 841 56.6 Working Formulae for Rydberg Collisions 56.6.1 Inelastic n,
-Changing Transitions ............................... 56.6.2 Inelastic n → n Transitions........ 56.6.3 Quasi-Elastic
-Mixing Transitions ............................... 56.6.4 Elastic n
→ n
 Transitions ...... 56.6.5 Fine Structure n
J → n
J Transitions ...............................

842 842 843 844 844 844

56.7 Impulse Approximation ........................ 56.7.1 Quantal Impulse Approximation.. 56.7.2 Classical Impulse Approximation . 56.7.3 Semiquantal Impulse Approximation..........................

845 845 849

56.8 Binary Encounter Approximation .......... 56.8.1 Differential Cross Sections .......... 56.8.2 Integral Cross Sections ............... 56.8.3 Classical Ionization Cross Section . 56.8.4 Classical Charge Transfer Cross Section ....................................

852 852 853 855

56.9 Born Approximation ............................ 56.9.1 Form Factors............................. 56.9.2 Hydrogenic Form Factors ............ 56.9.3 Excitation Cross Sections ............ 56.9.4 Ionization Cross Sections ............ 56.9.5 Capture Cross Sections................

856 856 856 858 859 859

851

855

References .................................................. 860

Part D 56

56.2 General Properties of Rydberg States..... 56.2.1 Dipole Moments ........................ 56.2.2 Radial Integrals ........................ 56.2.3 Line Strengths...........................

56.2.4 Form Factors............................. 838 56.2.5 Impact Broadening.................... 838

836

Part D

Scattering Theory

56.1 Rydberg Collision Processes Depolarization collisions:

(A) State-Changing Collisions Quasi-elastic
-mixing collisions: ∗

∗



A (n
) + B → A (n
) + B .

(56.1)

Quasi-elastic J-mixing collisions: Fine structure transitions with J = |
± 1/2| → J  = |
± 1/2| are ∗

∗



A (n
J ) + B → A (n
J ) + B .

(56.2)

Energy transfer n-changing collisions: ∗

∗





A (n) + B(β) → A (n ) + B(β ) ,

(56.3)

where, if B is a molecule, the transition β → β  represents an inelastic energy transfer to the rotationalvibrational degrees of freedom of the molecule B from the Rydberg atom A∗ . Elastic scattering: A∗ (γ) + B → A∗ (γ) + B ,

(56.4)

where the label γ denotes the set of quantum numbers n,
or n,
, J used.

A∗ (n
m) + B → A∗ (n
m  ) + B , A∗ (n
JM ) + B → A∗ (n
JM  ) + B . (B) Ionizing Collisions Direct and associative ionization:   A+ + B(β  ) + e− A∗ (γ) + B(β) →  BA+ + e− .

(56.5a) (56.5b)

(56.6)

Penning ionization: A∗ (γ) + B → A + B + + e− .

(56.7)

Ion pair formation: A∗ (γ) + B → A+ + B − .

(56.8)

Dissociative attachment: A∗ (γ) + BC → A+ B − + C .

(56.9)

56.2 General Properties of Rydberg States Table 56.1 displays the general n-dependence of a number of key properties of Rydberg states and some specific representative values for hydrogen.

 2n−7 1 1   2 25 1 2 − n 12  X 2 p→n  = 11 − .   144 n 3 1 1 2n+7 n2 + 2 n

56.2.1 Dipole Moments

(56.11c)

Definition. Di→ f = −eXi→ f where

Xi→ f = φ f |



eik·rj rj |φi  .

Asymptotic Expressions. For n
1, (56.10)

j

Hydrogenic Dipole Moments. See Bethe and Salpeter [56.1] and the references by Khandelwal and co-workers [56.2–5] for details and tables. Exact Expressions. In the limit |k| → 0, the dipole al-

lowed transitions summed over final states are

Part D 56.2

8 2n−5    X 1s→n 2 = 2 n 7 (n − 1) , (56.11a) 3 (n + 1)2n+5  2n−7 1   5 −1n   1 1 1  X 2s→n 2 = 2  2 1− , − 2n+7 4 n2 3n 3 1 n2 + 1n 2

(56.11b)

 2 5.731 13.163 24.295 + n 3  X 1s→n  ≈ 1.563 + 2 + n n4 n6 39.426 58.808 + + , (56.12a) n8 n 10  2 180.785 1435.854 n 3  X 2s→n  ≈ 14.658 + + n2 n4 9341.634 54 208.306 + + n6 n8 292 202.232 + , (56.12b) n 10  2 218.245 2172.891 n 3  X 2 p→n  ≈ 13.437 + + n2 n4 17 118.786 117 251.682 + + n6 n8 731427.003 + . (56.12c) n 10

Rydberg Collisions: Binary Encounter, Born and Impulse Approximations

56.2 General Properties of Rydberg States

837

Table 56.1 General n-dependence of characteristic properties of Rydberg states. After [56.6] Property

n−dependence

n = 10

n = 100

n = 500

n = 1000

Radius (cm) Velocity (cm/s) Area (cm2 ) Ionization potential (eV) Radiative lifetime (s)a Period of classical motion (s) Transition frequency (s−1 ) Wavelength (cm) √ 

a A = 8α3 /(3 3π) (v /a ) 0 B 0

n 2 a0 /Z vB Z/n πa02 n 4 /Z 2 Z 2 R∞ /n 2 n 5 (3 ln n − 14 )/(A0 Z 4 ) 2π/ωn,n±1 = hn 3 /(2Z 2 R∞ ) ωn,n±1 = 2Z 2 R∞ /(~n 3 ) λn,n±1 = 2πc/ωn,n±1

5.3 × 10−7 2.18 × 107 8.8 × 10−13 1.36 × 10−1 8.4 × 10−5 1.5 × 10−13 4.1 × 1013 4.6 × 10−3

5.3 × 10−5 2.18 × 106 8.8 × 10−9 1.36 × 10−3 17 1.5 × 10−10 4.1 × 1010 4.6

1.3 × 10−3 4.4 × 105 5.5 × 10−6 5.44 × 10−5 7.3 × 104 1.9 × 10−8 3.3 × 108 570

5.3 × 10−3 2.18 × 105 8.8 × 10−5 1.36 × 10−6 7.22 hours 1.5 × 10−7 4.1 × 107 4.5609 × 103

56.2.2 Radial Integrals Definition.  

n
Rn
≡

∞ Rn
(r)r Rn 
 (r)r 2 dr ,

(56.13)

0

where Rn
(r) are solutions to the radial Schrödinger equation. See Chapt. 9 for specific representations of Rn
for hydrogen. Exact Results for Hydrogen. For
 =
− 1 and n =

n  [56.7], 

n  −


n+n  −2
−2

a0 (−1) (4nn  )
+1 (n − n  )  Z 4(2
− 1)!(n + n  )n+n 

(n +
)!(n  +
− 1)! 1/2 × (n  +
 )!(n −
− 1)!  × 2 F1 (−n +
+ 1, −n  +
; 2
; Y )

n
−1 = Rn


(56.17a) (56.17b) (56.17c) (56.17d)

and Jn (y) is the Anger function. The energies of the states n
and n 
 are given in terms of the quantum defects by E n
= − Z 2 R∞ /n ∗2 , n ∗ = n − δ
, E n 
 = − Z R∞ /n 2

∗2

∗



, n = n − δ
 .

(56.18a) (56.18b)

Sum Rule. For hydrogen

   2    2 n
−1 n
+1 Rn
 = Rn
 n

  n−n  2  F (−n+
−1, −n +
; 2
; Y ) , − 2 1 n+n  (56.14)

where Y = −4nn  /(n − n  )2 . For n = n  ,   a0 3 n
−1 n n 2 −
2 . = Rn
Z 2

where n c = 2n ∗ n ∗ /(n ∗ + n ∗ ) , ∆ = n ∗ − n ∗ , ∆
=
 −
,
> = max(
,
 ) ,  x = e∆, e = 1 − (
> /n c )2 ,

(56.15)

Semiclassical Quantum Defect Representation [56.8].

(56.19b)

See §61 of [56.1] for additional sum rules.

56.2.3 Line Strengths Definition.

2  S(n 
 , n
) = e2 (2
+ 1) rn 
 ,n
    2  n
 = e2 max(
,
 ) Rn
 ,

(56.20a) (56.20b)

where
 =
± 1. For hydrogen   ea 2 (n − n  )2(n+n )−3 0 (nn  )6 S(n  , n) = 32  Z (n + n  )2(n+n )+4 

 2 × 2 F1 (−n  , −n + 1; 1; Y )

2  , (56.21) − 2 F1 (−n  + 1, −n; 1; Y ) where Y = −4nn  /(n − n  )2 .

Part D 56.2

   2  a 2  n 2  ∆

> 0  n
 c  J∆−1 (−x) 1− Rn
 = Z  2∆ nc  ∆

> J∆+1 (−x) − 1+ nc 2  2 (56.16) + sin(π∆)(1 − e)  , π

=

(56.19a)

n  n 2 a02 2 5n + 1 − 3
(
+ 1) . 2 Z2

838

Part D

Scattering Theory

Line Strength of Line n.

Semiclassical Representation [56.9].

32  ea0 2 (εε )3/2 S(n , n) = √ G(∆n) , (56.22) (ε − ε )4 π 3 Z 

where ε = 1/n 2 , ε = 1/n 2 , and the Gaunt factor G(∆n) is given by √    G(∆n) = π 3∆n  J∆n (∆n)J∆n (∆n) , (56.23) where the prime on the Anger function denotes differentiation with respect to the argument ∆n. Equation (56.23) can be approximated to within 2% by the expression 1 . 4|∆n|

1−

(56.24)

Relation to Oscillator Strength. 

S(n , n) =



S(n
, n
)

1 . k3

(56.29)

Born Approximation to Line Strength Sn [56.6].

 2R  1 1 Z ∞ 1 1− ln(1 + εe /ε) SnB = E 2 4k k4 k =0   0.60 1  4 εe  1− + 3 ε + εe k k3 k =0

  εe  1.47εe Z 2 R∞ = 0.82 ln 1 + , (56.30) + E ε ε + εe

where ε = |E n |Z 2 /R∞ and εe = ε/Z 2 R∞ .        n
m  eiQ·r n 
 m  2 . (56.31)
,m
 ,m 

(56.25)


,


Connection with Generalized Oscillator Strengths.

f n  n (Q) =

Connection with Radial Integral.

~ω max(
,
 )  n 
  Rn
. 3R∞ (2
+ 1)

S(n + k, n)

k =0

Fn  n (Q) =


,


− f n 
 ,n
=



56.2.4 Form Factors

 

R∞  = 3e2 a02 f n 
 ,n
. ~ω 

Sn ≡ S(n) =

(56.26)

Density of Line Strengths. For bound-free n
→ E


transitions in a Coulomb field, the semiclassical representation [56.6] is  d R∞ 2 S(n
, E) = 2n(2
+ 1) dE ~ω  2 2

 e a0 1 × J∆ (e∆)2 + 1 − 2 J∆ (e∆)2 , R∞ e (56.27)

Part D 56.2

 2  where ∆ = ~ωn 3 /2R∞ and e = 1 −
+ 12 /n 2 . Asymptotic expression for ∆
1:  4  2
+ 1 2 2(2
+ 1) R∞ d S(n
, E) = dE ~ω 3π 2 n3   e2 a2 0 2 2 × K 1/3 (η) + K 2/3 (η) , R∞

(56.28)

where η = (E/R∞ )(
+ 1/2)3 /6 and the K ν (x) are Bessel functions of the third kind.

Z 2 ∆E Fn  n (Q) . n 2 Q 2 a02

(56.32)

Semiclassical Limit.

lim f n  n (Q) =

Q→0

3  32 nn  3n 2 ∆n(n + n  )  × ∆n J∆n (∆n)J∆n (∆n) ,

(56.33)

where Jm (y) denotes the Bessel function. Representation as Microcanonical Distribution.

2Z 2 R∞ d p|gn
( p)|2 Fn  ,n
(Q) = (2
+ 1) n 3  ( p − ~ Q)2 p2 − − E n  − E n
, ×δ 2m 2m (56.34)

 2 d p dr Ze2 p ∞ − − E Fn  ,n (Q) = δ n r (nn  )3 (2π ~)3 2m  ( p − ~ Q)2 Ze2 ×δ − − En , 2m r 4Z 2 R

2

(56.35)

29 κ5 = , 2 )3 (κ 2 + κ 2 )3 3π(nn  )3 (κ 2 + κ+ − (56.36)

where κ = Qa0 /Z and κ± = |1/n ± 1/n  |.

Rydberg Collisions: Binary Encounter, Born and Impulse Approximations

56.2.5 Impact Broadening The total broadening cross section of a level n is   σn = πa02 /Z 4 n 4 Sn . (56.37) The width of a line n → n + k is [56.10]

 γn,n+k = n e vσn  + vσn+k  ,

56.3 Correspondence Principles

where n e is the number density of electrons, and  n4 vσn  = vσn+k,n  = 3 K n (56.39a) Z k =0 ∞ n 4 πa02 vB E dE = 3 3/2 e−E/kB T Sn 2 , Z θ (Z R∞ )2 0

(56.38)

839

where θ = kB broadening.

T/Z 2 R

(56.39b)

∞ . See Chapt. 59 for collisional line

56.3 Correspondence Principles Correspondence principles are used to connect quantum mechanical observables with the corresponding classical quantities in the limit of large n. See [56.11] for details on the equations in this section.

The first order S-matrix is iω S fi = − 2π ~

∞

2π/ω

V [R(t), r(t1 )] eisω(t1 −t) dt1 ,

dt

−∞

0

56.3.1 Bohr–Sommerfeld Quantization 

Ai = Ji ∆wi

pi dqi = 2π ~(n i + αi ) ,

(56.40)

where n i = 0, 1, 2, . . . and αi = 0 if the generalized coordinate qi represents rotation, and αi = 1/2 if qi represents a libration.

56.3.2 Bohr Correspondence Principle E n+s − E n = hνn+s,n ∼ s~ωn , s = 1, 2, . . .  n , (56.41)

where νn+s,n is the line emission frequency and ωn is the angular frequency of classical orbital motion. The number of states with quantum numbers in the range ∆n is D

D

(∆Ji ∆wi ) /(2π ~) D

∆n =

∆N = i=1 D

where R denotes the classical path of the projectile and r the orbital of the Rydberg electron.

56.3.3 Heisenberg Correspondence Principle For one degree of freedom [56.11], (q) Fmn (R) =

∞

∗ φm (r)F(r, R)φn (r) dr 2π/ω

ω = 2π

 F (c) r(t) eisωt dt .

(56.46)

0

The three-dimensional generalization is [56.11]

 1 (q) (c) Fn,n ∼ Fs (J) = F c r(J, w) eis·w dw , 8π 3 (56.47)

(56.42)

i=1

n,m

where the are the quantal matrix elements between time independent states.

where n, n denotes the triple of quantum numbers (n,
, m), (n  ,
 , m  ), respectively, and s = n − n . The correspondence between the three dimensional quantal and classical matrix elements in (56.47) follows from the general Fourier expansion for any classical function F (c) (r) periodic in r,

  (c) Fs (J) exp(−is · w) , (56.48) F (c) r(t) = s

where J, w denotes the action-angle conjugate variables for the motion. For the three dimensional Coulomb

Part D 56.3

for systems with D degrees of freedom, and the mean value F¯ of a physical quantity F(q) in the quantum state Ψ is  (q) ∗ am an Fmn eiωmn t , (56.43) F¯ = Ψ |F(q)|Ψ  = (q) Fmn

(56.45)

0

i=1

(∆ pi ∆qi )/(2π ~) D ,

=

(56.44)

840

Part D

Scattering Theory

problem, the action-angle variables are  ∂E t +δ , Jn = n ~ , wn = ∂Jn  1 J
=
+ ~, w
= ψE , 2 Jm = m ~ ,

wm = φE ,

S fi = −

iω 2π ~

∞

dte 0

The S-matrix is ω S fi = 2π

2π/ω

! dte exp i(sωte )

0

(56.49)

where ψE is the Euler angle between the line of nodes and a direction in the plane of the orbit (usually taken to be the direction of the perihelion or perigee), and is constant for a Coulomb potential. The Euler angle φE is the angle between the line of nodes and the fixed x-axis. See [56.11] for details. The first order S-matrix is 2π/ω

56.3.4 Strong Coupling Correspondence Principle

 dt V R(t), r(t + te ) eisωte ,

−∞

(56.50)

with s = i − f , R is the classical path of the projectile, and r(te ) is the classical internal motion of the Rydberg electron.

−

i ~

∞

 " V R(t), r(t + te ) dt .

(56.51)

−∞

See [56.11–14] for additional details.

56.3.5 Equivalent Oscillator Theorem 

an (t)V fn (t) e

iω fn t

n

=



ad+ f (t)Vd (t) e−idωt . (56.52)

d=− f

The S-matrix is Sn ,n = an (t → ∞) (56.53) ∞ 2π   i dw = exp is · w − V(w + ωt, t) dt . ~ 8π 3 −∞

0

56.4 Distribution Functions The function Wα (x) dx characterizes the probability (distribution) of finding an electron in a Rydberg orbital α within a volume dx centered at the point x in phase space. Integration of the distribution function Wα over all phase space volumes dx yields, depending upon the normalization chosen, either unity or the density of states appropriate to the orbital α.

Distribution over n: Wn (r)r 2 dr = g(n)

  r 2 1/2 r dr 2 1− 1− , π a a2 (56.56)

with g(n) = n 2 .

56.4.2 Momentum Distributions 56.4.1 Spatial Distributions Distribution over n,
, m [56.6]: (56.54) W n
m (r, θ)r 2 sin θ dr dθ r 2 sin θ dr dθ = 

 1/2 , 

π 2 a2 r e2 − (1 − r/a)2 sin2 θ − (m/
)2

Part D 56.4

where a = Ze2 /2|E| = n 2 ~2 /m Ze2 ~2 is the semimajor axis, and e2 = 1 − (
/n)2 is the eccentricity. Distribution over n,
: W n
(r, θ)r 2 sin θ dr dθ r 2 sin θ dr dθ = g(n
) 1/2 ,

2πa2r e2 − (1 − r/a)2 where g(n
) = 2
.

Distribution over n,
[56.6]: dx 4 Wn
( p) p2 d p = g(n
)   , π 1 + x2 2 where x = p/ pn and p2n = 2m|E|. Distribution over n: 32 x 2 dx Wn ( p) p2 d p = g(n)   . π 1 + x2 4

(56.57)

(56.58)

Sum Rules. (56.55)

2  na0 3 1   8  = G (k) n
m  4 , 2 2 Z n2 π x +1
,m (56.59a)

Rydberg Collisions: Binary Encounter, Born and Impulse Approximations

n−1  2 32na0 x 2 1  (2
+ 1)gn
(k) k2 =  4 , 2 n πZ x 2 + 1
=0

(56.59b)

where x = nka0 /Z, and

(56.62)

 n ∗ sin n ∗ (β − π) J(n ∗ , 0; X) = − sin(β − π)  ∗ 1  1 − s2 sn sin n ∗ π   ds , − π 1 − 2Xs + s2 0

(56.60b) ( j)

Quantum Defect Representation [56.15].

2 Γ (n ∗ −1) 1/2∗ n (a0 /Z)3/2 22(
+1) π Γ (n ∗ +
+1) (
+ 1)!(−ix)
∗ ×  
+2 J(n ,
+ 1; X) , 2 x +1 

gn
(k) = −

841

where n ∗ = n − δ, δ being the quantum defect, and x = n ∗ ka0 /Z. The function J is given by the recurrence relation ∂ 1 J(n ∗ ,
; X) , J(n ∗ ,
+ 1; X) = − 2(2
+ 2) ∂X

G n
m (k) = gn
(k)Y
m (kˆ ) , (56.60a) 1/2    3/2 a0 2 (n−
−1)! gn
(k) = 22(
+1) n 2
! π (n+
)! Z  2 x −1 (−ix)
(
+1) × , C 
+2 n−
−1 x2 + 1 x2 + 1 where Ci (y) is the associated Gegenbauer polynomial. See Chapt. 9 for additional details on hydrogenic wave functions.

56.5 Classical Theory

(56.63)



   where X = x 2 − 1 / x 2 + 1 , and β = cos−1 X. In the ∗ limit
 n , (56.61) becomes

  ∗ 3 1−(−1)
cos 2n ∗ (β − π) n a0 |gn
(k)|2 = 4 . Z πx 2 (x 2 + 1)2 (56.64)

Classical Density of States.



ρ(E ) =

δ [E − H( p, r)]

d p dr n 5 ~2 = . (2π ~)3 m Z 2 e4

(56.61)

(56.65)

The classical cross section for energy transfer ∆E between two particles, with arbitrary masses m 1 , m 2 and charges Z 1 , Z 2 , is given by [56.16]

fective cross section averaged over the direction nˆ 2 of v2 is 1 (eff) dnˆ 2 |v1 − v2 nˆ 2 |σ∆E (v1 , v2 ) . v1 σ∆E (v1 , v2 ) = 4π

56.5 Classical Theory

σ∆E (v1 , v2 ) =

2π(Z 1 Z 2 e2 V )2 (56.66)  3 v2 ∆E   ∆E cos θ¯ , × 1 + cos2 θ¯ + µvV

valid for −1 ≤ cos θ¯ − ∆E/(µvV ) ≤ 1, and σ∆E (v1 , v2 ) = 0 otherwise, where (56.67a)

V = (m 1 v1 + m 2 v2 )/M ,

(56.67b)

cos θ¯ =

1 v·V , vV

If v1 is also isotropic, then the average of (56.68), together with (56.66), gives for the special case of a Coulomb potential (eff) σ∆E (v1 , v2 )  2    π Z 1 Z 2 e2  2 v1 − v22 v22 − v12 vl−1 − vu−1 = 2 4|∆E|3v1 v2    1 + v12 + v22 + v12 + v22 (vu − v
) − vu3 − vl3 , 3

(56.69)

(56.67c)

and µ = m 1 m 2 /M, M = m 1 + m 2 . If particle 2 has an isotropic velocity distribution in the lab frame, the ef-

where

1/2  v1 = v12 − 2∆E/m 1 ,

(56.70)

Part D 56.5

v = v1 − v2 ,

(56.68)

842

Part D

Scattering Theory

1/2  v2 = v22 + 2∆E/m 2 ,

(56.71)

and vu , vl are defined below for cases 1.–4. With the definitions ∆ε12 = 4m 1 m 2 (E 1 − E 2 )/M 2 , ∆m 12 = |m 1 −m 2 | ,  4m 1 m 2 v2 v1 E , ∆˜ε12 = − E 1 2 v1 v2 M2 the four cases are 1. ∆E ≥ ∆ε12 + |∆˜ε12 | ≥ 0, and 2m 2 v2 ≥ ∆m 12 v1 : vl = v2 − v1 ,

vu = v1 + v2 ,

∆E ≥ 0 ; (56.72a)

vl = v2 − v1 ,

vu = v1 + v2 ,

∆E ≤ 0 . (56.72b)

2.

eff (v , v ) = 0, If 2m 2 v2 < ∆m 12 v1 , then σ∆E 1 2 ∆ε12 − ∆˜ε12 ≤ ∆E ≤ ∆ε12 + ∆˜ε12 , and m 1

vl = v2 − v1 ,

vu = v1 + v2 , vu = v1 + v2 ,

∆E ≥ 0 ;

∆E ≥ 0 ; (56.72e)

vl = v1 − v2 ,

vu = v1 + v2 ,

∆E ≤ 0 . (56.72f)

eff (v , v ) = 0, If 2m 1 v1 < ∆m 12 v2 , then σ∆E 1 2 4. ∆ε12 + ∆˜ε12 ≤ ∆E ≤ ∆ε12 − ∆˜ε12 , and m 1 < m 2 :

vl = v1 − v2 ,

vu = v1 + v2 ,

∆E ≥ 0 ; (56.72g)

vl = v1 − v2 ,

vu = v1 + v2 ,

∆E ≤ 0 . (56.72h)

If 2m 1 v1 < ∆m 12 v2 , then

which simply expresses the fact that the particle losing energy in the collision cannot lose more than its initial kinetic energy. The cross section (56.69) must be integrated over the classically allowed range of energy transfer ∆E and averaged over a prescribed speed distribution W(v2 ) before comparison with experiment can be made. See [56.16, 17] for details. Classical Removal Cross Section [56.18]. The cross

∞ σr (V ) =

f(v)σ∆E (v1 , v2 ) dv .

(56.74)

0

∆E ≤ 0

3. ∆E ≤ ∆ε12 − |∆˜ε12 | ≤ 0, and 2m 1 v1 ≥ ∆m 12 v2 : vu = v1 + v2 ,

(56.73)

section for removal of an electron from a shell is given by

(56.72d)

vl = v1 − v2 ,

1 1 − m 2 v22 ≤ ∆E ≤ m 1 v12 , 2 2

> m2:

(56.72c)

vl = v2 − v1 ,

Since v1 and v2 , given by (56.70) and (56.71) respectively, must be real, σ∆E (v1 , v2 ) = 0 for ∆E outside the range

eff (v , v ) = 0. σ∆E 1 2

Total Removal Cross Section [56.18]. In an independent

electron model, σrtotal (V ) = Nshell σr (V ) ,

(56.75)

where Nshell is the number of equivalent electrons in a shell. In a shielding model,

 (Nshell − 1) σ (V ) Nshell σr (V ) , σrtotal (V ) = 1 − r 4π r¯ 2 (56.76)

where r¯ 2 is the root mean square distance between electrons within a shell. Experiment [56.19] favors (56.76) over (56.75). See Fig. 4a–e of [56.18] for details. Classical trajectory and Monte-Carlo methods are covered in Chapt. 58.

56.6 Working Formulae for Rydberg Collisions Part D 56.6

56.6.1 Inelastic n,
-Changing Transitions A∗ (n
) + B → A∗ (n  ) + B + ∆E n  ,n
,

(56.77)

where ∆E = E − E n
is the energy defect. The cross section for (56.77) in the quasifree electron n  ,n


n

model [56.20] is σn  ,n
(V ) = 

2πas2 f n  ,n
(λ) , 2 V/vB n 3


n, (56.78)

where as is the scattering length for e− + B scattering, λ = n ∗ a0 ωn  ,n
/V , ωn  ,n
= |∆E n  ,n
|/~,

Rydberg Collisions: Binary Encounter, Born and Impulse Approximations

E n  = −R∞ /n 2 , and E n
= −R∞ /n ∗2 , with ∗ n = n − δ
. Also, vB is the atomic unit of velocity (see Chapt. 1), and  

 2 λ 4 2 −1 . f n  ,n
(λ) = tan − ln 1 + 2 π λ 2 λ (56.79)

Limiting  cases: f n  ,n
(λ) → 1 as λ → 0, and f n  ,n
(λ) ∼ 8/ 3πλ3 for λ
1. Then  2πas2   , λ→0,  (V/vB )2 n 3 σn  ,n
∼ (56.80) 2 3    16as Vn , λ
1. 3vB |δ
+ ∆n|3 Rate Coefficients.

(56.82b)

and erfc (x) is the complementary error function.

56.6.2 Inelastic n → n Transitions A∗ (n) + B → A∗ (n  ) + B + ∆E n  n .  (2
+ 1)

(56.87)

where σe− − B elastic is the elastic cross section for e− + B scattering. (B) Rate Coefficients.

K n 
 ,n
(T ) = Vσn 
 ,n
 ,  (2
+ 1) K n  n (T ) = K n 
 ,n
, n2 

(56.88a) (56.88b)

(56.83)

where

(56.88c)

&

λ  1  λ2 2 T eλT /8 erfc λT − √ T D−3 2 2 2π   5λT λT (56.89a) − √ D−4 √ π 2  1 − 8λ /3√π , λ  1 T T , (56.89b) ∼ 26 / √πλ5  , λ
1 T T λ2T /8

Φn  n (λT ) = e

where D−ν (y) denotes the parabolic cylinder function. Limiting cases:  elastic µR∞ 1/2 vB σe− −B   , λT → 0 ,   πm k T n3 e B %2 $ K n  n (T ) ∼ elastic n 7  26 vB σe−B 2kB T   , λT
1 .  π|∆n|5 µvB2 Born Results:

σn  n

2πas2 Fn  n (λ) , (V/vB )2 n 3

(56.85)

where λ = na0 ωn  n /V = |∆n|vB /(n 2 V ), and &  '  2λ 3λ2 + 20 2 2 −1 . Fn  n (λ) = tan −  2 π λ 3 4 + λ2 (56.86)

8π 1 = 2 2 k n

k+k 

Fn  n (Q) |k−k |

d(Qa0 ) . (Qa0 )3

(56.91)

(A) Electron–Rydberg Atom Collision.

σn  n =

 8πa02 R∞ 1 (εε )3/2 1 − ln (1 + εe /ε) 2 2 4∆n (∆ε)4 Z En    0.6 1 4 ε e (ε )3/2 + 1− + ∆n ε + εe (∆ε)2 3∆n ε (56.92)

Part D 56.6

(56.84)

n2

vB σ elastic − K n  n (T ) = √ 3 e −B Φn  n (λT ) , πn (VT /vB )

(56.90)

σn 
 ,n
,





σn  ,n (V ) =

Limiting cases:  2πas2   , λ1,   V/vB2 n  3 σn  n ∼ 3 7  256σ eelastic  − −B (V/vB ) n   , λ
1, 15π|∆n|5

(56.81a)

2πas2 = 2 ϕn  ,n
(λT ) , (56.81b) V T /vB n 3 √ where VT = 2kB T/µ, λT = n ∗ a0 ωn  ,n
/VT ,  ∆n = n − n, and µ is the reduced mass of A–B. The function ϕn  ,n
(λT ) in (56.81b) is given by  1 λ2T /4  λT erfc (56.82a) ϕn ,n
(λT ) = e 2 $ % ∞ λT 4 du − √ e−u ln 1 + 2 π u λT 0    λT  1 − √ ln 1/λ2T , λT → 0 π =  2/ √πλ3  , λT
1 T

σn  n =

843


,


σn  ,n
(V ) ≡ Vσn  ,n
(V )/V 

(A) Cross Sections.

56.6 Working Formulae for Rydberg Collisions

844

Part D

Scattering Theory

  for n  > n, where εe = E/ Z 2 R∞ , ε = 1/n 2 , ε = 1/n 2 , and ∆ε = ε − ε .

(B) Rate Coefficients [56.22] (Three Cases). With the

(B) Heavy Particle–Rydberg Atom Collision.

σn  n

 1 (εε )3/2 1− = ln(1 + εe /ε) 4∆n (∆ε)4    (ε )3/2 4 1 0.6 ε ε+εe + , + 1− e (∆ε)2 ∆n 3∆n ε 8πa02 Z2 Z 4 n 2 εe

(56.93)

where εe = mε/MZ 2 R∞ with heavy particle mass and charge denoted above by M and Z, respectively, and all other terms retain their meaning as in (56.92).

56.6.3 Quasi-Elastic
-Mixing Transitions (
−mixing)

σn


≡



σn 
 ,n


where K(k) denotes the complete elliptic integral of the first kind.

(56.94a)


 =


 σ = 4πa2 n 4 , n  n geo max , 0 ∼ 2πa2 v2 /V 2 n 3 , n
n max . s B

definitions

 νB = vB /vrms , vrms = (8kB T )/µπ , (56.99)   f(y) = y−1/2 1 − (1 − y) e−y + y3/2 Ei (y) , (

y = (νB as ) 4πa02 n ∗8 n 1 = (|as |νB /4a0 )1/4 , 2



(56.100)

,

 1/6 1/3 5/6  n 2 = 0.7 |as |νB / αd a03 ,

For Rydberg atom–noble gas atom scattering, n max = 8 to 20, while for Rydberg atom–alkali atom scattering n max = 15 to 30.

A∗ (n
) + B → A∗ (n
 ) + B .

(56.96)

(A) Cross Sections.

Part D 56.6

elastic σns (V ) =

2πCss as2 , (V/vB )2 n ∗4

(56.97)

valid for n ∗
[vB |as |/(4Va0 )]1/4 with Css =

8 π2

√ 1/ 2

[K(k)]2 dk , 0

(56.98)

(56.103)

(56.104)

56.6.5 Fine Structure n
J → n
J  Transitions A∗ (n
J ) + B → A∗ (n
J  ) + B + ∆E J  J . (56.105) (A) Cross Sections (Two Cases). 

n
J σn
J (V ) =

2J  + 1 cnorm 4πa02 n ∗4 , 2(2
+ 1)

(56.106)

valid for n ∗ ≤ n 0 (V ), and n
J  (V ) = σn
J

56.6.4 Elastic n
→ n
 Transitions

(56.102)

where αd is the dipole polarizability of A∗ , then   8πa02 n ∗4 , n∗ ≤ n1 ,     4π 1/2 a |a |ν f(y) , n1 ≤ n∗ ≤ n2 , 0 s B  ) *  2 2 el 4a ν σns ∼ 7(αd νB )2/3 + s∗4 B    n    2.7as2 νB2 (αd νB )1/3  − , n∗ ≥ n2 . a0 n ∗6

(56.94b)

The two limits correspond to strong (close) coupling for n  n max , and weak coupling for n
n max , and expressions (56.94b) are valid when the quantum defect δ
of the initial Rydberg orbital n
is small. n max is the principal quantum number, where the
-mixing cross section reaches a maximum [56.21],  vB |as | 2/7 n max ∼ . (56.95) Va0

(56.101)

% $ 2 2 2πC (
) n 80 (V ) J  J as vB (
) , ϕ J  J (ν J  J ) 1 − V 2 n ∗4 2n ∗8 (56.107)

n∗

valid for ≥ n 0 (V ), where the quantity n 0 (V ) is the effective principal quantum number such that the impact parameter ρ0 of B (moving with relative velocity V ) equals the radius 2n ∗2 a0 of the Rydberg atom A∗ . n 0 (V ) is given by the solution to the following transcendental equation   (2
+1)C (
) vB as 2 (
)  J J n 80 (V ) = ϕ J J ν J J [n 0 (V )] .  2(2J +1)cnorm Va0 (56.108)

Rydberg Collisions: Binary Encounter, Born and Impulse Approximations

The constant cnorm in (56.106) is equal to 5/8 if σgeo =  πr 2 n
, or 1 if σgeo = 4πa02 n ∗4 . The function ϕ(
) J J (ν J J ) in (56.107) is given in general by [56.23, 24] (
) (
)   ϕ(
) (56.109a) J  J (ν J J ) = ξ J  J (ν J J )/ξ J  J (0) , ∞
 ξ J(
) A(2s) js2 (z)Js2 (z)z dz ,  J (ν J  J ) = 
J ,
J s=0

νJ J

(56.109b)

vB ν J  J = |δ
J  − δ
J | ∗ , (56.109c) Vn where js (z) is the spherical Bessel function and (2s) the coefficients C (
) J  J and A
J  ,
J in (56.107) and (56.109b), respectively, are given in table 5.1 of Beigman and Lebedev [56.6]. The quantum defect of Rydberg state n
J is δ
J . For elastic scattering, ν JJ = 0, and ϕ(
) JJ (0) = 1. Symmetry relation: (
) ξ JJ  (ν J  J ) =

2J + 1 (
) ξ  (ν J  J ) . 2J  + 1 J J

(56.110)

56.7 Impulse Approximation

845

(B) Rate Coefficients.

%1/2 cnorm (2J  + 1)C (
) J J 2(2
+ 1)  vB |as | , × πa02 F(ζ) VT a0 $

n
J  σn
J =

where ζ = n 80 (VT )/n ∗8 , and

  1 F(ζ) ≡ ζ E2 (ζ) + (1 − e−ζ ) , ζ

(56.111)

(56.112)

where E2 (x) is an exponential integral. Limiting cases:  2J  + 1  2 ∗4 ∗ ∗   *  2(2
+ 1) cnorm 4πa0 n , n  n max , )  n
J = σn
J 2 2  2πC (
)  J  J as vB  , n ∗
n ∗max ,  VT2 n ∗4 (56.113)

where

n ∗max

= (3/2)1/8 n

0 (V )

if ν J  J  1.

56.7 Impulse Approximation 56.7.1 Quantal Impulse Approximation Basic Formulation [56.25] Consider a Rydberg collision between a projectile (1) of charge Z 1 and a target with a valence electron (3) in orbital ψi bound to a core (2). The full three-body wave function for the system of projectile + target is denoted by Ψi . The relative distance between 1 and the centerof-mass of 2 − 3 is denoted by σ, while the separation of 2 from the center-of-mass of 1 − 3 is ρ.

Projection Operators.

Formal Scattering Theory.

Ψi(+) = Ω (+) ψi ,

(56.114)

where the Möller scattering operator Ω (+) = 1 + G + Vi , and Vi = V12 + V13 . Let χm be a complete set of free-particle wave functions satisfying

(56.118a)

ωij+ = bij+ + 1 .

m +

G Vij |ψi  =



(56.118b) +

G Vij |χm χm |ψi 

(56.119a)

(56.115)

by

1 Vij χm , E m − H0 − Vij + i (56.116)

  = bij+ + G + V23 , bij+ 

 + G + V12 + V13 − Vij bij+ (m) |ψi  . (56.119b)

Part D 56.7

and define operators  ωij+ (m)χm = 1 +

bij+ (m) = ωij+ (m) − 1 ,  bij+ = bij+ (m)|χm χm | ,

m

(H0 − E m )χm = 0 , ωij+ (m)

where Vij denotes the pairwise interaction potential between particles i and j (i, j = 1, 2, 3). Then the action of the full Green’s function G + on the two-body potential Vij is   G + Vij = ωij+ (m) − 1

 + G + (E m − E)+ V12 + V13 + V23 − Vij   × ωij+ (m) − 1 . (56.117)

846

Part D

Scattering Theory

With the definitions

Möller Scattering Operator.

  

+ +  + Ω + = ω+ V23 , b+ 13 + ω12 − 1 + G 13 + b12  + (56.120) + G + V13 b+ 12 + V12 b13 . Exact T -Matrix.

     + Ti f = ψ f V f  ω+ 13 + ω12 − 1 ψi    

 +  + ψ f V f G + V23 , b+ 13 + b12 ψi     +  + ψ f V f G + V13 b+ 12 + V12 b13 ψi . (56.121)

The impulse approximation to the exact T -matrix element (56.121) is obtained by ignoring the second term involving the commutator of V23 .   imp + (56.122) Ψi −→ Ψi = ω+ 13 + ω12 − 1 ψi .

4αδ2 , (T − 2δ)(T − 2αδ) δ = iβK − p · K , z=

t1 = K /a + v , M1 a= , M1 + m e k = ak2 − (1 − a)k f , t = (K − p)/a , β = aZ 1 /n ,

T = β2 + P2 , ν = aZ 1 /K , N(ν) = eπν/2 Γ (1 − iν) , M2 b= , M2 + m e K = ak1 − (1− a)ki , p = ak f − ki ,

and n is the principal quantum number, the impulse approximation to the T -matrix becomes, in this case,     imp Ti f ∼ ψ f V23 ω+ (56.126) 13 ψi dK −1 = N(ν)gi (t1 )F ( f, K , p) , 2π 2 a3 t2 (56.127)

Impulse Approximation: Post Form. imp Ti f

   + = ψ f |V f  ω+ 13 + ω12 − 1 ψi .

(56.123)

The impulse approximation can also be expressed using incoming-wave boundary conditions by making use of the prior operators  1 ωij− (m)χm = 1 + Vij χm , E m − H0 − Vij − i ωij−

=



(56.124a)

ωij− (m)|χm χm | .

(56.124b)

m

The impulse approximation (56.123) is exact if V23 is a constant since the commutator of V23 vanishes in that case. Applications [56.25] (1) Electron Capture. X + + H(i) → X( f ) + H+ . imp

Ti f

     + = ψ f V12 + V23 ω+ 12 + ω13 − 1 ψi .

(56.125)

Part D 56.7

Wave functions: ψi = eiki ·σ ϕi (r), ψ f = eik f ·ρ ϕ f (x), χm = (2π)−3 exp [i(K · x + k · ρ)], where the ϕn are hydrogenic wave functions. If X above is a heavy particle, the V12 term in (56.125) may be omitted due to the difference in mass between the projectile 1 and the bound Rydberg electron 3. See [56.25] and references therein for details.

where, for a general final s-state,

 F ( f, K , p) = ϕ∗f (x) ei p·x 1 F1 iν, 1; i Kx  (56.128) − K · x dx , and gi (t1 ) denotes the Fourier transform of the initial state. The normalization of the Fourier transform is chosen such that momentum and coordinate space hydrogenic + wave functions are related by ϕn (r) = (2π)−3 exp(it1 · r)gn (t1 ) dt1 . For the case f = 1s, β 3/2 ∂ I(ν, 0, β, −K , p) F (1s, K , p) = − √ π ∂β

√ (1 − iν)β = 8 πβ 3/2 T2  iν  iν(β − iK) T + T(T − 2δ) T − 2δ (56.129)

evaluated at β = aZ 1 . For the case f = 2s,

 ∂2 β 3/2 ∂ +β 2 F (2s, K , p) = − √ ∂β ∂β π    × I ν, 0, β, −K , p

(56.130)

evaluated at β = aZ 1 /2. For a general final ns-state f , √  4 π T − 2αδ iν I(ν, α, β, K , p) = (56.131) T T − 2δ × (U cosh πν ± iV sinh πν) ,

Rydberg Collisions: Binary Encounter, Born and Impulse Approximations

where the complex quantity U + iV is 1  iν Γ 2 + iν U + iV = (4z) Γ(1 + iν) × 2 F1 (−iν, −iν; 1 − 2iν; 1/z) . (56.132)

where

(56.133)

Neglecting V12 and exchange yields the approximate T -matrix element     imp Ti f ∼ ψ f V13 ω+ 13 ψi 1 −Z 1 dx dr eiq·σ ϕ∗f (r) = 3 x (2πa) ×

dK N(ν)gi (t1 ) eit1 ·r 1 F1 (iν, 1; iKx − iK · x)

=

−Z 1 dK N(ν)gi (t1 )g∗f (t2 ) (2πa)3   × I ν, 0, 0, −K , −q ,

(56.134)

(56.135)

I(ν, 0, 0, −K , −q) iν  β2 + q2 4π = lim 2 β→0 β + q 2 β 2 + q 2 + 2q · K − 2iβK (56.136)

with A(cos θ) given by (56.138), and  K2 K q2 + ∆E cos θ , α = b2 + v2 + 2 + aq µ a

(56.142a)

&

2 '1/2 K q2 2 2 sin θ 4v q − + ∆E β= , (56.142b) aq µ 

δ = 4β, γ = 4α + D , (56.142c) 4bq (q + 2K cos θ) , D= (56.142d) a while ν and N(ν) retain their meaning from (56.127).

imp

Ti f

∼−

iν  4π q2 N(ν)g (k − bq) , i q2 q 2 − 2q · K (56.143)

where K = a(k − bq − v) and q = ki − k f and exchange and V12 are neglected.

(56.137)

(5) Rydberg Atom Collisions [56.11, 27].

A + B(n) → A + B(n  )

with

 1 , cos θ > −q/2K , A(cos θ) =  e−πν , cos θ < −q/2K ,

$

(4) Ionization. e− + H(i) → e− + H+ + e− .

where

−iν   2K 4π  cos θ  A(cos θ) , = 2 1 + q q

847

    8 3b2 − D αD D − 2b2  3/2 +  1/2 α2 − β 2 α2 − β 2 

48γ 2D2 b2 16D γD−(3γ +4α)b2 − 5/2 +  3/2 γ 2 −δ2 γ 2 − δ2  % 32 D − 3b2 +  (56.141) 1/2 A(cos θ) , γ 2 − δ2

2π , A(cos θ) = 4 D

(2) Electron Impact Excitation.

e− + H(i) → e− + H( f ) .

56.7 Impulse Approximation

+

(56.144) −

→ A+ B +e . (56.138)

and cos θ = Kˆ · q, ˆ t2 = t1 + bq and q = ki − k f . (3) Heavy Particle Excitation [56.26].

(56.140)

Consider a Rydberg collision between a projectile (3) and a target with an electron (1) bound in a Rydberg orbital to a core (2) [56.11, 27]. Full T -matrix element: T fi (k3 , k3 ) (56.146)   (+)   ik3 ·r3   V(r1 , r3 ) Ψi (r1 , r3 ; k3 ) , = φ f (r1 ) e with primes denoting quantities after the collision, and where the potential V is V(r1 , r3 ) = V13 (r) + V3C (r3 + ar1 ) ,

r = r1 − r3 , (56.147)

with a = M1 /(M1 + M2 ), while the subscript C labels the core. The impulse approximation to the full, outgoing

Part D 56.7

H+ + H(1s) → H+ + H(2s) . (56.139) 1 ∞ Z 1 215/2 b5 imp Ti f = − dK N(K)K 2 d(cos θ) πa3 q 2 0 −1 −iν    2K , cos θ  A(cos θ) , × 1 + q

(56.145)

848

Part D

Scattering Theory

wave function Ψi(+) is written imp Ψi = (2π)3/2 gi (k1 )Φ(k1 , k3 ; r1 , r3 ) dk1 , gi (k1 ) =

1 (2π)3/2



φi (r1 ) e−ik1 ·r1 dr1 .

(56.148)

T is the total cross section for 1–3 scatterwhere σ13 ing at relative speed v13 . The resultant cross section (56.156) is an upper limit and contains no interference terms.

(56.149)

(2) Plane-Wave Final State.

Impulse approximation: imp T fi (k3 , k3 ) = dk1 dk1 g∗f (k1 )gi (k1 )T13 (k, k ) 

(56.150) × δ P − (k1 − k1 ) ,

k1 = k1 + (k3 − k3 ) ≡ k1 + P , M3 k = (k1 + k3 ) − k3 = k + P . M1 + M3

(56.157)

g f (k1 ) = δ(k1 − κ1 ) ,

(56.158)

T fi (k3 , k3 ) = gi (k1 )T13 (k, k ) ,

k1 = κ1 − P , (56.159)

where T13 is the exact off-shell T -matrix for 1–3 scattering,   T13 (k, k ) = exp(ik · r)|V13 (r)|ψ(k, r) . (56.151) The delta function in (56.150) ensures linear momentum, K = k1 + k3 = k1 + k3 , is conserved in 1–3 collisions, with

φ f (r1 ) = (2π)−3/2 exp(iκ1 · r1 ) ,

dσi f = dkˆ  dk 3



1

M AB M13

2

2  2 k3  gi (k1 )  f 13 (k, k ) . k3 (56.160)

(3) Closure.



(56.152a)

g f (k1 )g∗f (k1 ) = δ(k1 − k1 ) ,

(56.161)

f

(56.152b)

Elastic scattering: Tii (k3 , k3 ) = g∗f (k1 )gi (k1 )T13 (k, k) dk1 ,

 2  2 k¯  M AB 2  dσiT gi (k1 )  f 13 (k, k ) dk1 , = 3 k3 M13 dkˆ  3

(56.153)

where k = (M3 /M )k1 + (M1 /M )k3 and M = M1 + M3 . Integral cross section: for 3–(1,2) scattering,    M AB 2 k3 g f (k1 + P) σi f (k3 ) = M13 k3 2  × f 13 (k, k )gi (k1 ) dkˆ 3 , (56.154) where M AB is the reduced mass of the 3–(1,2) system, M13 the reduced mass of 1–3. The 1–3 scattering amplitude f 13 is given by  1 2M13 (56.155) f 13 (k, k ) = − T13 (k, k ) . 4π ~2 Six Approximations to (56.150)

Part D 56.7

(1) Optical Theorem.

1 2M AB Tii (k3 , k3 ) k3 ~2  

 1 gi (k1 )2 v13 σ T (v13 ) dk1 , = 13 v3

σtot (k3 ) =

(56.156)

(56.162)

where k1 = (M3 /M )(k1 + k3 ) − k3 . Conditions for validity of (56.162): (a) k3 is high enough to excite all atomic bound and continuum states, and (b) k32 = (k32 − 2ε fi /M AB ) can be approximated by k3 , or by an averaged wavenumber k¯ 3 = (k32 − 2¯ε fi /M AB )1/2 , where the averaged excitation energy is ε¯ fi = lnε fi  =



f ij ln εij



j

−1 f ij

, (56.163)

j

and the f ij are the oscillator strengths. (4) Peaking Approximation.

(k3 , k3 ) = F fi (P)T13 (k, k ) ,

(56.164)

where F fi is the inelastic form factor F fi (P) = g∗f (k1 + P)gi (k1 ) dk1

(56.165)

peak

T fi

= ψ f (r) exp(iP · r)|ψi (r) .

(56.166)

(5) T13 = T13 (P).

T fi (k3 , k3 ) = T13 (P)F fi (P) .

(56.167)

Rydberg Collisions: Binary Encounter, Born and Impulse Approximations

(6)

f 13 ≡ as = constant scattering

T13 = constant.

length. 2πa2 σi f (k3 ) = 2 s k3





M AB M13

2 k 3 +k3    F fi (P)2 P dP , k3 −k3

 4πa2 , v3
v1 s σtot (k3 ) = v 4πa2 /v , v  v . 1 3 1 s 3

56.7.2 Classical Impulse Approximation

(56.169)

(A) Ionization. For electron impact on heavy par-

separately from 1 and 2, i. e., r12
A1,2 ; the relative separation of (1,2)
the scattering lengths of 1 and 2. (ii) λ13  r12 , i. e., the reduced wavelength for 1–3 relative motion  r12 . Interference effects of 1 and 2 can be ignored and 1, 2 treated as independent scattering centers. (iii) 2–3 collisions do not contribute to inelastic 1–3 scattering. (iv) Momentum impulsively transferred to 1 during collision (time τcoll ) with 3
momentum transferred to 1 due to V12 , i. e., (56.170)

For a precise formulation of validity criteria based upon the two-potential formula see the Appendix of [56.27]. Two classes of interaction in A–B(n) Rydberg collisions justify use of the impulse approximation for the T -matrix for 1–3 collisions: (i) quasiclassical binding with Vcore = const., and (ii) weak binding with E 3
∆E c ∼ ψn (r)|V1C (r)|ψn (r) , ψn (r1 )| −

~2

∇12 |ψn (r1 ) ∼ |εn | ,

(56.171a) (56.171b)

ticles [56.29], the cross section for ionization of a particle moving with velocity t by a projectile with velocity s is 1 2s Q(s, t) = 2 u m2 where A(z) =

s2

dz z2

 A(z) + B(z) , z

(56.174)

1

 1 1  3/2 3/2 x02 − x01 3 2st 3    1/2 1/2 − 2 s2 + t 2 x02 − x01

 2 2 2   s −t −1/2 −1/2 x , − x (56.175a) − 02 01 2ts3     −1/2  1 −1/2 (m 1 + m 2 ) s2 − t 2 x02 − x01 B(z) = 3 2m 2 st   1/2 1/2 . (56.175b) − (m 2 − m 1 ) x02 − x01 For electron impact, (56.175b) is evaluated at m 1 = 1. The remaining terms above are given by s2 = v22 /v02 ,

t 2 = v12 /v02 = E 1 /u , E 2 = m 2 v22 ,

u = v02 = Ionization potential of target , x0i = (s2 + t 2 − 2st cos θi ) , i = 1, 2 ,    κ0 ± κ1 , |κ0 ± κ1 | ≤ 1 cos θi = 1 , κ0 ± κ1 > 1 ,    −1 , κ0 ± κ1 < −1 -  m1 z z  z 1 ± 1+ 2 1− 2 . κ0 ± κ1 = − 1 − 2 m 2 st t s Equal Mass Case: (m 1 = m 2 )  4 1 2(s2 −1)3/2   , 1 ≤ s2 ≤ t 2 +1 ,  2 2 t 3s u  Q(s, t) = 3 4 1    2 2 2t 2 + 3 − 2 2 , s2 ≥ t 2 +1 . 3s u s −t (56.176)

Part D 56.7

2M12 where E 3 is the kinetic energy of relative motion of 3, and ∆E c is the energy shift in the core. The fractional error is [56.28]  f 13 ∆E c ~ + τdelay  1 , (56.172) λ ~ E3 whereλ ∼ k3−1 is the reduced wavelength of 3, f 13 is the scattering amplitude for 1–3 collisions and τdelay is the time delay associated with 1–3 collisions. Special Case: for nonresonant scattering with τdelay = 0 f 13 |εn | 1, (56.173) λ E3

849

which follows from (56.172) upon identifying the shift in the core energy ∆E c with the binding energy |ε|. Condition (56.173) is less restrictive than (56.171a) or (56.171b) since f 13 can be either less than or greater thanλ.

(56.168)

Validity Criteria (A) Intuitive Formulation [56.27]. (i) Particle 3 scatters

P
ψn
| − ∇V12 |ψn
τcoll .

56.7 Impulse Approximation

850

Part D

Scattering Theory

Integrating over the speed distribution (Sect. 56.4), 32 1 Q(s) = π u2

∞ 0

Q(s, t)t 2 dt , (t 2 + 1)4

i σcapture (V )

(56.177)

which is then numerically evaluated. For electrons, the integral can be done analytically with the result Q(y) =

8 3πy2 (y + 1)4

  × 5y4 + 15y3 − 3y2 − 7y + 6 (y − 1)1/2   + 5y5 + 17y4 + 15y3 − 25y2 + 20y × tan−1 (y − 1)1/2 √  √  y + y −1 3/2  ,  − 24y ln  √ √ y − y −1

with y = s2 . Thomson’s Result:  1 4 1 1 − . Q T (y) = y u2 y

(C) Capture Cross Section from Shell i [56.18].

(56.178)

C(+1) ri 25/2 Ni π = dr d(cos η ) [Pi (r)]2 3V 7 0 C(−1) 

2  2   2 1 + y 4ε − ε − y2 1 + ε2 + a2 − y2 × ,  3  r 3/2 ε9/2 1 + a2 y 2 − a2

(56.182)

where C denotes that the integration range [−1, +1] is restricted suchthat the integrand is real and positive and that |1 − ε| < y2 − a2 . The dimensionless variables a and y above are defined as y2 =

2 |V(R)|V 2 , me

a2 =

2 Ii V 2 , me

(56.183)

and with Pi (r)/r representing the Hartree–Fock–Slater radial wave function for shell i, with normalization ri [Pi (r)]2 dr = 1 . (56.179)

(56.184)

0

The ionization potential and number of electrons in shell i are denoted above, respectively, by Ii and Ni . (B) Electron Loss Cross Section [56.18].

A(V ) + B(u) → A+ + e− + B( f ) ,

(D) Total Capture Cross Section [56.18]. (56.180)

where B( f ) denotes that the target B is left in any state (either bound or free) after the collision with the projectile A. The initial velocity of the projectile is V while the velocity of the Rydberg electron relative to the core is u, and the ionization potential of the target B is I. 1 σloss = 3πν2

∞ τ/4ν

$ 8νx − 1 − (ν − x)2 − 2τ dx σT (x u) ¯

3 1 + (ν − x)2

1

%

+

2 , 1 + (ν + x)2 − τ

total σcapture (V ) =



i σcapture .

(56.185)

i

(E) Universal Capture Cross Section [56.18]. A universal curve independent of projectile mass M and charge Z is obtained from the above expressions for the capture cross section by plotting the scaled cross section total , σcapture =

E 11/4 M 11/4 Z 7/2 λ3/4

total σcapture

(56.186)

versus the scaled energy (56.181)

Part D 56.7

√ where ν = V/u, ¯ τ = I/ 12 m e u¯ 2 , u¯ = 2I/m e , and σT is the total electron scattering cross section at speed x u. ¯ The cross section (56.181) is valid only for particles being stripped (or lost from the projectile) which are not strongly bound. See [56.18, 30, 31] for details and numerous results.

, = me E , E M I

(56.187)

where m e is the mass of the particle captured, which is usually taken to be a single electron, and I is the ionization potential of the target. The term λ in (56.186) is the coupling constant in the target potential, V(R) = m e λ/R2 , which the electron being captured experiences during the collision. See Fig. 11 of [56.18] for details.

Rydberg Collisions: Binary Encounter, Born and Impulse Approximations

56.7.3 Semiquantal Impulse Approximation Basic Expression [56.27, 32, 33].

k12

k3



2

  dσ M3 gi (k1 )2 = dε dP dk1 dk dφ1 J55 k3 M13  2 ×  f 13 (k, k ) . (56.188) J55 is the 5-dimensional Jacobian of the transformation   (56.189a) (P, ε, k1 , k, φ1 ) → kˆ 3 , k1 , ∂ (P, ε, k1 , k, φ1 ) . J55 =  ∂ cos θ3 φ3 , k1 , cos θ1 , φ1 (56.189b)

| f 13 (k, k )|2 dg2  , 2 − g2 g2 − g2 g+ −  2 = 1 B ± 1 B 2 − C, and where g± 2 4 × 

851

Elemental Cross Sections (m-Averaged and φ 1 Integrated).

dσ =

dε dP Wn
(v1 ) dv1 | f 13 (P, g)|2 dg2   , 2 2 2v1 M13 v3 2 − g2 g2 − g2 g+ − (56.191)

where the speed distribution Wn
is given by (56.57). Two Representations for 1–3 Scattering Amplitude [56.27]. (i) f 13 = f 13 (P, g) is a function of

momentum transferred and relative speed. Then σ(v3 ) =

ε2

1 2 v2 M13 3

ε1



× g−

∞ dε

g+

Expression for Elemental Cross Section [56.27]. In the

(P, ε, k1 , k, φ1 ) representation, & ' dε dP |gi (k1 )|2 k12 dk1 dφ1 dσ = 2 2 v1 M13 v3

56.7 Impulse Approximation

v10

Wn
(v1 ) dv1 v1

P + dP P−

| f 13 (P, g)|2 dg2  , 2 − g2 g2 − g2 g+ −

(56.192)

2 (ε) = max [0, (2ε/M )], and the limits to the where v10 P integral are

(56.190)

B = B(ε, P, v1 ; v3 )

 a 2∆3 P2 2 2 2 2 + v + v + v + v + 1 1 3 3 2 M13 (1 + a)2 M13 4ε(ε + ∆3 ) , − P2 C = C(ε, P, v1 ; v3 )   v2 + av32 P 2  2 2 2 2 v = 1 + v − v − v 1 3 1 3 2 1 + a M13     2∆3 2 4∆3 + v1 + v32 + 2 v12 (ε + ∆3 ) − εv32 , M13 P M2 M3 , a= M1 (M1 + M2 + M3 ) ,1 = M1 (1 + M1 /M2 ) , M 2ε 2(ε + ∆3 ) v12 = v12 + , v32 = v32 − , ,1 M AB M =

Hydrogenic Systems gi (k1 ) = gn‘ (k1 )Y‘m (θ 1 ‚φ 1 ). The

gn
are the hydrogenic wave functions in momentum space. See Chapt. 9 for details on hydrogenic wave functions.

(56.193a) −

−

P = P (ε, v1 ; v3 )   

 = max M v1 − v1 , M AB v3 − v3  , (56.193b)

P+

P−,

and unless > the P integral is zero. (ii) f 13 = f 13 (g, ψ) is a function of relative speed and scattering angle. Then ε2 ∞ g+ 1 Wn
(v1 ) dv1 g dg dε σ(v3 ) = 2 v1 S(v1 , g; v3 ) v3 ε1

v10

g−

ψ+

| f 13 (g, ψ)|2 d(cos ψ) ×  ,  + − cos ψ cos ψ − cos ψ − cos ψ ψ− (56.194)

where

1/2   M13  (1 + a) v12 + av32 − ag2 . 1+a Scattering angle ψ-limits, S(v1 , g; v3 ) =

(56.195) cos ψ ± = cos ψ ± (ε, v1 , g; v3 ) !    g 1 2 2 2 α(α + ε ) ± β ω α =  2 + β ˜ g α + β2 1/2  , (56.196) − (α + ε˜ )2

Part D 56.7

and ∆3 is the change in internal energy of particle 3, while ε denotes the energy change in the target 1 − 2.

P + = P + (ε, v1 ; v3 ) 

= min M(v1 + v1 ), M AB (v3 + v3 ) ,

852

Part D

Scattering Theory

where

ω = g /g ,

ε˜ = ε +

α = α(v1 , g; v3 ) =

  1 1−a M13 v12 − v32 + g2 , 2 1+a

a ∆3 . 1+a

Special Case: f13 = f13 (P ).

σ(v3 ) =

β = β(v1 , g; v3 )

π 2 v2 M13 3

ε2

∞ dε

ε1

v10

Wn
(v1 ) dv1 v1

P + × | f 13 (P)|2 dP .

  2 1/2  1 2 2 2 2 2 2 , = M13 2v1 + 2v3 − g g − v1 − v3 2

(56.197)

P−

56.8 Binary Encounter Approximation The basic assumption of the binary encounter approximation is that an excitation or ionization process is caused solely by the interaction of the incoming charged or neutral projectile with the Rydberg electron bound to its parent ion. If, for example, the cross section depends only on the momentum transfer P to the Rydberg electron (as in the Born approximation), then the total cross section is obtained by integrating σ P over the momentum distribution of the Rydberg electron. The basic cross sections required are given in the following section. For further details see [56.34] and references therein.

Cross Section per Unit Momentum Transfer Let the masses, velocities and charges of the particles be (m 1 , v1 , Z 1 , e) and (m 2 , v2 , Z 2 , e), with v = |v1 − v2 | denoting the relative velocity and quantities after the collision are denoted by primes. Then for distinguishable particles,   8πZ 12 Z 22 e4 P  exp(iη P ) 2 (56.198) σP =  P2  , v2

where the phase shift η P is (56.199)

and with

Part D 56.8

µ=

Z 1 Z 2 e2 Γ(1 + iγ) m1m2 , e2iη0 = . ,γ= m1 + m2 ~v Γ(1 − iγ) (56.200)

For identical particles, σ P± =

  8πZ 12 Z 22 e4 P  eiη P eiη S 2 ± ,  P2 v2 S2 

η S = −2γ ln(S/2µv) + 2η0 + π ,

(56.201)

(56.202)

while η0 is given by (56.200). The momenta P and S transferred by direct and exchange collisions, respectively, are given by P = m 1 (v1 − v1 ) = m 2 (v1 − v2 ) , S = m 1 (v1 − v2 ) = m 2 (v1 − v2 ) .

(56.203a) (56.203b)

The collision rates (in cm3 /s) are αˆ P = v1 σ P ,

56.8.1 Differential Cross Sections

η P = −2γ ln(P/2µv) + 2η0 + π ,

where η P is given by (56.199) and η S is

± αˆ ± P = v1 σ P .

(56.204)

Cross Section per Unit Momentum Transferred per Unit Steradian Differential relationships:

α E,P =

d2 α dϕ d2 α dϕ = = α P,ϕ . (56.205) dP dE dP dϕ dE dE

For distinguishable particles, α P = 2πv1 σ P,ϕ = 2πα P,ϕ ,   4Z 12 Z 22 e4 P  eiη P 2 α P,ϕ =  P2  , v   8Z 12 Z 22 e4  eiη P 2 α E,P = v1 σ E,P = √  2  . v1 v2 X P

(56.206a) (56.206b)

(56.206c)

For identical particles,   4Z 12 Z 22 e4 P  eiη P eiη S 2 = ± , (56.207a) α± P,ϕ  P2 v S2    8Z 12 Z 22 e4  eiη P eiη S 2 ± α± = v σ = ± , √ 1 E,P E,P S2  v1 v2 X  P 2 (56.207b)

Rydberg Collisions: Binary Encounter, Born and Impulse Approximations

where

   X = − cos2 φ + 2 vˆ 1 · Pˆ vˆ 2 · Pˆ cos φ + 1  2  2 − vˆ 1 · Pˆ − vˆ 2 · Pˆ (56.208a)

= (cos φmin − cos φ)(cos φ − cos φmax ) (56.208b) 2  v = (E max − E)(E − E min ) , (56.208c) v1 v2 P with φ being the angle between the velocity vectors v1 and v2 . For the special case of electron impact, M2 = m e , Z 2 = −1, and

56.8 Binary Encounter Approximation

which is valid for 2v1 ≥ v2 + v2 , E ≤ 2m e v1 (v1 − v2 ), or  πZ 12 e4 1  3 2 4v1 − (v2 − v2 ) , (56.215) σE = 2 2 3v1 v2 E 3 which is valid for v2 − v2 ≤ 2v1 ≤ v2 + v2 , 2m e v1 (v1 − v2 ) ≤ E ≤ 2m e v1 (v1 + v2 ), or otherwise, σ E = 0 for E ≥ m e v1 (v2 + v2 ). For incident electrons (two cases): % $ 2m e v22 2m e v22 2Φ 2πe4 1 1 ± , + + 2+ ± σE = ED 3E 3 D 3D3 m e v12 E 2

8Z 12 e4 (56.209) √ , v12 v2 P 4 X √ ± σ E,P (φ) = 8e4 v12 v2 X $ %  1 2 cos η P − η S 1 , (56.210) × 4+ 4 P S P 2 S2   where η P − η S = −2γ ln(P/S) = 2e2 /~v ln(S/P ), and X is given by (56.208b).

which is valid for m e (v2 + v2 ) ≤ m e (v1 + v1 ), D ≥ 0, or  2m e v12 2πe4 1 1 + + 2 σE = 2 2 3 3E D m e v1 E 2 2m e v1 2Φ v1 + ± , E|D| v1 3|D|3

Integrated Cross Sections For incident heavy particles: π 4πZ 2 e4 1 σ E,P = σ E,P (φ) sin φ dφ = 2 1 . (56.211) 2 v1 v2 P 4

which is valid for m e (v2 − v2 ) ≤ m(v1 − v1 ), m e (v1 + v1 ) ≤ m e (v2 + v2 ), D ≤ 0. In the expressions above, the exchange energy D transferred during the collision is 1 1 1 1 D = m e v12 − m e v22 = m e v12 − m e v22 − E . 2 2 2 2

σ E,P (φ) =

0

For incident electrons: π 1 ± ± (φ) sin φ dφ (56.212a) σ E,P = σ E,P 2 0 $ v12 + v22 − P 2 /2m 2e − 2E 2 /P 2 4πe4 1 = 2 +  3 v1 v2 P 4 m 4e v12 − v22 − 2E/m e  % 2Φ   , (56.212b) ± m 2e P 2 v12 − v22 − 2E/m e  where Φ can be approximated [56.35] by $ %     R∞ 1/2  E  .  ln  Φ ∼ cos    E3 − E2 E3 − E2 − E  (56.213)

Cross Sections per Unit Energy For incident heavy particles (three cases): $ % 2πZ 12 e4 1 2m e v22 σE = + , E2 3E 2 m e v12

(56.214)

(56.216)

m e (v2 − v2 ) ≤ m e (v1 − v1 ),

(56.217)

(56.218)

56.8.2 Integral Cross Sections e− (T ) + A(E 2 ) → e− (E) + A+ + e− ,

(56.219)

where T is the initial kinetic energy of the projectile electron, while the Rydberg electron, initially bound in potential Ui to the core A+ , has kinetic energy E 2 . The cross section per unit energy E is denoted below by σE . See the review by Vriens [56.34] for details. For electron impact, there are two collision models: the unsymmetrical collision model of Thomson and Gryzinski assumes that the incident electron has zero potential energy, and the symmetrical collision model of Thomas and Burgess assumes that the incident electron is accelerated initially by the target (and thereby gains kinetic energy) while losing an equal amount of potential energy. Unsymmetrical model (two cases):  πe4 1 4E 2 1 4E 2 Φ σE = , + + + − T E 2 3E 3 D2 3D3 E D (56.220)

Part D 56.8

and E 3 is defined in [56.35].

853

854

Part D

Scattering Theory

which is valid for D = T − E 2 − E ≥ 0 or,  πe4 1 4T  1 4T  Φ + + + − σE = T E 2 3E 3 D2 3|D|3 E|D|   1/2 T × (56.221) E2 T

which is valid for D ≤ 0 and T ≥ E; and where ≡ T − E. Symmetrical model (two cases): $ % πe4 1 4E 2 1 4E 2 Φ σE = , + + + − Ti E 2 3E 3 X i2 3X i3 E X i (56.222)

which is valid for X i ≡ T + Ui − E ≥ 0, with Ti ≡ T + Ui + E 2 , and % $ 4Ti 4Ti πe4 1 1 Φ + + + − σE = Ti E 2 3E 3 X i2 3|X i |3 E|X i |   1/2 Ti × (56.223) E2 which is valid for 0 ≤ Ti ≤ E 2 , T ≥ 0, with Ti ≡ Ti − E, and where Φ is given by (56.213). For incident heavy particles, the unsymmetrical model (56.220) should be used. Single Particle Ionization The total ionization cross section per atomic electron for incident heavy particles is $ % 2πZ 12 e4 1 m e v22 1   , Qi = + − Ui m e v12 3Ui2 2m e v12 − v22

the unsymmetrical model for electrons to obtain % $ πe4 1 2E 2 1 dir + − (56.227) Qi = , T Ui 3Ui2 T − E 2 which is valid for T ≥ E 2 + Ui , or Q idir =

(56.225)

Part D 56.8

which is valid for 2m e v1 (v1 − v2 ) ≤ Ui ≤ 2m e v1 (v1 + v2 ), or otherwise Q i = 0 for Ui ≥ 2m e v1 (v1 + v2 ). For electron impact,  1  dir Q i + Q iex + Q iint . Qi = (56.226) 2 In the unsymmetrical model, Q iex diverges, hence the exchange and interference terms above are omitted in

(56.228)

which is valid for Ui ≤ T ≤ E 2 + Ui . In the symmetrical model, Q idir = Q iex

$ & %' πe4 1 1 2 E2 E2 = − + − , (56.229) Ti Ui T 3 Ui2 T 2  2Φ  πe4 T int , Qi = − ln (56.230) Ti T + Ui Ui

where Φ  can be approximated by [56.35] & ' 1/2 R∞ E1  Φ = cos ln . E 1 + Ui Ui

(56.231)

and E 1 is defined in [56.35]. The sum of (56.229) and (56.230) yields

 πe4 1 1 2 E2 E2 Qi = − + − 2 Ti Ui T 3 Ui T  Φ T , − ln (56.232) T + Ui Ui which is also obtained by integrating the expression (56.223) for σE , 1 2 (T +Ui )



Qi =

(56.224)

which is valid for Ui ≤ 2m e v1 (v1 − v2 ), or  πZ 12 e4 1 1 + Qi = m e v12 2m e v2 (v1 + v2 ) Ui     me  3 3 2 3/2 2v , + + v − 2U /m + v i e 2 1 2 3v2 Ui2

2πe4 (T − Ui )3/2 , √ 3T Ui2 E 2

σE dE .

(56.233)

Ui

Ionization Rate Coefficients. For heavy particle im-

pact [56.29], Q =

  a02 128  3 3 κ b − b3/2 2 9 κ  58 8 2 1 + λb 35 − b b 3 3 3

  3 2 + κab 5 − 4κ 2 3a2 + ab + b2 3 2     15 + 9a + 5b − 16κa4 ln 4κ 2 1 − λκ 2 

 35 5 , − κ2a + 3a + 4a2 + 8a3 +θ 6 2 (56.234)

Rydberg Collisions: Binary Encounter, Born and Impulse Approximations

κ = v1 /v0 ,

T

λ = κ − (4κ)−1 ,

Q ex e

θ = π + 2 tan−1 λ ,

64e4 v05

Scaling Laws. Given the binary encounter cross section

for ionization by protons of energy E 1 of an atom with binding energy u a , the cross section for ionization of an atom with different binding energy u b and scaled proton energy E 1 can be determined to be [56.18] $ %    u a2 σion (E 1 , u a ) , σion E 1 , u b = (56.236) u 2b (56.237)

where σion (E, u) is the ionization cross section for removal of a single electron from an atom with binding energy u by impact with a proton with initial energy E. Double Ionization. See [56.36] for binary encounter

cross section formulae for the direct double ionization of two-electron atoms by electron impact. Excitation. Excitation is generally less violent than ionization and hence binary encounter theory is less applicable. Binary encounter theory can be applied to exchange excitation transitions, e.g., e− + He(n 1 L) → e− + He(n 3 L), with the restriction of large incident electron velocities. The cross section is U n+1

&

1 Tn+1

1 2 − + Tn 3

$

E2 E2 − 2 2 Tn+1 Tn

$

E2 E2 − 2 2 Tn Ui

%' ,

%' , (56.238)

Applying the classical energy-change cross section result (56.69) of Gerjuoy [56.16] to the case of electronimpact ionization yields the four cases [56.17] ∆E u

σion (v1 , v2 ) ∼

eff σ∆E (v1 , v2 ; m 1 /m 2 ) d(∆E) ∆E


% 2 $  π Z 1 Z 2 e2 −2v23 6v2 , − = (∆E)2 m 2 ∆E 3v12 v2 (56.240)

which is valid for 0 < ∆E < b, or 2  π Z 1 Z 2 e2 σion (v1 , v2 ) = 3v2 v2 $ 1   % 4 v2 − 2v2 4 v1 − 2v1 ×  2 + 2  2 , m 21 v1 − v1 m 2 v2 − v2 (56.241)

which is valid for b < ∆E < a, or $ % π(Z 1 Z 2 e2 )2 −2v13 , σion (v1 , v2 ) = (∆E)2 3v12 v2

(56.242)

which is valid for ∆E > a, 2m 2 v2 > |m 1 − m 2 |v1 , or otherwise is zero for ∆E > a, 2m 2 v2 < |m 1 − m 2 |v1 . The limits ∆E
,u to the ∆E integration in each of the four cases is indicated in the appropriate validity conditions. The constants a and b above are given by

 1 4m 1 m 2 E v − E + v (m − m ) , a= 1 2 1 2 1 2 2 (m 1 + m 2 )2

 1 4m 1 m 2 E 1 − E 2 − v1 v2 (m 1 − m 2 ) . b= 2 (m 1 + m 2 )2 The expressions above for σion (v1 , v2 ) must be averaged over the speed distribution of v2 before comparison with experiment. See [56.17] for explicit formulae for the case of a delta function speed distribution.

Part D 56.8

σE,ex dE Un

1 2 1 − + Ui Tn 3

56.8.3 Classical Ionization Cross Section

where 12 m e v02 is the ionization energy of H(1s).

E 1 = (u b /u a )E 1 ,

&

(56.239)

valid for Un ≤ T ≤ Un+1 . Un and Un+1 denote the excitation energies for levels n and n + 1, respectively.

3v12 P 4 $ %−3  E P 2 2  ×  − v dP dE , P 2m e  0 (56.235)

πe4 = Ti

σE,ex dE

πe4 = Ti

b = (1 + λ2 )−1 . σE,P dP dE =

= Un

a = (1 + κ 2 )−1 ,

=

855

valid for T ≥ Un+1 , with Tn ≡ T + Ui − Un and Tn+1 ≡ T + Ui + Un+1 , or

where

Q ex e

56.8 Binary Encounter Approximation

856

Part D

Scattering Theory

56.8.4 Classical Charge Transfer Cross Section Applying the classical energy-change cross section result (56.69) of Gerjuoy [56.16] to the case of chargetransfer yields the four cases [56.17] ∆E u (eff) σCX (v1 , v2 ) ∼ σ∆E (v1 , v2 ) d∆E ∆E


πe4 = 2 3v1 v2

$

2v23 6v2 /m 2 − − ∆E (∆E)2

% , (56.243)

which is valid for 0 < ∆E < b, or  πe4  v1 /m 1 − v2 /m 2 σCX (v1 , v2 ) = 2 3 ∆E 3v1 v2   %  3 v2 − v23 − v13 + v13 + , (∆E)2 (56.244)

which is valid for b < ∆E < a, or $ % 2v12 πe4 − , σCX (v1 , v2 ) = 2 (∆E)2 3v1 v2

(56.245)

which is valid for ∆E > a, m e v2 > (m 1 − m e )v1 , or otherwise σCX = 0 when ∆E > a, m e v2 < (m 1 − m e )v1 . The above expressions for σCX (v1 , v2 ) must be averaged over the speed distribution W(v2 ) before comparison with experiment. See [56.17] for details. The constants a and b above are as defined in Sect. 56.8.3, and the limits ∆E
,u are given by 1 ∆E
= m e v12 + U A − U B , 2 1 ∆E u = m e v12 + U A + U B , 2  v2 = 2U A /m e ,

(56.246a) (56.246b) (56.246c)

where U A,B are the binding energies of atoms A and B. The expressions above for σCX diverge for some v1 > 0 if U A < U B . If U A = U B then σCX diverges at v1 = 0. To avoid the divergence, employ Gerjuoy’s modification, ∆E
= 12 m e v12 + U A .

56.9 Born Approximation See reviews [56.37,38], as well as any standard textbook on scattering theory, for background details on the Born approximation, and [56.39–42] for extensive tables of Born cross sections.

56.9.1 Form Factors

Part D 56.9

The basic formulation of the first Born approximation for high energy heavy particle scattering is discussed in Sect. 53.1. For the general atom–atom or ion–atom scattering process A(i) + B(i  ) → A( f) + B( f  ) , (56.247) with nuclear charges Z A and Z B respectively, let ~ K i and ~ K f be the initial and final momenta of the projectile A, and ~q = ~ K f − ~ K i be the momentum transferred to the target. Then (53.6) can be written in the generalized form t+  2 8πa02 dt   i f  A σi f = 2 δ − F (t) Z  A i f i f s t3 t−

 2   × Z B δi  f  − FiB f  (t) , (56.248) where the momentum transfer is t = qa0 , and s = v/vB is the initial relative velocity in units of vB . The form

factors are NA     exp(it · ra /a0 )  ΦiA , FiAf (t) = Φ Af 

(56.249)

k=1

where N A is the number of electrons associated with atom A, and similarly for FiBf (t). The limits of integration are t± = |K f ± K i |a0 . For heavy particle collisions, t+ ∼ ∞ and t− = (K i − K f )a0  m e ∆E i f ∆E i f , 1+  2s 4Ms2 where M = M A M B /(M A + M B ).

(56.250)

Limiting Cases. As discussed in Sect. 53.1, for the case i = f , FiAf (t) → N A as t → 0, so that Z A − FiAf (t) → Z A − N A . For the case i = f , FiAf (t) → 0 as t → 0 and t → ∞.

56.9.2 Hydrogenic Form Factors Bound–Bound Transitions. In terms of τ = t/Z,

16 |F1s,1s | =  2 , 4 + τ2

(56.251a)

Rydberg Collisions: Binary Encounter, Born and Impulse Approximations

τ2 |F1s,2s | = 217/2  3 , 4τ 2 + 9 3τ |F1s,2 p | = 215/2  3 , 2 4τ + 9   27τ 2 + 16 τ 2 |F1s,3s | = 24 37/2  4 , 9τ 2 + 16   2 11/2 3 27τ + 16 τ |F1s,3 p | = 2 3  4 , 9τ 2 + 16

(56.251b)

(56.251c)

(56.251d)

(56.251e)

τ2

|F1s,3d | = 217/2 37/2  4 . 9τ 2 + 16

(56.251f)

Bound–Continuum Transitions. In terms of the scaled

wave vector κ = ka0 /Z for the ejected electron,   28 κτ 2 1+3τ 2 + κ 2 exp (−2θ/κ) 2 |F1s,κ | =

3

3  3 1+(τ − κ)2 1 +(τ + κ)2 1− e−2π/κ

  an
= (n + 1) (n − 1)2 + n 2 τ 2 , 



 bn
= 2n (n − 1)2 + n 2 τ 2 (n + 1)2 + n 2 τ 2 ,   cn
= (n − 1) (n + 1)2 + n 2 τ 2 , and argument n2 − 1 + n2τ 2 x = 



. (n + 1)2 + n 2 τ 2 (n − 1)2 + n 2 τ 2 Summation over final
states: |F1s,n |2 =

× sin2 (n tan−1 x + tan−1 y) ,

2τ n(τ 2 + 1 − n −2 )

,

y=

2τ τ 2 − 1 + n −2




   1 2 n − 1 + n2τ 2 3 n−3

(n − 1)2 + n 2 τ 2 ×

n+3 . (n + 1)2 + n 2 τ 2

=2 n τ

(56.256)

which becomes for large n,    28 τ 2 3τ 2 + 1 −4 . exp  6 (τ 2 + 1) 3n 3 τ 2 + 1

For initial 2s and 2 p states, (56.253)

. (56.254)

For final n
states [56.46] F1s,n
(τ)

|F1s,n
|2

(56.257)

where x=



8 7 2

|F1s,n |2 ∼

General Expressions and Trends For final ns states n−1

24 n (n − 1)2 + n 2 τ 2 |F1s,ns |2 =

n+1 τ 2 (n + 1)2 + n 2 τ 2

857

with coefficients an
, bn
and cn
given by

(56.252)

  where θ = tan−1 2κ/ 1 + τ 2 − κ 2 . Expressions for the bound–continuum Form Factors for the L-shell (2
→ κ) and M-shell (3
→ κ) transitions can be found in [56.43] and [56.44], respectively. See also §4 of [56.45] for further details.

56.9 Born Approximation

Part D 56.9

 √ (n −
− 1)! 1/2 = (iτ)
23(
+1) 2
+ 1(
+ 1)! (n +
)!

(n−
−3)/2  2 2 2 (n−1) +n τ (
+2) × n
+1

(n+
+3)/2 an
Cn−
−1 (x) (n+1)2 +n 2 τ 2  (
+2) (
+2) − bn
Cn−
−2 (x) + cn
Cn−
−3 (x) , (56.255)

1 1 3 1 |F2s,n |2 = 24 n 7 τ 2 − + n 2 − n 4 + n 6 3 2 16 48  1 2 2 19 4 − n + n + n2τ 2 3 3 48   7 2 4 4 5 − n + n6τ 6 +n τ 3 6

 1  2 2 2 n−4 2n −1 +n τ ×  , (56.258)  2 1 2 2 n+4 2n +1 +n τ

7 24 n 9 τ 2 1 11 4 − n2 + n |F2 p,n |2 = 3 4 24 192   1 4 4 23 2 2 2 5 − n + n τ −n τ 6 24 4

 1 n−4 2 2 2 n −1 +n τ ×  2 . (56.259)  2 1 2 2 n+4 2n +1 +n τ

858

Part D

Scattering Theory

Power Series Expansion. τ 2  1 [56.3]

230 34 πa02  2 11 , 11s2 4t− +9   2 +9 229 32 44t− 1s,2 p σ1s,2s =  2 11 , 55s2 4t− +9 1s,2 p

|F1s,ns (τ)|2 = A(n)τ 4 + B(n)τ 6 + C(n)τ 8 + · · · , (56.260)

where 28 n 9 (n − 1)2n−6 , 32 (n + 1)2n+6   29 n 11 n 2 + 11 (n − 1)2n−8 , B(n) = − 32 5(n + 1)2n+8   28 n 13 313n 4 − 1654n 2 − 2067 C(n) = − 32 52 7(n + 1)2n+10 2n−10 × (n − 1) .

σ1s,2 p =

(56.266b)

(56.266c)



 

  2 = 9/ 16s 2 1 + 3m e / 4Ms2 + · · · . with t−

A(n) =

Ion–Atom Collisions. For the proton impact process

H+ + H(1s) → H+ + H(n
) ,

(56.267)

(56.248) reduces to

For analytical expressions for A(n), B(n) and C(n) for final n p and nd states see [56.47, 48]. General Trends in Hydrogenic Form Factors [56.49].

The inelastic form factor |Fn
→n 
 | oscillates with
 on an increasing background until the value % $  2(n + 3) 1/2 1  
max = min (n − 1), n − (n + 1) 2 (56.261)

is reached, after which a rapid decline for
>
max occurs. See [56.49] for illustrative graphs.

σ1s,n
=

8πa02 s2

∞

2 dt  F1s,n
(t) , 3 t

(56.268)

t−

  with t− = 1 − n −2 /(2s). Asymptotic Expansions  

4πa02 n 2 − 1 |X 1s→ns |2 1 Cs (n) − 2 σ1s,ns = 24s2 n 2 s  n 2 + 11 313n 4 − 1654n 2 − 2067 , + + 20n 2 s4 8400n 4 s6 (56.269)

56.9.3 Excitation Cross Sections

σ1s,n p =

Atom-Atom Collisions [56.50, 51] Single Excitation. For the process

A(i) + B → A( f ) + B ,

+

(56.262)

σi f

∞

 n 2 + 11 313n 4 − 1654n 2 − 2067 , + 10n 2 s2 5600n 4 s4 (56.270)

(56.248) reduces to 8πa2 = 20 s

210 πa02 n 7 (n − 1)2n−5 C p (n) + ln s2 3s2 (n + 1)2n+5

2 dt  A 2  Fi f Z B − FiB i   . 3 t

σ1s,nd (56.263)

t−

Double Excitation. For the process

Part D 56.9

H(1s) + H(1s) → H(n
) + H(n 
 ) , (56.264) ∞ 2  2 8πa02 dt  1s,n  l  F1s,n
  F1s,n 
  . (56.265) σ1s,n
= 2 s t3 t−

Special cases are [56.52]   4 + 396t 2 + 81 230 πa02 880t− − 1s,2s σ1s,2s = ,  2 11 495s2 4t− +9

   2 211 πa02 n 2 − 4 n 5 n 2 − 1 (n − 1)2n−7 = 32 5s2 (n + 1)2n+7 

1 11n 2 + 13 , (56.271) × Cd (n) − 2 + s 28n 2 s4

where Cs (2) = 16/5, Cs (3) = 117/32, Cs (4) ≈ 3.386, and 1.3026 1.7433 16.918 + + , (56.272) n7 n3 n5 2.0502 7.6125 γn Cd (n) = + , (56.273) n3 n5

γn C p (n) =

with (56.266a)

γn ≡

28 n 7 (n − 1)(2n−5) . 3(n + 1)(2n+5)

(56.274)

Rydberg Collisions: Binary Encounter, Born and Impulse Approximations

Further asymptotic expansion results can be found in [56.2–5]. A general expression for Born excitation and ionization cross sections for hydrogenic systems in terms of a parabolic coordinate representation (Chapt. 9) is given in [56.53]. Number of Independent Transitions Ni Between Levels n and n . [56.53]

  n   n  + . Ni = n 2 n  − 3 3

for transitions n → n  when n, n 
1 and |n − n  | ∼ 1. The constant Jn is undetermined (see [56.54] for details) but is generally taken to be the ionization potential of level n.

56.9.4 Ionization Cross Sections (56.277)

The general expression for the Born differential ionization cross section can be evaluated in closed form using screened hydrogenic wave functions. The differential cross section per incident electron scattered into solid angle dΩk , integrated over directions κ for the ejected electron (treated as distinguishable) is [56.55–57] I(θ, φ) dΩk dκ  =

4k kq 4 a0 Z˜ B4

Electron Capture.

A+ + B(n
) → A(n 
 ) + B + .

(56.280)

In the Oppenheimer–Brinkman–Kramers (OBK) approximation [56.58], σn
,n 


M2 v f = 2π ~3 vi

1

2  d(cos θ) Fn
→n 
  ,

−1

(56.281)

(56.276)

e− (k) + H → e− (k ) + H+ + e− (κ) .

   Fn
,κ  2 (q) dΩk dκ  , (56.278)

where vi = vi nˆ i , v f = v f nˆ f , θ is the angle between nˆ i and nˆ f , M = M A M B /(M A + M B ), and    Z A e2 ∗  Fn
,n 
  = dr ds ϕi (r)ϕ f (s) r i(α·r+β·s) , (56.282) ×e with MA , MA + m e MB β = −ki nˆ i − k f nˆ f , MB + m e v f M B (M A + m e ) , ki = ~ (M A + M B + m e ) v f (M B + m e )M A kf = . ~ (M A + M B + m e ) The Jackson–Schiff correction factor [56.59] is  56 32 1 γJS = 127 + 2 + 4 192 p p  tan−1 12 p 60 32 − 83 + 2 + 4 48 p p p 2  tan−1 12 p  32 16 31 + , (56.283) + + 24 p2 p2 p4 and the capture cross section is α = k f nˆ f + ki nˆ i

σ(n i
i , n f
f ) =

γJS πa02 C(n i
i , n f
f ) p2 ∞ × F(n i
i , n f
f ; x) dx , (56.284) x

B = σion 0

dκ 

k+k 

I(q, κ  ) dq ,

k−k

which is generally evaluated numerically.

with (56.279)

mvi ZA ZB , a= , b= , ~ ni nf  0  4 p2 . x = p2 + (a + b)2 p2 + (a − b)2 p=

Part D 56.9

where the form factor is given by (56.253)) for the case n
= 1s, with the ejected electron wavenumber κ and momentum transferred q in the collision, κ  = κa0 / Z˜ B , q = (k − k)a0 / Z˜ B , being scaled by the screened nuclear charge Z˜ B appropriate to the n
-shell from which the electron is ejected. The total Born ionization cross section per electron is κ max

859

56.9.5 Capture Cross Sections

(56.275)

Validity Criterion. The Born approximation is valid

provided that [56.54]  4E n E
ln Jn

56.9 Born Approximation

860

Part D

Scattering Theory

Table 56.2 Coefficients C(n i
i → n f
f ) in the Born capture cross section formula (56.284) nf
f

C(1s → n f
f )

1s 28 Z 5A Z 5B 2s 25 Z 5A Z 5B 2p 25 Z 5A Z 7B 3s 28 Z 5A Z 5B /33 3p 213 Z 5A Z 7B /36 3d 215 Z 5A Z 9B /39 4s 22 Z 5A Z 5B 4p 5Z 5A Z 7B 4d Z 5A Z 9B 4f Z 5A Z 11 B /20 C(2s → n f
f ) = C(1s → n f
f )/8 C(2 p → n f
f ) = C(1s → n f
f )/24

The coefficients C in (56.284) are given in Table 56.2, while the functions F are given in Table 56.3 [56.58]. In

Table 56.3 Functions F(n i
i → n f
f ; x) in the Born cap-

ture cross section formula (56.284) nf
f

F(ni
i , n f
f ; x)

1s x −6 2s (x − 2b2 )2 x −8 2p (x − b2 )x −8 16 4 2 −10 2 3s (x 2 − 16 3 b x+ 3 b ) x 2 3p (x − b )(x − 2b2 )2 x −10 3d (x − b2 )2 x −10 4s (x − 2b2 )2 (x 2 − 8b2 x + 8b4 )2 x −12 24 4 2 −12 2 4p (x − b2 )(x 2 − 24 5 b x+ 5 b ) x 2 2 2 2 −12 4d (x − b ) (x − 2b ) x 4f (x − b2 )3 x −12 F(2s, n f
f ; x) = (x − 2a2 )2 x −2 F(1s, n f
f ; x) F(2 p, n f
f ; x) = (x − a2 )x −2 F(1s, n f
f ; x)

Table 56.3, the appropriate value of a and b is indicated by the quantum numbers n i ,
i and n f ,
f .

References 56.1

56.2 56.3 56.4 56.5 56.6 56.7 56.8 56.9 56.10 56.11

56.12

Part D 56

56.13

56.14 56.15 56.16

H. A. Bethe, E. E. Salpeter: Quantum Mechanics of One- and Two-Electron Atoms (Springer, Berlin, Heidelberg 1957) G. S. Khandelwal, B. H. Choi: J. B. Phys. 1, 1220 (1968) G. S. Khandelwal, E. E. Fitchard: J. Phys. B 2, 1118 (1969) G. S. Khandelwal, J. E. Shelton: J. Phys. B 4, 109 (1971) G. S. Khandelwal, B. H. Choi: J. Phys. B 2, 308 (1969) I. L. Beigman, V. S. Lebedev: Phys. Rep. 250, 95 (1995) W. Gordon: Ann. Phys. (Leipzig) 2, 1031 (1929) V. A. Davidkin, B. A. Zon: Opt. Spectrosk. 51, 25 (1981) L. A. Bureeva: Astron. Zh. 45, 1215 (1968) H. Griem: Astrophys. J. 148, 547 (1967) M. R. Flannery: Rydberg States of Atoms and Molecules, ed. by R. F. Stebbings, F. B. Dunning (Cambridge Univ. Press, Cambridge 1983) Chap. 11 A. Burgess, I. C. Percival: Adv. At. Mol. Phys. 4, 109 (1968) I. C. Percival: Atoms and Molecules in Astrophysics, ed. by T. R. Carson, M. J. Roberts (Academic, New York 1972) p. 65 I. C. Percival, D. Richards: Adv. At. Mol. Phys. 11, 1 (1975) M. Matsuzawa: J. Phys. B 8, 2114 (1975) E. Gerjuoy: Phys. Rev. 148, 54 (1966)

56.17 56.18 56.19 56.20 56.21 56.22 56.23 56.24 56.25

56.26 56.27 56.28 56.29 56.30 56.31 56.32 56.33 56.34

56.35

J. D. Garcia, E. Gerjuoy, J. E. Welker: Phys. Rev. 165, 66 (1968) D. R. Bates, A. E. Kingston: Adv. At. Mol. Phys. 6, 269 (1970) J. M. Khan, D. L. Potter: Phys. Rev. 133, A890 (1964) V. S. Lebedev, V. S. Marchenko: Sov. Phys. JETP 61, 443 (1985) A. Omont: J. de Phys. 38, 1343 (1977) B. Kaulakys: J. Phys. B 17, 4485 (1984) V. S. Lebedev: J. Phys. B: At. Mol. Opt. Phys. 25, L131 (1992) V. S. Lebedev: Soviet Phys. (JETP) 76, 27 (1993) J. P. Coleman: Case Studies in Atomic Collision Physics I, ed. by E. W. McDaniel, M. R. C. McDowell (North Holland, Amsterdam 1969) Chap. 3 J. P. Coleman: J. Phys. B 1, 567 (1968) M. R. Flannery: Phys. Rev. A 22, 2408 (1980) M. L. Goldberger, K. M. Watson: Collision Theory (Wiley, New York 1964) Chap. 11 M. R. C. McDowell: Proc. Phys. Soc. 89, 23 (1966) D. R. Bates, J. C. G. Walker: Planetary Spac. Sci. 14, 1367 (1966) D. R. Bates, J. C. G. Walker: Proc. Phys. Soc. 90, 333 (1967) M. R. Flannery: Ann. Phys. 61, 465 (1970) M. R. Flannery: Ann. Phys. 79, 480 (1973) L. Vriens: Case Studies in Atomic Collision Physics I, ed. by E. W. McDaniel, M. R. C. McDowell (North Holland, Amsterdam 1969) Chap.6 L. Vriens: Proc. Phys. Soc. 89, 13 (1966)

Rydberg Collisions: Binary Encounter, Born and Impulse Approximations

56.36 56.37 56.38 56.39 56.40 56.41 56.42 56.43 56.44 56.45

56.46

B. N. Roy, D. K. Rai: J. Phys. B 5, 816 (1973) A. R. Holt, B. Moiseiwitsch: Adv. At. Mol. Phys. 4, 143 (1968) K. L. Bell, A. E. Kingston: Adv. At. Mol. Phys. 10, 53 (1974) L. C. Green, P. P. Rush, C. D. Chandler: Astrophys. J. Suppl. Ser. 3, 37 (1957) W. B. Sommerville: Proc. Phys. Soc. 82, 446 (1963) A. Burgess, D. G. Hummer, J. A. Tully: Phil. Trans. Roy. Soc. A 266, 255 (1970) C. T. Whelan: J. Phys. B 19, 2343, 2355 (1986) M. C. Walske: Phys. Rev. 101, 940 (1956) G. S. Khandelwal, E. Merzbacher: Phys. Rev. 144, 349 (1966) E. Merzbacher, H. W. Lewis: X-ray Production by Heavy Charged Particles. In: Handbuch der Physik, Vol. 34/2, ed. by E. Flügge (Springer, Berlin, Heidelberg 1958) p. 166 H. Bethe: Quantenmechanik der Ein- und ZweiElektronenprobleme. In: Handbuch der Physik, Vol. 24/1, ed. by E. Flügge (Springer, Berlin, Heidelberg 1933) p. 502

56.47 56.48 56.49 56.50 56.51 56.52 56.53 56.54 56.55 56.56 56.57

56.58 56.59

References

861

M. Inokuti: Argonne Nat. Lab. Radio Phys. Div. A Report No. ANL-7220, 1–10 (1965) M. Inokuti: Rev. Mod. Phys. 43, 297 (1971) M. R. Flannery, K. J. McCann: Astrophys. J. 236, 300 (1980) D. R. Bates, G. Griffing: Proc. Phys. Soc. 66A, 961 (1953) D. R. Bates, G. Griffing: Proc. Phys. Soc. 67A, 663 (1954) D. R. Bates, A. Dalgarno: Proc. Phys. Soc. 65A, 919 (1952) K. Omidvar: Phys. Rev. 140, A26 and A38 (1965) A. N. Starostin: Sov. Phys. JETP 25, 80 (1967) E. H. S. Burhop: Proc. Camb. Phil. Soc. 36, 43 (1940) E. H. S. Burhop: J. Phys. B 5, L241 (1972) N. F. Mott, H. S. W. Massey: The Theory of Atomic Collisions (Clarendon Press, Oxford 1965) pp. 489– 490 D. R. Bates, A. Dalgarno: Proc. Phys. Soc. 66A, 972 (1953) J. D. Jackson, H. Schiff: Phys. Rev. 89, 359 (1953)

Part D 56

863

57. Mass Transfer at High Energies: Thomas Peak

Thomas peaks correspond to singular secondorder quantum effects whose location may be determined by classical two step kinematics. The widths of these peaks (or ridges) may be estimated using the uncertainty principle. A second-order quantum calculation is required to obtain the magnitude of these peaks. Thomas peaks and ridges have been observed in various reactions in atomic and molecular collisions involving mass transfer and also ionization.

57.1

The Classical Thomas Process ................ 863

57.2 Quantum Description ........................... 864 57.2.1 Uncertainty Effects .................... 864

Transfer of mass is a quasiforbidden process. Simple transfer of a stationary mass M2 to a moving mass M1 is forbidden by conservation of energy and momentum. If M1 < M2 then M1 rebounds, if M1 = M2 then M1 stops and M2 continues on, and if M1 > M2 then M2 leaves faster than M1 . In none of these cases do M1 and M2 leave together. Thomas [57.1] understood this in 1927 and further realized that transfer of mass occurs only when a third mass is present and all three masses interact. The simplest allowable process is a two-step process now called a Thomas process [57.2, 3].

57.2.2 Conservation of Overall Energy and Momentum 864 57.2.3 Conservation of Intermediate Energy .............. 865 57.2.4 Example: Proton–Helium Scattering .......... 865 57.3 Off-Energy-Shell Effects ....................... 866 57.4 Dispersion Relations ............................ 866 57.5 Destructive Interference of Amplitudes ..................................... 867 57.6 Recent Developments........................... 867 References .................................................. 868

Quantum mechanically [57.4], the second Born term at high energies is the largest Born term and corresponds to the simplest allowed classical process, namely the Thomas process. While the classically forbidden first Born term is not zero (saved by the uncertainty principle), the first Born cross section varies at high v as v−12 , in contrast to the second Born cross section which varies as v−11 , thus dominating. Higher Born terms correspond to multistep processes that are unlikely in fast collisions where there is not enough time for complicated processes. The higher Born terms (n > 2) are also smaller than the second Born term.

57.1 The Classical Thomas Process The basic Thomas process is shown in Fig. 57.1. Here, the entire collision is coplanar since particles 1 and 2 go off together (that is what is meant by mass transfer). We assume that all the masses and the incident velocities v are known. Thus, there are six unknowns, v
, v f and v3 , each with two components. Conservation of momentum gives two equations of constraint for each collision. Conservation of overall energy gives a fifth constraint, and conservation of energy in the intermediate state (which only holds classically) gives a sixth

constraint. With six equations of constraint, all six unknowns may be completely determined. The allowed values of v
, v f and v3 depend on the masses M1 , M2 and M3 . For example, in the case of the transfer of an electron from atomic hydrogen to a proton, i. e., p+ = H → H + p+ , it is easily verified that the angles are α = (M2 /M1 ) sin 60◦ , β = 60◦ and γ = 120◦ , where m
= M2 = m e and M1 = M3 = Mp = 1836 me . For the case e+ + H → Ps + p+ , it may be verified that α = 45◦ , β = 45◦ and γ = 90◦ .

Part D 57

Mass Transfer

864

Part D

Scattering Theory

(1, 2) + 3

∞

Zf

M1 / M2

Mf vf M1 v

A α

Z1

3 Zf

β

mf vf

mf v

α

Zf γ

2

Two-step forbidden

Part D 57.2

1 + (2, 3)

M1

M2

M3

M2

B M1

A and C allowed

M3 C

M3 v3

M1

M3

M2

Z3

Fig. 57.1 Diagram for mass transfer via a Thomas two-step

process

1

C only

A and B allowed B only

m


In general, the intermediate mass may be equal to M1 , M2 or M3 . We shall regard these as different Thomas processes, and label them B, A, and C respectively. The standard Thomas process (the one actually considered by Thomas in 1927) is case A, and corresponds to the first example given above. Lieber diagrams for the Thomas processes A, B, and C [57.5] are illustrated in Fig. 57.2. In the Lieber diagram, mass regions in which solutions exist for processes A, B, and C are shown. (An equivalent diagram was given earlier by

Two-step forbidden 1

2

3

∞ M3 / M2

Fig. 57.2 Lieber diagram for mass transfer

Detmann and Liebfried [57.3].) There are some regions in which two-step processes are forbidden. In these regions the theory of mass transfer is not fully understood at present.

57.2 Quantum Description 57.2.1 Uncertainty Effects In quantum mechanics, energy conservation in the intermediate states may be violated within the limits of the uncertainty principle, ∆E ≥ ~/∆t, where ∆t is the uncertainty in time of mass transfer. It is not possible to determine if mass transfer actually occurs at the beginning, in the middle, or at the end of the collision. Thus, we choose ∆t = r¯ /¯v, where r¯ is the size of the collision region and v¯ is the mean collision velocity. Taking r¯ ≈ a0 /Z target and v¯ ≈ v, the projectile velocity, we have ~ Z target ~ ~v¯ ~v = ∆E ≈ = = . (57.1) r¯ ∆t a0 /Z target a0 Here, a0 is the Bohr radius and Z target is the nuclear charge of the target in units of the electron charge. Within this range of energy ∆E, the constraint of energy conservation in the intermediate state does not apply.

57.2.2 Conservation of Overall Energy and Momentum Conservation of overall energy and momentum then gives three equations of constraint on the four unknowns v f and v3 [57.6], namely, 
M1 v = M f + M˜ f v f + M3 v3 , (57.2)  2
2 2 ˜ M1 v = M f + M f v f + M3 v3 , (57.3)
 where M f M˜ f is the mass of the upper (lower) particle in the final state of the bound system shown in Fig. 57.1, in which m
is the mass of the intermediate particle M1 , M2 or M3 . From (57.2) and (57.3) it may be shown that the velocity of the recoil particle is constrained by the condition M1 + M2 + M3 v3 M2 v − . 2v3 · vˆ = 2 cos γ = M1 v M3 v3 (57.4)

Thus, the magnitude and the direction of v3 are not independent. Specifying either v3 or vˆ 3 is sufficient, to-

Mass Transfer at High Energies: Thomas Peak

57.2.3 Conservation of Intermediate Energy In a classical two-step process, the projectile hits a particle in the target, and the intermediate mass m
then propagates and subsequently undergoes a second collision. Quantum mechanically, this corresponds to a second-Born term represented by V1 G 0 V2 , where V represents an interaction and G 0 is the propagator of the intermediate state, namely, G 0 = (E − H0 + i)−1 1 , (57.5) E − H0 where ℘ is the Cauchy principal value of G 0 which excludes the singularity at E = H0 . This singularity corresponds to conservation of energy in the intermediate state. It is this singularity which gives rise to the weaker secondary ridge in Fig. 57.3 at v3 = v. The width of the secondary ridge is given approximately by ∆E = ~/∆t, as discussed above. The intersection of the ridges is the Thomas peak. At very high collision velocities, the Thomas peak dominates the total cross section for mass transfer. = −iπδ(E − H0 ) + ℘

10–23

p + He

The effect of the constraints of conservation of overall energy and momentum may be seen in Fig. 57.3, where a sharp ridge is evident in the reaction p+ + He → H + He++ + e− ,

(57.8)

and M1 = Mp , M2 = M3 = m e . Here v3 is the speed of the recoiling ionized target electron, and the target nucleus is not directly involved in the reaction. The width d2σ/dEd⍀ (cm2/sr eV)

10–25 10–28

10–33 10–26

, v3

v–10v0

57.2.4 Example: Proton–Helium Scattering

H + He+++ e

2

d σ (arb. unit) dv3

The constraint imposed by conservation of intermediate energy may be expressed by replacing the speed of the recoil particle v3 by the scaled variable K = M3 v3 /m
v. Then it may be shown that the conservation of intermediate energy may be expressed in the form [57.6]:   M3 v3 2 ≡ K2 = 1 . (57.6) m
v The constraints of conservation of overall energy and momentum, i. e., (57.4), may be easily written in terms of K as M2 1 m
K−
(57.7) , 2 cos γ = r M2 m K where r = (M1 + M2 + M3 )M2 /(M1 M3 ).

Rec oil sp

eed

v

10–27

v+ 10v0 π 4

π 2

3π 2 Recoil angle, γ

Fig. 57.3 Counting rate (or cross section) on the vertical

axis versus recoil speed v3 and recoil angle γ for a Thomas process in which a proton (projectile) picks up an electron from helium (target). The captured electron bounces off the second target electron

10–28

60

90

120 Emission angle (deg)

Fig. 57.4 Observation of a slice of the Thomas ridge structure in p+ + He → H + He++ + e− at 1 MeV by Palinkas et al. [57.7]

865

Part D 57.2

gether with the equations of constraint, to determine the energies and directions of all particles in the final state. Similarly, one may express the equations of constraint in terms of v f and vˆ f .

57.2 Quantum Description

866

Part D

Scattering Theory

Part D 57.4

of the sharp ridge is due to the momentum spread of the electrons in helium and may be regarded as being caused by the uncertainty principle since this momentum (or velocity) spread corresponds to ∆ p = ~/∆r where ∆r is taken as the radius of the helium atom. The locus of the sharp ridge in Fig. 57.3, corresponding to conservation of overall energy and momentum, is given by (57.8). The locus of the weaker ridge, corresponding to the conservation of energy in the intermediate state, is given by (57.7). The intersection of these two loci gives the unique classical result suggested

by Thomas. The width of these ridges may be estimated from the uncertainty principle as described above. Experimental evidence for the double ridge structure has been reported by Palinkas et al. [57.7] corresponding to the calculations given in Fig. 57.3, but at a collision energy of 1 MeV, as shown below. The data in Fig. 57.4 corresponds to a slice across the sharp ridge of Fig. 57.3 at v = v3 . The solid line is a second Born calculation [57.8, 9] at 1 MeV. The bump of data above a smooth backgroud is the indication of the ridge structure.

57.3 Off-Energy-Shell Effects In (57.6), the Green’s function G 0 contains an energyconserving term iπδ(E − H0 ) which is imaginary, and a real energy-nonconserving term ℘[1/(E − H0 )]. The latter does not occur classically; it is permitted by the uncertainty principle and represents the contribution of virtual (off-the-energy-shell or energy-nonconserving) states within ±∆E = ~/∆t about the classical value E = H0 . This quantum term also represents the effect of time-ordering in the second Born amplitude [57.10]. In plane wave second-Born calculations, the offenergy-shell term gives the real part of the scattering amplitude f 2 , while the on-shell (energy conserving) term gives the imaginary part of f 2 . These two contributions are shown in Fig. 57.5. Half of the Thomas peak comes from energy-nonconserving contributions which are not included in a classical description. Also, the energy-nonconserving contribution plays a significant role in determining the shape of the standard Thomas peak, which has been observed [57.11].

Scattering amplitude (10–6 a. u.) p+H 2.0

H+p

50 MeV

Thomas peak ImT2 (on-shell)

ReT2 (off-shell)

1.0

0.0

–1.0

1

2

3

4 5 λ = (4M sin θ–2 )2

Fig. 57.5 Energy-conserving (on-shell) and energy-nonconserving contributions to the second Born scattering amplitude

57.4 Dispersion Relations Because of the form of the Green’s function of (57.6), the second Born contribution f 2 to the scattering amplitude has a single pole in the lower half of the complex plane. Consequently it obeys the dispersion relation 1 Re[ f 2 (λ)] = − ℘ π

+∞

−∞

Im[ f 2 (λ)] =

1 ℘ π

+∞

−∞

Im[ f 2 (λ
)]
dλ , λ − λ


Re[ f 2 (λ
)]
dλ . λ − λ


(57.9)

where Re f 2 and Im f 2 denote the real and imaginary parts of f 2 . Thus the energy-nonconserving part of f 2 is related to an integral over the energy-conserving part and vice versa. In the case of the dielectric constant it is well known that the real and imaginary parts of  are also related by a dispersion relation, namely the Kramers–Kronig relatio [57.12]. Resonances are usually a function of energy E. The width of a resonance gives the lifetime τ of the resonance. Classically, τ is how long the projectile orbits the target before it leaves, corresponding to a delay or shift in time of the projectile during the interac-

Mass Transfer at High Energies: Thomas Peak

pact parameter of the scattering event [57.13]. However, unlike energy resonances, our Thomas resonance in the scattering angle seems to have no classical analog [57.14].

57.5 Destructive Interference of Amplitudes It has already been noted that the location of the Thomas peaks depends on the mass of the collision partners. For the process +

p + Atom → Atom + H ,

Thomas peak p+

(57.10)

there are two separate Thomas peaks [57.2, 15] corresponding to cases A and B in the Lieber diagram (Fig. 57.2). Experimental evidence exists for both peaks. The standard Thomas peak occurs at small forward angles [57.11], while the second peak [57.16] occurs at about 60◦ . If the mass of the projectile is reduced, the positions of these Thomas peaks move toward one another [57.17] as illustrated in Fig. 57.6. When M1 = M2 , then both Thomas peaks occur at 45◦ . This occurs in positronium formation where M1 = M2 = m e , i. e., e+ + He → Ps + He++ + e− .

dσ d⍀

60° Peak θT

60°

θ

dσ d⍀ e+ Destructive interference

(57.11)

In cases A and B of Fig. 57.2, the two V1 G 0 V2 second Born terms are of opposite sign because V2 is of opposite sign in diagrams A and B. This leads to destructive interference for 1s − 1s electron capture (which is dominant at high velocities) as was first discussed by Shakeshaft and Wadehra [57.17]. Consequently, the observed Thomas peak structure is expected [57.18, 19] to be quite different for e+ impact than for impact of p+ or other projectiles heavier than an electron. The double ridge structure for transfer ionization of helium by e+ is expected to differ significantly from

45°

θ

Fig. 57.6 Change of position and nature of the Thomas

peaks with decreasing projectile mass

the structure shown in Fig. 57.3. Understanding such destructive interference between resonant amplitudes may give deeper insight into the physical nature of the intermediate states in this special few-body collision system.

57.6 Recent Developments In the late 1990’s observations [57.20] of the Thomas peak in the case of transfer ionization (where one electron is ionized and another is transferred) differential in the momentum of the ejected electron provided new specific detail on the kinematics of the two step process [57.21]. In 2001 the Thomas peak was discussed [57.22] in the context of quantum time ordering.

In this case time ordering surprisingly is not significant at the center of the peak, in contradiction to the classical picture that there is a definite order in the two step process for transfer ionization. However, time ordering does contribute to the shape of the Thomas peak. In 2002 the Stockholm group [57.23] reported that at very high velocities, the ratio of trans-

867

Part D 57.6

tion. If the width of the resonance is ∆E, then the lifetime is τ = ~/∆E. E and τ are conjugate variables. The Thomas peak is an overdamped resonance in scattering angle, corresponding to a shift in the im-

57.6 Recent Developments

868

Part D

Scattering Theory

Part D 57

fer ionization to total transfer approaches the same asymptotic limit as in double to single ionization in (non-Compton) photoionization, namely 1.66%. This

was interpreted in terms of a common shake process occurring when the wavefunction collapses after a sudden collision.

References 57.1 57.2 57.3 57.4

57.5 57.6

57.7 57.8 57.9 57.10 57.11 57.12 57.13 57.14 57.15

L. H. Thomas: Proc. Soc. A 114, 561 (1927) R. Shakeshaft, L. Spruch: Rev. Mod. Phys. 51, 369 (1979) K. Detmann, G. Liebfried: Z. Phys. 218, 1 (1968) P. R. Simony: A second order calculation for charge transfer. Ph.D. Thesis (Kansas State University, Manhattan, KS, USA 1981) M. Lieber: private communication (1987) J. H. McGuire, J. C. Straton, T. Ishihara: The Application of Many-Body Theory to Atomic Physics, ed. by M. S. Pindzola, J. J. Boyle (Cambridge Univ. Press, Cambridge 1994) J. Palinkas, R. Schuch, H. Cederquist, O. Gustafsson: Phys. Rev. Lett. 22, 2464 (1989) J. H. McGuire, J. C. Straton, W. C. Axmann, T. Ishihara, E. Horsdal: Phys. Rev. Lett. 62, 2933 (1989) J. S. Briggs, K. Taulbjerg: J. Phys. B 12, 2565 (1979) J. H. McGuire: Adv. At. Mol. Opt. Phys. 29, 217 (1991) E. Horsdal-Pederson, C. L. Cocke, M. Stöckli: Phys. Rev. Lett. 57, 2256 (1986) J. D. Jackson: Classical Electrodynamics (Wiley, New York 1975) p. 286 O. L. Weaver, J. H. McGuire: Phys. Rev. A 32, 1435 (1985) J. H. McGuire, O. L. Weaver: J. Phys. 17, L583 (1984) J. H. McGuire: Indian J. Phys. 62B, 261 (1988)

57.16 57.17 57.18 57.19 57.20

57.21 57.22

57.23

E. Horsdal-Pederson, P. Loftager, J. L. Rasmussen: J. Phys. B 15, 7461 (1982) R. Shakeshaft, J. Wadehra: Phys. Rev. A 22, 968 (1980) A. Igarashi, N. Toshima: Phys. Rev. A 46, R1159 (1992) J. H. McGuire, N. C. Sil, N. C. Deb: Phys. Rev. A 34, 685 (1986) V. Mergel, R. Dörner, M. Achler, Kh. Khayyat, S. Lencinas, J. Euler, O. Jagutski, S. Nüttgens, M. Unverzagt, L. Spielberger, W. Ru, R. Ali, J. Ullrich, H. Cederquist, A. Salin, C. J. Wood, R. E. Olson, Dz. Belkic, C. L. Cocke, H. Schmidt-Böcking: Phys. Rev. Lett. 79, 387 (1977) S. G. Tolmanov, J. H. McGuire: Phys. Rev. A 62, 032771 (2000) A. L. Godunov, A. L. Godunov, J. H. McGuire, P. B. Ivanov, V. A. Shipakov, H. Merabet, R. Bruch, J. Hanni, Kh. Shakov: J. Phys. B 34, 5055 (2001) H. T. Schmidt, H. Cederquist, A. Fardi, R. Schuch, H. Zettergren, L. Bagge, A. Kallberg, J. Jensen, K. G. Resfelt, V. Mergel, L. Schmidt, H. SchmidtBoecking, C. L. Cocke: Photonic, Electronic and Atomic Collisions, ed. by J. Burgdoerfer, J. S. Cohen, S. Datz, C. R. Vane (Rinton, Princeton, NJ 2002) p. 720

869

58. Classical Trajectory and Monte Carlo Techniques

Classical Trajec 58.1 Theoretical Background ....................... 58.1.1 Hydrogenic Targets .................... 58.1.2 Nonhydrogenic One-Electron Models..................................... 58.1.3 Multiply-Charged Projectiles and Many-Electron Targets.........

869 869 870

870

58.2 Region of Validity ................................ 871 58.3 Applications ........................................ 58.3.1 Hydrogenic Atom Targets ............ 58.3.2 Pseudo One-Electron Targets ...... 58.3.3 State-Selective Electron Capture.. 58.3.4 Exotic Projectiles ....................... 58.3.5 Heavy Particle Dynamics ............

871 871 872 872 873 873

58.4 Conclusions ......................................... 874 References .................................................. 874 excitation, electron capture, and ionization channels. Vector- and parallel-processors now allow increasingly detailed study of the dynamics of the heavy projectile and target, along with the active electrons.

58.1 Theoretical Background 58.1.1 Hydrogenic Targets

and their corresponding momenta,

For a simple three-body collision system comprised of a fully-stripped projectile (a), a bare target nucleus (b), and an active electron (c), one begins with the classical Hamiltonian for the system, H = pa2 /2m a + p2b /2m b + p2c /2m c + Z a Z b /rab + Z a Z c /rac + Z b Z c /rbc , (58.1) where pi are the momenta and Z i Z f /ri f are the Coulomb potentials between the individual particles. From (58.1), one obtains a set of 18 coupled, first-order differential equations arising from the necessity to determine the time evolution of the Cartesian coordinates of each particle, dqi / dt = ∂H/∂ pi ,

(58.2)

d pi / dt = −∂H/∂qi .

(58.3)

Five random numbers, constrained by Kepler’s equation, are then used to initialize the plane and eccentricity of the electron’s orbit, and another is used to determine the impact parameter within the range of interaction [58.4, 5]. A fourth-order Runge–Kutta integration method is suitable because of its ease of use and its ability to vary the time step-size. This latter requirement is essential since it is not uncommon for the time step to vary by three orders of magnitude during a single trajectory. In essence, the CTMC method is a computer experiment. Total cross sections for a particular process are determined by σR = (NR /N )πb2max ,

(58.4)

Part D 58

The classical trajectory Monte Carlo (CTMC) method originated with Hirschfelder, who studied the H + D2 exchange reaction using a mechanical calculator [58.1]. With the availability of computers, the CTMC method was actively applied to a large number of chemical systems to determine reaction rates, and final state vibrational and rotational populations (see, e.g., Karplus et al. [58.2]). For atomic physics problems, a major step was introduced by Abrines and Percival [58.3] who employed Kepler’s equations and the Bohr–Sommerfield model for atomic hydrogen to investigate electron capture and ionization for intermediate velocity collisions of H+ + H. An excellent description is given by Percival and Richards [58.4]. The CTMC method has a wide range of applicability to strongly-coupled systems, such as collisions by multiply-charged ions [58.5]. In such systems, perturbation methods fail, and basis set limitations of coupled-channel molecular- and atomic-orbital techniques have difficulty in representing the multitude of active

870

Part D

Scattering Theory

Part D 58.1

where N is the total number of trajectories run within a given maximum impact parameter bmax , and NR is the number of positive tests for a reaction, such as electron capture or ionization. Angle and energy differential cross sections are easily generalized from the above. As in an experiment, the cross section given by (58.4) has a standard deviation of ∆σR = σ[(N − NR )/NNR ]1/2 ,

(58.5) 1/2 1/NR .

which for large N is proportional to Here lies one of the major difficulties associated with the CTMC method: it takes considerable computation time to determine minor or highly differential cross sections. Present day desktop workstations can provide a partial remedy of this statistics problem. To decrease the statistical error of a cross section by a factor of two, four times as many trajectories must be evaluated.

58.1.2 Nonhydrogenic One-Electron Models For many-electron target atoms, it is sometimes adequate to treat the problem within a one-electron model and employ the independent electron approximation to approximate atomic shell structure [58.6]. For an accurate calculation, it is necessary to use an interaction potential that simulates the screening of the target nucleus by the electrons. One can simply apply a Coulomb potential with an effective charge Z eff obtained from, for example, Slater’s rules. Then, the computational procedure is the same as for the hydrogenic case. However, the boundary conditions for the long- and short-range interactions are poorly satisfied. To improve the electronic representation of the target, potentials derived from quantum mechanical calculations are now routinely used. Here, the simple solution of Kepler’s equation cannot be applied. However, Peach et al. [58.7] and Reinhold and Falc´on [58.8] have provided the appropriate methods that yield a target representation that is correct under the microcanonical distribution. The method of Reinhold and Falc´on is popular because of its ease of use and generalizability. For the effective interaction potential, Garvey et al. [58.9] have performed a large set of Hartree–Fock calculations, and have parametrized their results in the form V(R) = −[Z − NS(R)]/R ,

(58.6)

with the screening of the core given by S(R) = 1 − {(η/ξ)[exp(ξR) − 1] + 1}−1/2 ,

(58.7)

where Z and N denote the nuclear charge and number of nonactive electrons in the target core, and η and ξ are

screening parameters. Screening parameters are given in [58.9] for all ions and atoms with Z ≤ 54. This potential can also be used for the representation of partially-stripped projectile ions.

58.1.3 Multiply-Charged Projectiles and Many-Electron Targets Multiple ionization and electron capture mechanisms in energetic collisions between multiply-charged ions and many-electron atoms is poorly understood because major approximations must be made to solve a manyelectron problem associated with transitions between two centers. For a representative collision system such as Aq+ + B → A(q− j)+ + Bi+ + (i − j)e− ,

(58.8)

it is essential that the theoretical method be able to predict simultaneously the various charge states of the projectile and recoil ions, and also the energy and angular spectra of the ejected electrons. Theoretical methods based on the independent electron model fail because the varying ionization energies of the electrons are not well represented by a constant value, especially for outer shells. To present, only the nCTMC method, which is a direct extension of the hydrogenic CTMC method to an n-electron system, has been able to make reasonable predictions of the cross sections and scattering dynamics of such strongly-coupled systems [58.10]. As such, the number of coupled equations rises to 6n + 12, where n is the number of electrons included in the calculation. However, computing time does not increase linearly, since modern vector-processors become very efficient with coupled equations for multiples of 64. In the nCTMC technique, all interactions of the projectile and target nuclei with each other and the electrons are implicitly included in the calculations. The inclusion of all the particles then allows a direct determination of their angular scattering, along with an estimate of the energy deposition to electrons and heavy particles. Post collision interactions are included between projectile and recoil ions with the electrons; however, electron–electron interactions are introduced only in the bound initial state via a screening factor in a centralfield approximation. This theoretical model has been very successful in predicting the single and double differential cross sections for the ionized electron spectra. Moreover, since a fixed target nucleus approximation is not used, this method has been the only one available to help understand and predict the results for the new field of recoil-ion momentum spectroscopy [58.11].

Classical Trajectory and Monte Carlo Techniques

from the quantum mechanical wave function for the ground state of H2 . Total cross sections for a variety of projectiles are in reasonable accord with experiment. The effect of the orientation of the molecular axis on the cross sections was also investigated, along with tests as to the validity of assuming that the cross sections for H2 are simply the product of twice the H values.

58.2 Region of Validity The CTMC method has a demonstrated region of applicability for ion–atom collisions in the intermediate velocity regime, particularly in the elucidation of both heavy-particle and electron collision dynamics. The method can be termed a semiclassical method in that the initial conditions for the electron orbits are determined by quantum mechanically determined interaction potentials with the parent nucleus. Since the method is most applicable to strongly-coupled systems, it has been applied successfully to a variety of intermediate energy multiply-charged ion collisions. Figure 58.1 describes pictorially the regions of validity of theoretical models. Both the atomic orbital (AO) and molecular orbital (MO) basis set expansion methods (Chapt. 50) work well until ionization strongly mediates the collision, since the theoretical description of the ionization continuum is not well-founded and relies on pseudostates to span all ejected electron energies and angles. We have arbitrarily limited these methods to a projectile charge to target charge ratio of, Z P /Z T ,  8, since above this value the number of terms in the basis set becomes prohibitively large. The CTMC method does not include molecular effects, and thus it is restricted from low velocities, except in the case of high-chargestate projectiles that capture electrons into high-lying Rydberg states which are well-described classically. Likewise, at high velocities the method is inapplicable in the perturbation regime where quantum tunneling is important, and thus is restricted to strongly cou-

100

Zp Zt

CTMC CDW 10 MO

BORN

1 AO 0.1 0.1

1

10

100 v/ve

Fig. 58.1 Approximate regions of validity of various theo-

retical methods. Z P /Z T is the ratio of the projectile charge to the target charge, and v/ve is the ratio of the collision velocity to the velocity of the active target electron. Theoretical methods: molecular orbital (MO), atomic orbital (AO), classical trajectory Monte Carlo (CTMC), continuum distorted wave (CDW), and first-order perturbation theory (BORN)

pled systems. The continuum distorted wave (CDW) method (Chapt. 52) greatly extends the region of applicability of first-order perturbation methods and has demonstrated validity in high-charge state ionization collisions.

58.3 Applications 58.3.1 Hydrogenic Atom Targets The original application of the CTMC method to atomic physics collisions were done on the H+ + H system [58.3]. Here, the electron capture and ionization

total cross sections were found to be in very good accord with experiment. The Abrines and Percival procedure casts the coupled equations into the c.m. coordinate system to reduce the three-body problem to 12 coupled equations. However, this reduction complicates exten-

871

Part D 58.3

Only a modest amount of work has been completed on molecular targets. At present, only an H2 target has been formulated for application to the CTMC method [58.12]. For H2 , a fixed internuclear axis is assumed which is then randomly orientated for each trajectory. The electrons are initialized in terms of two one-electron microcanonical distributions constructed

58.3 Applications

872

Part D

Scattering Theory

Part D 58.3

sions of the code to laser processes and collisions in electric fields or with many electrons. An ideal application for the CTMC method is for collisions involving excited targets. Such processes are well-described classically, and basis set expansion methods show limited applicability due to computer memory constraints. Considerable early work has been done on Rydberg atom collisions which includes state-selective electron capture, ionization, and electric fields [58.13–15]. Presently, there is a resurgence of work on Rydberg atom collisions because new crossed-field experimental techniques allow the production of these atoms with specific spatial orientations and eccentricities [58.16]. For hydrogenic ion–ion collision processes, one must be careful to apply the CTMC method only for projectile charges Z P ≥ Z T because after the initialization of the active electron’s orbit and energy, there is no classical constraint on the orbital energy of a captured electron. For a low-charge-state ion colliding with a ground state highly charged ion, one will obtain unphysical results because a captured electron will tend to preserve its original binding energy. Thus, excess probability will be calculated for electron orbits that lead to unrealistic deeply bound states of the projectile.

58.3.2 Pseudo One-Electron Targets Collisions involving alkali atoms are of interest because of their relevance to applied programs, such as plasma diagnostics in tokamak nuclear fusion reactors. They are also a testing ground for theoretical methods since experimental benchmarks are difficult to realize with hydrogenic targets, but are amenable for such cases as alkali atoms. In such collisions, it is essential that a theoretical formalism be used that correctly simulates the screening of the nucleus by the core electrons (Sect. 58.1.2), since a simple −Z eff /R Coulomb potential is inadequate for both large and small R. One can also apply the methods of Sect. 58.1.2 to partially or completely filled atomic shells. This works reasonably well for collisions with a low charge state ion such as a proton, but fails for strong collisions involving multiply-charged ions. The reason is that the independent electron approximation [58.6] must be applied to the calculated transition probabilities in order to simulate the shell structure. This latter method can only maintain its validity if the transition probability is low. Otherwise, it will greatly overestimate the multiple elec-

tron removal processes since the first ionization potential is inherently assumed for each subsequent electron that is removed from the shell, leading to an underestimate of the energy deposition.

58.3.3 State-Selective Electron Capture One of the powers of the CTMC method is that it can be applied to electron capture and excitation of high-lying states that are not accessible with basis set expansion techniques [58.17]. For the C4+ + Li system, where AO calculations and experimental data exist, the CTMC method agrees quite favorably with both [58.18]. The procedure is first to define a classical number n c related to the calculated binding energy E of the active electron to either the projectile (electron capture) or target nucleus (excitation) as
 E = −Z 2 / 2n 2c . (58.9) Then, n c is related to the principal quantum number n of the final state by the condition [58.19]   1/3  1/3  1 1 n n − (n − 1) < n c ≤ n n + (n + 1) . 2 2 (58.10)

From the electron’s normalized classical angular momentum lc = (n/n c )(r × k), lc is related to the orbital quantum number l of the final state by l < lc ≤ l + 1 .

(58.11)

The magnetic quantum number m l is then obtained from 2m l + 1 2m l − 1 l z ≤ < , 2l + 1 lc 2l + 1

(58.12)

where l z is the z-projection of the angular momentum obtained from calculations [58.20]. In principle, it is also possible to analyze the final-state distributions from the effective quantum number n ∗ = n − δl ,

(58.13)

where δl is the quantum defect. In this latter case, it is necessary to sort the angular momentum quantum numbers first, and then sort the principal quantum numbers. The CTMC method has been widely applied to collisions of multiply-charged ions and hydrogen targets in the nuclear fusion program. Here, the calculated nl

Classical Trajectory and Monte Carlo Techniques

58.3.4 Exotic Projectiles The study of collisions involving antimatter projectiles, such as positrons and antiprotons, is a rapidly growing field which is being spurred on by recent experimental advances. Such scattering processes are of basic interest, and they also contribute to a better understanding of normal matter-atom collisions. Antimatter-atom studies highlight the underlying differences in the dynamics of the collision, as well as on the partitioning of the overall scattering. In the Born approximation, ionization cross sections depend on the square of the projectile’s charge and are independent of its mass. Thus, the comparison of the cross sections for electron, positron, proton and antiproton scattering from a specific target gives a direct indication of higher-order corrections to scattering theories. Early work using the CTMC method concentrated on the spectra of ionized electrons for antimatter projectiles [58.24]. Later work focused on the angular scattering of the projectiles during electron removal collisions, such as positronium formation, on ratios of the electron removal cross sections, and on ejected electron ‘cusp’ and ‘anticusp’ formation. A recent review that compares various theoretical results and available experiments is given in [58.25].

58.3.5 Heavy Particle Dynamics A major attribute of the CTMC method is that it inherently includes the motion of the heavy particles after the collision. A straight-line trajectory for the projectile is not assumed, nor is the target nucleus constrained to be fixed. This allows one to compute easily the differential cross sections for projectile scattering or the recoil momenta of the target nucleus. As a computational note, the angular scattering of the projectile should be computed from the momentum components, not the position coordinates after the collision, since faster convergence of the cross sections using the projectile momenta is obtained. For recoil ion momentum transfer studies, one must ini-

tialize the target atom such that the c.m. of the nucleus plus its electrons has zero momentum so that there is no initial momentum associated with the target. A common error is to initialize only the target nucleus momenta to zero. Then the target atom after a collision has an artificial residual momentum that is associated with the Compton profile of the electrons because target–electron interactions are included in the calculations. Examples of recoil and projectile scattering cross sections are given in [58.10, 11]. The field of recoil ion momentum spectroscopy is rapidly expanding and the CTMC theoretical method has impacted the interpretation and understanding of experimental results because the method inherently provides a kinematically complete description of the collision products. For the studied systems, primarily He targets because of experimental constraints, it is necessary that a theoretical method be able to follow all ejected electrons and the heavy particles after a collision. As an example, it has been possible to observe the backward recoil of the target nucleus in electron capture reactions, which is due to conservation of momentum when the active electron is transferred from the target’s to the projectile’s frame of [58.26]. Theoretical methods are being tested further with the recent development of magneto-optical-taps (MOT) that provide frozen alkali metal atomic targets (T  1 mK) from which to perform recoil ion studies [58.27]. For three- and four-body systems, it is now possible to measure the momenta of all collision products. These observations provide a severe test of theory, since all projectile and target interactions must be included in calculations. The CTMC method includes all projectile interactions with the target nucleus and electrons. Thus, it is possible to calculate fully differential cross sections. It is of interest that recent triply differential cross sections calculated using the CTMC method compare very favorably with sophisticated continuum distorted wave methods [58.28]. The CTMC technique allows one to incorporate electrons on both the projectile and target nuclear centers. All interactions between centers are included. The only interaction that needs to be approximated is the electron–electron interactions on a given center. Here, simple screening parameters derived from Hartree–Fock calculations are employed to eliminate nonphysical autoionization. Within this many-electron model, the signatures of the electron–electron and electron–nuclear interactions on the dynamics of the collisions have been observed [58.29, 30]. Further, projectile ionization studies can be undertaken [58.31].

873

Part D 58.3

charge exchange cross sections are used to predict the resulting visible and UV line emissions arising after electron capture to high principle quantum numbers. These line emission cross sections are routinely used as a diagnostic for tokamak fusion plasmas [58.21]. Likewise, for low energy collisions, CTMC results have been used to provide an explanation for the X-ray emission discovered from comets as they orbit through our solar system [58.22, 23].

58.3 Applications

874

Part D

Scattering Theory

58.4 Conclusions

Part D 58

In many ways it is surprising that a classical model can be successful in a quantum mechanical world, especially since the classical radial distribution for the hydrogen atom is described so poorly. However, hydrogen’s classical momentum distribution is exactly equivalent to the quantum one, and since collision processes are primarily determined by velocity matching between projectile and electron, reasonable results can be expected. Moreover, the CTMC method pre-

serves conservation of flux, energy, and momentum; and Coulomb scattering is the same in both quantal and classical frameworks. Of significant importance is that the CTMC method is not restricted to one-electron systems and can easily be extended to more complicated systems involving electrons on both projectile and target. For these latter cases, multiple electron capture and ionization reactions can be investigated.

References 58.1 58.2 58.3 58.4 58.5 58.6 58.7 58.8 58.9 58.10 58.11 58.12 58.13 58.14 58.15 58.16 58.17 58.18

58.19

J. Hirschfelder, H. Eyring, B. Topley: J. Chem. Phys. 4, 170 (1936) M. Karplus, R. N. Porter, R. D. Sharma: J. Chem. Phys. 43, 3259 (1965) R. Abrines, I. C. Percival: Proc. Phys. Soc. 88, 861 (1966) I. C. Percival, D. Richards: Adv. At. Mol. Phys. 11, 1 (1975) R. E. Olson, A. Salop: Phys. Rev. A 16, 531 (1977) J. H. McGuire, L. Weaver: Phys. Rev. A 16, 41 (1977) G. Peach, S. L. Willis, M. R. C. McDowell: J. Phys. B 18, 3921 (1985) C. O. Reinhold: Phys. Rev. A 33, 3859 (1986) R. H. Garvey, C. H. Jackman, A. E. S. Green: Phys. Rev. A 12, 1144 (1975) R. E. Olson, J. Ullrich, H. Schmidt-Böcking: Phys. Rev. A 39, 5572 (1989) C. L. Cocke, R. E. Olson: Phys. Rep. 205, 153 (1991) L. Meng, C. O. Reinhold, R. E. Olson: Phys. Rev. A 40, 3637 (1989) R. E. Olson: J. Phys. B 13, 483 (1980) R. E. Olson: Phys. Rev. Lett. 43, 126 (1979) R. E. Olson, A. D. MacKellar: Phys. Rev. Lett. 46, 1451 (1981) D. Delande, J. C. Gay: Europhys. Lett. 5, 303 (1988) R. E. Olson: Phys. Rev. A 24, 1726 (1981) R. Hoekstra, R. E. Olson, H. O. Folkerts, W. Wolfrum, J. Pascale, F. J. de Heer, R. Morgenstern, H. Winter: J. Phys. B 26, 2029 (1993) R. C. Becker, A. D. MacKellar: J. Phys. B 17, 3923 (1984)

58.20

58.21

58.22

58.23 58.24 58.25 58.26

58.27

58.28 58.29

58.30 58.31

S. Schippers, P. Boduch, J. van Buchem, F. W. Bliek, R. Hoekstra, R. Morgenstern, R. E. Olson: J. Phys. B 28, 3271 (1995) H. Anderson, M. G. von Hellermann, R. Hoekstra, L. D. Horton, A. C. Howman, R. W. T. Konig, R. Martin, R. E. Olson, H. P. Summers: Plasma Phys. Control. Fusion 42, 781 (2000) P. Beiersdorfer, R. E. Olson, G. V. Brown, H. Chen, C. L. Harris, P. A. Neill, L. Schweikhard, S. B. Utter, K. Widmann: Phys. Rev. Lett. 85, 5090 (2000) P. Beiersdorfer, C. M. Lisse, R. E. Olson, G. V. Brown, H. Chen: Astrophys. J. 549, 147 (2001) R. E. Olson, T. J. Gay: Phys. Rev. Lett. 61, 302 (1988) D. R. Schultz, R. E. Olson, C. O. Reinhold: J. Phys. B 24, 521 (1991) V. Frohne, S. Cheng, R. Ali, M. Raphaelian, C. L. Cocke, R. E. Olson: Phys. Rev. Lett. 71, 696 (1993) J. W. Turkstra, R. Hoekstra, S. Knoop, D. Meyer, R. Morgenstern, R. E. Olson: Phys. Rev. Lett. 87, 123202 (2001) J. Fiol, R. E. Olson: J. Phys. B 35, 1759 (2002) H. Kollmus, R. Moshammer, R. E. Olson, S. Hagmann, M. Schulz, J. Ullrich: Phys. Rev. Lett. 88, 103202 (2002) J. Fiol, R. E. Olson, A. C. F. Santos, G. M. Sigaud, E. C. Montenegro: J. Phys. B 34, 503 (2001) R. E. Olson, R. L. Watson, V. Horat, K. E. Zaharakis: J. Phys. B 35, 1893 (2002)

875

59. Collisional Broadening of Spectral Lines

Collisional Bro 59.1 Impact Approximation ......................... 875 59.2 Isolated Lines...................................... 59.2.1 Semiclassical Theory .................. 59.2.2 Simple Formulae ....................... 59.2.3 Perturbation Theory................... 59.2.4 Broadening by Charged Particles .................. 59.2.5 Empirical Formulae ...................

876 876 877 878 879 879

59.3 Overlapping Lines ................................ 880 59.3.1 Transitions in Hydrogen and Hydrogenic Ions ........................ 880 59.3.2 Infrared and Radio Lines ............ 882 59.4 Quantum-Mechanical Theory ................ 59.4.1 Impact Approximation ............... 59.4.2 Broadening by Electrons ............ 59.4.3 Broadening by Atoms ................

882 882 883 884

59.5 One-Perturber Approximation .............. 59.5.1 General Approach and Utility ...... 59.5.2 Broadening by Electrons ............ 59.5.3 Broadening by Atoms ................

885 885 885 886

59.6 Unified Theories and Conclusions .......... 888 References .................................................. 888 General reviews of the theory of pressure broadening have been given [59.8–10], and Chapt. 2, Chapt. 10, Chapt. 14, Chapt. 19, Chapt. 45, Chapt. 47, and Chapt. 86 discuss topics relevant to the theory of collisional broadening of spectral lines. The International Conference on Spectral Line Shapes (ICSLS) is devoted exclusively to this subject.

59.1 Impact Approximation If the perturbers are rapidly moving, the broadening and shift of the line arise from a series of binary collisions between the atom and one of the perturbers. The theory assumes that although weak collisions may occur simultaneously, strong collisions are relatively rare and only occur

one at a time. The impact approximation is valid if wτ
1 ,

V¯ τ/~
1 ,

(59.1)

where w is the half width at half maximum, τ is the average time of collision and V¯ is the average emitter-

Part D 59

One-photon processes only are discussed and aspects of line broadening directly related to collisions between the emitting (or absorbing) atom and one perturber are considered. Molecular lines and bands are not considered here. Pointers to other aspects are included and a comprehensive bibliography of work on atomic line shapes, widths, and shifts already exists [59.1–7]. The perturber may be an electron, a neutral atom or an atomic ion and can interact weakly or strongly with the emitter. The emitter is either a hydrogenic or nonhydrogenic atom that is either neutral or ionized. In general, transitions in nonhydrogenic atoms can be treated as isolated, that is the separation between neighboring lines is much greater than the width of an individual line. When the emitter is hydrogen or a hydrogenic ion, the additional degeneracy of the energy levels with respect to orbital angular momentum quantum number means that lines overlap and are coupled. Pressure broadening is a general term that describes any broadening and shift of a spectral line produced by fields generated by a background gas or plasma. The term Stark broadening implies that the perturbers are atomic ions and/or electrons, and collisional broadening implies that the ‘collision’ model is appropriate; this term is often used to describe an isolated line perturbed by electrons. Neutral atom broadening indicates neutral atomic perturbers; this implies short-range emitter-perturber interactions which in turn influence the approximations made.

876

Part D

Scattering Theory

perturber interaction. It is not only widely applicable to electron and neutral atom broadening, but also, for certain plasma conditions, to broadening by atomic ions. The power radiated per unit time and per unit interval in circular frequency ω, in terms of the line profile I (ω), is

Part D 59.2

4 ω4 P (ω) = I (ω) . (59.2) 3 c3 For an isolated line produced by a transition from an upper energy level i to a lower level f , the line profile is Lorentzian with a shift d:  w 1
∗ i f |∆| i f ∗  I (ω) = , 2 π ω − ωi f − d + w2 (59.3)

and if the profile is for a transition between an upper set of levels i, i  and a lower set f, f  with quantum numbers Ji Mi , Ji  Mi  , J f M f , and J f  M f  , then 
∗  1 I (ω) = Re (59.4) i f |∆| i  f ∗ π ii  f f       −1   , × i  f ∗  w+id −i ω−ωi f I  i f ∗ where I is the unit operator and w and d are width and shift operators. In (59.4) ∆ is an operator corresponding to the dipole line strength defined by
 
∗ (59.5) i f |∆| i  f ∗ ≡ q 2 i |r| f  · i  |r| f  , where r represents the internal emitter coordinates, q 2 = e2 / (4π0 ) is the square of the electronic charge, and 0 is the permittivity of vacuum in SI units of J m. On taking the average over all degenerate magnetic sublevels, 
∗  1 i f ∆ i  f ∗ (59.6) I (ω) = Re π ii  f f 

    −1

∗  ∗ × i f w + id − i ω − ωi f I i f

in terms of reduced matrix elements that are independent of magnetic quantum numbers. They are defined by
∗ 
 i f |∆| i  f ∗ = Di fi  f  i f ∗ ∆ i  f ∗ , (59.7) where Di fi  f  =



(−1) Ji +Ji  −Mi −Mi 

µ



×

 Ji 1 J f −Mi µ M f

 Ji  1 J f  −Mi  µ M f 


 ∗ 

s i f w + id − i(ω − ωi f )I i f ∗  = Di fi  f 

, (59.8)

Mi ,Mi  ,M f ,M f 

 


s  × i  f ∗  w + id − i(ω − ωi f )I  i f ∗

(59.9)

with s = −1, 1. For the line profile (59.3), the width and each have a single matrix element: 

γ = 2w = i f ∗ ||2w|| i f ∗ , d = i f ∗ ||d|| i f ∗ , (59.10)

where γ is the full width at half maximum. Throughout the rest of this article it will be assumed that collisions only connect the set of upper levels i, i  or the set of lower levels f, f  which is valid when w
ω; pressure broadening of spectral lines is also assumed to be independent of Doppler broadening. However, for microwave spectra of molecules, w can be of the order of ω and collisions connecting the upper to the lower levels become important; for further details see Ben-Reuven [59.11, 12]. Also for microwave spectra, pressure broadening and Doppler broadening cannot be considered to be independent effects and a generalized theory has been developed by Ciuryło and Pine [59.13].

59.2 Isolated Lines 59.2.1 Semiclassical Theory The motion of the perturber relative to the emitter is treated classically and is assumed to be independent of the internal states of the emitter and perturber. This common trajectory is specified by an emitter–perturber separation R ≡ R (b, v, t) ,

b·v = 0 ,

(59.11)

where v is the relative velocity and b is the parameter. The time-dependent wave equation emitter-perturber system is dΨ i~ = HΨ dt and the eigenfunctions ψi for the unperturbed obey H0 ψi = E i ψi ,

i = 0, 1, 2, . . . .

impact for the (59.12)

emitter (59.13)

Collisional Broadening of Spectral Lines

If Ψ (r, R) is expanded in the form  a ji (t)ψ j (r) exp(−iE j t/~) , Ψ(r, R) =

(59.14)

(59.15)

j = i, f ,

(59.23)

(59.16)

then 

(59.24)

 

 Si  i (b, v) S∗f  f (b, v) = exp 2i ηi − η f , av

ψ ∗j (r) V

(r, R) ψk (r) dr

in which V (r, R) is the emitter-perturber interaction and

~ω jk = E j − E k .

(59.18)

Integration of equations (59.16) for −∞ ≤ t ≤ ∞ gives the unitary scattering matrix S, with elements S ji (b, v) ≡ a ji (∞) ,

(59.25)

(59.17)

i, j = 0, 1, 2, . . . . (59.19)

Then ∞ w + id = 2πN v f (v) dv 0

∞  × δi  i δ f  f − Si  i (b, v) S∗f  f (b, v) b db , av

0

(59.20)

where Ji  = Ji and J f  = J f , N is the perturber density and [· · · ]av denotes an average over all orientations of the collision and over the magnetic sublevels [see Eq. (59.9)]. In (59.20), f (v) is the Maxwell velocity distribution at temperature T for an emitter-perturber system of reduced mass µ:  3/2  µ µv2 f (v) = 4πv2 , exp − 2πkB T 2kB T ∞ f (v) dv = 1 . (59.21) 0

59.2.2 Simple Formulae These are useful for making quick estimates, but in individual cases may give results in error by a factor of two

where the phase shifts are 1 η j (b, v) = − 2~

∞ V jj (R) dt ,

j = i, f .

−∞

(59.26)

Equations (59.20)–(59.26) give    2/( p−1) β p C p  w + id = πN v¯ v¯   iπ p−3 exp ± α p , (59.27) ×Γ p−1 p−1 where

2 p − 3  π −1/( p−1) , αp = Γ p−1 4   1 √ Γ 2 ( p − 1)   βp = π , Γ 12 p 

C p = Ci − C f ,  ∞ 8kB T 1/2 . v¯ = v f (v) dv = πµ

(59.28)

0

In (59.27) the ± sign indicates the sign of C p , α p  1 for p ≥ 3, and Γ (· · · ) is the gamma function. The cases p = 3, 4, and 6 correspond to resonance, quadratic Stark and van der Waals broadening, respectively. This approximation is invalid for the dipole case ( p = 2) for which (59.27) is not finite. The dipole–dipole interaction ( p = 3) occurs when emitter and perturber are identical atoms (apart from isotopic differences). If the level i is connected to the ground state by a strong allowed transition with

Part D 59.2

V jk (R) =

V jj (R) = ~C j R− p ,

R = b + vt ,

k



(59.22)

and C j depends only on the state j of the emitter, and if the relative motion is along the straight line

and (59.12)–(59.14) give   da ji  = aki V jk exp iω jk t , i~ dt

where

Vij (R) = V jj (R) δij , where V jj (R) is a simple central potential

where initially at time t = −∞

i, j, k = 0, 1, 2, . . . ,

877

or more. If it is assumed in (59.17) that

j

a ji (−∞) = δ ji ,

59.2 Isolated Lines

878

Part D

Scattering Theory

absorption oscillator strength f gi , and the perturbation of the level f can be neglected by comparison, then q2

f  gi  , 2m e ωgi   √  1 cd = 1 + √ ln 2 + 3 = 1.380 173 . 2 3

C3 = cd

(59.29)

59.2.3 Perturbation Theory An approximate solution of (59.16) is given by [59.8– 10, 15, 16] i S ji (b, v) = δ ji − ~

Part D 59.2

C4 = −

q2



 αi − α f ,

(59.31)

2~ where αi and α f are the dipole polarizabilities of states i and f , respectively. Van der Waals broadening occurs when the emitter and perturber are nonidentical neutral atoms. If energy level separations of importance in the perturbing atom are much greater than those of the emitter (e.g., alkali spectra broadened by noble gases), C6 is given by  q2  2 αd ri − r 2f , (59.32) ~ where αd is the dipole polarizability of the perturber. The mean square radii can be calculated from the normalized radial wave functions 1r Pn ∗j l j (r) or estimated from C6 = −

r 2j

∞ = Pn2∗ l j (r) r 2 dr

  V ji (t) exp iω ji t dt

−∞

 ∞   1  − 2 V jk (t) exp iω jk t dt ~

Also, if g j is the statistical weight of level j, the constant cd may be replaced by an empirical value  4 gg 1/2 cd = , (59.30) π gi and this gives a width correct to about 10% [59.14]. Equation (59.27) does not predict a finite shift. Quadratic Stark broadening occurs when a nonhydrogenic emitter is polarized by electron perturbers. Then

∞

k

−∞

       Vki t exp iωki t dt . (59.35)

t × −∞

This gives a cross section for the collisional transition i → j: ∞ σij (v) = 2π



Pij (b, v) av b db ,

0

i, j = 0, 1, 2, . . . ,

(59.36)

where

 2 Pij (b, v) = δ ji − S ji (b, v) ,  2 Re [1 − Sii (b, v)] = Pij (b, v) ,

correct to second-order in V (r, R) on both sides. Using (59.10), (59.20) and (59.36)–(59.38), the full width is ∞ γ = N v f (v) dv 0    × σij (v) + σ f j (v) + σi f (v) , j=i

j= f

(59.39)

0

2  n ∗2   j a0 ∗2 5n l + 1 − 3l + 1 , j j j 2z 2 j = i, f

where the sums are taken over all energy-changing transitions, the tilde indicates an interference term, and (59.33)

in which the effective principal quantum numbers n ∗j

are

∞  σi f (v) = 2π

z2 Ih , n ∗2 j



i f (b, v) b db , P av

(59.40)

0

given by

Ej ≡ −

(59.38)

j

j



(59.37)

in which z = Ze + 1 ,

(59.34)

where Ih = hcR∞ is the Rydberg energy, Z e is the charge on the emitter and z = 1 in this case.

 ∞ 2    1

  . i f (b, v) =  P V − V dt (t) (t) ii ff ~    −∞

(59.41)

Collisional Broadening of Spectral Lines

59.2.4 Broadening by Charged Particles B (β, ξ) =

The total emitter-perturber interaction is Z pq2 z Z pq2 −  V0 (R) + V (r, R) , |R− r| R

(59.42)

where Z p is the charge on the perturber and V0 (R) =

Z e Z pq2 , R

V (r, R) = −Z p q 2

r·R . R3 (59.43)

Z e Z pq2 d2 R = −∇V = R, (59.44) 0 dt 2 R3 with the resulting hyperbola characterized by a semimajor axis a and an eccentricity , where      Z e Z p q2 b2 = a2 2 − 1 , a = . (59.45) µv2 µ

On using (59.17), (59.20), (59.36)–(59.41) and (59.43), i f (b, v) = 0 , P

Vii (t) = 0 ,

 σi f (v) = 0 , (59.46)

and

0

in (59.50) and (59.51). Approximation (59.48) breaks down at small values of b because of assumption (59.43) and the lack of unitarity of S as given by (59.35). This problem is discussed elsewhere [59.8, 15, 16], and all methods used involve choosing a cutoff at b = b0 , where b20
r 2f , and using (59.48) only for b > b0 . For b ≤ b0 , an effective constant probability is introduced and the method works well as long as the contribution from b ≤ b0 is small. For b > b0 (or β > β0 ), the contribution to σij (v) in (59.39) is evaluated using (59.47)–(59.50) and (59.52), where ∞ A (β, ξ)

db  = − e∓πξ β0 K iξ (β0 ) K iξ (β0 ) . b (59.53)

0

 j=i

(59.51)

where K iξ (β) is a modified Bessel function. In (59.50),  the ∓ sign corresponds to Z e Z p = ±  Z e Z p  , and in (59.51), ℘ indicates the Cauchy principal value. If Z e = 0, then   ωij  , δ=1 (59.52) ξ =0, β=b v

v f (v) dv

w + id = 2πN

×

0

  A β  , ξ dβ    , β 2 − β 2

b0

∞

 ∞

∞

Q ij (b, v) +



 Q f j (b, v) b db ,

j= f

(59.47)

where 4Z 2p Ih2 a02 f ij   [A (β, ξ) + iB (β, ξ)] , ~m e ωij  b2 v2

2 Re[Q ij (v)] = Pij (b, v) av . (59.48) Q ij (v) =

If

  a ωij  2 − 1 , β ≡ ξ , δ ≡ , (59.49) ξ≡ v 2 the functions A (β, ξ) and B (β, ξ) in (59.48) are given by A (β, ξ) = δ exp (∓πξ) β 2   2  2    × K iξ (β) + δ  K iξ (β)

(59.50)

879

A similar treatment exists for the quadrupole contribution to V (r, R) in (59.43) [59.8, 15, 16].

59.2.5 Empirical Formulae An empirical formula based on the theory of Sect. 59.2.4 for the width of an atomic line Stark broadened by electrons has been developed [59.17]. Konjevi´c [59.18] has reviewed the data available for nonhydrogenic lines and has provided simple analytical representations of the experimental results for widths and shifts. The full half-width is given by (59.39), where  σi f = 0 and  2 8π 2 ~ v f e (v) σ jk (v) dv = √ v−1 m 3 3 e k= j    jj  T × T j g(x j ) + g(x jj  ) , j = i, f ,

∞ 0



l j  =l j ±1

(59.54)

Part D 59.2

If Z e = 0, the relative motion is described by (59.24), but if Z e = 0, the trajectory is hyperbolic and is given by

2β ℘ π

59.2 Isolated Lines

880

Part D

Scattering Theory

  ∗ nl> − nl∗<
1. The effective Gaunt factors g (x) and  g (x) are given by

where v−1

 ∞ 2m e 1/2 −1 = v f e (v) dv = , πkB T

(59.55)

0

and f e (v) = f (v) with µ = m e . In (59.54),  ∗ 2  3n j 1  ∗2   Tj = n j + 3l j l j + 1 + 11 , (59.56) 2z 9   jj  =  l>  R2  nl∗ , nl∗ , l> , T (59.57) > < 2l j + 1 jj     l< = min l j , l j  , l> = max l j , l j  ,

Part D 59.3

with R

jj 

 ∗ ∗  nl> , nl< , l> ≡ a0−1

∞ Pnl∗

l > >

(r) r Pnl∗

l <
, nl∗< , l>     jj  nl∗ , l> φ nl∗ , nl∗ , l> =R > > < 3nl∗>  ∗2 2 1/2  ∗ ∗  ≡ n l> − l > φ n l> , n l< , l > (59.59) 2z   ∗ ∗ and φ nl> , nl< , l> is tabulated elsewhere [59.19]. The effective principal quantum numbers nl∗> and nl∗< of the states j and j  in (59.57)–(59.59) both correspond to principal quantum number n j and φ  1 for

 g (x) = 0.7 − 1.1/z + g (x) , x =

3kB T , 2∆E

(59.60)

where x g (x)

≤2 0.20

3 0.24

5 0.33

10 0.56

30 0.98

100 1.33

is used for x < 50, and for x > 50 √    3 1 4 |E|  g (x) = g (x) = + ln x , (59.61) π 2 3z Ih with (59.60) and (59.61) joined smoothly near x = 50. The energy E = E j is given by (59.34) and x j and x jj  in (59.54) are evaluated using ∆E j =

2z 2

Ih n ∗3 j

,

  ∆E jj  =  E j − E j   .

(59.62)

For Z e = 2 and 3, (59.54) is generally accurate to within ±30% and ±50%. For Z e ≥ 4, (59.54) is less accurate, as relativistic effects and resonances become more important. Accuracy increases for transitions to higher Rydberg levels as long as the line remains isolated. Tables in appendices IV and V of [59.8] give widths for atoms with Z e = 0, 1 and other semi-empirical formulas based on detailed calculations have been developed by Seaton [59.20, 21] for use in the Opacity Project where simple estimates of many thousands of line widths are required.

59.3 Overlapping Lines 59.3.1 Transitions in Hydrogen and Hydrogenic Ions The most important case is that of lines of hydrogenic systems emitted by a plasma with overall electrical neutrality, broadened by perturbing electrons and atomic ions. The line profile is given by (59.4)–(59.9) in which (59.20) is generalized to give


 ∗  i f w + id i f ∗ = 2πN

∞ v f (v) dv 0

∞   δi  i δ f  f − Si  i (b, v) S∗f  f (b, v) ×

av

b db .

0

(59.63)

The superscripts and suffices e and i will be used to denote electron and ion quantities, and em indicates that averaging over magnetic quantum numbers has not been carried out. In the impact approximation, electron and ion contributions evaluated using (59.63) are additive and the matrix to be inverted is of order n i n f . However, under typical conditions in a laboratory plasma, e.g., a hydrogen plasma with Ne = Ni = 1022 m−3 , perturbing atomic ions cannot be treated using the impact approximation. The ions collectively generate a static field at the emitter which produces first-order Stark splitting of the upper and lower levels. The ions are randomly distributed around the emitter and the field distribution W (F) used assumes that each ion is Debye screened by electrons; allowance is made for these heavy composite perturbers interacting with each other as well as with the emitter. If the ion field has a slow

Collisional Broadening of Spectral Lines

(59.64)

and the destruction of the degeneracy by the ion field means  that2the matrix to be inverted in (59.64) is of order n i n f . Inclusion of higher multipoles in V(r, R) in (59.43) introduces small asymmetries. The Stark representation for the hydrogenic wave functions is often used because it diagonalizes the shift matrix in (59.64). The transformation is given by (see Sect. 9.1.2 and 13.4.2) n j −1

    1/2 n j K j m j = (−1) K 2l j + 1 l j =|m j |     N N lj n j l j m j , × M1 M2 −m j j = i, i  , f, f  ,

where 0 γem

∞      ∗  0 ∗ ≡ i f 2we  i f = Ne v f e (v) dv 0      σijem (v) + σ em × f j (v) δii  δ f f  , j=i

γ˜em

j= f

(59.68)


 we | i f ∗ ≡ i  f ∗ |2 ∞ = Ne v f e (v) dv σ˜ iem  f  i f (v) .

(59.69)

0

    In (59.68), n i − n j  = 0 and n f − n j  = 0 in the first and second terms,  respectively, and in  (59.69) (n i  − n i ) = n f  − n f = 0 . The matrix element (59.68) can be evaluated using (59.36), (59.48), (59.50), and (59.52) as before. In (59.68), ∞   Pije (b, v) σijem (v) = 2π b db , av0

0

i, j = 0, 1, 2, . . . ,

(59.70)

and in (59.69) σ˜ iem  f i f

(v) = 2π

∞ 

 P˜ie f  i f (b, v)

av0

b db

(59.71)

0

(59.65)

where quantum number K j replaces l j and    1 nj −1 , n j = K j + K j + m j  + 1 , N = 2  1     K= 2K + m j + m j + 1 , 2  j  0 ≤ K j ≤ nj −1 ,  1 M1 = m j + K j − K j , 2  1 M2 = m j + K j − K j . (59.66) 2 For the electron impact broadening, it is convenient to separate the energy-changing and the zero energychange transitions, so that (59.39) is generalized to give 
0 + γ˜em ≡ i  f ∗ |2we | i f ∗ , (59.67) γem = γem

881

by analogy with (59.36) and (59.40), where av0 indicates an average over all orientations of the collision only. On using (59.48)–(59.50), (59.59) and (59.8) with ji = li , ji  = li  , j f = l f , and j f  = l f  ,   8Ih a02 Di fi  f  R˜ i  f  i f A(0, 0) , P˜ie f  i f (b, v) = av0 3m e b2 v2 (59.72)

where R˜ i  f  i f  ≡



R˜ ij2 (n i , li> ) +

l j =li ±1



R˜ 2f j



  n f , l f> 

l j =l f ±1

× δli  li δl f  l f   − 2 R˜ i  i (n i , li> ) R˜ f  f n f , l f> δli  li ±1 δl f  l f ±1 . (59.73)

From (59.45) and (59.49)–(59.53) A (0, 0) = δ ,

Part D 59.3

time variation, ion dynamic effects on the line are produced [59.8, 9, 22]. The shift produced by electron perturbers is very small and the usual model adopted is to assume that the ions split the line into its Stark components, and that each component is broadened by electron impact. In both cases, only the dipole interactions in (59.43) are included and the profile is symmetric. Then (59.4) takes the form  
 1 I (ω) = Re W (F) dF i f ∗ |∆| i  f ∗ π ii  f f       −1   ∗  × i f  w + id − i ω − ωi f I  i f ∗ ,

59.3 Overlapping Lines

882

Part D

Scattering Theory

b1

db = A (0, 0) b

b0



ln (1 /0 ) , Z e = 0 , ln (b1 /b0 ) , Z e = 0 .

(59.74)

Part D 59.4

(59.53). The impact approximation neglects electron– electron correlations and the finite duration of collisions, so the long-range dipole interaction leads to a logarithmic divergence at large impact parameters in (59.74). Therefore, a second cutoff parameter is introduced which is chosen to be the smaller of the Debye length bD and vτ:   1/2  kB T b1 = min bD ≡ , vτ , (59.75) 4πq 2 Ne but estimating τ in this case is not straightforward; it depends on the splitting of the Stark components [59.8].

59.3.2 Infrared and Radio Lines If the density Ni is low enough, the impact approximation becomes valid for the perturbing atomic ions, and since impact shifts are unimportant, (59.6) gives 
 1 I (ω) = Re i f ∗ ∆ i  f ∗ π ii  f f 

   −1 

× i  f ∗ we + wi − i ω − ωi f I i f ∗ (59.76)

where in (59.76)


0 γe,i = γe,i + γ˜e,i ≡ i  f ∗ 2we,i i f ∗ , (59.77) ∞ 

 

0 γe,i ≡ i  f ∗ 2w0e,i i f ∗ = Ne,i v f e,i (v) dv  0    ×  σije,i (v) + σ e,i f j (v) δi  i δ f  f , (59.78) j=i

j= f


γ˜e,i ≡ i  f ∗ 2w ˜ e,i i f ∗ ∞ = Ne,i v f e,i (v) dv σ˜ ie,i  f  i f (v) .

(59.79)

0

In general, cross sections for electron and heavy-particle impact are roughly comparable for the same velocity and hence different impact energies. Therefore, using (59.78) and (59.79), γe0  γi0 ,

γ˜e
γ˜i ,

(59.80)

and this result is consistent with approximation (59.64)   for high density plasmas. If n i − n f = 1, 2, say, as n f increases, the relative contributions from (59.78) and (59.79) decrease because there is increasing coherence, and hence cancellation in σ˜ ie f  i f (v) and σ˜ ii f  i f (v) between the effects of levels i, i  and f, f  . Radio lines of hydrogen are observed in galactic HII regions where principal quantum numbers are of the order of n f  100, temperatures are Te = Ti  104 K, and densities are Ne = Ni  109 m−3 . If γ is the full-half width and γ˜ is the full-half width when only contributions (59.79) are retained, the effect of cancellation is illustrated by ni − n f = 1

electrons

protons + electrons

5

γ˜ /γ 0.81

γ˜ /γ 0.99

10

0.44

0.95

15

0.21

0.87

20

0.11

0.75

25

0.06

0.61

50

0.00

0.32

100

0.00

0.16

nf

59.4 Quantum-Mechanical Theory 59.4.1 Impact Approximation The scattering amplitude for a collisional transition i → j is given in terms of elements of the transition matrix T = 1 − S by     f k j , ki ≡ f χ j M j k j , χi Mi ki      2πi  ∗ il−l Ylm = kˆ i Yl  m  kˆ j 1/2 ki k j lml  m    (59.81) × T χ j M j l  m  ; χi Mi lm ,

where the quantities ki lm and k j l  m  refer to the motion of the perturber relative to the emitter before and after the collision, and χi and χ j represent all nonmagnetic quantum numbers associated with the unperturbed states i and j of the emitter. The total energy of the emitter-perturber system is given by EJ = E j +εj ,

εj =

(J, j) = (I, i) , (F, f ) ,

~2 2 k , 2µ j (59.82)

Collisional Broadening of Spectral Lines

and for an isolated line, γ is given by (59.39), where σij (v) =

kj 1 ki 4πgi      f χ j M j k j , χi Mi ki 2 dkˆ i dkˆ j , × Mi M j

ki = µv/~ ,

(59.83)

and the interference term σ˜ i f (v) ≡ σ˜ i fi f (v) is given by   1  Di fi  f   f(χi Mi  k , χi Mi k) σ˜ i f (v) = 4π M M 2 − f(χ f M f  k , χ f M f k) dkˆ dkˆ  , (59.84)

where

1 − T jj (l, v) = S jj (l, v) = exp 2iη j (l, k) , j = i, f , (59.89) [cf. (59.23), (59.25), and (59.26)]. For the case of overlapping lines, (59.63) becomes
 ∗  i f w + id i f ∗ = π

 2 ∞ ~ 1 f (v) dv N µ v 0

×

∞ 

  (2l + 1) δi  i δ f  f − Si  i (l, v) S∗f  f (l, v)

l=0

(59.90)

with k = k = µv/~. From (59.8) and (59.20),  2 ∞ ~ 1 f(v) dv N w + id = π µ v 0

∞    × (2l + 1) 1 − Sii (l, v)S∗f f (l, v) , l=0

(59.85)

where (µvb)2 ⇒ ~2 l (l + 1)

(59.86)

and the integral over b has been replaced by a summation over l . In (59.85)), Sii (l, v) S∗f f (l, v) is given by Sii (l, v) S∗f f (l, v)    1 Di fi  f  S I χi  Mi  lm  ; χi Mi lm = (2l + 1)   Mi Mi M f M f mm 

  × S∗F χ f  M f  lm  ; χ f M f lm and subscripts I and F are introduced size that the S-matrix elements correspond total energies E I and E F defined by scattering by the emitter in a state j using a central potential, the amplitude scattering is

(59.87)

to emphato different (59.82). If is treated for elastic

∞   i  f k ,k = (2l + 1) T jj (l, v) Pl kˆ  · kˆ , 2k







l=0

(59.88)

883

on generalizing (59.85) and using (59.87). Formulae (59.85) and (59.90) have been obtained by assuming that a collision produces no change in the angular momentum of the relative emitter-perturber motion. This corresponds to the assumption of a common trajectory in semiclassical theory, and means that the total angular momentum of the emitter-perturber system is not conserved. This assumption is removed in the derivation of the more general expressions given in the following sections.

59.4.2 Broadening by Electrons Different coupling schemes can be used to describe the emitter-perturber collision. For L S coupling, χ j M j ⇒ χ 0j L j M j SM S ,

j = i, i  , f, f  , (59.91)

in (59.81), where χ 0j denotes all other quantum numbers required to describe state j that do not change during the collision. Then     L j lL Tj S 12 S T  L j M j SM S lm 1 m s = C C T  M j m MS M S m s M ST 2 T T T L j M j S MS

   1 ×  L j Sl L Tj M Tj S T M ST , 2

(59.92) j j j

where Cm11 m2 23m 3 is a vector coupling coefficient, the superscript T denotes quantum numbers of the emitterperturber system, and 12 , m s are the spin quantum numbers of the scattered electron. On using (59.92),

Part D 59.4

i i M f M f

59.4 Quantum-Mechanical Theory

884

Part D

Scattering Theory

(59.90) is replaced by 
 ∗ i f w + id i f ∗ = π (~/m e )2 N      × (−1) L i +L i  +l+l 2L iT + 1 2L Tf + 1

it is often sufficient to use energies defined by    2J j + 1    E L j SJ j , EL j S = 2L j + 1 2S j + 1 J

L iT L Tf S T ll 

Part D 59.4

  T  2S + 1 L Tf L iT 1 L Tf L iT 1 × 2 (2S + 1) L i L f l L i L f  l  ∞ 1 f e (v) dv δl  l δ L i  L i δ L f  L f × v 0  1 T T 1 T T  − S I L i  Sl L i S ; L i Sl L i S 2 2   1 1 , (59.93) × S∗F L f  Sl  L Tf S T ; L f Sl L Tf S T 2 2 where, for an isolated line, the width and shift are given by (59.93) with L i  = L i and L f  = L f . For hydrogenic systems, where states i, i  and f, f  with different angular momenta are degenerate, a logarithmic divergence occurs for large values of l and l  [cf. Eq. (59.74)], and must be removed by using (59.75) and (59.86). If a jj coupling scheme is used χ j M j ⇒ χ 0j J j M j , j = i, i  , f, f  in (59.81), and     J j M j lm 1 m s  2    J j jJ Tj l 12 j  T T , (59.94) C M m  M T Cmm =  J j l jJ j M j sm J Tj M Tj

jm 

j

j

then (59.93) becomes 
 ∗ i f w + id i f ∗ = π (~/m e )2 N  T 1 × (−1) Ji +Ji  +2J f + j+ j (2JiT + 1)(2J Tf + 1) 2 T T   Ji J f jj ll



×

J Tf JiT 1 Ji J f j

∞ × 0

−



and obtain the S-matrix elements in an L S coupling scheme. They are then transformed to jj coupling by using the algebraic transformation   S J j  l  j  J jT ; J j l jJ jT     1/2  = 2J j + 1 2J j  + 1 (2 j + 1) 2 j  + 1  × (2L Tj + 1)(2S T + 1) L Tj S T

   T T    L j l L j   Lj l Lj    × S 12 S T S 12 S T        J j j J jT J j  j  J jT  1 T T 1 T T  × S L j Sl L j S ; L j Sl L j S , 2 2

(59.97)

and introducing the splitting of the fine structure components in (59.4) or (59.6). If in L S coupling the line is isolated, but nevertheless the broadened fine structure components overlap significantly, then the interference terms in (59.6) must be included.

59.4.3 Broadening by Atoms The formal result is very similar to (59.95), but in this case, the relative motion only gives rise to orbital angular momentum. Thus
 ∗  i f w + id i f ∗ = π (~/µ)2 N  T  (−1) Ji +Ji  +2J f +l+l (2JiT + 1)(2J Tf + 1) × JiT J Tf ll 

 ×

J Tf JiT 1

J Tf JiT 1 Ji J f l



J Tf JiT 1 Ji  J f  l 

∞

Ji  J f  j 

1 f(v) dv δl  l δ Ji  Ji δ J f  J f v 0    

. − S I Ji  l  JiT ; Ji l JiT S∗F J f  l  J Tf ; J f l J Tf

×

 1 f e (v) dv δl  l δ j  j δ Ji  Ji δ J f  J f v

S I (Ji  l  j  JiT ; Ji l jJiT ) S∗F (J f  l  j  J Tf ;

(59.96)

j

 J f l jJ Tf )

,

(59.95)

where Ji  = Ji and J f  = J f for an isolated line. If the spectrum of the emitter is classified using L S coupling,

(59.98)

For many cases of practical interest, transitions of type Ji → J f are isolated and so have line profiles given by (59.3), (59.10), and (59.98), where Ji = Ji  and J f = J f  [59.14]. In order to obtain the S-matrix elements in

Collisional Broadening of Spectral Lines

(59.98), it is usually sufficient to use adiabatic potentials for the emitter-perturber system that have been calculated neglecting fine structure. Since T is typically a few hundred degrees, only coupling between adiabatic states that tend to the appropriate separatedatom limit are retained in the scattering problem. The

59.5 One-Perturber Approximation

885

coupled scattering equations are then solved with fine structure introduced by applying an algebraic transformation to the adiabatic potentials, and using the observed splittings of the energy levels. The Born– Oppenheimer approximation is valid, and details are given in [59.23].

59.5 One-Perturber Approximation with

59.5.1 General Approach and Utility

~ω IF = E I − E F ,

(59.99)

and av denotes an average over states I and a sum over states F [59.9]. Wave functions Ψ J are given by    ψ j (r) φ k j , k j0 ; R , (59.100) Ψ J (r, R) = O j

where J = I, F, and O is an operator that takes account of any symmetry properties of the emitter-perturber system. The energies E I and E F are given by (59.82). The perturber wave functions for initial state j0 and final state j are expanded in the form      −1/2 il j0 k j0 Yl∗j m j kˆ j0 φ k j , k j0 ; R = 2πi l j0 m j0 l j m j

0

0

 1   ˆ F Γ j , Γ j0 ; R × Yl j m j R R where Γ j denotes a channel characterized by j = 0, 1, 2, . . . ,

Ψ (r, R, t) = Ψ J (r, R) exp (−iE J t/~) , (59.105)   are integrated to give functions F Γ j , Γ j0 ; R . Using (59.100) and (59.101), (59.99) becomes 
 1 u i0 Γi Γ f ∗ |∆| Γi  Γ f  ∗ I (ω) = N 2 Γi0 Γi Γi  Γ f0 Γ f Γ f 

∞ × 0

1 f(v) dv F (Γ, v) , v

(59.106)

where ∞ F (Γ, v) =

    F ∗ Γi , Γi0 ; R F Γ f , Γ f 0 ; R dR

0

(59.101)

Γ j = χ j M jl jm j ,

 1 z  θ j = k j R − l j π − ln 2k j R 2 ki  z + arg Γ l j + 1 + i , kj µq 2 (59.104) z = 2 Ze Zp . ~ The coupled equations obtained by using (59.12), (59.13), (59.100), and (59.101), where

∞     × F Γi  , Γi0 ; R F ∗ Γ f  , Γ f0 ; R dR , 0

(59.102)

[see Eq. (59.81)]. In (59.101), the radial perturber wave function has the limiting forms   F Γ j , Γ j0 ; R R→0 Rl j +1 ∼    R→∞ k −1/2 δΓ Γ exp −iθ j j j0 j ∼     −S J Γ j ; Γ j0 exp iθ j , (59.103)

u i0 = gi0

( i 0

(59.107)

gi0 ,

v=

~ki0 . µ

(59.108)

The one-perturber approximation is valid when ∆ω ≡ |ω − ωi f |  w ;

V  V¯ ,

(59.109)

where V is the effective interaction potential required to produce a shift ∆ω . In the center of the line, many-body effects are always important and the one-perturber approximation diverges as ∆ω → 0 . In many cases, there

Part D 59.5

If only one perturber is effective in producing broadening, I (ω) can be obtained by considering a dipole transition between initial and final states I and F of the emitter-perturber system. Then P (ω) is given by (59.2), where


I (ω) = δ (ω − ω IF ) IF ∗ |∆| IF ∗ av ,

886

Part D

Scattering Theory

is a region of overlap where criteria (59.1) and (59.109) are all valid, but when ∆ω τ  1, (59.99) is a static approximation, since the average time between collisions is ∆ω−1 .


 where i  f ∗ w i f ∗ is given by (59.93). Line shape (59.113) is identical to that obtained from (59.6) when ∆ω  w . If the jj coupling scheme specified by (59.94) is used, and channel Γ j is defined by

If L S coupling is used, definition of channel Γ j in (59.104) is replaced by 1 Γ j = L j Sl j L Tj S T , j = i 0 , i, i  , f 0 , f, f  , 2

Equation (59.106) becomes [cf. (59.95)] I(ω) 1 = Ne 2

Part D 59.5

[cf. (59.91) and (59.92)]. Then assuming that the weights u i0 of all the levels i 0 that effectively contribute to the line are the same, (59.106) becomes 
 ∗  ∗  1 I (ω) = Ne L i S L f S ∆ L i  S L f  S 2 Γ ΓΓ i0 i i  Γf Γf Γf 0

× δli l f δli  l f  δ L T L T δ L T L T δ L T L T δ L T L T  i0 i f f i f0 f i 0 T 0  2S + 1 × (−1) L i +L i  +li +li  2 (2S + 1)    T T × 2L i + 1 2L f + 1   L Tf L iT 1 L Tf L iT 1 × Li L f l L i  L f  li  ∞ 1 f e (v) dv F (Γ, v) , (59.111) × v 0

where F (Γ, v)is defined by  (59.107) and (59.111). If the functions F Γ j  , Γ j0 ; R in (59.107) are replaced by their asymptotic forms (59.103), then F (Γ, v)  ∆ω−2 (~/m e )2      × δΓi0 Γi δΓ f0 Γ f − S I Γi ; Γi0 S∗F Γ f ; Γ f 0      × δΓi0 Γi  δΓ f0 Γ f  − S∗I Γi  ; Γi0 S F Γ f  ; Γ f 0 . (59.112)

On substituting (59.112) into (59.111), summing over Γi0 and Γ f0 and using the unitary property of the S-matrix,

=

1 π∆ω2


 ∗  ∗  L i S L f S ∆ L i  S L f  S L iT L Tf S T ll 


 × i  f ∗ w i f ∗ ,

(59.113)




 ∗  ∗  Ji J f ∆ Ji  J f 

Γi0 Γi Γi  Γ f0 Γ f Γ f 

(59.110)

I (ω)

j = i 0 , i, i  , f 0 , f, f  , (59.114)

Γ j = J j l j j j J jT ,

59.5.2 Broadening by Electrons

× δli l f δli  l f  δ ji j f δ ji  j f  δ J T J T δ J T i0 i

T f0 J f

δ J T J T δ J T i0 i

× (−1) Ji +Ji  +2J f + ji + ji    1 T 2Ji + 1 2J Tf + 1 × 2   J Tf JiT 1 J Tf JiT 1 × Ji J f ji Ji  J f  ji  ∞ 1 f e (v) dv F (Γ, v) , × v

T f0 J f 

T

(59.115)

0

where F (Γ, v) is given by (59.107) and (59.114).

59.5.3 Broadening by Atoms   In the wings of a line where ~ |∆ω|   E Ji − E Ji   , j = i 0 , i, i  , f 0 , f, f  , coupling between the fine structure levels is important. If channel Γ j is defined by Γ j = J j l j J jT ,

j = i 0 , i, i  , f 0 , f, f  ,

(59.116)

Equation (59.106) becomes [cf. (59.115)] 
 ∗  ∗  1 I (ω) = N Ji J f ∆ Ji  J f  2 Γi0 Γi Γi  Γ f0 Γ f Γ f 

× δli l f δli  l f  δ J T J T δ J T i0 i

T f0 J f

δ J T J T δ J T J T i i f0 f  0   T 2Ji + 1 2J Tf + 1

× (−1) Ji +Ji  +2J f +li +li    J Tf JiT 1 J Tf JiT 1 × Ji J f li Ji  J f  li  ∞ 1 f (v) dv F (Γ, v) , × v T

(59.117)

0

where F (Γ, v) is given by (59.107)  and (59.116).  In the far wings, where ~ |∆ω|   E Ji − E Ji   , j = i 0 , i, i  , f 0 , f, f  , an adiabatic approximation is valid.

Collisional Broadening of Spectral Lines

Adiabatic states of the diatomic molecule formed by the emitter-perturber system are considered in which the total spin is assumed to be decoupled from the total orbital angular momentum of the electrons. The coupling between rotational and electronic angular momentum can also be neglected, because typically, contributions to the line profile come from 0 ≤ l j  400, whereas Λ j  2. Therefore, transitions take place between channels defined by Γ j = Λ j L j SL j ,

j = i, f ,

59.5 One-Perturber Approximation

is the dipole moment, then using (59.119)), the free–free contribution is given by N  u Λi δli l f (2li + 1) I0 (ω) = 2 Γi Γ f ∞ 

f(v) dv G(Γ, εi , ε f ) , v

× 0

(59.124) (59.118)

j

where εi = 12 µv2 , and u Λi is the relative weight of state Λi [cf. Eqs. (59.106)–(59.108)]. The bound-free contribution is N  u Λi δli l f (2li + 1) I1 (ω) = 2 Γi Γ f    × g (εi ) G Γ, εi , ε f i

(59.119)

where k j = k j0 and ψ j (r; R) is the wave function for molecular state Λ j . In (59.119), the only molecular states retained are those that correlate with emitter states i and f . The scattering is described by     lj lj +1 d2 2z 2µ 2 − − − 2 VΛ j (R) + k j R dR2 R2 ~ × F j (R) = 0 , (59.120)

(59.125)

and the free–bound contribution is  ~∆ω 1  u Λi δli l f (2li + 1) exp − I2 (ω) = N 2 kB T Γi Γ f      × g ε f G Γ, εi , ε f , (59.126) f

where

 2 ~ f(v) µ v2  2  3/2  ε ~ µ , = 8π 2 exp − µ 2πkB T kB T

g(ε) = 2π

where k j = k j0 and (59.121)

for vibrational or free states, respectively, and VΛ j (R) is the potential energy of state Λ j . Free–free transitions always contribute to the line profile, but bound-free and free-bound transitions only contribute on the red and blue wings, respectively. On using (59.82), (59.99), and (59.109), ~∆ω = εi − ε f , and ε j becomes the energy of bound state j with vibrational quantum number v j when ε j < 0 . If ∞ 2       ∗  ¯ G Γ, εi , ε f ≡  Fi (R) ∆ (R) F f (R) dR   0

(59.122)

[cf. (59.107)], where  ¯ (R) = −q ψi∗ (r; R) r ψ f (r; R) dr ∆

(59.123)

(59.127) 1 2 2 µv .

on using (59.21) with ε = The full line profile is then given by (59.124)–(59.126), so that I (ω) =

2 

I j (ω) .

(59.128)

j=0

The satellite features that are often seen in line wings arise because turning

points in the difference potential VΛi (R) − VΛ f (R) produce a phenomenon analogous to the formation of rainbows in scattering theory. The JWKB approximation is often used for the functions Fk j l j (R) , and can be shown to lead to the correct static limit in which transitions take place at fixed values of R called ‘Condon points’, i.e., the Franck– Condon principle is valid. Further details are given in [59.9, 10, 24–26].

Part D 59.5

where the unperturbed emitter in state j has quantum numbers L j S, the quantum number Λ j represents the projection of the orbital angular momentum on the internuclear axis, and (59.100) is replaced by    Ψ J (r, R) = O ψ j (r; R) φ k j , k j0 ; R ,

F j (R) = Pv j l j (R) or   F j (R) ≡ F Γ j , Γ j ; R = Fki li (R)

887

888

Part D

Scattering Theory

59.6 Unified Theories and Conclusions

Part D 59

The pressure broadening of spectral lines is in general a time-dependent many-body problem and as such cannot be solved exactly. After all, even the problem of two free electrons scattered by a proton is still a subject of active research. There is no practical theory that leads to the full static profile in the limit of high density (or low temperature) and to the full impact profile in the limit of low density (or high temperature). As with so many problems in physics, it is the intermediate problem that is intractable because no particular feature can be singled out as providing a weak perturbation on a known physical situation. However, much progress has been made over the last thirty years in developing theories that take into account many of the key features of the intermediate problem and they are often successful in predicting line profiles for practical applications [59.8–10, 24–26].

More recently, time-dependent many-body problems have been tackled using computer-oriented approaches that invoke Monte Carlo and other simulation methods to study line broadening in dense, high-temperature plasmas, see for example [59.27]. In this chapter, the emphasis has been on aspects of the subject that relate directly to electron-atom and low-energy atom–atom scattering. Many experts in the fields of electron–atom and atom–atom collisions are still not exploiting the direct applicability of their work to line broadening. It is hoped that this contribution will encourage more research workers to study these fascinating problems that not only provide links with plasma physics and in particular with the physics of fusion plasmas, but also with a quite distinct body of laboratory-based experimental data.

References 59.1

59.2

59.3

59.4

59.5

59.6

59.7 59.8 59.9

J. R. Fuhr, W. L. Wiese, L. J. Roszman: Bibliography on Atomic Line Shapes and Shifts, Special Publication 366 (National Bureau of Standards, Washington, DC. 1972) J. R. Fuhr, L. J. Roszman, W. L. Wiese: Bibliography on Atomic Line Shapes and Shifts, Special Publication 366 (National Bureau of Standards, Washington, DC. 1974), Supplement 1 J. R. Fuhr, G. A. Martin, B. J. Specht: Bibliography on Atomic Line Shapes and Shifts, Special Publication 366 (National Bureau of Standards, Washington, DC. 1974), Supplement 2 J. R. Fuhr, B. M. Miller, G. A. Martin: Bibliography on Atomic Line Shapes and Shifts, Special Publication 366 (National Bureau of Standards, Washington, DC. 1978), Supplement 3 J. R. Fuhr, A. Lesage: Bibliography on Atomic Line Shapes and Shifts, Special Publication 366 (National Institute of Standards and Technology, Washington, DC. 1992), Supplement 4 A. Lesage, J. R. Fuhr: Bibliography on Atomic Line Shapes and Shifts (Observatoire de Paris, Meudon 1998), Supplement 5 http://www.physics.nist.gov/PhysRefData H. R. Griem: Spectral Line Broadening in Plasmas (Academic, New York 1974) G. Peach: Adv. Phys. 30, 367 (1981)

59.10 59.11 59.12 59.13 59.14 59.15 59.16 59.17 59.18 59.19 59.20 59.21 59.22 59.23 59.24 59.25 59.26 59.27

N. Allard, J. R. Kielkopf: Rev. Mod. Phys. 54, 1103 (1982) A. Ben-Reuven: Phys. Rev. 145, 7 (1966) A. Ben-Reuven: Adv. Atom. Molec. Phys. 5, 201 (1969) R. Ciuryto, A. S. Pine: J. Quant. Spectrosc. Radiat. Transfer 67, 375 (2000) E. L. Lewis: Phys. Rep. 58, 1 (1980) S. Sahal-Bréchot: Astron. & Astrophys. 1, 91 (1969) S. Sahal-Bréchot: Astron. & Astrophys. 2, 322 (1969) M. S. Dimitrijevi´c, N. Konjevi´c: J. Quant. Spectrosc. Radiat. Transfer 24, 451 (1980) N. Konjevi´c: Phys. Rep. 316, 339 (1999) G. K. Oertel, L. P. Shomo: Astrophys. J. Suppl. 16, 175 (1969) M. J. Seaton: J. Phys. B 21, 3033 (1988) M. J. Seaton: J. Phys. B 22, 3603 (1989) V. I. Kogan, V. S. Lisitsa, G. V. Sholin: Rev. Plasma Phys. 13, 261 (1987) P. J. Leo, G. Peach, I. B. Whittingham: J. Phys. B 28, 591 (1995) J. Szudy, W. E. Baylis: J. Quant. Spectrosc. Radiat. Transfer 15, 641 (1975) J. Szudy, W. E. Baylis: J. Quant. Spectrosc. Radiat. Transfer 17, 269 (1977) J. Szudy, W. E. Baylis: Phys. Rep. 266, 127 (1996) A. Calisti, L. Godbert, R. Stamm, B. Talin: J. Quant. Spectrosc. Radiat. Transfer 51, 59 (1994)

889

Part E

Scattering Part E Scattering Experiments

60 Photodetachment David J. Pegg, Knoxville, USA 61 Photon–Atom Interactions: Low Energy Denise Caldwell, Arlington, USA Manfred O. Krause, Oak Ridge, USA 62 Photon–Atom Interactions: Intermediate Energies Bernd Crasemann, Eugene, USA

64 Ion–Atom Scattering Experiments: Low Energy Ronald Phaneuf, Reno, USA 65 Ion–Atom Collisions – High Energy Lew Cocke, Manhattan, USA Michael Schulz, Rolla, USA

66 Reactive Scattering Arthur G. Suits, Stony Brook, USA Yuan T. Lee, Taipei, Taiwan 63 Electron–Atom and Electron–Molecule Collisions Sandor Trajmar, Redwood City, USA 67 Ion–Molecule Reactions William J. McConkey, Windsor, Canada James M. Farrar, Rochester, USA Isik Kanik, Pasadena, USA

891

Photodetachm 60. Photodetachment

60.1 Negative Ions ...................................... 891 60.2 Photodetachment................................ 60.2.1 Threshold Behavior ................... 60.2.2 Resonance Structure .................. 60.2.3 Higher Order Processes...............

892 892 892 893

60.3 Experimental Procedures...................... 60.3.1 Production of Negative Ions........ 60.3.2 Interacting Beams ..................... 60.3.3 Light Sources ............................ 60.3.4 Detection Schemes ....................

893 893 893 894 895

60.4 Results ............................................... 60.4.1 Threshold Measurements ........... 60.4.2 Resonance Parameters ............... 60.4.3 Lifetimes of Metastable Negative Ions......................................... 60.4.4 Multielectron Detachment ..........

895 895 896 897 898

References .................................................. 898 dersen [60.6], Andersen et al. [60.7] and Bilodeau and Haugen [60.8].

60.1 Negative Ions Interest in negative ions stems from the fact that their structure and dynamics are qualitatively different from those of isoelectronic atoms and positive ions. This can be traced to the nature of the force that binds the outermost electron. In the case of atoms and positive ions, the outermost electron moves asymptotically in the long range Coulomb field of the positively charged core. The relatively strong 1/r potential is able to support an infinite spectrum of bound states that converge on the ionization limit. In contrast, the outermost electron in a negative ion experiences the short-range induceddipole field of the atomic core. The relatively weak 1/r 4 polarization potential is shallow and typically can only support a single bound state. The weakness of the binding is reflected in the magnitudes of electron affinities of atoms, which are numerically equal to the binding energies of the outermost electron in the corresponding negative ion. Electron affinities are typically an order of magnitude smaller than the ionization energies of

atoms. Excited bound states of negative ions are rare. With the possible exception of Os− , all such states that exist have the same configuration, and therefore parity, as the ground state. A rich spectrum of unbound excited states, however, are associated with most ions. These discrete states are embedded in the continua lying above the first detachment limit. Electron correlation plays an important role in determining the structure and dynamics of many-electron systems [60.9]. Weakly bound systems such as negative ions are ideally suited for investigations of the effects of correlation. As a result of the more efficient shielding of the nucleus by the atomic core, the electron–electron interactions become relatively more important than the electron-nucleus interaction in negative ions. The goal of photodetachment experiments is to measure, in high resolution, correlation-sensitive quantities such as electron affinities and the energies and widths of resonant states. These quantities

Part E 60

Investigations of photon-ion interactions have grown rapidly over the past few decades due primarily to the increased availability of laser and synchrotron light sources. At photon energies below about 1 keV the dominant radiative process is the electric dipole induced photoelectric effect. In the gaseous phase the photoelectric effect is referred to as either photoionization (atoms and positive ions) or photodetachment (negative ions). This chapter reviews developments in the field of photodetachment that have taken place over the past decade. The focus will be on acceleratorbased investigations of the photodetachment of atomic negative ions. The monographs of Massey [60.1] and Smirnov [60.2] offer a good introduction to the subject of negative ions. Recent reviews of negative ions and photodetachment include those of Bates [60.3], Buckman and Clark [60.4], Blondel [60.5], An-

892

Part E

Scattering Experiment

provide sensitive tests of the ability of theorists to incorporate electron correlation into their calculations. The stimulating interplay between experiment and

theory continues to help elucidate the role of manyelectron effects in the structure and dynamics of atomic systems.

60.2 Photodetachment

Part E 60.2

Essentially all information about the structure and dynamics of negative ions comes from controlled experiments in which electrons are detached from the ions when they interact with photons or other particles. Photodetachment is the preferred method of studying negative ion structure and dynamics since the energy resolution associated with such measurements is typically much higher than that attainable in any particle-induced detachment process. Generally, one or more electrons are detached from a negative ion following the absorption of one or more photons in the photodetachment process. Most measurements to date, however, involve the simplest process of single electron detachment following single photon absorption. Cross sections for this process start at zero at threshold, rise to a maximum a few eV above threshold and then decrease monotonically. Photodetachment cross sections at their maximum have a typical magnitude of ≈ 10–100 Mb. Threshold behavior and resonance structure in detachment cross sections are of particular interest since they both involve a high degree of correlation between the electrons.

60.2.1 Threshold Behavior Cross sections for photodetachment are zero at threshold, in contrast to the finite value characteristic of photoionization cross sections. The threshold behavior is determined by the dynamics of two particles in the final continuum state. The Wigner law [60.10] governs the energy dependence of the near-threshold cross section for the photodetachment of a single electron from an atomic negative ion. The Wigner law can be written as σ = Ak2l+1 = B(E − E t )l+1/2 ,

(60.1)

where k represents the wavenumber of the detached electron, (E − E t ) is the excess energy of the electron above threshold and l is the smallest value of the orbital angular momentum quantum number. As a result of the electric dipole selection rules, the detached electron is represented, in general, by two

partial waves with l = l0 + 1 and l0 − 1, where l0 is the angular momentum of the bound electron in the negative ion prior to detachment. Wigner demonstrated that for a two-body final state the near-threshold cross section depends only on the dominant longrange interaction between the two product particles. In the case of photodetachment involving electrons with l > 0, this contribution arises from the centrifugal force. Shorter-range interactions, such as the polarization force, will not change the form of the threshold behavior but they will limit the range of validity of the Wigner law. There is no a priori way of determining the range of validity of the Wigner law in any particular experiment. It depends on the strengths of short-range interactions. Measured threshold data is usually fit to the Wigner law in order to determine the threshold energy. In principle, it is possible to extend the range of the fit beyond that of the Wigner law. O’Malley [60.11], for example, considered the effects of multipole forces on threshold behavior. O’Malley’s formalism, however, does not treat polarization explicitly. This is, however, accounted for in the modified effective range theory of Watanabe and Greene [60.12]. Recently, Sandstroem et al. [60.13] have used a modified effective range theory to fit photodetachment data taken at excited state thresholds of the alkali-metal atoms, Li and K. In these cases the dipole polarizability is very high and consequently the range of validity of the Wigner law is correspondingly small.

60.2.2 Resonance Structure Negative ion resonances correspond to states in which an electron and an atom are transiently associated. Such states are the subject of a review by Buckman and Clark [60.4]. In photodetachment they arise when more than one electron, or a core electron, is excited. These unbound discrete states are embedded in the continua above the first detachment limit and are therefore subject to decay via the spontaneous process of autodetachment. The allowed autodetachment process is induced by the relatively strong electrostatic interaction

Photodetachment

60.2.3 Higher Order Processes With the advent of high power, pulsed lasers it became possible to observe multiphoton detachment. In this process a single electron is ejected following the absorption of two or more photons. Early work in this area has been reviewed by Crance [60.17], Davidson [60.18] and Blondel [60.5]. More recently, Haugen and coworkers have used two photon E1 transitions to determine fine structure splittings in the ground state of negative ions and to measure the binding energies of excited states of negative ions that have the same parity as the ground state. Bilodeau and Haugen [60.8] have reviewed these measurements. Multielectron detachment involves the detachment of two or more electrons following the absorption of a single photon. This process, which appears to be initiated by the detachment of an inner shell electron, requires photons with energies higher than can be generated by lasers. Such measurements can be performed at synchrotron radiation sites.

60.3 Experimental Procedures 60.3.1 Production of Negative Ions Negative ions are created in exoergic attachment processes when an electron is captured by an atom or molecule. These quantum systems are weakly bound with diffuse outer orbitals. As a consequence, they are easily destroyed in collisions with other particles. Due to their fragility they are rarely observed in bulk matter. The production of negative ions with a density sufficiently high for spectroscopic studies poses a challenge to the experimentalist since processes involved in their creation must compete with more probable destruction processes. The most versatile source of production of negative ions for accelerator-based experiments is the Cs sputter ion source [60.19]. This source has been used to generate a wide variety of atomic, molecular, and cluster negative ions. Negative ions can be produced and maintained in ion traps [60.20]. In this case, the ions are produced inside the trap by electron-induced dissociative attachment collisions and photodetachment is investigated by monitoring the depletion of the negative ions. The most commonly used source for spectroscopic studies of negative ions is, however, a beam produced by an accelerator. In an accelerator-based apparatus the ions

893

are extracted from the ion source and focused to form a collimated beam that is accelerated to a desired energy, typically 1–10 keV. Mass analysis of the ions is used to produce an elementally and isotopically pure beam that is essentially mono–energetic and unidirectional. The directed particles then drift to the interaction region through a beam line that is maintained at low pressure to minimize destructive collisions between the ions and the residual gas. Recently, negative ions have been injected into storage rings. In this case the ions make repeated passes through the interaction region. The enhanced luminosity associated with multiple-pass experiments makes it possible to investigate relatively rare processes that would be impossible in single-pass experiments.

60.3.2 Interacting Beams The well-defined spatial dimensions of an ion beam readily permit an efficient overlap with a beam of photons. The two interacting beams are most often mated in either crossed or collinear beam geometries. The choice of geometry is typically determined by the types of particles to be detected, the detection geometry to be used, and the level of sensitivity and resolution required in

Part E 60.3

between the outermost electrons. This process causes discrete continuum states to be very short lived. If the selection rules on the allowed Coulomb-induced autodetachment process are violated, however, the state may live much longer. Metastable states eventually decay via autodetachment processes induced by the weaker magnetic interactions. The He− ion is the prototypical metastable negative ion. It is formed in the spin-aligned 1s2s2p 4P 0 state when an electron attaches itself to a He atom in the metastable 1s2s 3S . It is bound by 77.516 meV [60.14]. The decay of a discrete state in the continuum by autodetachment is manifested as a resonance structure in the detachment cross section. The shape of a resonance is determined by the interference between the two pathways for reaching the same final continuum state: direct detachment and detachment via the discrete state embedded in the continuum. A resonance can be parametrized by fitting it to a Fano [60.15] or Shore profile [60.16]. The energy and width of the discrete continuum state are extracted from the fit.

60.3 Experimental Procedures

894

Part E

Scattering Experiment

Part E 60.3

the experiment. The crossed beam arrangement is best suited for spectroscopic studies of the photoelectrons ejected following photodetachment, since the electrons can most easily be collected from a spatially well defined interaction region. In a collinear beam arrangement it is better to detect the residual heavy particles produced in the photodetachment process since they all travel in the same direction, the direction of motion of the ion beam, and can be collected with high efficiency. Both the sensitivity and energy resolution attainable using a collinear beam apparatus are typically much higher than for a crossed beam apparatus. Nowadays, most experiments employ an apparatus in which the photon and ions are collinearly merged and the present chapter will focus on this arrangement. Figure 60.1 shows a typical collinear beam apparatus that was designed by Hanstorp [60.21]. The signal is enhanced when the photon and ion beams are collinearly merged due to the extended interaction region and the high collection and detection efficiencies of the heavy residual particles. The major source of background noise in collinear beam experiments is associated with the production of atoms or positive ions by collisions of the beam ions with the atoms or molecules of the residual gas in the vacuum chamber. By maintaining a high vacuum, typically 10−9 mTorr or better, one can keep the background contribution to a tolerable level. In interacting beam experiments involving the detection of the heavy residual particles, the energy resolution that is attainable is usually limited by kinematic broadening. The amount of broadening is determined by the properties of the ion and photon beams and how they are overlapped. The longitudinal velocity distribution of the ions in a beam is compressed when the ions undergo acceleration after leaving the ion source [60.22]. If the photon beam is merged collinearly with the “cooled” ion beam, the photons sample the narrowed velocity distribution, thus significantly reducing the contribution from Doppler broadening. If kinematic broadening is rendered negligible, the energy resolution is usually determined by the bandwidth of the light source.

60.3.3 Light Sources Since the particle density in the ion beam is typically low, it is important to have a light source that generates an intense beam of photons. In addition, the output of the light source must be tunable. Pulsed lasers are most often used in photodetachment experiments. Their time structure is often used to advantage in timeof-flight schemes to enhance the signal-to-background

σ1, 2

Quadrupole deflector

+

–

–

+

Ion optics

Wien filter

Ion source


3 mm

Interaction region


3 mm –

+

Field ionizer Faraday cup

Positive ion detector σ2, 1

Fig. 60.1 A schematic of a collinear laser-negative ion beam apparatus. The quadrupole deflector is used to merge the laser and ion beam in the interaction region. The first laser is used to photodetach electrons from the ions. A second laser beam is directed along the common path of the first laser beam and the ion beam. This laser is used in the stateselective detection scheme based on resonance ionization. The positive ions produced in the sequential interaction of the negative ions with both laser beams and the external electric field are detected in a channel electron multiplier. The directions of the laser beams can be reversed

ratio. The large peak powers characteristic of pulsed lasers are required in multiphoton experiments. Lasers or laser-based sources used in photodetachment experiments span the wavelength range from the ultraviolet to the infrared. Second harmonic generation in a nonlinear crystal is the conventional method of producing UV radiation. The generation of tunable infrared radiation with wavelengths of a few µm has proven to be more difficult. Recently, however, Haugen and coworkers have performed experiments using infrared radiation produced in a laser-pumped Raman conversion cell [60.8]. Commercial optical parametric oscillators are also becoming more readily available. In order to investigate inner shell excitation and detachment pro-

Photodetachment

cesses it is necessary to access the VUV or X-ray region. These regions are currently outside the limits of lasers and can only be accessed at synchrotron radiation facilities.

60.3.4 Detection Schemes Photodetachment events can be monitored by either measuring the attenuation of the negative ions or by detecting the particles (electrons, atoms or positive ions) produced in the breakup of the ion. In accelerator-based measurements the ion beam is too tenuous to be able to

60.4 Results

895

monitor attenuation and particle detection must be employed. The heavy residual particles, atoms or positive ions, are usually detected in experiments that employ collinearly merged beams of photons and negative ions. The selectivity and sensitivity of a measurement is improved significantly if the residual particles are stateselectively detected. In the case of residual excited atoms, the method most often employed is based on the use of a second laser to excite the atoms to a state near the ionization limit. This resonance step is followed by electric field ionization. The resulting positive ions constitute the signal.

60.4 Results

60.4.1 Threshold Measurements A measurement of a threshold energy using photodetachment allows one to determine the binding energy of the extra electron in the negative ion or, equivalently, the electron affinity of the parent atom. Andersen et al. [60.7] have recently published a review of the methods currently used to measure the binding energies of atomic negative ions. The article includes an up-to-date compilation of recommended electron affinities. The simplest, and potentially the most accurate, method of determining binding energies is the laser photodetachment threshold (LPT) method. In this technique, the normalized yield of residual atoms is recorded as a function of the photon energy in the near-threshold region of the cross section. In most cases, the Wigner law can be fitted to the data and the threshold energy is determined by extrapolation. The Wigner law demonstrates that not all thresholds have the same energy dependence. The most accurate measurements to date involve detachment into an s-wave continuum. In the case of l = 0, the threshold energy dependence of E 1/2 is more pronounced

than for cases with l > 0. S-wave photodetachment requires that a p-orbital electron be ejected. Haugen and coworkers have used tunable infrared spectroscopy to measure the binding energies of negative ions with open p-shells [60.23–25]. In these experiments the detachment process left the residual atom in its ground state so that state-selective detection was not needed. Considerable experimental and theoretical effort have gone into investigating the negative ions of the alkaline earth elements since the experimental discovery [60.26] and subsequent theoretical confirmation [60.27] of the existence of a stable Ca− ion in 1987. Prior to this time it was generally accepted that the closed s-shell configurations of the alkaline earth atoms would inhibit the production of stable negative ions. Andersen et al. [60.28] have reviewed progress in this field. Andersen and coworkers used the LPT method combined with state-selective detection to determine the binding energies of the negative ions of the heavier alkaline earths Ca− , Sr− and Ba− [60.29–31]. No stable negative ions of Be and Mg have been found, but the Be− ion is known to be metastable. These heavier ions are weakly bound but tunable infrared sources were not available at the time to detach them into the ground state of the parent atom. Instead, UV radiation was used to access an excited state threshold. In the case of Ca− the 4s5s 3 S threshold was used since it allowed access to an s-wave continuum. In order to suppress the background noise in the experiment, the Ca atoms left in this excited state following detachment were selectively detected by a method based on resonance ionization. Before the excited Ca atom could radiatively decay, a second laser was used to induce a transition from the excited state to a high lying Rydberg state. The Rydberg atoms were efficiently

Part E 60.4

There have been several new developments in accelerator-based photodetachment measurements during the past decade. Tunable infrared radiation has been used in single photon and multiphoton experiments. State-selective detection schemes based on resonance ionization have been successfully employed in measurements of thresholds and resonances. The lifetimes of long-lived negative ions have been determined by the use of magnetic storage rings. Synchrotron radiation sources have been instrumental in the pioneering studies of inner shell processes in negative ions.

896

Part E

Scattering Experiment

Part E 60.4

ionized in an electrostatic field applied to the beam. The Ca+ ion thus produced were used as the signal that, once normalized, was proportional to the photodetachment cross section. The structures of the heavy alkaline earths are very difficult to calculate. Three relatively loosely bound electrons move in the field of a highly polarizable core. Electron correlation and relativistic effects must be included in a theoretical description of their structure. As calculations became more sophisticated it became clear that correlations between the core electrons and between the valence and core electrons had to be taken into account in addition to the correlations between the valence electrons [60.32]. The negative ions of the alkali-metal elements have a closed s-shell configuration. In this case it is necessary to access an excited state threshold in order to detach into an s-wave continuum. Hanstorp and coworkers have used the LPT method combined with state-selective detection to measure the electron affinities of Li [60.33] and K [60.34]. Since accelerator-based measurements involve the use of fast and unidirectional beams of ions, one must take into account Doppler shifts in accurate measurements of threshold energies. In the K− experiment [60.34], two separate sets of data were accumulated, one with the laser and ion beams co-propagating and the other with them counterpropagating. The Doppler shift can be eliminated to all orders by taking the geometric mean of the measured red-shifted and blue-shifted threshold energies [60.35]. LPT measurements can be used to selectively suppress one isotope relative to other isotopes of the same element, thereby changing the relative abundances from their natural values. This technique could be applied, for example, to the problem of sensitivity enhancement in mass spectrometry by suppressing unwanted isotopic interferences. Sandstroem et al. [60.36] recently performed a proof-of-principle experiment using the 34 S and 32 S isotopes. The goal of the experiment was to enrich the 34 S isotope relative to the more abundant 32 S isotope. Due to the large differential Doppler shifts associated with the fast moving ions of the two isotopes of different masses, it was possible to selectively photodetach one isotope and leave the other untouched. In this feasibility experiment, the 34 S/32 S ration was enhanced by a factor of > 50 over its natural value. With a better vacuum and the selection of a more suitable laser, it is predicted that the enhancement ratio could be significantly improved. The application of LPT to mass spectrometry clearly has the potential for enhancing the sensitivity in measurements of the abundances of rare and ultra-rare isotopes.

60.4.2 Resonance Parameters The simplest negative ion is the two-electron H− ion. This three-body Coulomb system is fundamentally important in our understanding of the role played by electron correlation in atomic structure. The pioneering measurements of the photodetachment of one and two electrons from the H− ion were performed by Bryant and coworkers [60.37–39] several decades ago. The ASTRID (Aarhus storage ring Denmark) heavy ion storage ring has been used in two new measurements of the resonance structure in the vicinity of the H(n = 2) threshold [60.40, 41]. The energy resolution of these storage ring experiments was much higher than that attained in previous experiments. As a consequence, Andersen et al. [60.41] were able to observe, for the first time, a second resonance below the H(n = 2) threshold. In principle, the 1/r 2 dipolar potential should support an infinite series of resonances below each excited state of the H atom [60.42]. Calculations, however, indicate that the series will be truncated after the third member by relativistic and radiative interactions [60.43]. Detachment continua contain a wealth of structure and many measurements of Feshbach resonances in non-hydrogenic negative ions have been reported during the past decade. The dipole polarizability of an atom increases with the degree of excitation, making it easier for electrons to attach to the excited parent atom. Series of Feshbach resonances containing several members are often found below excited state thresholds. Resonances in the photodetachment spectra of the metastable He− ion [60.44] and the alkali-metal negative ions [60.45–49] have been studied extensively by Hanstorp and coworkers using the collinear beam apparatus shown in Fig. 60.1. R-matrix calculations [60.50–53] have generally been successful in predicting the energies and widths of most of the resonances observed in the experiments. There has been keen interest in the similarities and differences between the photodetachment spectra of Li− and H− . The He− ion is a metastable negative ion but it is sufficiently long lived to pass from the ion source to the interaction region with relatively little attenuation via autodetachment. Electric dipole selection rules limit photon-induced transitions from the 1s2s2p 4 P0 ground state to excited states with 4 S, 4 P and 4 D symmetry. The spectra of Feshbach resonances that lie below the He(n = 3, 4, 5) thresholds have been investigated using the collinear beam apparatus shown in Fig. 60.1 [60.44,54,55]. Resonance ionization was used to state selectively detect the residual excited He atoms.

Photodetachment

Figure 60.2 shows a high resolution spectrum of the resonance structure in the range 3.7–4.0 eV, a range that encompasses the H(n = 4) thresholds. In this relatively small energy range Kiyan et al. [60.44] found many resonances exhibiting a variety of different shapes. The resonances labeled a,c,e are members of the 4 P series with dominant configurations of 1s4pnp (n = 4, 5, 6). The resonances labeled b,d appear to be the n = 5, 6 members of the 1s4sns 4 S series. Recent studies using synchrotron radiation have revealed resonances in photodetachment cross sections in the X-ray and VUV regions that can be associated with the excitation of inner shell electrons. Resonances arising from K-shell excitation in the Li− ion have been reported by Kjeldsen et al. [60.56] and Berrah et al. [60.57]. Similarly, resonances were found in a study of He− [60.58] and C− [60.59]. Resonances associated with L-shell excitation of the Na− ion were observed by Covington et al. [60.60]. Figure 60.3 shows

60.4 Results

Cross section (Mb) 7.0 6.0 5.0

2p53s2 2P0

2p53s(3P0)4s2P0

4.0 3.0 2.0 1.0 0.0 30

35

40

45

50 Photon energy (eV)

the range 30–51 eV. Thresholds are indicated by vertical lines. The peaks are resonances associated with the excitation of a 2p core electron accompanied, in most cases, by the excitation of a 3s valence electron

Normalised He+ signal 43P

60

d 3.95

e 3.96

43S

3.97 43P 43D

40

60.4.3 Lifetimes of Metastable Negative Ions

20

ab 0

part of the spectrum in which the dominant feature is a resonance at ≈ 36 eV that arises from the excitation of a pair of electrons – a 2p core electron and a 3s valence electron. Absolute cross sections were measured in most of the experiments. R-matrix calculations of the cross sections at energies corresponding to K-shell excitation have successfully accounted for most of the observations [60.61–63].

3.75

3.80

cd e 3.85

3.90

f

3.95 4.00 Photon energy (eV)

Fig. 60.2 Partial cross section for the photodetachment of

He− via the He(1s3p 3 P)+e(kp) continuum channel in the energy range 3.73–4.00 eV. The open circles represent the measured data. The fits to the sum of Shore profiles are shown by the solid lines. The energies of the resonances obtained from the fits are shown as short vertical lines. The inset shows the region near the He(1s4p 3 P) threshold in finer detail

Heavy ion storage rings are well suited for studies of the radiative or autodetaching decay of long lived excited states of negative ions. They have also been used to investigate the effect of blackbody radiation on weakly bound stable negative ions [60.64]. Andersen and coworkers have used the ASTRID facility to measure the lifetimes of the metastable negative ions Be− [60.65] and He− [60.66] against autodetaching decay. The decay rate was measured by simply detecting the neutral atoms produced in the ring as a function of time after injection. The range of autodetaching lifetimes that can be measured in a storage ring depends on the size of the ring and on the destruction rate of the ions by collisional detachment with the residual gas in

Part E 60.4

Fig. 60.3 Total cross section for the photodetachment of Na− over 80

897

898

Part E

Scattering Experiment

the ring. More recently, Ellman et al. [60.67] have used the CRYRING facility (at the Manne Siegbahn Laboratory in Stockholm) to measure the radiative lifetime of a bound excited state of a negative ion. In this proof-ofprinciple experiment, the lifetime of the 5p 5 2 P1/2 level of Te− was measured to be 0.42( 5) s. This value is in excellent agreement with the result of a multi-configuration Dirac Hartree-Fock (MCDHF) calculation. The J = 1/2 level radiatively decays to the J = 3/2 ground level, primarily via M1 transitions. The idea of the experiment was to monitor the population of the J = 1/2 level as a function of time after injection of the Te− into the ring. This was accomplished by selectively photodetaching ions in the J = 1/2 level as the Te− ions repeatedly passed through the field of a laser beam situated along one arm of the ring. The neutral Te atoms thus produced were used as the signal. Corrections were made for collisionally-induced detachment and repopulation. Data was taken at four different ring pressures. A lin-

ear fit to this data yielded the zero-pressure radiative lifetime of the excited J = 1/2 level.

60.4.4 Multielectron Detachment Synchrotron radiation has been used over the past few years in order to study how negative ions respond to the absorption of high-energy photons. Photons in the VUV and X-ray regions will excite and/or detach inner shell electrons. Multiple electron detachment appears to be initiated by the detachment of a core electron. This process triggers the ejection of one or more valence electrons either by shake off or by interactions of the detached core electron with the valence electrons as it leaves the atom. The ALS (Advanced Light Source) has been used to investigate multiple electron detachment. Measurements of the absolute cross sections for the detachment of two electrons from the closed shell ions Cl− [60.68] and F− [60.69] have been reported.

Part E 60

References 60.1 60.2 60.3 60.4 60.5 60.6 60.7 60.8 60.9 60.10 60.11 60.12 60.13

60.14 60.15 60.16 60.17 60.18 60.19

H. S. W. Massey: Negatives Ions (Cambridge Univ. Press, London 1976) B. M. Smirnov: Negative Ions (Mcgraw-Hill, New York 1972) D. R. Bates: Adv. At. Mol. and Opt. Phys. 27, 1 (1991) S. J. Buckman, C. W. Clark: Rev. Mod. Phys. 66, 539 (1994) C. Blondel: Physica Scripta T58, 31 (1995) T. Andersen: Physica Scripta T43, 23 (1991) T. Andersen, H. K. Haugen, H. Hotop: J. Phys. Chem. Ref. Data 28, 1511 (1999) R. C. Bilodeau, H. K. Haugen: Photonic, Electronic and Atomic Collisions (Rinto Press, New York 2002) U. Fano: Rep. Prog. Phys. 46, 96 (1983) E. P. Wigner: Phys. Rev. 73, 1002 (1948) T. F. O’Malley: Phys. Rev. 137, 1668 (1965) S. Watanabe, C. H. Greene: Phys. Rev. A 22, 158 (1980) J. Sandstroem, G. Haeffler, I. Kiyan, U. Berzinsh, D. Hanstorp, D. J. Pegg, J C. Hunnell, S. J. Ward: Phys. Rev. A 70, 052707 (2004) P. Kristensen, U. V. Pedersen, V. V. Petrunin, T. Andersen, K. T. Chung: Phys. Rev. A 55, 978 (1997) U. Fano: Phys. Rev.A 124, 1866 (1961) B. W. Shore: Phys. Rev. 171, 43 (1968) M. Crance: Comments At. Mol. Phys. 2, 95 (1990) M. D. Davidson, H. G. Muller, H. B. van Linden van den Heuvell: Comments At. Mol. Phys. 29, 65 (1993) R. Middleton: Nucl. Instrum. Methods 214, 139 (1983)

60.20 60.21 60.22 60.23 60.24 60.25 60.26 60.27 60.28 60.29 60.30

60.31

60.32 60.33

D. J. Larson C. J. Edge, R. E. Elmquist, N. B. Mansour, R. Trainham: Physica Scripta T22, 183 (1988) D. Hanstorp, M. Gustafsson: J. Phys. B 25, 1773 (1992) S. L. Kauffman: Opt. Comm. 17, 309 (1976) M. Scheer, R. C. Bilodeau, H. K. Haugen: Phys. Rev. Lett 80, 2562 (1998) M. Scheer, R. Bilodeau, J. Thägersen, H. K. Haugen: Phys. Rev. A 57, 1493 (1998) M. Scheer, R. Bilodeau, C. A. Brodie, H. K. Haugen: Phys. Rev. A 58, 2844 (1998) D. J. Pegg, J. S. Thompson, R. N. Compton, G. D. Alton: Phys. Rev. Lett. 59, 2267 (1987) C. Froese Fischer, J. B. Lagowski, S. H. Vosko: Phys. Rev. Lett. 59, 2263 (1987) T. Andersen, H. H. Andersen, P. Balling, P. Kristensen, V. V. Petrunin: J. Phys. B 30, 3317 (1997) V. V. Petrunin, H. H. Andersen, P. Balling, T. Andersen: Phys. Rev. Lett. 76, 744 (1996) P. Kristensen, C. A. Brodie, U. V. Pedersen, V. V. Petrunin, T. Andersen: Phys. Rev. Lett. 78, 2329 (1997) V. V. Petrunin, J. D. Voldstad, P. Balling, P. Kristensen, T. Andersen, H. K. Haugen: Phys. Rev. Lett. 75, 3317 (1995) S. Salmonson, H. Warston, I. Lindgren: Phys. Rev. Lett. 76, 3092 (1996) G. Haeffler, D. Hanstorp, I. Yu. Kiyan, A. E. Klinkmüller, U. Ljungblad, D. J. Pegg: Phys. Rev. A 53, 4127 (1996)

Photodetachment

60.34

60.35

60.36

60.37

60.38

60.39

60.40

60.42 60.43 60.44 60.45

60.46 60.47

60.48 60.49 60.50 60.51 60.52

60.53 60.54

60.55

60.56 60.57 60.58

60.59 60.60

60.61 60.62 60.63 60.64

60.65

60.66

60.67

60.68

60.69

899

C-N. Liu, A. F. Starace: Phys Rev. A 59, 3643 (1999) A. E. Klinkmüller, G. Haeffler, D. Hanstorp, I. Yu. Kiyan, U. Berzinsh, C. W. Ingram, D. J. Pegg, J. Peterson: Phys. Rev. A 56, 2788 (1997) A. E. Klinkmüller, G. Haeffler, D. Hanstorp, I. Yu. Kiyan, U. Berzinsh, D. J. Pegg: J. Phys. B 31, 2549 (1998) H. K. Kjeldsen, P. Andersen, F. Folkmann, B. Kristensen, T. Andersen: J. Phys. B 34, L353 (2001) N. Berrah: Phys. Rev. Lett. 87, 253002 (2001) N. Berrah, J. D. Bozek, G. Turi, G. Akerman, B. Rude, H.-L. Zhou, S. T. Manson: Phys. Rev. Lett. 88, 093001 (2002) N. D. Gibson et al.: Phys. Rev. A 67, 03070 (2003) A. M. Covington, A. Aguilar, V. T. Davis, I. Alvarez, H. C. Bryant, C. Cisneros, M. Halka, D. Hanstorp, G. Hinojosa, A. S. Schlacter, J. S. Thompson, D. J. Pegg: J. Phys. B 34, L735 (2001) H.-L. Zhou, S. T. Manson, L. Voky, N. Feautrier, A. Hibbert: Phys. Rev. Lett. 87, 02301 (2001) H.-L. Zhou, S. T. Manson, L. Voky, A. Hibbert, N. Feautrier: Phys, Rev. A 64, 012714 (2001) O. Satsarinny, T. W. Gorczyca, C. Froese Fischer: J. Phys. B 35, 4161 (2002) H. K. Haugen, L. H. Andersen, T. Andersen, P. Balling, N. Hertel, P. Hvelplund, S. D. M‘ller: Phys. Rev. A 46, R1 (1992) P. Balling, L. H. Andersen, T. Andersen, H. K. Haugen, P. Hvelplund, K. Taulbjerg: Phys. Rev. Lett. 69, 1042 (1992) T. Andersen, L. H. Andersen, P. Balling, H. K. Haugen, P. Hvelplund, W. W. Smith, K. Taulbjerg: Phys. Rev. A 47, 890 (1993) A. Ellmann, P. Schef, P. Lundin, P. Royen, S. Mannervik, K. Fritioff, P. Andersson, D. Hanstorp, C. Froese Fischer, F. Österdahl, D. J. Pegg, N. D. Gibson, H. Danared, A. Källberg: Phys. Rev. Lett. 92, 253002 (2004) A. Aguillar, J. S. Thompson, D. Calabrese, A. M. Covington, C. Cisneros, V. T. Davis, M. S. Gulley, M. Halka, D. Hanstorp, J. Sandström, B. M. McLaughlin, D. J. Pegg: Phys. Rev.A 69, 022711 (2004) V. T. Davis, A. Aguilar, J. S. Thompson, D. Calabrese, A. M. Covington, C. Cisneros, M. S. Gulley, M. Halka, D. Hanstorp, J. Sandström, B. M. Mclaughlin, G. F. Gribakin, D. J. Pegg: J. Phys. B (to be published)

Part E 60

60.41

K. T. Andersen, J. Sandström, I. Yu. Kiyan, D. Hanstorp, D. J. Pegg: Phys. Rev. A 62, 022503 (2000) P. Juncar, C. R. Bingham, J. A. Bounds, D. J. Pegg, H. K. Carter, R. L. Mlekodaj, J. D. Cole: Phys. Rev. Lett. 54, 11 (1985) J. Sandstroem P. Andersson, K. Fritioff, D. Hanstorp, R. Thomas, D. J. Pegg, K. Wendt: Nucl. Instrum. Methods B 217, 513 (2004) H. C. Bryant, B. D. Dieterle, J. Donahue, H. Sarifian, H. Tootoonchi, D. M. Wolfe, P. A. M. Gram, M. A. Yates-Williams: Phys. Rev. Lett. 38, 228 (1977) D. W. MacArthur, K. B. Butterfield, D. A. Clark, J. B. Donahue, P. A. M. Gram, H. C. Bryant, C. J. Harvey, W. W. Smith, G. Comtet: Phys. Rev. A 32, 1921 (1985) P. G. Harris, H. C. Bryant, A. H. Mohagheghi, R. A. Reeder, C. Y. Tang, J. B. Donahue, C. R. Quick: Phys. Rev. A 42, 6443 (1990) P. Balling, P. Kristensen, H. H. Andersen, U. V. Pedersen, V. V. Petrunin, L. Præstegaard, H. K. Haugen, T. Andersen: Phys. Rev. Lett. 77, 2905 (1996) H. H. Andersen, P. Balling, P. Kristensen, U. V. Pedersen, S. A. Aseyev, V. V. Petrunin, T. Andersen: Phys. Rev. Lett. 79, 4770 (1997) M. Gailitis, R. Damburg: Sov. Phys. JETP 17, 1107 (1963) E. Lindroth, A. Burgers, N. Brandefelt: Phys. Rev. 57, 685 (1998) I. Yu. Kiyan, U. Berzinsh, D. Hanstorp, D. J. Pegg: Phys. Rev. Lett. 81, 2874 (1998) U. Berzinsh, G. Haeffler, D. Hanstorp, A. Klinkmüller, E. Lindroth, U. Ljungblad, D. J. Pegg: Phys. Rev. Lett. 74, 4795 (1995) U. Ljungblad, D. Hanstorp, U. Berzinsh, D. J. Pegg: Phys. Rev. Lett. 77, 3751 (1996) G. Haeffler, I. Yu. Kiyan, U. Berzinsh, D. Hanstorp, N. Brandefelt, E. Lindroth, D. J. Pegg: Phys. Rev. A 63, 053409 (2001) G. Haeffler, I. Yu. Kiyan, D. Hanstorp, B. J. Davies, D. J. Pegg: Phys Rev. A 59, 3655 (1999) I. Yu Kiyan, U. Berzinsh, J. Sandström, D. Hanstorp, D. J. Pegg: Phys. Rev. Lett. 84, 5979 (2000) C. Pan, A. F. Starace, C. H. Greene: J Phys. B 27, 137 (1994) C. Pan, A. F. Starace, C. H. Greene: Phys. Rev. A 53, 840 (1996) C-N. Liu, A. F. Starace: Phys Rev. A 58, 4997 (1998)

References

901

Photon–Atom 61. Photon–Atom Interactions: Low Energy

Theoretical and experimental aspects of the atomic photoelectric effect at photon energies up to about 1 keV are presented. Relevant formulae and interpretations are given for the various excitation and decay processes. Techniques and results of photoelectron spectrometry in conjunction with synchrotron radiation are emphasized.

61.1

Theoretical Concepts ............................ 61.1.1 Differential Analysis .................. 61.1.2 Electron Correlation Effects ......... 61.2 Experimental Methods ......................... 61.2.1 Synchrotron Radiation Source ..... 61.2.2 Photoelectron Spectrometry ....... 61.2.3 Resolution and Natural Width..... 61.3 Additional Considerations .................... References ..................................................

901 901 904 907 907 908 910 911 912

61.1 Theoretical Concepts

γ(hν, jγ =1, πγ =−1) + A(E i , Ji , πi ) → A+ (E f , J f , π f ) + e− [ε, s j, πe =(−1) ] . (61.1)

Conservation laws require that hν + Ei = ε + E f , Ji + jγ = J f + s +  , πi · πγ = π f · πe = (−1) · π f .

(61.2)

Since E f − E i becomes quite large for inner shells or deep core levels, scattering of low-energy photons involves the removal of an electron from a valence or shallow core level. In the low-energy regime, from the first ionization threshold to hν ≈ 1 keV, the photoelectric effect accounts for more than 99.6% of the photon interactions in the elements, with elastic scattering contributing the remainder [61.1, 2]. Ionization by inelastic scattering, the Compton effect, assumes increasing importance with the higher photon energies and the lower Z elements. Above the first ionization potential, the

total photoabsorption cross section and the photoionization cross section are essentially equivalent at the lower photon energies. The cross section σi f for producing a given final ionic state in the photoionization process is given by 4π 2 α2
(61.3) | Ψ f | Tˆ | Ψi |2 , k where k is the photon momentum, Tˆ is the transition operator, Ψi and Ψ f are the wave functions of the initial and final states, and the summation includes an average over all initial states and a summation over all the unobserved variables in the final state. A detailed derivation of this expression, including the different forms for Tˆ , is given in the articles by Fano and Cooper [61.3] and Starace [61.4] and in Chapt. 24. The total cross section is given by the sum of all these different partial cross sections, σi f . σi f =

61.1.1 Differential Analysis Detailed information about the photoionization process can be obtained most directly in emission measurements, especially those involving the photoelectron. The resulting photoelectron spectrum yields the energy and intensity for a given interaction. Further differentiation is obtained by varying the angle of observation and by a spin analysis of the photoelectron. Hence, electron emission analysis can reveal all energetically allowed photoprocesses connecting an initial atomic state i to a final ionic state f and yield their dynamic prop-

Part E 61

Scattering of low-energy photons proceeds predominantly through the photoelectric effect. In this process a photon γ of energy hν, angular momentum jγ = 1, and parity πγ = −1 interacts with a free atom or molecule A, having total energy E i , angular momentum Ji , and parity πi to produce an electron of energy ε, spin s = 1/2, orbital angular momentum , total angular momentum j, and parity πe = (−1) and an ion A+ with final total energy E f , angular momentum J f , and parity π f . This process can be written as the reaction

902

Part E

Scattering Experiment

erties. When averaged over the spin, the differential cross section dσi f / dΩ is given in terms of the partial cross section σi f and an expression involving an expansion in Legendre polynomials of order n with the coefficients Bn : σ 
dσi f if = Bn Pn (cos θ) , (61.4) dΩ 4π n where the angle θ is measured between the direction of the emitted electron and the unpolarized incoming photon beam. In the dipole approximation, which describes the dominant process at low energy, only the terms containing P0 and P2 contribute. Then (61.4) reduces, for a photon beam with linear polarization p, to  σ   dσi f βi f if = (1 + 3 p cos 2θ) , (61.5) 1+ dΩ 4π 4

Part E 61.1

where the angle θ lies in the plane perpendicular to the direction of propagation and is measured with respect to the major axis of the polarization ellipse [61.5]. Then, the differential cross section, or photoelectron angular distribution, is characterized by the single angular distribution or anisotropy parameter βi f for a particular process i → f . For observation at the so-called pseudomagic angle θm , defined as   1 −1 , (61.6) θm = cos−1 2 3p the differential cross section dσi f / dΩ becomes proportional to the angle-integrated, or partial, cross section σi f . In the absence of correlation effects, the partial cross section σi f for the production of an individual final state and the corresponding anisotropy parameter βi f are given by simple expressions derived from a single-particle model [61.1, 4]. For the central field potential, σi f =

4π 2 α 2 a Nn hν 3 o       +1 2 2 × R− + R+ , 2 + 1 2 + 1

(61.7)

where Nn is the occupation number of the subshell, and βi f =

2 + ( + 1)( + 2)R2 ( − 1)R− +  2 2 (2 + 1) R− + ( + 1)R+

−

6( + 1)R+ R− cos ∆ .  2 2 (2 + 1) R− + ( + 1)R+

(61.8)

The subscripts + and − refer to the ( + 1) and ( − 1) channels respectively, and ∆ = δ+ − δ− is the difference in phase shift between these two allowed outgoing waves. The parameter R± is the radial dipole matrix element connecting the electron in the bound orbital with orbital angular momentum  with the outgoing wave having orbital angular momentum  ± 1. Effects of the electron correlation on the direct photoionization process can result in values for β which are not reproduced by the Cooper–Zare expression (61.8) [61.4]. The contribution of the different partial waves to the outgoing wave function can, however, be ascertained through the angular momentum transfer formalism developed by Fano and Dill [61.6]. In this approach, one defines the angular momentum transferred from the photon to the unobserved variables jt as jt = jγ −
= J f + s − Ji ,

(61.9)

where the second portion of the equality results from the conservation of angular momentum. For each allowed value of jt the associated transfer can be defined as either parity favored or parity unfavored according to whether the product πi π f is equal to +(−1) jt or −(−1) jt respectively. (All symbols have the same definition as in (61.1).) Calculation of the partial cross section for the production of a given final state characterized by the values J f and s is then determined from the cross section corresponding to each angular momentum transfer according to
σi f = σ( jt ) . (61.10) jt

The associated anisotropy parameter βi f is derived from a similar sum:

σ( jt )fav β( jt )fav − σ( jt )unfav . σi f βi f = jt =fav

jt =unfav

(61.11)

The second equation derives from the fact that β( jt ) for each parity-favored value must be calculated separately, whereas for the parity-unfavored case β( jt ) = −1 always. The physical effect described by the angular momentum transfer approach is the interaction between an electron and the anisotropic distribution of the other electrons in the atom. Thus, it becomes most useful in the case of ionization from an open-shell atom having an extra electron or a hole in a shell with  = 0. An illustrative example is the 3s ionization of chlorine. Here βi f = 2 identically in (61.8) because only the single

Photon–Atom Interactions: Low Energy

L f Sf Jf

R

[J f ][L f ][S f ] g(, L i , Si , L f , S f ) [1/2][]  2   j=+1/2  1/2 j  
[ j] L i Si Ji . (61.12) ×     j=−1/2 L f Sf Jf =

Here the term in curly brackets is a 9–j symbol, and the notation [J] = 2J + 1 is used. The quantities g(, L i , Si , L f , S f ) are weighting factors determined solely by the initial-state wave function. For the case in which  represents a closed shell, these factors are equal to unity [61.8]. In situations in which the target atoms possess an initial orientation, i. e., have an average value  Jz  = 0, or if the ionization is performed with circularly polarized radiation, the electrons which are produced have a net spin [61.9, 10]. It is also possible that unpolarized atoms which are ionized by unpolarized photons can have a net spin, provided that the detection is carried out at a specific angle, and the ionization is from a given fine-structure component of the initial state to a given fine-structure component of the final state. In the latter

903

case, the transverse spin polarization is given by P=

−2ξ sin θ cos θ , 1 + βP2 (cos θ)

(61.13)

for linearly polarized radiation, and by P=

2ξ sin θ cos θ , 2 − βP2 (cos θ)

(61.14)

for unpolarized radiation [61.11, 12]. The angle θ is the same as in the angular distribution measurement; the parameter ξ is the spin parameter analogous to β; and P2 (cos θ) is the Legendre polynomial of order 2. Yet another parameter which describes the differentiation inherent in the photoionization process is the alignment A, which reflects an anisotropy in the quadrupole distribution of the angular momentum J f of the ion [61.12]. For the cylindrically symmetric coordinate system appropriate to dipole photon excitation, only one moment A0 of the distribution is nonzero. This is defined by    2 m j 3m j − J f (J f + 1) σ(m j )  , (61.15) A0 = J f (J f + 1) m j σ(m j ) where σ(m j ) is the partial cross section for production of a given m j component of J f . A very useful approach to the interpretation of the alignment can be obtained through the angular-momentum-transfer formalism [61.13]. In this approach the angular momentum transfer jt is defined as jt = jγ − J f .

(61.16)

In contrast to the case of the electron angular distribution, it is possible to derive an alignment for each value of jt as a function of the angular momentum J f of the ion. The net alignment is then the incoherent sum of the contributions corresponding to each: 

A0 ( jt )σ( jt ) σ( jt ) (61.17) A0 = jt

jt

If the photoionization produces an ion in an excited state which decays by photoemission, the parameter A0 is reflected either in the angular distribution of the fluorescence photons I(θ) or the linear polarization P measured at one angle, typically 90◦ , according to  1 I(θ) = I0 1 − h (2) A0 P2 (cos θ) 2  3 + h (2) A0 sin2 θ cos(2χ) cos(2η) (61.18) 4

Part E 61.1

value  = 1 is allowed in the single-particle model. However, the three possible values, jt = 0, 1, 2 are allowed, of which only the first corresponds to the Cooper–Zare or single-particle, central field result [61.4]. That β = 2 for 3s ionization of atomic chlorine has been demonstrated experimentally [61.7]. It is generally the case for ionization of elements which are found naturally in the atomic state that there is an equal population in all the fine-structure components of the initial state. This is because of the relatively small energies associated with the fine-structure splitting. (This does not necessarily apply to atomic species generated through a process of molecular dissociation or high-temperature metal vaporization.) Thus, the determination of all cross sections and angular distributions involves an average over these fine-structure components. However, it is possible to generate atoms in which one of the fine-structure components is preferentially populated. In this case there can also be a preferential ionization to a particular J-component of the final ionic state even in the limit in which the electron correlation is neglected, i. e., the geometrical limit. The partial intensities for the production of a given ionic state characterized by the angular momenta L f S f J f by removal of an electron from an orbital  of a state characterized by L i Si Ji are given by

61.1 Theoretical Concepts

904

Part E

Scattering Experiment

or, for θ = π/2 and χ = 0, P=

61.1.2 Electron Correlation Effects 3h (2) A

I(η = 0) − I(η = π/2) 0 = , I(η = 0) + I(η = π/2) 4 + h (2) A0 (61.19)

respectively. The angle θ is the angle at which the fluorescence is determined, and the angle χ is measured between the axis of the polarization selected by the detector and the quantization axis. The polarization of the fluorescence is given by ζ = (cos η, i sin η, 0). The quantity h (2) is a ratio of 6–j symbols depending on the angular momenta J f of the intermediate ion and the final state J f : 

h (2) = (−1) J f −J f    Jf Jf 2 Jf Jf 2 . × 1 1 J f 1 1 Jf (61.20)

Part E 61.1

When it is energetically allowed, a hole in a shallow inner-shell will preferentially undergo Auger decay, emitting an electron with an energy εA determined by the energy difference between the energy E f of the ion and E f of the state of the doubly-charged ion to which the decay occurs. Angular analysis of the Auger electrons reflects the alignment of the intermediate ionic state, which is different from, and does not bear a one-to-one relationship to, the angular distribution parameter β of the photoelectrons. Normally, Auger decay is regarded as a two-step process in which the first step is the production of the hole and the release of the primary photoelectron, followed by the decay and the release of the second electron. Within this approximation [61.14], the angular distribution of the Auger electrons takes on the simple form   I0 I(θ) = [1 + α2 A0 P2 (cos θ)] . (61.21) 4π Here P2 (cos θ) is the second-order Legendre polynomial, and α2 is the matrix element corresponding to the Auger decay. For the specific case in which the Auger decay is to a final ionic state of 1S0 symmetry, α2 is purely geometric, and a measurement of the angular distribution leads directly to a determination of the alignment. Correspondingly, if the alignment of a specific state can be determined through such a decay, then analysis of the angular distribution of the decay to other states provides a value for α2 .

The primary focus of advanced studies in photoionization is to determine the role played by electron correlation in the structure and dynamics of electron motion above the lowest ionization threshold. Because the form of the interaction potential for the Coulomb interaction is very well known, theory can focus on the many-body aspects of the process (Chapt. 23). Electron correlation manifests itself in many ways. Most prominent are the appearance of autoionization structure due to the excitation of one or two electrons, the production of correlation satellites due primarily to the ionization of one electron accompanied by the excitation of another, and the creation of two continuum electrons in a double ionization process. Autoionization resonances are perhaps the oldest known features associated with electron correlation (Chapt. 25). These features arise when the absorption of a photon creates a localized state which lies in energy above at least one ionization limit. This state is then degenerate in energy with a state of an electron in the continuum, and the interaction between these states results in the decay of the quasi-localized state into the continuum. Such resonance states appear in an absorption spectrum in the form of strong, localized variations over an energy range characteristic of the width Γ of the feature, which is in turn related to the lifetime τ of the state by Γ = ~/τ .

(61.22)

In contrast to absorption features between bound states, autoionization resonances are characterized by having an asymmetric line shape. When only one localized state and one continuum are involved, these line shapes can be derived analytically, as first shown by Fano [61.15] and later by Shore [61.16], resulting in simple parametrized forms which are suitable for numerical calculation of overlap integrals for determining widths. For the Fano profile σ() = σa

( + q)2 + σb , 2 + 1

(61.23)

with =

E − Er , (Γ/2)

(61.24)

the parameter q describes the asymmetry of the line, E r is the resonance position, and Γ is the width of the line. The parameters σa and σb reflect the relative contributions to states in the continuum which do and do not interact with

Photon–Atom Interactions: Low Energy

the autoionizing state respectively. The energy E r does not correspond to the peak energy E m of the resonance feature but is related to the maximum through Em = Er +

Γ . 2q

RM =

The Shore profile,

E n = E ∞ − R∞ /(n − µs )2

(61.27)

where n is the principal quantum number and µs is the quantum defect characteristic of a given series and reflecting the short-range electrostatic interactions of the

R∞ 1 + 5.485 799 × 10−4 /(MA − m e )

(61.28)

should be used instead of the value R∞ for infinite nuclear mass (Chapt. 1). The atomic mass MA and the electron mass m e are in a.u. A process closely related to the autoionization phenomenon is resonant Auger decay. This process differs from the ordinary Auger process [61.24] in that an electron from an inner shell is not ionized but excited to either a partially filled or an empty subshell. It may be viewed either as an Auger process or as autoionization. Such an inner-shell excited state of a neutral atom (molecule) lies above one or more of the ionization limits of the singly ionized species and consequently must decay by electron emission unless the decay is forbidden by selection rules. As a result, resonance structure will appear superimposed on the continua of direct photoionization from the various subshells. From a most general point of view, the resonant Auger process can be considered as resonances in the continua of single photoionization, while the ordinary Auger process can be regarded as resonances in the continua of double photoionization. If excitation proceeds to a partially filled subshell within a principal shell, as, for example, Mn 3p → 3d [61.20], interference between the direct photoionization channels and the indirect resonance channel may be strong, and the lineshapes are

30

Cross section (Mb) N+(3P)

25 20 15 10 5 n=3 0 17.5

18.0

4 18.5

19.0

5

6 78

19.5 20.0 20.5 Photon energy (eV)

Fig. 61.1 Autoionization resonances 2s2 2p3 (4S) → 2s2p3 np in atomic nitrogen

Part E 61.1

A + B , (61.26) 2 + 1 describes the same phenomenon except that the interpretation of the parameters A and B is different. In this case, they represent products of dipole and Coulomb matrix elements. C() is the continuum contribution. From an experimental point of view, the parametrized forms for the Fano and Shore profiles are very useful as a basis for fitting autoionization spectra. However, they both have the limitation that they only describe the interaction of an isolated state with the continuum. While they can be extended to include several continuua [61.4], they do not allow for an interaction among two or more localized states [61.17, 18]. Nevertheless, it is possible to use these functions to achieve often good fits of states which do interact with each other, as these functions are mathematical representations of localized resonances in a continuous spectral distribution. If this is done, the parameters no longer have the physical meaning which they have for the noninteracting case. Mixing of discrete ionization channels with competing continuum channels adds complexity to the photoionization process, not just in the classical autoioinization regime but also in the vicinity of inner shells [61.19–21]. In a rigorous application of the Mies formalism [61.17], feasible with modern computer power, even complex experimental spectra can now be satisfactorily interpreted and reproduced. A case in point is the excitation spectrum from the 2p level of the open-shell chlorine atom [61.19]. The process of autoionization is discussed in more detail in Chapt. 25. In Fig. 61.1 an example is shown of the set of 2s2 p3 (4S) → 2s2p3 np, n ≥ 3 autoionization resonances which decay into the 3P ground state of the N+ ion [61.22]. The energies E n of these resonances are related to the ionization limit E ∞ of the series by the Rydberg formula

905

electron with the ion core. Values of µs for s, p, d, and f electrons have been calculated for atoms and ions up to Z = 50 [61.23]. For high precision work, the reduced Rydberg constant

(61.25)

σ() = C() +

61.1 Theoretical Concepts

906

Part E

Scattering Experiment

Part E 61.1

given by (61.23) with arbitrary q values and σa /σb ratios. If, however, the excitation proceeds to an empty shell, as, for example, Mg 2p → ns or n d, n ≥ 4, interference with the direct channels is likely to be negligible, and the resulting resonances are distinguished by essentially Lorentzian line shapes (as for normal Auger lines) with q 1 and σa /σb 1 in (61.23). For a given excitation state a number of resonance peaks may arise because more than one ionization channel is usually available and, in addition, the excited electron can change its orbital from n to n  = n ± 1, 2, . . . in a shakeup or shakedown process [61.25]. As a consequence of the electron–electron interactions which occur simultaneously with the electron– photon interaction, ions are produced in states which do not correspond to those which would be expected based on an interpretation using an independent particle model, which allows for only a single-electron transition (Chapt. 24). Evidence for these states appears as correlation satellites in the photoelectron spectrum, the Auger electron spectrum or the X-ray spectrum [61.26]. Figure 61.2 [61.27] presents as an example the photoelectron spectrum of argon produced by photons with hν =60.6 eV. In addition to the 3s main line of single electron photoionization (and the 3p main lines not shown) numerous satellite lines are seen as the manifestation of two-electron transitions involving ionization-with-excitation correlations. It is convenient to categorize the satellites in a photoelectron spectrum according to various electron correlations, as, for example, initial state interactions which mix dif103 counts / channel 55° 4

3s 13

3

22

10

×3 2

16

1

1

σ ∝ E (2µ−1)/4 ,

27

4

29

6

0 28

ferent configurations into the initial state, and final state interactions, which include core relaxation and electron–electron interactions in the final ionic state, electron–continuum, and continuum–continuum interactions. While initial-state correlations are essentially independent of the photon energy, final-state correlations depend on the energy of the photon through the interactions with the continuum channels. However, the heuristic value of placing correlation effects into a strict classification scheme is limited by the fact that their relative strengths depend on the basis set used in a particular theoretical model and its expansion into a “fully correlated” system within a given gauge [61.3, 4, 28, 29] (see also Chapt. 24). Another manifestation of double-electron processes is the simultaneous excitation of two electrons to bound states. These states may decay by electron or photon emission and are seen as resonance structures above the thresholds of inner-shell ionization or near autoionizing members of Rydberg series. As single or double ionization continua are usually strong in the spectral range of the double excitations, interference occurs, and the lineshapes can display dispersion forms. Typically, the cross section for the sum of all correlation processes is between 10 and 30% of that for single photoionization, but may exceed this range considerably in special cases. Photon scattering near thresholds is complex because of the possibility of strong interactions between the various particles created and the different modes of deexcitation (Chapt. 62). In the case of ionizationwith-excitation processes, the threshold cross section is finite, as it is for single electron photoionization, in accord with Wigner’s theorem [61.30] (Sect. 60.2.1). In the case of double photoionization, the cross section is zero at threshold and then rises according to Wannier’s law [61.31] (Chapt. 52). For the motion of two electrons with essentially zero kinetic energies in the field of the ionic core,

where µ depends on the value of the nuclear charge Z through µ=

31

34

37

40 43 46 Binding energy (eV)

Fig. 61.2 Photoelectron spectrum (PES) of 3s, 3p satellites

in argon at a photon energy of 60.6 eV. Note the reduced intensity of the satellites compared to the 3s main line

(61.29)

1 2



(100Z − 9) (4Z − 1)

1 2

.

(61.30)

For Z = 1 the Wannier exponent has the value 1.127. In the case of Auger decay following ionization at threshold, interaction between the two electrons results in a shift in the energy of the Auger

Photon–Atom Interactions: Low Energy

electron and a corresponding shift in that of the photoelectron (to conserve energy), as well as an asymmetry in the shape of the Auger electron peak and a corresponding asymmetry in the photoelectron peak shape. In this so-called post-collision interaction (Chapt. 62), the lineshape, averaged over angles, has the form [61.32] K(ε) =

(Γ/2π) f(ε) (ε − εA )2 + (Γ/2)2

(61.31)

with f(ε) =

61.2 Experimental Methods

907

   ε − εA πψ exp 2ψ tan−1 . sinh(πψ) (Γ/2) (61.32)

In the above equations, ε is the energy of the Auger line, εA is the nominal Auger energy, Γ is√the initial√holestate width, and the parameter ψ = 1/ 2εe − 1/ 2εA , with εe being the energy of the photoelectron, and εA ≥ εe .

61.2 Experimental Methods An overview of the experimental approaches to the study of photon interactions at low energies is given in Fig. 61.3. The sketch emphasizes the interaction of a polarized photon beam with a small static or particlebeam target of atoms, ions, molecules, or clusters, and the detection of the reaction products at various angles in a plane perpendicular to the direction of propagation of the photon beam, where the general equation (61.5) is valid. Emission products, such as electrons, ions, or photons, may be studied by way of the total Photon monitor (flux, polarization)

yˆ

Total yield detector (ions, electrons, photons) E

Photon source (laser, lamp, plasma, synchrotron)

θ

xˆ

Dispersive element and detector (electron, fluorescence, or ion spectrometer, spin or polarization analyzer) Target source (oven, gas discharge, plasma, static gas, particle beam)

Fig. 61.3 Generic arrangement for detection of particles in an emission measurement. The incoming radiation is assumed to be linearly polarized along the z-axis

61.2.1 Synchrotron Radiation Source The primary source of photons over a broad energy range for experiments in the VUV and soft X-ray region of the spectrum is the synchrotron radiation source [61.39, 40]. In a synchrotron or electron storage ring, radiation is produced as the electrons are bent to maintain the closed orbit. Such bending magnet radiation is emitted in a broad continuous spectrum which begins in the infra-red and ends sharply at a critical photon energy given by hνc = κE e2 , where E e is the energy of the electrons in the ring and κ is a constant characteristic of the ring. Synchrotron radiation can also be generated by introducing additional magnetic field structures [61.41]

Part E 61.2

Dispersive element and detector ˆz

yields, which can be related to the total photoionization cross section, or by differential analysis in a spectrometer according to energy, intensity, emission angle, and polarization. The various particles may be measured independently, simultaneously, or in coincidence. The photon monitor provides the information for normalization of the data with regard to flux and polarization. The photon monitor can also be used for a measurement of the total photoionization cross section, equivalent to the photoabsorption or photoattenuation coefficient at low photon energies. For this purpose, the size of the target source is advantageously increased in the direction of the photon beam. While experimental apparatus differs, sometimes drastically, for the photon sources as well as for the spectrometry of electrons, ions, and fluorescence photons, many features are common, and the relationships of the measured quantities to basic properties of the atoms and the photon–atom interaction are similar. Thus, the following will place emphasis only on the roles of the synchrotron radiation source and photoelectron spectrometry, whereas specific references to other methods can be found elsewhere [61.5, 33–38].

908

Part E

Scattering Experiment

Part E 61.2

into the ring, such as undulators or wigglers, which produce a deviation of the electron motion from a straight path in a well-defined manner. Wiggler radiation has the same spectrum as a bending magnet, except that the critical energy is generally much higher because the effective magnetic fields can be larger than those of the bending magnet. Undulator radiation is very different in that it consists of a sharp spiked profile of about a 1% bandwidth at energies determined by the magnetic field within the undulator and by the electron beam energy in the storage ring. Synchrotron radiation, no matter what the magnetic field structure of the source, requires monochromatization before it can be used for experiments. For the wavelengths of interest in low-energy photon scattering, this can be achieved by using grating instruments with a metallic coating on the grating surface. The highest resolution possible is obtained through the use of a normal incidence monochromator (NIM) with a plane grating set at normal incidence. However, because the reflectivity of the metallic coating at normal beam incidence decreases drastically as the photon energy increases, use of a NIM has an upper limit of about 40 eV. At energies above this, up to about 1 keV, gratings can still be used but must be mounted at grazing incidence. There is a number of functional designs for these grazing-incidence instruments which vary in the shape of the grating – spherical, toroidal, or plane surfaces (SGM, TGM, PGM) – and the associated optics. Above 1 keV, gratings are no longer suitable, and crystal diffraction must be used. While the radiation emerging from a beamline which couples the monochromator to a bending magnet, wiggler, or undulator has a high degree of linear polarization, varying from 80 to 99% in the plane of the electron orbit, a useful flux of circularly polarized radiation can be derived from out-of-plane radiation [61.33], by the use of multiple reflection optics, or from a helical undulator.

61.2.2 Photoelectron Spectrometry The primary particle emitted in photoionization is the photoelectron. Hence, a photoelectron spectrum provides a detailed view of the photon interaction by (a) specifying the individual processes from an initial state i to a final state f by way of the electron energy, (b) determining their differential and partial cross sections by recording the number of electrons as a function of emission angle, and (c) measuring the polarization of the electrons by a spin analysis (spin polarimetry). The experimental approach is governed largely by the rela-

tions (61.3), (61.5), and (61.6). The number of electrons Ni f (e) detected per unit time at an angle θ within an energy interval dε and within a solid angle dΩ is given by dσi f dε (61.33) dΩ where G is a geometry factor, which includes the source dimensions, N(hν) the number of photons, N(A) the number of atoms in the source, f(hν) and f(ε) efficiency factors depending, respectively, on photon and electron kinetic energies, and dσi f / dΩ is the differential cross section for a particular transition i → f . Equation (61.33) assumes that dΩ and dε are sufficiently small that integration over the pertinent parameters is not needed. Since Ni f ∝ dσi f / dΩ, a measurement at two angles, e.g., θ = 0◦ and 90◦ , yields the electron angular distribution parameter βi f according to (61.5), and a measurement at θm (61.6) yields the partial cross section σi f . In the case of closed-shell atoms, n j notation is sufficient to designate single ionization to an ε continuum, e.g., 3p1/2,3/2 → εs or ε d in argon, but for open-shell atoms L SJ notation is required, e.g., 3p5 (2 Po3/2 ) → 3p4 (3 Pe2,1,0 , 1D e2 , 1S e0 )ε(2D e , 2P e , 2S e ) in chlorine. Similarly, for ionization-with-excitation transitions, the final state requires an open-shell designation. The sum of the partial cross sections is equal to the total photoionization or absorption cross section
σtot = σi f (61.34) Ni f (e) = G N(hν)N(A) f(hν) f(ε)

i, f

where the σi f encompass (a) single ionization events in all energetically accessible subshells n j, or the L SJ multiplet components, (b) ionization-with-excitation events (shakeup or shakedown), and (c) double ionization events (shakeoff). All σi f can be determined from a photoelectron spectrum from its discrete peaks (cf. Fig. 61.2) and from the continuum distribution of multiple ionization. However, the latter process is measured more readily by observing the multiply charged ions in a mass spectrometer. The differentiation afforded by measuring the various partial cross sections, and the associated β parameters, can be augmented by differentiating the continuum channels according to the spin using a spin polarimeter [61.33]. In closed-shell atoms, this allows for an experimental determination of the relevant matrix elements and phase shifts, and hence for a direct comparison with theory at the most basic level [61.34, 35]. In a more global measurement, the cross section σtot is

Photon–Atom Interactions: Low Energy

obtained by ion or mass spectrometry from σtot = σ(A+ ) + σ(A2+ ) + σ(A3+ ) + · · · .

(61.35)

Generally, the charge states can be correlated with the various initial photoionization processes if allowance is made for Auger transitions and the fluorescence yield upon exceeding the binding energies of core levels. If the charge states are not distinguished, as in a total ion yield measurement, σtot is obtained directly. Similarly, a direct measurement of the global quantity σtot is obtained by the total electron yield, although care must be exercised to avoid discrimination by angular distribution effects. At photon energies below about 1 keV, ionization, absorption, and attenuation are virtually equivalent, and σtot can also be determined in an ion chamber setup [61.36] or in a photoabsorption measurement in which the number of photons ∆N absorbed in a source of length d and having an atom density n is given by ∆N = N(ph)[1 − exp(−σtot nd)]

(61.36)

61.2 Experimental Methods

geometry of the source volume and the number density. However, once a single absolute value of σtot or any σi f is available, all relative values of the other quantities can be converted to absolute values. An electron spectrometric experiment can be carried out in three different operational modes, as defined in Fig. 61.4. In the most conventional mode, PES, the photon energy is fixed, and a scan of the electron kinetic energy reveals all the electron-emission processes possible and yields their properties. The CIS (constant ionic state) mode is especially suited to follow continuously a selected process as a function of photon energy by locking onto a given state E f − E i (denoted by E B ) which requires a strict synchronization of the photon energy (hν) and electron kinetic energy (ε) during a scan. This mode is particularly advantageous to elucidate resonance features, such as autoionization resonances. Finally, a CKE (constant kinetic energy) scan allows one to access various processes sequentially or, most importantly, follow a process of fixed energy, such as an Auger transition, as a function of photon energy. This description also includes the technique of zero-

with N(ph) being the flux of incident photons. As a rule, in all experiments employing the relation (61.5), the total, partial, or differential cross sections are determined on a relative rather than an absolute scale because it is very difficult to know accurately such factors as the

PES

3s3p6np(1p10) n = 100

E

3s23p5 ε l 3p1⁄2

3p3⁄2

ε2

ε1

n=6 n=5

CIS (EB = hv – ε = const.)

n=4

CKE (ε = hv – EB = const.)

3s np 26.62 eV

EB

CIS – σtot

3p

εl

PES ( hv = EB + ε = const.)

Electron kinetic energy ε

Fig. 61.4 Energy relationship among the three different

operational modes of the technique of photoelectron spectrometry. E B is the binding energy of the level

3s23p6(fs0)

Fig. 61.5 Connection between the PES and CIS techniques as illustrated by the 3s → np autoionization resonances in argon

Part E 61.2

Photon energy, hv

909

910

Part E

Scattering Experiment

kinetic-energy measurements. Most frequently, the PES and CIS modes are employed, and Fig. 61.5 gives a selfexplanatory example of an actual experiment directed at the characterization of the argon 3s → np autoionizing resonances. It should be stressed that the cross section σtot can be partitioned into its components by CIS scans that differentiate between the 3p1/2 and 3p3/2 doublet states. Energy analysis can be performed either by electrostatic energy analyzers or by time-of-flight techniques. The latter is well-suited to those electrons which have very low kinetic energy, including threshold electrons with ε ≈0 eV. Of the electrostatic energy analyzers now in use, two designs are prevalent, the cylindrical mirror analyzer (CMA) and the hemispherical analyzer, where the latter readily lends itself to the application of multichannel detectors.

61.2.3 Resolution and Natural Width

Part E 61.2

The details that can be gleaned from an experiment using photons depend on the resolution achievable with the particular photon source and spectrometry used, the particular excitation or analysis modes and the target conditions chosen, and, ultimately, on the natural width of either the levels or transitions examined as well as any fine structure present. Generally, the instrumental and operational resolution should approach, but need not exceed by much, the natural width inherent in the photoprocess under scrutiny. The demands are most severe for processes involving outer levels because of their typically very narrow widths, and are relatively mild for processes involving inner levels [61.5,35]. It is desirable that in the former case the resolving power (the inverse of the resolution) of the instrument exceed 105 , while in the latter case 104 may suffice. If the target atoms move randomly, a resolution limit is set by the thermal motion, namely ∆ε = 0.723 (εT/M )1/2 (meV) ,

(61.37)

where, in an experiment involving photoelectron spectrometry, ε is the kinetic energy of the photoelectron in eV , T the temperature in K, and M the mass in a.u. of the target atom. This contribution can be limited in first order by employing a suitably directed atomic beam. The experimental peak-width and shape generally contain the natural width; the extent to which instrumental factors enter depends on the specific experiment. In a measurement of the total or partial photoionization

cross section in which either the fluorescence photons, the electrons, or the ions are monitored, the resolution of the photon source (often called the bandpass) is the only instrumental contributor. This applies specifically to the CIS mode of electron spectrometry, in which features are scanned in photon energy and the electron serves solely as a monitor. However, in such a CIS study, or a corresponding fluorescence study, the resolution of the electron or fluorescence spectrometer must be adequate to be able to distinguish adjacent processes. For the example of Fig. 61.5, the 3p3/2 and 3p1/2 levels of Ar need to be separated if the partial photoionization cross sections are to be determined across the resonances. In such a case of more than one open ionization channel, the natural widths of the features will be identical in all channels [61.4], but the shapes may be different. In emission processes subsequent to initial photoionization, namely electron (Auger) decay or photon (X-ray) emission, the resolution of only the spectrometer performing the detection counts on the instrumental side. In the PES mode (Fig. 61.4) the observed lines contain contributions from all sources, the photon source or photon monochromator, the electron analyzer, thermal broadening, and the natural level width. In the special case of photoprocesses near inner thresholds, the postcollision interaction influences the position and shape of photoelectron and Auger lines (Chapt. 62). Excluding threshold regions and the resonant Raman effect, the line profile observed in the various experiments is given by the Voigt function  V(ω, ω0 ) = L(ω − ω)G(ω − ω0 ) dω . (61.38) In this the Lorentzian function L represents the natural level or transition profile, and the Gaussian function G is representative of the window functions of the dispersive apparatus. Although the integral representing the Voigt profile has no analytic form, it can be represented for practical purposes by the analytic Pearson-7 function [61.42] −C  (ε − ε0 )2 P7 (ε) = A 1 + , (61.39) B2 C where A is the peak height, ε0 the peak position, B the nominal half-width-half-maximum of the peak, and C the shape of the peak. In the limit in which C = 1, this function is identically a Lorentzian; in the limit C → ∞, the function is essentially Gaussian. Use of this function allows one to fit the resulting photoelectron spectrum using standard numerical techniques. If the width of

Photon–Atom Interactions: Low Energy

the feature is the only quantity of interest, the simple approximate expression  2 ΓL ΓG = 1− ΓV ΓV

(61.40)

which relates the Voigt width ΓV with the Lorentzian width ΓL and the Gaussian width ΓG can be used to determine either ΓL or ΓG from the measured ΓV [61.5]. Often the observed feature exhibits a dispersive shape given by the Fano or Shore profile. In this instance the instrumental function must be convoluted with the resonance profiles given by (61.23) or (61.26) in order to fit the data and to extract the parameters [61.43].

61.3 Additional Considerations

911

For the special case of resonant Auger decay in which the bandpass of the exciting radiation is very narrow compared with the natural width of the excited state, the experimental linewidth is governed by the width of the exciting radiation, and will be more narrow than the natural width of the line. The resulting lineshape in this resonant Raman effect is then the simple product [61.24] L  (ω, ω0 ) = L(ω)G(ω − ω0 ) ,

(61.41)

where L(ω) is the line profile as determined by the natural width, typically Lorentzian for resonant Auger decay, and G(ω − ω0 ) is the, usually, Gaussian function representing the bandpass of the exciting radiation.

61.3 Additional Considerations

σ   dσi f if = 1 + βP2 (cos θ) dΩ 4π  + (δ + γ cos2 θ) sin θ cos φ ,

using linearly polarized radiation, where P2 (cos θ) is the second-order Legendre polynomial, θ is the angle between the electron emission direction and the electric vector, φ is the angle between the electron and photon directions, β is the angular distribution parameter related to B2 , and the parameters δ and γ are related to B1 and B3 in (61.4). Figure 61.6 shows the geometry for the relationship between the photoelectron momentum vector, the polarization vector, and the photon propagation vector as used in (61.42). It serves as the template for the arrangement and motion of the electron detector, or, for added efficiency and accuracy, several detectors [61.51–53, 56]. The parameters β, δ, and γ have been calculated for most subshells of the noble gases for photoelectron energies between 100 eV and 5 keV [61.55] z Electron (p) Polarization (E) θ

y Photon (k) Φ x

(61.42)

Fig. 61.6 Geometry of the relationship between the photoelectron momentum p, the polarization vector E, and the photon momentum k. (Courtesy of O. Hemmers)

Part E 61.3

Although low-energy photon interactions are well described nonrelativistically in the dipole approximation, relativistic and higher multipole effects which become increasingly important at higher energies cannot be ignored even below 1 keV. Spin-orbit effects [61.44] and relativistic effects [61.45] are of special significance even at low energies in Cooper minima, where one of the transition matrix elements becomes zero. Moreover, the use of intermediate coupling, which includes both the spin-orbit and electrostatic interactions [61.46], is required in open-shell systems, as exemplified for the halogen atoms and atomic oxygen [61.47]. Level energies of heavy elements also require a relativistic treatment [61.48] (Chapt. 22), and it is natural to employ relativistic formulations for calculating the spin parameters appearing in photoionization [61.49]. Although low energy photon scattering is dominated by the dipole contribution, experiments and theory have shown higher multipole effects to be present at hν 10 keV). See also recent comprehensive monographs by Starace [62.22] and Amusia [62.23]. In the present volume, the theory of atomic photoionization for photon energies below 1 keV is summarized in Chapt. 24. Experimental aspects have been reviewed, e.g., by Samson [62.22, 24] and by Siegbahn and Karlsson [62.25]. An extensive compilation of measured X-ray attenuation coefficients or total absorption cross sections, as well as calculated photoelectric cross sections (computed with relativistic Hartree-Slater wave functions), has been published by Saloman et al. [62.26]. Based on the National Bureau of Standards (now the National Institute of Standards and Technology) data base, this work covers the photon energy range from 0.1–100 keV and includes all elements with atomic numbers 1 ≤ Z ≤ 92; an extensive bibliography is included. Very useful tables of theoretical subshell photoionization cross sections covering all elements and the photon energy range 1–1500 keV have been computed by Scofield [62.27]; the atomic electrons are treated relativistically as moving in a Hartree-Slater central potential, and all relevant multipoles as well as retardation effects are included. Accuracy of the results is borne out by systematic comparisons with experiment and detailed tests [62.28]. A useful energy range for applications in electron spectrometry is covered by a tabulation of subshell cross

sections calculated by Yeh and Lindau [62.29] in the dipole approximation with nonrelativistic Hartree-FockSlater wave functions. Because they were obtained in a frozen-core model, these cross sections like those of Scofield [62.27] automatically include all multiple excitations [62.30]. Such multi-electron processes (Sect. 62.4.4) can make substantial contributions, e.g., ≈ 40% of the 1s photoionization in the threshold region of Kr [62.31]. A very useful family of codes being developed by I Grant’s Oxford group is “GRASP – A General Purpose Relativistic Atomic Structure Program”, several versions of which are being distributed through the Computer Physics Communications Program Library (Elsevier Science). The usefulness of photoexcitation and ionization for experimental studies of atomic structure and processes has been greatly enhanced by the advent of synchrotron radiation sources beginning in the 1960’s [62.1, 32, 33]. At first, experiments were performed mostly “parasitically”. Later, they were done with dedicated sources, taking advantage of the high brightness, narrow bandwidth and wide tunability of the radiation (with suitable monochromators, from the visible to the hard X-ray regime), its high degree of polarization and sharp time structure [62.34]. In the 1990’s, “third-generation” sources are being commissioned in which radiation is primarily derived from insertion devices, wigglers and undulators [62.35], entailing further increase of the power of this experimental tool. There is a growing literature on research applications of synchrotron radiation in atomic and molecular physics, including handbooks [62.36–38], proceedings of the series of conferences on X-ray and atomic inner-shell physics [62.39], and mono-

Photon–Atom Interactions: Intermediate Energies

graphs [62.40–43]. Much of the physics described in the remainder of this chapter is being explored with synchrotron radiation.

62.3.2 Compton Scattering Compton scattering denotes the scattering of photons from free electrons. The term is also used for inelastic photon scattering from bound electrons, which approaches the free electron case when the photon energy greatly exceeds the electron binding energy [62.1]. The theory of the process is discussed in standard textbooks [62.17, 44, 45]. Photon scattering from a free electron at rest is expressed, within lowest-order relativistic quantum electrodynamics, by the Klein–Nishina formula. A low energy approximation to the Klein– Nishina cross section is   2~ω σ KN = σ T 1 + + · · · , ~ω  m e c2 , m e c2 (62.7) where σ T is the classical total Thomson cross section, σ T = (8π/3)re2 .

(62.8)

The energy ~ω of a photon scattered through an angle θ is related to the incident photon energy ~ω by the

62.4 Atomic Response to Inelastic Photon-Atom Interactions

919

Compton relation ~ω  .  ~ω = (62.9) 1 + ~ω/m e c2 1 − cos θ The differential cross section for Compton scattering from free electrons, averaged over initial and summed over final photon and electron polarizations, is     dσ KN re2 ω 2 ω ω 2 = − sin + θ . (62.10) dΩ 2 ω ω ω In the limit ω → ω, this reduces to the Thomson differential cross section,  dσ T re2  = 1 + cos2 θ (62.11) dΩ 2 contained in (62.4). Compton scattering from bound electrons has extensive applications in the study of electron momentum distributions in atoms and solids [62.46]. An exact second-order S-matrix code for the relativistic numerical calculation of cross sections for Compton scattering of photons by bound electrons within the independent particle approximation has been developed [62.47]. A systematic treatment of elastic and inelastic photon scattering, starting from many-body formalism, has been described by Pratt, Kissel, and Bergstrom [62.48].

62.4 Atomic Response to Inelastic Photon-Atom Interactions 62.4.1 Auger Transitions

∆L = ∆S = ∆M L = ∆M S = 0 , ∆J = ∆M = 0 , Πi = Π f .

(62.12)

Thus, for example, 2784 electron-electron interaction matrix elements can contribute to the radiationless decay of a 2p3/2 hole state in Hg.

where the quantum numbers n, , j characterize electrons that are identified schematically in Fig. 62.3. The

Part E 62.4

Atomic inner shell vacancy states tend to decay predominantly through radiationless or Auger transitions, which are autoionization processes (see also Chapt. 25 and Chapt. 61) that arise from the Coulomb interaction between electrons [62.1,1,22,49–55]. Radiationless transitions dominate over radiative ones, except for 1s vacancies in atoms with atomic numbers Z < 30, primarily because of the magnitude of the matrix elements and because far fewer channels are allowed by electric dipole X-ray emission selection rules (Sect. 62.4.2, 62.4.3) than by the selection rules for Auger transitions, which in pure L S coupling are

In the simplest approach, ionization and subsequent Auger decay are treated as two distinct steps. This approximation is valid if the electron which is ejected in the ionization process is sufficiently energetic so that it does not interact significantly with the Auger electron, and the interaction of the core hole state with the Auger continuum is weak. Then the hole state can be considered to be quasi-stationary and the decay rate can be expressed according to Wentzel’s ansatz, formulated in 1927 and later known as Fermi’s Golden Rule No. 2 of time dependent perturbation theory [62.49]. In the independent-electron central-field approximation, this leads to the following nonrelativistic matrix element for a direct Auger transition:  2 

 e  ∗ ∗  D= ψn f (1)ψ∞ A j A (2)  r1 − r2  × ψn j (1)ψn j (2) dτ1 dτ2 , (62.13)

920

Part E

Scattering Experiment

∞

lA

jA

n⬘

l⬘

j⬘

n

l

j

n⬙

l⬙

j⬙

D

Table 62.1 Nomenclature for vacancy states. The sub-

scripts in the column headings are the values of j = l ± s, and n is the principal quantum number

E

n

s1/2

1

K

p1/2

p3/2

d3/2

d5/2

2

L1

L2

L3

3

M1

M2

M3

M4

M5

4

N1

N2

N3

N4

N5

f5/2

f7/2

N6

N7

Fig. 62.3 Energy levels involved in the direct (D) and ex-

change (E) Auger processes, and notation for the principal, orbital angular momentum, and total angular momentum quantum numbers that characterize the pertinent electron states

state of the continuum (Auger) electron is labeled by ∞ A jA . In the physically indistinguishable exchange process, described by a matrix element E, the roles of electrons n j and n j are interchanged (Fig. 62.3). The radiationless transition probability per unit time is w fi =

2   2π  D − E ρ E f , ~

(62.14)

Part E 62.4

where ρ(E f ) is the density of final states for the energy E f that satisfies energy conservation. With the continuum wave function normalized to one electron ejected per unit time, we have ρ(E f ) = (2π ~)−1 ([62.22, 52], Appendix E). The matrix elements D and E in (62.14) can be separated into radial and angular factors. Evaluation of the angular factors depends upon the choice of an appropriate angular momentum coupling scheme [62.1, 51, 53], ranging from the (L SJM) representation of RussellSaunders coupling for the lightest atoms in which the spin-orbit interaction can be neglected, through intermediate coupling, to j– j coupling in the high-Z limit. The coupling scheme particularly affects intensity ratios among multiplet components of Auger spectra, which can vary by more than an order of magnitude depending upon its choice. Furthermore, realistic calculations of Auger energies and rates call for the use of relativistic wave functions and inclusion of correlation and relaxation effects [62.1,54]. It is this extreme sensitivity of Auger transitions to the details of the atomic model that makes them an exceedingly useful probe of atomic structure (see also Chapt. 23 and Chapt. 61). Classification of Auger transitions in the central field model is conventionally based on the nomenclature summarized in Table 62.1. The spectroscopic notation s, p, d, f, . . . is employed for orbital angular momentum quantum numbers

= 0, 1, 2, 3, . . . , and shells with principal quantum numbers n = 1, 2, 3, . . . are denoted by K, L, M, . . . . Let an inner shell vacancy be created initially in a subshell X ν and consider a radiationless transition in which an electron from subshell Yµ fills that vacancy and a Z ξ -electron is ejected, in the direct process. This transition is denoted by X ν -Yµ Z ξ , which stands for (n j ) -(n j)(n j ) in terms of the vacancies indicated in Fig. 62.3 Coster–Kronig transitions are a subclass of Auger transitions in which a vacancy “bubbles up” among subshells of the same shell, i. e., X ν -X µ Yξ . These are exceptionally fast because the low energy of the ejected electron tends to prevent cancellations of its wave function overlap with the other factors in the matrix element (62.13) and the overlap of the bound state wave functions can be large owing to their similarity. Coster–Kronig transitions can therefore lead to hole-state widths in excess of 10 eV (lifetimes < 10−16 s) for the L 1,2 subshells of the heaviest elements, for example. McGuire has coined the name super-Coster–Kronig transitions for the (even faster) type X ν -X µ X ξ [62.50]. Here the initial state can no longer be considered quasi-stationary, the width becomes energy-dependent and the two-step model breaks down. The Auger transition energy or kinetic energy of the ejected electron is, within the central field model, E A = E n j − E n j,n j ,

(62.15)

where, in the notation of Fig. 62.3, E n j is the absolute value of the energy of the atom with an n j vacancy, with reference to the neutral atom energy, and E n j,n j is the energy of the atom with two vacancies in the states indicated by the subscripts. This latter energy can be approximated by a mean of measured singleionization energies of the atom under consideration and that with the next higher atomic number (the “Z + 1 rule” [62.49]); with present-day computers it is readily calculated through one of the nonrelativistic [62.56, 57] or relativistic [62.58] self-consistent-field codes.

Photon–Atom Interactions: Intermediate Energies

Photoionization and, even more so, ionization by charged particle impact, often produces more than one vacancy in the target atom (see Sect. 61.1.2 and Sect. 62.4.4). The ensuing Auger spectrum can then exhibit satellite lines shifted in energy, due to the presence of spectator vacancies and electrons, with respect to the diagram lines that arise from the decay of a singly ionized atom [62.1, 50, 53].

62.4.3 Widths and Fluorescence Yields The energy profile of an atomic hole state that decays exponentially with time is found to be the same from quantum as from classical theory [62.68]; it has the resonance or Lorentz shape (~ω − ~ω0 )2 + 14 Γ 2

The decay rate τ −1 = Γ/~ of an atomic hole state is commonly given in units of eV/~ or in atomic units (a.u.) of inverse time, with the corresponding level width Γ in eV or a.u. Thus, for the decay rate, 1 eV = 1 a.u. = 4.134 14 × 1016 s−1 = 27.2116 . τ ~

(62.17)

.

(62.16)

The full width Γ at half maximum of this energy distribution is proportional to the total decay rate τ −1 of the state, in accordance with Heisenberg’s uncertainty principle Γ τ = ~.

Γ (1 s) = 1.73 Z 3.93 × 10−6 eV .

(62.18)

L-subshell widths, as functions of atomic number, exhibit sharp discontinuities that correspond to energetic cutoffs and onsets of intense Coster–Kronig channels (Sect. 62.4.1) [62.69]. If there are several decay channels, their partial widths Γi add:  Γ= Γi . (62.19) i

The main decay channels for inner shell hole states are radiative, with a partial width Γ R , and radiationless or Auger, with a partial width Γ A . The fluorescence yield ωi of a hole state i is defined as the relative probability that the state decays radiatively: ωi ≡ ΓR (i)/Γ (i) .

(62.20)

Macroscopically, for example, the K-shell fluorescence yield ω K is the number of characteristic K X-ray photons emitted from a sample divided by the number of primary K-shell vacancies created in the sample. This ratio rises from 0.04 for Al to 0.97 for U [62.49]. The definition of fluorescence yields of shells above the K-shell is more complicated because the higher shells consist of several subshells and Coster–Kronig transitions (Sect. 62.4.1) can shift primary vacancies to higher subshells before they are filled radiatively. The concept of fluorescence yields is useful in many applications, particularly where the transport of radiant energy through matter is an issue, as in space sciences, medical radiology, and some fields of engineering as well as in physics. Tables of fluorescence yields can be found in the review by Bambynek et al. [62.49] and in the compilations by Krause [62.70] and Hubbell et al. [62.71]. Considerable uncertainty persists in the fluorescence yields of higher shells.

Part E 62.4

The emission of X-rays by atoms ionized in inner shells was studied long before the discovery of the much more probable radiationless transitions, and has played a crucial role in the investigation of atomic structure and interactions with the environment [62.59]. At present, X-ray spectrometry is providing important insights into the structure of highly ionized species produced, e.g., in electron beam ion traps [62.60] or plasmas [62.61], and for the very precise experimental evaluation of inner-vacancy level energies [62.1, 62]. A classical treatment of multipole radiation by atomic and nuclear systems is given by Jackson [62.44], who shows by a simple argument that for atoms the electric dipole transition rate is ≈ (137/Z eff )2 times more intense than electric quadrupole or magnetic dipole emission, where Z eff is the effective (screened) nuclear charge. The quantum theory of atomic radiation is presented in the treatises by Sakurai [62.63], Mizushima [62.64], Berestetskii et al. [62.17], and Sobel’man [62.65], for example. X-ray emission has been reviewed by Scofield [62.50, 66] and, with emphasis on relativistic effects, by Chen [62.1, 54, 60, 67]. In the present volume, radiative transitions are treated in Chapt. 33 to which the reader is referred for details.

I0 (Γ/2π) d(~ω)

921

K -level (1s hole-state) widths increase monotonically with atomic number, from 0.24 eV for Ne to 96 eV for U, closely following the approximate relation [62.49]

62.4.2 X-Ray Emission

I(ω) dω =

62.4 Atomic Response to Inelastic Photon-Atom Interactions

922

Part E

Scattering Experiment

Part E 62.4

62.4.4 Multi-Electron Excitations

62.4.5 Momentum Spectroscopy

In atomic inner-shell photoionization, more than one electron can be excited with significant probability. Final states are thus produced that can be described approximately by removal of a core electron and excitation of additional electrons to higher bound states (shakeup) or to the continuum (shakeoff). These multiple excitation processes exhibit themselves through satellites in photoelectron spectra (Sect. 61.1.2) and in the Auger and X-ray spectra emitted when the excited states decay, as well as in some cases by features in absorption spectra [62.72]. As the photon-electron interaction is described by a one-electron operator, the frozen-core, central field model does not predict changes of state under photon impact by more than one electron. Direct multiple excitation processes are thus a result of electron-electron correlation (Chapt. 23, Sect. 61.1.2). Cross sections for photoexcitation and ionization of two inner shell electrons have been evaluated [62.72] by the multichannel multiconfiguration Dirac–Fock (MMCDF) method [62.73]. Within this model, the correlation effects that cause direct two-electron processes are (i) relaxation or core rearrangement, (ii) initial state configuration interaction, (iii) final ionic state configuration interaction, and (iv) final continuum state configuration, or final state channel interaction. Mechanisms that contribute to two-electron photoionization of the outermost shells of noble gas atoms have been separated within many-body perturbation theory [62.74]. In first order of the combined perturbations by the photon field and electron correlations, the important contributions are core rearrangement, initial state correlations, virtual Auger transitions, and “direct collisions” by the photoelectron with another orbital electron. If treated nonperturbatively, these mechanisms belong within the MMCDF scheme to categories (i)–(iv) described above. The different mechanisms have been found to be of varying importance, depending on the photon energy relative to the double ionization threshold energy, the orbitals which are ionized, and the relative final state energies of the active electrons. Interest in the theory [62.75, 76] has been enhanced by experiments with synchrotron radiation on the double ionization of He [62.77], a process that epitomizes the problem. Above ≈ 5 keV photon energy, the double ionization of He has been shown to be dominated by Compton scattering, not photoeffect [62.78].

We close with a few examples of fruitful recent advances in approaches to the study of electron-atom interactions. A step-function advance in viewing atomic collision dynamics, including photon-electron interactions, arose from the development (first in the University of Frankfurt by Schmidt-Böcking and his group) of Cold-Target Recoil-Ion Momentum Spectroscopy (COLTRIMS), a “momentum microscope” to view the dynamics of photon-atom collisions [62.79–81]. This novel momentum space imaging technique allows the investigation of the dynamics of ion, electron, or photon impact reactions with atoms or molecules. Studies performed with this technique yield kinematically complete pictures of the fragments of atomic and molecular breakup processes, unprecedented in resolution, detail and completeness. The multiple-dimensional momentum-space images often directly unveil the physical mechanism underlying the many-particle transitions that are being investigated [62.82]. Important applications are being developed [62.83, 84].

62.4.6 Ultrashort Light Pulses Advances in laser technology have made it possible to obtain high-intensity (up to 1014 W/cm2 ), short (5 fs) pulses, making laser-atom [62.85] and laser-molecule interactions [62.86–89] amenable to experimental study. In the latter category, nonlinear phenomena such as above-threshold ionization and laser-induced molecular potentials have already been explained theoretically [62.90–92]. Ultrashort light pulses make it possible to follow ultrafast relaxation processes on never-before-accessed time scales and to study light-matter interactions at unprecedented intensity levels [62.85, 93]. Ultrafast optics has permitted the generation of light wave packets comprising only a few oscillation cycles of the electric and magnetic fields. The spatial extension of these wave packets along the direction of their propagation is limited to a few times the wavelength of the radiation. The pulses can be focused to a spot size comparable to the wavelength. Radiation can thus be temporarily confined to a few cubic micrometers, forming a “light bullet”. The extreme temporal and spatial confinement allows moderate pulse energies of the order of one microjoule to result in peak intensities higher than 1015 W/cm2 . These field strengths exceed those of the static field seen by outer-shell electrons in atoms. The laser field consequently is strong enough to suppress the

Photon–Atom Interactions: Intermediate Energies

binding Coulomb potential in atoms and triggers optical field ionization [62.94, 95].   Attosecond 10−18 s trains from high-harmonic generation have been discovered [62.90–92]. High harmonics, generated in a gas jet by a femtosecond laser pulse, have a well-defined phase relation with respect to each other. The relative phases result in strong amplitude beating between the various harmonics. The emerging radiation thus consists of attosecond pulses. The process of harmonic generation can be thought of as consisting of three steps [62.92]: a (rate-limiting) field ionization of the atom at times near the electric field maxima, followed by acceleration of the free electron which, due to the ac character of the driving light, eventually makes the electron recollide at high energy with its parent ion. Radiative recapture of this fast electron into the ground state, strongly favored due to the remaining coherence between the two parts of what was initially the same wave function, completes the harmonic generation process without any change to the atom. In summary, intense few-cycle light pulses open up never yet accessed parameters in high-field physics [62.85]. Strong-field processes induced by fewcycle laser fields may permit access to the phase of the carrier wave and hence to the light fields for the first time. Phase-controlled light pulses may allow control of high-intensity light-matter interaction on a subcycle time scale. The search for laser-driven ultrafast X-ray sources is also drawing benefits from this research area. Substantial major progress can be anticipated in this new field.

A powerful tool for exploring photon-atom interactions has been photoionization of atoms and molecules [62.96]. Analysis of photoelectron angular distribution measurements was routinely performed in the electric-dipole approximation, in which all higher-order multipoles are neglected [62.97]. While breakdown of the dipole approximation at higher photon

energies (above 5 keV) was known, so that a proper description requires inclusion of many multipoles [62.98], experimental limitations appeared to make such extension unnecessary. A notable exception was a hint found by Krause, who noticed deviations from the dipole approximation in measurements on rare gases using unpolarized X-rays [62.99]. Only when more sophisticated radiation sources became available, notably synchrotron radiation, and advances in detector technology took place, did it become possible to gain data on nondipolar photoionization [62.98]. Intense experimental and theoretical work ensued. Photoelectron angular distributions depend upon dynamical parameters of the target atoms and upon the polarization of the photon beam. Peshkin developed a systematic analysis of the restrictions imposed by symmetry, through a geometric description of polarization that is easily visualized and has been found useful in planning experiments [62.100]. Earlier theories predicted nondipolar asymmetries of 1s photoelectron distributions in the point-Coulomb potential. More recently, photoionization amplitudes are calculated in screened potentials, predicting richer Z-dependent and subshell-dependent energy variations of nondipole asymmetries [62.97]. A full relativistic multipole expansion, incuding screening effects, in the independent-particle approximation was discussed by Tseng et al. [62.97]. In addition, leading correction terms to the dipole approximation can be included by adopting the “retardation expansion” of the exponential factor exp(ik · r) [62.101, 102] which, unlike the multipole expansion, includes a finite number of multipoles within the long-wavelength limit [62.103]. A lucid summary of the development of the subject can be found in the paper by Krässig et al.on nondipole asymmetries of Kr 1s photoelectrons [62.104]. Extending the subject, experiments and theory on electric-octupole and pure-electricquadruple effects on soft-X-ray photoemission have been described by Derevianko et al. [62.105].

62.5 Threshold Phenomena During the creation of a core vacancy by photoexcitation or ionization, the escape of the photoelectron from an inner shell involves complex dynamics of electron excitations with multiple correlational aspects. The entire process of inner shell photoionization and

subsequent de-excitation has drastically different characteristics near threshold from that in the high energy limit. In the latter case, discussed in the preceding sections, if an X-ray photon promotes an inner shell electron to an energetic continuum state, the atom first relaxes in

923

Part E 62.5

62.4.7 Nondipolar Interactions

62.5 Threshold Phenomena

924

Part E

Scattering Experiment

its excited (hole) state. In a practically distinct second step, the hole is then filled—most often by a radiationless transition, under emission of an Auger electron with diagram energy that can readily be calculated from the wave function of the stationary, real intermediate state [62.1, 53]. On the other hand, in the vicinity of core-level energy thresholds, atomic photoexcitation and the ensuing X-ray or Auger electron emission occur in a single second-order quantum process, the resonant Raman effect. Here, the intermediate states are virtual and there is no relaxation phase. The resonant Raman regime, comprising the energy region just below threshold, is linked by the post-collision interaction regime (Sect. 62.5.2) above threshold to the (asymptotically) two-step regime. The primary process of photoionization and Auger electron emission can be viewed as resonant double photoionization mediated by a complete set of intermediate states; these correspond to an intermediate virtual inner shell hole and an electron in excited bound or continuum states. A unified treatment of the process, as well as of resonant X-ray Raman scattering, has been developed within the framework of relativistic time independent resonant scattering theory [62.48, 106–108]. As the energy of the incoming photons is increased, Electron energy ε A (eV) 0 10 20 Ii(L3) 3385 5d spectator

Part E 62.5

3380

30

Eexc (eV) 40

Theory Experiment

Energy shift ∆ (eV) 15

5d

10 3375 6d

6d spectator

5

3370 Diagram line

4780

0

εA0

3365 4790

4800

4810

4820 4830 X-ray energy ω (eV)

Fig. 62.4 Energy of the Xe L3 –M4 M5 (1G 4 ) Auger electron peak, as

a function of exciting photon energy. Near threshold Auger satellites, caused by the spectator photoelectron in bound orbits, exhibit linear Raman dispersion. The post-collision interaction shift (right-hand scale) vanishes only in the asymptotic limit (After [62.83])

sweeping through an inner shell threshold, continuous asymmetric electron distributions associated with different two-hole multiplets evolve into Auger-electron lines with characteristic energies. The shapes and energies of the characteristic lines continue to change until the energy of the photoelectron exceeds that of the Auger electron. The nonresonant two-electron emission amplitude is usually negligible in this region, compared with the resonant amplitude.

62.5.1 Raman Processes Resonant Raman transitions to bound final states are the inelastic analog of resonance fluorescence [62.68], with which they share two characteristic features: (1) the emitted radiation can have a narrower bandwidth than the natural lifetime width of the corresponding diagram line, and (2) the energy of the emitted radiation exhibits linear dispersion with excitation energy. Radiative resonant Raman scattering was first observed by Sparks in 1974 [62.109] and subsequently explored with synchrotron radiation by Eisenberger et al.(see [62.48, 48, 108, 110]). Radiationless resonant Raman scattering was identified by Brown et al. [62.111] and in subsequent studies [62.112] of the deformation of Auger-electron lines near threshold (Fig. 62.4). The connection between resonant Raman and Compton scattering has been pointed out by Bergstrom et al. [62.47]. According to a generalization of time independent resonant scattering theory [62.106, 107], a transition matrix element can be constructed that accounts for photoexcitation and either radiative or radiationless deexcitation, including resonance structure [62.48, 108]. The wave functions represent both the atomic and photon fields and include all electrons throughout the process. The conventional two-step model of X-ray fluorescence and Auger electron emission is but a special case of the transition amplitude  

 
(H − E)|ψτν ψτν |Vγ  −  Tβα = ψβεε Vγ + dτ ψα   E − E τν (E) ν (62.21)

where H is the total Hamiltonian, Vγ the photon-electron interaction operator, and E = ~ω1 + E α is the total energy. The first term in Tβα represents direct nonresonant Raman scattering; the resonance behavior is embedded in the second amplitude. The complex eigenenergies E τν (E) satisfy a complicated secular equation that involves diagonal and nondiagonal matrix elements of the level shifts and widths [62.1, 3, 48, 108]. The initial state

Photon–Atom Interactions: Intermediate Energies

wave function ψα is a direct product of the initial atomic state wave function and a one-photon state |k1 ,e1 , where k1 is the wave vector and e1 , the polarization vector. The − final state scattering wave function ψβεε corresponds, in the radiative case, to a scattered photon in a definite state |k2 ,e2  and an electron plus ion in a given quantum state, which may involve the linear momentum ~ ke of the electron if the latter is in the continuum. In the radiationless case, there are an ion and two electrons, both of which may be in a continuum state |ke , ke . The wave − function ψβεε fulfills the ingoing-wave boundary condition with respect to the electron(s) and accounts for final state channel interaction effects. Work on inner shell Raman processes is as yet in its infancy. Synchrotron radiation experiments, interpreted in light of the theory outlined above, promise to become a useful tool in atomic physics and materials science [62.113, 114].

62.5.2 Post-Collision Interaction

ple terms, the attractive potential well in which the slow photoelectron moves is deepened suddenly when the residual singly ionized atom undergoes Auger decay and its net charge changes from +e to +2e. The photoelectron sinks down into this deeper well, there being no time for it to speed up, and the energy lost by the photoelectron is transferred to the Auger electron. The photoelectron slows down and may even be recaptured by the atom from which it was emitted [62.117]. Quantum theoretically, post-collision interaction can be treated from the point of view of resonant scattering theory (Sect. 62.5.1). A lowest-order line shape formula, corresponding to a shakedown mechanism, can be shown to emerge from approximations of the general multichannel transition matrix element [62.118]; the phenomenon thus arises as a consequence of a resonant rearrangement collision in which a photon and an atom in the initial channel turn into an ion and two electrons (one of which has nearly characteristic energy) in the final channel. In this lowest order, the results can be interpreted in terms of an analytic line shape formula based on asymptotic Coulomb wave functions [62.118]. The line shape depends only on the excess photon energy, the lifetime of the initial state of the Auger process, and the change of the ionic charge during Auger electron emission. The interaction between the photoelectron and the Auger electron in the final state can be included by reinterpreting the ionic charge seen by the photoelectron on the basis of asymptotic properties of the continuum wave function pertaining to the two outgoing electrons [62.119, 120]. This results in a procedure which is consistent with semiclassical models that account for the time required for the Auger electron to overtake the photoelectron. Calculations based on this approach agree very well with measurements performed with synchrotron radiation [62.120].

References 62.1

62.2 62.3 62.4 62.5 62.6

L. Kissel, R. H. Pratt: In: Atomic Inner-Shell Physics, ed. by B. Crasemann (Plenum, New York 1985) Chap. 11 L. Kissel, R. H. Pratt, S. C. Roy: Phys. Rev. A 22, 1970 (1980) J. Tulkki, T. Åberg: [62.1, Chap. 10] V. A. Bushuev, R. N. Kuz’min: Sov. Phys. Usp. 20, 406 (1977) K.-I. Hino, P. M. Bergstrom Jr., J. H. Macek: Phys. Rev. Lett. 72, 1620 (1994) For up-to-date atomic data see Photon Interaction Data (EPDL97); Electron Interaction Data

62.7

(EEDL), Atomic Relaxation Data (EADL). Contact for users within the U.S.A.: National Nuclear Data Center (NNDC), Brookhaven National Laboratory, Vicki McLane, [email protected]. Contact for users outside the U.S.A.: Nuclear Data Section (NDS), International Atomic Energy Agency (IAEA), Vienna, Austria, Vladimir Pronyaev, [email protected]. J. H. Hubbell: Radiation physics. In: Encyclopedia of Physical Sciences and Technology, 3rd edn., ed. by R. Meyers (Academic, San Diego 2001)

925

Part E 62

In the radiationless decay of an atom that has been photoionized near an inner shell threshold, the Coulomb field of the receding photoelectron perturbs the Auger electron energy and line shape. This post-collision interaction provides continuity in the energy evolution of inner shell dynamics between the Raman and asymptotic two-step regimes. In Auger decay, following photoionization, the Auger electron initially screens the ionic Coulomb field seen by the receding photoelectron. This screening subsides when the (usually fast) Auger electron passes the slow photoelectron. Distortion of the Auger line shape results, and the Auger energy is raised at the expense of the photoelectron energy (Fig. 62.4). A semiclassical potential curve model of postcollision interaction leads to intuitive insight and surprisingly accurate predictions [62.115, 116]. In sim-

References

926

Part E

Scattering Experiment

62.8

62.9

62.10

62.11

62.12 62.13 62.14 62.15 62.16 62.17

62.18 62.19 62.20 62.21

Part E 62

62.22 62.23 62.24 62.25 62.26 62.27 62.28

62.29 62.30

62.31

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62.32 62.33

62.34 62.35 62.36

62.37 62.38

62.39

62.40 62.41 62.42

62.43

62.44 62.45 62.46 62.47 62.48

62.49 62.50

62.51 62.52

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Photon–Atom Interactions: Intermediate Energies

62.53 62.54 62.55

62.56 62.57 62.58 62.59

62.60

62.61 62.62 62.63 62.64

62.65 62.66 62.67 62.68 62.69 62.70 62.71

62.73 62.74 62.75 62.76 62.77

62.78 62.79

62.80

62.81 62.82

62.83

62.84

62.85 62.86 62.87 62.88 62.89 62.90

62.91 62.92 62.93 62.94 62.95

62.96

62.97 62.98 62.99 62.100

62.101 62.102 62.103 62.104

62.105 62.106 62.107

basch, M. Vollmer, H. Giessen, R. Dörner: Phys. Rev. Lett. 84, 443 (2000) R. Mosshammer: Phys. Rev. A 65, 035401 (2002) L. Cocke: Momentum Imaging in Atomic Collisions. In: ICPEAC XXIII (2003), Physica Scripta Vol. T110, 9 (2004) G. B. Armen, T. Åberg, J. C. Levin, B. Crasemann, M. H. Chen, G. E. Ice, G. S. Brown: Phys. Rev. Lett. 54, 1142 (1985) J. Ullrich, R. Dörner, R. Moshammer, H. Rottke, W. Sander: In: Proceedings of the XVIII International Conference on Atomic Physics, ed. by H. R. Sadgehpour, E. J. Heller, D. E. Pritchard (World Scientific, New Jersey 2003) p. 219 T. Brabec, F. Krausz: Rev. Mod. Phys. 72, 545 (2000) M. Gavrila: Atoms in Intense Laser Fields (Academic, New York 1992) A. D. Bandrauk: Molecules in Laser Fields (Dekker, New York 1994) A. D. Bandrauk, E. Constant: J. Phys. 1, 1033 (1991) A. D. Bandrauk: Phys. Rev. A 67, 013407 (2003) H. G. Muller: Characterization of attosecond pulse trains from high-harmonic generation. In: Proceedings of the XVIII International Conference on Atomic Physics, ed. by H. R. Sadgehpour, E. J. Heller, D. E. Pritchard (World Scientific, New Jersey 2003) p. 209 M. Ferray, A. L’Huillier, X. F. Li, L. A. Lompre, G. Mainfray, C. Manus: J. Phys. B 21, L31 (1988) P. B. Corkum: Phys. Rev. Lett. 71, 1993 (1994) C. Wunderlich, E. Kobler, H. Figger, Th. W. Hänsch: Phys. Rev. Lett. 78, 2333 (1997) L. V. Keldysh: Sov. Phys. JETP 20 (1965) L. V. Keldysh: Electron Spectrometry of Atoms Using Synchrotron Radiation (Cambridge Univ. Press, Cambridge 1997) V. Schmidt: Electron Spectrometry of Atoms Using Synchrotron Radiation (Cambridge Univ. Press, Cambridge 1997) H. K. Tseng, R. H. Pratt, S. Yu, A. Ron: Phys. Rev. A 17, 1061 (1978) B. Krässig: Phys. Rev. Lett. 75, 4736 (1995) M. O. Krause: Phys. Rev. 177, 151 (1969) M. Peshkin: Photon beam polariziation and nondipolar angular distributions. In: Atomic Physics with Hard X-Rays from High-Brilliance Synchrotron Light Sources, ed. by S. Southworth, D. Gemmell (Argonne National Laboratory, Argonne 1996) A. Bechler, R. H. Pratt: Phys. Rev. A 39, 1774 (1989) A. Bechler, R. H. Pratt: Phys. Rev. A 42, 6400 (1990) M. Peshkin: Adv. Chem. Phys. 18, 1 (1970) B. Krässig, J. C. Levin, I. A. Sellin, B. M. Johnson, D. W. Lindle, R. D. Miller, N. Berrah, Y. Azuma, H. G. Berry, D.-H. Lee: Phys. Rev. A 67, 022707 (2003) A. Derevianko: Phys. Rev. Lett. 84, 2116 (2000) T. Åberg: Phys. Scr. 21, 495 (1980) T. Åberg: Phys. Scr. T41, 71 (1992)

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W. Mehlhorn: [62.1[Chap. 4]] M. H. Chen: [62.1[Chap. 2]] W. Mehlhorn: Atomic auger spectroscopy: historical perspective and recent highlights. In: X-Ray and Inner-Shell Processes (American Institute of Physics, Melville 2000) C. Froese Fischer: Comput. Phys. Commun. 14, 145 (1978) C. Froese Fischer: Comput. Phys. Commun. 64, 369 (1991) I. P. Grant: Comput. Phys. Commun. 21, 207 (1980) A. Meisel, G. Leonhardt, R. Szargan: X-Ray Spectra and Chemical Binding (Springer, Berlin, Heidelberg 1989) P. Beiersdorfer: X-ray spectroscopy of high-A ions. In: American Institute of Physics Conference Proceedings, ed. by T. A. Carlson, M. O. Krause, S. T. Manson (American Institute of Physics, New York 1990) p. 648 E. Källne, J. Källne: Phys. Scr. T17, 152 (1987) R. D. Deslattes, E. G. Kessler Jr.: [62.1[Chap. 5]] J. J. Sakurai: Advanced Quantum Mechanics (Addison-Wesley, Reading 1967) M. Mizushima: Quantum Mechanics of Atomic Spectra and Atomic Structure (Benjamin, New York 1970) I. I. Sobel’man: An Introduction fo the Theory of Atomic Spectra (Pergamon Press, Oxford 1972) J. H. Scofield: In: [62.50] Vol. 1, Chap. 6. M. H. Chen: In: [62.60] p. 391 W. Heitler: The Quantum Theory of Radiation, 3rd edn. (Clarendon, Oxfordrd 1954), Chap. V. M. H. Chen, B. Crasemann, K.-N. Huang, M. Aoyagi, H. Mark: At. Data Nucl. Data Tables 19, 97 (1977) M. O. Krause: J. Phys. Chem. Ref. Data 8, 307 (1979) J. H. Hubbell, P. N. Trehan, N. Singh, B. Chand, D. Mehta, M. I. Garg, R. R. Garg, S. Singh, S. Puri: J. Phys. Chem. Ref. Data 23, 339 (1994) S. J. Schaphorst, A. F. Kodre, J. Ruscheinski, B. Crasemann, T. Åberg, J. Tulkki, M. H. Chen, Y. Azuma, G. S. Brown: Phys. Rev. A 47, 1953 (1993) J. Tulkki, T. Åberg, A. Mäntykenttä, H. Aksela: Phys. Rev. A 46, 1357 (1992) T. N. Chang, T. Ishihara, R. T. Poe: Phys. Rev. Lett. 27, 838 (1971) C. Pan, H. P. Kelly: Phys. Rev. A 41, 3624 (1990) K.-I. Hino, T. Ishihara, F. Shimizu, N. Toshima, J. H. McGuire: Phys. Rev. A 48, 1271 (1993) J. C. Levin, I. A. Sellin, B. M. Johnson, D. W. Lindle, R. D. Miller, N. Berrah, Y. Azuma, H. G. Berry, D.H. Lee: Phys. Rev. A 47, R16 (1993) J. A. R. Samson, C. H. Greene, R. J. Bartlett: Phys. Rev. Lett. 71, 201 (1993) R. Dörner, V. Mergel, O. Jagutzki, L. Spielberger, J. Ullrich, R. Moshammer, H. Schmidt-Böcking: Phys. Rep. 330, 96 (2000) Th. Weber, M. Weckenbrock, A. Staudte, L. Spielberger, O. Jagutzki, V. Mergel, F. Afaneh, G. Ur-

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62.114 62.115 62.116

B. Crasemann, T. Åberg: [62.48] C. J. Sparks Jr.: Phys. Rev. Lett. 33, 262 (1974) P. L. Cowan: [62.48] G. S. Brown, M. H. Chen, B. Crasemann, G. E. Ice: Phys. Rev. Lett. 45, 1937 (1980) G. B. Armen, T. Åberg, J. C. Levin, B. Crasemann, M. H. Chen, G. E. Ice, G. S. Brown: Phys. Rev. Lett. 54, 1142 (1985) H. Wang, J. C. Woicik, T. Åberg, M. H. Chen, A. Herrera-Gomez, T. Kendelewicz, A. Mänttykenttä, K. E. Miyano, S. Southworth, B. Crasemann: Phys. Rev. A 50, 1359 (1994) B. Craseman: Can. J. Phys. 76, 251 (1998) A. Niehaus: J. Phys. B 10, 1845 (1977) A. Niehaus, C. J. Zwakhals: J. Phys. BL 16, L135 (1983)

62.117 W. Eberhardt, S. Bernstorff, H. W. Jochims, S. B. Whitfield, B. Crasemann: Phys. Rev. A 38, 3808 (1988) 62.118 J. Tulkki, G. B. Armen, T. Åberg, B. Crasemann, M. H. Chen: Z. Phys. D 5, 241 (1987) 62.119 W. Eberhardt, G. Kalkoffen, C. Kunz: Phys. Rev. Lett. 41, 156 (1978) 62.120 G. B. Armen, J. Tulkki, T. Åberg, B. Crasemann: Phys. Rev. A 36, 5606 (1987) 62.121 Nucl. Instrum. Methods A 261, 1 (1987) 62.122 Nucl. Instrum. Methods A 266, 1 (1988) 62.123 Nucl. Instrum. Methods A 282, 369 (1989) 62.124 Nucl. Instrum. Methods A 291, 1 (1990) 62.125 Nucl. Instrum. Methods A 303, 397 (1991) 62.126 Nucl. Instrum. Methods A 319, 1 (1991)

Part E 62

929

Electron–Atom

63. Electron–Atom and Electron–Molecule Collisions

Electron–atom and electron–molecule collision processes play a prominent role in a variety of systems ranging from discharge or electronbeam lasers and plasma processing devices to aurorae and solar plasmas. Early studies of these interactions contributed significantly to the understanding of the quantum nature of matter. Experimental activities in this field, initiated by Franck and Hertz [63.1], flourished in the 1930s and, after a dormant period of about a quarter of a century, have had a renaissance in recent years. When electrons collide with atomic or molecular targets, a large variety of reactions can take place (see Sect. 63.1.1). We limit our discussion to electron collisions with gaseous targets, where single collision conditions prevail. Furthermore, we discuss only low-energy (threshold to few hundred eV) impact processes where the interaction between the valence-shell electrons of the target and the free electron dominates. Comprehensive discussions on electron–atom (molecule) collision physics can be found in the books of Massey et al. [63.2], McDaniel [63.3] and the volumes of Advances in Atomic and Molecular (since 1990 Atomic, Molecular and Optical) Physics. The latest developments are usually published in

63.1 Basic Concepts..................................... 63.1.1 Electron Impact Processes .......... 63.1.2 Definition of Cross Sections......... 63.1.3 Scattering Measurements ...........

929 929 929 930

63.2 Collision Processes ............................... 63.2.1 Total Scattering Cross Sections..... 63.2.2 Elastic Scattering Cross Sections... 63.2.3 Momentum Transfer Cross Sections ................................... 63.2.4 Excitation Cross Sections ............ 63.2.5 Dissociation Cross Sections ......... 63.2.6 Ionization Cross Sections ............

933 933 933 933 933 935 935

63.3 Coincidence and Superelastic Measurements..................................... 936 63.4 Experiments with Polarized Electrons .... 938 63.5 Electron Collisions with Excited Species ............................ 939 63.6 Electron Collisions in Traps ................... 939 63.7 Future Developments ........................... 940 References .................................................. 940 Physical Review A, Journal of Physics B, Journal of Chemical Physics and Journal of Physical and Chemical Reference Data, and are presented at the biannual International Conference on Photonic, Electronic and Atomic Collisions (ICPEAC).

63.1.1 Electron Impact Processes Lox energy electrons can very effectively interact with the valence-shell electrons of atoms and molecules, in part because they have similar speeds. In elastic scattering, the continuum electron changes direction and transfers momentum to the target. Inelastic collisions include also a transfer of kinetic energy to the target, such as excitation of valence electrons to discrete energy levels, to the ionization continuum, and, in the case of molecules, excitation of nuclear motion (rotational,

vibrational) and excitation to states which dissociate into neutral or ionic fragments. Various combinations of these processes are also possible, e.g., dissociative attachment, excitation or ionization. Excitation of more than one valence electron at the same time, or excitation of electrons from intermediate and inner shells, may also occur but these processes are more likely at impact energies of a few keV. These excitations lie above the first ionization limit and lead, therefore, with high probability, to autoionization, except for heavy elements where X-ray emission is an important competing process.

Part E 63

63.1 Basic Concepts

930

Part E

Scattering Experiment

63.1.2 Definition of Cross Sections The parameters which characterize collision processes are the cross sections. Electron collision cross sections depend on impact energy E 0 and scattering polar angles θ and φ. The differential cross section, for a specific well-defined excitation process indicated by the index n is defined as dσn (E 0 , Ω) k f = | f n (E 0 , Ω)|2 , (63.1) dΩ ki where Ω is the polar angle of detection, ki and k f are the initial and final electron momenta, and f n is the complex scattering amplitude (n = 0 refers to elastic scattering). Integration over the energy-loss profile is assumed. If the energy-loss spectrum is broad, differentiation with respect to energy loss also has to be included. For certain processes it may be necessary to define differential cross sections with respect to angle and energy for both primary and secondary particles. Integration over all scattering angles yields the integral cross sections
2π
π σn (E 0 ) = 0

dσn (E 0 , Ω) sin θ dθ dφ . dΩ

(63.2)

0

In the case of elastic scattering, the momentum transfer cross section is defined as
2π
π σ0M (E 0 ) =

dσ0 (E 0 , Ω) (1 − cos θ) sin θ dθ dφ . dΩ

0 0

an averaging over the finite energy and angular resolution of the apparatus. It is important, therefore, to specify clearly the nature of the measured cross section, otherwise their utilization and comparison with other experimental and theoretical cross sections become meaningless. We denote the conventionally measured differential and integral cross sections by Dn (E 0 , θ) and Q n (E 0 ), with the various averagings implied. Similarly, Q M (E 0 ) and Q tot (E 0 ) are the corresponding momentum transfer and total scattering cross sections. The collision strength for a process i → j, which is the particle equivalent of the oscillator strength, is defined by Ωij (E 0 ) = qi E 0 σij (E 0 ) ,

(63.5)

where qi is the statistical weight of the initial state [qi = (2L i + 1)(2Ji + 1)], E 0 is in Rydbergs and σij is in units of πa02 . The rate for a specific collision process (e.g., excitation) for electrons of energy E 0 is given as Rij (E 0 ) = NI(E 0 )σij (E 0 ) ,

(63.6)

 −3  where N is the density of the target particles m ,  and I(E 0 ) is the electron flux m−2 s−1 ; σij is in m2 , yielding Rij in units of m−3 s−1 . For nonmonoenergetic electron beams, (63.6) must be integrated over E 0 to get the average rate.

63.1.3 Scattering Measurements Most scattering experiments, are carried out in a beam– beam arrangement (Fig. 63.1). A beam of nearly

(63.3)

Part E 63.1

The total electron scattering cross section is obtained by summing all integral cross sections:   σtot (E 0 ) = σn (E 0 ) + σm (E 0 ) , (63.4) n

Energy analyzer Detector

m

where σm are the cross sections for other possible channels. Experimental cross sections typically represent averages over indistinguishable processes (e.g., magnetic sublevels, hyperfine states, rotational states etc.). The cross section obtained this way corresponds to an average over initial and sum over final indistinguishable states with equal weight given to the initial states. (This may not always be true, as discussed later.) If the target molecules are randomly oriented, the cross section averaged over these orientations is independent of φ. In addition, there is

Signal Optics

Optics θ Energy selector Gun

Photons Electrons Ions

Fig. 63.1 A schematic diagram for electron scattering measurements

Electron–Atom and Electron–Molecule Collisions

931

c/s He 90° Er = 1.2 eV

250 200

4

n=2

150 5

6

IE

100 ×8

50

3

0 20

21

22

23

24 25 Energy loss (eV)

Fig. 63.2 Energy-loss spectrum of He at a constant residual energy of 1.2 eV and scattering angle of 90◦ . The inelastic features with the corresponding principal quantum numbers are shown. IE is the ionization potential of He (24.58 eV). No background is subtracted and true signal zero is indicated by a dotted line under the expand portion of the spectrum m. (Taken from Allen [63.6])

ple, taken from the work of Allan [63.6], ist shown in Fig. 63.2. The energy loss spectrum becomes equivalent to the photoabsorption spectrum in the limit of small momentum transfer K , where K = k f − ki (i. e., high impact energy, small scattering angle). The equivalence of electrons and photons in this limit follows from the Born approximation, and it can be used to obtain optical absorption and ionization cross sections. The correspondence is defined through the Limit Theorem opt

lim f nG (K ) = f n

K →0

f nG (K ) =

,

∆E ki 2 dσn (k) , K 2 kf dΩ

(63.7) (63.8)

where f nG is the generalized oscillator strength for opt excitation process n, and f n is the corresponding optical f -value. Equation (63.7) was originally derived by Bethe [63.7] from the Born cross section. It was extended later by Lassettre et al. [63.8] to cases where the Born approximation does not hold. In app this case, f nG (K ) is replaced by f n (K ), the apparent generalized oscillator strength. The Limit Theorem implies that, in the limit of small K , optical selection rules apply to electron impact excitation. The practical problem is that the limit is nonphysical and the extrapolation to zero K involves some arbitrariness. When K is significantly different from zero,

Part E 63.1

monoenergetic electrons is formed by extracting electrons from a hot filament and selecting a narrow segment of the thermal energy distribution. For the formation and control of the electron beam, electrostatic lenses are used and the energy selection is achieved with electrostatic energy selectors. A magnetic field may also be applied to obtain a magnetically collimated electron beam. The target beam is formed by letting the sample gas effuse from an orifice, tube or capillary array with various degrees of collimation. Target species which are in the condensed phase at room temperature need to be placed in a crucible and evaporated by heating. Extensive discussion of this technique has been given by Scoles [63.4]. The electron beam intersects the target beam at a 90◦ angle and electrons scattered direction, over  into a specific  a small solid angle ≈ 10−3 sr , are detected. However the scattered electron is not necessarily the same as the incoming electron. Exchange with the target electrons may occur, and is required for spin-forbidden transitions in light elements. The detector system consists of electron lenses and energy analyzers similar to those used in the electron gun. The actual detector is an electron multiplier which generates a pulse for each electron. In the scattering process, secondary species (electrons, photons, ions, neutral fragments) may also be generated and can be detected individually or in various coincidence schemes. The experiments are carried out in a vacuum chamber and it is important to minimize stray electric and magnetic fields. More details about the apparatus and procedures can be found in a review by Trajmar and Register [63.5]. The primary information gained in these experiments is the energy and angular distribution of the scattered electrons. There are several methods used to carry out scattering measurements. In the most commonly used energy-loss mode, the impact energy and scattering angle are fixed, and the scattering signal as a function of energy lost by the electron is measured by applying pulse counting and multichannel scaling techniques. The result of such an experiment is an energy-loss spectrum. The elastic scattering feature appears at zero energy loss; the other features correspond to various excitation processes and to ionization. Energy-loss spectra can also be generated in the constant residual energy mode. In this case, the detector is set to detect only electrons with a specific residual energy E R = E 0 − ∆E at a fixed scattering angle, and E 0 and ∆E are simultaneously varied. Each feature in the energy-loss spectrum is obtained then at the same energy above its own threshold. An exam-

63.1 Basic Concepts

932

Part E

Scattering Experiment

103 counts / channel

Intensity (arb. units) He E0 = 40.1 eV 6

4

125°

He θ = 90° ∆E = OeV

2 × 50

50°

0 19.2

× 20

5° 0.6

0.8

20

0.4

0.6

0.8

19.4

19.5 E0 (eV)

Fig. 63.4 The 19.37 eV He resonance observed in the elas-

tic channel at 90◦ scattering angle

× 20 0.2

19.3

21 0.2 0.4 Energy loss (eV)

Fig. 63.3 Variation of energy-loss spectra (and DCSs) with

scattering angle for He at 40 eV impact energy. Spectra are shown at 5◦ , 50◦ and 125◦ scattering angles

Part E 63.1

optical-type selection rules do not apply to electronimpact excitation. As can be seen from Fig. 63.3, spin and/or symmetry forbidden transitions then readily occur and can be an efficient way of producing metastable species. Selection rules for electron impact excitations can be derived from group theoretical arguments [63.9, 10]. For atoms, the selection rule Sg ↔ Su applies in general and scattering to 0◦ and 180◦ is forbidden if (L i + Πi + L f + Π f ) is odd. Here, L i and L f are the angular momenta and Πi and Π f are the parities. For molecules, selection rules can be derived under two special conditions: (a) rules concerning 0◦ and 180◦ scattering for arbitrary orientation of the molecule, and (b) rules concerning scattering to any angle but for specific orientation of the molecule. An important example of the first case is the Σ − ↔ / Σ + selection rule for linear molecules at 0◦ and 180◦ scattering angles.

The energy dependence of cross sections is obtained by fixing the energy-loss value (scattering channel) and studying the variation of scattering signal with impact energy at a given angle or integrated over all scattering angles. In general, cross sections associated with spin forbidden and optically allowed transitions peak near and at several times the threshold impact energy respectively, and they usually vary smoothly with impact energy. However, resonances may appear at certain specific impact energies. These sudden changes are associated with temporary electron capture and are the result of quantum mechanical interference between two indistinguishable paths. An example is shown in Fig. 63.4 for He in the elastic channel at 19.37 eV impact energy. Integral cross sections can be obtained from extrapolation of the measured Dn (E 0 , θ) to 0◦ and 180◦ scattering angles and integration over all angles. Recently, incorporation of an “angle-changing” device has enabled measurements to be extended over the whole range of scatering angles [63.11, 12]. In certain cases it is possible to measure integral cross sections directly by detecting secondary products such as photons and ions. These procedures and the resulting cross sections will be discussed in some detail in Sect. 63.2.

Electron–Atom and Electron–Molecule Collisions

63.2 Collision Processes

933

63.2 Collision Processes In addition to the basic elastic and inelastic processes defined in Sect. 63.1.2, we now also explicitly include dissociation (to neutral and charged fragments) cross sections Q D (E 0 ); and ionization cross sections Q I (E 0 ). Each of these is now considered separately.

Cross section (10–16 cm2) 10 Helium Qr Qo Ql Qn = 2

1

63.2.1 Total Scattering Cross Sections Total electron scattering cross sections represent the sum of all integral cross sections:  Q n (E 0 ) + Q I (E 0 ) + Q D (E 0 ) , Q tot (E 0 ) =

0.1

21P 23S + 21S 0.01

n

(63.9)

Q tot (E 0 ) are useful for checking the validity of scattering theories, and the consistency of available data, for normalization of integral and differential cross sections, and as input to the Boltzmann equation. At low impact energies, elastic scattering dominates, while at intermediate and high impact energies, electronic excitations and ionization become major contributors to Q tot . Figure 63.5 shows the various cross sections for electron–helium collisions. The data are from the recommended values of Trajmar and Kanik [63.13]. Two methods are commonly used for measuring Q tot (E 0 ): the transmission method and the target recoil method (for details see [63.5,14]). Total scattering cross sections measured by these techniques are, in general, accurate to within a few percent. The extensive reviews by Zecca and co-workers [63.15–17] should be noted.

63.2.2 Elastic Scattering Cross Sections

1

10

100

1000 Energy (eV)

Fig. 63.5 Cross sections for various processes in the electron–helium collision (see text for data sources)

malize the D0 (E 0 , θ) to the absolute scale. We briefly outline here only the most commonly used procedure. The most practical and reliable method of obtaining the absolute D0 (E 0 , θ) is the relative flow technique in which scattering signals for a known standard gas and an unknown test gas are compared at each energy and angle [63.5, 18–20]. The He elastic cross section is the natural choice of standard since it is known accurately over a wide energy and angular range, and He is experimentally easy to handle. Only the relative electron beam flux and molecular beam densities (and their distributions) need be known in the two measurements. The flow rate of the test gas is adjusted so that the flux and density distribution patterns of the two gases are identical, and all geometrical factors cancel in the scattering intensity ratios. The absolute D0 (E 0 , θ) for the sample gas is obtained from the measured scattering intensity, target density, and electron beam intensity ratios and the standard D0 (E 0 , θ) value. See [63.5, 20, 21] for a detailed discussion of this technique.

63.2.3 Momentum Transfer Cross Sections Q M (E 0 ) can be obtained both from the elastic DCSs and from swarm measurements. At low electron-impact energies (from 0.05 to a few eV), where only a few collision channels are open, the electron swarm technique is the most accurate (≈ 3%) way to determine the momentum transfer cross sections. Beam–beam experiments are mandatory at higher energies. A detailed

Part E 63.2

Elastic scattering cross sections Q 0 (E 0 ) are not as readily available as Q tot (E 0 ). They are obtained from differential scattering experiments over limited angular ranges by extrapolation and integration of the measured values. Typical error limits are 5 to 20%. For molecular species rotational excitation is usually not resolved but is included in the D0 (E 0 , θ) and Q 0 (E 0 ) values. In order to obtain the absolute D0 (E 0 , θ) directly from the scattering signal, one has to know the electron flux, the number of scattering species, the scattering geometry and the overall response function of the apparatus. A direct measurement of these parameters can be made at high energies (> 100 eV). However, at low electron energies, this approach is not feasible. A number of methods have been devised to derive relative D0 (E 0 , θ) from the measured scattering intensities and then to nor-

0.001

934

Part E

Scattering Experiment

discussion of these techniques is given by Trajmar and Register [63.5].

63.2.4 Excitation Cross Sections Dn (E 0 , θ) and Q n (E 0 ) can be derived from energy-loss spectra obtained in beam–beam scattering experiments. The relative Dn (E 0 , θ) is usually normalized to D0 (E 0 , θ) which in turn is normalized to the helium D0 (E 0 , θ) by the relative flow technique described in Sect. 63.2.2. There are, however, complications and uncertainties associated with this technique because of the sensitivity of the instrument response function to the residual energy of the scattered electrons. For more details, see Trajmar and McConkey [63.21]. Data obtained by this procedure are rather limited, partly due to experimental difficulties and partly due to the time required to carry out such measurements. For cases where an excited state j is formed which can radiatively decay by means of a short-lived (dipoleallowed) transition to a lower lying state i, the intensity of the resultant radiation is directly related to the cross section for production of the excited state in the original collision process. An optical emission cross section, Q ji (E 0 ), is defined by Q ji (E 0 ) =

N j Γ ji , In 0 τ j

(63.10)

Part E 63.2

where N j and n 0 are the densities in the excited and ground states, respectively, Γ ji is the branching ratio for radiative decay from state j to state i, I is the electron beam flux, and τ j is the natural radiative lifetime of state j. Since the excited state may be produced either by direct electron impact or by cascade from higherlying states k, also formed in the collision process, we may define the direct excitation cross section Q dj (E 0 ) by   Q ji (E 0 ) − Q k j (E 0 ) . (63.11) Q dj (E 0 ) = i

k

The last term subtracts the cascade contribution from  higher lying states. The quantity Q aj (E 0 ) = i Q ji (E 0 ) is known as the apparent excitation cross section for level j. Clearly, to obtain Q dj (E 0 ) from Q aj (E 0 ), the cascade contribution must be known. In (63.10), N j Γ ji /τ j gives the steady state number of j → i photons per unit time per unit volume emitted from the interaction region. Since observation is made in a particular direction care must be taken to correct for any anisotropy in the radiation pattern. Alternatively, if observation is made at the so-called “magic” angle

(54◦ 44 ) to the electron beam direction, the emission intensity per unit solid angle is equal to the average intensity per unit solid angle irrespective of the polarization of the emitted radiation. However, even at this magic angle, care must be taken to avoid problems with polarization sensitivity of the detection equipment [63.22, 23]. The phenomenon of radiation trapping is often a problem if the radiative decay channels of the excited state include a dipole allowed channel to the ground state. Repeated absorption and re-emission of the radiation can occur and can lead to a diffuse emitting region much larger than the original interaction region, and the polarization of the emitted light can also be altered. Often a study of the variation of the emitted intensity or polarization with the target gas pressure is sufficient to reveal the presence of radiation trapping or other secondary effects. The emission cross sections of certain lines have been measured with great care and now serve as bench-marks for other work. Examples of these are the measurements of van Zyl et al. [63.24] on the n 1S levels of He in the visible spectral region or the measurements of the cross section for production of Lyman α from H2 in the VUV region (see [63.25] for a full discussion of this including many references). Use of secondary standards is particularly important when crossed-beam measurements are being carried out because of the cancellation of geometrical and other effects which occur. The Bethe–Born theory [63.26] provides a convenient calibration of the detection system for optically allowed transitions of known oscillator strength. At sufficiently high energies, the excitation cross section, Q n , of level n is given by opt 4πa02 fn ln(4cn E 0 /R∞ ) . (63.12) E 0 /R∞ ∆E n /R∞ Here ∆E n is the excitation energy, and cn is a constant dependent on the transition. A plot of Q n E 0 versus ln E 0 opt is a straight line with a slope proportional to f n and the intercept with the ln E 0 axis yields an experimental value for cn independent of the normalization. For example, the He n 1P –11S optical oscillator strengths are very accurately known, as are cascade contributions. Thus accurate normalization of the slope of the Bethe plot can be made, yielding accurate excitation cross sections. As mentioned above, the excitation cross sections display characteristic shapes as a function of energy. For optically allowed transitions, the cross section rises relatively slowly from threshold to a broad maximum approximately five times the threshold energy. At higher energies the (ln E 0 )/E 0 dependence of the cross section

Qn =

Electron–Atom and Electron–Molecule Collisions

63.2 Collision Processes

predicted by (63.12) is observed. For exchange processes, e.g., a triplet excited state from a singlet ground state, the cross section peaks sharply close to threshold and falls off at high energy as E −3 . If the excitation is spin allowed but optically forbidden, e.g., He n 1D from 11S , then the Bethe theory predicts an E −1 dependence of the cross section at high energies. When excitation occurs to a long-lived (metastable or Rydberg) state following electron impact, it is often possible to detect the excited particle directly. Timeof-flight (T.O.F.) techniques are used to distinguish the long lived species from other products, e.g., photons, produced in the collisions.

clei the transition occurs vertically between potential energy curves. Since dissociation rapidly follows a vertical transition to the repulsive part of a potential energy curve, compared with the period of molecular rotation, the dissociation products tend to move in the direction of vibrational motion. Since the excitation probability depends on the relative orientation of the electron beam and the molecule, dissociation products often demonstrate pronounced anisotropic angular distributions. The angular distributions have been analyzed by Dunn [63.33] using symmetry considerations.

63.2.5 Dissociation Cross Sections

Tate and Smith [63.34] some 60 years ago developed the basic techniques for measuring total ionization cross sections. These were later improved by Rapp and Englander-Golden [63.35]. Full details of the experimental methods are given in the reviews and books already cited. Märk and Dunn [63.36] reviewed the situation as it existed in the mid 1980s. In the basic “parallel plate” method, the electron beam is directed through a beam or a static target gas between collector plates which detect the resultant ions. Unstable species can be studied by the “fast neutral beam” technique [63.37], in which the neutral target species is formed by charge neutralization from a fast ion beam, and is subsequently ionized by a crossed electron beam. For the determination of partial ionization cross sections specific to a given ion species in a given ionization stage, mass spectroscopic (quadrupole mass spectrometer, electrostatic or magnetic charged particle analyzer or time of flight) methods are used. Fourier Transform Mass Spectrometry (FTMS) has also been used effectively to study fragmentation with formation of both positive and negative ions. Reference [63.38] is a recent example of this. Absolute total ionization cross sections have been measured for a large number of species with an accuracy of better than 10%. Christophorou and colleagues have presented helpful compilations of ionization and other data of particular relevance to the plasma processing industry, [63.39, and earlier references in this journal]. A large number of mechanisms can contribute to the ionization of atoms and molecules by electron impact. For targets with only a few atomic electrons, the dominant process is single ionization of the outer shell, with the resultant ion being left in its ground state. The process is direct and is characterized by large impact parameters b and small momentum transfers. The cross section varies with incident electron energy in a way very similar to the optically allowed

63.2.6 Ionization Cross Sections

Part E 63.2

Dissociation of a molecular target can result in fragments which may be excited or ionized. Such processes may be studied using the techniques discussed in the previous section or in the following section, where charged particle detection is considered. Because a repulsive state of the molecule is accessed, the fragments can leave the interaction region with considerable kinetic energy (several eV). If the fragment is in a long-lived metastable or Rydberg state, T.O.F. techniques may be used to distinguish the long-lived species from other products such as photons, and also to measure the energies of the excited fragments, and thus provide information on the repulsive states responsible for the dissociation. For further discussion see the reviews by Compton and Bardsley [63.27], Freund [63.28], and Zipf [63.29]. If the detector can be made sensitive to a particular excited species, its excitation can be isolated and studied. Examples are the work of McConkey and co-workers [63.30,31] on O(1S ) and S(1S 0 ) production from various molecules. The detection of unexcited neutral fragments is more challenging. One early method was to trap selectively the dissociation products using a getter and measure the resulting pressure decrease. In a more sophisticated approach, Cosby [63.32] produced a fast (≈ 1 keV) target molecular beam by resonant charge exchange and subjected it to electron impact dissociation. The fast dissociation products were detected by conventional particle detectors in a time correlated measurement. Laser techniques, such as laser-induced fluorescence or multiphoton ionization, have also been used recently to detect the dissociation products. The Franck–Condon principle largely governs molecular dissociation. The principle states that if the excitation takes place on a time scale which is short compared with vibrational motion of the atomic nu-

935

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Scattering Experiment

excitation processes discussed in Sect. 63.2.4. Processes involving ionization of more than one outer shell electron become more important as the size of the target increases. These events are associated with small b and electron–electron correlations are usually strong. Autoionization increases in significance for heavier targets. Here also, collisions with small b dominate and electron–electron correlations are strong. For heavier targets, inner shell effects, such as Auger electron or X-ray emission, become progressively more important. For molecular targets, dissociative ionization (either di-

rectly or through a highly excited intermediate state) and ion pair formation also play a significant role. In addition to measurement of gross ion production, it is also possible to study the ionization process by monitoring the electron(s) ejected or scattered inelastically. Conventional electron spectroscopic techniques are used for this purpose. The addition of coincidence techniques (e–2e measurements) in which the momenta of all the electrons involved are completely specified has allowed many of the fine details of the ionization process to be extracted [63.40].

63.3 Coincidence and Superelastic Measurements

Part E 63.3

The cross section measurements described so far do not yield complete information on electron scattering processes. As mentioned in Sect. 63.1.2, these cross sections do not distinguish for magnetic sublevels, electron spin etc.and represent summation of cross sections over these experimentally indistinguishable processes (summation of the square moduli of the corresponding scattering amplitudes). The quantum mechanical description of a scattering process is given in terms of scattering amplitudes and under certain conditions requires summing up amplitudes and squaring the sum. This leads to interference terms which arise from the coherent nature of the scattering process. A complete characterization of a scattering process, therefore, requires knowledge of the complex scattering amplitudes. Sophisticated experimental techniques have been developed in recent years, which go beyond the conventional scattering cross section measurements and yield information on magnetic sublevel specific scattering amplitudes and the polarization (alignment and orientation) of the excited atomic ensemble. The experimental techniques fall into two main categories: a) electron–photon coincidence measurements, and b) superelastic scattering measurements involving coherently excited species. (We still consider unpolarized electron beams in the description of these two techniques here and will address the question of spin polarization in the following section.) In electron–photon coincidence measurements, the radiation pattern emitted by the excited atom is determined for a given direction of the scattered electron. A scattering plane is defined by ki and k f , and hence the symmetry is lowered from cylindrical (around the incident beam direction) to planar, (Fig. 63.6). It is now possible to determine, at least in principle, both the atomic alignment (i. e., the shape of the excited

state charge cloud and its alignment in space) and its orientation (i. e., the angular momentum transferred to the atom during the course of the collision). Complete sets of excitation amplitudes for the coherently excited atomic states and their relative phases have been measured in some cases. A comparison with theory can then be made at the most fundamental level. See [63.41] for full discussion and analysis. z + – kin

L⊥

y

w kout

γ θcol

x h

l

Fig. 63.6 Schematic illustration of a collisionally induced

charge cloud in a p-state atom. The scattering plane is fixed by the direction of incoming kin and outgoing kout momentum vectors of the electrons. The atom is characterized by the relative length (l), width (w), and height (h) of the charge cloud, by its alignment angle γ , and by its inherent angular momentum L ⊥ . The coordinate frame is the natural frame with the z-axis perpendicular to the scattering plane and with the x- and y-axes defined as shown in the figure relative to kin and kout

Electron–Atom and Electron–Molecule Collisions

937

1/2  P + = P12 + P22 ,

L+ ⊥ = −P3 , (1 + P1 )(1 − P4 ) . ρ00 = 4 − (1 − P1 )(1 − P4 )

(63.14)

+ The total polarization Ptot , which is defined as

 2 1/2 + L+ ⊥ 1/2  2 2 = P1 + P2 + P32 ,

+ = Ptot



P +

2

(63.15)

is a measure for the degree of coherence in the excitation process. In the absence of atomic depolarizing effects due to, for example, fine and/or hyperfine interactions, + a value of Ptot = +1 for the emitted radiation indicates total coherence of the excitation process. Much of the earlier work involved excitation of helium n 1P state. Here the situation is simplified as L–S coupling applies strictly: P4 = 1 and ρ00 is zero. Excitation of the 21P state, for example, is fully coherent and hence the excitation is completely specified two    by just1/2 parameters, γ and L ⊥ or P since P = 1 − L 2⊥ . More recently, the techniques have been applied to heavier targets and more complicated excitation processes [63.42–46]. The superelastic scattering experiments could be looked at as time inverse electron–photon coincidence experiments (although this is not exactly the case). In these experiments, a laser beam is utilized to prepare a coherently excited, polarized ensemble of target atoms for the electron scattering measurement. The superelastic scattering intensity is then measured as a function of laser-beam polarization and/or angle with respect to a reference direction. Linearly polarized laser light produces an aligned target (uneven population in magnetic sublevels for quantum numbers of different |M J | value). Circularly polarized laser light produces oriented targets (uneven population in M J = +m and M J = −m magnetic sublevels). From these measurements the same electron impact coherence parameters can be deduced as from the coincidence experiments. This approach has been applied to atomic species (mainly metal atoms) which are conveniently excited with available lasers. Detailed descriptions of the experimental techniques, the underlying theoretical background, and the interpretation of the experimental data are given in [63.47–56]. It should be noted that electron scattering by coherently excited atoms can be utilized not only for obtaining electron impact coherence parameters for inelastic processes originating from ground state but for

Part E 63.3

The electron–photon coincidence measurements can be carried out in two ways: (1) measuring polarization correlations, and (2) measuring angular correlations. In (1), polarization analysis of the emitted photon in a given direction occurs, while in (2), the angular distribution of the emitted photons is determined without polarization analysis. We will describe here only method (1) in some detail. Method (1) has the advantage that it measures directly the angular momentum (perpendicular to the scattering plane) transferred in the collision. For P-state excitation from a 1S 0 ground state, four parameters plus a cross section are needed to describe fully the collisionally excited P-state. The natural parameters introduced by Andersen et al. [63.41] are defined as follows (Fig. 63.6): γ is the alignment angle of the excited state charge cloud relative to the electron beam axis, P + is the linear polarization in the scattering plane, L+ ⊥ is the orbital angular momentum perpendicular to the scattering plane that is transferred to the atom in the collision, and ρ00 is the relative height of the charge cloud perpendicular to the scattering plane at the point of origin. The + superscript indicates positive reflection symmetry with respect to the scattering plane. In polarization correlation experiments, one typically measures two linear (P1 , P2 ) and one circular (P3 ) polarization correlation parameters perpendicular to the scattering plane. One additional linear polarization correlation parameter P4 is measured in the scattering plane. Each parameter is the result of two intensity measurements for different orientations of the polarization analyzer: I(0◦ ) − I(90◦ ) −1 β , P1 = I(0◦ ) + I(90◦ ) I(45◦ ) − I(135◦ ) −1 β , P2 = I(45◦ ) + I(135◦ ) I R − I L −1 β , P3 = R I + IL I(0◦ ) − I(90◦ ) −1 P4 = β . (63.13) I(0◦ ) + I(90◦ ) Here I(α) denotes the photon intensity measured for a polarizer orientation α with respect to the electron beam axis, I R and I L refer to right- and left-handed circularly polarized light and β denotes the polarization sensitivity of the polarization analyzer. The relationships between the experimentally determined polarization correlation parameters and the natural parameters are given by 1 γ = tan−1 (P2 /P1 ) , 2

63.3 Coincidence and Superelastic Measurements

938

Part E

Scattering Experiment

elastic, inelastic, and superelastic transitions involving excited states. These measurements yield information on creation, destruction, and transfer of alignment and ori-

entation in electron collision processes which is needed, e.g., in the application of plasma polarization spectroscopy [63.57].

63.4 Experiments with Polarized Electrons So far we have considered the utilization of unpolarized electron beams which yield spin averaged cross sections. Little information on spin dependent interactions is gained from these experiments. However, these interactions can be studied using polarized electron beam techniques. Developments on both the production and detection of spin-polarized electron beams have resulted in a wide range of elegant experiments probing these effects. The theory is also highly developed. For a detailed discussion see the works of Kessler [63.58, 59], Blum and Kleinpoppen [63.60], Hanne [63.61–63] and Andersen et al. [63.44, 45] and the references therein. Some basic concepts are presented here. The degree of polarization P of an electron beam is given by P=

N(↑) − N(↓) , N(↑) + N(↓)

(63.16)

Part E 63.4

where N(↑) and N(↓) are the numbers of electrons with spins respectively parallel and antiparallel to a particular quantization direction. Measurements of P both before and after the collision enable one to probe directly for specific spin dependent processes. For example, in elastic scattering from heavy spinless atoms any changes in the polarization of the electrons must be caused by spin– orbit interactions alone since, in this instance, it is not possible to alter the polarization of the electron beam by electron exchange. Measurements have been carried out for Hg and Xe and both direct (f) and spin-flip (g) scattering amplitudes, as well as their phase differences, have been determined [63.59]. The spin–orbit interaction for the continuum electron caused by the target nucleus leads to different scattering potentials and consequently to different cross sections for spin-up and spin-down electrons (called Mott scattering). Consequently, an initially unpolarized electron beam can become spin polarized after scattering by a specific angle according to P  = S p (θ)nˆ ,

(63.17)

where n is the unit vector normal to the scattering plane, S p (θ) is the polarization function and P  is the polarization of the scattered beam. For the same reasons, when

a spin-polarized electron beam is scattered by an angle θ to the left and to the right, an asymmetry is found in the scattering cross sections. Furthermore, an existing polarization P  can be detected through the left–right asymmetry A in the differential cross section, which is given by σl (θ) − σr (θ) = SA (θ)P  · nˆ , (63.18) A≡ σl (θ) + σr (θ) where σl (θ) and σr (θ) are the differential cross sections for scattering at an angle θ relative to the incident beam axis to the left and to the right, respectively. For elastic scattering, the polarization function SP , and the asymmetry function SA are identical and are called the Sherman function. When electron exchange is studied under conditions where other explicitly spin-dependent forces can be neglected, the cross sections for scattering of polarized electrons from polarized atoms depend on the relative orientation of the polarization vectors. According to [63.59] σ(θ) = σu (θ)[1 − Aex (θ)Pe · PA ] , (63.19) where Pe and PA are the electron and atom polarization vectors, and σu (θ) is the cross section for unpolarized electrons. Hence, an “exchange asymmetry” Aex (θ) can be defined by σ↑↓ (θ) − σ↑↑ (θ) , (63.20) Aex (θ) Pe · PA = σ↑↓ (θ) + σ↑↑ (θ) where σ↑↑ (θ) and σ↑↓ (θ) denote the cross sections for parallel and antiparallel polarization vectors respectively. As Bartschat [63.64] points out, an asymmetry can occur even if the scattering angle is not defined. In this case the function Aex (θ) is averaged over all angles. Differential and integral measurements of this kind have been performed for elastic scattering, excitation and ionization. For heavy target systems it is necessary to consider a combination of effects together with a description of the target states in the intermediate or fully coupled scheme. Consequently, the number of independent parameters can become very large and the “complete” experiments, which disentangle the various contributions to any observed asymmetry in the scattering, are rarely possible.

Electron–Atom and Electron–Molecule Collisions

Even for very light target atoms, where conventional Mott scattering is negligible, Hanne [63.61] has shown that the “fine structure” effect, in which electron scattering from individual fine structure levels of a multiplet occurs, can lead to polarization effects. In fact it can be a dominating effect for inelastic collision processes. For full details of these various mechanisms and how density matrix theory and other theoretical techniques

63.6 Electron Collisions in Traps

939

have been applied to scattering involving polarization effects, the reader is referred to the review articles cited, particularly Andersen et al. [63.44]. In certain cases, experiments involving spin polarized electron beams coupled with coincidence (or superelastic) measurements allow one to extract the maximum possible information for a given process, and are termed as complete or perfect in the sense defined by Bederson [63.65–67].

63.5 Electron Collisions with Excited Species Laser excitation is more involved but very well defined. Specific fine and hyperfine levels of individual isotopes can be excited. When laser excitation is used in conjunction with superelastic electron scattering, an energy resolution of 10−8 eV is easily achieved, compared with the 10−2 eV resolution possible in conventional electron scattering. Depending on the method of preparation, the population distribution in the magnetic sublevels of the target atoms may be uneven and some degree of polarization (alignment or orientation) may be present. The scattering will then be φ-dependent. For polarized target atoms the measured electron collision cross sections do not correspond to the conventional cross sections (which are summed over final and averaged over initial experimentally indistinguishable states, with equal populations in the initial states). One, therefore, has to characterize precisely the state of the target beam in order to be able to deduce a well defined, meaningful cross section. Polarization of atoms can be conveniently controlled, in the case of excitation with laser light, through the control of the laser light polarization (as discussed in Sect. 63.3). Since the atomic ensemble is coherently excited in this case, the scattering cross sections will depend on the azimuthal scattering angle φ. These considerations also come into play when one tries to relate measured inelastic and corresponding inverse superelastic cross sections by the principle of detailed balancing.

63.6 Electron Collisions in Traps A technique which has recently begun to be exploited involves collisions with trapped atoms. Pioneered by Lin and colleagues [63.73, 74], using Rb targets the technique has many advantages over more conventional techniques, not least of which is the fact that the absolute number density of the target need not be known. Cross

section data are obtained from measurements of trap loss and electron beam current density. Because up to half of the atoms in the trap can be in the excited state, it is possible to make measurements of cross sections involving excited states as well [63.75]. Measurements involving Cs targets have also been reported [63.76].

Part E 63.6

There are many plasma systems where electron collisions with excited atoms and molecules play a prominent role, e.g., electron beam and discharge pumped lasers, planetary and astrophysical plasmas. Especially important are electron collisions with metastable species because of the long lifetime, large cross section and large amount of excitation energy associated with them. Electron collision studies and cross section data in this area are scarce mainly due to experimental difficulties associated with the production of target beams with sufficiently high densities of excited species. With the application of lasers for the preparation of the excited species this problem can be overcome. However, this approach has not yet been extensively exploited. Reviews of this field are given by Lin and Anderson [63.68], Trajmar and Nickel [63.69] and Christophorou and Olthoff [63.70]. Since electron collisions with excited species necessarily involve a method of preparation, they are two step processes. Excitation and ionization in these cases are frequently referred to as stepwise excitation and ionization. The target preparation leads to mixed beams containing both ground and excited atoms or molecules. Preparation of excited atoms is achieved by electron impact or photoabsorption. Fast metastable beams can be produced by near-resonance charge exchange. For more details see [63.69, 71, 72]. Electron impact excitation is simple and effective but highly nonspecific, and characterization of the composition of the mixture is difficult.

940

Part E

Scattering Experiment

63.7 Future Developments Electron-driven processes have been identified as being of fundamental importance in a wide range of environmentally and technologically significant areas [63.77]. Boudaiffa et al. [63.78] have shown that electron attachment is a significant process in bond-breaking in DNA. Electron-initiated dissociation of large molecules can act as a catalyst for reactive chemistry in environmentally sensitive situations. Developments in large

scale computing have opened the door to calculations involving large molecules which could not even have been contemplated a few years ago. Electron collisions in intense laser field situations is an exciting new field which is rapidly expanding [63.79, 80]. Electron– cluster interactions allow one to probe how interactions change as one progresses from the gaseous to the solid phase.

References 63.1 63.2

63.3

63.4 63.5

63.6 63.7 63.8 63.9 63.10 63.11 63.12 63.13

Part E 63

63.14 63.15 63.16 63.17 63.18

J. Franck, G. Hertz: Verh. dtsch. Phys. Ges. 16, 457 (1914) H. S. N. Massey, E. H. S. Burhop, H. B. Gilbody: Electronic and Ionic Impact Phenomena, Vol. 1 & 2 (Clarendon Press, Oxford 1969) E. W. McDaniel: Atomic Collisions, Electron and Photon Projectiles (John Wiley and Sons, New York 1989) G. Scoles: Atomic and Molecular Beam Methods, Vol. 1 (Oxford Univ. Press, New York 1988) S. Trajmar, D. F. Register: Experimental techniques for cross section measurements. In: Electron Molecule Collisions, ed. by I. Shimamura, K. Takayanagi (Plenum, New York 1984) M. Allen: J. Phys. B 25, 1559 (1992) H. A. Bethe: Ann. Phys. 5, 325 (1930) E. N. Lassettre, A. Skerbele, M. A. Dillon: J. Chem. Phys. 50, 1829 (1969) D. C. Cartwright, S. Trajmar, W. Williams, D. L. Huestis: Phys. Rev. Lett. 27, 704 (1971) W. A. Goddard, D. L. Huestis, D. C. Cartwright, S. Trajmar: Chem. Phys. Lett. 11, 329 (1971) M. Zubek, B. Mielewska, J. Channing, G. C. King, F. H. Read: J. Phys. B 32, 1351 (1996) J. Channing, F. H. Read: Rev. Sci. Instr. 67, 2372 (1996) S. Trajmar, I. Kanik: Elastic and excitation electron collisions with atoms. In: Atomic and Molecular Processes in Fusion Edge Plasmas, ed. by R. K. Janev, H. P. Winter, W. Fritsch (Plenum, New York 1995) B. Bederson, L. J. Kieffer: Rev. Mod. Phys. 43, 601 (1971) A. Zecca, G. P Carwasz, R. S. Brusa: La Rivista del Nuovo Cimento 19(3), 1 (1996) G. P Carwasz, R. S. Brusa, A. Zecca: La Rivista del Nuovo Cimento 24(1), 1 (2001) G. P. Carwasz, R. S. Brusa, A. Zecca: La Rivista del Nuovo Cimento 24(4), 1 (2001) S. Srivastava, A. Chutjian, S. Trajmar: J. Chem. Phys. 63, 2659 (1975)

63.19 63.20 63.21 63.22 63.23 63.24 63.25

63.26 63.27

63.28

63.29

63.30 63.31 63.32 63.33 63.34 63.35 63.36 63.37

J. C. Nickel, C. Mott, I. Kanik, D. C. McCollum: J. Phys. B 21, 1867 (1988) J. C. Nickel, P. Zetner, G. Shen, S. Trajmar: J. Phys. E 22, 730 (1989) S. Trajmar, J. W. McConkey: Adv. At. Mol. Opt. Phys. 33, 63 (1994) P. N. Clout, D. W. O. Heddle: J. Opt. Soc. Am. 59, 715 (1969) F. G. Donaldson, M. A. Hender, J. W. McConkey: J. Phys. B 5, 1192 (1972) B. van Zyl, G. H. Dunn, G. Chamberlain, D. W. O. Heddle: Phys. Rev. A 22, 1916 (1980) A. R. Filippelli, C. C. Lin, L. W. Anderson, J. W. McConkey: Adv. At. Mol. Opt. Phys. 33, 1 (1994) M. Inokuti: Rev. Mod. Phys. 43, 297 (1971) R. N. Compton, J. N. Bardsley: Dissociation of molecules by slow electrons. In: Electron Molecule Collisions, ed. by I. Shimamura, K. Takayanagi (Plenum, New York 1984) R. S. Freund: Rydberg States of Atoms and Molecules, ed. by R. F. Stebbings, F. B. Dunning (Cambridge Univ. Press, Cambridge 1983) E. C. Zipf: Dissociation of molecules by electron impact. In: Electron-Molecule Interactions and Their Applications, Vol. 1, ed. by L. G. Christophorou (Academic, New York 1984) L. R. LeClair, J. W. McConkey: J. Chem. Phys. 99, 4566 (1993) W. Kedzierski, J. Borbely, J. W. McConkey: J. Phys. B 34, 4027 (2001) P. C. Cosby: J. Chem. Phys. 98, 7804, 9544 and 9560 (1993) G. H. Dunn: Phys. Rev. Lett. 8, 62 (1962) J. T. Tate, P. T. Smith: Phys. Rev. 39, 270 (1932) D. Rapp, P. Englander-Golden: J. Chem. Phys. 43, 1464 (1965) T. D. Märk, G. H. Dunn: Electron Impact Ionization (Springer, Vienna, New York 1985) A. J. Dixon, M. F. A. Harrison, A. C. H. Smith: J. Phys. B 9, 2617 (1986)

Electron–Atom and Electron–Molecule Collisions

63.38 63.39 63.40 63.41 63.42 63.43 63.44 63.45

63.46

63.47 63.48 63.49 63.50 63.51 63.52 63.53 63.54 63.55 63.56

63.57

63.58

63.61

63.62

63.63

63.64 63.65 63.66

63.67 63.68 63.69 63.70 63.71 63.72 63.73 63.74 63.75 63.76

63.77

63.78 63.79

63.80

G. F. Hanne: Coherence in Atomic Collision Physics, ed. by H. J. Beyer, K. Blum, R. Hippler (Plenum, New York 1988) p. 41 G. F. Hanne: Collisions of polarized electrons with atoms and molecules. In: Proceedings of the 17th ICPEAC, Electronic and Atomic Collisions, ed. by W. R. MacGillivray, I. E. McCarthy, M. C. Standage (Hilger, Bristol 1992) p. 199 K. Bartschat: Comments At. Mol. Phys. 27, 239 (1992) B. Bederson: Comments At. Mol. Phys. 1, 41 (1969) D. H. Yu, J. F. Williams, X. J. Chen, P. A. Hayes, K. Bartschat, V. Zeman: Phys. Rev. A 67, 032707 (2003) H. M. Al-Khateeb, B. G. Birdsey, T. J. Gay: Phys. Rev. Lett 85, 4040 (2000) C. C. Lin, L. W. Anderson: Adv. At. Mol. Opt. Phys. 29, 1 (1992) S. Trajmar, J. C. Nickel: Adv. At. Mol. Opt. Phys. 30, 45 (1993) L. G. Christopherou, J. K. Olthoff: Adv. At. Mol. Opt. Phys. 44, 156 (2001) J. B. Boffard, M. E. Lagus, L. W. Anderson, C. C. Lin: Rev. Sci. Inst 67, 2738 (1996) J. B. Boffard, M. F. Gehrke, M. E. Lagus, L. W. Anderson, C. C. Lin: Europhys. J. D 8, 193 (2000) R. S. Schappe, T. Walker, L. W. Anderson, C. C. Lin: Europhys. Lett 29, 439 (1995) R. S. Schappe, T. Walker, L. W. Anderson, C. C. Lin: Phys. Rev. Lett 76, 4328 (1996) M. L. Keeler, L. W. Anderson, C. C. Lin: Phys. Rev. Lett 85, 3353 (2000) J. A. MacAskill, W. Kedzierski, J. W. McConkey, J. Domyslawska, I. Bray: J. Elect. Spect. Rel. Phen 123, 173 (2002) K H. Becker, C. W. McCurdy, T. M. Orlando, T. N. Resigno: Electron-Driven Processes: Scientific Challenges and Technological Opportunities (US DOE Report, 2000) B. Boudaiffa, P. Cloutier, D. Hunting, M. A. Huels, L. Sanche: Science 287, 1658 (2000) H. Niikura, F. Légaré, R. Hasbani, M. Y. Ivanov, D. M. Villeneuve, P. B. Corkum: Nature 421, 826 (2003) M. Weckenbrock, A. Becker, A. Staudte, S. Kammer, M. Smolarski, V. R. Bhardwaj, D. M. Rayner, D. M. Villeneuve, P. B. Corkum, R. Dorner: Phys. Rev. Lett 91, 123004 (2003)

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63.59 63.60

C. Q. Jaio, C. A. DeJoseph, A. Garscadden: J. Phys. Chem 107, 9040 (2003) L. G. Christophorou, J. K. Olthoff: J. Phys. Chem. Ref. Data 31, 971 (2002) H. Ehrhardt, K. Jung, G. Knoth, P. Schlemmer: Z. Phys. D 1, 3 (1986) N. Andersen, J. W. Gallagher, I. V. Hertel: Phys. Rep. 165, 1 (1988) K. Becker, A. Crowe, J. W. McConkey: J. Phys. B 25, 3885 (1992) M. Sohn, G. F. Hanne: J. Phys. B 25, 4627 (1992) N. Andersen, K. Bartschat, J. T. Broad, I. V. Hertel: Phys. Rep 279, 251 (1997) N. Andersen, K. Bartschat: Polarization, Alignment, and Orientation in Atomic Collisions (Springer, Berlin 2001) C. Herting, G. F. Hanne: Correlation and polarization in photonic, electronic and atomic collisions, AIP Conf. Proc. 697, 181 (2004) I. V. Hertel, W. Stoll: Adv. At. Mol. Phys. 13, 113 (1977) P. W. Zetner, S. Trajmar, G. Csanak: Phys. Rev. A 41, 5980 (1990) U. Fano: Rev. Mod. Phys. 29, 74 (1957) K. Blum: Density Matrix Theory and Applications (Plenum Press, New York 1981) U. Fano, J. H. Macek: Rev. Mod. Phys. 45, 553 (1973) J. H. Macek, I. V. Hertel: J. Phys. B 7, 2173 (1974) W. R. MacGillivray, M. C. Standage: Phys. Rep. 168, 1 (1988) V. Karaganov, I. Bray, P. J. O. Teubner: Phys. Rev. A 57, 208 (1998) K. A. Stockman, V. Karaganov, I. Bray, P. J. O. Teubner: J. Phys. B 34, 1105 (2001) P. W. Zetner, P. V. Johnson, Y. Li, G. Csanak, R. E. H. Clark, J. Abdallah: J. Phys. B 34, 1619 (2001) S. A. Kazantsev, J.-C. Hénoux: Polarization Spectroscopy of Ionized Gases (Kluwer, Dordrecht 1995) J. Kessler: Polarized Electrons, 2nd edn. (Springer, Berlin, Heidelberg 1985) p. 2 J. Kessler: Adv. At. Mol. Opt. Phys. 27, 81 (1991) K. Blum, H. Kleinpoppen: Adv. At. Mol. Phys. 19, 187 (1983) G. F. Hanne: Phys. Rep. 95, 95 (1983)

References

943

Ion–Atom Scat

64. Ion–Atom Scattering Experiments: Low Energy

This chapter outlines the physical principles and experimental methods used to investigate low energy ion–atom collisions. A low energy collision is here defined as one in which the initial ion–atom relative velocity is less than the mean orbital velocity
ve  of the electrons affected by the collision. For outer or valence electrons,
ve   vB , where vB = 2.1877 × 108 cm/s is the Bohr velocity. In terms of the energy of a projectile ion, vB corresponds to 24.8 keV/N, where N is the projectile nuclear number (e.g., 16 for O+ ). The theory and results of ion–atom scattering studies are further discussed in Chapts. 37, 38, and 47 to 51. The focus here is on the experimental techniques. Since several of these depend on the characteristics of a specific process, the following section presents a summary of the physics of low-energy ion–atom collisions. See Chapts. 50 and 51 for more detailed information.

64.1 Low Energy Ion–Atom Collision Processes 943 64.2 Experimental Methods for Total Cross Section Measurements .... 64.2.1 Gas Target Beam Attenuation Method ......... 64.2.2 Gas Target Product Growth Method ............. 64.2.3 Crossed Ion and Thermal Beams Method ....... 64.2.4 Fast Merged Beams Method ........ 64.2.5 Trapped Ion Method .................. 64.2.6 Swarm Method ......................... 64.3 Methods for State and Angular Selective Measurements .... 64.3.1 Photon Emission Spectroscopy .... 64.3.2 Translational Energy Spectroscopy 64.3.3 Electron Emission Spectroscopy ... 64.3.4 Angular Differential Measurements .......................... 64.3.5 Recoil Ion Momentum Spectroscopy ............................

945 945 945 945 946 946 947 947 947 947 948 948 948

References .................................................. 948

64.1 Low Energy Ion–Atom Collision Processes The most important and widely studied inelastic ion– atom collision process in the low energy region is electron capture (also referred to as charge exchange, charge transfer or electron transfer) represented by A+q + B → A+q−k + B k+ + Q ,

(64.1)

A+q + B → A+q−k + B m+ + (m − k)e− + Q . (64.2)

At low energies, transfer ionization is particularly important in collisions of highly charged ions with multi-electron atoms.

Part E 64

where Q is the potential energy difference between the initial and final states. For an exoergic process, Q > 0 and this energy appears as excess kinetic energy of the products after the collision. For an endoergic process, Q < 0 and must be provided by the initial kinetic energy of the reactants, so that the corresponding cross section is usually small at low collision energies. Cross sections for electron capture are appreciable even at very low energies if Q is zero or very small (resonant or nearresonant process). Electron capture by multiply charged ions from atoms is predominantly an exoergic process, for which cross sections may also be large at low energies. In this case, electrons are preferentially captured

into excited levels of A+q−k . The typical cross section behavior for single electron capture (k = 1) by a multiply charged ion from atomic hydrogen is shown in Fig. 64.1. The initial ionic charge is the major determinant of the cross section at intermediate and high collision energies, whereas the cross section at low energies depends strongly on the structure of the transient quasimolecule formed during the collision. Multiple electron capture (k > 1) from multielectron atoms occurs predominantly into multiply excited levels, which stabilize either radiatively, leading to stabilized or “true” double capture, or via autoionization. The latter process is usually referred to as transfer ionization,

944

Part E

Scattering Experiment

Cross section (cm2) 10–13

Potential energy (a. u.) 2 q × 10–15 cm2

10–14

1

v0

10–15

A+q–1 + B+

10–16

0

10–17

–1

A+q–1 + B+

A +Η

(q–1)+

q+

10–18

A+q + B

A

A+q–2 + B 2+ A+q–2 + B 2+

+Η

+

–2

1

2

4

A+q–1 + B+

10 20 40 Internuclear separation (a. u.)

–19

10

101

102

103

104

105 106 107 Collision energy (eV/ u)

Fig. 64.1 Typical cross section variation with collision

energy for electron capture by a multiply charged ion from hydrogen. The low energy behavior depends on the structure of the quasimolecule formed during the collision

Part E 64.1

Adiabatic potential energy curves representing the collision of a multiply charged ion with a neutral atom are presented in Fig. 64.2. For such collisions, the Coulomb repulsion in the final state produces avoided crossings of the initial and final state potential curves, the positions of which are determined by the binding energy of the atomic electron and electronic energy level structure of the product ion. Generally, reaction channels that are moderately exoergic produce curve crossings at internuclear separations where there is sufficient overlap of the electron clouds for electron capture to be a likely process. The Landau–Zener curve-crossing model [64.1] (Chapt. 49), the classical over-barrier model for single capture [64.2] and the extended classical over-barrier model for multiple capture [64.3], are useful in predicting the important final product states, and in providing a semiquantitative interpretation. In the case of single electron capture by a bare multiply charged ion (of charge q = Z) from a hydrogen atom, the principal quantum number n p of the most probable final ionic state is given in this model [64.2] by
2Z 1/2 + 1 np = . (64.3) Z + 2Z 1/2

Fig. 64.2 Schematic representation of potential energy

curves for the collision of a multiply charged ion Aq+ with a multielectron atom B

The internuclear separation Rp at which the potential curves cross in this case is given by Rp =

2(Z − 1) . Z 2 /n 2p − 1

(64.4)

There has been much discussion of the role of electron correlation in the multiple electron capture process. At issue is the relative importance of the mechanism whereby several electrons are transferred (in a correlated manner) at a single curve crossing compared with that whereby single electrons are transferred successively at different curve crossings. Experimental evidence exists for both mechanisms, with the relative importance depending on the electronic structure of the transient quasimolecule that is formed during the collision. Measurements of the distribution of final ion product electronic states provides the major insight into such collision mechanisms [64.4]. Other inelastic ion–atom collision processes, such as direct electronic excitation and ionization, are endothermic, with relatively small cross sections that fall off with decreasing energy below a few tens of keV/N. Exceptions are collisions involving Rydberg atoms and collisional excitation of fine structure transitions, for which the required energy transfer is relatively small. Relatively little experimental data are available for direct excitation and ionization processes at low collision energies.

Ion–Atom Scattering Experiments: Low Energy

64.2 Experimental Methods for Total Cross Section Measurements

945

64.2 Experimental Methods for Total Cross Section Measurements In the present context, a total cross section measurement refers to an integration or summation over scattering angles, product kinetic energies and (frequently) electronic states. The total cross section is usually measured as a function of relative collision energy or velocity.

64.2.1 Gas Target Beam Attenuation Method The attenuation of a collimated ion beam of incident intensity I0 in a differentially pumped gas target cell or gas jet is related to the collision cross section σ by I = I0 e−σNL ,

(64.5)

The product growth method is similar to the beam attenuation method; the major difference that the growth of reaction products is measured rather than the loss of reactant projectiles. The products may be derived from either the projectile beam or the gas target, or both. The main advantage of this method is its higher degree of selectivity of a specific collision process. In addition, the reactants and products can usually be registered simultaneously, or in some cases in coincidence, eliminating the sensitivity of the measurement to temporal variation of ion beam intensity. An important criterion is that the target gas density be low enough that single collision conditions prevail (i. e. that the likelihood of an ion passing through the gas target and interacting with more than one target atom is negligibly small). This must in general be satisfied in order for (64.5) to relate correctly the measured attenuation to the collision cross section of interest, and is critical to the product growth method [64.5]. In this case, under single collision conditions, one may set the number of products Ip = I0 − I, and (64.5) then may be written as Ip = 1 − e−σNL ≈ σNL . (64.6) I0 The approximate expression is useful for σNL  1, which is a requirement for single collision conditions. It is also important that the products not be lost in a subsequent collision in the gas cell, so the magnitudes of cross sections for loss of products in the target gas must also be considered. The products in such an experiment may be derived from either the projectile beam or the target gas (e.g., collection of slow product ions in a gas cell), or from both in coincidence to enhance the specificity of the method. The method may in principle be used to determine either total or differential cross sections, depending on the degree of selectivity of collision products.

64.2.3 Crossed Ion and Thermal Beams Method Replacement of the gas target cell by an effusive thermal beam is advantageous for studying collisions of ions with reactive species such as atomic hydrogen, as well as for collecting slow ion products. The use of accelerating electrodes or grids for slow charged products allows coincident detection of fast and slow products, permitting

Part E 64.2

where I is the intensity after traversing an effective length L of the target gas, and N is the number density of target atoms. For a gas target cell with entrance and exit apertures of diameter d1 and d2 which are much less than the physical length z of the gas cell, L is given to a good approximation by z + (d1 + d2 )/2. This is valid under molecular flow conditions, for which the mean free path between collisions of target atoms is much larger than the dimensions of the gas cell. In designing the gas cell for measurements of total cross sections, d2 and the beam detector must be large enough that elastic scattering may be eliminated as a contributor to the measured attenuation. Usually d2 is made larger than d1 . Measurement of the gas pressure in a target cell is usually made using a capacitance manometer connected to the cell via a tube whose conductance is much larger than that of the gas cell apertures so that, to a good approximation, the pressure will be the same in both the manometer and the gas cell. For gas jet targets, the effective target thickness NL is usually determined by in situ normalization to some well-known cross section. The quantity σ in (64.5) refers to an effective cross section for removing projectile ions from the incident beam, which is the sum of cross sections for all such processes. In many cases, a single process (e.g., electron capture) is dominant, and σ primarily describes that process. Whether a collision process removes a projectile ion from the reactant beam or not depends on the configuration of the experiment. For example, the projectile particle may remain physically in the beam after passing through the target, but with a changed charge due to a collision. This would be registered as an attenuation of the primary ion beam if the beam is charge analyzed after the reaction.

64.2.2 Gas Target Product Growth Method

946

Part E

Scattering Experiment

measurement of ionization as well as electron-capture cross sections. Use of an effusive source or gas jet precludes accurate measurement of the effective target thickness, and in situ normalization to the cross section for some well known process is usually employed. A comprehensive discussion of such methods as applied to collisions of multiply charged ions with atomic hydrogen is given by Gilbody [64.6].

64.2.4 Fast Merged Beams Method Cross sections for ion–atom collision processes at very low energies have been measured by merging fast beams of ions and neutral atoms [64.7], as in Fig. 64.3. In this case, σ is determined from experimental parameters by σ=

v+ v0 R e F,  I+ I0 |v+ − v0 |

(64.7)

where R is the number of products detected per second,  is the product detection efficiency, e is the electronic charge, I+ is the ion current, I0 is the flux of atoms, v+ and v0 are the laboratory velocities of the ion and atom beams, vrel is their relative velocity, and F is the form factor that describes the spatial overlap of the two beams. If the z-axis is chosen to be the direction of propagation of the beams, the form factor has units of length and is given by   I+ (x, y, z) dx dy I0 (x, y, z) dx dy  F= . I+ (x, y, z)I0 (x, y, z) dx dy dz (64.8)

The two-dimensional integrals in the numerator represent the total intensities of the two beams, which are independent of z, so that F is also independent of z. xq+

Neutral beam detector Collision region (50 cm)

H

H H+

x(q–1)+

xq+

The relative collision energy E rel in eV/u is given by
E+ E0 E+ E0 E rel = + −2 cos θ , µ m+ m0 m+ m0

(64.9)

where E + , m + and E 0 , m 0 are the energies and masses of the ion and atom, respectively, and µ is their reduced mass. For collinear merged beams, θ = 0, and E rel can be reduced to zero by making the two beam velocities the same. In practice, the finite divergences of the beams place a lower limit on the energy and the energy resolution. The fast neutral atom beam is created by neutralizing an accelerated ion beam by electron capture by a positive ion beam in a gas, by electron detachment of a negative ion beam either in a gas, or using a laser beam. In gas collisions, a small fraction of the neutral beam is produced in excited n-levels (with an n −3 distribution), which may influence the measurements. With fast colliding beams, the maximum effective beam densities are invariably much smaller than the background gas density, even under ultrahigh vacuum conditions. For example, a typical 10 keV proton beam with a circular cross section of diameter 3 mm and a current I = 10 µA has an average effective density n = I/(eAv) = 1.6 × 106 cm−3 (A is the beam cross sectional area and v is its velocity). It is therefore necessary either to modulate the beams or to use coincidence techniques to separate signal events due to beam–beam collisions from background events produced by collisions of either beam with background gas. A typical two-beam modulation scheme is shown in Fig. 64.4. To eliminate the production of spurious signals, the detector gates are delayed for a short time after the beams are switched, and the beam modulation period is made much shorter than the pressure time constant of the vacuum system. Absolute electron-capture cross sections have been measured for O5+ + H collisions to energies below 1 eV/N, where the attractive ion-induced-dipole (polarization) interaction is expected to play a role [64.7]. The inverse velocity dependence of the cross section in this region is suggestive of the classical Langevin orbiting model for ion–neutral collisions [64.8].

Part E 64.2

Faraday cup Channel electron multiplier

Fig. 64.3 Schematic of the merged beams arrangement

used by Havener et al. [64.7] to study low energy electron capture collisions of multicharged ions with H atoms

64.2.5 Trapped Ion Method The trapped ion method is used to determine rate coefficients and effective cross sections for ion–atom collisions at near thermal energies. The technique involves storing ions in an electrostatic or electromagnetic

Ion–Atom Scattering Experiments: Low Energy

On Atom beam

Off

Ion beam

On

22 µ s Scaler No. 1

Off On

Off On

Scaler No. 2

Off

Fig. 64.4 Fast two-beam modulation scheme to separate the signal due to beam–beam collisions from events due to beam collisions with residual gas or surfaces [64.7]

trap, and measuring the rate of loss of ions from the trap after a small quantity of neutral gas is admitted [64.9]. Like the beam attenuation method, the trap technique

64.3 Methods for State and Angular Selective Measurements

947

cannot distinguish different processes that cause ions to be lost from the trap. The mean collision energy is estimated from an analysis of the ion dynamics in the trap.

64.2.6 Swarm Method The swarm method, using the flowing afterglow, drift tube or selected-ion flow tube, has been used successfully to study ion–atom collisions at very low energies [64.10]. Ions are injected into a homogeneous electric field and drift through a suitable low-density buffer gas such as helium. The ions move as a swarm whose mean energy depends on the applied electric field and on collisions with the buffer gas, and can be varied from the near-thermal region to tens of eV. The method involves measuring the additional attenuation of the directed ion swarm by a known quantity of added reactant gas, and is the major technique that has been used for the study of ion–atom collisions at near-thermal energies [64.11]. As with ion beam attenuation and ion trap methods, the drift tube is not selective of the process that leads to attenuation of the ion swarm.

64.3 Methods for State and Angular Selective Measurements Three principal methods have been developed and applied to the measurement of partial cross sections for population of specific product electronic states in ion–atom collisions involving electron capture [64.12]. These are based on spectroscopic measurements of photon emission, translational energy spectroscopy and electron emission in collisions of ion beams with gas targets.

brated using a standard photon source or detector, or by using a reference ion or electron beam and well established cross section data for photon emission [64.14]. Cascading from higher populated levels must also be considered whenever measured spectral line emission intensities are used to infer cross sections for populating specific energy levels. Successful state-selective cross section measurements have been made for transitions in the visible, VUV and X-ray spectral regions.

64.3.1 Photon Emission Spectroscopy 64.3.2 Translational Energy Spectroscopy Translational energy loss or energy gain spectroscopy provides a convenient method to determine the distribution of final states in low energy ion–atom collisions that are either endoergic or exoergic (Q = 0). For example, this method has been used extensively by the Belfast group [64.15] to study the predominantly exoergic process of electron capture by multiply charged ions from hydrogen atoms. An ion beam with a well defined energy is directed through a gas target, and the energy of the product ion beam is energy analyzed after the collision. Since the energy gain or loss to be measured is only a very small fraction of the initial kinetic

Part E 64.3

Since electron capture from atoms by multiply charged ions populates excited levels, photon emission spectroscopy may be employed to determine state-selective partial cross sections [64.13]. An ion beam is directed through a gas cell or jet, and a photomultiplier or suitable detector registers photons analyzed by an optical filter, a grating or a crystal spectrometer. The measured photon signal at a given angle depends on the detection solid angle, the polarization of the emitted radiation, the absolute efficiencies of the dispersive device and detector, and (for emission by fast ions) the lifetime of the radiating state. Depending on the spectral region, the photon detection system may be absolutely cali-

948

Part E

Scattering Experiment

energy, it is usually necessary to reduce the initial energy spread to a few tenths of an eV by decelerating the reactant ion beam prior to energy selection by an electrostatic analyzer. If the scattering angle of the ion is very small, its energy change is approximately equal to Q. Since the ion beam is attenuated by deceleration and energy analysis, cross sections for collisionally populating specific states are determined by normalizing the measured product-state distributions to total cross section data. The attainable state resolution is not as good as for photon emission spectroscopy.

64.3.3 Electron Emission Spectroscopy As noted in Sect. 64.1, multiple electron capture by multiply charged ions from atoms at low energies occurs primarily into multiply excited states, which decay either radiatively or via autoionization (with a branching ratio) [64.4]. The latter decay pathway (transfer ionization) provides an experimental method to determine the product ionic states by ejected-electron spectroscopy. Analysis of electrons emitted into the forward (ion beam) direction (zero degree spectroscopy) offers significant advantages for analysis of low energy electrons with high resolution [64.16]. Since a gas jet is often employed and absolute electron collection and spectrometer efficiencies are difficult to determine, some normalization procedure is usually employed to determine state-selective cross sections by this method. Electrons and product projectile or recoil ions have also been detected in coincidence to increase the specificity of the method.

64.3.4 Angular Differential Measurements The measurement of angular distributions of scattered ions in low energy ion–atom collisions has been facilitated by the availability of position-sensitive particle detectors consisting of a microchannel plate and a resistive or segmented anode [64.17]. The method for

processes that have forward-peaked angular distributions involves directing a highly collimated ion beam through a gas target cell, and counting the scattered projectile ions on a position-sensitive particle detector. Product ions produced by electron capture can be selected by the use of electrostatic retarding grids mounted immediately in front of the detector, to reject the primary ion beam.

64.3.5 Recoil Ion Momentum Spectroscopy Perhaps the most significant experimental development of the last decade, cold target recoil-ion momentum spectroscopy (COLTRIMS) [64.18] has been made possible by advances in position-sensitive particle detection. This technique, based on momentum imaging and also called a “reaction microscope”, has yielded important new insights into the dynamics of ion–atom collision processes [64.19] as well as other types of interactions involving cold atoms and molecules. The method is particularly suited to studies of charge-changing and molecular fragmentation processes. In this method, an ion beam intersects a cold supersonic atomic beam in an interaction volume within which small electric and magnetic fields are imposed in order to guide slow ions and ejected electrons to fast position-sensitive detectors. The charge and the transverse and longitudinal components of the momentum of the slow (recoil) ion are measured by a combination of position and time-of-flight measurements, permitting the Q-value of the collision and the final electronic states of the projectile and recoil ions to be uniquely determined. The position measurement additionally provides information about the angular scattering during the collision. Coincident measurement of the scattered projectile ion and determination of its charge state by electrostatic deflection, and/or time-of-flight measurements of ejected electrons have provided new insights into complex multielectron processes occurring in low-energy ion–atom collisions.

References

Part E 64

64.1 64.2 64.3 64.4 64.5

F. W. Meyer, A. M. Howald, C. C. Havener, R. A. Phaneuf: Phys. Rev. Lett. 54, 2663 (1985) H. Ryufuku, K. Sasaki, T. Watanabe: Phys. Rev. A 23, 745 (1980) A. Niehaus: J. Phys. B 19, 2925 (1986) M. Barat, P. Roncin: J. Phys. B 25, 2205 (1992) H. B. Gilbody: Adv. At. Mol. Phys. 22, 143 (1986)

64.6 64.7 64.8 64.9

F. W. Meyer, A. M. Howald, C. C. Havener, R. A. Phaneuf: Phys. Rev. A 32, 3310 (1985) C. C. Havener, M. S. Huq, H. F. Krause, P. A. Schulz, R. A. Phaneuf: Phys. Rev. A 39, 1725 (1989) G. Gioumoussis, D. P. Stevenson: J. Chem. Phys. 29, 294 (1958) D. A. Church: Phys. Rep. 228, 254 (1993)

Ion–Atom Scattering Experiments: Low Energy

64.10 64.11 64.12 64.13 64.14 64.15

Y. Kaneko: Comm. At. Mol. Phys. 10, 145 (1981) W. Lindinger: Phys. Scrip. T 3, 115 (1983) R. K. Janev, H. Winter: Phys. Rep. 117, 266–387 (1985) D. ´Ciri´c, A. Brazuk, D. Dijkkamp, F. J. de Heer, H. Winter: J. Phys. B 18, 3639 (1985) B. Van Zyl, G. H. Dunn, G. Chamberlain, D. W. O. Heddle: Phys. Rev. A 22, 1916 (1980) H. B. Gilbody: Adv. At. Mol. Opt. Phys. 32, 149 (1994)

64.16 64.17

64.18

64.19

References

949

N. Stolterfoht: Phys. Rep. 146, 316 (1987) L. N. Tunnell, C. L. Cocke, J. P. Giese, E. Y. Kamber, S. L. Varghese, W. Waggonner: Phys. Rev. A 35, 3299 (1987) R. Dörner, V. Mergel, O. Jagutzki, L. Spielberger, J. Ullrich, R. Moshammer, H. Schmidt-Böcking: Phys. Rep. 330, 95 (2000) J. Ullrich, R. Moshammer, A. Dorn, R. Dörner, L. Ph. H. Schmidt, H. Schmidt-Böcking: Rep. Prog. Phys. 66, 1463 (2003)

Part E 64

951

This chapter deals with inelastic processes which occur in collisions between fast, often highly charged, ions and atoms. Fast collisions are here defined to be those for which V/ve ≥ 1, where V is the projectile velocity and ve the orbital velocity of this electron. For processes involving outer shell target electrons, this implies V
1 a.u., or the projectile energy
25 keV/a.m.u. For inner shell electrons, typically, V
Z2 /n a.u., where Z2 is the target nuclear charge and n the principal quantum number of√ the active electron. A useful relationship is V = 6.35 E/M, where V is in a.u., E is in MeV, and M is in a.m.u. Fast collisions involving outer shell processes can be studied using relatively small accelerators, while those involving inner shell processes require larger van de Graaffs, LINACs, etc. Because the motion of the inner shell electrons is dominated by the nuclear Coulomb field of the target, and because transitions involving these electrons take place rather independently of what transpires with the outer shell electrons, it has proven somewhat easier to understand one electron processes involving inner shell electrons. Thus, for a long time, a great deal of the work on fast ion–atom collisions has concentrated on inner shell processes involving heavy target atoms. However, more recently, new experimental techniques have led to a shift of this focus to

65.1 Basic One-Electron Processes ................ 951 65.1.1 Perturbative Processes ............... 951 65.1.2 Nonperturbative Processes ......... 955 65.2 Multi-Electron Processes ...................... 957 65.3 Electron Spectra in Ion–Atom Collisions . 959 65.3.1 General Characteristics............... 959 65.3.2 High Resolution Measurements ... 960 65.4 Quasi-Free Electron Processes in Ion–Atom Collisions ......................... 65.4.1 Radiative Electron Capture ......... 65.4.2 Resonant Transfer and Excitation 65.4.3 Excitation and Ionization ...........

961 961 961 961

65.5 Some Exotic Processes .......................... 962 65.5.1 Molecular Orbital X-Rays ............ 962 65.5.2 Positron Production from Atomic Processes ............... 962 References .................................................. 963 inelastic processes involving light target atoms. Furthermore, present investigations go beyond the one-electron picture to include the influence of the electron–electron interaction. The present chapter outlines some of the developments in this area over a very active past few decades. The literature is vast, and only a small sampling of references is given. Emphasis is on experimental results (for the theory see Chapts. 45–57)

65.1 Basic One-Electron Processes 65.1.1 Perturbative Processes Inner Shell Ionization of Heavy Targets For ion–atom collisions involving projectile and target nuclear charges param
 Z 1 and 2 Z 2 respectively,
 the  2 eters η1 = ~V/ Z 1 e2 ) and η2 = ~ve / Z 2 e2 are useful in characterizing the strength of the interaction between Z 1 , Z 2 , and the target electron. If η2  η1 , (i. e., Z 1 /Z 2  V/ve ), the effect of the projectile on the target wave function can be treated perturbatively. Perturbation treatments of inner shell ionization by lighter projectiles have been extensively studied and reviewed

[65.1–8]. Two well-known formulations have been used: the plane wave Born approximation (PWBA) [65.1–4], and the semiclassical approximation (SCA) [65.9, 10]. The former represents the nuclear motion with plane waves, while the latter is formulated in terms of the impact parameter b with the nuclear motion treated classically. For straight line motion of the nuclei, the results are equivalent [65.11]. The total cross section for ionizing the K-shell of a target of charge Z 2 by a projectile of charge Z 1 is given within the PWBA by   σi = 8πZ 12 /Z 24 η2 f(θ2 , η2 ) a02 ,

(65.1)

Part E 65

Ion–Atom Coll 65. Ion–Atom Collisions – High Energy

952

Part E

Scattering Experiment

Part E 65.1

where θ2 = 2u k n 2 /Z 22 and u k is the target binding energy. The function f rises rapidly for V < ve , reaching a value near unity near V = ve and falling very slowly thereafter. Tables of f for K- and L-shell ionization are given in [65.3, 4]. Figure 65.1 shows a comparison of experimental data for K vacancy production by protons with PWBA calculations, and with a classical binary encounter approximation [65.12] for a large range of proton data [65.6]. For larger Z 1 , corrections to the PWBA and SCA must be made for the effective increase of u k due to the presence of the projectile during the ionization, for nuclear projectile deflection, for relativistic corrections, and for the polarization of the electron cloud, as reviewed in [65.13–17]. Total cross section measurements for inner shell vacancy production in the perturbative region are reviewed in [65.15, 16]. In the SCA treatment, the heavy particle motion is taken to be classical, and the evolution of the electronic wave function under the influence of the projectile field is calculated by time-dependent perturbation theory. The assumption of classical motion is valid if the U2k σk / Z12 (keV2 cm2)

Bohr parameter K = 2Z 1 Z 2 e2 /(~V ) is much larger than unity [65.18]. If this condition is satisfied, the projectile scattering angle can be associated with a particular b through a classical deflection function. For K-shell ionization, the action occurs typically at sufficiently small b that a screened Coulomb potential is sufficient for calculating the deflection. In the absence of screening, θ = r0 /b, where r0 = Z 1 Z 2 e2 /E with θ and E expressed in either the laboratory or c.m. system. Calculations for K- and L-shell ionization have been carried out [65.10]. The typical ionization probability P(b) for V ∼ ve and b = 0 is P(0) ∼ (Z 1 /Z 2 )2 . For V < ve , P(b) decreases with increasing b with a characteristic scale length of rad = V/ω, the adiabatic radius, where ω is the transition energy. For V > ve , P(b) cuts off near the K-shell radius of the target. A more sophisticated relativistic SCA program has been written [65.19], and is widely used for calculating P(b), cross sections, and probabilities differential in final electron energy and angle. Experimentally, the probability P(b) for inner shell ionization can be determined from   1 Y P(b) = (65.2) , ω∆Ω N[θ(b)] where Y is the coincidence yield for the scattering of N(θ) ions into a well-defined angle θ(b) accompanied by X-ray (or Auger electron) emission with fluorescence yield ω into a detector of efficiency  and solid angle ∆Ω [65.20]. The necessary ω can be obtained from calculations for neutral targets [65.21] (Chapt. 62). However, they must be corrected for changes due to extensive outer shell ionization during the collision. Such corrections are particularly important for targets with low fluorescence yields, for Z 2 below 30, and for collisions in which the L-shell is nearly depleted in the collision [65.15,16]. Values of P(b) have been measured for many systems and generally show good agreement (better than 10%) with the SCA for fast light projectiles such as protons, with increasing deviation as higher Z 1 or slower V are used [65.22]. Examples of P(b) for K vacancy production for several systems are shown in Fig. 65.2, showing the evolution away from the SCA as the collision becomes less perturbative.

10–19

10–20

10–21

10–22

10–23

10–24

10–25 10–3

10–2

10–1

100

101 E/λUk

Fig. 65.1 Comparison of experimental cross sections for K-shell vacancy production with PWBA (dashed) and binary encounter (solid) theories. Uk is the target binding energy in keV and λ the projectile/electron mass ratio [65.6]

Ionization of Light Target Atoms Ionization of light target atoms by bare ion impact is a particularly suitable process to study the atomic fewbody problem. In the case of an atomic hydrogen target the collision represents a three-body system, i. e., the simplest system for which the Schrödinger equation is not analytically solvable. However, because of the ex-

Ion–Atom Collisions – High Energy

Projectile / Target 10–3

Z2 / Z1

F(5+)/Ar

2.0

0.52

C(4+)/Ar

3.0

0.52

O(6+)/Cu

3.62

0.35

Be/Cu

7.25

0.35

10–3

–3

10

10–3

p/Cu

29

0.35

SCA 1.0

2.0

3.0

b/rad

Fig. 65.2 P(b) for K-shell vacancy production versus b/rad

for several systems (see text). The ratio V/ve is designated as “V” in this figure. For protons p, agreement with the SCA theory is found [65.10], while for higher Z 1 /Z 2 , P(b) moves to larger impact parameters as one leaves the perturbative region [65.22]

perimental difficulties associated with atomic hydrogen, measurements with this target species are rare [65.23] and experimental studies have focused on helium targets. Here, the collision still constitutes a relatively simple four-body system. With regard to the few-body problem, studies of ionization processes have the important advantage that, in contrast to pure excitation and capture processes, the final state involves at least three independently moving particles. Detailed information about the few-body dynamics in a collision can be extracted from multiply differential measurements. This can be accomplished by measuring the kinematic properties (e.g., energy, momentum, ejection angle) of one or more of the collision fragments. The first experimental multiply differential single ionization cross sections were obtained by studying the ionized electron spectra as a function of energy and ejection angle. Such studies were reviewed by

Rudd et al. [65.24] and are discussed in more detail in Sect. 65.3. More recently, complementary multiply differential data were obtained by measuring projectile energy-loss spectra as a function of scattering angle in p + He collisions [65.25, 26]. A comprehensive picture of ionizing collisions can be obtained from kinematically complete experiments. In such a study the momentum vectors of all collision fragments need to be determined. However, in the case of single ionization it is sufficient to directly measure the momentum vectors of any two particles in the final state; the third one is then readily determined by momentum conservation. For ionization by electron impact, this has been accomplished by momentum-analyzing the scattered and the ionized electrons (for a review see [65.27]). For ion impact, this approach is difficult because of the very small scattering angles and energy losses (relative to the initial collision energy) resulting from the large projectile mass. Consequently, the only kinematically complete experiments involving a direct projectile-momentum analysis were reported for light ions at relatively low projectile energies [65.28]. For heavy-ion impact at high projectile energies, in contrast, the complete determination of the final space state is only possible through a direct measurement of the ionized electron and recoil-ion momenta [65.29]. The technology to measure recoil-ion momenta with sufficient resolution, and therefore to perform kinematically complete experiments for heavy-ion impact, has only become available over the last decade (for reviews, see [65.30–32]). Figure 65.3 shows measured (top) and calculated (bottom) three-dimensional angular distributions of electrons ionized in 100 MeV/a.m.u. C6+ + He collisions for fully determined kinematic conditions [65.33]. The arrows labeled po and q indicate the direction of the initial projectile momentum and the momentum transfer defined as the difference between p0 and the final projectile momentum pf . This plot is rich in information about the dynamics of the ionization process. The main feature is a pronounced peak in the direction of q. It can be explained in terms of a binary interaction between the projectile and the electron, i. e., a first-order process, and is thus dubbed the “Binary Peak”. A second, significantly smaller, structure is a contribution centered on the direction of −q (called the “Recoil Peak”). This has been interpreted as a two-step mechanism where the electron is initially kicked by the projectile in the direction of q and then backscattered by the residual target ion by 180◦ . Although this process involves two interactions of the electron, it is nevertheless a first-order process in the projectile–target atom inter-

953

Part E 65.1

P(b)/Z 21

65.1 Basic One-Electron Processes

954

Part E

Scattering Experiment

Part E 65.1

P0 (z-axis)

x

y

q

Fig. 65.3 Three-dimensional angular distribution for fully determined kinematic conditions of electrons ionized in 100 MeV/a.m.u. C6+ + He collisions. Top, experimental data; bottom, CDW calculation (see text)

action. Therefore, as expected for this very large value of η1 = 100 (in a.u.), the ionization cross sections are dominated by first-order contributions. The basic features of the data in Fig. 65.3 are well reproduced even by the relatively simple first Born approximation (FBA). Furthermore, the calculation shown in the bottom of Fig. 65.3, which is based on the more sophisticated continuum distorted wave approach (CDW)([65.35–37] see also Chapt. 52), yields practically identical results to the FBA. In the CDW method, higher-order contributions are accounted for in the final-state scattering wavefunction. Apart from this good overall agreement, a closer inspection of the comparison between experiment and theory also reveals some significant discrepancies. While in the calculation the Binary and Recoil peaks are sharply separated by a minimum near the origin, in the data this minimum is almost completely filled up giving rise to a “ringlike” shape of the recoil peak. This was explained by a higher-order ionization mechanism involving an interaction between the projectile and the residual target

ion [65.33,37,38]. Although the contribution of this process to the total cross section is negligible, it is a very surprising result that for selected kinematic conditions higher-order processes can be important even at large projectile energies. A sobering conclusion of recent research on ionization of light target atoms is that even well inside the perturbative regime the atomic few-body problem is not nearly as well understood as was previously assumed based on studies for restricted collision geometries. At large perturbation, the lack of understanding is dramatic [65.39]. Excitation Inner shell excitation can be treated within the same perturbative framework, which leads to a cross section given in terms of the generalized oscillator strength for the transition [65.40–42]. For inner shell vacancy production by light projectiles, the excitation is generally much smaller than the ionization, since the strongest oscillator strengths are to low-lying occupied orbitals, as reviewed by Inokuti [65.41]. Excitation cross sections can be deduced from photon production cross sections and from inelastic energy loss experiments. An example of the cross section for excitation of the n = 2 level of H by protons, measured by the latter technique, is shown in Fig. 65.4 [65.34]. Contributions of individual terms in dσ/ dξ

σ (n = 3) Continuum terms

σ (n = 2)

σ (n = 4) 0

10

20

30 40 Energy loss (eV)

Fig. 65.4 Energy loss spectrum for 50 keV protons in atomic hydrogen, showing excitation to discrete states in H proceeding smoothly into ionization at the continuum limit [65.34]

Ion–Atom Collisions – High Energy

σOBK = 29 π(Z 1 Z 2 )5 /5 V 2 ν5 n 3 β 5 a02 ,

(65.3)

from a filled shell ν to all final states n, where   1  β = V 2 V 4 + 2V 2 Z 22 /ν2 + Z 12 /n 2 4  2  + Z 22 /ν2 − Z 12 /n 2 .

(65.4)

Both the PWBA/SCA and the OBK cross sections maximize near the matching velocity, but the OBK falls off much more strongly with increasing V beyond this, eventually falling as V −12 , while the ionization cross section only falls as V −2 ln V . The OBK amplitude for capture is simply the momentum space overlap of the initial wave function with the final state wave function, where the latter is simply a bound state on the projectile but moving at a velocity V relative to the initial bound state. The integral is done only over the transverse momentum, since the longitudinal momentum transfer is fixed by energy conservation [65.11]. This capture amplitude thus depends heavily on there being enough momentum present in the initial and/or final wave function to enable the transfer, and the loss of this match is what leads to the steep decrease in the OBK cross section above velocity matching. Cross sections for K-shell capture have been measured by detection of K Auger electrons and K X-rays in coincidence with charge capture by the projectiles [65.22, 45, 46]. On the basis of these and many other data on electron capture, the OBK is a factor of approximately three too large [65.45–48]. This factor comes from a fundamental failure of first-order perturbation theory for electron capture. As pointed out already in 1927 by Thomas [65.49], who proposed a classical two-collision mechanism for capture, it is essential that the electron interacts with both nuclei during the collision in order to be captured (Chapt. 57). In quantum theories, this corresponds to the fundamental need to include second-order terms (and higher) in the capture amplitude. In the limit of large V , the second-order cross section decreases more slowly than the OBK term, as V −10 , and thus is asymptotically larger than the first-order term [65.50]. At large V , the

coefficient of the V −12 term, the dominant one at most experimentally reachable V , is 0.29 times the OBK cross section when the theory is carried out to second-order in the projectile potential [65.50,51]. Roughly speaking, this provides an explanation for the factor of three. Much more sophisticated treatments of high velocity capture are now available [65.52–60]. The underlying role of the second-order scattering process was confirmed experimentally by the detection of the Thomas peak in the angular distribution of protons capturing electrons from He and H [65.61, 62] (Chapt. 57). In spite of the basic importance of second-order amplitudes in perturbative capture, the OBK gives an excellent account of the relative contributions from and to different final shells over a large range of V above ve , and is thus, when appropriately reduced, still useful as an estimate for perturbative capture cross sections between well defined ν and n for large V . For electron capture, as in the case of ionization (see previous section), the development of recoil-ion momentum spectroscopy (RIMS) has enabled much more detailed studies of the collision dynamics. The transverse (perpendicular to the beam direction) recoil-ion momentum component p⊥ reflects the closeness of the collision both relative to the target nucleus and the electrons. The longitudinal (parallel to the beam direction) component pz , on the other hand, is related to the internal energy transfer Q in the collision by (in a.u.) pz = −Q/V − nV/2 ,

(65.5)

where n is the number of captured electrons. A measurement of pz is therefore equivalent to a measurement of Q. The advantage over measuring Q from the projectile energy loss is that at large collision energies a much better energy resolution is achievable. A sample Q measurement with RIMS is shown in Fig. 65.5 [65.63]. Very recently, RIMS was applied to study capture processes in collisions with an atomic hydrogen target [65.64]. This could be an important breakthrough in advancing our understanding of the atomic few-body problem as it opens the possibility to perform kinematically complete experiments on the true three-body system X Z+ + H, where X can be any bare projectile.

65.1.2 Nonperturbative Processes Fano–Lichten Model When the collision becomes increasingly perturbative, either due to a decreased V or increased Z 1 /Z 2 , higher-order effects become generally more important. One approach to account for such contributions

955

Part E 65.1

Capture As Z 1 /Z 2 is raised, the probability for direct transfer of inner shell electrons from projectile to target becomes competitive with, and can even exceed, that for ionization of the target electron into the continuum. The first-order perturbation treatment for electron capture, given by Oppenheimer [65.43] and by Brinkman and Kramers [65.44] (OBK) ([65.11], p. 379) results in a cross section per atom

65.1 Basic One-Electron Processes

956

Part E

Scattering Experiment

Part E 65.1

Counts (arb. units) He2++ He– – He1+ (n) + He1+(n)

0.25 MeV

σ π δ

4fσ

3d

3p Cl 3s Cl

(1, 2) & (2, 1) 3p

(n, n) = (1, 1)

(1,3) & (3, 1)

0.26 a.u. FWHM

3s

(2, 2)

2pπ 2p

–3

–2

–1

0

1 P

rec (a. u.)

2p Ar 2s Ar 2p Cl 2s Cl

Rot coup 2pσ

2s

Fig. 65.5 Longitudinal momentum spectrum of recoil ions

from 0.25 MeV He2+ capturing a single electron from a cold He target, showing clear resolution of capture to n = 1 from that which leaves target or projectile excited [65.63]

is the continuum distorted wave–eikonal initial state (CDW-EIS)model ([65.35, 36, 65], see also Chapt. 52 and Sect. 65.1.1). The range of validity of CDW-EIS is roughly given by Z 1 /V 2  1 [65.35]. Therefore, if the perturbation is large due to the projectile charge, the collision may still be treated perturbatively provided that the collision energy is sufficiently large. Otherwise, the perturbation treatment is replaced by a molecular orbital treatment. Fano and Lichten [65.67] pointed out that the ratio V/ve can be small for inner orbitals even for V of several a.u., and thus an adiabatic picture of the collision holds. K vacancy production cross sections become much larger than the perturbation treatments above predict and extend to much larger b. In the molecular orbital picture, the collision system is described in terms of time-dependent molecular orbitals (MO) formed when the inner shells of the systems overlap. Vacancy production occurs due to rotational, radial, and potential coupling terms between these orbitals during the collision. The independent electron model is used, but the results in any specific collision are quite sensitive to the occupation numbers (or vacancies) in the initial orbitals. These are very difficult to control in ion–atom collisions in solids and even problematic in gases, since outer shell couplings can produce vacancies at large internuclear distances which then enable transfers at smaller distances. Numerous reviews of the subject are available, including [65.68–72]. The most famous MO ionization mechanism involves the

Radial coup

ls Cl ls Ar

lsσ

ls R=O

R=∞

Fig. 65.6 Schematic correlation diagram for the Cl−Ar

system, indicating the rotational coupling and radial couplings important for K vacancy production and the 4fσ orbital whose promotion leads to L vacancy production [65.66]

promotion of the 4fσ orbital in a symmetric collision (Fig. 65.6), which promotes both target and projectile L electrons to higher energies where they are easily lost to the continuum during the collision. There are now many treatments of inner shell vacancy production mechanisms based on MO expansions (Chapts. 50, 51). For the case of K vacancy production in quasisymmetric collisions, an important MO mechanism is the transfer of L vacancies in the projectile to the K-shell of the target through the rotational coupling between 2pπ and 2pσ orbitals which correlate to the L- and K-shells respectively of the separated systems (Fig. 65.6) [65.73, 74]. The process can be dynamically altered by the sharing at large b between L vacancies of target and projectile through a radial coupling mechanism [65.75, 76]. This sharing mechanism can also give rise to the direct transfer of K vacancies from projectile to target (KK sharing). In symmetric systems, the KK sharing results in an oscillation of the K vacancy back and forth between target and projectile during the collision, and leads to an oscillatory behavior of the transfer prob-

Ion–Atom Collisions – High Energy

tive to ionization. While the MO correlation diagrams and mechanisms are qualitatively useful, actual close coupling calculations for both inner and outer shell processes are often carried out using atomic orbitals instead of molecular orbitals, as well as other basis sets (Chapts. 50, 51).

65.2 Multi-Electron Processes In a single collision between multi-electron partners, two or more electrons may be simultaneously excited or ionized. The electric fields created during a violent ion–atom collision are so large that the probability of such multi-electron processes can be of order unity. While there are many similarities between ion–atom collisions and the interaction of atoms with photons (X-rays or short laser pulses) or electrons, the dominance of multi-electron processes is very much less common in the photon and electron cases. As an example, when a K-shell electron is removed from a target atom by the passage of a fast highly charged ion through its heart, the probability that L-shell electrons will be removed at the same time can be large. This gives rise to target X-ray and Auger-electron spectra which are dominated by satellite structure [65.20]. For example, the spectrum Number of counts 104 Kα1.2(2p)6 103

Kα3.4(2p)5

2

10

of X-rays from Ti bombarded by 30 MeV oxygen shows that the production of the K vacancy is accompanied by multiple L vacancy production, and that the dominant K X-rays are those of systems which are missing several L electrons [65.79] (Fig. 65.7). When a gas target is used, the recoil target ion is heavily ionized and/or excited electronically without receiving much translational kinetic energy. In the impulse approximation the transverse momentum ∆ p⊥ received by the target from a projectile passing at impact parameter b is given in a.u. by ∆ p⊥ = 2Z 1 Z 2 /bV . This expression ignores the exchange of electronic translational momentum but gives a good estimate. The resulting recoil energies are typically quite small, ranging from thermal to a few eV. This subject has been reviewed in [65.63,80]. These slow moving recoils have

H1 + Ti Ep = 0.8 MeV

Kβ1.3(2p)6 Kβ5 Kβ(2p)5

vp = vα

101 104

Kα3.4(2p)5 Kα3.6(2p)4

Kα1.2(2p)5

3

10

102

He4 + Ti Eα = 3.2 MeV

Kβ1.3(2p)6

Kβ3 Kβ(2p)5 Kβ(2p)4

101 Kα(2p)4

105 104

3

Kα(2p) Kα(2p)6

Kα(2p)3 Kα(2p)2 Kα(2p)1

O16 + Ti Eα = 30 MeV

Kβ(2p)4 5

Kβ(2p) Kβ(2p) 6

103

Kβ(2p)3 Kβ(2p)2

Kβ(2p)1

102 101

2.80

2.70

2.60

2.50

2.40 Wavelength (Å)

Fig. 65.7 K X-ray spectrum of Ti for various projectiles, showing dominance  of multi-electron transitions when

K vacancies are collisionally produced by heavily ionizing projectiles

16 O

beam [65.79]

957

Part E 65.2

ability with V and b [65.77, 78]. Both of the above vacancy production mechanisms are electron transfer processes rather than direct ionization processes, in that no inner shell electron need be liberated into the continuum. Between the perturbation region and the full MO region the importance of transfer increases rela-

65.2 Multi-Electron Processes

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Part E 65.2

been used to provide information about the primary collision dynamics, and as secondary highly charged ions from a fast-beam-pumped ion source. Such an ion source has, for moderately charged ions, a high brightness and has been used extensively for energy-gain measurements. The primary recoil production process is difficult to treat without the independent electron model, and even in this model the nonperturbative nature of the collision makes the theory difficult. The most successful treatments have been the CTMC (see Chapt. 58) and a solution of the Vlasov equation [65.64]. Studies of many-electron transitions in collisions of bare projectiles with a He target are particularly suitable to investigate the role of electron–electron correlation effects because such collisions represent the simplest systems where the electron–electron interaction is present. Such studies have been performed extensively for a variety of processes, such as double ionization, transfer-ionization, double excitation, transfer-excitation, or double capture (for reviews see [65.82–84]). It is common to distinguish (somewhat artificially) between such correlations in the initial state, the final state, and during the transition (dynamic correlation). From a theoretical point of view, the biggest challenge is to describe electron–electron correlation effects and the dynamics of the two-center potential generated by the projectile and the target nucleus simultaneously with sufficient accuracy. In the case of double ionization, an experimental method, based on the so-called correlation function [65.81], was developed to analyze electron–electron correlations independently of the collision dynamics. Here, a measured two-electron spectrum (for example the momentum difference spectrum of both ionized electrons) is normalized to the corresponding spectrum one would obtain for two independent electrons. An example of such a correlation function R is shown in Fig. 65.8 for three very different collision systems (η1 ranging from 0.05 to 100 and η2 from 0.01 to 0.5 in a.u.). The similarity in these three data sets illustrates that R is remarkably insensitive to the collision dynamics. Rather, the shape of R is determined predominantly by correlations in the final state [65.81, 85]. However, for selected kinematic conditions, R can also be sensitive to initial-state correlations [65.86]. Clear signatures of initial-state correlations were found in the recoil-ion momentum spectra for transfer-ionization [65.87]. Early attempts to identify dynamic correlations were based on measurements of the ratio of double to single ionization cross sections [65.88, 89]. From such studies, it was found that at small V double ionization

R 100 MeV/amu C6+ + He 3.6 MeV/AMU Au53++ He 3.6 MeV/AMU Au53++ Ne

0.8

0.4

0.0

– 0.4

– 0.8 0

2

4

6

8
p1–p2
(a. u.)

Fig. 65.8 Correlation function R for double ionization in the collisions indicated in the legend as a function of the momentum difference between the two electrons [65.81]. R is defined as R = Iexp /IIEM − 1, where Iexp is the directly measured momentum spectrum and IIEM the one obtained for independent electrons

is dominated by an uncorrelated mechanism involving two independent interactions of the projectile with both electrons. In contrast, at large V the double to single ionization ratio asymptotically approaches a common value for all collision systems [65.90]. This is indicative of the dominance of first-order double ionization mechanisms, where the projectile interacts with only one electron and the second electron is ionized through an electron–electron correlation effect. This may either be a rearrangement process of the target atom adjusting to a new Hamiltonian (shake-off, an initial-state correlation), or a direct interaction with the first electron (i. e., dynamic correlation). However, a recent nearly kinematically complete experiment on double ionization in p + He collisions revealed that even at large V higher-order contributions are not negligible [65.91]. In Fig. 65.9 the ejection angles of both electrons are plotted against each other for almost completely determined kinematics. For comparison, the bottom part of Fig. 65.9 shows the corresponding spectra for electron impact at the same V [65.92]. For both projectiles, the basic features of these spectra are determined by the

Ion–Atom Collisions – High Energy

b) ϑ 2

c) ϑ 2

225

225

225

180

180

180

135

135

135

90

90

90

45

45

45

270

270

270

0

0

0

–45

–45

–45

–90 –90 –90 –90 –45 0 45 90 135 180 225 270 –90 –45 0 45 90 135 180 225 270 –90 –45 0 45 90 135 180 225 270 ϑ 1 e) ϑ 1 f) ϑ1 d) ϑ ϑ2 ϑ2 2 270 270 270 225

225

225

180

180

180

135

135

135

90

90

90

45

45

45

0

0

0

–45

–45

–45

–90 –90 –90 –90 –45 0 45 90 135 180 225 270 –90 –45 0 45 90 135 180 225 270 –90 –45 0 45 90 135 180 225 270 ϑ1 ϑ1 ϑ1

0 2 4 6 8 10 12 14 16 18

0 6 12 18 24 30 36 42 48 54 60

Fig. 65.9a–f Differential double ionization cross sections in 6 MeV p + He (top) and 2 keV e− + He (bottom) collisions

as a function of the polar emission angle of both electrons, which are emitted into the scattering plane. The electrons have equal energy and data are shown for small (left), medium (center), and large momentum transfers (right) [65.91]

electric dipole selection rules, which again is indicative of dominating first-order contributions. However, a closer inspection of the comparison between the proton and electron impact data shows some non-negligible

differences. Since in a first-order treatment the cross sections should be identical for both projectile species, this demonstrates that higher-order contributions cannot be ignored.

65.3 Electron Spectra in Ion–Atom Collisions 65.3.1 General Characteristics An ionizing collision between a single ion and a neutral atom ejects electrons into the continuum via two major processes. Electrons ejected during the collision form broad features or continua, and are traditionally referred to as delta rays; electrons ejected after the collisions from the Auger decay of vacancies created during the collision form sharp lines in the spectra. The distributions of energy and angle of all electrons determine the electronic stopping power and characteristics of track formation of ions in matter (Chapt. 91), and the study

of these distributions in the binary encounter of one ion with one atom form the basis of any detailed understanding of these averaged quantities. Figure 65.10 shows a typical electron spectrum from the collision of a fast O ion with O2 [65.93, 94]. Electrons from the projectile can be identified in the cusp peak (electron loss, P) or ELC, and the O-K-Auger (P) peak. Electrons from the target include the soft (large b) collision electrons (T) which are ejected directly by Coulomb ionization by the projectile, the binary collision (or encounter) electrons coming from hard collisions between projectile and quasifree target electrons, and the target O-K-Auger (T)

959

Part E 65.3

a) ϑ 2

65.3 Electron Spectra in Ion–Atom Collisions

960

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Part E 65.3

6

Fig. 65.10 Electron spectrum from 30 MeV O on molecular

Cross section × Electron energy (10–17cm2/sr)

oxygen. See text for explanation of features [65.93, 94]

30 MeV 05+on O2

Soft collisions (T) 5 O – K Auger (T) 4

Binary collisions (T)

Electron loss (P)

3

O– K Auger (P)

2

25°

1 0 1

30°

0 1 40° 0 1 90° 0

6

1

2

3 4 Electron energy (keV)3

d2σ/ddε (10–20 cm2/eV sr) 3.91 MeV B4+ + H2

2l2l 5 4

2p2 1D

2s2p 1P

2lnl Series limit

3 2s2p 3P 2s2 1S

2p2 1S

2l3l

2

2l4l 2l5l

1

65.3.2 High Resolution Measurements

0 dσ/d (10–18 cm2/sr)

180

electrons. The electron loss peak is widely called the cusp peak because the doubly differential cross section in the laboratory d2 σ/ dE dΩ becomes infinite, in principle, if it is finite in the projectile frame. In general, this peak may also contain capture to the continuum. All of these features have been heavily studied; some reviews are [65.95, 96]. Capture to the continuum [65.96] is an extension of normal capture into the continuum of the projectile, and is not a weak process. Both it and ELC produce a heavy density of events in the electron momentum space centered on the projectile velocity vector, and thus appear strongly only at or near zero degrees in the laboratory and at ve  V . The binary encounter electrons at forward angles occur at ve ∼ = 2V . For relatively slow collisions it was found that the ELC and binary peaks are just part of a more general and complex structure of the electron spectra [65.98]. Additional peaks in the forward direction were found for ve ∼ = nV , where n in principle can be any integer number. These structures reflect a “bouncing back and forth” (known as Fermi shuttle) between the projectile and the target core before the electron eventually gets ejected from the collision system. In electron spectra for molecular targets, additional structures were found that were not observed for atomic targets [65.99]. These were initially interpreted as an interference effect. The electronic wavefunction has maxima at the atomic centers of the molecule. Since in the experiment it cannot be distinguished from which center the electron is ionized, both possibilities have to be treated coherently. However, more recent studies showed that at small electron energies the structures in the electron spectra reflect vibrational excitation of the molecule [65.100].

The Auger electron spectra provide detailed information about inner shell vacancy production mechanisms.

e– + B4+(1s)

190

200

210

220

230

240 250 260 Electron energy (eV)

Fig. 65.11 High resolution Auger electron spectrum from H-like B on H2 , showing resolved lines from doubly excited projectile states lying on top of a continuum due to electron elastic scattering [65.97]. The bottom part shows an R-matrix calculation which does not account for the experimental resolution. The smooth line in the upper figure is the R-matrix calculation convoluted with the experimental spectrometer resolution

Ion–Atom Collisions – High Energy

angle of the spectrometer and velocity of the emitter, but at 0◦ to the beam this problem vanishes, and the resolution in the emitter frame is actually enhanced by the projectile motion, such that for electrons with eV energies in the projectile frame, resolutions in the meV region are possible [65.101,102]. The highest resolution Auger lines from ion–atom collisions has been done on the projectiles. A sample spectrum is shown in Fig. 65.11. From such high resolution spectra, one-electron processes in which one electron is excited, captured or ionized can be distinguished from the configuration of the emitting state.

65.4 Quasi-Free Electron Processes in Ion–Atom Collisions At sufficiently low V , those electrons not actively involved in a transition play only a passive role in screening the Coulomb potential between the nuclei, and thereby create a coherent effective potential for their motion. However, at high V the colliding electrons begin behaving as incoherent quasifree particles capable of inducing transitions directly via the electron–electron interaction. Such processes signal their presence through their free-particle kinematics, as if the parent nucleus were not present. For example, a projectile ionization process requiring energy U has a threshold at 12 m e V 2  U, in collisions with light targets where the quasifree picture is meaningful. The threshold is not sharp, due to the momentum distribution or Compton profile of the target electrons. Within the impulse approximation, the cross section for any free electron process can be related to the corresponding cross section for the ion–atom process by folding the free electron cross section into the Compton profile [65.97, 103, 104].

65.4.1 Radiative Electron Capture The first quasifree electron process to be observed was radiative electron capture (REC), the radiative capture of a free electron by an ion. Conservation of energy and momentum is achieved by the emission of a photon which carries away the binding energy. The cross section exceeds that for bound state capture at high V . Radiative electron capture was observed through the X-ray spectra from fast heavy projectiles for which the electrons of light targets appear to be ‘quasi-free’. The corresponding free electron process was seen [65.105] and has recently been heavily studied in EBIT [65.106], cooler [65.107], and storage rings [65.108]. Total cross

sections for REC have also been deduced from measured total capture cross sections at large V where REC dominates bound state capture [65.109]. At high velocities, the cross section for radiative capture to the K-shell of a bare projectile is given approximately by
  σn = n/ κ −2 + κ −4 × 2.1 × 10−22 cm2 , (65.6) √ where [65.110] κ = E B /E 0 , E B is the binding energy of the captured electron, E 0 the energy of the initial electron in the ion frame and n the principal quantum number of the captured electron. The theory seems to be in good agreement with experiment for capture to all shells of fast bare projectiles, although a small unexplained discrepancy between theory and experiment exists for capture to the K-shell [65.108].

65.4.2 Resonant Transfer and Excitation Dielectronic recombination in electron–ion collisions is the process whereby an incident electron excites one target electron and, having suffered a corresponding energy loss, drops into a bound state on the projectile (Chapt. 55). If the doubly excited state so populated decays radiatively, resonant radiative recombination is achieved (DR); if it Auger decays, resonant elastic scattering has occurred. The process has long been known to be important as a recombination process in hot plasmas [65.111], but was not observed in the laboratory until 1983 [65.112–114]. The corresponding ion–atom process, known as resonant transfer and excitation (RTE) was seen a bit earlier by Tanis et al. [65.115]. (See [65.116, 117] for reviews of both DR and RTE.)

961

Part E 65.4

When coupled with fluorescence yields, Auger electron production probabilities and cross sections can be converted into the corresponding quantities for vacancy production [65.15, 95]. This is best done when sufficient resolution can be obtained to isolate individual Auger lines. The Auger spectra in ion–atom collisions are often completely different from those obtained from electron or photon bombardment because of the multiple outer shell ionization which attends the inner shell vacancy producing event, in close analogy to X-ray spectra (see previous section). Projectile Auger electron spectra suffer from kinematic broadening due to the finite solid

65.4 Quasi-Free Electron Processes in Ion–Atom Collisions

962

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Part E 65.5

65.4.3 Excitation and Ionization Excitation and ionization of inner shells of fast projectile ions by the quasifree electrons of light targets (usually He or H2 ) have been identified and studied. This process competes with excitation and ionization by the target nucleus [65.118–120], and special signatures must be sought to distinguish the processes. In the case of excitation, the e–e excitation populates states through the exchange part of the interaction which is excluded for the nuclear excitation, and this has been used to separate this mechanism [65.121]. For ionization, enhancements of the ionization cross section above the Born result for nuclear ionization [65.122], coincident charge exchange measurements [65.123], and

projectile [65.124] and recoil ion momentum spectroscopy [65.125, 126] have been used to distinguish the two processes. Rapid development in the production of good sources of beams of highly charged ions (EBIS/T, ECR; see [65.127]) have made these studies possible. Continued study of this field in heavy ion storage rings is now achieving resolutions of meV and opening broad new opportunities for data of unprecedented high quality for electron-ion collisions [65.128]. Recently, the first kinematically complete experiment on projectile ionization by quasifree electrons was reported [65.129]. The observed features are qualitatively similar to those found for ionization of neutral target atoms by free electron impact.

65.5 Some Exotic Processes 65.5.1 Molecular Orbital X-Rays A typical time duration for an ion–atom encounter is ∼ 10−17 s, which is much shorter than Auger and X-ray lifetimes, so that hard characteristic radiation is emitted by the products long after the collision. There remains, however, a small but finite probabiltiy that X-rays or Auger electrons can be emitted during the collision, in which case the radiation proceeds between the time-dependent molecular orbitals formed in the collision and reflects the time evolution of the energies and transition strengths between the orbitals. Such molecular orbital X-radiation (MOX) has now been observed in many collision systems [65.130] and is reviewed in [65.131–133]. MOX spectra have been studied in total cross sections as well as a function of impact parameter. In the latter case, oscillating structures in the MOX spectra are seen [65.132, 134], due to the interference between amplitudes for the emission of X-rays with the same energy on the incoming and outgoing parts of the trajectory. The formation of transient molecular orbitals in close collisions between highly charged ions provides opportunities for studying the electrodymamics of very highly charged systems [65.135]. For example, two uranium nuclei passing within one K-shell radius of each other form a transient molecule whose energy levels resemble those of an atom of charge 184.

65.5.2 Positron Production from Atomic Processes Investigating the MOX interference patterns in such exotic systems offers the interesting prospect of performing spectroscopy on superheavy ions. Reinhart et al. [65.136] predicted that, for such highly charged species, the binding energy of the united atom K-shell exceeds twice the rest mass energy of the electron, and that if a K vacancy is either brought into the collision or created during it, spontaneous electronpositron pair production occurs (the decay of the charged vacuum) with the electron filling the K-hole. However, further analysis showed that the dominant mechanism for positron production (other than those resulting from the decay of nuclear excitations) likely results from the dynamic time dependence of the fields during the collision [65.137]. Experiments showed evidence for such positron production in collisions at 6 MeV/u [65.138], but reported sharp lines in the positron spectra were later attributed to an error in the data analysis. Present theories for the production of lepton pairs in the close collision of two highly charged systems predict that the cross section grows rapidly with collision energy. Electrons produced in such a process may end up in bound states on either collision partner, and thus represent a new charge changing mechanism. At highly relativistic velocities, the cross section for this process exceeds that for any other charge changing process. In a heavy ion collider, such as RHIC,

Ion–Atom Collisions – High Energy

with theory [65.140]. The bound state capture has been measured at lower energy by Belkacem et al. [65.141], with similarly good agreement. The extension of ion– atom collisions to such extreme velocities has opened the field for the study of processes not even imagined a short time ago.

References 65.1 65.2

65.3 65.4 65.5 65.6 65.7

65.8 65.9 65.10 65.11

65.12 65.13 65.14 65.15 65.16 65.17 65.18 65.19 65.20 65.21

65.22 65.23

E. Merzbacher, H. W. Lewis: Handbuch der Physik, Vol. 34th (Springer, Berlin 1958) p. 166 D. Madison, E. Merzbacher: Theory of charge– particle excitation. In: Atomic Inner Shell Processes, ed. by B. Craseman (Academic, New York 1975) B. S. Khandelwal, B. H. Choi, E. Merzbacher: At. Data 15, 103 (1969) B. S. Khandelwal, B. H. Choi, E. Merzbacher: At. Data 5, 291 (1973) B. Crasemann (Ed.): Atomic Inner Shell Processes (Academic, New York 1975) J. D. Garcia, R. J. Fortner, T. M. Kavanagh: Rev. Mod. Phys. 45, 111 (1973) P. Richard: Atomic physics: Accelerators. In: Meth. Expt. Physics, Vol. 17, ed. by L. Williams (Academic, New York 1980) D. Williams: Spectroscopy. In: Meth. Expt. Physics, Vol. 13 (Academic, New York 1976) J. Bang, J. M. Hansteen: Kgl. Dan. Viden. Mat.-Fys. Medd. 31, 13 (1973) J. M. Hansteen, O. M. Johnsen, L. Kocbach: At. Data Nucl. Data Tables 15, 305 (1975) M. R. C. McDowell, J. P. Coleman: Introduction to the Theory of Ion-Atom Collisions (North Holland, Amsterdam 1970) J. D. Garcia: Phys. Rev. A 4, 955 (1971) R. Anholt: Phys. Rev. A 17, 983 (1978) G. Basbas, W. Brandt, R. Laubert: Phys. Rev. A 7, 983 (1973) J. Gray T. Meth. Expt. Physics, Vol. 17 (Academic, New York 1980) p. 193 H. Paul, J. Muhr: Phys. Rep. 135, 47 (1986) W. Brandt, G. Lapicki: Phys. Rev. A 10, 474 (1974) N. Bohr: K. Dan. Vidensk Selsk. Mat. Fys. Medd. 18 (1948) No. 8 M. Pauli, D. Trautmann: J. Phys. B 11, 667, 2511 (1978) R. L. Kauffman, P. Richard: Meth. Expt. Physics, Vol. 17 (Academic, New York 1980) p. 148 W. Bambynek, C. Crasemann, R. W. Fink, H.-U. Freund, H. Mark, C. D. Swift, R. E. Price, P. V. Rao: Rev. Mod. Phys. 44, 716 (1972) C. L. Cocke: Meth. Expt. Physics, Vol. 17 (Academic, New York 1980) p. 303 J. T. Park, J. E. Aldag, J. M. George, J. L. Peacher, J. H. McGuire: Phys. Rev. A15, 508 (1977)

65.24 65.25 65.26

65.27 65.28 65.29

65.30

65.31

65.32 65.33

65.34 65.35 65.36 65.37

65.38 65.39 65.40 65.41 65.42 65.43 65.44

M. E. Rudd, Y.-K. Kim, D. H. Madison, T. J. Gay: Rev. Mod. Phys. 64, 441 (1992) M. Schulz, T. Vajnai, A. D. Gaus, W. Htwe, R. E. Olson: Phys. Rev. 54, 2951 (1996) T. Vajnai, A. D. Gaus, J. A. Brandt, W. Htwe, D. H. Madison, R. E. Olson, J. L. Peacher, M. Schulz: Phys. Rev. Lett. 74, 3588 (1995) H. Ehrhardt, K. Jung, G. Knoth, P. Schlemmer: Z. Phys. D1, 3 (1986) L. An, Kh. Khayyat, M. Schulz: Phys. Rev. A63, 030703(R) (2001) R. Moshammer, J. Ullrich, M. Unverzagt, W. Schmidt, P. Jardin, R. E. Olson, R. Mann, R. Dörner, V. Mergel, U. Buck, H. Schmidt-Böcking: Phys. Rev. Lett. 73, 3371 (1994) R. Dörner, V. Mergel, O. Jagutzki, L. Spielberger, J. Ullrich, R. Moshammer, H. Schmidt-Böcking: Physics Reports 330, 95 (2000) J. Ullrich, R. Moshammer, A. Dorn, R. Dörner, L. P. H. Schmidt, H. Schmidt-Böcking: Rep. Prog. Phys. 66, 1463 (2003) C. L. Cocke: Physics Scripta T110, 9 (2004) M. Schulz, R. Moshammer, D. Fischer, H. Kollmus, D. H. Madison, S. Jones, J. Ullrich: Nature 422, 48 (2003) J. T. Park, J. E. Aldag, J. M. Geroge, J. L. Peacher: Phys. Rev. A 14, 608 (1976) D. S. F. Crothers, J. F. McCann: J. Phys. B16, 3229 (1983) D. H. Madison, M. Schulz, S. Jones, M. Foster, R. Moshammer, J. Ullrich: J. Phys. B35, 3297 (2002) D. H. Madison, D. Fischer, M. Foster, M. Schulz, R. Moshammer, S. Jones, J. Ullrich: Phys. Rev. Lett. 91, 253201 (2003) M. Schulz, R. Moshammer, D. Fischer, J. Ullrich: J. Phys. B36, L311 (2003) M. Schulz, R. Moshammer, A. N. Perumal, J. Ullrich: J. Phys. B35, 161L (2002) N. F. Mott, S. N. Massey: The Theory of Atomic Collisions (Oxford Univ. Press, Oxford 1965) M. Inokuti: Rev. Mod. Phys. 43, 297 (1971) H. Bethe: Ann. Phys. 5, 325 (1930) J. R. Oppenheimer: Phys. Rev. 31, 349 (1928) H. S. Brinkman, H. A. Kramers: Proc. Acad. Sci. Amsterdam 33, 973 (1930)

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Part E 65

this process could limit the ultimate storage time for the counter-propagating beams, since charge-exchanged ions are lost. The cross section has been measured recently by Vane et al. [65.139], for 6.4 TeV S on several targets (the highest energy ion–atom collision experiment performed to date), and are in good agreement

References

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Part E 65

65.45 65.46 65.47 65.48 65.49 65.50 65.51 65.52 65.53 65.54 65.55 65.56 65.57 65.58 65.59 65.60 65.61 65.62 65.63

65.64 65.65 65.66 65.67 65.68 65.69 65.70 65.71 65.72

65.73 65.74 65.75 65.76 65.77 65.78

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65.85 65.86 65.87

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65.89 65.90

65.91

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65.93 65.94 65.95

65.96 65.97

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Ion–Atom Collisions – High Energy

65.101

65.102

65.103

65.104 65.105

65.106 65.107 65.108

65.109 65.110

65.111 65.112

65.113 65.114

65.115

65.116

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65.121 T. Zouros, D. H. Lee, P. Richard: Proceedings of the XVI International Conference on the Physics of Electronic and Atomic Collisions, New York, 1989, AIP Conference Proceedings No. 205, ed. by A. Dalgarno, R. S. Freund, M. S. Lubell, T. B. Lucatorto (AIP, New York 1990) p. 568 65.122 W. E. Meyerhof, H.-P. Hülskötter, Q. Dai, J. H. McGuire, Y. D. Wang: Phys. Rev. A. 43, 5907 (1991) 65.123 E. C. Montenegro, W. S. Melo, W. E. Meyerhof, A. G. de Pinho: Phys. Rev. Lett. 69, 3033 (1992) 65.124 E. C. Montenegro, A. Belkacem, D. W. Spooner, W. E. Meyerhof, M. B. Shah: Phys. Rev. A 47, 1045 (1993) 65.125 W. Wu, K. L. Wong, R. Ali, C. Y. Chen, C. L. Cocke, V. Frohne, J. P. Giese, M. Raphaelian, B. Walch, R. Dörner, V. Mergel, H. Schmidt-Böcking, W. E. Meyerhof: Phys. Rev. Lett. 72, 3170 (1994) 65.126 R. Dörner, V. Mergel, R. Ali, U. Buck, C. L. Cocke, K. Froschauer, O. Jagutzki, S. Lencinas, W. E. Meyerhof, S. Nüttgens, R. E. Olson, H. Schmidt-Böcking, L. Spielberger, K. Tökesi, J. Ullrich, M. Unverzagt, W. Wu: Phys. Rev. Lett. 72, 3166 (1994) 65.127 C. L. Cocke: Progress in atomic collisions with multiply charged ions. In: Review of Fundemental Processes and Application of Ions and Atoms, ed. by C. D. Lin (World Scientific, Singapore 1993) p. 138 65.128 R. Schuch: Cooler storage rings: New tools for atomic physics. In: Review of Fundemental Processes and Application of Ions and Atoms, ed. by C. D. Lin (World Scientific, Singapore 1993) p. 169 65.129 H. Kollmus, R. Moshammer, R. E. Olson, S. Hagmann, M. Schulz, J. Ullrich: Phys. Rev. Lett. 88, 103202–1 (2002) 65.130 F. W. Saris, W. F. van der Weg, H. Tawara, R. Laubert: Phys. Rev. Lett. 28, 717 (1972) 65.131 P. O. Mokler: Quasi molecular radiation. In: Topics in Current Physics, Vol. 5, ed. by I. A. Sellin (Springer, Berlin 1978) p. 245 65.132 R. Schuch, M. Meron, B. M. Johnson, K. W. Jones, R. Hoffmann, H. Schmidt-Böcking, I. Tserruya: Phys. Rev. A. 37, 3313 (1988) 65.133 R. Anholt: Rev. Mod. Phys. 57, 995 (1985) 65.134 I. Tserruya, R. Schuch, H. Schmidt-Böcking, J. Barrette, W. Da-Hai, B. M. Johnson, M. Meron, K. W. Jones: Phys. Rev. Lett. 50, 30 (1983) 65.135 W. Pieper, W. Geiner: Z. Phys. 218, 126 (1969) 65.136 J. Reinhardt, U. Müller, B. Müller, W. Greiner: Z. Phys. A 303, 173 (1981) 65.137 G. Soff, J. Reinhard, B. Müller, W. Greiner: Z. Phys. A 294, 137 (1980) 65.138 U. Müller-Nehrer, G. Soff: Electron excitations in superheavy quasi-molecules, Phys. Rep. 246 (1994) 65.139 C. R. Vane, S. Datz, P. F. Dittner, H. F. Krause, C. Bottcher, M. Strayer, R. Schuch, H. Gao, R. Hutton: Phys. Rev. Lett. 69, 1911 (1992) 65.140 C. Bottcher, M. Strayer: Phys. Rev. A 39, 4 (1989)

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J. A. Tanis, M. E. Galassi, R. D. Rivarola: Phys. Rev. Lett. 87, 023201 (2001) C. Dimopoulou, R. Moshammer, D. Fischer, C. Höhr, A. Dorn, P. D. Fainstein, J. R. C. L. Urrutia, C. D. Schröter, H. Kollmus, R. Mann, S. Hagmann, J. Ullrich: Phys. Rev. Lett. 93, 123203 (2004) A. Itoh, T. Schneider, G. Schiwietz, Z. Roller, H. Platten, G. Nolte, D. Schneider, N. Stolterfoht: J. Phys. B. 16, 3965 (1983) D. H. Lee, P. Richard, T. J. M. Zouros, J. M. Sanders, J. L. Shinpaugh, H. Hidmi: Phys. Rev. A. 41, 4816 (1990) C. L. Cocke: Recent trends in ion–atom collisions. In: Electronic and Atomic Collisions, Invited Papers, XVII ICPEAC, Book of Abstracts, ed. by I. E. MacCarty, W. R. MacGillivray, M. C. Standage (Adam Hilger, Bristol, Philadelphia and New York 1992) p. 49 I. A. Sellin: , Vol. 376, ed. by D. Berenyi, D. Hock (Springer, Berlin 1990) H. W. Schnopper, H. D. Betz, J. P. Delvaille, K. Kalata, A. R. Sohval, K. W. Jones, H. E. Wegner: Phys. Rev. Lett. 29, 898 (1972) R. W. Marrs: Phys. Rev. Lett. 60, 1757 (1988) L. H. Andersen, J. Bolko: J. Phys. B 23, 3167 (1990) T. Stöhlker, C. Kozhuharov, A. E. Livongston, P. H. Mokler, Z. Stachura, A. Warczak: Z. Phys. D. 23, 121 (1992) H. Gould, D. Greiner, P. Lindstrom, T. J. M. Symons, H. Crawford: Phys. Rev. Lett. 52, 180 (1984) H. A. Bethe, E. E. Salpeter: Encyclopidia of Physics, Vol. 35, ed. by S. Fluegge (Springer, Berlin 1957) p. 408 A. Burgess: Astrophy. J. 139, 776 (1964) J. B. A. Mitchell, C. T. Ng, J. L. Forand, D. P. Levac, R. E. Mitchell, A. Sen, D. B. Miko, J. Wm. McGowan: Phys. Rev. Lett. 50, 335 (1983) D. S. Belic, G. H. Dunn, T. J. Morgan, D. W. Mueller, C. Timmer: Phys. Rev. Lett. 50, 339 (1983) P. F. Dittner, S. Datz, P. D. Miller, C. D. Moak, P. H. Stelson, C. Bottcher, W. B. Dress, G. D. Alton, N. Neskovic, C. M. Fou: Phys. Rev. Lett. 51, 31 (1983) J. A. Tanis, S. M. Shafroth, J. E. Willis, M. Clark, J. Swenson, E. N. Strait, J. R. Mowat: Phys. Rev. Lett. 47, 828 (1981) W. G. Graham, W. Fritsch, Y. Hahn, J. H. Tanis (Eds.): Recombination of Atomic Ions, NATO ASI Ser. B (Plenum, New York 1992) p. 296 J. A. Tanis: Resonant transfer excitation (RTE) associated with single X-ray emission. In: Recombination of Atomic Ions, NATO ASI Series B, Vol. 296, ed. by W. G. Graham, W. Fritsch, Y. Hahn, J. H. Tanis (Plenum, New York 1992) p. 241 D. R. Bates, G. Griffing: Proc. Phys. Soc. London A 66, 961 (1955) D. R. Bates, G. Griffing: Proc. Phys. Soc. London A 67, 663 (1954) D. R. Bates, G. Griffing: Proc. Phys. Soc. London A 68, 90 (1955)

References

966

Part E

Scattering Experiment

Part E 65

65.141 A. Belkacem, H. Gould, B. Feinberg, R. Bossingham, W. E. Meyerhof: Phys. Rev. Lett. 71, 1514 (1993)

65.142 W. König, F. Bosch, P. Kienle, C. Kozhukarov, H. Tsertos, E. Berdermann, S. Muckler, W. Wagner: Z. Phys. A. 328, 129 (1987)

967

Reactive Scatt 66. Reactive Scattering

66.1 Experimental Methods ......................... 66.1.1 Molecular Beam Sources............. 66.1.2 Reagent Preparation.................. 66.1.3 Detection of Neutral Products ..... 66.1.4 A Typical Signal Calculation.........

967 967 968 969 971

66.2 Experimental Configurations ................ 971 66.2.1 Crossed-Beam Rotatable Detector 971

66.2.2 Doppler Techniques ................... 973 66.2.3 Product Imaging ....................... 973 66.2.4 Laboratory to Center-of-Mass Transformation ......................... 975 66.3 Elastic and Inelastic Scattering ............. 66.3.1 The Differential Cross Section ...... 66.3.2 Rotationally Inelastic Scattering................................. 66.3.3 Vibrationally Inelastic Scattering................................. 66.3.4 Electronically Inelastic Scattering.................................

976 976

66.4 Reactive Scattering .............................. 66.4.1 Harpoon and Stripping Reactions ................................. 66.4.2 Rebound Reactions ................... 66.4.3 Long-lived Complexes ...............

978

977 977 978

978 979 979

66.5 Recent Developments........................... 980 References .................................................. 980

66.1 Experimental Methods 66.1.1 Molecular Beam Sources The development of molecular beam methods in the past two decades has transformed the study of chemical physics [66.1]. Supersonic molecular beam sources allow one to prepare reagents possessing a very narrow velocity distribution with very low internal energies, ideal for use in detailed studies of intermolecular interactions. Early experiments generally employed continuous beam sources, but in recent years intense pulsed beam sources have come into common usage [66.2]. The advantages of pulsed beams primarily arise from the lower gas loads associated with their use, hence reduced demands on the pumping system. If any component of the experiment is pulsed (pulsed laser detection, for example) then considerable advantage may be obtained by also pulsing the beam. Although the theoretical descriptions of pulsed and continuous expansions are essentially equivalent, in practice some care is required in employing pulsed beams because the temperature and

velocity distributions may change dramatically through the course of the pulse. Free jet expansions are supersonic because the dramatic drop in the local temperature in the beam is associated with a drop in the local speed of sound. A detailed description of the supersonic expansion may be found in [66.3–5]. In practice, many of the detailed features associated with a supersonic expansion may be ignored and one may assume an isentropic expansion into the vacuum. For an isentropic nozzle expansion of an ideal gas, the maximum terminal velocity is given by
vmax = 2Cˆ p T0 , (66.1) where, for an ideal gas, the heat capacity is   R γ ˆ Cp = , γ −1 m

(66.2)

R is the gas constant, m is the molar molecular mass, T0 the temperature in the stagnation region, and γ the

Part E 66

This chapter presents a résumé of the methods commonly employed in scattering experiments involving neutral molecules at chemical energies, i. e., less than about 10 eV. These experiments include the study of intermolecular potentials, the transfer of energy in molecular collisions, and elementary chemical reaction dynamics. Closely related material is presented in Chapts. 35, 37, and 38 as well as in other chapters on quantum optics.

968

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heat capacity ratio. For ideal gas mixtures, and assuming Cp independent of temperature for the range encountered in the expansion, one may use    γi  ¯ Cp = X i Cpi = Xi (66.3) R, γi − 1 i

i

Part E 66.1

and the average molar mass  m X i mi , ¯ =

V R (66.4)

i

where X i is the mole fraction of component i, to obtain an estimate of the maximum velocity for a mixture:
(66.5) vmax = 2C¯p T0 /m¯ . By seeding heavy species in light gases one may accelerate them to superthermal energies. Supersonic beams are characterized by the speed ratio, i. e., the mean velocity divided by the velocity spread: v S≡ √ , (66.6) 2kT/m where T is the local translational temperature, or by the Mach number v (66.7) . M≡√ γkT/m For the purpose of order-of-magnitude calculations, the number density on axis far from the nozzle may be estimated as n ≈ n 0 (d/x)2 ,

(66.8)

where n 0 is the number density in the stagnation region, d is the nozzle diameter, and x is the distance from the nozzle. The number density versus speed distribution of a nozzle beam is well described as a Gaussian characterized by the speed ratio S and a parameter α = v0 /S, where v0 is the most probable velocity:   (66.9) n(v) = v2 exp − (v/α − S)2 . Cooling efficiencies for the various internal degrees of freedom correlate with the efficiency of coupling of these modes with translation, hence they vary widely. Coupling of modes A and B is expressed by the collision number Z A−B : Z A−B ≡ Zτ A−B ,

Table 66.1 Collision numbers for coupling between different modes. V, R, T refer to vibrational, rotational, and translational energy, respectively. Each entry is the typical range of Z A−B

(66.10)

where τ A−B is the bulk relaxation time, and Z the collision frequency. This represents the number of collisions between effective inelastic events. Typical values are

V

R

T

100.5−3

103−4

105−6

100−1

102−3

summarized in Table 66.1. R–T coupling is relatively efficient, while V–T coupling is quite inefficient, so that vibrational excitation may not be effectively cooled in the expansion.

66.1.2 Reagent Preparation Molecular beam methods may be used in conjunction with a variety of other techniques to prepare atoms or molecules in excited or polarized initial states (Chapt. 46), to generate unstable molecules or radicals [66.6,7] or to produce beams of refractory materials such as transition metals or carbon [66.8,9]. Some of the common techniques are outlined below. Optical pumping of atoms to excited electronic states is a useful means of reagent preparation, and this topic is presented in detail in Chapt. 10. This technique further allows one, using polarized lasers, to explore the influence of angular momentum polarization in the reagents on the collision dynamics [66.10]. Most of these studies have been performed using alkali and alkaline earth metals since there exist strong electronic transitions and convenient narrow-band visible lasers suitable for use with these systems. Laser excitation may also be used to generate vibrationally excited molecules in their ground electronic states. The techniques employed include direct IR excitation using an HF chemical laser [66.11], population depletion methods [66.12] and various Raman techniques [66.13]. Metastable atoms may also be prepared by laser  photolysis of a suitable precursor. O 1D preparation is readily prepared by photolysis of ozone or N2 O, for example [66.14]. Alternatively, rf or microwave discharges may be used to produce metastable species or reactive atoms or radicals [66.15]. These techniques may also be used to prepare ground state atoms; for example, hot H atom beams are frequently produced by photolysis of HI or H2 S [66.16]. Such atomic or molecular radical beams may also be generated by pyrolysis in the nozzle. In this case care must be taken to minimize recombination through careful choice of the temperature, nozzle geometry, and transit time through the heated region.

Reactive Scattering

66.1.3 Detection of Neutral Products Broadly speaking, detection of neutral molecules is accomplished either by optical (spectroscopic) or nonoptical techniques. Nonoptical methods usually involve nonspecific ionization of neutral particles, most commonly by electron impact, followed by mass selection and ion counting. Thermal detectors such as cryogenic bolometers are also finding widespread application in molecular beam experiments owing to their remarkable sensitivity [66.20]. In general, optical methods may rely on resonant or nonresonant processes, hence they may or may not enjoy quantum state selectivity. Both photoionization and laser-induced fluorescence methods are now in common usage, usually in applications where quantum state resolved information is desired. The advantage of nonoptical methods is primarily one of generality: all neutral molecules may be detected, and branching into different channels readily measured. Quantum state resolution is more difficult to achieve using nonoptical detection methods, but both vibrationally- and rotationally-resolved measurements have been obtained by these means [66.21, 22]. The primary advantage of spectroscopic detection is the aforementioned possibility of quantum state specificity. Another unique opportunity afforded by spectroscopic probes is the measurement of product

969

aligment and orientation. In addition, in some cases background interference may be reduced or eliminated using state-specific probes, thereby affording enhanced signal-to-noise ratios. Nonoptical Techniques Detectors based on nonspecific ionization remain the most commonly used in molecular beam experiments, owing to the ease of subsequent mass selection, and the convenience and sensitivity of ion detection. Surface ionization is a sensitive means of detecting alkali atoms and other species exhibiting low ionization potentials [66.23]. Surface ionization occurs when a neutral atom or molecule with a low ionization potential sticks on a surface with a high work function and is subsequently desorbed. Typically these detectors employ a hot platinum or oxidized tungsten wire or ribbon for formation and subsequent desorption of the ions, which is surrounded by an ion collector. They are very efficient for the detection of alkali atoms and molecules whose ionization potentials are
6 eV. All neutral gas molecules may be ionized by collision with energetic electrons, and electron beam ionizers may be produced that couple conveniently to quadrupole mass spectrometers [66.24]. Collision of a molecule with a 100–200 eV electron leads predominantly to formation of the positive ion and a secondary electron. Other processes also occur and can be very significant: doubly or triply charged ions may be formed and, importantly, molecules can fragment yielding many daughter ions in addition to the parent ion. These fragmentation patterns vary with different molecules, and may further show a strong dependence on molecular internal energy, so particular care must be taken to determine the role of these phenomena in each particular application. It is often necessary to record data for the parent ion and daughter ions for a given product channel and compare them to eliminate contributions arising from cracking of the parent molecule or other species [66.25]. Electron impact ionization probabilities for most species exhibit a similar dependence on electron energy, rising rapidly from the ionization potential to a peak at 80–100 eV, then falling more slowly with increasing collision energy. The ionization cross section for different species scales with molecular polarizability according to a well-established empirical relation [66.26]: √ σion = 36 α − 18 , (66.11)

where σion is in Å2 and α, the molecular polarizability, is in Å3 . This relation can be used to estimate branching ratios in the absence of any other means of calibrating

Part E 66.1

Beams of refractory materials are now commonly generated using laser ablation sources [66.8, 9]. Typically these employ a rod or disk of the substrate of interest which is simultaneously rotated and translated to provide a fresh surface for ablation at each laser pulse. A laser beam is focused on the substrate and timed to fire just as a carrier gas pulse passes over. Laser power and wavelength must be optimized for a given substrate. Lasers operating in the IR, visible, and UV have all been employed. Aligned or oriented molecules have been prepared using multipole focussing [66.17, 18], and more recently using strong electric fields (“brute force”) [66.19]. In the former case, specific quantum states are focused by the field. In the latter case, so called pendular states are prepared from the low rotational levels of molecules possessing large dipole moments and small rotational constants. The ability to orient these molecules can be estimated on the basis of the Stark parameter ω = µE/B, where µ is the dipole moment, E the electric field strength, and B the rotational constant. Orientation is feasible for low rotational levels of molecules when the Stark parameter is on the order of 10 or higher [66.19].

66.1 Experimental Methods

970

Part E

Scattering Experiment

the relative contributions of two different channels. The ionization rate is given by

Part E 66.1

d[M + ] = Ie σ[M] , (66.12) dt where Ie is the electron beam intensity, typically 10 mA/cm2 or 6 × 1016 electrons/cm2 s, and [M] is the number density of molecules M in the ionizer. If one assumes an ionization cross section σion of 10−16 cm2 for collision with 150 eV electrons (a typical value for a small molecule), the ionization probability for molecules residing in the ionizer is then d[M + ] 1 = Ie σ = 6 × 1016 × 10−16 = 6 s−1 . dt [M] (66.13)

However, product molecules arriving in the detector are not stationary. Typically, product velocities are on the order of 500 m/s. If the ionization region has a length of 1 cm, the residence time τ of a product molecule is on the order of 2 × 10−5 s. Consequently, the ionization probability of product molecules passing through the ionizer is d[M + ] τ = 2 × 10−5 × 6 = 1.2 × 10−4 . (66.14) dt [M] Although this does not appear very efficient (indeed, it is 4 orders of magnitude less so than surface ionization), nevetheless, if the background count rate is sufficiently low, then good statistics may be obtained with signal levels as low as 1 Hz. Thus, for detection based on electron impact ionization, a key factor determining the sensitivity of the experiment is the background count rate at the masses of interest. Spectroscopic Detection Spectroscopic detection methods usually involve either laser-induced fluorescence (LIF) or resonant photoionization (REMPI) (Chapt. 44). Alternative techniques such as laser-induced grating methods and nonresonant VUV photoionization are also being applied to scattering experiments. Essential to the use of spectroscopic methods for reactive scattering studies is an understanding of the spectrum of the species of interest. This may be challenging for many reactive systems because the products may be produced in highly excited vibrational or electronic states that may not be well characterized. Additional spectroscopic data may be required. Franck– Condon factors are necessary to compare the intensities of different product vibrational states, while a calibration of the relative intensities of different electronic bands requires a measure of the electronic transition moments.

In some cases, one must include the specific dependence of the electronic transition moment on the internuclear distance by integrating over the vibrational wave function. Populations corresponding to different rotational lines may be compared after the appropriate correction, which is represented by the Hönl–London factors, only for isotropic irradiation and detection. This is certainly not the case for most laser-based experiments. Generally, the detailed dependence of the excitation and detection on the relevant magnetic sublevels must be considered [66.27–29]. Caution is required in using any spectroscopic method involving a level that is predissociated. This may lead to a dramatic decrease in the associated fluorescence or photoionization yield if the predissociation rate approaches or exceeds the rate of fluorescence or subsequent photoionization. An important question in any experiment based on spectroscopic detection is whether product flux or number density is probed. This question is considered in detail in several articles [66.13,30]. It depends on the lifetime of the state that is probed, the relative time that the molecule is exposed to the probe laser field, and its residence time in the interaction region. Saturation phenomena are also important, yet not necessarily easily anticipated. Complete saturation does not readily occur because excitation in the wings of the laser beam profile becomes more significant as the region in the center of the beam becomes saturated [66.31]. LIF is currently the most widely used spectroscopic technique in inelastic and reactive scattering experiments [66.27, 32, 33]. It has been used to measure state-resolved total cross sections [66.34] and differential cross sections in electronic [66.35], vibrational and rotationally inelastic scattering [66.12] as well as reactive scattering [66.36]. With the development of high-power tunablebreak lasers and the discovery of useful photoionization schemes, REMPI is becoming a more general technique [66.37,38]. REMPI has the advantages associated with ion detection, namely considerable convenience in mass selection and efficient detection, in addition to the capability for quantum state selectivity. Disadvantages associated with REMPI arise primarily from higher laser power employed compared with LIF. Caution is required in attempting to extract quantitative information from REMPI spectra if one or several of the steps involved in the ionization process are saturated. This is of particular concern at the high laser powers necessary for multiple photon transitions. An alternative to direct photoionization involves excitation of products to metastable Rydberg states, followed by field ionization

Reactive Scattering

66.1.4 A Typical Signal Calculation For a crossed-beam system in which a beam of atoms A collides with a beam of molecules B yielding products C and D, the rate of formation of C is given by dNC = n A n B σr g∆V , (66.15) dt where n A and n B are the number densities of the respective reagents at the interaction region, σr is the reaction cross section, g the magnitude of the relative velocity between the reactants, and ∆V the volume of intersection of the beams. For a typical experiment employing continuous supersonic beams, the number densities of the atomic and molecular reactants are ∼ 1011 –1012 cm−3 and the scattering volume 10−2 cm3 . For g = 105 cm/s and σr = 10−15 cm2 , the rate of product formation dNC / dt = 1011 molecules/s. The kinematics and energetics of the reaction then determines the range of laboratory angles into which the products scatter, and the magnitude of the scattered signal. If the products scatter into 1 sr of solid angle, and the detector aperture is 3 × 10−3 sr (roughly 1 degree in both directions perpendicular to the detector axis), then the detector receives 3 × 107 product molecules/s.

Given the detection probability obtained above, 3600 product ions/s are detected. This is adequate to obtain very good statistics in a short time as long as the background count rate is not considerably higher. For a nonspecific detection technique, such as electron bombardment ionization coupled with mass filtering,  −10 it is necessary to use ultrahigh vacuum torr in the detector region to minimize interfer10 ence from background gases. The residual gases are then primarily H2 and CO, with number densities on the order of 106 cm3 . Differential pumping stages, each of which may reduce the background by 2 orders of magnitude, are generally used to lower the background from gases whose partial pressures are lower than the ultrahigh vacuum limit of the detector chamber. However, this differential pumping helps only for those molecules that do not follow a straight trajectory through the detector. The contribution from the latter is given by nA , (66.16) 4πx 2 where n is the number density of molecules effusing from an orifice of area A, and n  is their number density at a distance x on axis downstream. For a distance of 30 cm and a main chamber pressure of 3 × 10−7 torr, this corresponds to a steady state density of 105 molecules/cm3 at the ionizer, a reduction of 6 orders of magnitude. Three stages of differential pumping are thus the maximum useful under these conditions, since the primary source of background is then molecules following a straight trajectory from the main chamber. A liquid helium cooled surface opposite the detector entrance may then be useful to minimize scattering of background molecules into the ionizer. n =

66.2 Experimental Configurations 66.2.1 Crossed-Beam Rotatable Detector The configuration illustrated in Fig. 66.1 represents a standard now widely used [66.24], usually with two continuous beams fixed at 90 degrees. The molecular beam sources are differentially pumped and collimated to yield an angular divergence of about 2 degrees. The beams cross as close as possible to the nozzles, with a typical interaction volume of 3 mm3 . Scattered products pass through an aperture on the front of the detector, thence through several stages of differential pumping before reaching the ionizer. Ions formed

by electron impact on the neutral products are then extracted into a quadrupole mass spectrometer with associated ion counter. A chopper wheel is generally used at the entrance to the detector to provide a time origin for recording time-of-flight spectra. Pseudorandom sequence chopper disks provide optimal counting statistics while maintaining a high duty cycle (50%) [66.40]. The detector may be rotated about the interaction region, typically through a range of 120◦ or so, allowing one to examine products scattered at a range of laboratory angles. In addition to time-of-flight detection, one of the beams may be gated on and off for background sub-

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Part E 66.2

some distance from the interaction region. This technique has the advantage of very low background and is capable of extraordinary time-of-flight resolution. Remarkable results have recently been obtained for the reaction D + H2 using this method [66.39]. Photoionization techniques are becoming more widely used in scattering experiments as the basis for product imaging detection schemes discussed below.

66.2 Experimental Configurations

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Part E

Scattering Experiment

Velocity sector

Cold trap D2 beam source Liquid nitrogen feedline

Part E 66.2

10–4

10–4 Heater 10–6

Effusive F atom source

Skimmer Turning fork chopper

10–7

Synchronous motor Cross-correlation chopper

Ultrahigh vacuum differentially pumped mass spectrometer detector

10–9 10–10

10–11

Fig. 66.1 Experimental arrangement for F + D2 → DF + F reactive scattering. Pressure (in torr) indicated in each region. Components are (1) effusive F atom source; (2) velocity selector; (3) cold trap; (4) D2 beam source; (5) heater; (6) liquid nitrogen feedline; (7) skimmer; (8) tuning fork chopper; (9) synchronous motor; (10) cross-correlation chopper; (11) ultrahigh vacuum differentially pumped mass spectrometer detector

traction and the detector moved to record the integrated signal at each laboratory angle. Two kinds of measurements are typically made in these experiments: time-of-flight spectra and angular distributions. Usually one is interested in obtaining the complete product-flux vs. velocity contour map, since this contains full details of the scattering process. This is obtained by measuring a full angular distribution as well as time-of-flight data at many laboratory angles. The results are then simulated using a forward convolution fitting procedure to obtain the underlying contour map [66.41–43]. Because scattering of isotropic reagents exhibits cylindrical symmetry about the relative velocity vector, it is sufficient to measure products scattered in any plane containing this vector to determine the product distribution. This is not true for structured particles (e.g. involving atoms in P states); however, this azimuthal anisotropy has been used to explore the impact parameter dependence of the reaction dynamics [66.44]. In a typical reactive scattering experiment, A + BC → AB + C, either of the two products may be detected. Conservation of linear momentum requires that the cm frame momenta of the two products must sum

to zero. It is thus only necessary to obtain the contour map for one of the products. The choice of detected product is usually dictated by kinematic considerations, although one may choose to detect a product that is kinematically disfavored if its partner happens to have a mass with a large natural background in the detector. Kinematic considerations can be critical in assessing the suitability of a given system for study. It is very important that one of the products be scattered entirely within the viewing range of the detector in order to obtain a complete picture of the reaction dynamics. The advantages of crossed-beams employed in conjunction with an electron impact ionizer-mass spectrometer detector derive primarily from the universality of the detector. No spectroscopic information is required and there are no invisible channels, such as may occur with spectroscopic detection methods. In addition, the resolution of these machines may be increased almost arbitrarily; indeed, even rotationally inelastic scattering has been studied [66.45]. The disadvantages are complementary to the advantages: the universal detector implies that quantum state resolution is not achieved directly, al-

Reactive Scattering

66.2.2 Doppler Techniques Spectroscopic detection methods in crossed-beam experiments allows the measurement of state-resolved differential cross sections, and thus the ultimate level of insight into the reaction dynamics. A method developed by Kinsey and others [66.47, 48] determines differential cross sections by measurement of product Doppler profiles using LIF (called ADDS for Angular Distribution by Doppler Spectroscopy). For a laser directed parallel to the relative velocity vector, a particle scattered with a cm velocity of u perceives the photon as having the Doppler shifted frequency   ν = ν 1 − (u + Vcm ) · n/c , (66.17) ˆ where ν is the laser frequency and Vcm is the velocity of the center of mass, both in the laboratory frame of refernce, and nˆ is the unit vector in the probe laser direction. For the case of a single possible recoil speed, one may obtain the full differential cross section directly in the cm frame by reconstruction of a single Doppler profile [66.47]. In this case, the angular resolution is a maximum for the sideways scattered products, and a minimum at the poles. An alternative approach is to measure the Doppler profile with a laser perpendicular to the relative velocity vector. This approach (PADDS for Perpendicular ADDS) affords complementary angular resolution, but folds the forward and backward scattered products into a single symmetric component [66.48]. For the case in which the detected product does not possess a known recoil speed (for example if the thermodynamics of the process is not known, or if one probes the atomic fragment in an A + BC → AB + C reaction), a single Doppler profile is insufficient to reconstruct the double differential cross sections. Nevertheless, Kinsey’s earliest experimental results were for one such example: the reaction H + NO2 → OH + NO [66.49]. More recently Mestdagh et al. [66.50] have studied electronically inelastic collision processes using this approach by measuring the Doppler profiles over a range

of probe laser angles, as illustrated in Fig. 66.2. A beam of barium atoms is crossed at 90 degrees by a beam of some molecular perturber. At the interaction region, the barium atoms are electronically excited using a narrow band dye laser. Scattered barium atoms that have undergone a specific electronic transition as a result of the collision are probed at the interaction region using a second dye laser, which is scanned across the Doppler profile. The product-flux vs. velocity contour map is then reconstructed by means of a forward convolution simulation procedure analogous to that described in the preceding section. In addition to state-resolved detection, another difference between the Doppler methods and the traditional crossed-beam configuration is that the kinematic considerations favor detection of fast particles, and almost any system that is spectroscopically suitable may be considered. The primary disadvantage of Doppler methods is the limited angular and translational energy resolution possible. Often, however, modest angular resolution is sufficient to achieve a global picture of the reaction dynamics. Much current work involves the study of photoinitiated reactions in cells, relying on the short excited state lifetimes to guarantee single collision conditions, and using iterative fitting procedures to probe product velocity distributions and angular momentum polarization [66.14]. Angular momentum polarization can have a profound effect on the measured distributions and can afford a powerful additional means of exploring the collision dynamics. Examples of 3- and 4-vector correlation experiments approach a “complete description” of the scattering process (Chapt. 46) [66.50–52].

66.2.3 Product Imaging Another spectroscopic technique is based on direct imaging of the scattered product distribution. The technique was first used to record state-resolved angular distributions of methyl radicals from the photodissociation of methyl iodide [66.53]. The method has since been widely employed to study photodissociation, and more recently to record state-resolved inelastic scattering in a crossed-beam experiment [66.54]. Recently it has been applied to a crossed-beam reactive scattering system [66.55]. The crossed-beam configuration used by Houston and coworkers is shown schematically in Fig. 66.3. The two skimmed supersonic beams cross at right angles, and scattered products are state-selectively ionized on the axis of a time-of-flight mass spectrometer using resonant photoionization. The ion cloud thus formed continues to expand with its nascent recoil vel-

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Part E 66.2

though in favorable cases the product vibrational states may be resolved in the translational energy distributions [66.21, 46]. In addition, if the product of interest represents a mass that receives interference from one of the beam masses, background interference may be problematic. Kinematic considerations mentioned above may also preclude study of certain systems. However, the kinematic requirements for the Doppler and imaging approach discussed below are complementary to those of the rotatable-detector configuration.

66.2 Experimental Configurations

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Part E

Scattering Experiment

Mass spectrometer

Collection optics

Pilot tube

Single mode optical fiber

Part E 66.2

Time-of-flight arm

Laser beam

Chopper Atomic beam Oven for effusive beam

Molecular beam source

Collision zone

Scale 0

0.5 m

Fig. 66.2 Schematic view of crossed molecular beam apparatus with LIF-Doppler detection

ocity as it drifts through the flight tube. The ions then strike a microchannel plate coupled to a phosphor screen. The latter is viewed by a video camera gated to record the signal at the mass of interest. The images are thus two-dimensional projections of the nascent three dimensional product distributions.

Video camera

Flight tube

Phosphor screen MCP Pulsed nozzle

Pulsed nozzle Ar beam

Probe laser

Skimmers

NO/He beam

Fig. 66.3 Schematic view of crossed molecular beam ap-

paratus with product imaging detection

There now exist two alternatives for regenerating the three dimensional distribution from the projection. The first, a tomographic reconstruction using an inverse Abel transform, is widely used in photodissciation studies [66.56,57]. It is a direct inversion procedure feasible for cases in which the image is the projection of a cylindrically symmetric object, with its axis of symmetry parallel to the image plane. This analysis yields a unique product contour map directly from the image, but it is difficult to incorporate apparatus functions, and is sensitive to noise in the data. The second alternative is a forward convolution fitting method. A Monte Carlo based simulation has the advantage that one may treat the averaging over experimental parameters quite rigorously. The advantages of the imaging method again derive from its reliance on a spectroscopic probe, so that quantum state resolution is possible and background interference may be avoided. In addition, it possesses a multiplexing advantage since the velocity distribution is recorded for all angles simultaneously. Imaging relies exclusively on photoionization, unlike the Doppler methods which may use either photoionization or LIF. This is somewhat disadvantageous since the available photoionization schemes are limited and often high laser

Reactive Scattering

66.2 Experimental Configurations

power is necessary to achieve adequate signal intensity. As a result, background ions can be a problem. In general, resonantly enhanced two-photon ionization, i. e. [1 + 1], detection schemes are thus preferable.

gle of the cm velocity vector with respect to A is given by

66.2.4 Laboratory to Center-of-Mass Transformation

For an arbitrary Newton diagram with angle α between the two beams the magnitude of the relative velocity is

Vcm =

MF vF + MD2 vD2 . MF + MD2

(66.18)

Vcm divides the g into two segments corresponding to the cm velocities of the two reagents. The magnitude of these vectors, u F and u D2 are inversely proportional to the respective masses. If scattered DF products are formed with a laboratory scattering angle Θ and a laboratory velocity vDF as shown in Fig. 66.4, this corresponds to DF backscattered with respect to the incident F atom, in the cm system. It is common to refer the scattering frame direction to the atomic reagent in an A + BC → AB + C reaction, for example, to make clear the dynamics of the process. In this case the backscattered DF arises as a result of a direct rebound collision. Some useful kinematic quantities are summarized here. For beams A and BC intersecting at 90 degrees, the anv=1 v=2 0°

v=3 vF

vH2

Fig. 66.4 Newton diagram for collision of F with H2 with superimposed c.m. flux vs. velocity contour map

M BC v BC . MAvA

g2 = v2A + v2BC − 2v A v BC cos α ,

(66.19)

(66.20)

the relative velocity vector is g = v A − v BC ,

(66.21)

and the collision energy is 1 (66.22) E coll = µi g2 , 2 where µi is the reduced mass of the initial collision system. The magnitude of the cm frame velocity of particle A before collision is m BC uA = g. (66.23) m A + m BC The final relative velocity is g = v AB − vC ,

(66.24)

with magnitude g = 2E avail /µF ,

(66.25)

where the available energy E avail is E avail = E coll + E int,reac + E exo − E int,prod , (66.26)

in which E int,reac is the internal energy of the reactants, E exo is the exoergicity of the reaction, and E int,prod is the internal energy of the products. One must transform the  laboratory intensity I(Ω) ≡ d2 σ/ d2 Ω into I(ω) ≡ d2 σ/ d2 ω , the corresponding cm quantity. For the crossed-beam configuration described in Sect. 66.2.1, the laboratory distributions are distorted by a transformation Jacobian that arises because the laboratory detector views different cm frame solid angles depending on the scattering angle and recoil velocity. For the spectroscopic experiments described in Sects. 66.2.2 and 66.2.3, the Jacobian is unity (the cm velocity represents a simple frequency offset of the Doppler profiles, for example); however, the transformation of the scattering distributions from the recorded quantities (2-dimensional projections or intensity vs. wavelength) to recoil velocity distributions may be complex. Two cases must be considered for the

Part E 66.2

Angular and velocity distributions measured in the laboratory frame must be transformed to the cm frame for theoretical interpretation. Accounts of this transformation and details concerning the material presented below may be found in [66.58–61], among others. The Newton diagram is useful to aid in visualizing the transformation, and in understanding the kinematics of a given collision system. For the scattering of F + D2 for example, shown in Fig. 66.4, a beam of fluorine atoms with a velocity vF is crossed by a beam of D2 , velocity vD2 , at 90 degrees. The relative velocity between the two reactants is g = vF − vD2 , and the velocity of the cm of the entire system is

Θcm = arctan

975

976

Part E

Scattering Experiment

Part E 66.3

configuration discussed in Sect. 66.2.1: one in which discrete velocities result (such as elastic or state-resolved scattering experiments), and one in which continuous final velocities are measured. For the first case, the laboratory and cm differential cross sections are independent of the respective product velocities v and u and these quantities are related by d2 σ d2 σ =J 2 , (66.27) 2 d Ω d ω so that the transformation Jacobian is given by d2 ω J= 2 . (66.28) d Ω For discrete recoil velocities, the cm solid angle is dA d2 ω = 2 , (66.29) u where dA is a surface element of the product Newton sphere. The laboratory solid angle corresponding to this quantity is cos(u, v) d2 Ω = dA , (66.30) v2 so that the Jacobian for the first case is given by v2 J= 2 (66.31) . u cos(u, v) For the case of continuous final velocities, the σ are velocity-dependent and are related by d3 σ d3 σ =J 2 , (66.32) 2 d Ω dv d ω du so that here the Jacobian is given by d2 ω du J= 2 . (66.33) d Ω dv

In this case we consider a recoil volume element dτ (in velocity space), which must be the same in both coordinate frames: dτcm = u 2 du d2 ω = dτlab = v2 dv d2 Ω ,

(66.34)

so that the Jacobian is J = v2 /u 2 .

(66.35)

The laboratory intensity is then related to that in the cm frame by (66.36) Ilab (v, Θ) = (v2 /u 2 ) Icm (θ, u) . For a mass spectrometer detector with electron bombardment ionizer, one measures number density of particles rather than flux, so that the recorded signal is given by

Nlab (v, Θ) =

v Ilab (v, Θ) = 2 Icm (u, θ) . v u

(66.37)

The usual flux vs. velocity contour map is a polar plot of the quantity Icm (u, θ). The product velocity distributions are then

I(u) = I(θ, u) sin θ dθ dφ

π = 2π I(u, θ) sin θ dθ ,

(66.38)

0

and the translational energy distributions are    du   . I(E T ) = I(u)  dE T 

(66.39)

66.3 Elastic and Inelastic Scattering When particles collide, they may exchange energy or recouple it into different modes, they may change their direction of motion, and they may even change their identity. The study of these processes reveals a great deal of information about the forces acting between the particles and their internal structure. It is useful to begin with a summary of the dominant features of elastic and inelastic scattering.

66.3.1 The Differential Cross Section Figure 66.5 illustrates the relation between the deflection function χ and the impact parameter b for a realistic potential containing an attractive well and a repulsive

core. For large b there is no interaction, hence no deflection. At smaller values of b, the attractive part of the potential is experienced and some positive deflection results. At a smaller value of b, br , the influence of the attractive component of the potential reaches a maximum, giving the greatest positive deflection: this is the rainbow angle by analogy with the optical phenomenon. There is another value of the b for which point the attractive and repulsive parts of the potential balance, yielding no net deflection. This is the glory impact parameter bg . For yet smaller values of b, the interaction is dominated by the repulsive core and rebound scattering gives a negative deflection function.

Reactive Scattering

br* * 1 bg

66.3.2 Rotationally Inelastic Scattering π

+

Fig. 66.5 Schematic diagram showing the relation between impact parameter b and deflection function χ

The important expressions related to the differential cross section are summarized here [66.62]. For scattering involving an isotropic potential, the deflection angle is Θ = |χ|. The differential cross section dσ gives the rate of all collisions leading to deflection angles in the solid angle element dω: dN(θ) ∝ I(θ) dω = I(θ)2π sin(θ) dθ . (66.40) dt The incremental cross section is dσ = I(θ) dω = 2πb db, so b . I(θ) = (66.41) sin θ ( dθ / db) For classical particles, the relation between the deflection function and the potential is  −1/2

∞ V(R) b2 dR χ = π − 2b 1 − − , ET R2 R2 R0

(66.42)

where V(R) is the potential as a function of interparticle distance R, R0 is the turning point of the collision, and E T the collision energy. In the high energy limit, for large b ≈ R0 , χ(b, E T ) ∝ V(b)/E T .

(66.43)

For a long-range potential V(R) proportional to R−s , 2/s

E T θ 2(1+1/s) I(θ) = const .

(66.44)

For a potential exhibiting a minimum, the rainbow angle θr is proportional to the collision energy, and clearly resolved when the collision energy is 3 to 5 times the well depth. In addition, supernumery rainbows and quantum

Classical scattering involving an anisotropic potential results in another rainbow phenomenon, distinct from that seen in pure elastic scattering and notable in that it does not require an attractive component in the potential. These rotational rainbows are equivalently seen in a plot of integral cross section against change in rotational angular momentum ∆ j, or in the differential cross section for a particular value of ∆ j. The rotational rainbow peaks arise from the range of possible orientation angles γ in a collision involving an anisotropic potential. When there is a minimum in dγ/ dθ for a given ∆ j, the differential cross section reaches a maximum [66.64]. The rotational rainbow peak occurs at the most forward classically allowed value of the scattering angle, and dσ drops rapidly at smaller angles. The rainbow moves to more backward angles with increasing ∆ j because the larger j-changing collisions require greater momentum transfer, hence must arise from lower impact parameter collisions. For heteronuclear molecules, two rainbow peaks may be observed, corresponding to scattering off either side of the molecule. One can relate the location of the rainbow peak to the shape of the potential using a classical hard ellipsoid model [66.65]:   j θr,cl −1 A− B = 2 sin , (66.45) p0 2 where j is the rotational angular momentum, p0 is the inital linear momentum, θr,cl is the classical rainbow position, and A and B are the semimajor and semiminor axes of a hard ellipse potential. The classical rainbow positions occur somewhat behind the quantum mechanical and experimental rainbow positions, so the classical rainbow may be estimated as the point at which the peak has fallen to 44% of the experimental value. Real molecular potentials may be far from ellipsoids, however, so detailed quantitative insight into the potential requires a comparison of scattering data with trajectory calculations.

66.3.3 Vibrationally Inelastic Scattering There has been no direct observation of the differential cross section of T–V or V–T energy transfer involving neutral molecules owing to the small cross sections for

Part E 66.3

– 0 + χ

977

mechanical “fast osillations” occur in the dσ, and these provide a sensitive probe of the interaction. Accurate interatomic potentials are routinely obtained from elastic scattering experiments [66.60, 63].

2

0

66.3 Elastic and Inelastic Scattering

978

Part E

Scattering Experiment

Part E 66.4

these processes. Integral cross section data are available, however. Above threshold, the latter has shown a linear dependence of σ on collision energy for ∆ν = 1, quadratic for ∆ν = 2 and cubic for ∆ν = 3 [66.66]. In addition, a great deal of information on vibrational relaxation processes has been obtained in cell experiments [66.67].

66.3.4 Electronically Inelastic Scattering A wealth of information is available on electronically inelastic scattering systems, since these in general exhibit much larger cross sections than V–T processes [66.68, 69]. In addition, spectroscopic methods

may be used to overcome some of the background problems that hamper the study of the latter. Often, quenching of electronically excited states involves curve crossing mechanisms, so that very effective coupling of electronic to vibrational energy   may occur. Spin-orbit changing collisions of Ba 1P with O2 or NO, for example, occur by a near-resonant process and result in almost complete conversion of electronic energy to vibrational excitation of the product [66.70]. The analogous collisions with N2 and H2 , however, reveal very repulsive energy release with little concomitant vibrational excitation. Both processes likely occur via curve crossings of the relevant electronic states, but the near-resonant mechanism occurs by way of an ionic intermediate.

66.4 Reactive Scattering Reactive differential cross sections reveal several distinct aspects of the chemical encounter. The angular distributions themselves may be used to infer the lifetime of the collision intermediate: long-lived complexes exhibit forward-backward symmetry along the relative velocity vector. In this case “long-lived” means on the order of several rotational periods. The rotational period of the complex may thus be used as a clock to study the energy dependence of the intermediate’s lifetime. The angular distributions further reveal the relation of initial and final orbital angular momentum. Sharply peaked angular distributions generally indicate strongly correlated initial and final orbital angular momentum vectors. Finally, the product translational energy release contains the details of the energy disposal, and reveals a wealth of information about the thermodynamics of the process, the existence of barriers, and sometimes even the geometry of the transition state. Together, the angular and tranlational energy distributions reveal many of the details of the potential energy surface. The dynamics of reactive collisions fall broadly into three main categories characterized by distinct angular and energy distributions. The three categories are harpoon/stripping reactions, rebound reactions, and long-lived complex formation. Some reactions may exhibit more than one of these mechanisms at once, or the dynamics may change from one to another as the collision energy is varied.

66.4.1 Harpoon and Stripping Reactions It was known in the 1930s that collisions of alkali atoms with halogen molecules exhibit very large cross

sections and yield highly excited alkali halide products. These observations were accounted for by the harpoon mechanism proposed by M. Polanyi. Because alkali atoms have low ionization potentials and halogen molecules large electron affinities, as the alkali atom approaches the molecule, electron transfer may occur at long range. These processes are considered in detail in Chapt. 49 [66.71, 72]. The harpooning distance Rc at which this curve crossing takes place may be estimated simply as the distance at which the Coulomb attraction of the ion pair is sufficient to compensate for the endoergicity of the electron transfer: Rc = e2 /(IP − Ae ) ,

(66.46)

where IP and Ae represent the ionization potential and electron affinity of the electron donor and recipient, respectively. For R in Å and E in eV, this relation is Rc = 14.4/(IP − Ae ) .

(66.47)

Owing to the large Coulombic attraction between the ion pair, reaction proceeds immediately following electron transfer. The crossing distance may then be used to estimate the effective reaction cross section. The vertical Ae is not necessarily the appropriate value to use in estimating these crossing distances; stretching of the halogen bond may occur during approach, so the effective Ae is generally somewhere between the vertical and adiabatic values. Often there exists some repulsion between the atoms in the resulting halogen molecular ion, so that electron transfer is accompanied by dissociation of the molecule in the strong field of the ion

Reactive Scattering

uAB = −MC u BC /M AB ,

(66.48)

where u BC is the initial cm velocity of the BC molecule. This spectator stripping mechanism may occur in systems other than harpoon reactions, and is useful to remember as a limiting case. The likelihood of electron transfer at these crossings may be estimated using a simple Landau–Zener model [66.72] (Chapt. 49). For relative velocity g, impact parameter b and crossing distance Rc , the probability for undergoing a transition from one adiabatic curve to another (that is, the probability for remaining on the diabatic curve) is given by p = 1 − e−δ ,

(66.49)

where δ=

2 R2 2πH12 c g

− 12  b2 1− 2 , Rc

(66.50)

and H12 is the coupling matrix element between the two curves. H12 may be estimated from an empirical relation which is accurate within a factor of three over a range of 10 orders of magnitude [66.73]. In atomic units ∗ H12 = I1 I2 Rc∗ e−0.86Rc , (66.51) where Rc∗ =



I1 +

 I2

(66.52)

is the reduced crossing distance, and I1 and I2 are the initial and final ionization potentials of the transferred electron. One finds electron transfer probabilities near unity for curve crossing distances below about 5 Å, dropping to zero for crossing at distances greater than about 8 Å. These estimates are based on electron transfer in atom–atom collisions, and it is important to remember that atom–molecule collisions occur on surfaces rather than curves, so the crossing seam may cover a broad range of internuclear distances.

979

66.4.2 Rebound Reactions Another common direct reaction mechanism is the rebound reaction exemplified by F + D2 → DF + D [66.21]. The cm product flux vs. velocity contour map obtained for this reaction is shown in Fig. 66.4. Owing to the favorable kinematics and energetics in this case, the FD product vibrational distribution is clearly resolved, and peaks at v = 2. The dominant v = 2 product peaks at a cm angle of 180 degrees (referred to the direction of the incident F atom). This rebound scattering is characteristic of reactions exhibiting a barrier in the entrance channel. Rebound scattering implies small b collisions, and this serves to couple the translational energy efficiently into overcoming the barrier. Small b collisions have necessarily smaller cross sections however, since cross section scales quadratically with b.

66.4.3 Long-lived Complexes A third important reaction mechanism involves the formation of an intermediate that persists for some time before dissociating to give products. If the collision com  plex survives for many rotational periods ∼ 10−11 s , then the cm angular distribution exhibits a characteristic forward–backward symmetry, usually with peaking along the poles. The latter occurs because the initial and final orbital angular momenta tend to be parallel (and perpendicular to the initial relative velocity vector). When there exist dynamical constraints enforcing some other relation, as in the case F + C2 H4 , then sideways scattering may be observed, despite a lifetime of several rotational periods [66.74–76]. For some systems exhibiting this long-lived behavior, the rotational period may be used as a molecular clock to monitor the lifetime of the complex. By increasing the collision energy until the distribution begins to lose its forward–backward symmetry, one can investigate the internal energy of the system just when its lifetime is on the order of a rotational period. Systems that have an inherent symmetry may exhibit this forward–backward symmetry in the scattering distributions despite lifetimes that are considerably shorter than a rotational period. This is the case for O(1D ) reacting with H2 , for example [66.77]. This reaction involves insertion of the O atom into the H2 bond resulting in an intermediate that accesses the deep H2 O well and contains considerable vibrational excitation. Trajectory calculations show that the complex dissociates after a few vibrational periods, but the distribution exhibits forward–backward symmetry because the O atom is equally likely to depart with either H atom.

Part E 66.4

pair. The alkali ion, having sent out the electron as the “harpoon”, then reels in the negative ion, leaving the neutral halogen atom nearly undisturbed as a spectator. Because these events occur at long range, there is no momentum transfer to the spectator atom, and it is a simple matter to estimate the anticipated angular and translational energy distributions in this spectator stripping limit. The product molecule is scattered forward (relative to the direction of the incident atom) and for the reaction A + BC → AB + C, the final cm velocity for the product AB is given by

66.4 Reactive Scattering

980

Part E

Scattering Experiment

66.5 Recent Developments

Part E 66

Astonishing progress in reactive scattering methods has continued in the past decade, and a few highlights are summarized here. These advances have taken the form of improvements in detection methods or, in some cases, entirely new experimental approaches. One of the most important of these is the H atom Rydberg time-of-flight (HRTOF) method [66.78, 79] pioneered by the late Karl Welge and coworkers for the hydrogen exchange reaction. This approach employs a conventional scattering geometry, and is suitable only for experiments yielding product H or D atoms. Despite this narrow focus, owing to the general importance of hydrogen elimination reactions and the remarkable resolution of the technique, this has been an important development. The H or D atom products are excited to long-lived high-n Rydberg states in a 1 + 1 excitation scheme in the interaction region. The atoms fly through a field free region and impinge upon a rotatable field-ionization detector. The result is very high velocity resolution, largely because the spreads in the beam velocities make a negligible contribution to the product velocity spread since the H atoms are moving so fast. In addition, the dimensions of the scattering volume and ionization region may easily be made small relative to the flight length. Using this technique, Welge and coworkers achieved fully rotationally-resolved differential cross sections for the hydrogen exchange reaction. A second, widely-used approach is a variation of the state-resolved Doppler probe in a bulb configuration. This strategy, pioneered by Hall at Brookhaven [66.80] and Brouard et al. in Oxford [66.81], has been applied most notably to study excited oxygen atom reactions. Although not a true crossed-beam approach, through appropriate exploitation of kinematic constraints, energy conservation, and careful analysis, state-resolved doubly differential cross sections may be obtained, sometimes with additional vector properties as well [66.82].

Another significant new direction in detection strategies is the use of near-threshold VUV product ionization. This is a universal approach, in that little advance spectroscopic information is required, but it is selective in that dissociative ionization is minimized and sometimes isomer-selective detection may be achieved. This approach has been used in synchrotron-based studies of Cl atom reactions [66.83], in transition metal reactions [66.84], and in product imaging studies of oxygen and chlorine atom reactions using the F2 excimer at 157 nm [66.85]. Inspired by the threshold VUV detection methods, Casavecchia and coworkers have recently advanced the use of near-threshold electron impact ionization in a conventional universal crossedbeam configuration [66.86]. Their recent results show the great promise of this technique to deliver higher signal-to-noise and to minimize fragmentation processes in the detection step that may obscure the underlying dynamics. A final note concerns advances in imaging techniques applied to reactive scattering. Mention has been made of the successful application of the VUV excimer probe for imaging radical products of reaction of Cl and O(3P) reactions with alkanes. Two significant advances in imaging strategies have made it a very powerful technique. The first of these is “velocity mapping”, developed by Eppink and Parker at Nijmegen [66.87], a simple but important strategy that eliminates spatial blurring in the images. The second is “slicing”, or 3-D, methods that allow the velocity-flux contour map to be recorded directly [66.88–90]. This has seen its most beautiful illustration in recent work by Liu and coworkers at IAMS in state-resolved detection of methyl radicals following reaction of F atoms with methane [66.91]. Their images provide quantum state correlated differential cross sections for this reaction directly, allowing comparison to theory at an unprecedented level of detail.

References 66.1 66.2

66.3

Y. T. Lee: Science 236, 793 (1987) W. R. Gentry: Atomic and Molecular Beam Methods, ed. by G. Scoles (Oxford Univ. Press, Oxford 1988) Chap. 3, p. 54 J. B. Anderson, R. P. Andres, J. B. Fenn: Adv. Chem. Phys. 10, 275 (1966)

66.4

66.5 66.6

D. R. Miller: Atomic and Molecular Beam Methods, ed. by G. Scoles (Oxford Univ. Press, Oxford 1988) Chap. 2, p. 14 R. Campargue: J. Phys. Chem. 88, 275 (1984) H. F. Davis, B. Kim, H. S. Johnston, Y. T. Lee: J. Phys. Chem. 97, 2172 (1993)

Reactive Scattering

66.7

66.8 66.9

66.13

66.14 66.15 66.16 66.17 66.18 66.19 66.20 66.21

66.22 66.23 66.24 66.25

66.26 66.27 66.28 66.29 66.30 66.31 66.32 66.33 66.34 66.35

66.36 66.37 66.38 66.39

66.40 66.41 66.42 66.43 66.44 66.45

66.46 66.47 66.48 66.49 66.50

66.51

66.52

66.53 66.54 66.55 66.56 66.57 66.58 66.59 66.60

66.61 66.62

S. D. Jons, J. E. Shirley, M. T. Vonk, C. F. Giese, W. R. Gentry: J. Chem. Phys. 92, 7831 (1992) D. L. Feldman, R. K. Lengel, R. N. Zare: Chem. Phys. Lett. 52, 413 (1977) E. E. Marinero, C. T. Rettner, R. N. Zare: J. Chem. Phys. 80, 4142 (1984) L. Schneider, K. Seekamp-Rahn, F. Liedeker, H. Stewe, K. H. Welge: Farad. Discuss. Chem. Soc. 91, 259 (1991) G. Comsa, R. David, B. J. Schumacher: Rev. Sci. Instrum. 52, 789 (1981) E. Entemann, D. R. Herschbach: J. Chem. Phys. 55, 4872 (1971) R. Buss: Ph. D. thesis (University of California, Berkeley 1972) R. T. Pack: J. Chem. Phys. 81, 1841 (1984) A. G. Suits, H. Hou, H. F. Davis, Y. T. Lee: J. Chem. Phys. 95, 8207 (1991) M. Faubel, K.-H. Kohl, J. P. Toennies, K. T. Tang, Y. Y. Yung: Faraday Discuss. Chem. Soc. 73, 205 (1982) R. E. Continetti, B. A. Balko, Y. T. Lee: J. Chem. Phys. 93, 5719 (1990) J. L. Kinsey: J. Chem. Phys. 66, 2560 (1977) J. A. Serri, J. L. Kinsey, D. E. Pritchard: J. Chem. Phys. 75, 663 (1981) E. L. Murphy, J. H. Brophy, G. S. Arnol, W. L. Dimpfl, J. L. Kinsey: J. Chem. Phys. 70, 5910 (1979) J. M. Mestdagh, J. P. Visticot, P. Meynadier, O. Sublemontier, A. G. Suits: J. Chem. Soc. Faraday Trans. 89, 1413 (1993) C. J. Smith, E. M. Spain, M. J. Dalberth, S. R. Leone, J. P. J. Driessen: J. Chem. Soc. Faraday Trans. 89, 1401 (1993) H. A. J. Meijer, T. J. C. Pelgrim, H. G. M. Heideman, R. Morgenstern, N. Andersen: J. Chem. Phys. 90, 738 (1989) D. W. Chandler, P. L. Houston: J. Chem. Phys. 87, 1445 (1987) L. S. Bontuyan, A. G. Suits, P. L. Houston, B. J. Whitaker: J. Phys. Chem. 97, 6342 (1993) T. A. Kitsopoulos, D. P. Baldwin, M. A. Buntine, R. N. Zare, D. W. Chandler: Science 260, 1605 (1993) R. N. Strickland, D. W. Chandler: Appl. Opt. 30, 1811 (1990) K. R. Castleman: Digital Image Processing (Prentice Hall, Englewood Cliffs 1979) T. T. Warnock, R. B. Bernstein: J. Chem. Phys. 49, 1878 (1968) R. K. B. Helbing: J. Chem. Phys. 48, 472 (1968) J. P. Toennies: in Physical Chemistry, an Advanced Treatise, Vol. VIA, ed. by W. Jost (Academic, New York 1974) Chap. 5 M. Faubel, J. P. Toennies: Adv. At. Mol. Phys. 13, 229 (1978) R. D. Levine, R. B. Bernstein: Molecular Reaction Dynamics and Chemical Reactivity (Oxford, New York 1987) Chap. 3

981

Part E 66

66.10 66.11 66.12

P. Chen: Unimolecular and Bimolecular Reaction Dynamics, ed. by C. Y. Ng, T. Baer, I. Powis (John Wiley and Sons, New York 1994) Chap. 8, p. 371 R. E. Smalley: Laser Chem. 2, 167 (1983) M. D. Morse, J. B. Hopkins, P. R. R. LangridgeSmith, R. E. Smalley: J. Chem. Phys. 79, 5316 (1983) I. V. Hertel, W. Stoll: Adv. At. Mol. Phys. 13, 113 (1978) J. G. Pruett, R. N. Zare: J. Chem. Phys. 64, 1774 (1976) K. Bergmann, U. Hefter, J. Witt: J. Chem. Phys. 72, 4777 (1980) K. Bergmann: Atomic and Molecular Beam Methods, ed. by G. Scoles (Oxford Univ. Press, Oxford 1988) Chap. 12 M. Brouard, S. Duxon, P. A. Enriquez, J. P. Simons: J. Chem. Soc. Faraday Trans. 89, 1435 (1991) S. J. Sibener, R. J. Buss, P. Cassavechia, T. Hirooka, Y. T. Lee: J. Chem. Phys. 72, 4341 (1980) R. E. Continetti, B. A. Balko, Y. T. Lee: J. Chem. Phys. 93, 5719 (1990) P. R. Brooks: Science 193, 11 (1976) S. Stolte: Ber. Bunsenges. Phys. Chem. 86, 413 (1982) B. Friedrich, D. R. Herschbach: Nature 353, 412 (1991) D. Bassi, A. Boschetti, M. Scotoni, M. Zen: Appl. Phys. B 26, 99 (1981) D. M. Neumark, A. M. Wodtke, G. N. Robinson, C. C. Hayden, K. Shobatake, R. K. Sparks, T. P. Schaefer, Y. T. Lee: J. Chem. Phys. 82, 3067 (1985) M. Faubel: Adv. At. Mol. Phys. 19, 345 (1983) T. R. Touw, J. W. Trischka: J. Appl. Phys. 34, 3635 (1963) Y. T. Lee, J. D. McDonald, P. R. LeBreton, D. Herschbach: Rev. Sci. Intstrum. 40, 1402 (1969) Y. T. Lee: Atomic and Molecular Beam Methods, ed. by G. Scoles (Oxford Univ. Press, Oxford 1988) Chap. 22, p. 563 R. E. Center, A. Mandl: J. Chem. Phys. 57, 4104 (1972) C. H. Greene, R. N. Zare: J. Chem. Phys. 78, 6741 (1983) D. A. Case, G. M. McLelland, D. R. Herschbach: Mol. Phys. 35, 541 (1978) G. Hall, P. Houston: Ann. Rev. Phys. Chem. 40, 375 (1989) D. M. Sonnenfroh, K. Liu: Chem. Phys. Lett. 176, 183 (1991) N. Billy, B. Girard, G. Gouédard, J. Vigué: Mol. Phys. 61, 65 (1987) R. N. Zare, P. J. Dagdigian: Science 185, 739 (1974) R. Altkorn, R. N. Zare: Ann. Rev. Phys. Chem. 35, 265 (1984) H. Joswig, P. Andresen, R. Schinke: J. Chem. Phys. 85, 1904 (1986) J. M. Mestdagh, J. P. Visticot, A. G. Suits: The Chemical Dynamics and Kinetics of Small Radicals, ed. by K. Liu, W. Wagner (World Scientific, Singapore 1994)

References

982

Part E

Scattering Experiment

66.63

66.64

Part E 66

66.65 66.66 66.67 66.68 66.69 66.70

66.71

66.72

66.73 66.74 66.75

J. M. Farrar, T. P. Schafer, Y. T. Lee: AIP Conference Proceedings No. 11, Transport Phenomena, ed. by J. Kestin (American Institute of Physics, New York 1973) R. Schinke, J. Bowman: Molecular Collision Dynamics, ed. by J. Bowman (Springer, Berlin, Heidelberg 1983) S. Bosanac: Phys. Rev. A 22, 2617 (1980) G. Hall, K. Liu, M. J. McAuliffe, C. F. Giese, W. R. Gentry: J. Chem. Phys. 81, 5577 (1984) X. Yang, A. Wodtke: Int. Rev. Phys. Chem. 12, 123 (1993) I. V. Hertel: Adv. Chem. Phys. 50, 475 (1982) W. H. Breckenridge, H. Umemoto: Adv. Chem. Phys. 50, 325 (1982) A. G. Suits, P. de Pujo, O. Sublemontier, J. P. Visticot, J. Berlande, J. Cuvellier, T. Gustavsson, J. M. Mestdagh, P. Meynadier, Y. T. Lee: J. Chem. Phys. 97, 4094 (1992) J. Los, A. W. Kleyn: Alkali Halide Vapors, ed. by P. Davidovits, D. L. McFadden (Academic, New York 1979) Chap. 8 E. A. Gislason: Alkali Halide Vapors, ed. by P. Davidovits, D. L. McFadden (Academic, New York 1979) Chap. 13 R. E. Olson, F. T. Smith, E. Bauer: Appl. Optics 10, 1848 (1971) W. B. Miller, S. A. Safron, D. R. Herschbach: Discuss. Faraday Soc. 44, 108 (1967) W. B. Miller: . Ph.D. Thesis (Harvard Univ., Harvard 1969)

66.76 66.77 66.78 66.79 66.80 66.81 66.82 66.83 66.84 66.85 66.86 66.87 66.88 66.89 66.90 66.91

J. M. Farrar, Y. T. Lee: J. Chem. Phys. 65, 1414 (1976) R. J. Buss, P. Casavecchia, T. Hirooka, Y. T. Lee: Chem. Phys. Lett. 82, 386 (1981) L. Schnieder, K. Seekamp-Rahn, E. Wrede, K. H. Welge: J. Chem. Phys. 107, 6175 (1997) B. Strazisar, C. Lin, H. F. Davis: Science 290, 958 (2000) H. L. Kim, M. A. Wickramaaratchi, X. Zheng, G. E. Hall: J. Chem. Phys. 101, 2033 (1994) M. Brouard, S. P. Duxon, P. A. Enriquez, J. P. Simons: J. Chem. Phys. 97, 7414 (1992) T. P. Rakitzis, S. A. Kandel, R. N. Zare: J. Chem. Phys. 107, 9382 (1997) D. A. Blank, N. Hemmi, A. G. Suits, Y. T. Lee: Chem. Phys. 231, 261 (1998) P. A. Willis, H. U. Stauffer, P. Z. Hinrichs, H. F. Davis: J. Chem. Phys. 108, 2665 (1998) M. Ahmed, D. S. Peterka, A. G. Suits: Phys. Chem. Chem. Phys. 2, 861 (2000) G. Capozza, E. Segoloni, F. Lenori, G. G. Volpi, P. Casavecchia: J. Chem. Phys. 120, 4557 (2004) A. T. J. B. Eppink, D. H. Parker: Rev. Sci. Instrum. 68, 3457 (1997) T. N. Kitsopoulos, C. R. Gebhardt, T. P. Rakitzis: Rev. Sci. Instrum. 72, 3848 (2001) D. Townsend, M. P. Minitti, A. G. Suits: Rev. Sci. Instrum. 74, 2530 (2003) J. J. Lin, J. Zhou, W. Shiu, K. Liu: Rev. Sci. Instrum. 74, 2495 (2003) J. Zhou, J. J. Lin, K. Liu: J. Chem. Phys. 119, 8289 (2003)

983

Ion–Molecule 67. Ion–Molecule Reactions

A number of different physical processes can be categorized as ion-molecule reactions, with examples such as A± + BC → A + BC ± ,

charge transfer

A± + BC → A + (B + C)± ,

dissociative charge transfer

A± + BC → AB ± + C .

particle transfer

The ± superscript indicates charge appropriate to anions and cations. The parentheses indicate that the charge can reside on either the B or C fragment. Particle transfer reactions often involve the transfer of a hydrogen atom or a proton, but heavy particle 0, while Im[n(ω)] is a Lorentzian curve peaked at ω = ω0 . The intensity absorption coefficient a(ω) is

Part F 68.3

a(ω) = 2 Im [n(ω)] ω/c (68.26)   1/2   iγω+ ω20 −ω2 Ne2 2ω Im 1 + = .  c m 0 ω2 −ω2 2 +γ 2 ω2 0 For atomic vapors, the corrections to the vacuum index of refraction are normally small, so that the square root in (68.26) can be expanded to first order, giving   γω2 N e2 . (68.27) a(ω) =  2 0 mc ω − ω2 2 + γ 2 ω2 0 The intensity of a monochromatic field propagating along the z-direction through a gas of Lorentz atoms is therefore attenuated according to Beer’s law given by I(ω, z) = I(ω, 0) e−a(ω)z .

(68.28)

If the index of refraction at a given frequency becomes purely imaginary, no electromagnetic wave can propagate inside the medium. This is the case for field frequencies smaller than the plasma frequency

Ne2 ωp = . (68.29) m 0 While the Lorentz atom model gives an adequate description of absorption and dispersion in a weakly excited absorbing medium, it fails to predict the occurence

of important phenomena such as saturation and light amplification. This is because, in this model, the phase of the induced atomic dipoles with respect to the incident field is always such that the polarization field adds destructively to the incident field. The description of light amplification requires a quantum treatment of the medium, which gives a greater flexibility to the possible relative phases between the incident and polarization fields.

68.2.4 Slowly-Varying Envelope Approximation Light–matter interactions often involve quasi-monochromatic fields for which the electric field (taken to propagate along the z-axis) can be expressed in the form 1 E(R, t) = E + (R, t) ei(kz−ωt) + c.c. , 2 such that

+

+

∂E





 ω E + , ∂E  k E + .

∂z

∂t

(68.30)

(68.31)

It is further consistent within this approximation to assume that the polarization takes the form 1 P(R, t) = P + (R, t) ei(kz−ωt) + c.c. , 2 with

∂P +

+



∂t  ω P .

(68.32)

(68.33)

Under these conditions, known as the slowly-varying envelope approximation ([68.5]), Maxwell’s wave equation reduces to   1∂ k ∂ + E + (z, t) = − P + (z, t) . (68.34) ∂z c ∂t 2i 0 Hence, in the slowly-varying envelopes approximation we ignore the backward propagation of the field [68.12]. The slowly-varying amplitude and phase approximation is essentially the same, except that it expresses the electric field envelope in terms of a real amplitude and phase.

68.3 Two-Level Atoms A large number of optical phenomena can be understood by considering the interaction between a quasimonochromatic field of central frequency ω and

a two-level atom, which simulates a (dipole-allowed) atomic transition [68.5, 7, 10, 11, 13–17]. This approximation is well justified for near-resonant interactions;

Light–Matter Interaction

i. e., ω  ω0 . The next three sections discuss the model Hamiltonian for this system in the semiclassical approximation where the electromagnetic field can be described classically. The formal results are then extended to the case of a quantized field, where the electric field is treated as an operator.

68.3.1 Hamiltonian In the absence of dissipation mechanisms, the dipole interaction between a quasi-monochromatic classical field and a two-level atom is H = ~ωe |ee| + ~ωg |gg| − d · E(R, t) ,

(68.35)

where |e and |g label the upper and lower atomic levels, of frequencies ωe and ωg , respectively, with ωe − ωg = ω0 , and R is the location of the center of mass of the atom. The electric dipole operator (68.8), couples the excited and ground levels, and may be expressed as

where d is a unit vector in the direction of the dipole and d the matrix element of the electric dipole operator between the ground and excited state, which we take to be real for simplicity. We also neglect the vector character of d and E(R, t) in the following, assuming, for example, that both d and  are parallel to x-axis. The Hamiltonian (68.35) may then be expressed as H = ~ωe |ee| + ~ωg |gg| − d (|eg| + |ge|)   (68.37) × E + (R, t) + E − (R, t) , where we have generalized the notation of (68.20) in an obvious way. One can introduce the pseudo-spin operators = |eg| ,

to lowest-order in dE/~ω. The neglect of these terms is the Rotating Wave Approximation (RWA). Note that it is normally inconsistent to regard an atom as a two-level system and not to perform the RWA. In the RWA, the atomic system is described by the Hamiltonian   H = ~ω0 sz − d s+ E + (R, t) + s− E − (R, t) , or, in a frame rotating at the frequency ω of the field,  1  H = ~∆sz − d s+ E eik · R + h.c. , (68.42) 2 where ∆ = ω0 − ω is the atom-light detuning. (Note that the alternate definition δ = ω − ω0 is frequently used in the literature.) In the rest of this chapter, we consider atoms placed at R = 0.

68.3.3 Rabi Frequency The dynamics of the two-level atom is conveniently expressed in terms of its density operator ρ, whose evolution is given by the Schrödinger equation i dρ = − [H, ρ] , dt ~

sz = (|ee| − |gg|)/2 , † s+ = s−

of the field. The corresponding contributions to the atomic dynamics oscillate at frequencies ω0 − ω and ω0 + ω, respectively, and their contributions to the probability amplitudes involve denominators containing this same frequency dependence. For near-resonant atomfield interactions, the rapidly oscillating contributions lead to small corrections, the first-order one being the Bloch–Siegert shift, whose value near resonance, is ω  ω0 [68.17].  + 2 d E /~ δωeg = − (68.40) 4ω

(68.41) (68.36)

(68.38)

and redefine the zero of atomic energy to introduce the commonly used form   H = ~ω0 sz − d (s+ + s− ) E + (R, t) + E − (R, t) . (68.39)

68.3.2 Rotating Wave Approximation Under the influence of a monochromatic electromagnetic field of frequency ω, atoms undergo transitions between their lower and upper states by interacting with either the positive or the negative frequency part

1001

(68.43)

where ρee = e|ρ|e and ρgg = g|ρ|g are the upper and lower state populations Pe and Pg , respectively, while  the off-diagonal matrix elements ρeg = e|ρ|g = ρge are called the atomic coherences, or simply coherences, between levels |e and |g. These coherences play an essential role in optical physics and quantum optics, since they are proportional to the expectation value of the electric dipole operator. The evolution of Pg (t) and Pe (t) = 1 − Pg (t), is characterized by oscillations at the generalized Rabi frequency 1/2  Ω = Ω12 + ∆2 , (68.44)

Part F 68.3

d = d d (|eg| + |ge|) ,

68.3 Two-Level Atoms

1002

Part F

Quantum Optics

where the Rabi frequency Ω1 is Ω1 = dE/~, (or Ω1 = dE( d · )/~ when the vector character of the electric field and dipole moment are included). Specifically, assuming that the atom is initially in its ground state |g, the probability that it is in the excited state |e at a subsequent time t is given by Rabi’s formula Pe (t) = (Ω1 /Ω)2 sin2 (Ωt/2) .

(68.45)

At resonance (∆ = 0), the generalized Rabi frequency Ω reduces to the Rabi frequency Ω1 . (In addition to the texts on quantum optics already cited, see also [68.18].)

68.3.4 Dressed States

Part F 68.3

Semiclassical Case The atomic dynamics can alternatively be described in terms of a dressed states basis instead of the bare states |e and |g (see especially [68.17]). The dressed states |1 and |2 are eigenstates of the Hamiltonian (68.42), and, by convention, the state |1 is the one with the greatest energy. They are conveniently expressed in terms of the bare states via the Stückelberg angle θ/2 as

|1 = sin θ|g + cos θ|e , |2 = cos θ|g − sin θ|e ,

(68.46)

where sin(2θ) = −Ω1 /Ω, cos(2θ) = ∆/Ω. The corresponding eigenenergies are 1 E 1 = + ~Ω , 2 1 E 2 = − ~Ω . (68.47) 2 These energies are illustrated in Fig. 68.1 as a function of the field frequency ω. The dressed levels repel each other and form an anticrossing at resonance ω = ω0 . As the detuning ∆ varies from positive to negative values, state |1 passes continuously from the excited state |e to the bare ground state |g, with both bare states having equal weights at resonance. The distances between the perturbed levels and their asymptotes for |∆|  Ω1 represent the ac Stark shifts, or light shifts, of the atomic states when coupled to the laser. From Fig. 68.1, the ac Stark shift of |g is positive for ∆ < 0 and negative for ∆ > 0, while the |e state shift is negative for ∆ < 0 and positive for ∆ > 0.

field are still described by the Hamiltonian (68.41), except that the positive and negative frequency components of the field are now operators, and the free field Hamiltonian must be included. The Hamiltonian of the total atom-field system becomes    1 † H = ~ω0 sz + ~ω a a + + ~g s+ a + a† s− , 2 (68.48)

a†

where the creation and annihilation operators and   a obey the boson commutation relation a, a† = 1 (Chapt. 6), and the coupling constant  ω (68.49) g=d 2 0 ~V is the vacuum Rabi frequency, with V being a photon normalization volume. This Hamiltonian defines the Jaynes–Cummings model, [68.7, 19] which is discussed in more detail in Chapt. 78. The dressed states of the atom-field system are the eigenstates of the Jaynes–Cummings model. Since, in the RWA, the dipole interaction only couples states of same “excitation number”, e.g., |e, n and |g, n + 1, where |n is an eigenstate of the photon number operator, a† a|n = n|n, with n an integer, the diagonalization of the Jaynes–Cummings model reduces to that of the semiclassical driven two-level atom in each of these manifolds. Hence, the dressed states are |1, n = sin θn |g, n + 1 + cos θn |e, n , |2, n = cos θn |g, n + 1 − sin θn |e, n ,

(68.50)

with

√ tan(2θn ) = −2g n + 1/∆ .

(68.51)

E1 ⱍg冭

∆

ⱍe冭

Quantized Field The concept of dressed states can readily be generalized to a two-level atoms interacting with a single-mode quantized field in the dipole and rotating wave approximations. The atom and its dipole interaction with the

E2

Fig. 68.1 Dressed levels of a two-level atom driven by a classical monochromatic field as a function of the detuning ∆ = ω0 − ω

Light–Matter Interaction

(The factor of 2 difference between this and the semiclassical case is due to the use of a running waves quantization scheme, while the semiclassical discussion was for standing waves.) The corresponding eigenenergies are E 1n = ~(n + 1)ω − ~ Rn , E 2n = ~(n + 1)ω + ~ Rn ,

(68.52)

where

68.3.5 Optical Bloch Equations Introducing the density operator matrix elements ρab = a|ρ|b, where a, b can be either e or g, as well as the real quantities U = ρeg eiωt + c.c. , V = iρeg eiωt + c.c. , W = ρee − ρgg ,

(68.54)

the equations of motion for the density matrix elements ρij = i|ρ| j may be expressed, with (68.43), as dU = −∆V , dt dV = ∆U + Ω1 W , dt dW = −Ω1 V . (68.55) dt These are the optical Bloch equations, as discussed extensively in [68.5, 17]. Physically, U describes the component of the atomic coherence in phase with the driving field, V the component in quadrature with the field, and W the atomic inversion. The optical Bloch equations have a simple geometrical interpretation offered by thinking of U, V and W as the three components of a vector called the Bloch vector U, whose equation of motion is dU = Ω ×U , (68.56) dt where Ω = (−Ω1 , 0, ∆). Thus U precesses about Ω, of length Ω, while conserving its length. The evolution of a two-level atom driven by a monochromatic field is thus mathematically equivalent to that of a spin- 12 system in two magnetic fields B0 and 2B1 cos ωt which are parallel to the z- and x-axis, respectively, and whose amplitudes are such that the Larmor spin precession frequencies around them are ω and 2Ω1 cos ωt, respectively. In optics, this vectorial picture is often referred to as the Feynman– Vernon–Hellwarth picture [68.21]. It is very useful in discussing the coherent transient phenomena discussed in Chapt. 73.

68.4 Relaxation Mechanisms In addition to their coherent interaction with light fields, atoms suffer incoherent relaxation mechanisms, whose origin can be as diverse as elastic and inelastic collisions and spontaneous emission. Collisional broadening is discussed in Chapt. 59, while a QED microscopic discussion of spontaneous emission is described in Chapt. 78 in terms of reservoir theory. One advantage of describing the atomic state in terms of a density operator ρ is that the physical interpretation of its elements allows us phenomenologically to add various relaxation terms directly to its elements.

1003

68.4.1 Relaxation Toward Unobserved Levels If the relaxation mechanisms transfer populations or atomic coherences toward uninteresting or unobserved levels, their description can normally be given in terms of a Schrödinger equation, but with a complex Hamiltonian. In contrast, if all levels involved in the relaxation mechanism are observed, a more careful description, e.g. in terms of a master equation, is required. Specifically, in the case of relaxation to unobserved levels, the evolution of the atomic density operator, restricted to the

Part F 68.4

 1 ∆2 + 4g2 (n + 1) . (68.53) 2 Chapter 78 shows that by including the effects of spontaneous emission, this picture yields a straightforward interpretation of a number of effects, including the Burshtein–Mollow resonance fluorescence spectrum. Dressed states also help to elucidate the interaction between two-level atoms and quantized single-mode fields, as occur for example in cavity QED ([68.19] and Chapt. 79). Their generalization to the case of moving atoms offers simple physical interpretations of several aspects of laser cooling, see Chapt. 75 and [68.20]. Rn =

68.4 Relaxation Mechanisms

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levels of interest, is of the general form [68.16]  i  dρ † =− (68.57) Heff ρ − ρHeff , dt ~ where Heff = H + Γˆ , (68.58) H being the atom-field Hamiltonian and Γˆ the nonHermitian relaxation operator, defined by its matrix elements ~ n|Γˆ |m = γn δnm . (68.59) 2i Both inelastic collisions and spontaneous emission to unobserved levels can be described by this form of evolution. In the framework of this chapter, inelastic, or strong, collisions are defined as collisions that can induce atomic transitions into other energy levels.

68.4.2 Relaxation Toward Levels of Interest Part F 68.4

A master equation description is necessary when all involved levels are observed [68.7, 17]. This master equation can rapidly take a complicated form if more than two levels are involved. We give results only for the case of a two-level atom and upper to lower-level spontaneous decay and elastic or soft collisions; i. e., collisions that change the separation of energy levels during the collision, but leave the level populations unchanged. In that case, the atomic master equation takes the form i Γ dρ = − [H, ρ] − (s+ s− ρ + ρs+ s− − 2s− ρs+ ) dt ~ 2 1 − γph ρ + 2γph sz ρsz , (68.60) 2 where the free-space spontaneous decay rate Γ is found from QED to be 1 4d 2 ω30 , (68.61) 4π 0 3~c3 and γph is the decay rate due to elastic collisions. It is possible to express the classical decay rate (68.17) in terms of the quantum spontaneous emission rate (68.61) as Γ=

Γ = Γcl f ge ,

(68.62)

where f ge is the oscillator strength of the transition. The various oscillator strengths characterizing the dipoleallowed transitions from a ground state |e to excited levels |e obey the Thomas–Reiche–Kuhn sum rule  f ge = 1 , (68.63) e

where the sum is on all levels dipole-coupled to |g. Assuming that d and the polarization of the field are both parallel to the x-axis, this gives f ge =

2mω0 |g|x|e|2 . ~

(68.64)

68.4.3 Optical Bloch Equations with Decay In general, the optical Bloch equations cannot be generalized to cases where relaxation mechanisms are present. There are, however, two notable exceptions corresponding to situations where 1. the upper level spontaneously decays to the lower level only, while the atom undergoes only elastic collisions; 2. spontaneous emission between the upper and lower levels can be ignored in comparison with decay to unobserved levels, which occur at equal rates γe = γg = 1/T1 . Under these conditions, (68.55) generalizes to dU = −U/T2 − ∆V , dt dV = −V/T2 + ∆U + Ω1 W , dt dW = −(W − Weq )/T1 − Ω1 V , dt

(68.65)

where we have introduced the longitudinal and transverse relaxation times T1 and T2 , with T1 = 1/Γ and T2 = (1/2T1 + γph )−1 in the first case, and T2 = (1/T1 + γph )−1 in the second case. The equilibrium inversion Weq is equal to zero in the second case since the decay is to unobserved levels.

68.4.4 Density Matrix Equations In the general case, it is necessary to consider the density operator equation (68.43) instead of the optical Bloch equations. The equations of motion for the components of ρ become, for the general case of complex Ω1 , 1 dρee = −γe ρee − iΩ1∗ ρ˜ eg + c.c. , dt 2 dρgg 1 = −γg ρgg + iΩ1∗ ρ˜ eg + c.c. , dt 2 dρ˜ eg Ω1  = −(γ + i∆)ρ˜ eg − i ρee − ρgg , (68.66) dt 2

Light–Matter Interaction

where γ = (γe + γg )/2 + γph , and ρ˜ eg = ρeg eiωt . In the case of spontaneous decay from the upper to the lower level, these equations become dρee 1 = −Γρee − iΩ1∗ ρ˜ eg + c.c. , dt 2

68.5 Rate Equation Approximation

1005

dρgg 1 = +Γρee + iΩ1∗ ρ˜ eg + c.c. , dt 2 dρ˜ eg Ω1  = −(γ + i∆)ρ˜ eg − i ρee − ρgg , (68.67) dt 2 where γ = Γ/2 + γph . Equations (68.67) are completely equivalent to the optical Bloch equations (68.65).

68.5 Rate Equation Approximation If the coherence decay rate γ is dominated by elastic collisions, and hence is much larger than the population decay rates γe and γg , ρ˜ eg can be adiabatically eliminated from the equations of motion (68.66) and (68.67) to obtain the rate equations (Sect. 68.2)

(68.68)

and  dρee = −Γρee − R ρee − ρgg , dt  dρgg = +Γρgg + R ρee − ρgg , dt

(68.69)

(68.70)

The transitions between the upper and lower state are thus described in terms of simple rate equations. Adding phenomenological pumping rates Λe and Λg on the right-hand side of (68.68) provides a description of the excitation of the upper and lower levels from some distant levels, as would be the case in a laser. The equations then form the basis of conventional, singlemode laser theory. In the absence of such mechanisms, the atomic populations eventually decay away.

68.5.1 Steady State In the case of upper to lower-level decay, the state populations reach a steady state with inversion [68.5,17] 1 Γ =− , Γ + 2R 1+s

(68.72)

(68.73)

In steady state, the other two components of the Bloch vector U are given by   2∆ s (68.74) Ust = − Ω1 1 + s

Vst =

and we have introduced the dimensionless Lorentzian  L(∆) = γ 2 / γ 2 + ∆2 . (68.71)

Wst = −

Ω12 /2 . Γ 2 /4 + ∆2

and

respectively, where the transition rate is R = |Ω1 |2 L(∆)/(2γ) ,

s=

Γ Ω1



s 1+s

 .

(68.75)

Ust varies as a dispersion curve as a function of the detuning ∆, while Vst is a Lorentzian of power-broadened  1/2 half-width at half maximum Γ 2 /4 + Ω12 .

68.5.2 Saturation As the intensity of the driving field, or Ω12 , increases, Ust and Vst first increase linearly with Ω1 , reach a maximum, and finally tend to zero as Ω1 → ∞. The inversion Wst , which equals −1 for Ω1 = 0, first increases quadratically, and asymptotically approaches Wst = 0 as Ω1 → ∞. At this point, where the upper and lower state populations are equal, the transition is said to be saturated, and the medium becomes effectively transparent, or bleached. (This should not be confused with self-induced transparency discussed in Chapt. 73.) The inversion is always negative, which means in particular that no steady-sate light amplification can be achieved in this system. This is one reason why external pump mechanisms are required in lasers.

68.5.3 Einstein A and B Coefficients When atoms interact with broadband radiation instead of the monochromatic fields considered so far, (68.69)

Part F 68.5

 dρee = −γe ρee − R ρee − ρgg , dt  dρgg = −γg ρgg + R ρee − ρgg , dt

where s is the saturation parameter. In the case of pure radiative decay, γph = 0, s is given by

1006

Part F

Quantum Optics

still apply, but the rate R becomes R → Beg (ω) ,

(68.76)

where (ω) is the spectral energy density of the inducing radiation. Einstein’s A and B coefficients apply to an atom in thermal equilibrium with the field, which is described by Planck’s black-body radiation law (ω) =

~ω3 1 , π 2 c3 e~ω/kB T − 1

(68.77)

where T is the temperature of the source and kB is Boltzmann’s constant. Invoking the principle of detailed balance, which states that at thermal equilibrium, the average number of transitions between arbitrary states |i and |k must be equal to the number of transitions between |k and |i, one finds

~ω3 Aki = 2 3, (68.78) Bki π c where Aki is the rate of spontaneous emission from |k to |i, and Γk = i Aki is the level width.

68.6 Light Scattering

Part F 68.6

Far from resonance, the approximation of a two- or fewlevel atom is no longer adequate. Two limiting cases, which are always far from resonance, are Rayleigh scattering for low frequencies, and Thomson scattering for high frequencies.

68.6.1 Rayleigh Scattering Rayleigh scattering is the elastic scattering of a monochromatic electromagnetic field of frequency ω, wave vector k and polarization  by an atomic system in the limit where ω is very small compared with its excitation energies [68.4, 17]. To second-order in perturbation theory, the Rayleigh scattering differential cross section into the solid angle Ω about the wave vector k with k = k, and polarization is  2  f ge dσ 2 4 2 = r0 ω ( ·  ) , (68.79) dΩ ω2eg e where r0 is the classical electron radius, the sum is over all states |e, f ge is the E1 oscillator strength (68.64) and ωge is the transition frequency. The corresponding total cross section is 2  8πr02 ω4  f ge σ= . (68.80) 3 ω2eg e

68.6.2 Thomson Scattering Thomson scattering is the corresponding elastic photon scattering by an atom in the limit where ω is very large compared with the atomic ionization energy, yet small enough compared to αmc2 /~, that the dipole approximation can be applied. The differential cross section for

this process is [68.4, 17]  2 dσ = r02  ·  , dΩ and the total cross section is 8 σ = πr02 . 3

(68.81)

(68.82)

This is a completely classical result, which exhibits no frequency dependence.

68.6.3 Resonant Scattering We finally consider elastic scattering in the limit where ω is close to the transition frequency ω0 between |g and |e. Provided that no other level is near-resonant with the ground state, the resonant scattering differential cross section is [68.6, 17] 9 2 dσ (Γ/2)2 2  ·  = λ , dΩ 16π 2 0 ∆2 + (Γ/2)2

(68.83)

where λ0 = 2πc/ω0 is the wavelength of the transition and Γ is the spontaneous decay rate (68.61). The total elastic scattering cross section is σ=

3 2 λ . 2π 0

(68.84)

In contrast to the nonresonant Rayleigh and Thomson scattering cross sections, which scale as the square of the classical electron radius, the resonant scattering cross section scales as the square of the wavelength. For optical to frequencies, λ0 /r0  104 , giving a resonant enhancement of about eight orders of magnitude. This illustrates why near resonant phenomena, which form the bulk of the following chapters, are so important in optical physics and quantum optics.

Light–Matter Interaction

References

1007

References 68.1 68.2

68.3 68.4 68.5 68.6 68.7 68.8 68.9

68.11 68.12 68.13 68.14 68.15 68.16 68.17

68.18 68.19 68.20

68.21

M. S. Zubairy, M. O. Scully: Quantum Optics (Cambridge Univ. Press, Cambridge 1997) Y. R. Shen: The Principles of Nonlinear Optics (Wiley, New York 1984) M. Orszag: Quantum Optics (Springer, Berlin, Heidelberg 2000) W. P. Schleich: Quantum Optics in Phase Space (Wiley-VCH, Weinheim 2001) M. Sargent III, M. O. Scully Jr., W. E. Lamb: Laser Physics (Wiley, Reading 1977) S. Stenholm: Foundations of Laser Spectroscopy (Wiley-Interscience, New York 1984) C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg: Atom-Photon Interactions: Basic Processes and Applications (Wiley-Interscience, New York 1992) P. L. Knight, P. W. Milonni: Phys. Rep. 66, 21 (1980) P. P. Berman (Ed.): Cavity QED (Academic, Boston 1994) C. Cohen-Tannoudji: Atomic motion in laser light. In: Fundamental Systems in Quantum Optics, ed. by J. Dalibard, J. M. Raimond, J. Zinn-Justin (NorthHolland, Amsterdam 1992) R. P. Feynman, F. L. Vernon, R. W. Hellwarth: J. Appl. Phys. 28, 49 (1957)

Part F 68

68.10

B. W. Shore: The Theory of Coherent Atomic Excitation (Wiley-Interscience, New York 1990) C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg: Photons and Atoms: Introduction to Quantum Electrodynamics (Wiley-Interscience, New York 1989) A. Sommerfeld: Optics (Academic, New York 1967) J. D. Jackson: Classical Electrodynamics (Wiley, New York 1975) L. Allen, J. H. Eberly: Optical Resonance and TwoLevel Atoms (Dover, New York 1987) P. W. Milonni, J. H. Eberly: Lasers (Wiley-Interscience, New York 1988) P. Meystre, M. Sargent III: Elements of Quantum Optics, 3rd edn. (Springer, Berlin, Heidelberg 1999) M. Born, E. Wolf: Principles of Optics, 4th edn. (Pergamon, Oxford 1970) R. J. Glauber: Optical coherence and photon statistics. In: Quantum Optics and Electronics, ed. by C. DeWitt, A. Blandin, C. Cohen-Tannoudji (Gordon and Breach, New York 1965) L. Mandel, E. Wolf: Optical Coherence and Quantum Optics (Cambridge Univ. Press, Cambridge 1995)

1009

Absorption an 69. Absorption and Gain Spectra

This chapter develops theoretical techniques to describe absorption and emission spectra, using concepts introduced in Chapt. 68, and density matrix methods from Chapt. 7. The simplest cases are treated, compatible with the physics involved, and more realistic applications are referred to in other chapters. Vector notation is not used, but it can be inserted as required. Only steady-state spectroscopy is covered; for time-resolved transient techniques see Chapt. 73. Laser technology has greatly expanded the potential of atomic and molecular spectroscopy, but the same techniques for describing the interaction of light with matter also apply to the traditional arc lamps and flash discharges,

69.2 Density Matrix Treatment of the Two-Level Atom......................... 1010 69.3 Line Broadening .................................. 1011 69.4 The Rate Equation Limit ....................... 1013 69.5 Two-Level Doppler-Free Spectroscopy ... 1015 69.6 Three-Level Spectroscopy ..................... 1016 69.7 Special Effects in Three-Level Systems ... 1018 69.8 Summary of the Literature ................... 1020 References .................................................. 1020 and the more recent synchrotron radiation sources.

At the other extreme, the spectroscopy of dilute gases is well characterized by ensemble averages over the properties of the individual particles, interrupted by occasional brief collisions. Ensemble averages, however, may no longer apply to recent experiments probing a single atomic particle in a trap, as discussed in Chapt. 75.

69.1 Index of Refraction As discussed in Sect. 68.2.2, the complex index of refraction for a medium containing harmonically bound charges (electrons) [69.1] with natural frequency ω0 is
Nα(ω) Nα(ω) n(ω) = 1 + ≈ 1+ ε0 2ε0     2 2 iγω + ω0 − ω2 Ne = 1+  2 2 2mε0 ω − ω2 + γ 2 ω2 = n  + i n  .

0

(69.1)

The expansion is valid when the density of atoms N is low. A plane wave can be written in the form 

E ∝ eikz = eiωnz/c = eiωn z/c e−ωn

 z/c

.

(69.2)

The absorption of light through the medium then shows a resonant behavior near ω ≈ ω0 determined by   Ne2 γω  n =  2 2 2mε0 ω0 − ω2 + γ 2 ω2   γ/2π πNe2 . (69.3) ≈ 4mε0 ω0 (ω − ω0 )2 + γ 2 /4 This is called an absorptive lineshape. When the single electron is harmonically bound, its interaction with radiation is found in this response. For a real atom, the response of the electron is divided among the various transitions to other states. The fraction assigned to one single transition is characterized by the oscillator strength f n as discussed in Sect. 68.4.2. In ordinary linear spectroscopy, the laser is tuned through the resonance ω ≈ ω0 , and the value of ω0 is

Part F 69

In many cases, departures from the thin sample limit, such as beam attenuation, light scattering and radiation trapping (Sect. 69.2) may be important. However, the properties of laser devices themselves depend in an essential way on these effects, making a self-consistent treatment of their properties necessary.

69.1 Index of Refraction .............................. 1009

1010

Part F

Quantum Optics

determined from the lineshape (69.3). Several closely spaced resonances can be resolved if their spacing is larger than their widths    (1) (2)  (69.4) ω0 − ω0  > γ . This defines the spectral resolution (Chapt. 10). The velocity of light in the medium is seen to be given by the expression  

ω20 − ω2 c Ne2 ceff =  ≈ c 1 − .   n 2mε0 ω2 − ω2 2 + γ 2 ω2 0

(69.5)

This expression shows a dispersive behavior around the position ω = ω0 , where the modification of the velocity disappears. Below resonance ω < ω0 , the velocity of light is lower than in vacuum. This derives from the fact that the polarization is in phase with the driving field. Thus, by storing the incoming energy, the driving field retards the propagation of the radiation. For a harmonically bound charge, the refractive index (69.1) always stays absorptive and it is independent of the intensity of the laser radiation. This no longer holds for discrete level atomic systems. In order to see this, we consider the two-level atom in Sect. 69.2 (Sect. 68.3).

69.2 Density Matrix Treatment of the Two-Level Atom

Part F 69.2

The response of atoms to light is conveniently expressed in terms of the density matrix ρ. In addition to the direct physical meaning of the density matrix elements discussed in Sect. 68.3.3, the density matrix formalism is advantageous because the various relaxation mechanisms effecting the atomic resonances can be introduced phenomenologically into its equations of motion (Sect. 68.4), and theoretical derivations often provide master equations for the density matrix (Chapt. 7). The two-level Hamiltonian (68.35) can be written as

~ω0 /2 −dE(R, t) H= (69.6) . −dE(R, t) −~ω0 /2 The equation of motion for the density matrix (68.67) is then   idE d ρee = − Γρee + cos ωt ρge − ρeg , dt ~   idE d ρgg = Γρee − cos ωt ρge − ρeg , dt ~   idE d ρeg = − (γ + iω0 )ρeg + cos ωt ρgg − ρee . dt ~ (69.7)

Here Γ is the spontaneous decay rate given by (68.61) and γ = Γ/2 + γph ,

(69.8)

where γph derives from all processes that tend to randomize the phase between the quantum states |e and |g, such as collisions (Chapts. 7 and 19), noise in the laser fields and thermal excitation of the environment

in solid state spectroscopy. The Greek letters Γ and γ correspond to the longitudinal relaxation rate (T1 process) and the transverse relaxation rate (T2 -process), respectively (Sect. 68.4.3). In the rotating wave approximation (RWA), discussed in Sect. 68.3.2, the density matrix equations (69.7) become identical to (68.67) with ρeg = ρ˜ eg e−iωt .

(69.9)

Using the condition of conservation of probability ρee + ρgg = 1 ,

(69.10)

the steady state solutions to (68.67) are [69.2]   Ω12 γ 1 , ρee = 2Γ ∆2 + γ 2 + Ω12 γ/Γ   iΩ1 ρgg − ρee ρ˜ eg = 2 γ + i∆   γ − i∆ iΩ1 = , 2 ∆2 + γ 2 + Ω12 γ/Γ

(69.11)

(69.12)

where Ω1 is the Rabi frequency from (68.44), and ∆ = ω0 − ω is the detuning. The induced polarization is then   ˆ = N Tr P = N Tr dρ



0 d d 0   = Nd e−iωt ρ˜ eg + eiωt ρ˜ ge  = N αE (+) + α∗ E (−) ,



ρee ρeg ρge ρgg



(69.13)

Absorption and Gain Spectra

where N is the density of active two-level atoms. Setting 1 E (+) = E e−iωt , 2 the complex polarization is   d2 iγ + ∆ α(ω) = , ~ ∆2 + γ 2 + Ω12 γ/Γ

(69.14)

(69.15)

and from (69.1), the complex index of refraction is Nα(ω) 2ε0   f0 iγ + ∆ πNe2 , = 1+ 4ε0 mω0 π ∆2 + γ 2 + Ω12 γ/Γ

n(ω) = 1 +

(69.16)

where f 0 = 2d 2 mω0 /~e2 is the oscillator strength. Summing over all possible transitions yields the f-sum rule (68.63) (Chapt. 21). The imaginary part of (69.16) shows exactly the same absorptive behavior as in the harmonic oscillator model of Sect. 69.1 [see (69.3)]. However, the additional  factor of Ω12 γ/Γ in the denominator makes the line

69.3 Line Broadening

appear broader than in the harmonic case; the line is power broadened. Physically, this derives from a saturation of the two-level system in which the population of the upper level becomes an appreciable fraction of that of the lower level. In the limit Ω1 → ∞, (69.16) shows that n(ω) → 1 and the atom-field interaction effectively vanishes. In this limit, ρee → 12 (69.11), and the field induces as many upward transitions as downward transitions. When radiation at the frequency ω propagates in a medium of two-level atoms, the energy density is 

 2  γz Nd ω . I(z) ∝ E ∗ E ∝ exp − ~ε0 c (ω − ω0 )2 + γ 2 (69.17)

Far from resonance (|ω − ω0 |  γ ), the medium is transparent; but near resonance, damping is observed. The impinging radiation energy is deposited in the medium and propagation is impeded. This is called radiation trapping. In spectroscopy, the phenomenon is seen as a prolongation of the radiative decay time; the spontaneously emitted energy is seen to emerge from the sample more slowly than the single atom lifetime implies.

(69.18)

The various contributions are as follows [69.3]. The term γ contains all the transverse relaxation mechanisms. If decay to additional levels occurs, these must be included (Sect. 69.4). The term γph contains all perturbing effects effecting each single atom. For low enough pressures, collisional perturbations are proportional to the density of perturbing atoms so that γph = ηcoll p ,

(69.19)

where p is the pressure of the perturbing gas and ηcoll is a constant of proportionality. This is called collision or pressure broadening (Chapt. 59), whose order of magnitude can be estimated to be the inverse of the average free time between collisions. For high pressure (usually of the order of torrs), the linearity in (69.19) breaks down. When identical atoms collide, resonant

exchange of energy may also take place. The third term in (69.18) is the power broadening term. It derives from the effect of the laser field on each individual atom. All such relaxation processes that are active on each and every individual atom separately are called homogeneous broadening processes. For a detailed discussion of line broadening, consult Chapt. 19. In contrast to homogeneous broadening, Doppler broadening is characterized by an atomic velocity parameter v which varies over the observed assembly. In a thermal assembly at temperature T (e.g., a gas cell), the velocity distribution is   1 v2 P(v) = √ exp − 2 , (69.20) 2u 2πu 2 where u 2 = kB T/M. A particular atom with velocity v in the direction of the optical beam with wave vector k then experiences the Doppler-shifted frequency ω − kv relative to a stationary atom, and the effective detuning becomes ∆ = ∆ + kv ,

(69.21)

Part F 69.3

69.3 Line Broadening The effective width of a spectral line from (69.16) is   Ω2 1 γ γeff = γ 2 + Ω12 Γ + γph + 1 + O Ω14 . Γ 2 2Γ

1011

1012

Part F

Quantum Optics

replacing ∆. The population in the lower level from (69.11) is then   Ω12 γ 1 . ρgg = 1 − 2Γ (∆ + kv)2 + γ 2 + Ω12 γ/Γ (69.22)

The atoms in the lower level, originally distributed according to (69.20), are now depleted from the velocity group around v = (ω − ω0 )/k .

(69.23)

The width of the depleted region is given by γeff of (69.18). This region is called a Bennett hole. When the laser frequency ω is tuned, the hole sweeps over the velocity distribution of the atoms. The atomic response is saturated at the velocity group of the hole, indicating that spectral hole burning has occurred. The observed spectrum is obtained by averaging the single atom response (69.15) over the velocity distribution. From the imaginary part, the absorption response is d2 α (ω) = √ ~ 2πu 2

+∞

Part F 69.3

−∞

γ e−v /2u dv . (∆ + kv)2 + γ 2 + Ω12 γ/Γ 2

2

(69.24)

In the limits Ω1 → 0 (no saturation), and γ  ku (the Doppler limit), the Lorentzian line shape sweeps over the entire velocity profile, finding a resonant velocity group according to (69.23) as long as v ≤ u. Thus, the linear spectroscopy sees a Doppler broadened line of width ku. This is called inhomogeneous broadening. In the unsaturated regime, the atomic response function (69.24) is proportional to the imaginary part of the function   +∞ exp −x 2 /2σ 2 1 dx , V(z) = √ (69.25) z−x 2πσ 2 −∞

at z = −∆ − iγ and σ = ku. This is the Hilbert transform of the Gaussian, and its shape is called a Voigt profile [69.4]. For γ  ku, it traces over the Gaussian, but for large detunings it always goes to zero as slowly as the Lorentzian, i. e., as ∆−1 . The profile has been widely used to interpret the data of linear spectroscopy. The function is tabulated [69.5] and its expansion is √ n  ∞  π  iz/ 2σ V(z) = − 2 , (69.26) Γ (1 + n/2) 2σ n=0

and it has the continued fraction representation, 1 (69.27) V(z) = . σ2 z− 2σ 2 z− 3σ 2 z− 2 z − z 4σ −··· In addition to velocity, any other parameter shifting the individual atomic resonance frequencies ω0 by different amounts for the different individuals leads to inhomogeneous broadening. The detuning ∆ is then different for different members of the observed assembly, and a line shape similar to (69.24) applies. The distribution function must be replaced by the one relevant for the problem. In practice a Gaussian is almost always assumed. An example of inhomogeneous broadening is the influence of the lattice environment on impurity spectroscopy in solids. The resonant light selectively excites atoms at those particular positions which make the atoms resonant. Thus only these spatial locations are saturated, and the phenomenon of spatial hole burning occurs. This has been investigated as a method for storing information, signal processing, and volume holography. It is, however, possible that the effects of collisions can counteract the inhomogeneous broadening. In order to see this we observe that the induced atomic dipole is proportional to the induced density matrix element, from (69.9), ρeg = ρ˜ eg ei(kz(t)−ωt) ∝ eikvt .

(69.28)

If the atoms now experience collisions characterized by an average free time of flight τ, the phase kvt cannot build up coherently for times longer than this duration, the atomic velocity is quenched on the average, and the full Doppler profile cannot be observed. To see how this comes about, consider a time t  τ. During this period the atom experiences on the average n¯ = t/τ collisions. Assuming a Poisson distribution, the probability of n collisions in time t is e−n¯ n pn = n¯ . (69.29) n! Taking the average of (69.28) over the time t = nτ with the distribution pn yields   ∞  e−n¯ n ikvτn n¯ e ρ¯ eg ∝ n! n=0  = e−n¯ exp n¯ eikvτ   1 ≈ eikvt exp − tk2 v2 τ . (69.30) 2

Absorption and Gain Spectra

This heuristic derivation suggests that for long enough interaction times (t  τ), the large velocity components are suppressed. This tends to prevent the tails of the velocity distribution from contributing to the observed spectral profile. The effect is

69.4 The Rate Equation Limit

1013

called collisional narrowing or Dicke narrowing. It is an observable effect, but the narrowing cannot be very large. To overcome the Doppler broadening one has to turn to nonlinear laser methods (Sect. 69.5).

69.4 The Rate Equation Limit Consider now a generalized theory for the case of several incoming electromagnetic fields of the form 1 Ei (R) e−iωi t+iϕi + c.c. . (69.31) E(R, t) = 2

This can be surmised to hold when the phase relaxation contributions to γ are large (69.8) or the detuning |∆| is large. Insertion of (69.32) into (69.35) for the single mode case gives

The index i may range over several laser sources, the output of a multimode laser or the multitude of components of a flashlight or a thermal source. Each component carries its own amplitude Ei . In steady state, the generalization of (69.12) becomes   i  dEi ρgg − ρee −iωi t+iϕi e , (69.32) ρeg = 2 ~ γ + i∆i

(69.37)

i

i

where the detuning is ∆i = ω0 − ωi .

(69.33)

i

This resolution is of key importance to the theory. With the multimode field, (68.67) for the level occupation probabilities become d d ρee = − ρgg dt dt i  = − Γρee + 2 i   dEi −iωi t+iϕi × e ρge − c.c. . ~

where the rate coefficient is given by  2 γ/π dE . W = 2π 2 2~ ∆ +γ2

(69.38)

Multiplying (69.37) by the density of active atoms N then produces the conventional rate equations for the populations Nee = Nρee and N gg = Nρgg . Two physical effects can be discerned in (69.37): induced and spontaneous emission. The term with Γ gives the spontaneous emission which forces the entire population to the lower level. The terms proportional to W describe induced emission, with upward transitions proportional to Wρgg and downward transitions proportional to Wρee . In the absence of spontaneous emission, they strive to equalize the population of the two levels. Using (69.10), the steady-state solution is W Γ + 2W   1 1 dE 2 γ . = 2 ~ Γ ∆2 + γ 2 + (dE/~)2 γ/Γ

ρee =

(69.39)

(69.35)

Insertion of the steady state result (69.32) into this equation yields a closed set of equations for the level occupation probabilities. These are called rate equations. To justify the above steps, consider the single frequency case again. The off-diagonal time derivatives can be neglected when    d   ρeg   |γ + i∆||ρeg | . (69.36)  dt 

This is clearly seen to agree with the solution (69.11) as is expected in steady state. Although the rate equations were derived in the limit (69.36), the rate coefficient (69.38) has a special significance in the limit γ → 0. In this limit, the factor   γ/π 1 1 = δ(∆) lim ~ γ →0 ∆2 + γ 2 ~ (69.40) = δ(~ω − ~ω0 ) , enforces energy conservation in the transition. Using the field in (69.14), and the interaction from (69.6), the

Part F 69.4

The response of the atom now separates into individual contributions oscillating at the various frequencies ωi according to  (i) −iωi t+iϕi ρeg e . (69.34) ρeg =

  d ρee = −Γρee − W ρee − ρgg , dt

1014

Part F

Quantum Optics

off-diagonal matrix element is 1 (69.41) | e|H|g| = dE . 2 With these results, (69.38) can be written in the form of Fermi’s Golden Rule 2π | e|H|g|2 δ(~ω − ~ω0 ) , (69.42) W= ~ usually derived from time dependent perturbation theory. Returning to the multimode rate equations, an incoherent broad band light source has many components that contribute to the sum over field frequencies. In the case of flash pulses, thermal light sources, or freerunning multimode lasers the spectral components are uncorrelated. Inserting (69.32) into (69.35) yields d d ρee = − ρgg dt dt

 d2  = − Γρee + 2 ρgg − ρee 2~  Ei E j ei(ω j −ωi )t ei(ϕi −ϕ j ) × i, j

γ ∆2j + γ 2

.

(69.43)

Part F 69.4

The contributions from the different terms i = j average to zero either by beating at the frequencies |ωi − ω j | or by incoherent effects from the random phases ϕ j . Thus, only the coherent sum survives to give d d ρee = − ρgg dt dt    = − Γρee × W (i) ρgg − ρee ,

(69.44)

i

where the rate coefficients W (i) are given by (69.38) with the appropriate detunings ∆ j = ω0 − ω j . This is a rate equation in the limit of many uncorrelated components of light, i. e., for a broad band light source. In this case the incoherence between the different components justifies the use of a rate approach, and no assumption like (69.36) is needed. Thus, the limit γ → 0 (69.40) is legitimate, and the W (i) can be calculated in time dependent perturbation theory from Fermi’s Golden Rule. In the limit of an incoherent broad band light source, the sum in (69.44) can be replaced by an integral. In particular, this is allowed for incandescent light sources as used in optical pumping experiments [69.6]. Pumping of lasers by strong lamps or flashes are also describable by the same rate equations. In amplifiers and lasers, the atoms must be brought into states far from equilibrium by incoherent optical

excitation or resonant transfer of excitation energy in collisions (Chapt. 70). In the two-level description, the atomic levels are constantly replenished. The normalization condition (69.10) is then no longer appropriate; often the density matrix is normalized so that Tr(ρ) directly gives the density of active atoms. With pumping into the levels, one must allow for decay out of the two-level system in order to prevent the atomic density from growing in an unlimited way. This decay takes the atom to unobserved levels. In the rate equation approximation, the pumping and decay processes can be described by terms added to the equations of the form (68.68) d ρee = λe − γe ρee , dt d ρgg = λg − γg ρgg . (69.45) dt In a laser, the level |g is usually not the ground state of the system. From (69.45), the steady state population is (0) (0) ρgg − ρee = λg /γg − λe /γe .

(69.46)

A population inversion exists when this is negative. The population difference (69.46) is modified when the effects of spontaneous and induced processes are added, as in (69.37). Then the transitions saturate because of the induced processes. Using (69.46) in (69.12), the calculated polarizability without saturation is  2  d iγ + ∆  (0) (0) α(ω) = . (69.47) ρ − ρ gg ee ~ γ 2 + ∆2 According to (69.1), the index of refraction is Nα (ω) 2ε  02  (ω − ω0 )  (0) Nd (0) ρ . = 1− − ρ gg ee 2ε0 ~ (ω − ω0 )2 + γ 2

n  (ω) = 1 +

(69.48)

For weakly excited atoms, ρgg ≈ 1, and (69.47) agrees with the unsaturated limit of (69.16). The dispersion of a light signal behaving according to (69.48) is called normal, i. e., according to the harmonic model in Sect. 69.1. Below resonance (ω < ω0 ), n  is larger than unity, implying a reduction of the velocity of light. As a function of ω, the curve (69.48) starts above unity, and passes below unity for ω > ω0 . This is normal dispersion.  (0) (0)   However, for an inverted medium ρgg , n is < ρee less than unity for low frequencies and goes through

Absorption and Gain Spectra

unity with a positive slope. This is called anomalous dispersion and signifies the presence of a gain profile. In such a medium, α = Im[α(ω)] is of opposite sign, as seen from (69.47); in the inverted medium, α becomes negative near ∆ = 0. From (69.2), this indicates a growing electromagnetic field, i. e., an amplifying medium. The amplitude grows, and the assumption of a small

69.5 Two-Level Doppler-Free Spectroscopy

1015

signal becomes invalid. Then saturation has to be included, either at the rate equation level or by performing a full density matrix calculation. This regime describes a laser with saturated gain. In steady state, the two levels become nearly equally populated (ρee ≈ ρgg ), and the operation is stable. The theory of the laser is discussed in detail in Chapt. 70.

69.5 Two-Level Doppler-Free Spectroscopy

∆2 = ∆2 − kv

(69.49)

(as compared with ∆1 = ∆1 + kv for E1 ). Since ω1 ω2 , the two k-vectors are nearly equal in magnitude. The linear response now becomes     1 idE2 (2) ρgg − ρee . = ρeg 2~ γ + i(∆2 − kv) (69.50)

With only the signal E1 present, the population difference follows directly from (69.22)). The linear response at frequency ω2 is then   Ω12 γ dE2 iγ + (∆2 − kv) (2) × 1 − ρeg = 2~ γ 2 + (∆2 − kv)2 Γ  

1 × . (69.51) (∆1 + kv)2 + γ 2 + Ω12 γ/Γ This is the linear response of atoms moving with velocity v. To obtain the polarization of the whole sample, we must average over the velocity distribution using the Gaussian weight (69.20). The first term in (69.51) gives the linear response in the form of a Voigt profile, as discussed in Sect. 69.3. This part of the response carries no Doppler-free information. The second terms contain the nonlinear response. This shows the details of the homogeneous features under the Doppler line shape. For simplicity we assume the Doppler limit, γ  ku, and neglect the variation of the Gaussian over the atomic line shape. We also neglect the power broadening due to the field E1 and obtain, using (69.13), (69.20), and (69.51), α (ω) = −



d2 ~



Ω2γ 2 √1 2πΓu

+∞ dv ×    2 2 (∆2 − kv) + γ (∆1 + kv)2 + γ 2 −∞  2 √ 2πΩ12 γ d =− . ~ 4Γku (ω − ω0 )2 + γ 2 (69.52)

This denotes the energy absorbed from the field E1 , as induced nonlinearly by the intensity E12 . The resonance is still at ω = ω0 , but with a homogeneous atomic line shape. In the Doppler limit, the Doppler broadening is

Part F 69.5

The linear absorption of a scanned laser signal defines linear spectroscopy and gives information characterizing the sample. However, Sect. 69.3 shows that inhomogeneous broadening masks the desired information by dominating the line shape. The availability of laser sources has made it possible to overcome this limitation, and to use the saturation properties of the medium to perform nonlinear spectroscopy. This section discusses how Doppler broadening can be eliminated to achieve Doppler-free spectroscopy. Similar techniques may be used to overcome other types of inhomogeneous line broadening; a general name is then hole-burning spectroscopy (see the discussion in Sect. 69.3). Other aspects of nonlinear matter-light interaction are found in Chapt. 72. Equation (69.24) shows that a single laser cannot resolve beyond the Doppler width. However, if a strong laser is used to pump the transition, a weak probe signal can see the hole burned into the spectral profile by the pump. This technique is called pump-probe spectroscopy. Because the probe is taken to be weak, perturbation theory may be used to calculate the induced polarization to lowest order in the probe amplitude only. In the field expansion (69.31), define the strong pump amplitude to be E1 and the weak probe E2 at frequency ω2 . From the resolution (69.34), the com(2) carries the information about the linear ponent ρeg response at frequency ω2 . If the field E2 propagates in a direction opposite to that of E1 , its detuning is

1016

Part F

Quantum Optics

only seen in the prefactor   Ω12 γ Ω12 = , Γku Γγ ku

(69.53)

which shows that only the fraction (γ/ku) of all atoms can contribute to the resonant response. The first factor on the right-hand side of (69.53) is the dimensionless saturation factor. The Doppler-free character of this spectroscopy derives from the fact that the two fields burn their two separate Bennett holes at the velocities kv1 = −∆1 ,

and kv2 = ∆2 .

(69.54)

Part F 69.6

When these two groups coincide, i. e., when v1 = v2 , the probe E2 sees the absorption saturated by the pump field E1 and a decreased absorption is observed. For ω1 = ω2 , the two holes meet in the middle at zero velocity. With two different frequencies, one can make the holes meet at a nonzero velocity to one side of the Doppler profile. The decreased absorption seen in these experiments is called an inverted Lamb dip Chapt. 70. The results derived here are based on a simplified view of the pump-probe response which in turn is based on the rate equation approach. Certain coherent effects are neglected, which would considerably complicate the treatment [69.2]. Section 69.6 discusses these effects in the three-level system where they are more important. In addition to measuring the probe absorption induced by the pump field, it is also possible to observe

the dispersion of the probe signal caused by the saturation induced by the pump. Assuming that the pump introduces the velocity dependent population difference ρee − ρgg = ∆ρ(v) ,

(69.55)

then from (69.50), Re(n) is  ∆2 − kv Nd 2 ∆ρ(v) dv . n = 1 + 2~ε0 (∆2 − kv)2 + γ 2 (69.56)

From (69.2), the phase of the electromagnetic signal feels the value of n  through the factor E ∝ exp(iωn  z/c) .

(69.57)

Thus, by modifying ∆ρ(v), a pump laser can control the phase acquired by light traversing the sample, and thereby control the optical length of the sample. A real atom has magnetic sublevels, which are coupled to light in accordance with dipole selection rules (see the discussion in Chapt. 33). If a pump laser is used to affect populations in the various sub-levels differently, the optical paths experienced by the differently circularly polarized components of a linearly polarized probe signal are different. Thus, its plane of polarization will turn, corresponding to the Faraday effect. By tuning the pump and the probe over the spectral lines of the sample, the turning of the probe polarization provides a signal to investigate the atomic level structure. This method of polarization spectroscopy can be used both with two-level and three-level systems [69.7].

69.6 Three-Level Spectroscopy New nonlinear phenomena appear when one of the levels in the two-state configuration, |e say, is coupled to a final level | f  by a weak probe. This may be above the level |e (the cascade configuration denoted by Ξ) or below |e (the lambda configuration Λ) (if the third level were coupled to |g we would talk about the inverted lambda or V configuration) [69.8]. For simplicity, only the Ξ configuration is discussed here. Assume that the level pair |g ↔ |e is pumped by the field E1 and its effect probed by the field E2 coupling |e ↔ | f . The dipole matrix element is d2 = f |H|e. The RWA is now achieved by introducing slowly varying quantities through the definitions ρ fe = ei(k2 z−ω2 t) ρ˜ fe , ρ fg = ei[(k1 +k2 )z−(ω1 +ω2 )t] ρ˜ fg ,

(69.58)

and omitting all components oscillating at multiples of the optical frequencies. From the equation of motion for the density matrix, the steady state equations are [69.2] d2 E2 d1 E1 ρ˜ fg , ρee − 2~ 2~ [(∆1 + ∆2 ) + (k1 + k2 )v − iγ fg ]ρ˜ fg

(∆2 + k2 v − iγ fe )ρ˜ fe =

=

d2 E2 d1 E1 ρ˜ eg − ρ˜ fe , 2~ 2~

(69.59)

where the second step detuning is now ∆2 = ω fe − ω2 .

(69.60)

These equations contain the lowest order response proportional to E2 , including some coherence effects. The coherence effects remain to lowest order, even when the last term in (69.59) is neglected; in that case, the

Absorption and Gain Spectra

solution is ρ˜ fe =





ρee d2 E2 2~ ∆2 + k2 v − iγ fe    d2 E2 d1 E1 − 2~ 2~ ρ˜ eg × (∆2 + k2 v − iγ fe ) 1  . (69.61) × (∆1 + ∆2 ) + (k1 + k2 ) v − iγ fg

From (69.61), these matrix elements give     2γeg A ρ˜ fe = − ∆1 + k1 v − iγeg γee 1   × ∆1 + k1 v + iγeg ∆2 + k2 v − iγ fe 1  − ∆2 + k2 v − iγ fe

1  , × (∆1 + ∆2 ) + (k1 + k2 ) v − iγ fg (69.63)

where A=−



d2 E2 2~



d1 E1 2~

2 .

(69.64)

The imaginary part of this yields the absorptive part of the polarization at the frequency ω2 . The first term

1017

becomes the product of two Lorentzians, and is an incoherent rate contribution. It dominates when the induced population ρee decays much more slowly than the induced coherence ρ˜ eg , i. e., when γee  γeg . The significance of the second term in (69.63) is evident in the limit when no phase perturbing processes intervene, so that 1 γeg = Γ , 2 1 1 γ fg = γ f f , γ fe = (γ f f + Γ ) . (69.65) 2 2 For v = 0, ρ˜ fe becomes   A 1 . ρ˜ fe = − 2 ∆1 + (Γ/2)2 ∆1 + ∆2 − i 12 γ f f γee = Γ ,

(69.66)

Neglecting the decay of the final level | f , γ f f → 0, the absorption becomes proportional to   Im ρ˜ fe =

π|A| ∆21 + (Γ/2)2

δ (∆1 + ∆2 ) .

(69.67)

In this limit, strict energy conservation between the ground state and the final state must prevail; the final state must be reached by the absorption of exactly two quanta. The delta function in (69.67) indicates precisely this: ∆1 + ∆2 = ω fe + ωeg − (ω1 + ω2 ) = ω fg − ω1 − ω2 =0.

(69.68)

The detuning and the width of the intermediate state affect the total transition rate, but not the condition of energy conservation. The presence of the second term in (69.63) makes the resonance contributions at ∆2 = ω fg − ω2 = 0

(69.69)

cancel approximately. Only a two-photon transition remains; in the Λ configuration this would be a Raman process (Chapts 62 and 72). If the velocity dependence in (69.63) is retained, the nonlinear response of an atomic sample must be averaged over the velocity distribution given by the Gaussian (69.20). The computations become involved, but the results show more or less well resolved resonances around the two positions (69.68) and (69.69), i. e., the coherent two-photon process and the single step rate process |e → | f  appearing due to a previous single step process |g → |e; this is the two-step process.

Part F 69.6

The result to lowest order in E2 follows by replacing the density matrix elements for the two-level system |e ↔ |g by their results calculated without this field. This shows that the polarization induced at the frequency ω2 consists of two parts: one induced by the population excited to level |e by the field E1 proportional to ρee , and the other one is induced by the coherence ρ˜ eg created by the pump. Retaining only the former produces a rate equation approximation, called a two-step process. This misses important physical features which are included in the second term called a two-photon or coherent process. In order to see the effects of the two terms most clearly, consider the two-level matrix elements from (69.11) and (69.12) in lowest perturbative order with respect to the pump field E1 , i. e.,   1 d1 E1 2 γeg , ρee = 2 ~ 2γee (∆1 + k1 v)2 + γeg d1 E1 1 ρ˜ eg = (69.62) 2~ ∆1 + k1 v − iγeg

69.6 Three-Level Spectroscopy

1018

Part F

Quantum Optics

A special situation arises when the intermediate step is detuned so much that no velocity group is in resonance, i. e., |∆1 | ≈ |∆2 |  kv for all velocities contributing significantly to the spectrum. Then the second coherent term of (69.63) can be written in the form   A 1 ρ˜ fe = − 2 . ∆1 (∆1 + ∆2 ) + (k1 + k2 ) v − iγ fg (69.70)

If the two fields E1 and E2 have the same frequency but are counterpropagating, then k1 + k2 = 0, and no velocity dependence occurs in (69.70) for the two-photon resonance. All atoms in the sample contribute to the strength of the resonance, and then polarization is obtained directly by multiplication with the total atomic

density. Equation (69.13) then gives   d22 d1 E1 2 γ fg  . α2 =  2 ~ 2~∆1 2ω − ω fg + γ 2fg

(69.71)

This is a sharp Doppler-free resonance on the twophoton transition ω fg . The advantage is that all atoms contribute with the sharp line width γ fg , which is not easily affected by phase perturbations because of the two-photon nature of the transition. The disadvantage is the weakness of the transition, which is caused by the large detuning, making d1 E1 /|∆1 |  1 in most cases. With tunable lasers, however, this Doppler-free spectroscopy method has been used successfully in many cases.

69.7 Special Effects in Three-Level Systems We continue our considerations of a three-level system but this time in the V -configuration. Thus, we have a ground state |g coupled to a doublet of excited states {|e, | f } through the interaction V = Ω1 |g e| + Ω2 |g f | + h.c. ,

(69.72)

Part F 69.7

where the couplings are due to radiation fields and given by di Ei Ωi = . (69.73) 2~ We use the rotating wave approximation and set the detunings to ∆ωe = ωe − ω1

and ∆ω f = ω f − ω2 ,

(69.74)

where the frequency ωi derives from the coupling field Ei . The energy E g = 0. The time-dependent Schrödinger equation for this system is then written as i˙cg = Ω1 ce + Ω2 c f ,

(69.75)

i˙ce = ∆ωe ce + Ω1 cg ,

(69.76)

i˙c f = ∆ω f c f + Ω2 cg .

(69.77)

We now introduce the two new variables   ¯ −1 Ω1 ce + Ω2 c f cC = Ω   ¯ −1 Ω2 ce − Ω1 c f , cNC = Ω

(69.82)

We notice that cNC is not coupled to the ground state, but, in general, its coupling to the state cC provides an indirect coupling. This indirect coupling, however, can be made to disappear, if we set ∆ωe = ∆ω f ≡ ∆ω. Then we find the equations i˙cNC = ∆ωcNC , ¯ g, i˙cC = ∆ωcC + Ωc

(69.78) (69.79)

where ¯ 2 = Ω22 + Ω12 . Ω

The equations of motion follow:   Ω2 Ω1 ∆ωe ce − ∆ω f c f i˙cNC = ¯ ¯ Ω Ω   1  Ω1 Ω2 ∆ωe − ∆ω f cC . = 2 ¯ Ω   + ∆ωe Ω22 + ∆ω f Ω12 cNC , (69.81)   Ω1 Ω2 ¯ g ∆ωe ce + ∆ω f c f + Ωc i˙cC = ¯ ¯ Ω Ω 1  ∆ωe Ω12 + ∆ω f Ω22 cC = ¯2 Ω    ¯ g. + Ω1 Ω2 ∆ωe − ∆ω f cNC + Ωc

(69.80)

¯ C. i˙cg = Ωc

(69.83)

This describes a pair of coupled quantum levels and a single uncoupled level. Thus, if we start in the ground state, this latter level can never be populated. It remains unpopulated and is called a dark state. This state

Absorption and Gain Spectra

corresponds to the superposition |NC =

1 (Ω2 |e − Ω1 | f ) . ¯ Ω

(69.84)

Using the coupling operator (69.72), we find the matrix element

g|V |NC = 0 .

(69.85)

We also notice that the dark state can be found if the states |e and | f  are the lower ones, i. e., we have the Λ-configuration. Alternatively, if we start the system off in the dark state, it will never be able to get out of this state. This is also taken to hold true if we let the couplings depend slowly on time. In this case, we may let Ω1 come on later than Ω2 . Then we may have   Ω 1 =0, lim   t→−∞ 2 2 Ω2 + Ω1   Ω 2 =1 (69.86) lim   t→−∞ Ω22 + Ω12 and

Ω1 =1, lim   Ω22 + Ω12   Ω 2 =0. lim   t→∞ 2 Ω2 + Ω12 t→∞

(69.87)

Both couplings are thus pulses, but they occur with a slight time delay. If we now start the system in the state |e, we find from (69.84) that lim |NC = |e ,

t→−∞

The dark state has found a wide range of applications in laser physics. As we may use the method to pass radiation through a medium without any absorption, it has led to the phenomenon of light-induced transparency. It can also be used to affect the index of refraction without having the accompanying absorption. The absorptive part of a resonance normally manifests itself as a quantum noise; utilizing the dark state idea one may reduce the noise in quantum devices. The dressing of the levels due to the special features of the interaction has also made it possible to achieve lasing without an inversion of the bare levels. These topics, however, will not be treated here. A special application of the method to affect the refractive index deserves a more detailed consideration. We look at the relationship between the electric field vectors in the medium: D(ω) = ε0 E + Nα(ω)E(ω) = χ(ω)ε0 E(ω) , (69.90)

where χ(ω) is the susceptibility of the medium. From Maxwell’s equations we find the relation (68.24) k2 = n 2 (ω)



(69.88)

ω2 ω2 = [1 + χ(ω)] . c2 c2

(69.91)

If we take the real parts of the quantities, this describes the propagation of waves in the medium. Now we may use the relations (69.16) to estimate the function α(ω) in the case when we have two weak fields exciting the three-level system in the V -configuration. We assume that both couplings have the same frequency, ω1 = ω2 = ω. We write    ω−ωf + ; χ(ω) ≈ Λ 2 (ω − ωe )2 + γ 2 ω−ωf +γ2 

(ω − ωe )

(69.92)

and lim |NC = − | f  .

t→+∞

(69.89)

Thus, by keeping the system in this uncoupled state we can adiabatically transfer its population beteen the states without involving any population of the intermediate state |g. Especially if this is an upper state, which may decay and dephase rapidly, the proposed population transfer may be greatly advantageous. Because it is usually applied in the Λ-configuration, it is termed Stimulated Raman Adiabatic Passage (STIRAP).

1019

in the weak field limit we may assume the two processes to add independently. The parameter Λ=

Nd 2 2~ε0

(69.93)

has the dimension of a frequency and indicates the strength of the interaction. It is clear that tuning the frequency between the levels may give rise to a value of zero for the susceptibility. For a large enough γ , we

Part F 69.7



69.7 Special Effects in Three-Level Systems

1020

Part F

Quantum Optics

write, in the neighbourhood of the zero, χ(ω) ≈

 2Λ  ω − ω¯ , 2 γ

(69.94)

where ω¯ = 12 (ωe + ω f ). We now have the relation (69.91) to determine the dispersion relation, and assuming the effect of the medium to be substantial, we may derive an expression for the group velocity in the medium v−1 g =

∂k . ∂ω

(69.95)

We find 2ω ω2 2k = 2 (1 + χ) + 2 vg c c



2Λ γ2

 .

(69.96)

Even though χ = 0 near the point ω, ¯ we still have ω ∼ ck, so that  2 c γ vg = c. ≈ c (69.97) Λω Λω 1+ 2 γ

The last inequality follows from the fact that in all cases γ  ω. We have thus found that utilizing the interference of two near quantum levels, the refractive index may acquire a very strong dependence on frequency. This may manifest itself in an exceedingly slow propagation of light pulses. Such slow light has been shown to travel at only a few kilometers per hour, which is a most remarkable result. The drawback is, however, that this can only occur over a very narrow frequency range, as follows from the assumption of a strong dependence on frequency.

69.8 Summary of the Literature

Part F 69

Much of the material needed to formulate the basic theory of interaction between light and matter can be found in the text book [69.2]. A comparison between the harmonic model and the two-level model is given by Feld [69.1]. The density matrix formulation is presented in detail in [69.2]. The influence of various line broadening mechanisms on laser spectroscopy is discussed in the book [69.3]. The Voigt profile is related to the error function, which is treated in the compilation [69.5]. The numerical evaluation of the Voigt profile is discussed in [69.4]. Rate equations are commonly used in laser theory and they are derived for optical pumping and laser-induced processes in the lectures [69.6]. The Doppler-free spectroscopy was developed in the 1960s and 1970s by many authors following the initial discovery of the Benett hole by Bill R. Bennett Jr. and the Lamb dip by Willis E. Lamb Jr. Much of the pioneering

work can be found in the book [69.3]. The three-level work has been reviewed by Chebotaev [69.8]. Various applications of lasers in spectroscopy are treated by Levenson and Kano [69.7]. For references to other topics, we refer to the specialized chapters of the present book. Many features of the quantum dynamics of a fewlevel system are found in [69.9, 10]. The theoretical methods to treat such systems are presented in detail in [69.11]. The ensuing physical processes are presented in [69.12] with much additional material on quantum optics phenomena. The basic theory of the dark state and many of its applications in spectroscopy and laser physics are found in [69.13]. A rather complete review of adiabatic processes induced by delayed pulses is the article [69.14]. The slowing down and stopping of light is reviewed in [69.15]. A very recent article with earlier references is [69.16].

References 69.1

69.2 69.3 69.4

M. S. Feld: Frontiers in Laser Spectroscopy, ed. by R. Balian, S. Haroche, S. Liberman (North-Holland, Amsterdam 1977) p. 203 S. Stenholm: Foundations of Laser Spectroscopy (Wiley, New York 1984) V. S. Letokhov, V. P. Chebotaev: Nonlinear Laser Spectroscopy (Springer, Berlin, Heidelberg 1977) W. J. Thompson: Comp. Phys. 7, 627 (1993)

69.5 69.6

69.7

M. Abramowitz, I. E. Stegun: Handbook of Mathematical Functions (Dover, New York 1970) C. Cohen-Tanoudji: Frontiers in Laser Spectroscopy, ed. by R. Balian, S. Haroche, S. Liberman (NorthHolland, Amsterdam 1977) p. 1 M. D. Levenson, S. S. Kano: Introduction to Nonlinear Laser Spectroscopy (Academic, New York 1988)

Absorption and Gain Spectra

69.8

69.9

69.10

69.11

69.12

V. P. Chebotaev: High-Resolution Spectroscopy, ed. by K. Shimoda (Springer-Verlag, Berlin, Heidelberg 1976) B. W. Shore: The Theory of Coherent Atomic Excitation: Vol. 1. Simple Atoms and Fields (Wiley, New York 1990) B. W. Shore: The Theory of Coherent Atomic Excitation: Vol. 2. Multilevel Atoms and Incoherence (Wiley, New York 1990) C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg: Atom-Photon Interactions, Basic Processes and Applications (Wiley, New York 1992) L. Mandel, E. Wolf: Optical Coherence and Quantum Optics (Cambridge Univ. Press, Cambridge 1995)

69.13 69.14

69.15

69.16

References

1021

M. O. Scully, M. S. Zubairy: Quantum Optics (Cambridge Univ. Press, Cambridge 1997) N. V. Vitanov, M. Fleischauer, B. W. Shove, K. Bergmann: In: Advances of Atomic, Molecular and Optical Physics, Vol. 46, ed. by B. Bederson, H. Walther (Academic, New York 2001) p. 55 A. B. Matsko, O. Kocharovskaya, Y. Restoutsev, G. R. Welch, A. S. Zibrov, M. O. Scully: Advances in Atomic, molecular, and Optical Physics, Vol. 46, ed. by B. Bederson, H. Walther (Academic, New York 2001) p. 191 M. Bajcsy, A. S. Zibrov, M. D. Lukin: Nature 26, 368 (2003)

Part F 69

1023

Laser Principle 70. Laser Principles

Despite their great variety and range of power, wavelength, and temporal characteristics, all lasers involve certain basic concepts, such as gain, threshold, and electromagnetic modes of oscillation [70.1–3]. In addition to these universal characteristics are features, such as Gaussian beam modes, that are important to such a wide class of devices that they must be included in any reasonable compendium of important laser concepts and formulas. We have therefore included here both generally applicable results as well as some more specific but widely applicable ones.

70.1

Gain, Threshold, and Matter–Field Coupling ................... 1023

70.2 Continuous Wave, Single-Mode Operation ........................ 1025 70.3 Laser Resonators ................................. 1028 70.4 Photon Statistics ................................. 1030 70.5 Multi-Mode and Pulsed Operation ........ 1031 70.6 Instabilities and Chaos......................... 1033 70.7 Recent Developments........................... 1033 References .................................................. 1034

70.1 Gain, Threshold, and Matter–Field Coupling All lasers involve some medium that amplifies an electromagnetic field within some band of frequencies. At the simplest level of description, the amplifying medium changes the intensity I of a field according to the equation

Part F 70

dI = gI , (70.1) dz where z is the coordinate along the direction of propagation and g is the gain coefficient, typically expressed in cm−1 . Amplification occurs as a consequence of stimulated emission of radiation from the upper state (or band of states) of a transition for which a population inversion exists; i. e., for which an upper state has greater likelihood of occupation than a lower state. Different types of lasers may be classified by the pump mechanisms used to achieve population inversion (Chapt. 71). In the case that the amplifying transition involves two discrete energy levels, E 1 and E 2 > E 1 , the gain coefficient at the frequency ν is given by
 λ2 A g2 N g(ν) = − N (70.2) 2 1 S(ν) . g1 8πn 2   Here λ = c/ν is the transition wavelength, A s−1 is the Einstein A coefficient for spontaneous emission for the transition, and g2 , g1 are the degeneracies of the upper and lower energy levels. These quantities in nearly

every case are fixed characteristics of the medium, independent of the laser intensity or the pump   mechanism. N2 and N1 are the population densities cm−3 of the upper and lower levels, respectively, and S(ν) is the normalized transition lineshape function (Chapts. 19 and 69). n is the refractive index at frequency ν of the “background” host medium and in general has contributions from all nonlasing transitions. Equation (70.2) describes either amplification or absorption, depending on whether N2 − (g2 /g1 )N1 is positive (amplification) or negative (absorption). By far the most common configuration is that in which the gain medium is contained in a cavity bounded on two sides by reflecting surfaces. The mirrors allow feedback; i. e., the redirection of the field back into the gain medium for multipass amplification and sustained laser action. The two mirrors allow the field to build up along the directions parallel to the “optical axis” and to form a pencil-like beam of light. In order to sustain laser action, the gain in intensity due to stimulated emission must equal or exceed the loss due to imperfect mirror reflectivities, scattering, absorption in the host medium, and diffraction. Typically the imperfect mirror reflectivities dominate the other sources of loss. If the mirror reflectivities are r1 and r2 , then the intensity I is reduced by the factor r1r2 in a round trip pass through the cavity, while

1024

Part F

Quantum Optics

according to (70.1) the gain medium causes the intensity to increase by a factor exp(2g) in the two passes through the gain cell of length . Equating of the gain and loss factors leads to the threshold condition for laser oscillation: g ≥ gt , where the threshold gain is gt = −

1 ln(r1r2 ) + α , 2

(70.3)

α being an attenuation coefficient associated with any loss mechanisms that may exist in addition to reflection losses at the mirrors. Suppose, for example, that a laser has a 50 cm gain cell and mirrors with reflectivities 0.99 and 0.97, and that absorption within the host medium of the gain cell is negligible. Then the threshold gain is gt = 4 × 10−4 cm−1 . If the lasing transition is the 6328 Å Ne transition of the He–Ne laser, we have A ∼ = 1.4 × 106 s−1 , n ∼ = 1.0 and, assuming a pure Doppler lineshape,
 4 ln 2 1/2 1 S(ν) = (70.4) π δνD at line center, where δνD is the width (FWHM) of the Doppler lineshape (Sect. 69.3). For T = 400 K and the Ne atomic weight, δνD ∼ = 1500 MHz and S(ν) ∼ = 6.3 × 10−10 s. Then the threshold population difference required for laser oscillation is
 g2 8πn 2 gt ∼ N2 − N1 = 2 = 2.8 × 109 cm−3 . g1 λ AS(ν) t (70.5)

Part F 70.1

This is a typical result: the population inversion required for laser oscillation is small compared with the total number of active atoms. Calculations of population inversions and other properties of the gain medium are based on rate equations, or more generally, the density matrix ρ. In many instances, the medium is fairly well described in terms of two energy eigenstates, other states appearing only indirectly through pumping and decay channels. In this case, ρ is a 2 × 2 matrix whose elements satisfy [70.3] (68.66) and (68.67)  1  ρ˙ 22 = −(Γ2 + Γ )ρ22 − i Ω1∗ρ˜ 21 − Ω1ρ˜ 12 , 2  1  ρ˙ 11 = −Γ1 ρ11 + Γρ22 + i Ω1∗ρ˜ 21 − Ω1 ρ˜ 12 , 2 1 ρ˙˜ 21 = −(γ + i∆)˜ ρ21 − iΩ1 (ρ22 − ρ11 ) , (70.6) 2 ∗ . Here, Γ and Γ are, respectively, the with ρ˜ 12 = ρ˜ 21 2 1 rates of decay of the upper and lower states due to

all processes other than the spontaneous decay from state 2 to state 1 described by the rate Γ = A. γ , which is 2π times the homogeneous linewidth (HWHM) of the transition, is the rate of decay of off-diagonal coherence due to both elastic and inelastic processes; in general, γ ≥ (Γ1 + Γ2 + Γ )/2. Ω1 = d21 · E /~ is the Rabi frequency (The Rabi frequency is often defined as 2d21 · E /~.) (Sect. 68.3.3 and Chapt. 73), with E the complex amplitude of the electric field; i. e., the electric field is   E(r, t) = Re E (r, t) ei(k·r−ωt)   ∼ (70.7) = Re
E(r, t) eikz e−iωt . It is assumed that E is slowly varying in time compared with the oscillations at frequency ω (= 2πν), and that the wave vector k is approximately kˆz = (nω/c)ˆz , where zˆ is a unit vector pointing in the direction of propagation. Finally, ∆ = ω0 − ω in (70.6) is the detuning of ω from the central transition frequency ω0 = (E 2 − E 1 )/~ of the lasing transition. Rapidly oscillating terms involving ω0 + ω are ignored in the rotating wave approximation that pervades nearly all of laser theory (Sect. 68.3.2). In most lasers, γ is so large compared with the diagonal decay rates that the off-diagonal elements of ρ may be assumed to relax quickly to the quasisteady values obtained by setting ρ˙ 12 = 0 in (70.6). Then the diagonal density matrix elements satisfy the rate equations (69.37) |Ω1 |2 γ/2 (ρ22 − ρ11 ) , ∆2 + γ 2 |Ω1 |2 γ/2 ρ˙ 11 = −Γ1 ρ11 + Γρ22 + 2 (ρ22 − ρ11 ) . ∆ +γ2 ρ˙ 22 = −(Γ2 + Γ )ρ22 −

(70.8)

Such rate equations, usually expressed equivalently in terms of population densities N2 , N1 rather than occupation probabilities ρ22 , ρ11 , are the basis of most practical computer models of laser oscillation. These equations, or, more generally, the density matrix equations, must also include terms accounting for the pump mechanism. In the simplest model of pumping, one adds a constant pump rate Λ2 to the right-hand side of the equation for ρ22 to obtain (69.45) ρ˙ 22 = Λ2 − (Γ2 + A)ρ22 −

|Ω1 |2 γ/2 (ρ22 − ρ11 ) . ∆2 + γ 2 (70.9)

In the case of an inhomogeneously broadened laser transition (Sect. 69.3), equations of the type (70.6) and (70.8)

Laser Principles

apply separately to each detuning ∆ arising from the distribution of atomic or molecular transition frequencies. In writing these equations, we have assumed a nondegenerate electric dipole transition. The generalization to magnetic or multiphoton transitions, or to a case where the amplification is due, for instance, to a Raman process, is straightforward but of less general interest. A more realistic treatment of the electromagnetic field than that based on (70.1) proceeds from the Maxwell equations, which, for a homogeneous and nonmagnetic medium, lead to the equation
 1∂ 1 2 ∂ 4πiω ∇T E + + Nµ∗ ρ21 . E= 2ik ∂z c ∂t nc (70.10)

Here N is the density of active atoms, µ ≡ (d12 ·
)∗ , and ∇T2 ≡ ∂ 2 /∂x 2 + ∂ 2 /∂y2 . The result (70.10) assumes the validity of the rotating wave approximation as well as the assumption that E is slowly varying compared with exp(ikz) and exp(−iωt). In the plane wave approximation, (70.10) becomes (More generally, the velocity c on

70.2 Continuous Wave, Single-Mode Operation

1025

the left sides of (70.10) and (70.11) should be replaced by the group velocity vg associated with nonresonant transitions. If there is substantial group velocity dispersion, it is sometimes necessary to include a term involving the second derivative of E with respect to t)
 1∂ 4πiω ∂ + E= Nµ∗ ρ21 . (70.11) ∂z c ∂t nc Equations (70.6) or (70.8) and (70.10) or (70.11) are coupled matter–field equations whose self consistent solutions determine the operating characteristics of the laser. The density matrix or rate equations must be modified to include pumping, as in (70.9), and the field equations must be supplemented by boundary conditions and loss terms. With these modifications, the equations are the basis of semiclassical laser theory, wherein the particles constituting the gain medium are treated quantum mechanically whereas the field is treated according to classical electromagnetic theory [70.4]. Aside from fundamental linewidth considerations and photon statistics (see Sects.70.2 and 70.4), very few aspects of lasers require the quantum theory of radiation.

70.2 Continuous Wave, Single-Mode Operation In the case of steady state, continuous wave (cw) operation, the appropriate matter–field equations are those obtained by setting all time derivatives equal to zero. Equation (70.11), for instance, becomes (70.12)

4πωN|d|2 γ dI = (ρ22 − ρ11 )I dz 3n ~c ∆2 + γ 2 λ2 A (N2 − N1 )S(ν)I = g(ν)I (70.13) = 8πn 2 for the nondegenerate case under consideration. Here,   |d|2 = 3|d12 ·
|2 , N j = Nρ jj , S(ν) = γ/ ∆2 + γ 2 is the Lorentzian lineshape function for homogeneous broadening, and A = 4ω3 |d|2 n/3~c3 is the spontaneous emission rate in the host medium of (real) refractive index n. Local (Lorentz–Lorenz) field corrections will in general modify these results, but such corrections are ignored here [70.5]. The steady state solution of the density matrix or rate equations gives, similarly, g0 (ν) g(ν) = , (70.14) 1 + I/Isat

I = Isat [g0 (ν)/gt − 1] .

(70.16)

If I is assumed to be the sum of the intensities of waves propagating in the +z and −z directions, i. e. I = I+ + I− , then the output intensity from the laser is Iout = t2 I+ + t1 I− ,

(70.17)

Part F 70.2

2πωN|d|2 1 dE = (ρ22 − ρ11 )E , dz 3n ~c γ + i∆ or, in terms of the intensity I,

where the saturation intensity Isat , like the small signal gain coefficient g0 (ν), depends on decay rates and other characteristics of the lasing species. Thus, in the plane wave approximation, the growth of intensity in a homogeneously broadened laser medium is typically described by the equation g0 (ν)I dI = . (70.15) dz 1 + I/Isat This equation, supplemented by boundary conditions at the mirrors, and possibly other terms on the right side to account for any distributed losses within the medium, determines the intensity in cw, single-mode operation. The simplest model for calculating output intensity assumes that the intensity is uniform throughout the laser cavity. In steady state, the gain exactly compensates for the loss; i. e., g(ν) = gt , the gain clamping condition for cw lasing. Equation (70.14) then implies that the steady state intracavity intensity is

1026

Part F

Quantum Optics

where t2 , t1 are the mirror transmissivities at the right and left mirrors, respectively. The uniform intensity approximation implies that I+ = I− = I/2 and 1 Iout = (t2 + t1 )Isat [g0 (ν)/gt − 1] . (70.18) 2 Suppose one of the mirrors is perfectly transmitting, so that t1 = 0 and t2 = t > 0. Furthermore, if the reflectivity r of the transmitting mirror is close to unity, then gt ∼ = (1/2)(1 − r) = (1/2)(t + s), where s is the fraction of the incident beam power that is scattered or absorbed at the output mirror. Then 
2g0 (ν) 1 −1 , (70.19) Iout ∼ = Isat t 2 t +s and it follows that the optimal output coupling, i. e., the transmissivity that maximizes the output intensity, is  (70.20) topt = 2g0 (ν)s − s . This output coupling gives the output intensity 2   max = Isat g0 (ν) − s/2 . (70.21) Iout g0 (ν)Isat is the largest possible power per unit volume extractable as output laser radiation at the frequency ν. More generally, when mirror reflectivities are not necessarily close to unity, I+ = I− and both I+ and I− vary with the axial coordinate z. In this more general case, (70.14) and (70.15) are replaced by g(ν, z) =

Part F 70.2

g0 (ν) 1 + [I+ (z) + I− (z)]/Isat

(70.22)

based on several assumptions and approximations: the gain medium is assumed to be homogeneously broadened and to saturate according to the formula (70.14); g0 and Isat are taken to be constant throughout the medium; the field is approximated as a plane wave; field loss processes occur only at the mirrors; and interference between the left- and right-going waves is ignored. Interference of the counterpropagating waves in a standing wave, single-mode laser modifies the gain saturation formula (70.14) as follows: g(ν, z) =

g0 (ν) 1 + (2I+ /Isat ) sin2 kz

in the case of small output coupling, where I+ ∼ = I− as assumed in (70.18). (The general case of arbitrary output coupling with spatial interference of counterpropagating waves is somewhat complicated and is not considered here.) The sin2 kz term is responsible for spatial hole burning: “holes” are “burned” in the curve of g(ν, z) versus z at points where sin2 kz is largest. This spatially dependent saturation acts to reduce the output intensity, typically by as much as about 30%, compared with the case where interference of counterpropagating waves is absent or ignored. Spatial hole burning tends to be washed out by atomic motion in gas lasers, and is absent entirely in pure traveling wave ring lasers. If complete spatial hole burning based on (70.25) is assumed, the output intensity is

 t 1 g0 (ν) g0 (ν) 1 Iout = Isat − − + 2 gt 4 2gt 16 (70.26)

and dI− dI+ = g(ν, z)I+ , = −g(ν, z)I− , (70.23) dz dz where it is assumed that g0 (ν) is independent of z, and that all cavity loss processes occur at the mirrors. The solution of these equations with the boundary conditions I− (L) = r2 I+ (L), I+ (0) = r1 I− (0) for mirrors at z = 0 and z = L gives, for Iout = t1 I− (0) + t2 I+ (L), the formula [70.3, 6]


 √ r1 r2  t1 √ Iout = Isat t2 + √ √  r1 r1 + r2 1 − r1r2   √ (70.24) × g0 (ν) + ln r1r2 . Analysis of this result gives an optimal output coupling that reduces to (70.19) in the limit t + s  1. Curves for optimal output coupling and Iout as a function of g0 and s are given by Rigrod [70.6]. These results are

(70.25)

in the case where one mirror is perfectly reflecting. In inhomogeneously broadened media, the gain coefficient is obtained by integrating the contributions from all possible values of ν0 . The different contributions saturate differently, depending on the detuning of ν0 from the cavity mode frequency ν. If spatial hole burning and power broadening (see Sect. 69.3) are ignored, then, to a good approximation, the gain saturates as g0 (ν) g(ν) = √ 1 + I/Isat

(70.27)

in typical inhomogeneously broadened media. Oscillation on a single longitudinal mode (see Sect. 70.3) may be realized simply by making the cavity length L small enough that the mode spacing c/2n L (70.28) exceeds the spectral width of the gain curve. This is possible in many gas lasers where the spectral

Laser Principles

ν=N

c 2n L

(N an integer) ,

(70.28)

for the cavity mode frequencies in the plane wave approximation. If the gain medium of refractive index n does not fill the entire length L between the mirrors, then (70.28) must be replaced by ν=

Nc/2 , n + (L − )

(70.29)

1027

or  [n(ν) − 1]ν = νN − ν , (70.30) L where  (≤ L) is the length of the gain medium and νN = Nc/2L is an empty cavity mode frequency. The laser oscillation frequency will therefore be different, in general, from any of the allowed empty-cavity mode frequencies. If the refractive index n(ν) is attributable primarily to the lasing transition, as opposed to the host material, or other nonlasing transitions, then the following relation between n(ν) and g(ν) may be used in the case of homogeneous broadening [70.3]: n(ν) − 1 = −

λ0 ν0 − ν g(ν) , 4π δν0

(70.31)

where λ0 , ν0 , and δν0 are the wavelength, frequency, and homogeneous linewidth (HWHM), respectively, of the lasing transition. This implies that ν=

ν0 δνc + νN δν0 δνc + δν0

(70.32)

for the laser oscillation frequency ν, where cg(ν) (70.33) 4πL is the cavity bandwidth. Thus, the actual lasing frequency is not simply one of the allowed empty-cavity frequencies νN , but rather is “pulled” away from ν N toward the center of the gain profile. This is called frequency pulling. If spatial hole burning is absent or ignored, then g(ν) = gt in steady state oscillation, and the cavity bandwidth δνc = cgt /4πL is largest for lossy cavities. Most lasers fall into the “good cavity” category, that is δνc  δν0 , so that (70.32) may be approximated by δνc ≡

ν∼ = νN + (ν0 − νN )δνc /δν0 .

(70.34)

Similar results apply to inhomogeneously broadened lasers. For a Doppler broadened medium, for instance, the frequency pulling formula is  ν∼ (70.35) = νN + (ν0 − νN )(δνc /δνD ) 4 ln 2/π for good cavities. These results show that frequency pulling is most pronounced in lasers with large peak gain coefficients and narrow gain profiles, as observed experimentally. Spectral hole burning leads to especially interesting consequences in Doppler broadened gas lasers. Since two traveling waves propagating in opposite (±z) directions will strongly saturate spectral packets of atoms

Part F 70.2

width is small, and in semiconductor lasers, where L is very small. More generally the gain clamping condition g(ν) = gt implies that the cavity mode frequency having the largest small-signal gain g0 (ν) saturates the gain g(ν) down to the threshold value gt , while the gain at all other mode frequencies then lies below gt . In other words, the gain clamping condition implies single-mode oscillation. However, this conclusion assumes homogeneous broadening and also that spatial hole burning is unimportant, so that the gain saturates uniformly throughout the cavity. High pressure gas lasers, where the line broadening is due primarily to collisions and therefore is homogeneous, tend to oscillate on a single mode because spatial hole burning is largely washed out by atomic motion. On the other hand, homogeneously broadened solid state lasers can be multi-mode as a consequence of spatial hole burning. Single-mode oscillation in inhomogeneously broadened media is generally more difficult to achieve because of spectral hole burning (see Sect. 69.3), which makes the simple gain clamping argument inapplicable. However, single-mode oscillation can be enforced in any case by introducing, in effect, an additional loss mechanism for all mode frequencies except one. This is commonly done with a Fabry–Perot etalon having a free spectral range that is large compared with the spectral width of the gain curve. By choosing the tilt angle appropriately, a particular resonant frequency of the etalon can be brought close to the center of the gain curve while all other resonance frequencies lie outside the gain bandwidth. Laser oscillation at a fixed polarization can likewise be achieved by discriminating against the orthogonal polarization, as is done when Brewster windows are employed. Laser oscillation, in general, does not occur precisely at one of the allowed cavity mode frequencies. Associated with the sin kz dependence of the intracavity field is the condition kL = Nπ, or

70.2 Continuous Wave, Single-Mode Operation

1028

Part F

Quantum Optics

with oppositely Doppler shifted frequencies, a standing wave field will burn two holes on opposite sides of the peak of the Doppler profile. When the mode frequency is exactly at the center of the Doppler profile, however, the two holes merge, the field now being able to saturate strongly only those atoms with zero velocity along the z-direction. In this case, since the field “feeds” off a single spectral packet of more strongly saturated atoms, there is a dip in the output power compared with the case when the mode frequency is detuned from line center. This dip in the output power at line center is called the Lamb dip. It can be used to determine whether a gas laser is Doppler broadened, and more importantly, to stabilize the laser frequency to the center of the dip. Lamb dip frequency stabilization employs a feedback circuit to control the bias voltage across a piezoelectric element used to vary the cavity length, and thereby sweep the laser frequency. The above semiclassical laser theory suggests that cw laser radiation should be perfectly monochromatic, since the amplitude and phase of the field given by (70.7) are time independent. However, when quantum electrodynamical considerations are built into laser theory, it is found that spontaneous emission, which adds to the number of photons put in the lasing mode by stimulated emission, causes a phase diffusion that results in a Lorentzian linewidth (HWHM) ∆ν =

N2 8πhν (δνc )2 , N2 − N1 Pout

(70.36)

Part F 70.3

where Pout is the output laser power. This is the Schawlow–Townes linewidth, which implies a fundamentally quantum mechanical, finite linewidth that persists no matter how small various sources of “technical noise”, such as mirror jitter, are made. Although the Schawlow–Townes linewidth has been observed in

highly stabilized gas lasers, it is negligible compared with technical noise in conventional lasers. But in semiconductor lasers, L is very small and consequently δνc is large, and quantum noise associated with the Schawlow– Townes formula can be the dominant contribution to the laser linewidth. However, the 10–100 MHz linewidths typically observed in semiconductor lasers are too large to be explained by the Schawlow–Townes formula (70.36), and two modifications to this formula are necessary, each of them involving a multiplication of the Schawlow– Townes linewidth ∆ν by a certain factor:   ∆ν → ∆ν  = 1 + α2 K ∆ν , (70.37) where α is called the “Henry α parameter” and is associated with a coupling between phase and intensity fluctuations above the laser threshold [70.7]. Values of α between about 4 and 6 are typical in semiconductor injection lasers, and consequently, the correction to the Schawlow-Townes linewidth due to the Henry α parameter is substantial. The K factor [70.8–10] arises as a consequence of the deviation from the spatially uniform intracavity intensity assumed in the derivation of the Schawlow–Townes formula [70.11, 12]. (Intracavity intensities along the optical axis are approximately uniform only in the case of low output couplings.) The fundamental quantum mechanical linewidth under consideration can be associated with vacuum field fluctuations, which, according to general fluctuation–dissipation ideas, will increase as the cavity loss increases. This explains why the “Petermann K factor” deviates increasingly from unity as the output coupling (cavity loss) increases. Values of K between 1 and 2 appear to be typical for lossy, stable resonators [70.13, 14], but much larger values are possible for unstable resonators [70.15] (see also Siegman [70.2]).

70.3 Laser Resonators The assumption that the complex field amplitude E(r, t) is slowly varying in z compared with exp(ikz) leads to the paraxial wave equation (70.10) ∇T2 E + 2ik

∂E =0 ∂z

(70.38)

for a monochromatic field in vacuum. If E(x, y, z) satisfies the paraxial wave equation and is specified in the

plane (x, y, z = 0), it follows that  i E(x, y, z) = − dx  dy E(x  , y , 0) λz  2 +(y−y  )2 ]/2z

× eik[(x−x )

.

(70.39)

Thus, in the case of a laser resonator, the field E(x, y, L) at the mirror at z = L is related to the field E(x, y, 0) at

Laser Principles

the mirror at z = 0 by  i E(x, y, L) = − dx  dy E(x  , y , 0) λL  2

 2

× eik[(x−x ) +(y−y ) ]/2L  ≡ dx  dy K(x, y; x  , y )E(x  , y , 0) . (70.40)

Similarly, the field at z = 0 after one round trip pass through the resonator is E(x, y, 0)  = dx  dy K(x, y; x  , y )E(x  , y , L)   = dx  dy K(x, y; x  , y ) dx  dy

1029

Among stable configurations, the symmetric confocal resonator with R1 = R2 = L has the smallest mode spot sizes at the mirrors, while the concentric resonator with R1 = R2 slightly greater than L/2, has the smallest beam waist (Fig. 70.1). The widely used hemispherical resonator (R1 = ∞, R2 slightly greater than L) is relatively easy to keep aligned and allows the spot size at mirror 2 to be adjusted by slight changes in L. The fundamental Gaussian beam modes of stable resonators may be constructed from the free space solutions of the paraxial wave equation. The most important (lowest-order) solution for this purpose is E(x, y, z) =

× K(x  , y ; x  , y )E(x  , y , 0)  ≡ dx  dy K˜ (x, y; x  , y )E(x  , y , 0) . (70.41)

By definition, a mode of the resonator is a field distribution that does not change on successive round-trip passes through the resonator. More precisely, since an empty cavity is assumed, a mode will be such that the field spatial pattern remains the same except for a constant decrease in amplitude per pass. A longitudinal mode is defined by the value of k in exp(ikz), whereas a transverse mode is defined by the corresponding (x, y) dependence and satisfies the integral equation  γ E(x, y, z) = dx  dy K˜ (x, y; x  , y )E(x  , y , z) , where z defines any plane between the mirrors and |γ | < 1. Iterative numerical solutions of this equation for laser resonator modes were first discussed by Fox and Li [70.16]. Laser resonators may be classified as stable or unstable according to whether a paraxial ray traced back and forth through the resonator remains confined in the resonator or escapes. This leads to the condition

A e−iφ(z) eik(x

2 +y 2 )/2R(z)

e−(x  1 + z 2 /z 20

, (70.44)

where A is a constant, φ(z) = tan−1 (z/z 0 ), and R(z), w(z), and z 0 are the radius of curvature of surfaces of constant phase, the spot size, and the Rayleigh range, respectively. The confocal parameter, 2z 0 , is also used to characterize Gaussian beams. Here z 0 ≡ πw20 /λ, where w0 is the spot size at the beam waist at z = 0 (Fig. 70.1), and R(z) and w(z) vary with the distance z as follows:  R(z) = z + z 20 /z , w(z) = w0 1 + z 2 /z 20 . (70.45)

The divergence angle of a Gaussian beam (Fig. 70.1) is given by θ = λ/πw0 . The ABCD law for Gaussian beams allows the effects of various optical elements on Gaussian beam propagation to be calculated in a relatively simple fashion [70.1–3]. For instance, a Gaussian beam incident on a lens of focal length f at its waist isfocused to a new waist at a distance d = f/ 1 + f 2 /z 20 behind the lens, (x, y)

(x, y) z0

(70.43)

for stability, where the g parameters are defined in terms of the (spherical) mirror curvatures Ri and the mirror separation L by gi ≡ 1 − L/Ri . Ri is defined as positive or negative depending on whether the mirror i is concave or convex, respectively, with respect to the interior of the resonator. Plane–parallel mirrors (Ri → ∞) can be used, but they are difficult to keep aligned and have much larger diffractive losses.

2 +y 2 )/w2 (z)

w0

θ

w0 √2

w(z)

z=0 Beam waist Intensity

Intensity

Fig. 70.1 Variation of the spot size w(z) of a Gaussian beam with the propagation distance z from the beam waist

Part F 70.3

(70.42)

0 ≤ g1 g2 ≤ 1

70.3 Laser Resonators

1030

Part F

Quantum Optics

and the spot size at the new waist is w0 =

fλ 1  , πw0 1 + f 2 /z 2 0

(70.46)

which is approximately fλ/πw0 = fθ for tight focusing. This shows that a Gaussian beam can be focused to a very small spot. On the other hand, beam expanders consisting of two appropriately spaced lenses may be used to increase the spot size by the ratio of the focal lengths. Gaussian beam modes of laser resonators have radii of curvature that match in magnitude those of the mirrors. The spot sizes at the mirrors and the location of the beam waist with respect to the mirrors may be expressed in terms of λ, L, and g1 g2 . The empty-cavity mode frequencies are given by 
1 c √ N + cos−1 g1 g2 , νN = (70.47) 2L π

Part F 70.4

where N is an integer. For a host medium of refractive index n, c/2L is replaced by c/2nL, and (70.47) then generalizes the plane wave result (70.28) to account for both longitudinal and transverse effects in the determination of the cavity mode frequencies. The assumption of Gaussian modes presupposes that the resonator mirrors are large enough to intercept the entire beam without any “spillover”; i. e., that a w1 , w2 , where w1 , w2 are the spot sizes at the mirrors and a is an effective mirror cross sectional radius. This implies that the Fresnel number NF ≡ a2 /λL 1. Diffraction losses generally increase with decreasing Fresnel numbers. Higher-order Gaussian modes, where the Gaussian functions of x and y in (70.44) are replaced by higherorder Hermite polynomials, are often more difficult to realize than the fundamental lowest-order mode because their larger spot sizes imply higher diffractive losses, and beyond a certain mode order the spot sizes are too large

to satisfy the no-spillover condition for a pure Gaussian mode. It is not possible in general to write closed form expressions for laser modes. There are at least two reasons for this, the first being that the gain medium cannot in general be regarded as a simple amplifying element that preserves the basic empty-cavity mode structure. In low power gas lasers, the spatial variations of the gain and refractive index are sufficiently mild that the lasing modes can be accurately described as Gaussians, but more generally, there can be strong gain and index variations that themselves play an important role in determining the modes of the laser, as in “index-guided” and “gain-guided” semiconductor lasers. Secondly, the resonator structure itself may introduce complications that preclude closed form solutions even for the empty cavity. This is generally true, for instance, of unstable resonators, where iterative numerical solutions of the integral equation (70.42) are necessary for accurate predictions of modes. In such computer simulations, the mode structure must be determined self consistently with the numerical solutions of density matrix or, more commonly, rate equations for the gain medium. Unstable resonators, though inherently lossy, offer some important advantages for high-gain lasers. Thus, whereas the Gaussian modes of stable resonators typically involve very small beam spot sizes, the distinctly non-Gaussian modes of unstable resonators can have large mode volumes and make more efficient use of the available gain volume. Unstable resonators also tend to yield higher output powers when they oscillate on the lowest loss transverse mode, whereas in stable resonators higher output powers are generally associated with multitransverse mode oscillation. For very high power lasers, the fact that unstable resonators involve all-reflective optics, as opposed to transmissive output coupling, can be important in avoiding optical damage.

70.4 Photon Statistics Optical fields may be characterized and distinguished by their photon statistical properties (Chapt. 78 and [70.17–21]). In a photon counting experiment, the number of photons registered at a photodetector during a time interval T is measured and used to infer the probability Pn (T ) that n photons are counted in a time interval T . If the probability of counting a photon at the time t in the interval dt is denoted αI(t) dt, where I(t) is the intensity

of the field and α is a factor depending on the microscopic details of the photoelectric process and on the phototube geometry, then it may be shown from largely classical considerations that [70.3, 17–21] n   T   1     −α 0T dt  I(t  ) α dt I(t ) e , Pn (T ) = n! 0

(70.48)

Laser Principles

where the average  · · ·  is over the intensity variations during the counting interval. This is Mandel’s formula. In the simplest case of constant intensity, Pn follows the Poisson distribution: Pn = n n e−n /n! ,

(70.49)

where n = αIT . Since a single-mode laser field can be thought of approximately as a “classical stable wave” [70.20,21], it is not surprising that its photocount distribution is found both theoretically and experimentally to satisfy (70.49), which is characteristic of a coherent state of the field (Chapt. 78). A thermal light source, by contrast, follows the Bose–Einstein distribution, Pn = n n /(1 + n)n+1 ,

(70.50)

70.5 Multi-Mode and Pulsed Operation

1031

if the time interval T is short compared with the coherence time of the light; i. e., if T ∆ν  1, where ∆ν is the bandwidth. If T ∆ν 1, Pn is again Poissonian. Thus, if a quasi-monochromatic beam of light is made from a natural source by spatial and spectral filtering, it has measurably different photon counting statistics from a single-mode laser beam of exactly the same bandwidth and average intensity. Laser radiation approaches the ideal coherent state that, of all possible quantum mechanical states of the field, most closely resembles the “classical stable wave”. These differences are exhibited in other experiments, such as the measurement of intensity correlations of the Brown–Twiss type [70.3, 17–21]. In such experiments, thermal photons have a statistical tendency to arrive in pairs (photon bunching), whereas the photons from a laser arrive independently.

70.5 Multi-Mode and Pulsed Operation tion, and in particular that the spontaneous emission lifetime be relatively long. The pulse duration can be reduced to approximately a single cavity round trip time by Q-switching from low reflectivity mirrors to 100% reflectivities, and then switching the reflectivity of the outcoupling mirror from 100% to 0% at the peak of the amplified pulse. In addition to this pulsed transmission mode is cavity dumping, where both mirrors have nominally 100% reflectivity and the intracavity power is “dumped” by an acousto-optic or electro-optic intracavity element that deflects the light out of the cavity. The pulse duration achieved in this way is again roughly a round trip time. Cavity dumping can be employed with cw lasers and does not require the long energy storage times necessary for ordinary Q-switching. Shorter pulses can be realized by mode-locking, where the phases of N longitudinal cavity modes are locked together. In the simplest model, assuming equal amplitudes and phases of the individual modes, the net field amplitude is proportional to X(t) =

(N−1)/2 

X 0 sin[(ω0 + n∆)t + φ0 ]

n=−(N−1)/2

sin(N∆t/2) . (70.51) sin(∆t/2) The temporal variation described by this function for large N is a train of “spikes” of amplitude NX 0 at times tm = 2πm/∆, m = 0, ±1, ±2, · · · , the width of each spike being 2π/(N∆). In the case of a mode-locked laser, = X 0 sin(ω0 t + φ0 )

Part F 70.5

Multi-mode laser theory is generally much more complicated than single-mode theory, particularly in the case of inhomogeneous broadening with both spectral and spatial hole burning. In certain situations, however, considerable simplification is possible. For instance, when the cavity-mode frequency spacing is small compared with the homogeneous linewidth δν0 , the gain tends to saturate homogeneously, and the total output power on all modes is well described by the Rigrod analysis outlined in Sect. 70.2. Pulsed laser operation adds the further complication of temporal variations to the cw theory outlined in the preceding sections. It is possible, nevertheless, to understand some of the most important types of multi-mode and pulsed operation using relatively simple models. One method of obtaining short, high power laser pulses is Q-switching. In a very lossy cavity, the gain can be pumped to large values before the threshold condition is met and gain saturation occurs. If the cavity loss is suddenly decreased, there will be a rapid buildup of intensity because the small-signal gain is far above the (now reduced) threshold value. The switching of the cavity loss is called Q-switching, the “quality factor” Q being defined as ν/2δνc . This type of Q-switching produces intense pulses of duration typically in the range 10–100 ns, as dictated by the fact that a light pulse must make several passes through the gain cell for amplification. Q-switching requires that the gain medium be capable of retaining a population inversion over a time much larger than the Q-switched pulse dura-

1032

Part F

Quantum Optics

∆ = 2π(c/2L), and the output field consists of a train of pulses separated in time by T = 2π/∆ = 2L/c. The peak amplitude of the spikes is proportional to N, and the duration of each spike is approximately 2L/cN. The maximum number Nmax of modes that can actually be phase-locked is limited by the spectral width ∆νg of the gain curve: Nmax =

∆νg 2L = ∆νg . c/2L c

(70.52)

Similarly, the shortest pulse duration is τmin =

2L 1 = . cNmax ∆νg

(70.53)

Part F 70.5

Mode-locking thus requires a gain bandwidth large compared with the cavity mode spacing, and the shortest and most intense mode-locked pulse trains are obtained in gain media having the largest gain bandwidths. Trains of picosecond pulses are routinely obtained with liquid dye and solid gain media having gain bandwidths ∆νg ≈ 1012 s−1 or more. Various techniques, employing acoustic or electrooptic modulation or saturable absorbers, are used to achieve mode-locking [70.1–3]. The different methods all rely basically on the fact that a modulation of the gain or loss at the mode separation frequency c/2L tends to cause the different modes to oscillate in phase. Such a modulation is achieved “passively” when a saturable absorber is placed in the laser cavity: the multi-mode intensity oscillates with a beat frequency that is impressed on the saturated loss coefficient. Dye lasers, with their large gains across a broad range of optical frequencies, are often employed to generate mode-locked picosecond pulses. Ultrashort pulse generation is possible with additional nonlinear or frequency chirping techniques [70.22]. The colliding-pulse laser [70.23] is a three-mirror ring laser in which two mode-locked pulse trains propagate in opposite senses and overlap in a very thin (≈ 10 µm) saturable absorber placed in the ring in addition to the gain cell. The cavity loss is least when the two pulses synchronize to produce the highest intensity, and therefore the lowest loss coefficient in the saturable absorber. The short length of the absorbing cell forces the pulses to overlap within a very small distance and therefore to produce very short pulses (10 µm/c ≈ 30 fs pulse duration with c/L ≈ 100 MHz repetition rate). Another method of ultrashort pulse generation relies on frequency chirping; i. e., a time dependent shift of the frequency of an optical pulse [70.24–26]. In a medium

(e.g., a glass fiber) with linear and nonlinear refractive index coefficients n 0 and n 2 , the refractive index is n = n0 + n2 I ,

(70.54)

so that there is an instantaneous phase shift φ(t) that depends on the instantaneous intensity. Therefore, as I(t) increases toward the peak intensity of the pulse, φ(t) increases (assuming n 2 > 0), whereas φ(t) decreases as I(t) decreases from its peak value. The frequency ˙ is such that the instantaneous frequency of the shift φ(t) pulse is smaller at the leading edge and larger at the trailing edge of the pulse (Fig. 70.2), resulting in a stretching of the pulse bandwidth. Following this spectral broadening by the nonlinear medium, the pulse can be compressed in time by means of a frequency dependent delay line such that the smaller frequencies, say, are delayed more than the higher frequencies. The trailing edge of the pulse can therefore “catch up” to the leading edge, resulting in a shorter pulse whose duration is given by the inverse of the chirp bandwidth. The delay line can be realized with a pair of diffraction gratings (see Fig. 70.2). Using this pulse compression technique, 40 fs amplified pulses from a colliding pulse laser have been compressed to 8 fs, corresponding to about four optical cycles [70.27]. In chirped-pulsed-amplification (CPA) lasers [70.28] a laser pulse is chirped, temporally stretched, and then passed through an amplifier. The lengthening in time of the pulse prior to amplification allows greater energy extraction from the amplifier. After amplification, a)

Chirped pulse Glass fiber

b) Chirped input pulse

Grating

Blue

Red

Output pulse

Grating

Fig. 70.2a,b Nonlinear pulse compression by frequency chirping. In (a), the nonlinear refractive index of a glass fiber results in a time dependent frequency of the transmitted pulse, and in (b) a pair of diffraction gratings is used to produce frequency dependent path delays such as to temporally compress the pulse

Laser Principles

pulse compression is performed with a grating pair. The Ti :sapphire (Ti : Al2 O3 ) amplifier is particularly attractive for femtosecond CPA because of its very large spectral width and high saturation fluence. The wavelength dependence of the linear refractive index n 0 in (70.54) results in a group velocity vg = c[n 0 − λ dn 0 / dλ]−1

(70.55)

that, if dn 0 / dλ < 0, is such that higher frequencies propagate more rapidly than lower frequencies. Assuming

70.7 Recent Developments

1033

n 2 > 0, on the other hand, the nonlinear part of the index causes a delay of higher frequencies with respect to lower ones, as discussed above. This leads to soliton solutions of the wave equation, such that the opposing effects of the linear and nonlinear dispersion are balanced and the pulse propagates without distortion. Soliton lasers, with pulse durations ranging from picoseconds down to ≈ 100 fs, depending on the fiber length, have been made with solid state lasers and intracavity optical fibers [70.29].

70.6 Instabilities and Chaos threshold. In particular, the cw output of the laser gives way to an oscillatory intensity, even though the pumping and loss terms in the equations are assumed to be time independent. As the pumping and loss parameters are varied, this self-pulsing instability can give way to chaotic behavior, and numerical studies of the set of coupled matter–field equations reveal period doubling, two-frequency, and intermittency routes to chaos in different regimes [70.30]. This unstable behavior of single-mode inhomogeneously broadened lasers was first discovered experimentally and analyzed by Casperson [70.36–38] for low pressure, 3.51 µm He−Xe lasers. Arecchi et al. [70.39] reported the first observation and characterization of chaotic behavior in a laser system; by modulating the cavity loss of a CO2 laser they observed a period doubling route to chaos as the modulation frequency was varied. Extensive experimental and theoretical work on unstable and chaotic behavior in a wide variety of other laser systems has been reported [70.30–34, 40], including work aimed at the control of chaotic laser oscillation by the so-called occasional proportional feedback technique [70.41, 42]. Instabilities of single transverse mode dynamics have also been studied, especially in connection with spontaneous spatial pattern formation [70.40].

70.7 Recent Developments Recent developments in the basic physics of lasers include the application of cavity QED techniques to produce a single-atom laser [70.43] that emits < 105 photons/s, and a two-photon laser [70.44] that operates on the basis of amplification on a two-photon

transition. Recent progress in the development of ultrashort pulses includes the generation of attosecond pulses by high-order harmonic generation [70.45, 46], and the application of such pulses to a measurement of the photoionization time of Auger electrons [70.47].

Part F 70.7

Mode-locked pulses and solitons exemplify an ordered dynamics, as opposed to the erratic and seemingly random intensity fluctuations that are sometimes observed in the output of a laser. In fact, it is possible, under certain circumstances, for laser oscillation to exhibit deterministic chaos; i. e., an effectively random behavior that can nevertheless be described by purely deterministic equations of motion [70.30–34]. Lasers are nonlinear and dissipative systems, and as such exhibit essentially all the modes of behavior characteristic of such systems. It was shown by Haken [70.35] that (70.6) and (70.11), with ∆ = 0 and with pumping and field loss terms included, can be put into the form of the Lorenz model for chaos. For a single-mode, homogeneously broadened ring laser, the Lorenz model instability requires a “bad cavity” in the sense that the field loss rate is larger than the sum of the homogeneous linewidth of the lasing transition and the population decay rate. It also requires the gain medium to be pumped at least nine times above the threshold gain value, a condition sometimes referred to as a “second laser threshold.” When the single-mode laser equations corresponding to the Lorenz model are generalized to the case of inhomogeneous broadening, an instability occurs at small–signal gain values much less than nine times

1034

Part F

Quantum Optics

References 70.1 70.2 70.3 70.4 70.5 70.6 70.7 70.8 70.9 70.10 70.11 70.12 70.13 70.14 70.15 70.16 70.17 70.18 70.19 70.20 70.21

Part F 70

70.22 70.23 70.24 70.25 70.26 70.27

A. Yariv: Quantum Electronics, 2nd edn. (Wiley, New York 1989) A. E. Siegman: Lasers (University Science Books, Mill Valley. 1986) P. W. Milonni, J. H. Eberly: Lasers (Wiley, New York 1988) P. W. Milonni: Phys. Rep. 25, 1 (1976) B. DiBartolo: Optical Interactions in Solids (Wiley, New York 1968) p. 405 W. W. Rigrod: J. Appl. Phys. 36, 27 (1965) C. H. Henry: IEEE J. Quantum Electron. 18, 259 (1982) K. Petermann: IEEE J. Quantum Electron. 19, 1391 (1979) A. E. Siegman: Phys. Rev. A 39, 1253 (1989) A. E. Siegman: Phys. Rev. A 39, 1264 (1989) P. Goldberg, P. W. Milonni, B. Sundaram: Phys. Rev. A 44, 1969 (1991) P. Goldberg, P. W. Milonni, B. Sundaram: Phys. Rev. A 44, 4556 (1991) W. A. Hamel, J. P. Woerdman: Phys. Rev. Lett 64, 1506 (1990) M. A. Eijkelenborg, A. M. Lindberg, M. S. Thijssen, J. P. Woerdman: Phys. Rev. A. 55, 4556 (1997) K.-J. Cheng, P. Mussche, A. E. Siegman: IEEE J. Quantum Electron. 30, 1498 (1994) A. G. Fox, T. Li: Bell System Tech. J. 40, 453 (1961) R. J. Glauber: Phys. Rev. 130, 2529 (1963) R. J. Glauber: Phys. Rev. 131, 2766 (1963) L. Mandel, E. Wolf: Rev. Mod. Phys. 37, 231 (1965) R. Loudon: The Quantum Theory of Light, 3rd edn. (Clarendon Press, Oxford 2000) p. 32 P. Meystre, M. Sargent III: Elements of Quantum Optics, 2nd edn. (Springer, Berlin, Heidelberg 1991) p. 32 E. B. Treacy: IEEE J. Quantum Electron. 5, 454 (1969) R. L. Fork, B. I. Greene, C. V. Shank: Appl. Phys. Lett. 38, 671 (1981) D. Grischkowsky, A. C. Balant: Appl. Phys. Lett 41, 1 (1982) L. F. Mollenauer, R. H. Stolen, J. P. Gordon, W. J. Tomlinson: Opt. Lett. 8, 289 (1983) J. G. Fujimoto, A. M. Weiner, E. P. Ippen: Appl. Phys. Lett. 44, 832 (1984) C. V. Shank: Science 233, 1276 (1986) and references therein

70.28 70.29 70.30

70.31 70.32 70.33 70.34

70.35 70.36 70.37

70.38 70.39 70.40 70.41 70.42

70.43 70.44 70.45

70.46

70.47

P. Maine, D. Strickland, P. Bado, M. Pessot, G. Mourou: IEEE J. Quantum Electron. 24, 398 (1988) L. F. Mollenauer, R. H. Stolen: Opt. Lett. 9, 13 (1984) P. W. Milonni, M.-L. Shih, J. R. Ackerhalt: Chaos in Laser–Matter Interactions (World Scientific, Singapore 1987) D. K. Bandy, A. N. Oraevsky, J. R. Treddice: Special issues of J. Opt. Soc. Am. 5, 5 (1988) N. B. Abraham, W. J. Firth: Special issues of J. Opt. Soc. Am. 7, 6 (1990) N. B. Abraham, W. J. Firth: Special issues of J. Opt. Soc. Am. 7, 7 (1990) N. B. Abraham, F. T. Arecchi, L. A. Lugiato (Eds.): Instabilities and Chaos in Quantum Optics II (Plenum, New York 1988) H. Haken: Phys. Lett. A 53, 77 (1975) L. W. Casperson: IEEE J. Quantum Electron, Vol. 14 (Springer, Berlin, Heidelberg 1978) p. 756 L. W. Casperson: Spontaneous pulsations in lasers. In: Laser Physics, ed. by J. D. Harvey, D. F. Walls (Springer, Berlin, Heidelberg 1983) P. Chenkosol, L. W. Casperson: J. Opt. Soc. Am. B 20, 2539 (2003) F. T. Arecchi, R. Meucci, G. Puccione, J. Tredicce: Phys. Rev. Lett. 49, 1217 (1982) L. A. Lugiato, W. Kaige, N. B. Abraham: Phys. Rev. A 49, 2049 (1994) and references therein. R. Roy, T. W. Murphy Jr., T. D. Maier, Z. Gills: Phys. Rev. Lett. 68, 1259 (1992) Z. Gills, C. Iwata, R. Roy, I. B. Schwartz, I. Triandaf: Phys. Rev. Lett. 69, 3169 (1992) see also Optics and Photonics News (May,1994) J. McKeever, A. Boca, A. D. Boozer, J. R. Buck, H. J. Kimble: Nature 425, 268 (2003) D. J. Gauthier: Progress in Optics 45, 205 (2003) M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, F. Krausz: Nature 414, 509 (2001) P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H. G. Muller, P. Agostini: Science 292, 1689 (2001) M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinizi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, F. Krausz: Nature 419, 803 (2002)

1035

71. Types of Lasers

Types of Laser The availability of coherent light sources (i. e., lasers) has revolutionized atomic, molecular, and optical science. Since its invention in 1960, the laser has become the basic tool for atomic and molecular spectroscopy and for elucidating fundamental properties of optics and optical interactions with matter. The unique properties of laser light have spawned new types of spectroscopy, as discussed in Chapt. 72 to Chapt. 80. There are now literally hundreds of different types of lasers. However, only a few of these are commercially available, and lasers tailor made with operational properties optimized for specific applications are often needed. Chapter 70 describes the principles of laser operation leading to specific output characteristics. This chapter summarizes the current status of the development of different types of lasers, emphasizing those that are commercially available. There are several ways to categorize types of lasers; for example, in terms of spectral range, temporal characteristics, pumping mechanism, or lasing media.

Gas Lasers ........................................... 1036 71.1.1 Neutral Atom Lasers................... 1036 71.1.2 Ion Lasers................................. 1036 71.1.3 Metal Vapor Lasers .................... 1037 71.1.4 Molecular Lasers ....................... 1037 71.1.5 Excimer Lasers .......................... 1038 71.1.6 Nonlinear Mixing ...................... 1038 71.1.7 Chemical Lasers......................... 1039

71.2

Solid State Lasers................................. 1039 71.2.1 Transition Metal Ion Lasers ......... 1040 71.2.2 Rare Earth Ion Lasers ................. 1040 71.2.3 Color Center Lasers .................... 1042 71.2.4 New Types of Solid State Laser Systems ........................... 1043 71.2.5 Frequency Shifters..................... 1043

71.3

Semiconductor Lasers........................... 1043

71.4

Liquid Lasers ....................................... 1044 71.4.1 Organic Dye Lasers..................... 1044 71.4.2 Rare Earth Chelate Lasers ........... 1045 71.4.3 Inorganic Rare Earth Liquid Lasers1045

71.5

Other Types of Lasers ........................... 1045 71.5.1 X-Ray and Extreme UV Lasers ...... 1045 71.5.2 Nuclear Pumped Lasers .............. 1046 71.5.3 Free Electron Lasers ................... 1046

71.6

Recent Developments........................... 1046

References .................................................. 1048

diffraction limited beam quality with coherence lengths up to 10 m can be obtained. Temporal pulse widths of a few femtoseconds have been generated. Some lasers can produce peak powers of over 1013 W and average powers of 105 W with pulse energies greater than 104 J. The important operational characteristics, such

Table 71.1 Categories of lasers Gas lasers

Special lasers

Solid State lasers

Liquid lasers

Miscellaneous

Atomic

Metal vapor

Transition metal ion

Organic dye

X-ray

Ionic

Chemical

Rare earth ion

Rare earth chelate

Nuclear pumped

Molecular

Color center

Inorganic solvents with rare earth ions

Free electron

Excimer

Semiconductor

Part F 71

For this chapter, the types of lasers are categorized in terms of the lasing media, as shown in Table 71.1. The variety of different laser types offers a wide range of beam parameters. The spectral output of lasers covers the X-ray to far IR regions as shown in Fig. 71.1. Spectral linewidths can be as narrow as 20 Hz and

71.1

1036

Part F

Quantum Optics

as frequency range and output power, are given for each of the types of lasers described in the following sections. Extensive tables of laser properties have been published in several different handbooks [71.1–5] and all of these details for every laser are not repeated here due to space limitations. The data that are quoted

come from these reference books and from recent proceedings of conferences such as CLEO (Conference on Lasers and Electro-Optics) and ASSL (Advanced Solid State Laser Conference). The reader is referred to these sources for further details of specific laser operation parameters.

71.1 Gas Lasers Gas lasers can be separated into subclasses based on the lasing media: neutral atoms; ions; and molecules. In addition, molecular lasers contain the special classes of excimer lasers and chemical lasers. Except for chemical lasers, the pumping mechanism is generally an electrical discharge in a gas filled tube. This discharge causes the acceleration of electrons that transfer their kinetic energy to the lasing species through collisions, leaving them in a variety of excited states. These relax back to the ground state with different rates, resulting in the possibility of a population inversion for some transitions. These can be electronic, vibrational, or rotational transitions with wavelengths ranging from the near ultraviolet (UV) through the far infrared (IR) spectral regions. Systems lasing at short wavelengths generally operate only in the pulsed mode because of the short radiative lifetimes of the transitions involved. Because of their low gain, gas lasers usually have relatively long linear cavity designs. Gas lasers generally have narrow spectral emission lines with the possibility of lasing at several different wavelengths. They are broadened both by collisions (homogeneous broadening) and the Doppler effect of the motion of the atoms or molecules (inhomogeneous broadening). To date, 51 elements in the periodic table have shown either ionic or neutral ion gas laser emission.

71.1.1 Neutral Atom Lasers Part F 71.1

These lasers generally emit in the visible and near IR spectral range. The first and most common laser of this class is the He–Ne laser. Its most prominent emission line is at 632.8 nm. It is usually operated in the continuous wave (cw) configuration with typical power outputs between 0.5 to 50 mW, although powers of over 100 mW have been achieved. Excitation occurs through electrical discharge which pumps both the He and Ne atoms to excited states. The more abundant He atoms transfer their energy to several excited states of Ne atoms. Several radiative transitions of Ne are available for laser transitions. These provide emission at 543.36, 632.8,

Semiconductor Dye Solid state 500

1000

Atoms

5000

10 000 Wavelength (nm)

Gas lasers

Ions Molecules

Molecules

Fig. 71.1 Spectral range of laser emission

1152.27, and 3391.32 nm. Final relaxation back to the ground state occurs through collisions with the walls of the gas tube. The desired laser line can be selected by adjusting the reflectivity of the cavity mirrors to discriminate against unwanted transitions. In addition, it is possible to adjust the discharge current, gas ratio, and pressure to optimize a specific emission transition. Using an external magnetic field to produce a Zeeman effect can also be helpful in tuning the laser emission. In addition to the normal red He–Ne lasers, green lasers are now available. Typical He–Ne lasers operate with a coherence length of 0.1–0.3 m, a beam divergence of 0.5–2 mrad, and a stability of 5%/h. The gain at the red line is 0.5 dB/m. This line can be operated as a single mode with a linewidth of 0.0019 nm and a coherence length between 20 and 30 cm.

71.1.2 Ion Lasers The most important ion lasers are based on noble gas ions such as Ar, Kr, Ne, or Xe in various states of ionization. These operate in either a pulsed or cw mode, and their emission covers the wavelength range from

Types of Lasers

71.1.3 Metal Vapor Lasers These lasers can operate with either neutral atoms or ions. Their excitation process begins with vaporizing a solid or liquid to produce the gas for lasing, followed by normal electrical discharge pumping. Either cw or pulsed operation can be obtained, with laser emission lines in the near UV and visible spectral regions.

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One important ion laser of this class is the Helium– Cadmium laser. For the excitation processes, metallic Cd is evaporated and mixed with He. Then a d.c. electric discharge excites the He ions and ionizes the Cd. The excited He atoms transfer their energy to the Cd atoms and the laser transitions take place between electronic levels of the Cd atom. The main emission line of a He–Cd laser is the blue line at 441.6 nm. This typically has a cw output from 130 mW for single-mode operation up to 150 mW for multimode operation. The laser linewidth can be as narrow as 0.003 nm. This system also has an important laser emission at 325.029 nm, which typically has cw powers between 5 and 10 mW single-mode and 100 mW for multimode emission. The wall plug efficiency is between 0.002% and 0.02%. The most important neutral atom metal vapor laser of this type is the copper vapor laser. This has important emission lines in the green at 510.55 nm and in the yellow at 578.21 nm. These lasers operate in the pulsed mode with temporal pulse widths between 10 and 20 ns at pulse repetition rates of up to 20 000 pps. Typical pulse energies are ≈ 1 mJ, yielding average powers of 20 W. It is possible to increase the repetition rate significantly to achieve average powers of 120 W or even higher. However, this laser is self-terminating since the lower levels of the laser transitions are metastable. This restricts the pulse sequencing of the laser and requires fast discharge risetimes. Copper vapor lasers have high gain (10% to 30%/cm), and very high wall plug efficiency (≈ 1.0%). Gold vapor lasers have similar properties but operate in the red at 624 nm at several watts of power.

71.1.4 Molecular Lasers There are several types of molecular lasers that can be classified with respect to their spectral emission range, their mode of excitation, or the energy levels involved in the lasing transition. In the far IR, molecular lasers operate on transitions between rotational energy levels. These include water vapor lasers that emit between 17 and 200 µm, cyanide lasers at 337 µm, methyl fluoride lasers emitting between 450 and 550 µm, and ammonia lasers that operate at 81 µm. These are generally excited through optical pumping by a CO2 laser. They are built with a metal or dielectric wave guide cavity. The former design results in lower thresholds but gives multimode, mixed polarization output, while the latter design results in propagation losses, giving higher thresholds but linearly polarized outputs. CO2 lasers are some of the most widely used, with a variety of medical and industrial applications. They

Part F 71.1

the near UV through the visible part of the spectrum. In the electrical-discharge excitation process, electrons collide with neutral atoms in their ground states, transferring enough energy to ionize them and leave the ions in several possible excited states. For example, low discharge currents produce Ar+ giving rise to visible emission lines, while high discharge currents produce Ar2+ giving rise to UV emission lines. Radiative emission then occurs to lower excited levels of the ions, followed by subsequent spontaneous emission to the ground state of the ion, and then radiationless relaxation back to the neutral atom ground state. This transition scheme limits the wall plug efficiency to about 0.1% for visible and 0.01% for UV operation. Heat management is accomplished through either water cooling or air cooling. In the visible region of the spectrum, argon lasers have blue and green emission lines with the strongest ones at 488 and 515 nm, respectively. Krypton lasers have several strong emission lines in the green and red, with the most prominent ones at 521, 568, and 647 nm. Mixed gas lasers can produce all of these lines. In the near UV, argon has a strong laser line at 351 nm as well as several other emission lines down to 275 nm. The same methods for selecting specific laser lines on neutral atom lasers, discussed in Sect. 71.1.1, are used for ion lasers. These lasers can operate at powers of over 20 W of cw emission in the visible and at powers of several watts cw in the UV. The Doppler-limited linewidth of noble gas ion lasers is generally ≈ 5–10 GHz. By using special techniques for stabilization of the cavity, linewidths of ≈ 500 MHz with drifts of 100 MHz/hr can be obtained. By mode-locking argon or krypton lasers with 10 GHz linewidths, it is possible to produce trains of pulses with pulse lengths of 100 ps and peak powers of 1 kW, at a pulse repetition frequency of 150 MHz. Cavity-dumping produces narrow pulses at pulse repetition frequencies of ≈ 1 MHz with peak powers over 100 times the cw power. Lower power cw lasers of this type typically have beam divergences of 1.5 mrad and a stability of 5%, while high power lasers have beam divergences of ≈ 0.4 mrad with a stability of 0.5%.

71.1 Gas Lasers

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Part F 71.1

emit in the mid-IR range at 10.6 and 9.6 µm. The CO2 molecules are excited by electrical discharge, and it is common to mix CO2 with other gases, such as nitrogen, to enhance the efficiency of the excitation process through energy transfer, or with helium, to keep the average electron energy high and to depopulate the lower levels of the laser transition. Both the initial and final states of the lasing transition are vibrational levels that have many rotational sublevels. This allows discrete tuning of the output within the 9.4 and 10.4 µm bands. Using high pressures of the gas broadens the laser line into a continuum so that the emission can be continuously tuned over several microns near 10 µm. Without any frequency selective element in the cavity, the system oscillates on the transition with highest gain, which is near 10.6 µm. Under pulsed operating conditions, CO2 lasers can also emit at bands near 4.3 µm and at several bands between 11 and 18 µm. These laser systems typically operate at a few milliwatts of power to over 100 kW cw. Waveguide and slab cavity designs have been developed for heat removal. In the pulsed or Q-switched mode of operation, pulse widths are between a few microseconds and a few milliseconds, with energies as high as 10 000 J/pulse. This leads to peak powers more than 100 times higher than cw powers. It is also possible to mode-lock these systems to get a train of nanosecond pulses. In the transverseexcitation-atmospheric-pressure (TEA) configuration, CO2 lasers are pumped very rapidly compared with the lifetime of the metastable state, resulting in a large population inversion in the gain medium when the electromagnetic field builds up in the cavity. This results in an intense “gain-switched pulse” of between 100 and 200 ns in duration followed by a lower intensity emission due to continued pumping of the upper state. These systems provide up to 3 J/pulse at pulse repetition frequencies of up to 50 Hz. At much lower pulse repetition frequencies, energies as high as 1000 J/pulse have been obtained. The typical beam divergence of these lasers is less than 3 mrad. Carbon monoxide lasers operate on the vibrational levels of CO. They can be excited either through electrical discharge or chemical reaction (as discussed in Sect. 71.1.7). CO lasers have a tunable emission between 5 and 7 µm, operating with powers as high as 1 kW cw. In the pulsed mode, the typical energy emission is 10 mJ/pulse with 1 µs pulses at a pulse repetition frequency of 10 Hz. N2 O lasers extend the molecular laser wavelength range to beyond 10 µm. Nitrogen lasers operate in the near UV at 337.1 nm. These are based on transitions between electronic energy

levels of the N2 molecule excited by electrical discharge. The typical emission from an N2 laser is a single pulse of 10 ns in duration. The peak pulse power can be as high as 1 MW with 10 mJ/pulse at a pulse repetition frequency of less than 100 Hz. It is also possible to design these systems to obtain picosecond pulses. H2 operates in a similar way in the 120 to 160 nm region of the UV.

71.1.5 Excimer Lasers The most important lasers in the near UV to VUV for industrial and medical applications are based on rare-gas halide excimers such as XeF at 351 nm, XeCl at 308 nm, KrF at 248 nm, ArF at 193 nm, and F2 at 153 nm. The major problem with these systems is the corrosive nature of the gases. They can be pumped by electric discharges, electron or proton beams, or optical excitation. Using electrical discharge excitation, the electrons ionize the noble gas molecules, and these react by pulling an atom off the halide molecule to create an excited state dimer molecule (excimer) that radiates to an unstable lower state where dissociation occurs. The short radiative lifetimes of excimers result in laser pulses of 10 to 50 ns duration. Systems can be configured to have pulse durations ranging from picoseconds to microseconds. These lasers typically operate at high pulse repetition frequencies between 200 and 1000 Hz. The energy per pulse of discharge-pumped excimer lasers ranges from several mJ to 0.1 J, with typical average powers up to 200 W. Single shot, electron-beam-pumped excimer lasers can have as high as 104 J/pulse for an average power of almost 1 kW. Typical emission bands for excimer lasers are approximately 100 cm−1 wide due to vibrational sublevels. With the use of etalons and gratings, the laser line can be narrowed to 0.3 cm−1 . These frequency-selective elements can be used to tune the emission over several nanometers. Special configurations of excimer lasers have been developed to obtain specific operational parameters. For example, pulses as short as 45 fs have been obtained by mode-locking or by using excimer amplifiers to amplify frequency shifted dye laser pulses [71.3, 4]. Stimulated Brillouin scattering has been used as a phase conjugate mirror to minimize phase-front distortion in excimer lasers. Both master oscillator power amplifiers and injection locked resonator techniques have been used to achieve low spatial divergence with narrow bandwidths [71.3,4]. Input pulses for the latter system can be from either frequency shifted Nd:YAG or dye lasers.

Types of Lasers

One interesting recent development has been the demonstration of laser operation of solid state excimer systems [71.6–8] . This is discussed in Sect. 71.2.4.

71.1.6 Nonlinear Mixing Another way to produce tunable VUV laser transitions is to use nonlinear four-wave sum mixing in atomic vapors and molecular gases. This requires optical pumping with two sources and can be achieved with any of a variety of laser combinations such as excimer lasers with dye lasers, or Nd:YAG lasers with dye lasers. This technique has been used to obtain laser emission in the 57 to 195 nm spectral range with visible to VUV conversion efficiencies as high as 10−3 .

71.1.7 Chemical Lasers It is possible for some chemical reactions between molecules to leave the final molecule in an excited state. This type of “pumping” can result in a population inversion with respect to one of the lower states and laser transitions can occur between vibrational or rotational states of the molecule. Using mixtures of different types of molecules, pumping can be enhanced through energy transfer. Both pulsed and cw laser operation have been obtained with chemical lasers. Typical emission is of the order of a few hundred watts in the IR with a tunable output wavelength. Visible emission from chemical lasers has been demonstrated [71.9–11]. This type of laser provides the possibility of a system with a self contained chemical power supply for use in remote environments. The problems associated with handling hazardous chemicals have restricted the applications of chemical lasers. The HF chemical laser is associated with an exothermic chain reaction between H2 and F2 molecules to yield vibrationally excited HF [71.4]. The fluoride atoms are

71.2 Solid State Lasers

generated in an electrical discharge tube from the dissociation of SF6 . These are injected into the optical resonator along with the H2 or D2 gas which flows perpendicular to the lasing direction. Controlling the gas flow for mixing the reactants is critical to the laser design. The chemical reaction for direct excitation is F + H2 → HF∗ + H. For this reaction, ∆H = −32 kcal/mole, resulting in laser emission energies of between 100 and 400 kJ/kg, at wavelengths between 2.5 and 3.7 µm. This corresponds to vibrational-rotational transitions in the HF molecule. It is possible to select a single line for the laser output and then tune the laser output wavelength by selecting different lines. Single line output for cw operation can produce up to 100 W of power. If D2 replaces H2 , the emission shifts to between 3.6 and 4.2 µm, and the single line cw output power drops to about 50 W. The output power for a cw laser of this type, operating in the multiple line mode, can be as high as 2.2 MW, while in the pulsed mode of operation pulse energies of 5 kJ can be obtained with multiple line emission. An example of energy transfer pumping of chemical lasers is the DFCO2 system. Pumping of vibrationalrotational transitions of DF occurs through multiple chemical reactions of fluorine and deuterium, followed by energy transfer to excited states of the CO2 molecules. This exhibits laser emission with kilowatts of power at 10.6 µm, as described above. Another important laser system using energy transfer pumping is the chemicaloxygen-iodine laser (COIL). This is based on transitions between electronic levels in which singlet oxygen is excited and transfers its energy to a metastable state of iodine. Emission occurs at 1.3 µm and cw powers of up to 25 kW have been obtained. One example of chemical laser action in the visible region is excited GeO transferring its energy to atomic Tl, which lases at 535 nm. Only a few systems of this type have been demonstrated, and none are developed to the level of commercial availability [71.9–11].

defects that produce color centers are generally produced by post-growth radiation or heat treatments. The excitation mechanism is through optical pumping by either another laser or lamps. The spectral range covered by solid state lasers spans the visible and near IR. The variety of combinations of active centers and hosts provides the ability for both pulsed and cw operation with either narrow band or broad band emission. The latter type can provide frequency tunable output

Part F 71.2

71.2 Solid State Lasers Solid state lasers are based on luminescence centers randomly distributed in a crystalline or glass host material. These can be classified in terms of the type of their laser active centers: transition metal ion lasers; rare earth ion lasers; and color center lasers. The use of dye molecules in plastic host media is a new type of solid state laser that is discussed in Sect. 71.4.1. The active ions are substitutionally “doped” into the host during crystal growth or glass melting, whereas the lattice

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Quantum Optics

with the appropriate frequency-selective element in the cavity. Temperature tuning of narrow emission lines is also possible over a limited range. The spectral lines are broadened by internal strains in the host lattice (inhomogeneous broadening) and by radiationless relaxation and scattering processes involving thermal vibrations of the host (homogeneous broadening). Standard Q-switching and mode-locking techniques can be used with most solid state lasers.

71.2.1 Transition Metal Ion Lasers

Part F 71.2

The laser ions of this type have optically active electrons in unfilled 3dn electron configurations. They include the positively charged ions Cr3+ , Cr4+ , Co2+ , V2+ , Ni2+ , and Ti3+ . The transitions involved in pumping and lasing are associated with the optically active electrons. Because these are outer shell electrons, they are sensitive to their local crystal field environment. Typical host materials are ionic crystals formed from oxides such as sapphire, emerald, chrysoberyl, forsterite, and garnets, or fluorides such as MgF2 , KMgF3 , LiCaAlF6 , and LiSrAlF6 . The host determines the bulk optical, mechanical, and thermal properties of the laser material, and it influences the spectroscopic properties of the active ions. The most successful transition metal laser ion is Cr3+ . It has been made to lase in many different types of host crystals with both strong and weak crystal field environments. The most common strong field host is sapphire, and Al2 O3 :Cr3+ , commonly known as ruby, which was the first laser invented. The typical characteristic of a strong field laser material is a sharp laser line associated with a spin-flip electronic transition between states of the same crystal field configuration. In ruby this occurs at 699.7 nm. Because of its strong, broad absorption bands, ruby can be efficiently pumped by lamps and operates in either a pulsed or cw mode. Typical cw power output is a few watts with an efficiency of ≈ 0.1%. Ruby can be Q-switched to produce 10 ns pulses with several joules of energy per pulse, and mode-locked to produce pulses that are 5 ps in duration. The typical characteristic of a weak field laser material is a broad gain curve associated with vibronic transitions between states of different crystal field configurations. Although chrysoberyl is a host with intermediate crystal field, BeAl2 O4 :Cr3+ , commonly known as alexandrite, is sufficiently close to a weak field case to operate as a laser in this regime. Using frequency selective elements in the cavity, alexandrite lasers can be tuned in the 700 to 820 nm range. The gain of alexandrite increases with temperature, with the cross section

at the peak of the gain curve near 10−20 cm2 . Typical laser outputs are 4.5 J/pulse at 20 Hz pulse repetition rate and 90 W average power with 2% overall efficiency. In the Q-switched mode, 40 ns pulses with 2 J/pulse are obtained and the pulse width can be stretched to much longer values. Alexandrite lasers can also be mode-locked to obtain 28 ps pulses with 0.5 mJ/pulse. Fluoride crystals, such as LiCaAlF6 and LiSrAlF6 , are also weak field hosts for tunable Cr3+ lasers in the near IR. The latter is termed Cr:LiSAF and has a peak stimulated emission cross section of 0.4 × 10−19 cm2 with a tuning range from 780 to 1020 nm. This system can be either flashlamp pumped or diode laser pumped. Single pulses with energies of 75 J have been generated by these lasers. Kerr lens mode-locking has produced pulses shorter than 100 fs. One of the most interesting tunable solid state laser systems is Ti-sapphire (Al2 O3 :Ti3+ ) because it has the broadest tuning range of any ion, extending from about 660 nm to about 1180 nm. Pumping can be provided by either lasers or flashlamps, resulting in either cw or pulsed operation. This results in a versatile source of excitation in this spectral region and sub-picosecond pulses can be obtained through modelocking. The single 3d electron of Ti3+ gives a simplified energy level scheme which minimizes losses due to excited state absorption, which can be a problem in Cr3+ lasers. However, the metastable state lifetime of Ti3+ is significantly shorter than that of Cr3+ , and therefore Cr3+ lasers have much greater energy storage capability. Thus, Ti-sapphire lasers are difficult to pump by flashlamp, and are therefore generally pumped by argon lasers or frequency-doubled Nd-YAG lasers. Tisapphire has a very high peak gain cross section of about 4 × 10−19 cm2 . On the other hand, it is difficult to grow Ti-sapphire crystals with high concentrations of Ti3+ ions because of valance state stability.

71.2.2 Rare Earth Ion Lasers All of the trivalent lanthanide ions (Ce3+ , Pr3+ , Nd3+ , Pm3+ , Sm3+ , Eu3+ , Gd3+ , Tb3+ , Dy3+ , Ho3+ , Er3+ , Tm3+ , Yb3+ ) and the divalent ions Sm2+ , Dy2+ , Tm2+ have been used as active ions in solid state lasers. These ions are characterized by unfilled 4fn electron configurations and the most common source of laser emission comes from electronic transitions among their energy levels. Because the inner shell 4f electrons are shielded by outer shell electrons, the energy levels are not strongly affected by the environment of the local host material. This leads to sharp lines in both absorption and emission.

Types of Lasers

ion is excited to a low-lying metastable state by a photon from the pump source and then re-excited to a higher metastable state either by another photon from the pump source or by energy transfer interaction with a neighboring ion that has also been excited to the low-lying metastable state. Avalanche pumping [71.16] relies on thermal fluctuations to populate a low-lying energy level of an ion which can then be re-excited by a pump photon to a high energy metastable state. Another method involves the addition of a second dopant ion to the host. This “sensitizer” ion is one with broad pump bands (such as Cr3+ ) that can efficiently absorb the energy from the pump lamp. The excited sensitizer then interacts with the “activator” (lasing) ion through a radiationless, resonant energy transfer process. This deactivates the sensitizer and excites the activator ion. Nd3+ has been made to lase in a greater number of host materials than any other active ion. Y3 Al5 O12 :Nd3+ , commonly referred to as Nd-YAG, is one of the most successful commercially available lasers. Although it is possible for Nd-YAG lasers to operate at several different wavelengths around 1 µm, the standard lasers emit at 1.06 µm. Continuous wave powers of 250 W are available and pulsed performance of several megawatts at 10 Hz and 1 J/pulse can be obtained. To obtain visible (532 nm) and near UV (354 and 266 nm) emission, nonlinear optical crystals are used to modify the near IR output through second, third, and fourth harmonic generation. The optimum concentration of Nd3+ in YAG is a few percent, and above this amount, concentration quenching of the emission occurs through energy transfer and cross-relaxation processes. Attempts to co-dope Nd-YAG with Cr3+ to enhance pumping through energy transfer have not been successful [71.17, 18]. However, in other garnet crystal hosts such as Gd3 Sc2 Ga3 O12 (GSGG), the Cr–Nd energy transfer is strong enough to enhance pumping efficiency. Diode laser pumping of Nd-YAG has significantly increased pumping efficiency. Other crystal hosts such as the pentaphosphates do not exhibit concentration quenching, so materials with 100% Nd3+ can be used as “stoichiometric laser materials”. These can be important for some mini-laser applications. In glass hosts, Nd3+ ions can produce pulses with energies of ≈ 100 kJ/pulse, but these systems must operate at very low pulse repetition rates of one pulse every few minutes to allow for heat dissipation. New athermal glass compositions decrease the problems with thermal lensing for high power laser operation. Nd:YVO lasers have been operated in a microchip configuration pumped by diode lasers.

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Part F 71.2

One exception to this is vibronic emission from Ho3+ which can produce tunable laser emission in the near IR. In some cases, transitions between 5d and 4f levels are involved in the laser emission. Since the 5d levels are broadened by the environment, broadly tunable laser emission can be obtained. Examples are Ce3+ in the UV and Sm2+ in the near IR The only actinide ion that has been made into a laser is U3+ . Both crystals and glasses can be used as host materials for rare earth ion lasers. Common oxide crystal hosts include the garnets such as Y3 Al5 O12 (commonly referred to as YAG) and a typical fluoride crystal host is YLiF3 . A wide variety of glass hosts has been used, including silicates, phosphates, heavy metal fluorides, and mixtures of these. The major difference between crystal and glass hosts is that crystals provide similar crystal field sites for every dopant ion, leading to a minimum of inhomogeneous broadening, while the disorder associated with glass structure gives many different types of local crystal field sites for the dopant ions and thus significant inhomogeneous broadening. Because of the abundance of their energy levels, many trivalent rare earth ions have more than one metastable state, and laser emission is possible from several transitions. This results in over 100 possible laser emission lines ranging from the near UV through the visible and near IR. Both pulsed and cw operation can be obtained. The standard configuration for a rare earth solid state laser is a rod of laser material pumped by a lamp. Other configurations are used for special situations, such as a slab of laser material for high power glass lasers where heat management is a problem, and microchip lasers for photonics applications. Glass fiber lasers and amplifiers are becoming important configurations for some applications. A major problem is the inefficiency of coupling the excitation energy of a lamp source with a broad spectral output into the spectrally sharp absorption bands of the trivalent rare earth ions. This can be overcome by using a laser as a pump source. One of the major advances in solid state laser technology has been the development of bars of high power diode laser arrays as pump sources. This has significantly increased the efficiency and decreased the thermal problems in these lasers. Currently available diode laser pump sources cover a limited range of wavelengths and thus can only be used to excite a limited set of metastable states. Several schemes have been adapted to excite other metastable states. One of these is up-conversion pumping [71.12–15] in which an

71.2 Solid State Lasers

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A major area of recent development involves rare earth doped crystal lasers operating at specific wavelengths in the near IR [71.19–21]. This includes both the “eye safe” region between 1.35 and 2.2 µm for applications involving atmospheric transmission, and the 2 to 3 µm range to match water overtone absorption bands for medical applications. The ions of most interest are Er3+ , Tm3+ , and Ho3+ . Er3+ can lase on 13 transitions at wavelengths ranging from 0.56 to 4.8 µm. Both Tm3+ and Ho3+ exhibit laser transitions near 2 µm, but they are difficult to pump efficiently. One successful laser in this spectral region is triply doped Cr;Tm;Ho:YAG. The flashlamp energy is absorbed by the Cr3+ ions and transferred to Tm3+ ions. Cross-relaxation between two thulium ions doubles the quantum efficiency by leaving two Tm3+ ions in the excited state. Energy transfer among the thulium ions with transfer to Ho3+ then occurs and the lasing transition occurs on the holmium ions. Fiber lasers have been developed that consist of trivalent rare earth ions doped in either oxide or fluoride glass fibers and pumped by other lasers [71.22]. A useful application of active fibers is Er3+ and Pr3+ optical amplifiers for fiber communication systems. The extended length of the gain media and the nonlinear dispersion effects in fiber transmission allow precise tailoring of laser emission properties. Heavy metal fluoride fibers give improved IR transmission in the 2 to 3 µm spectral region. An important recent development involves writing laser-induced gratings in fibers to produce distributed feedback lasers with stable, single-mode operation [71.23]. Nd-doped fiber lasers pumped by diode lasers have produced 5 W of power. Mode-locked fiber ring lasers have produced solitons. Efficient up-conversion laser operation has been achieved in fibers. In addition, the efficient nonlinear optical properties of fibers has led to fiber Raman lasers.

Part F 71.2

71.2.3 Color Center Lasers In color center lasers, the optically active center is a point defect in the lattice. For example, in alkali halide host crystals, such as NaCl, a typical color center consists of an electron trapped at a halide ion vacancy. Similar color centers occur in oxide host crystals such as diamond and sapphire. Color centers can be produced by thermal treatment or exposure to radiation. Many times, these centers are stable only at low temperatures due to ion and electron mobility. In some cases, impurity ions act to stabilize the de-

fect center. A neutral Tl atom at a cation site next to a anion vacancy in KCl is an example of this. A major recent advance in color center lasers involves the development of room temperature stable pulsed laser systems. Systems based on the vibrational transitions of molecular defects such as CN− have been demonstrated to operate as lasers in the 5 µm spectral region. Color center absorption generally occurs in the visible, and they are optically pumped by Ar, Kr, or Nd:YAG lasers. Typical color center emission occurs as a broad band in the near IR between 0.8 and 4.0 µm. The emission is based on allowed transitions with high oscillator strengths leading to high gain cross sections. The homogeneously broadened emission band of color centers allows for efficient, tunable laser emission and single mode operation. Optically pumped cw output powers of ≈ 2 W have been obtained, and modelocked pulses of less than 100 fs and 1 MW peak power at repetition rates of 100 MHz and hundreds of milliwatts average power have been generated. Laser linewidths of less than 4 kHz have been obtained. One problem with a high gain medium such as a color center crystal is a tendency for multimode laser operation. In a linear standing wave cavity, a primary oscillating mode reaches gain saturation and burns spatially periodic holes in the population inversion of the gain medium. The high gain allows secondary modes to oscillate with peaks at the nodes of the primary mode. A grating/etalon combination can be used to select and tune the laser output frequency [71.4]. The etalon selects one cavity mode and the grating selects one order of the etalon. This results in single mode tunable output. The single mode power output is 70% of the multimode laser power due to energy loss in the hole burning mode. A ring laser configuration in a traveling wave operation can be used to give uniform saturation of the gain medium, and thus no hole burning. A ring laser cavity needs additional optics such as a Faraday rotator and an optically active plate to force oscillation in only one direction. Active frequency stabilization circuits are necessary to obtain linewidths of less than 4 kHz. Synchronously pumping a color center laser with a mode-locked Nd:YAG laser gives mode-locked output with pulses typically between 5 and 15 ps. If passive mode-locking is obtained through use of a saturable absorber, pulses of ≈ 200 fs can be obtained. Additive pulse mode-locking consists of two coupled cavities, one with a color center gain medium and the other with a single mode optical fiber. Self phase modulation in the fiber gives a broader frequency spectrum to the pulses

Types of Lasers

and thus shorter time widths. This scheme has generated pulses of about 75 fs [71.24]. Soliton lasers are also obtained by coupling a color center laser with a fiber laser.

71.2.4 New Types of Solid State Laser Systems One type of new solid state laser system being studied involves the fourth and fifth row transition metal ions. So far, stimulated emission and gain have been reported for Rh3+ [71.25] under strong pumping conditions, but no laser operation has been achieved. Solid state dye lasers [71.26] consist of organic dye molecules doped in crystal or glass host materials. The first systems of this type used the same class of organic dyes as in liquid lasers (such as rhodamine 6G) and host materials such as sol-gels or polymethylmethacrylate. These systems have traditionally had a problem with photo-degradation of the material after a limited number of shots. However, recent combinations of new dyes and new host materials have produced outputs of tens of millijoules per pulse, over 50% slope efficiency, and a degradation to 60% of the initial output after 30 000 pulses. One example of these new materials is pyrromethene-BF2 complex dye doped in an acrylic plastic host [71.26]. Another material system that has exhibited good laser performance involves xerogel hosts doped with perylene or pyrromethene dyes [71.27]. The performance of some of these new systems has reached the point that they may be useful for tunable laser applications in the visible spectral region. Another new type of system can be described as a solid state excimer laser [71.6–8]. These materials

71.3 Semiconductor Lasers

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consist of noble gas solids as host crystals, such as Ar and Ne, doped with excimer molecules, such as XeF. The emission lines occur in the UV and visible, with major lines at 286, 411, and 540 nm. These systems have very high stimulated emission cross sections and large gain coefficients.

71.2.5 Frequency Shifters Since solid state lasers are commercially available at only a limited number of wavelengths, it is sometimes easier to use nonlinear optical techniques to shift the frequency of an available laser than to develop a new primary laser system. The techniques for this include harmonic frequency generation, frequency mixing, optical parametric oscillators and amplifiers, and Raman shifting. Significant advances have been made recently [71.28] in developing new types of materials for these applications. For frequency mixing, harmonic generation and OPOs in the visible and near UV, important new crystals include KTiOPO4 (KTP), BaB2 O4 (BBO), and LiB3 O5 (LBO). In the 3 to 5 µm region, new materials for frequency mixing and OPOs include KTA (the arsinate analog of KTP) and ZnGeP2 . Gas phase Raman cells have been commercially available for solid state laser systems for many years. Recently, it has been demonstrated that crystals such as Ba(NO3 )2 can be used as efficient solid state Raman shifters. New waveguide configurations with periodic poling for quasiphase matching have greatly enhanced the efficiency of harmonic generation [71.29]. Optical damage threshold is still the limiting parameter for nonlinear optical materials.

71.3 Semiconductor Lasers lattice defects (inhomogeneous broadening) and radiationless relaxation and scattering processes associated with the thermal vibration of the host (homogeneous broadening). Temperature tuning can be used to change the output wavelength over a narrow spectral range. Direct modulation of the laser output can be achieved by modulating the external current. This is an important feature of electric current pumped semiconductor lasers, and leads to applications where high frequency modulation is required. Modulation bandwidths in excess of 11 GHz have been obtained. The ability to design and grow specialized structures one atomic layer at a time using techniques such as molecular beam epitaxy (MBE) has led to the design of

Part F 71.3

The light emission from semiconductor diode lasers is generally associated with the radiative recombination of electrons and holes. This occurs at the junction of an n-type material with excess electrons and a p-type material with excess holes. The excitation is provided by an external electric field applied across the p-n junction that causes the two types of charges to come together. The most common semiconductor laser emission lines occur in the near or mid-IR. These are generally made of III-V compounds such as gallium arsenide in the red and near IR, and lead salts in the mid-IR region. Wide bandgap II-VI materials are currently being explored for use in the green and blue spectral regions. The spectral lines are generally narrow with broadening due to

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Quantum Optics

Part F 71.4

quantum well lasers with enhanced properties, as well as more esoteric designs of quantum wires and quantum boxes [71.30, 31]. Quantum well microlasers can be as small as 100 µm and operate with milliamps of current at a few volts. These are generally based on material systems such as GaAs/GaAlAs or InP/InGaAsP. Heterostructure lasers have layers of aluminum, indium, and phosphorus on the sides of the junction to confine the electronic current to the junction region to minimize the amount of current required, and thus minimize heat dissipation compared with homostructures of the same materials. Special device structures can be fabricated to produce gain-guiding and index-guiding to enhance the operating characteristics of the lasers. Grating structures can be fabricated to give distributed feedback lasers that narrow the laser linewidths to ≈ 1 MHz. External cavity lasers with gratings have achieved linewidths as low as 1 kHz. cw operating powers of up to 10 mW have been achieved from a single p-n junction, while phased arrays have reached combined powers of well over 10 W. Gallium arsenide (GaAs) was the first compound semiconductor diode laser. It can produce laser emission at wavelengths between 750 and 870 nm. The development of strained-layer technology has allowed the use of mixed compounds of gallium aluminum arsenide (Ga1x Alx As) to fabricate different types of laser structures, and the concentration of aluminum determines the laser emission wavelength. Wavelengths from 620 nm to 905 nm have been obtained. The most common diode laser structures are simple double-heterostructure lasers, and monolithic arrays of laser stripes can be fabricated for higher power. In a typical GaAlAs laser, the active layer is sandwiched between two layers having larger bandgaps and lower refractive indices. The former characteristic produces electrical confinement and the latter produces optical confinement. This improves the efficiency and allows cw laser operation. In the conventional horizontal-cavity structure, cleaved end facets

of the chip produce the optical feedback required for laser oscillation. Fabricating quantum well structures in the active layer produces improved confinement, and thus higher efficiency operation. For low power lasers, high beam quality is achieved through an index-guiding structure that concentrates the optical beam in the laser stripe. In high power lasers, the current is concentrated in the laser stripe to achieve gain-guiding. Laser arrays can generate cw powers of the order 20 W and in a quasicw mode they can produce peak powers of 100 W. Stacking diode bars in planar arrays can generate kilowatts of power. Another approach to obtaining high powers is a master oscillator power amplifier (MOPA) configuration. This has the advantage of maintaining high beam quality, and gallium arsenide MOPAs have produced single frequency operation at 1 W of power. Long wavelength IR diode lasers are made of IV–VI compounds such as PbS. These lasers operate at cryogenic temperatures and provide tunable emission from 4 to 32 µm. The tunability is achieved by changing temperature or current. One of the most important areas of research in semiconducting lasers is the development of new device configurations such as vertical cavity surface emitting lasers (VCSELs) [71.32]. These have lower round trip gain but significantly reduced divergence of the output beam. This configuration allows for the fabrication of two dimensional arrays of independently modulated lasers. Another major research area is generating new laser wavelengths. Using strained layer technology, a variety of different combinations of direct bandgap materials can be made into semiconductor lasers [71.33]. The range of available bandgaps can conceivably result in lasers with emission wavelengths spanning the visible and near IR spectral regions. There is currently special emphasis on the development of lasers in the blue and green spectral regions using wide bandgap II–VI materials such as ZnSe [71.34].

71.4 Liquid Lasers There are three classes of liquid lasers. The most widely used are based on organic solvents with organic dye molecules as the active laser species. The other two types are based on rare earth ions for the lasing entity. In one case, the lasing system involves rare earth chelates in organic solvents, while in the other, the rare earth ions are in inorganic solvents. These systems are optically pumped with either flashlamps or other lasers.

71.4.1 Organic Dye Lasers Dye lasers are generally based on fluorescent dyes in liquid solvents, using optical pumping by either flashlamps or other lasers as the mechanism for excitation. They can operate in either a pulsed or cw mode at wavelengths as short as 310 nm out to about 1.5 µm. Dye lasers provide the versatility of

Types of Lasers

Several new technological advances have increased the spectral coverage of dye lasers. These include better pump sources, such as the increased power of argon pump lasers and the availability of UV pump lines, and the use of Ti-sapphire pump lasers. Combining these pump sources with new dyes has provided extended dye laser output in both the blue and near IR [71.4]. Also, improved nonlinear crystals have allowed coverage of the near UV from 260 to 960 nm through harmonic frequency generation and frequency mixing the dye laser output with pump laser wavelengths. Another developing technology is solid state dye lasers. As mentioned in Sect. 71.2.4, significant progress has been made recently in decreasing the photodegradation problems associated with dye molecules doped in solid host materials [71.26, 27].

71.4.2 Rare Earth Chelate Lasers In these systems, the active lasing center is a rare earth complex with organic molecules in an organic solvent. The chelate ligands are organic phosphates, carboxylate ions, or β-diketonate. The optical pump energy is absorbed by the ligand and efficiently transferred to the rare earth ion. Energy transfer quenching from the rare earth ion to the organic molecule vibrational levels decreases the efficiency of these lasers. This is especially true for Nd3+ . The three ions that have been most effective in these systems are Eu3+ , Tb3+ , and Nd3+ .

71.4.3 Inorganic Rare Earth Liquid Lasers In these systems, the active lasing center is a rare earth ion inorganic complex of heavy metal halides or oxyhalides (the optical pumping is directly into the rare earth ion). Nd3+ lasers can produce several hundred joules of energy per pulse in the long pulse mode and peak powers of 180 MW in a Q-switched mode. These lasers have also been mode-locked to obtain 3 ps long pulses having 1 GW of peak power. Self mode-locking and self Q-switching is also observed in these systems.

71.5 Other Types of Lasers Several more complex laser systems have been developed that have significant interest for scientific studies, but so far have had limited applications outside the laboratory. These include X-ray lasers, particle-beampumped lasers, and free electron lasers.

1045

71.5.1 X-Ray and Extreme UV Lasers These systems are based on highly ionized ions produced by powerful laser sources. Examples are krypton at 93 nm, molybdenum at 13 nm, and carbon at 18 nm.

Part F 71.5

varying wavelength, bandwidth, and pulse length as desired. The fluorescence emission of dye molecules appears as broad spectral bands due to coupling of the electronic energy levels with molecular vibrations. This gives a broadband gain curve for lasing, and thus with a dispersive element such as a grating, prism, filter, or etalon in the cavity, dye laser outputs can be tuned over a range of several hundred angstroms (30–60 nm). There are now over 200 organic laser dyes. One of the most successful dyes is rhodamine 6G which covers the spectral range 570–630 nm. Alcohol is a typical solvent. The various types of cavity designs include folded cavities and ring cavities. Flowing dye configurations are useful for heat management. Oscillator/amplifier configurations are used to suppress amplified spontaneous emission. Dispersive elements plus frequency stabilization have been used for spectral line narrowing to single mode cw operation. Frequencies as narrow as 10 GHz or less have been obtained. For cw operation, output powers of a few watts can be obtained, while in the pulsed mode the energy per pulse can be up to 100 mJ in 10 ns pulses. With mode-locking, trains of femtosecond pulses with intervals of 20 ns can be produced. Mode-locking takes advantage of the broad emission spectrum of the dye molecules to get short pulses. The standard techniques of synchronous pumping, active, and passive mode-locking have been used. In addition, colliding pulse mode-locking has been used in a ring configuration. In this case, pump beams going the opposite direction in the cavity collide in an absorber jet dye to produce interference fringes and saturation. The gain dye is located half way around the ring from the absorber dye. Fiber compression techniques have also been used with dye laser systems. The shortest pulses obtained so far are 6 fs [71.4]. Hybrid mode-locking utilizing synchronous pumping plus a saturable absorber and prism dispersion compensators has been employed to achieve powers of 350 mW and greater tunability than colliding pulse systems [71.4].

71.5 Other Types of Lasers

1046

Part F

Quantum Optics

There is significant interest in developing X-ray lasers for applications in lithography and medical imaging, but so far the lack of reliable X-ray optics for the laser cavities has limited the technology.

71.5.2 Nuclear Pumped Lasers These are gas lasers excited by high energy charged particles or gamma rays resulting from nuclear reactions. Either nuclear reactors or nuclear explosives are used as pump sources. These can operate in either a pulsed or cw mode and produce emission covering the spectral range from the UV through the IR [71.4]. Typical gases range from Xe∗2 at 170 nm to CO at 5.4 µm. Other gases that have been used include argon and nitrogen. Typical pulsed outputs produce 10 ns pulses with energies from 2 × 10−7 J to 3 J. There are significant problems with radiation damage of the laser components in these systems.

71.5.3 Free Electron Lasers These lasers are based on a high-energy beam of electrons in a spatially varying magnetic field. The varying field causes the electrons to oscillate, and thus to emit radiation at the oscillation frequency. The stimulated emission produced under these conditions provides the laser output. Since the electrons are making transitions between continuum states instead of discrete states, these systems can give high power output over the entire spectral range from the VUV to the far IR [71.35, 36]. Powers as high as 1 GW and efficiencies as high as 35%

have been obtained. Beam spread is controlled by the use of tapered instead of uniform wigglers. Both cw and picosecond pulsed emission can be obtained. There are three types of free electron laser configurations [71.35, 36]. The first is a master oscillator power amplifier (MOPA), in which an electron beam is injected into a wiggler in synchronism with the signal to be amplified. The external radiation source to drive the amplifier is a master oscillator such as a conventional laser system. This is a single pass, high gain system. The second configuration is an oscillator. This is designed with reflection at the ends of the wiggler so that the signal makes multiple passes in the cavity. This can operate with low gain. Since it amplifies spontaneous noise, no injected signal is necessary. The third configuration is a superradiant amplifier. In this configuration, shot noise is amplified over a single pass through the wiggler. These systems require high current accelerators to drive them. Because their operation is based on broad band shot noise, super-radiant amplifier radiation has a broader band than radiation from a master oscillator power amplifier. Different accelerator configurations can be used to produce the electron beam, storage rings, induction linacs, pulse line accelerators, etc. [71.35, 36]. These give different beam properties such as quality, current, and energy. They also each give a limited range of wavelengths and temporal structure of the output. Advances have been made recently in designing smaller and less complex free electron lasers [71.35,36]. As this trend continues, these systems will find important applications in medicine and industry.

71.6 Recent Developments

Part F 71.6

Over the past eight years, the designs of all types of lasers have continued to evolve, driven by applications requirements for lasers with specific operating parameters. Some of the major advances in this time period are summarized here. The requirements for laser outputs of several kilowatts with near-diffraction-limited beam quality in the infrared wavelength region has lead to improved designs of CO2 lasers. This includes a diffusion cooled, annular discharge design with free-space propagation instead of waveguiding [71.37]. The push for solid state lasers in the ultraviolet has led to recent progress in cerium-doped fluoride crystal lasers that provides direct laser emission tunable in the 280 to 330 nm spectral region. Using host crystals such as LiCaAlF6 , LiSrAlF6 or LiLuF4 has helped

to overcome the problems with excited state absorption and color center formation that has been a major problem with more common hosts such as YAG and YLF crystals [71.38]. These lasers are pumped by either excimer lasers or frequency-quadrupled Nd:YAG lasers and have produced up to 60 mJ per pulse and 60% slope efficiency. The output can be pulse compressed to 115 fs. Thin-disk solid state laser configurations using rare earth doped crystals and semiconductor saturable absorber mirrors for passive mode-locking have produced ultrashort (femtosecond) pulse trains with high average powers [71.39]. Yb:YAG thin disks reduce the problem of thermal lensing and have achieved output powers of up to 100 W in cw mode locked operation with neardiffraction-limited beam quality [71.40]. To achieve

Types of Lasers

tor. In this case the frequency doubling crystal is outside the cavity. These configurations have reached 20 mW cw operation with high beam quality. One enabling technology is the use of microelectromechanical systems (MEMS) for mirrors to tune VCELs [71.56]. New material configurations offer some advantages. Photonic crystals can be used to produce nanocavity lasers [71.57] while band-structure engineering can be used to design quantum-cascade lasers [71.58]. The latter are multiple-quantum-well heterostructures based on intersub-band transitions. They operate in the infrared to terahertz spectral region. The broad spectral band available for laser gain is the major distinguishing feature of liquid organic dye lasers. This provides the ability to have tunable output over a broad range of wavelength, the ability to generate ultrashort pulses, narrow linewidth cw operation, and high average power operation. This variety of operating parameters keeps this class of lasers competitive in a variety of applications. They have been especially useful for laser spectroscopy in the visible region of the spectrum and in the area of laser cooling. They have achieved energies of up to 800 J per pulse and average powers greater than 1 kW. The discovery of highly stable water-soluble dyes is an important advancement in this field [71.59]. Free-electron lasers (FELs) can produce coherent emission at a wide range of wavelengths. Output formats include ultrashort pulses and high powers [71.60]. Using a photocathode electron gun in a single pass, selfamplified spontaneous emission mode, FELs can operate at wavelengths where there are no mirrors with high reflectivity. This type of laser emission has been extended to the vacuum ultraviolet and the hard X-ray regions. The technique of using a subharmonic seed laser has been developed to improve the spectral purity of FEL emission. The use of an energy-recovering accelerator produces FELs with high average power (over 1 MW). In the spectral region around 1 mm, FELs have produced over 300 W of picosecond pulses. Laser systems using various nonlinear optics techniques continue to be developed. There has been significant research on high intensity, ultrashort pulse lasers because of their special characteristics for atmospheric propagation. Using femtosecond pulses above a critical power level produces “light strings” that propagate without dispersion for many kilometers due to the balance between Kerr self-focusing and air ionization [71.61]. Stimulated Brillouin scattering in water has been shown to be effective in pulse compression and the production of nondiffracting laser beams [71.62].

1047

Part F 71.6

shorter pulses, a thin-disk Yb:KYW laser has obtained 22 W with 240 fs pulses [71.41]. Advances continue to be made in the materials for solid state organic dye lasers. The use of polymer materials, organic-inorganic matrices, and nanoparticles has provided advances in this type of laser [71.42–46]. Use of semiconductor excitation and electrical excitation may play important roles in the future of solid state organic dye lasers. Commercially available solid state Raman lasers have been developed for frequency shifting to a wide variety of wavelengths and for pulse compression to the 0.1–1 ns region [71.47–49]. Both internal and external cavity designs have been demonstrated. New materials, such as KGd(WO4 )2 and KY(WO4 )2 , have been used for Raman lasers along with Ba(NO3 )2 , which has excellent properties for this application. An externalresonator Raman laser using Ba(NO3 )2 has reached 1.3 W of power [71.50]. The major advancement in fiber lasers has been in cladding geometry to allow for higher powers. Double clad fiber lasers operating in single transverse modes have exceeded 100 W of output for four-level systems and over 1 W for three-level systems [71.51]. These involve a variety of geometric shapes of cladding. A new breakthrough in fiber delivery systems that will impact the future of fiber lasers is the use of holey fibers (photonic crystal fibers) [71.52]. These are glass fibers that have a periodic array of air holes running their entire length. These fibers can be engineered to produce a photonic bandgap and allow for dispersion control and minimized nonlinear effects compared to standard fibers. This is useful for short pulse delivery. In the field of semiconductor lasers, vertical-cavity surface-emitting lasers (VCSELs) have developed as a competitive alternative to the conventional edgeemitting semiconductor lasers. The most recent advance involves designs that have a horizontal laser cavity but emits from the surface [71.53]. This combines the ease of packaging of VCSELs with the high power and good stability properties of edge-emitting lasers. Several new designs of high power semiconductor lasers that can be frequency shifted to the blue and green spectral regions have been developed as rugged, efficient sources [71.54, 55]. One of these is a GaAs-based vertical external cavity surface-emitting laser (VECSEL) optically pumped with an 808 nm semiconductor laser. This emits at 976 nm with a cavity that includes a wavelength selector and a doubling crystal. The second configuration uses a semiconductor material as a gain medium in an external cavity with a wavelength selec-

71.6 Recent Developments

1048

Part F

Quantum Optics

References 71.1 71.2 71.3 71.4 71.5 71.6 71.7 71.8 71.9 71.10 71.11 71.12 71.13 71.14 71.15 71.16 71.17 71.18 71.19 71.20 71.21 71.22 71.23

Part F 71

71.24 71.25 71.26 71.27

71.28

M. J. Weber (Ed.): Handbook of Laser Science and Technology, Vol. 1 (CRC, Boca Raton 1982) M. J. Weber (Ed.): Handbook of Laser Science and Technology (CRC, Boca Raton 1991) M. Bass, M. L. Stitch (Eds.): Laser Handbook, Vol. 5 (North Holland, Amsterdam 1985) R. A. Meyers (Ed.): Encyclopedia of Lasers and Optical Technology (Academic Press, San Diego 1991) P. K. Cheo (Ed.): Handbook of Solid State Lasers (Marcel Dekker, New York 1989) N. Schwentner, V. A. Apkarian: Chem. Phys. Lett 154, 413 (1989) G. Zerza, G. Sliwinski, N. Schwentner: Appl. Phys. B55, 331 (1992) G. Zerza, G. Sliwinski, N. Schwentner: Appl. Phys. A56, 156 (1993) W. H. Crumly, J. L. Gole, D. A. Dixon: J. Chem. Phys. 76, 6439 (1982) S. H. Cobb, J. R. Woodward, J. L. Gole: Chem. Phys. Lett. 143, 205 (1988) S. H. Cobb, J. R. Woodward, J. L. Gole: Chem. Phys. Lett. 157, 197 (1989) R. M. Macfarlane, F. Tong, A. J. Silversmith, W. Lenth: Appl. Phys. Lett. 52, 1300 (1988) R. A. Macfarlane: Appl. Phys. Lett. 54, 2301 (1989) T. Hebert, R. Wannemacher, W. Lenth, R. M. Macfarlane: Appl. Phys. Lett. 57, 1727 (1990) R. A. Macfarlane: Opt. Lett. 16, 1397 (1991) M. E. Koch, A. W. Kueny, W. E. Case: J. Appl. Phys. 56, 1083 (1990) N. Karayianis, D. E. Wortman, C. A. Morrison: Solid State Comm. 18, 1299 (1976) W. F. Krupke, M. D. Shinn, J. E. Marion, J. A. Caird, S. E. Stokowski: J. Opt. Soc. Am. B3, 102 (1986) M. J. Weber, M. Bass, G. A. deMars: J. Appl. Phys. 42, 301 (1971) G. J. Quarles, A. Rosenbaum, C. L. Marquardt, L. Esterowitz: Opt. Lett. 15, 42 (1990) L. Esterowitz: Opt. Eng. 29, 676 (1990) P. Urquhart: IEE Proc. J135, 385 (1988) I. M. Jauncey, L. Reekie, R. J. Mears, D. N. Payne, C. J. Rowe, D. C. J. Reid, I. Bennion, C. Edge: Electron. Lett. 22, 987 (1986) L. F. Mollenhauer, R. H. Stolen: Opt. Lett. 9, 12 (1984) R. C. Powell, G. J. Quarles, J. J. Martin, C. A. Hunt, W. A. Sibley: Opt. Lett. 10, 212 (1985) R. E. Hermes, T. H. Allik, S. Chandra, J. A. Hutchinson: Appl. Phys. Lett. 63, 877 (1993) B. Dunn, F. Nishida, R. Toda, J. I. Zink, T. H. Allik, S. Chandra, J. A. Hutchinson: Mat. Res. Soc. Symposium, Proc. 329, 267 (1994) V. G. Dmitriev, G. G. Girzodyan, D. N. Nikogosyan: Handbook of Nonlinear Optical Crystals (Springer, Berlin, Heidelberg 1991)

71.29 71.30 71.31 71.32 71.33

71.34 71.35 71.36 71.37 71.38 71.39

71.40 71.41

71.42 71.43

71.44 71.45 71.46 71.47

71.48

71.49 71.50 71.51

71.52 71.53 71.54

E. J. Lim, M. M. Fejer, R. L. Byer, W. J. Kozlovsky: Electron. Lett. 25, 731 (1989) Y. Arakawa, K. Vahala, A. Yariv: Surf. Sci. 174, 155 (1986) K. Vahala: IEEE J. Quantum Electron. 24, 523 (1988) R. E. Slusher: Opt., Photon. News 4, 8 (1993) W. W. Chow, S. W. Koch, M. II. I. Sargent: Semiconductor Laser Physics (Springer, Berlin, Heidelberg 1994) M. A. Hasse, J. Qui, J. M. DePuydt, H. Cheng: Appl. Phys. Lett. 58, 1272 (1991) C. A. Brau: Free Electron Lasers (Academic Press, San Diego 1990) H. P. Freund, G. R. Neil: Proc. IEEE. 87, 782 (1999) A. Lapucci, F. Rossetti, P. Burlamacchi: Opt. Com 111, 290 (1994) A. J. S. McGonigle, D. W. Coutts: Laser Focus World 39, 127 (2003) E. Innerhofer, T. Südmeyer, F. Brunner, R. Häring, A. Aschwanden, R. Paschotta, U. Keller, C. Hönninger, M. Kumkar: Opt. Lett. 28, 376 (2003) A. Giesen, H. Hügel, A. Voss, K. Wittig, U. Brauch, H. Popwer: Appl. Phys. B 58, 363 (1994) F. Brunner, T. Südmeyer, E. Innerhofer, R. Paschotta, F. Morier-Genoud, J. Gao, K. Contag, A. Giesen, V. E. Kisel, V. G. Shcherbitsky, N. V. Kuleshov, U. Keller: Opt. Lett. 27, 1162 (2002) A. Costela, I. Garcia-Moreno, J. M. Figuera, F. Amat-Guerri, R. Sastre: Laser Chem. 18, 63 (1998) I. Braun, G. Ihlein, J. U. Nöckel, G. Schulz-Ekloff, F. Schüth, U. Vietze, D. Wöhrle: Appl. Phys. B 70, 335 (2000) F. J. Duarte: Appl. Opt. 38, 6347 (1999) W. J. Wadsworth, I. T. McKinnie, A. D. Woolhouse, T. G. Haskell: Appl. Phys. B 69, 163 (1999) X. Zhu, S. K. Lam, D. Lo: Appl. Opt. 39, 3104 (2000) J. T. Murray, R. C. Powell, N. Peyghambarian, D. Smith, W. Austin, R. A. Stolzenberger: Opt. Lett. 20, 1017 (1995) A. A. Kaminskii, H. G. Eichler, K. Ueda, N. V. Klassen, B. S. Redkin, L. E. Li, J. Findeisen, D. Jaque, J. Garcia-Sole, J. Fernandez, R. Balda: Appl. Opt. 38, 4533 (1999) P. G. Zverev, T. T. Basiev, A. M. Prokhorov: Opt. Materials 11, 335 (1999) H. M. Pask, S. Myers, J. A. Piper, J. Richards, T. McKay: Opt. Lett. 28, 435 (2003) L. A. Zenteno, J. D. Minelly, A. Liu, A. J. G. Ellison, S. G. Crigler, D. T. Walton, D. V. Kuksenkov, M. J. Dejneka: Electron. Lett. 37, 819 (2001) H. Sabert, J. Knight: Photonics Spectra 37, 92 (2003) N. Anscombe: Photonics Spectra 37 60 (2003) E. H. Wahl, B. A. Richman, C. W. Rella, G. M. H. Knippels, B. A. Paldus: Opt., Photonics News 14, 36 (2003)

Types of Lasers

71.55 71.56 71.57 71.58

71.59

S. Lutgen, T. Albrecht, P. Brick, W. Reill, J. Luft, J. Späth: Appl. Phys. Lett. 82, 3620 (2003) J. Hecht: Laser Focus World 37, 121 (2001) G. G. Park, J. K. Hwang, J. Huh, H. Y. Ryu, Y. H. Lee: Appl. Phys. Lett. 79, 3032 (2001) R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, S. S. Dhillon, C. Sirtori: Appl. Phys. Lett. 82, 1518 (2003) F. J. Duarte: Opt., Photonics News 14, 20 (2003)

71.60 71.61

71.62 71.63

References

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H. P. Freund, P. O’Shea: Science 292, 1853 (2001) F. Courvoisier, V. Boutou, J. Kasparian, E. Salmon, G. Mejean, J. Yu, J.-P. Wolf: Appl. Phys. Lett. 83, 213 (2003) F. Brandi, I. Velchev, D. Neshev, W. Hogervorst, W. Ubachs: Rev. Sci. Inst. 74, 32 (2003) A. Costela, I. Garcia-Moreno, C. Gomez, O. Garcia, R. Sastre: Phys. Lett. 369, 656 (2003)

Part F 71

1051

Nonlinear Opt 72. Nonlinear Optics

Nonlinear optics is concerned with the propagation of intense beams of light through a material system. The optical properties of the medium can be modified by the intense light beam, leading to new processes that would not occur in a material that responded linearly to an applied optical field. These processes can lead to the modification of the spectral, spatial, or polarization properties of the light beam, or the creation of new frequency components. More complete accounts of nonlinear optics including the origin of optical nonlinearities can be found in references [72.1–4]. Both the Gaussian and MKS system of units are commonly used in nonlinear optics. Thus, we have chosen to express the equations in this chapter in both the Gaussian and MKS systems. Each equation can be interpreted in the MKS system as written or in the Gaussian system by omitting the prefactors (e.g., 1/4πε0 ) that appear in square brackets at the beginning of the expression on the right-hand-side of the equation.

72.1

72.2

Nonlinear Susceptibility ....................... 1051 72.1.1 Tensor Properties ...................... 1052 72.1.2 Nonlinear Refractive Index ......... 1052 72.1.3 Quantum Mechanical Expression for χ(n) .................................... 1052 72.1.4 The Hyperpolarizability .............. 1053 Wave Equation in Nonlinear Optics........ 1054 72.2.1 Coupled-Amplitude Equations .... 1054

72.2.2 Phase Matching ........................ 1054 72.2.3 Manley–Rowe Relations ............. 1055 72.2.4 Pulse Propagation ..................... 1055 72.3

Second-Order Processes ....................... 1056 72.3.1 Sum Frequency Generation......... 1056 72.3.2 Second Harmonic Generation ..... 1056 72.3.3 Difference Frequency Generation ............... 1056 72.3.4 Parametric Amplification and Oscillation.......................... 1056 72.3.5 Focused Beams ......................... 1056

72.4 Third-Order Processes .......................... 1057 72.4.1 Third-Harmonic Generation ....... 1057 72.4.2 Self-Phase and Cross-Phase Modulation .............................. 1057 72.4.3 Four-Wave Mixing ..................... 1058 72.4.4 Self-Focusing and Self-Trapping . 1058 72.4.5 Saturable Absorption ................. 1058 72.4.6 Two-Photon Absorption ............. 1058 72.4.7 Nonlinear Ellipse Rotation .......... 1059 72.5 Stimulated Light Scattering .................. 1059 72.5.1 Stimulated Raman Scattering...... 1059 72.5.2 Stimulated Brillouin Scattering ... 1060 72.6 Other Nonlinear Optical Processes ......... 1061 72.6.1 High-Order Harmonic Generation 1061 72.6.2 Electro-Optic Effect.................... 1061 72.6.3 Photorefractive Effect ................ 1061 72.6.4 Ultrafast and Intense-Field Nonlinear Optics ....................... 1062 References .................................................. 1062

72.1 Nonlinear Susceptibility frequency components such that
˜ t) = E(r, E(r, ωl ) e−iωl t ,

(72.1)

l

˜ t) = P(r,




P(r, ωl ) e−iωl t ,

(72.2)

l

where the summations are performed over both positive and negative frequencies. The reality of E˜ and P˜ is then

Part F 72

In linear optics it is customary to describe the response of a material in terms of a macroscopic polarization P˜ (i. e., dipole moment per unit volume) which is linearly related to the applied electric field E˜ through the linear susceptibility χ (1) . In order to extend the relationship between P˜ and E˜ into the nonlinear regime, the polarization is expanded in a power series of the electric field strength. We express this relationship mathematically by first decomposing the field and the polarization into their

1052

Part F

Quantum Optics

assured by requiring that E(r, ωl ) = E∗ (r, −ωl ) and P(r, ωl ) = P ∗ (r, −ωl ). In this case the general expression for the Cartesian component i of the polarization at frequency ωσ is given by 
Pi (ωσ ) = [ε0 ] χij(1) (ωσ )E j (ωσ ) j

+





(2) χijk (ωσ ; ωm , ωn )

jk (mn)

× E j (ωm )E k (ωn )

(3) + χijkl (ωσ ; ωm , ωn , ωo )E j (ωm ) jkl (mno)

× E k (ωn )El (ωo )  +··· ,

(72.3)

where ijkl refer to field components, and the notation (mn), for example, indicates that the summation over n and m should be performed such that ωσ = ωm + ωn is held constant. Inspection of (72.3) shows that the χ (n) can be required to satisfy intrinsic permutation symmetry, i. e., the Cartesian components and the corresponding frequency components [e.g., ( j, ω j ) but not (i, ωσ )] associated with the applied fields may be permuted without changing the value of the susceptibility. For example, for the second-order susceptibility, (2) (2) χijk (ωσ ; ωm , ωn ) = χik j (ωσ ; ωn , ωm ) .

(72.4)

If the medium is lossless at all the field frequencies taking part in the nonlinear interaction, then the condition of full permutation symmetry is necessarily valid. This condition states that the pair of indices associated with the Cartesian component and the frequency of the nonlinear polarization [i. e., (i, ωσ )] may be permuted along with the pairs associated with the applied field components. For example, for the second-order susceptibility, this condition implies that (2) χijk (ωσ ; ωm , ωn ) = χk(2) j i (−ωn ; ωm , −ωσ ) . (72.5)

Part F 72.1

If full permutation symmetry holds, and in addition all the frequencies of interest are well below any of the transition frequencies of the medium, the χ (n) are invariant upon free permutation of all the Cartesian indices. This condition is known as the Kleinman symmetry condition.

72.1.1 Tensor Properties The spatial symmetry properties of a material can be used to predict the tensor nature of the nonlinear susceptibility. For example, for a material that possesses inversion symmetry, all the elements of the evenordered susceptibilities must vanish (i. e., χ (n) = 0 for n even). The number of independent elements of the nonlinear susceptibility for many materials can be substantially fewer than than the total number of elements. For example, in general χ (3) consists of 81 elements, but for the case of isotropic media such as gases, liquids, and glasses, only 21 elements are nonvanishing and only three of these are independent. The non(3) vanishing elements consist of the following types: χiijj , (3) (3) χijij , and χijj i , where i = j. In addition, it can be shown that (3) (3) (3) (3) = χiijj + χijij + χijj χiiii i .

(72.6)

72.1.2 Nonlinear Refractive Index For many materials, the refractive index n is intensitydependent such that n = n0 + n2 I ,

(72.7)

where n 0 is the linear refractive index, n 2 is the nonlinear refractive index coefficient , and I = [4πε0 ]n 0 c|E|2 /2π is the intensity of the optical field. For the case of a single, linearly polarized light beam traveling in an isotropic medium or along a crystal axis of a cubic material, n 2 is related to χ (3) by   12π 2 (3) 1 χ (ω; ω, ω, ω, −ω) . n2 = 16π 2 ε0 n 20 c iiii (72.8)

For the common situation in which n 2 is measured in units of cm2 /W and χ (3) is measured in Gaussian units, the relation becomes  2 12π 2 × 107 (3) cm = χiiii (ω; ω, ω, ω, −ω) . n2 W n 20 c (72.9)

There are various physical mechanisms that can give rise to a nonlinear refractive index. For the case of induced molecular orientation in CS2 , n 2 = 3 × 10−14 cm2 /W. If the contribution to the nonlinear refractive index is electronic in nature (e.g., glass), then n 2 ≈ 2 × 10−16 cm2 /W.

Nonlinear Optics

72.1.3 Quantum Mechanical Expression for χ (n) The general quantum mechanical perturbation expression for the χ (n) in the nonresonant limit is (Under conditions of resonant excitation, relaxation phenomena must be included in the treatment, and the density matrix formalism must be used [72.4]. The resulting equation for the nonlinear susceptibility is then more complicated) χi(n) (ωσ ; ω1 , . . . 0 ···i n 

=

1 ε0



, ωn )
ρ0 (g)

N PF ~n ga

1 ···an

1 × (ωa1 g − ω1 − · · · − ωn )

72.1.4 The Hyperpolarizability The nonlinear susceptibility relates the macroscopic polarization P to the electric field strength E. A related

1053

microscopic quantity is the hyperpolarizability, which relates the dipole moment p induced in a given atom or molecule to the electric field Eloc (the Lorentz local field) that acts on that atom or molecule. The relationship between p and Eloc is pi (ωσ ) 
αij (ωσ )E loc = [ε0 ] j (ωσ ) j

+





loc βijk (ωσ ; ωm , ωn )E loc j (ωm )E k (ωn )

jk (mn)

+





γijkl (ωσ ; ωm , ωn , ωo )

jkl (mno)

 loc loc (ω )E (ω )E (ω ) + · · · , × E loc m n o j k l

(72.12)

where αij is the linear polarizability, βijk is the first hyperpolarizability, and γijkl is the second hyperpolarizability. The nonlinear susceptibilities and hyperpolarizabilities are related by the number density of molecules N and by local-field factors, which account for the fact that the field Eloc that acts on a typical molecule is not in general equal to the macroscopic field E. Under many circumstances, it is adequate to relate Eloc to E through use of the Lorentz approximation   1 4π loc P(ω) . (72.13) E (ω) = E(ω) + 4πε0 3 To a good approximation, one often needs to include only the linear contribution to P(ω), and thus the local electric field becomes Eloc (ω) = L(ω)E(ω) , (72.14)   −1

where L(ω) = ε0 ε(ω) + 2 /3 is the local field correction factor and ε(ω) is the linear dielectric constant. Since P(ω) = N p(ω), (72.3) and (72.12) through (72.14) relate the χ (n) to the hyperpolarizabilities through χij(1) (ωσ ) = L(ωσ )Nαij (ωσ ) , (72.15) (2) χijk (ωσ ; ωm , ωn ) = L(ωσ )L(ωm )L(ωn )

× Nβijk (ωσ ; ωm , ωn ) , (72.16) (3) χijk (ωσ ; ωm , ωn , ωo ) = L(ωσ )L(ωm )L(ωn )L(ωo )

× Nγijkl (ωσ ; ωm , ωn , ωo ) . (72.17)

For simplicity, the analysis above ignores the vector character of the interacting fields in calculating L(ω). A generalization that does include these effects is given in [72.5].

Part F 72.1

i n -1 i1 in 0 µiga 1 µa1 a2 · · · µan -1 an µan g (72.10) × (ωa2 g − ω2 − · · · − ωn ) · · · (ωan g − ωn ) where ωσ = ω1 + · · · + ωn , N is the density of atoms or molecules that compose the material, ρ0 (g) is the probability that the atomic or molecular population is initially in the state g in thermal equilibrium, µia11 a2 is the i 1 th Cartesian component of the (a1 a2 ) dipole matrix element, ωa1 g is the transition frequency between the states a1 and g, and P F is the full permutation operator which is defined such that the expression that follows it is to be summed over all permutations of the pairs (i 0 , ωσ ), (i 1 , ω1 ) · · · (i n , ωn ) and divided by the number of permutations of the input frequencies. Thus the full expression for χ (2) consists of six terms and that for χ (3) consists of 24 terms. In the limit in which the frequencies of all the fields are much smaller than any resonance frequency of the medium, the value of χ (n) can be estimated to be    1 2µ n (n) Nµ, (72.11) χ u ε0 ~ω0 where µ is a typical value for the dipole moment and ω0 is a typical value of the transition frequency between the ground state and the lowest-lying excited state. For the case of χ (3) in Gaussian units, the predicted value is χ (3) = 3 × 10−14 , which is consistent with the measured values of many materials (e.g., glass) in which the nonresonant electronic nonlinearity is the dominant contribution.

72.1 Nonlinear Susceptibility

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Part F

Quantum Optics

72.2 Wave Equation in Nonlinear Optics 72.2.1 Coupled-Amplitude Equations The propagation of light waves through a nonlinear medium is described by the wave equation   4π ∂ 2 1 ∂2 1 2˜ ˜ ∇ E− 2 2E = P˜ . (72.18) 4πε0 c2 ∂t 2 c ∂t For the case in which E˜ and P˜ are given by (72.1), the field amplitudes associated with each frequency component can be decomposed into their plane wave components such that
An (r, ωl ) eikn ·r , E(r, ωl ) = l

P(r, ωl ) =




Pn (r, ωl ) eikn ·r ,

(2) A(z, ω3 )A∗ (z, ω2 ) e−i∆kz , P NL (z, ω1 ) = [ε0 ]2χeff

(72.22)

P

NL

(2) (z, ω2 ) = [ε0 ]2χeff A(z, ω3 )A∗ (z, ω1 ) e−i∆kz

,

(72.23) (72.19)

l

(2) P NL (z, ω3 ) = [ε0 ]2χeff A(z, ω1 )A(z, ω2 ) ei∆kz ,

(72.24)

where kn = n(ωl )ωl /c is the magnitude of the wavevector kn . The amplitudes An and Pn are next decomposed into vector components whose linear optical properties are such that the polarization associated with them does not change as the field propagates through the material. For example, for a uniaxial crystal these eigenpolarizations could correspond to the ordinary and extraordinary components. In order to describe the propagation and the nonlinear coupling of these eigenpolarizations, the vector field amplitudes are expressed as An (r, ωl ) = uˆ ln An (r, ωl ) , Pn (r, ωl ) = uˆ ln Pn (r, ωl ) ,

processes. Equation (72.21)) is used to determine the set of coupled-amplitude equations describing a particular nonlinear process. For example, for the case of sum-frequency generation , the two fields of frequency ω1 and ω2 are combined through second-order nonlinear interaction to create a third wave at frequency ω3 = ω1 + ω2 . Assuming full permutation symmetry, the amplitudes of the nonlinear polarization for each of the waves are

(72.20)

where uˆ ln is the unit vector associated with the eigenpolarization of the spatial mode n at frequency ωl . If the fields are assumed to travel along the z-direction, and the slowly-varying amplitude approximation ∂ 2 An /∂z 2  2kn ∂An /∂z is made, the change in the amplitude of the field as it propagates through the nonlinear medium with no linear absorption is described by the differential equation   dAn (ωl ) i2πωl NL 1 =± P (ωl ) , (72.21) dz 4πε0 n(ωl )c n

Part F 72.2

where PnNL is the nonlinear contribution to the polarization amplitude Pn , n(ωl ) is the linear refractive index at frequency ωl , and the plus (minus) sign indicates propagation in the positive (negative) z-direction. Sections 72.3 and 72.4 give expressions for the PnNL for various second- and third-order nonlinear optical

where ∆k = k1 + k2 − k3 is the wavevector mismatch (2) is given by (see Sect. 72.2.2) and χeff (2) = χeff




(2) ∗ χijk (uˆ 1 )i (uˆ 2 ) j (uˆ 3 )k ,

(72.25)

ijk

where (uˆ l )i = uˆ l · ˆı. For simplicity, the subscripts on each of the field amplitudes have been dropped, since only one spatial mode at each frequency contributed. The resulting coupled amplitude equations are   (2) 1 i4πω1 χeff dA(ω1 ) = A(ω3 )A∗ (ω2 ) e−i∆kz , dz 4π n(ω1 )c 

dA(ω2 ) 1 = dz 4π 

dA(ω3 ) 1 = dz 4π



(72.26) (2) i4πω2 χeff

n(ω2 )c 

A(ω3 )A∗ (ω1 ) e−i∆kz , (72.27)

(2) i4πω3 χeff

n(ω3 )c

A(ω1 )A(ω2 ) ei∆kz . (72.28)

72.2.2 Phase Matching For many nonlinear optical processes (e.g., harmonic generation) it is important to minimize the wave vector mismatch in order to maximize the efficiency. For example, if the field amplitudes A(ω1 ) and A(ω2 ) are constant, the solution to (72.28) yields for the output

Nonlinear Optics

intensity



I(L, ω3 ) =

×

1 64π 3 ε0   (2) 2 2 32π 3 χeff ω3 I(ω1 )I(ω2 )L 2

n(ω1 )n(ω2 )n(ω3 )c3 × sinc (∆kL/2) ,

1055

be described by the Manley–Rowe relations. For example, for the case of sum-frequency generation, one can deduce from (72.26, 27, 28) that       d I(ω2 ) d I(ω3 ) d I(ω1 ) = =− . dz ω1 dz ω2 dz ω3



2

72.2 Wave Equation in Nonlinear Optics

(72.31) (72.29)

in terms of sinc x = (sin x)/x, where I(L, ω3 ) = (4πε0 )n(ω3 )c|A(L, ω3 )|2 /2π, and I(ω1 ) and I(ω2 ) are the corresponding input intensities. Clearly, the effect of the wavevector mismatch is to reduce the efficiency of the generation of the sum frequency wave. The maximum propagation distance over which efficient nonlinear coupling can occur is given by the coherence length 2 Lc = . (72.30) ∆k As a result of the dispersion in the linear refractive index that occurs in all materials, achieving phase matching over typical interaction lengths (e.g., 5 mm) is nontrivial. For the case in which the nonlinear material is birefringent, it is sometimes possible to achieve phase matching by insuring that the interacting waves possess some suitable combination of ordinary and extraordinary polarization. Other techniques for achieving phase matching include quasiphase matching [72.5] and the use of the mode dispersion in waveguides [72.6]. However, the phase matching condition is automatically satisfied for certain nonlinear optical processes, such as two-photon absorption (see Sect. 72.4.6) and Stokes amplification in stimulated Raman scattering (see Sect. 72.5.1). One can tell when the phase matching condition is automatically satisfied by examining the frequencies that appear in the expression for the nonlinear susceptibility. For a nonlinear susceptibility of the sort χ (3) (ω1 ; ω2 , ω3 , ω4 ) the wave vector mismatch is given in general by ∆k = k2 + k3 + k4 − k1 . Thus, for the example of Stokes amplification in stimulated Raman scattering, the nonlinear susceptibility is given by χ (3) (ω1 ; ω1 , ω0 , −ω0 ) where ω0 (ω1 ) is the frequency of the pump (Stokes) wave, and consequently the wave vector mismatch vanishes identically.

Under conditions of full permutation symmetry, there is no flow of power from the electromagnetic fields to the medium, and thus the total power flow of the fields is conserved. The flow of energy among the fields can

72.2.4 Pulse Propagation If the optical field consists of ultrashort ( ω1 ) of two incident lasers. Consider the case in which a strong (undepleted) pump wave at frequency ω3 and a weak (signal) wave at ω1 are incident on a nonlinear medium described by χ (2) (ω2 ; ω3 , −ω1 ) = χ (2) (ω1 ; ω3 , −ω2 ). The amplitude A(ω3 ) of the strong wave can be taken as a constant,

and thus the interaction can be described by finding simultaneous solutions to (72.26) and (72.27) for A(ω1 ) and A(ω2 ). In the limit of perfect phase matching (i. e., ∆k = 0), the solutions are A(z, ω1 ) = A(0, ω1 ) cosh κz ,

(72.36)

 n 1 ω2 A(ω3 ) ∗ A (0, ω1 ) sinh κz , A(z, ω2 ) = i n 2 ω1 |A(ω3 )| (72.37)

where  κ2 =

1 16π 2





2 16π 2 χ (2) ω21 ω22 |A(ω3 )|2 . (72.38) k1 k2 c4

Equation (72.37) describes the spatial growth of the difference frequency signal.

72.3.4 Parametric Amplification and Oscillation For the foregoing case of a strong wave at frequency ω3 and a weak wave with ω1 < ω3 incident on a secondorder nonlinear optical material, the lower frequency input wave is amplified by the nonlinear interaction; this process is known as parametric amplification. Difference frequency generation is a consequence of the Manley– Rowe relations, as described above in Sect. 72.2.3. Since ω3 = ω1 + ω2 , the annihilation of an ω3 photon must be accompanied by the simultaneous creation of photons ω1 and ω2 . An optical parametric oscillator can be constructed by placing the nonlinear optical material inside an optical resonator that provides feedback at ω1 and/or ω2 . When such a device is excited by a wave at ω3 , it can produce output frequencies ω1 and ω2 that satisfy ω1 + ω2 = ω3 . Optical parametric oscillators are of considerable interest as sources of broadly tunable radiation [72.7].

72.3.5 Focused Beams For conceptual clarity, much of the discussion so far has assumed that the interacting beams are plane waves. In practice, the incident laser beams are often focused into the nonlinear material to increase the field strength within the interaction region and consequently to increase the nonlinear response. However, it is undesirable to focus too tightly, because doing so leads

Nonlinear Optics

to a decrease in the effective length of the interaction region. In particular, if w0 is the radius of the laser beam at the beam waist, the beam remains focused only over a distance of the order b = 2πw20 /λ where λ is the laser wavelength measured in the non-

72.4 Third-Order Processes

1057

linear material. For many types of nonlinear optical processes, the optimal nonlinear response occurs if the degree of focusing is adjusted so that b is several times smaller than the length L of the nonlinear optical material.

72.4 Third-Order Processes A wide variety of nonlinear optical processes are possible as a result of the nonlinear contributions to the polarization that are third-order in the applied field. These processes are described by χ (3) (ωσ ; ωm , ωn , ωo ) (72.3) and can lead not only to the generation of new field components (e.g., third-harmonic generation) but can also result in a field affecting itself as it propagates (e.g., self-phase modulation). Several examples are described in this section.

Assuming full-permutation symmetry, the nonlinear polarization amplitudes for the fundamental and thirdharmonic beams are (3) A(z,3ω)[A∗(z, ω)]2 e−i∆kz , P NL (z, ω) = [ε0 ]3χeff

(72.39)

(3) χeff

where ∆k = 3k(ω) − k(3ω) and is the effective third-order susceptibility for third-harmonic generation (2) and is defined in a manner analogous to the χeff in (72.25). If the intensity of the fundamental wave is not depleted by the nonlinear interaction, the solution for the output intensity I(L, 3ω) of the third-harmonic field for a crystal of length L is  2   2 ω2 χ (3) 48π eff 1 I(L, 3ω) = 256π 4 ε20 n(3ω)n(ω)3 c4 × I(ω)3 L 2 sinh2 [∆kL/2] ,

The nonlinear refractive index leads to an intensitydependent change in the phase of the beam as it propagates through the material. If the medium is lossless, the amplitude of a single beam at frequency ω propagating in the positive z-direction can be expressed as A(z, ω) = A(0, ω) eiφ

72.4.1 Third-Harmonic Generation

(3) P NL (z, 3ω) = [ε0 ]χeff [A(z, ω)]3 ei∆kz ,

72.4.2 Self-Phase and Cross-Phase Modulation

(72.40)

,

(72.41)

φNL (z)

where the nonlinear phase shift is given by ω (72.42) φNL (z) = n 2 Iz , c and I = [4πε0 ]n 0 c|A(0, ω)|2 /2π is the intensity of the laser beam. If two fields at different frequencies ω1 and ω2 are traveling along the z-axis, the two fields can affect each other’s phase; this effect is known as cross-phase NL (z) for each modulation. The nonlinear phase shift φ1,2 of the waves is given by ω1,2 NL n 2 (I1,2 + 2I2,1 )z . (z) = (72.43) φ1,2 c For the case of a light pulse, the change in the phase of the pulse inside the medium becomes a function of time. In this case the solution to (72.33) for the timevarying amplitude A(z, τ) shows that in the absence of group-velocity dispersion (GVD) (i. e., β2 = 0) that the solution for A(z, τ) is of the form of (72.41), except that the temporal intensity profile I(τ) replaces the steady-state intensity I in (72.42). As the pulse propagates through the medium, its frequency becomes time dependent, and the instantaneous frequency shift from the central frequency ω0 is given by ωn 2 z ∂I ∂φNL (τ) =− . (72.44) ∂τ c ∂t This time-dependent self-phase modulation leads to a broadening of the pulse spectrum and to a frequency chirp across the pulse. δω(τ) = −

Part F 72.4

where I(ω) is the input intensity of the fundamental (3) field. As a result of the typically small value of χeff in crystals, it is generally more efficient to generate the third harmonic by using two χ (2) crystals in which the first crystal produces second harmonic light and the second crystal combines the second harmonic and the fundamental beams via sum-frequency generation. It is also possible to use resonant enhancement of |χ (3) | in gases to increase the efficiency of third-harmonic generation [72.8].

NL (z)

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Part F

Quantum Optics

If the group velocity dispersion parameter β2 and the nonlinear refractive index coefficient n 2 are of opposite sign, the nonlinear frequency chirp can be compensated by the chirp due to group velocity dispersion, and (72.33) admits soliton solutions . For example, the fundamental soliton solution is    τ 1 eiz/2L D , A(z, t) = sech (72.45) LD τp where τ p is the pulse duration and L D = τ 2p /|β2 | is the dispersion length. As a result of their ability to propagate in dispersive media without changing shape, optical solitons show a great deal of promise in applications such as optical communications and optical switching. For further discussion of optical solitons see [72.9].

72.4.3 Four-Wave Mixing Various types of four-wave mixing processes can occur among different beams. One of the most common geometries is backward four-wave mixing used in nonlinear spectroscopy and optical phase conjugation. In this interaction, two strong counterpropagating pump waves with amplitudes A1 and A2 and with equal frequencies ω1,2 = ω are injected into a nonlinear medium. A weak wave, termed the probe wave, (with frequency ω3 and amplitude A3 ) is also incident on the medium. As a result of the nonlinear interaction among the three waves, a fourth wave with an amplitude A4 is generated which is counterpropagating with respect to the probe wave and with frequency ω4 = 2ω − ω3 . For this case, the third-order nonlinear susceptibilities for the probe and conjugate waves are given by χ (3) (ω3,4 ; ω, ω, −ω4,3 ). For constant pump wave intensities and full permutation symmetry, the amplitudes of the nonlinear polarization for the probe and conjugate waves are given by

P NL (z, ω3,4 ) = ±[ε0 ]6χ (3) |A1 |2 + |A2 |2 A3,4

+ A1 A2 A∗4,3 ei∆kz , (72.46)

Part F 72.4

where ∆k = k1 + k2 − k3 − k4 is the phase mismatch, which is nonvanishing when ω3 = ω4 . For the case of optical phase conjugation by degenerate four-wave mixing (i. e., ω3 = ω4 = ω and A4 (L) = 0), the phase conjugate reflectivity RPC is |A4 (0)|2 = tan2 (κL) , (72.47) |A3 (0)|2



√ where κ = 1/16π 2 ε0 24π 2 ωχ (3) /(n 0 c)2 I1 I2 and I1,2 are the intensities of the pump waves. PhaseRPC =

conjugate reflectivities greater than unity can be routinely achieved by performing four-wave mixing in atomic vapors or photorefractive media.

72.4.4 Self-Focusing and Self-Trapping Typically a laser beam has a transverse intensity profile that is approximately Gaussian. In a medium with an intensity-dependent refractive index, the index change at the center of the beam is different from the index change at the edges of the beam. The gradient in the refractive index created by the beam can allow it to self-focus for n 2 > 0. For this condition to be met, the total input power of the beam must exceed the critical power Pcr for self-focusing which is given by π(0.61λ)2 , (72.48) 8n 0 n 2 where λ is the vacuum wavelength of the beam. For powers much greater than the critical power, the beam can break up into various filaments, each with a power approximately equal to the critical power. For a more extensive discussion of self-focusing and self-trapping see [72.10, 11]. Pcr =

72.4.5 Saturable Absorption When the frequency ω of an applied laser field is sufficiently close to a resonance frequency ω0 of the medium, an appreciable fraction of the atomic population can be placed in the excited state. This loss of population from the ground state leads to an intensity-dependent saturation of the absorption and the refractive index of the medium (see Sect. 69.2 for more detailed discussion) [72.4]. The third-order susceptibility as a result of this saturation is given by   δT2 − i 1 |µ|2 T1 T2 α0 c (3) χ =

2 , (72.49) ε0 3πω0 ~2 1 + (δT2 )2 where µ is the transition dipole moment, T1 and T2 are the longitudinal and transverse relaxation times, respectively (see Sect. 68.4.3), α0 is the line-center weak-field intensity absorption coefficient, and δ = ω − ω0 is the detuning. For the 3s ↔ 3p transition in atomic sodium vapor at 300 ◦ C, the nonlinear refractive index n 2 ≈ 10−7 cm2 /W for a detuning δT2 = 300.

72.4.6 Two-Photon Absorption When the frequency ω of a laser field is such that 2ω is close to a transition frequency of the material, it is

Nonlinear Optics

possible for two-photon absorption (TPA) to occur. This process leads to a contribution to the imaginary part of χ (3) (ω; ω, ω, −ω). In the presence of TPA, the intensity I(z) of a single, linearly polarized beam as a function of propagation distance is I(0) , I(z) = (72.50) 1 + βI(0)z



where β = 1/16π 2 ε0 24π 2 ω Im χ (3) /(n 0 c)2 is the TPA coefficient. For wide-gap semiconductors such as ZnSe at 800 nm, β ≈ 10−8 cm/W.

72.4.7 Nonlinear Ellipse Rotation The polarization ellipse of an elliptically polarized laser beam rotates but retains its ellipticity as the beam propagates through an isotropic nonlinear medium. Ellipse

72.5 Stimulated Light Scattering

1059

rotation occurs as a result of the difference in the nonlinear index changes experienced by the left- and right-circular components of the beam, and the angle θ of rotation is 1 θ = ∆nωz/c 2   12π 2 (3) 1 = χ 16π 2 ε0 n 20 c xyyx × (ω; ω, ω, −ω)(I+ − I− )z , (72.51) where I± are the intensities of the circularly polarized components of the beam with unit vectors σˆ ± = (xˆ ± √ i yˆ )/ 2. Nonlinear ellipse rotation is a sensitive technique for determining the nonlinear susceptibility (3) element χxyyx for isotropic media and can be used in applications such as optical switching.

72.5 Stimulated Light Scattering Stimulated light scattering occurs as a result of changes in the optical properties of the material that are induced by the optical field. The resulting nonlinear coupling between different field components is mediated by some excitation (e.g., acoustic phonon) of the material that results in changes in its optical properties. The nonlinearity can be described by a complex susceptibility and a nonlinear polarization that is of third order in the interacting fields. Various types of stimulated scattering can occur. Discussed below are the two processes that are most commonly observed.

72.5.1 Stimulated Raman Scattering

P NL (z, ω0,1 ) = [ε0 ]6χR (ω0,1 ) × |A(z, ω1,0 )|2 A(z, ω0,1 ) , (72.52) where χR (ω0,1 ) ≡ χ (3) (ω0,1 ; ω0,1 , ω1,0 , −ω1,0 ), the Raman susceptibility, actually depends only on the fre-

(72.53)

where the minus (plus) sign is taken for the ω0 (ω1 ) susceptibility, µ M is the reduced nuclear mass, and (∂α/∂q)0 is a measure of the change of the polarizability of the molecule with respect to a change in the intermolecular distance q at equilibrium. If the intensity of the pump field is undepleted by the interaction with the ω1 field and is assumed to be constant, the solution for the intensity of the ω1 field at z = L is given by I(L, ω1 ) = I(0, ω1 ) eG R ,

(72.54)

where the SRS gain parameter G R is   ω1 1 48π 2 GR = Im[χR (ω1 )]I0 L 2 16π ε0 (n 1 c)2 (72.55) = gR I0 L , gR is the SRS gain factor, and I0 is the input intensity of the pump field. For ω1 < ω0 (ω1 > ω0 ), the ω1 field is termed the Stokes (anti-Stokes) field, and it experiences exponential amplification (attenuation). For sufficiently large gains (typically G R
25), the Stokes wave can be seeded by spontaneous Raman scattering and can grow to an appreciable fraction of the pump field. For a complete discussion of the sponta-

Part F 72.5

In stimulated Raman scattering (SRS), the light field interacts with a vibrational mode of a molecule. The coupling between the two optical waves can become strong if the frequency difference between them is close to the frequency ωv of the molecular vibrational mode. If the pump field at ω0 and another field component at ω1 are propagating in the same direction along the z-axis, the steady-state nonlinear polarization amplitudes for the two field components are given by

quency difference Ω = ω0 − ω1 and is given by   1 1 N(∂α/∂q)20 , χR (ω0,1 ) = ε0 6µ M ω2v − Ω 2 ∓ 2iγΩ

1060

Part F

Quantum Optics

neous initiation of SRS see [72.12]. For the case of CS2 , gR = 0.024 cm/MW. Four-wave mixing processes that couple a Stokes wave having ω1 < ω0 and an anti-Stokes wave having ω2 > ω0 , where ω1 + ω2 = 2ω0 , can also occur [72.4]. In this case, additional contributions to the nonlinear polarization are present and are characterized by a Raman susceptibility of the form χ (3) (ω1,2 ; ω0 , ω0 , −ω2,1 ). The technique of coherent anti-Stokes Raman spectroscopy is based on this fourwave mixing process [72.13].

72.5.2 Stimulated Brillouin Scattering In stimulated Brillouin scattering (SBS), the light field induces and interacts with an acoustic wave inside the medium. The resulting interaction can lead to extremely high amplification for certain field components (i. e., Stokes wave). For many optical media, SBS is the dominant nonlinear optical proccess for laser pulses of duration > 1 ns. The primary applications for SBS are self-pumped phase conjugation and pulse compression of high-energy laser pulses. If an incident light wave with wave vector k0 and frequency ω0 is scattered from an acoustic wave with wave vector q and frequency Ω, the wave vector and frequency of the scattered wave are determined by conservation of momentum and energy to be k1 = k0 ± q and ω1 = ω0 ± Ω, where the (+) sign applies if k0 · q > 0 and the (−) applies if k0 · q < 0. Here, Ω and q are related by the dispersion relation Ω = v|q| where v is the velocity of sound in the material. These Bragg scattering conditions lead to the result that the Brillouin frequency shift ΩB = ω1 − ω0 is zero for scattering in the forward direction (i. e., in the k0 direction) and reaches its maximum for scattering in the backward direction given by ΩB = 2ω0 vn 0 /c ,

(72.56)

Part F 72.5

where n 0 is the refractive index of the material. The interaction between the incident wave and the scattered wave in the Brillouin-active medium can become nonlinear if the interference between the two optical fields can coherently drive an acoustic wave, either through electrostriction or through local density fluctuations resulting from the absorption of light and consequent temperature changes. The following discussion treats the more common electrostriction mechanism. Typically, SBS occurs in the backward direction (i. e., k0 = k0 zˆ and k1 = −k1 z), ˆ since the spatial overlap

between the Stokes beam and the laser beam is maximized under these conditions and, as mentioned above, no SBS occurs in the forward direction. The steadystate nonlinear polarization amplitudes for backward SBS are P NL (z, ω0,1 ) = [ε0 ]6χB (ω0,1 ) × |A(z, ω1,0 )|2 A(z, ω0,1 ) , (72.57) where χB (ω0,1 ) ≡ χ (3) (ω0,1 ; ω0,1 , ω1,0 , −ω1,0 ), the Brillouin susceptibility, depends only on Ω = ω0 − ω1 and is given by   ω20 γe2 1 1 , χB (ω0,1 ) = ε0 24π 2 c2 ρ0 Ω2B − Ω2 ∓ iΓB Ω (72.58)

where the minus (plus) sign is taken for the ω0 (ω1 ) susceptibility, γe is the electrostrictive constant, ρ0 is the mean density of the material, and ΓB is the Brillouin linewidth given by the inverse of the phonon lifetime. If the pump field is undepleted by the interaction with the ω1 field and is assumed to be constant, the solution for the output intensity of the ω1 field at z = 0 is given by I(0, ω1 ) = I(L, ω1 ) eG B ,

(72.59)

where the Brillouin gain coefficient G B is given by   ω1 1 48π 2 GB = Im[χB (ω1 )]I0 L, 16π 2 ε0 (n 0 c)2 ΩΩB ΓB2 = g0 I0 L

2 Ω2B − Ω2 + (ΩΓB )2 = gB I0 L ,

(72.60)

gB is the SBS gain factor, I0 is the input intensity of the pump field, and   ω20 γe2 1 (72.61) g0 = 2 ε0 n 0 c3 ρ0 vΓB is the line-center (i. e., Ω = ±ΩB ) SBS gain factor. For Ω > 0 (Ω < 0), the ω1 field is termed the Stokes (anti-Stokes) field, and it experiences exponential amplification (attenuation). For sufficiently large gains (typically G B
25), the Stokes wave can be seeded by spontaneous Brillouin scattering and can grow to an appreciable fraction of the pump field. For a complete discussion of the spontaneous initiation of SBS see [72.14]. For CS2 , g0 = 0.15 cm/MW.

Nonlinear Optics

72.6 Other Nonlinear Optical Processes

1061

72.6 Other Nonlinear Optical Processes 72.6.1 High-Order Harmonic Generation

72.6.2 Electro-Optic Effect

If full permutation symmetry applies and the fundamental field ω is not depleted by nonlinear interactions, then the intensity of the qth harmonic is given by   1 I(z, qω) = 4π(4πε0 )(q−1)/2   2πq 2 ω2 2πI(ω) q × 2 n (qω)c n(ω)c  (q) 2  × χ (qω; ω, . . . , ω)Jq (∆k, z 0 , z) ,

The electro-optic effect corresponds to the limit in which the frequency of one of the applied fields approaches zero. The linear electro-optic effect (or Pockels effect) can be described by a second-order susceptibility of the form χ (2) (ω; ω, 0). This effect produces a change in the refractive index for light of certain polarizations which depends linearly on the strength of the applied low-frequency field. More generally, the linear electro-optic effect induces a change in the amount of birefringence present in an optical material. This electrically controllable change in birefringence can be used to construct amplitude modulators, frequency shifters, optical shutters, and other optoelectronic devices. Materials commonly used in such devices include KDP and lithium niobate [72.17]. If the laser beam is propagating along the optic axis (i. e., z-axis) of the material of length L and the low-frequency field E z is also applied along the optic axis, the nonlinear index change ∆n = n y − n x between the components of the electric field polarized along the principal axes of the crystal is given by   1 n 3r63 E z ∆n = (72.66) 4π 0

(72.62)

where ∆k = [n(ω) − n(qω)]ω/c, z Jq (∆k, z 0 , z) = z0





ei∆kz dz , (1 + 2iz  /b)q−1

(72.63)

z = z 0 at the input face of the nonlinear medium, and b is the confocal parameter Sect. 72.3.5 of the fundamental beam. Defining L = z − z 0 , the integral Jq can be easily evaluated in the limits L  b and L b. The limit L  b corresponds to the plane-wave limit in which case   2 2 2 ∆kL |Jq (∆k, z 0 , z)| = L sinc . (72.64) 2 The limit L b corresponds to the tight-focusing configuration in which case   0, ∆k ≤ 0 ,     q−2  πb b∆k Jq (∆k, z 0 , z) = e−b∆k/2 ,  2  (q − 2)!    ∆k > 0 . (72.65)

72.6.3 Photorefractive Effect The photorefractive effect leads to an optically induced change in the refractive index of a material. In certain ways this effect mimics that of the nonlinear refractive index described in Sect. 72.1.2, but it differs from the nonlinear refractive index in that the change in refractive index is independent of the overall intensity of the incident light field, and depends only on the degree of spatial modulation of the light field within the nonlinear material. In addition, the photorefractive effect can occur only in materials that exhibit a linear electro-optic effect, and contain an appreciable density of trapped electrons and/or holes that can be liberated by the application of a light field. Typical photorefractive materials include lithium niobate, barium titanate, and strontium barium niobate.

Part F 72.6

Note that in this limit, the qth harmonic light is only generated for positive phase mismatch. Reintjes et al. [72.15, 16] observed both the fifth and seventh harmonics in helium gas which exhibited a dependence on I(ω) which is consistent with the I q (ω) dependence predicted by (72.62). However, more recent experiments in gas jets have demonstrated the generation of extremely high-order harmonics which do not depend on the intensity in this simple manner (see Chapt. 74 for further discussion of this nonperturbative behavior).

where r63 is one of the electro-optic coefficients. The quadratic electro-optic effect produces a change in refractive index that scales quadratically with the applied dc electric field. This effect can be described by a third-order susceptibility of the form χ (3) (ω; ω, 0, 0).

1062

Part F

Quantum Optics

A typical photorefractive configuration might be as follows: two beams interfere within a photorefractive crystal to produce a spatially modulated intensity distribution. Bound charges are ionized with greater probability at the maxima than at the minima of the distribution and, as a result of the diffusion process, carriers tend to migrate away from regions of large light intensity. The resulting modulation of the charge distribution leads to the creation of a spatially modulated electric field that produces a spatially modulated change in refractive index as a consequence of the linear electro-optic effect. For a more extensive discussion see [72.18].

72.6.4 Ultrafast and Intense-Field Nonlinear Optics Additional nonlinear optical processes are enabled by the use of ultrashort (< 1 ps) or ultra-intense laser pulses. For reasons of basic laser physics, ultra-intense pulses are necessarily of short duration, and thus these effects normally occur together. Ultrashort laser pulses possess a broad frequency spectrum, and therefore the dispersive properties of the optical medium play a key role in the propagation of such pulses. The three-dimensional nonlinear Schrödinger equation must be modified when treating the propagation of these ultrashort pulses by including contributions that can be ignored under other circumstances [72.19,20]. These additional terms lead to processes such as space-time coupling, self-steepening, and shock wave formation [72.21, 22]. The process of self-focusing is significantly modified under short-pulse (pulse duration shorter than approximately 1 ps) excitation. For example, temporal splitting of a pulse into two components can occur; this pulse splitting lowers the peak intensity, and can lead to the arrest of the usual

collapse of a pulse undergoing self-focusing [72.23]. Moreover, optical shock formation, the creation of a discontinuity in the intensity evolution of a propagating pulse, leads to supercontinuum generation, the creation of a light pulse with an extremely broad frequency spectrum [72.24]. Shock effects and the generation of supercontinuum light can also occur in one-dimensional systems, such as a microstructure optical fiber. The relatively high peak power of the ultrashort pulses from a mode-locked laser oscillator and the tight confinement of the optical field in the small (≈ 2 µm) core of the fiber yield high intensities and strong self-phase modulation, which results in a spectral bandwidth that spans more than an octave of the central frequency of the pulse [72.25]. Such a coherent octave-spanning spectrum allows for the stabilization of the underlying frequency comb of the mode-locked oscillator, and has led to a revolution in the field of frequency metrology [72.26]. Multiphoton absorption [72.27] constitutes an important loss process that becomes important for intensities in excess of ≈ 1013 W/cm2 . In addition to introducing loss, the electrons released by this process can produce additional nonlinear effects associated with their relativistic motion in the resulting plasma [72.28, 29]. For very large laser intensities (greater than approximately 1016 W/cm2 ), the electric field strength of the laser pulse can exceed the strength of the Coulomb field that binds the electron to the atomic core, and nonperturbative effects can occur. A dramatic example is that of high-harmonic generation [72.30–32]. Harmonic orders as large as the 341-st have been observed, and simple conceptual models have been developed to explain this effect [72.33]. Under suitable conditions the harmonic orders can be suitably phased so that attosecond pulses are generated [72.34].

References 72.1 72.2 72.3

Part F 72

72.4 72.5 72.6

N. Bloembergen: Nonlinear Optics (Benjamin, New York 1964) Y. R. Shen: Nonlinear Optics (Wiley, New York 1984) P. N. Butcher, D. Cotter: The Elements of Nonlinear Optics (Cambridge Univ. Press, Cambridge 1990) R. W. Boyd: Nonlinear Optics (Academic, Boston 1992) J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan: Phys. Rev. 127, 1918 (1962) G. I. Stegeman: Contemporary Nonlinear Optics, ed. by G. P. Agrawal, R. W. Boyd (Academic, Boston 1992) Chap. 1

72.7

72.8 72.9 72.10

72.11 72.12

See for example the Special Issue on: Optical Parametric Oscillation and Amplification, J. Opt. Soc. Am. B 10 (1993) No. 11 R. B. Miles, S. E. Harris: IEEE J. Quant. Electron. 9, 470 (1973) G. P. Agrawal: Nonlinear Fiber Optics (Academic, Boston 1989) S. A. Akhmanov, R. V. Khokhlov, A. P. Sukhorukov: Laser Handbook, ed. by F. T. Arecchi, E. O. SchulzDubois (North-Holland, Amsterdam 1972) J. H. Marburger: Prog. Quant. Electr. 4, 35 (1975) M. G. Raymer, I. A. Walmsley: Prog. Opt., Vol. 28, ed. by E. Wolf (North-Holland, Amsterdam 1990)

Nonlinear Optics

72.13 72.14 72.15

72.16

72.17 72.18

72.19 72.20 72.21 72.22 72.23

M. D. Levenson, S. Kano: Introduction to Nonlinear Spectroscopy (Academic, Boston 1988) R. W. Boyd, K. Rzazewski, P. Narum: Phys. Rev. A 42, 5514 (1990) J. Reintjes, C. Y. She, R. C. Eckardt, N. E. Karangelen, R. C. Elton, R. A. Andrews: Phys. Rev. Lett. 37, 1540 (1976) J. Reintjes, C. Y. She, R. C. Eckardt, N. E. Karangelen, R. C. Elton, R. A. Andrews: Appl. Phys. Lett. 30, 480 (1977) I. P. Kaminow: An Introduction to Electro-optic Devices (Academic, New York 1974) P. Günter, J.-P. Huignard (Eds.): Photorefractive Materials and Their Applications (Springer, Berlin, Heidelberg, Part I (1988), Part II (1989)) T. Brabec, F. Krausz: Phys. Rev. Lett. 78, 3283 (1997) J. K. Ranka, A. L. Gaeta: Opt. Lett. 23, 534 (1998) J. E. Rothenberg: Opt. Lett. 17, 1340 (1992) G. Yang, Y. R. Shen: Opt. Lett. 9, 510 (1984) J. K. Ranka, R. Schirmer, A. L. Gaeta: Phys. Rev. Lett. 77, 3783 (1996)

72.24 72.25 72.26

72.27 72.28 72.29 72.30 72.31 72.32 72.33 72.34

References

1063

A. L. Gaeta: Phys. Rev. Lett. 84, 3582 (2000) J. K. Ranka, R. S. Windeler, A. J. Stentz: Opt. Lett. 25, 25 (2000) D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, S. T. Cundiff: Science 288, 635 (2000) W. Kaiser, C. G. B. Garrett: Phys. Rev. Lett. 7, 229 (1961) P. Sprangle, C.-M. Tang, E. Esarez: IEEE Transactions on Plasma Science 15, 145 (1987) R. Wagner, S.-Y. Chen, A. Maksemchak, D. Umstadter: Phys. Rev. Lett. 78, 3125 (1997) P. Agostini, F. Fabre, G. Mainfray, G. Petite, N. K. Rahman: Phys. Rev. Lett. 42, 1127 (1979) Z. Chang: Phys. Rev. Lett. 79, 2967 (1997) Z. Chang: Phys. Rev. Lett. 82, 2006 (1999) P. B. Corkum: Phys. Rev. Lett. 71, 1994 (1993) H. R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, F. Krausz: Nature 414, 509 (2001)

Part F 72

1065

Coherent optical transients are excited in atomic and molecular systems when a stable phase relation persists between an exciting light field and the system’s electronic response. The extreme sensitivity of phase-dependent effects is responsible for the many applications of optical transient techniques in atomic and molecular physics [73.1–7]. The theory of coherent transients distinguishes carefully between two types of relaxation: homogeneous and inhomogeneous. Relaxation occurs whenever the environment of a physical system fluctuates randomly. By random environment one means the combination of all interactions that are too complex to be treated fundamentally, and that can be seen to lead to degradation of the degree of coherence of a particular interaction of main interest. The time scale of environmental fluctuations then provides the division between the two types. When environmental fluctuations are sufficiently rapid that all dynamical systems in a macroscopic sample experience the whole range of fluctuations in a time short compared with the time of an experiment, the resultant relaxation is called homogeneous. If environmental fluctuations exist randomly over a macroscopic sample, but change relatively slowly in time, then the relaxation is called inhomogeneous. For example, weak distant collisions are experienced constantly by all atoms at thermal equilibrium in a vapor cell, and give rise to homogeneous relaxation. If the vapor is sufficiently dilute, the same atoms may nevertheless retain for long times their own individual velocities. These velocities are relatively fixed in time, but they are random over the Maxwellian distribution of velocities and so give rise to inhomogeneous relaxation. Fundamentally,

73.1 Optical Bloch Equations........................ 1065 73.2 Numerical Estimates of Parameters ....... 1066 73.3 Homogeneous Relaxation..................... 1066 73.3.1 Rabi Oscillations........................ 1067 73.3.2 Bloch Vector and Bloch Sphere.... 1067 73.3.3 Pi Pulses and Pulse Area ............ 1067 73.3.4 Adiabatic Following ................... 1068 73.4 Inhomogeneous Relaxation .................. 1068 73.4.1 Free Induction Decay ................. 1068 73.4.2 Photon Echoes .......................... 1069 73.5 Resonant Pulse Propagation ................. 1069 73.5.1 Maxwell–Bloch Equations .......... 1069 73.5.2 Index of Refraction and Beers Law .......................... 1070 73.5.3 The Area Theorem and Self-Induced Transparency .. 1070 73.6 Multi-Level Generalizations.................. 1071 73.6.1 Rydberg Packets and Intrinsic Relaxation............. 1071 73.6.2 Multiphoton Resonance and Two-Photon Bloch Equations ........................ 1072 73.6.3 Pump–Probe Resonance and Dark States......................... 1073 73.6.4 Three-Level Transparency........... 1074 73.7 Disentanglement and “Sudden Death” of Coherent Transients ......................... 1074 References .................................................. 1076 the distinction between homogeneous and inhomogeneous relaxation is artificial, depending on a separation of time scales that may not always exist. Nevertheless, when it exists, the distinction provides an extremely useful way to classify coherent transients. It is one of the foundations of the subject. The presence of quantum entanglement leads to nonintuitive effects in coherent transients.

73.1 Optical Bloch Equations A very weakly excited dipole transition in an atom responds linearly to an applied time-dependent electric field. This is the basis of classical Lorentzian dielec-

tric theory, but because any transition can be inverted, an atom is more than a classical linear oscillator [73.8]. The three atomic variables that describe the primary co-

Part F 73

Coherent Tran 73. Coherent Transients

1066

Part F

Quantum Optics

Part F 73.3

herent optical transients in a dipole-allowed transition include the intrinsically quantum mechanical inversion variable, as well as the components of the expectation value of the atomic dipole moment that are in-phase and in-quadrature with the field. We write the time-dependent atomic dipole moment of a transition excited by light near exact resonance in the form −
ex(t) = −
Ψ(t)|ex|Ψ(t) 
= Re d(U − iV ) e−iωt ,

little to a first discussion of the principles of coherent transients. Section 68.3.5 shows that U and V are dynamically coupled to each other and to the inversion W through the optical Bloch equations (OBE). When relaxation terms are included, the OBEs are given by (68.55). These are dU = −∆V − U/T2 , dt dV = ∆U + Ω1 W − V/T2 , dt dW = −Ω1 V − (W − Weq )/T1 , dt

(73.1)

where d is the transition dipole matrix element, ω is the frequency of the impressed optical field, and U and V are the time-dependent amplitudes of the in-phase and in-quadrature dipole components. The impressed field is taken in the quasi-monochromatic form  1  −iωt Ee + c.c. (73.2) E(t) = 2 Both dipole moment and field will be taken to be real scalars because the complications of vector notation add

(73.3)

the key equations of the theory of optical transients [73.1, 2]. As in Chapt. 68, ∆ = (E e − E g )/~ − ω is the detuning and Ω1 = dE/~ is the Rabi frequency. It is the dipole interaction energy in frequency units, but has a significance beyond this, as discussed in Sect. 73.3.1.

73.2 Numerical Estimates of Parameters The nature of the coherent interaction between an atom or molecule and an optical field is controlled by the relative size of a number of frequencies or rates. In the case of single photon transitions they include: ∆ and Ω1 , the detuning and Rabi frequency defined above, 1/T2 the transverse, and 1/T1 the longitudinal damping rates, 1/T ∗ the inhomogenenous linewidth, and 2π/τp the transform bandwidth of the optical pulse. All of these frequencies with the exception of the last are defined in Chapt. 68. In the case of multiphoton transitions and simultaneous excitation by a number of resonant laser fields, the appropriately generalized versions of these same parameters apply. A laser pulse with τp ≥ 1 ns and with an intensity less than about 1 GW/cm2 can be tuned to an isolated atomic resonance and the interaction can be described in terms of a simple two-level theory. Laser pulses as short as a few fs in duration, or with intensities as

high as 1022 W/cm2 , have been produced and such extreme pulses quasi-resonantly excite more than one upper level. A 1 ps pulse has a bandwidth of approximately 20 cm−1 , while a 1 fs pulse has a bandwidth of about 20 000 cm−1 . If a 1 ps pulse were tuned so that it resonantly excited the n = 95 Rydberg state of an atom, it would simultaneously and coherently excite all the levels from n = 67 to the continuum limit, while a 10 fs pulse could excite all the levels from n = 4 to the continuum. Similarly, when laser pulses are intense enough so that the electric field amplitude approaches that of the Coulomb field holding the atom together – in hydrogen this occurs at an intensity of 3.6 × 1016 W/cm2 – a Rabi frequency on the order of 200 000 cm−1 is generated, again much more than enough to excite a coherent superposition of all atomic bound states [73.9].

73.3 Homogeneous Relaxation Homogeneous relaxation is dominant in well-collimated atomic and molecular beams as well as in high-pressure vapor cells. In the absence of a laser field (Ω1 = 0) the

solutions of the OBEs are (U − iV ) = (U − iV )0 e−(1/T2 +i∆)t ,

Coherent Transients

(73.4)

where the subscript denotes values at t = 0. The roles of T1 and T2 as relaxation times are clear. They are homogeneous because they apply to each atom individually.

1.0

W V

0.5 U 0.0

73.3.1 Rabi Oscillations –0.5

The OBEs predict coherent damped oscillations of the inversion with the angular Rabi frequency Ω1 if Ω1 is large enough, such that Ω1 T1  1 and Ω1 T2 1. These oscillations were originally called optical nutations following the terminology of nuclear magnetic resonance, however they are now usually called Rabi oscillations. Figure 73.1 shows the behavior of the atomic variables undergoing Rabi oscillations in a representative case.

73.3.2 Bloch Vector and Bloch Sphere Coherent dynamical behavior is simplest for times much shorter than the relaxation times T1 and T2 . In this case, the damping terms can be dropped from the OBEs and the resulting equations written in the form (Sect. 68.3.5) dU = Ω ×U , dt

(73.5)

where U = (U, V, W ) is the Bloch vector, and Ω = (−Ω1 , 0, ∆) acts as a torque vector defining the axis and rate of precession. By conservation of probability, U · U = 1. All possible quantum states of the two-level atom are mapped onto a unit sphere in U–V –W space. Conventionally, W defines the polar axis with the atomic ground state the south pole, and the excited state the north pole. Points on the sphere between the poles are coherent superpositions of the two states. The azimuthal angle φ represents the phase between the expectation value of the dipole moment and the optical field. In Fig. 73.2 the solutions to (73.5) are shown for the case of a square pulse applied to an atom in its ground state at t = 0. The solutions in this case are Ω1 sin Ωt , Ω ∆Ω1 V(t, ∆) = − 2 (1 − cos Ωt) , Ω Ω2 W(t, ∆) = −1 + 12 (1 − cos Ωt) . Ω

–1.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0 t / T2

Fig. 73.1 Damped Rabi oscillations of the atomic variables

after sudden turn-on of the field. In this example T1 = T2 , ∆T2 = 1, and Ω1 T2 = 15

For any ∆ the solution orbit is a circle on the surface of the sphere with the orbit passing through the south pole. The rate at which the system precesses about the circle √ is given by the generalized Rabi frequency Ω ≡ ∆2 + Ω12 .

73.3.3 Pi Pulses and Pulse Area The exactly resonant (∆ = 0) undamped OBEs can be solved analytically even for arbitrarily time dependent laser pulse envelopes. The solutions are U(t, 0) = 0 , V(t, 0) = − sin θ(t) , W(t, 0) = − cos θ(t) ,

(73.7)

W 1⁄4 1⁄2 –V –U 1 2

U(t, ∆) =

(73.6)

1067

Part F 73.3

W = −1 + (W0 + 1) e−t/T1 ,

73.3 Homogeneous Relaxation

Fig. 73.2 Orbits of the Bloch vector on the unit sphere for various ratios of the detuning ∆ to the Rabi frequency Ω1

1068

Part F

Quantum Optics

Part F 73.4

a rotation of the Bloch vector in the V –W plane. The angle θ is called the pulse area, defined by t θ(t) ≡ −∞

d dt Ω1 (t ) = ~



t

dt E(t ) .

(73.8)

−∞

The area under the envelope of the Rabi frequency is thus the same as the angle through which an exactlyresonant Bloch vector turns due to the pulse. If θ = π, the atom is driven from the ground state exactly to the excited state. This is “π-pulse” inversion. A “2π-pulse” takes the atom from the ground state through the excited state and back to the ground state. For ∆ = 0, the Bloch vector rotation angle does not depend upon the shape of the field pulse, only on the area of the pulse.

As shown in Fig. 73.3, the Bloch vector then precesses in a very small circle about the torque vector. If the field frequency is now slowly changed (chirped) so that ∆ goes from a large negative value to a large positive value then every atomic Bloch vector will continue to precess rapidly around the torque vector, and follow it as it proceeds from pointing straight down to pointing straight up. In this way the population is transferred between the two levels. W

–V

73.3.4 Adiabatic Following The Bloch vector picture is used for semiquantitative predictions. These are reliable even if ∆ and Ω1 are time-dependent, if the parameters change slowly (adiabatically). For example, if Ω is moved slowly the Bloch vector follows closely. It is possible to achieve complete inversion smoothly in this way. If the field is initially tuned far below resonance so ∆ Ω1 then Ω points approximately toward the south pole of the Bloch sphere.

–U

U




Fig. 73.3 In adiabatic inversion the Bloch vector of each

atom precesses in a small cone about the torque vector as the torque vector goes from straight down to straight up

73.4 Inhomogeneous Relaxation The fact that the various atoms in a sample may have different resonance frequencies produces a number of novel phenomena. Given a distribution g(∆) of detunings in a dilute gas of density N, the macroscopic polarization can be written P(t) = −N
ex(t)  
= Nd g(∆) Re (U − iV ) e−iωt d∆ , (73.9) where U − iV generally depends on both t and ∆.

73.4.1 Free Induction Decay Free induction refers to evolution of the polarization in the absence of a laser field. For Ω1 = 0, the Bloch vector of an atom with ∆ < 0 precesses counterclockwise in the U–V plane. In a macroscopic sample, there are many values of ∆ and about as many are positive as negative. Thus an oriented collection of Bloch vectors, all pointing in the V di-

rection at t = 0, will rapidly fan out in the U–V plane due to differing precession rates, and after a short time the net V value will be zero, as will the net U value. This is free induction decay (FID) of polarization. More precisely, if all atoms are first exposed to a θ0 pulse, so that at t = 0 U(0, ∆) = 0 , V(0, ∆) = − sin θ0 , W(0, ∆) = − cos θ0 ,

(73.10)

then if E = 0 for t > 0, an individual atom with detuning ∆ evolves according to (73.4): U − iV = i sin θ0 e−(1/T2 −i∆)t .

(73.11)

The macroscopic polarization is found by summing the individual (U − iV ) values over the detuning distribution g(∆). For simplicity, in this subsection we will ignore competition from homogeneous decay (take 1/T2 ≈ 0)

Coherent Transients

T ∗ −(∆−∆0 )2 T ∗2 /2 e g(∆) = √ , 2π

Free induction decay – dipoles dephase B A

(73.12) A

where 1/T ∗ is here defined as the width (standard devitation) of the Doppler distribution and ∆0 is the detuning of the zero-velocity atoms. The collective result is P(t)= Nd sin θ0 sin ωt e

i∆0 t

  t2 exp − ∗2 . (73.13) 2T

The detuning “inhomogeneity” in the sample leads to dephasing of the collective dipole moment, and the inhomogeneous relaxation time is obviously T ∗ . This is illustrated by the decrease of collective alignment of Bloch vectors in the top row of Fig. 73.4. For a typical room temperature gas a visible transition has a width given by 1/(2πT ∗ ) ≈ 1.5 GHz so that T ∗ ≈ 10−10 s.

73.4.2 Photon Echoes A photon echo is generated by pulse-induced recovery of a nonzero P(t) after P(t) → 0 due to free induction decay (FID). This analog of the spin echo effect is possible because each atom retains its own detuning for a relatively long time, usually up to an average collision time T2 . During FID, the U–V projection of every atom’s Bloch vector precesses steadily clockwise or counterclockwise depending on the sign of its ∆. Thus the Bloch vectors could be rephased if they could all be forced at the same moment to reverse their relative sense of precession. The prototypical echo scenario has FID beginning at t = 0, with P(t) → 0 for t  T ∗ , followed by a π–pulse at the time t , where t  T ∗ . The effect of the π-pulse is to reverse the sign of V and W [recall (73.10)], in effect flipping the equatorial plane of the Bloch sphere upside down. Thus for t  t we have Bloch vectors fanning back together. The macroscopic

t = t– ε

t=0

B

After π pulse – dipoles rephase B A

t = t + ε

A

B

t = 2t

Fig. 73.4 The ensemble of dipole moments spreads due to the distribution of resonance frequencies. The distribution of Bloch vectors in the U–V plane is shown at various times after the initial short pulse excitation. By the time t , the dipoles have spread uniformly around the unit circle. A πpulse then flips the relative orientation of the dipoles so that they subsequently rephase

polarization obeys

 (t − 2t )2 −t/T2 e P(t) = Nd sin θ0 sin ωt exp − , 2T ∗2

(73.14)

where the last factor recovers the effect of homogeneous dipole damping. We require T2  t  T ∗ for a strong echo signal in the neighborhood of t = t . The result is illustrated in Fig. 73.4. An “echo” of the initial excitation at t = 0 appears at the time t = 2t . After this, FID occurs again, and this second decay can also be reversed by applying another π–pulse, and so on, until t ≈ T2 , at which time the inevitable and irreversible homogeneous relaxation cannot be avoided. The scenario of π–pulse reversal is only the most ideal, leading to the most complete echo, and other pulse areas will also lead to echos; a more important factor is that the reversing pulse must be short enough that negligible dephasing takes place during its application.

73.5 Resonant Pulse Propagation 73.5.1 Maxwell–Bloch Equations Time-dependent atomic dipole moments created by applied fields are themselves a source of fields, another

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Part F 73.5

and assume the most common inhomogeneous lineshape (i.e., Doppler-Maxwellian):

73.5 Resonant Pulse Propagation

form of coherent transient. We limit discussion to planewave propagation in the z–direction. Note that we use z rather than Z for convenience, although in the dipole approximation the coordinate entering our equations is the

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Part F 73.5

coordinate of the center of mass of the atom rather than the internal electron coordinate. The field is generalized from (73.2) to  1

E(t, z) = (73.15) E(t, z) ei(Kz−ωt) + c.c. , 2 and the macroscopic polarization is the correspondingly generalized form of (73.9): 

 P(t, z) = Nd g(∆) Re (U − iV ) ei(Kz−ωt) d∆ . (73.16)

The difference between K and k = ω/c indicates that the refractive index is nonzero. The traveling-pulse rotating frame is also obtained by replacing ωt by ωt − Kz. If the field envelope E is slowly varying, its second derivatives can be dropped when E(t, z) is substituted into Maxwell’s wave equation. The resulting dispersive and absorptive reduced wave equations are

K 2 − k2 E = 4πk2 Nd  × g(∆)U(t, z, ∆) d∆ , 

(73.17)

 ∂ ∂ K +k E = 2πk2 Nd ∂z ∂(ct)  × g(∆)V(t, z, ∆) d∆ .

The Bloch equations along with these reduced wave equations form the self-consistent Maxwell–Bloch equations that are used to treat most resonant propagation problems in quantum optics and laser theory [73.1, 2, 5, 8].

73.5.2 Index of Refraction and Beers Law If a weak pulse of duration τ propagates in a medium of ground-state atoms (W ≈ −1), the Bloch equations have simple quasisteady-state solutions (τ  T2 ) Ω1 ∆ U= , ∆2 + 1/T22 −Ω1 /T2 V= . (73.18) ∆2 + 1/T22 When U and V are substituted back into the reduced wave equations (73.17), the dispersive equation gives the index of refraction n = K/k due to the ground-state atoms:  4πNd 2 ∆g(∆) 2 n −1 = d∆ , (73.19) ~ ∆2 + 1/T22

and the absorptive equation predicts steady state field attenuation during propagation: 1 ∂ E = − αB E . (73.20) ∂z 2 The constant αB given by  4πNd 2 ω g(∆) αB = d∆ (73.21) 2 ~cT2 ∆ + 1/T22 is called the extinction coefficient, or the reciprocal Beers length. Since field intensity I is proportional to E 2 , the solution to the absorptive equation is I(z, t) = I(0, t) e−αB z .

(73.22)

This relation is called Beers Law. Both the dispersive and absorptive results are familiar from classical physics [73.10], with the important distinction that here the ~-dependent oscillator strength enters naturally rather than as an empirical parameter from Lorentzian dielectric theory [73.8, 10].

73.5.3 The Area Theorem and Self-Induced Transparency A form of pulse propagation with no classical analog arises in the short-pulse limit (τ T2 , T1 , but τ  T2∗ ). By integration over the entire pulse, the absorptive Maxwell equation becomes an equation for ∂θ/∂z, where θ is the pulse area defined in (73.8). In the shortpulse limit, the relaxation terms in the OBEs can be ignored and when substituting from them we obtain the McCall-Hahn Area Theorem [73.1]: 1 ∂ θ(z) = − α B sin θ(z) . (73.23) ∂z 2 This predicts the same exponential attenuation as (73.20) in the case of a small area pulse, θ(z) π, but in the case of larger area pulses the behavior is quite different. In general, the area decreases during propagation for areas in the range 0 < θ(z) < π, but it increases for areas π < θ(z) < 2π. As seen from Fig. 73.5, this change of area with propagation causes the pulse area to evolve to one of the stable values 0, 2π, 4π, . . . . There is one special pulse, a soliton solution with area exactly 2π, which propagates without shape change in the short pulse limit, given by   2~ t − z/v E(t, z) = sech , (73.24) τd τ where τ is the pulse duration, which is arbitrary but must satisfy the short-pulse inequality τ T1 , T2 . The soliton’s group velocity is determined by the corresponding

Coherent Transients

δθ δz

soliton solutions to the OBE’s and the dispersive v=

c 1 + 12 αB cτ

,

(73.25)

0

π

2π

3π

where αB is to be taken in the limit T2∗ T2 . The group velocity can be slower than the speed of light by orders of magnitude if αB cτ  1.

4π

θ

73.6 Multi-Level Generalizations 73.6.1 Rydberg Packets and Intrinsic Relaxation A short laser pulse can populate a band of excited states whose probability amplitudes will exhibit mutual coherence. This single-atom coherence is transient, even without collisions or other external perturbations to disrupt it, and its decay can be called intrinsic relaxation. The decay is basically a dephasing. The dipole moments associated with the excited band interfere due to the wide variety of resonance frequencies of the states in the superposition. Because of the discreteness of the energy levels of any bounded quantum system, this relaxation has its own unique characteristics, including similarities with both homogeneous and inhomogeneous decay. The wave function for a coherently excited atom can be expressed in the interaction picture in the form [73.9]  Ψ(r, t) = a(t)ψg (r) + bn (t) e−iωn t ψn (r) , (73.26) n

where ψg (r) is the ground state wave function, and n labels the states in a band with excitation frequencies ωn ≈ ω. If |ωn − ω| ωn , the transition frequency ωn can be expanded about the principal quantum number n of the resonant excited state E n = ~ω to obtain 1 ∂ωn ∂ 2 ωn + (n − n)2 2 + · · · ∂n 2! ∂n 2π 2 2π + (n − n) + · · · . (73.27) = ωn + (n − n) TK TR Thus 2π/TK is the mean frequency separating neighboring levels, i. e., TK /2π = ~ρ(E), where ρ(E) is the density of excited states, and 2π/TR is the mean change in this frequency separation. TK is the same as the Kepler period a classical orbit, and TR is the revival time. ωn = ωn + (n − n)

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Part F 73.6

Fig. 73.5 McCall-Hahn area theorem for an absorbing medium. On propagation the area of the pulse will follow the arrows toward one of the stable values, 0, 2π, 4π, . . .

73.6 Multi-Level Generalizations

Substituting the Bohr frequencies into the definitions for TK and TR yields TR = nTK /3. For times t ≤ TR the expansion can be truncated after the third term. Then the wave function is given approximately by  bn+m (t) e−i2πmt/TK Ψ(r, t) ≈ a(t)ψg (r) + e−iωn t m −i2πm 2 t/TR

× e

ψn+m (r) ,

(73.28)

where m = n − n. For high Rydberg states, n  1, the time scales associated with the two exponentials inside the sum are quite different. For times t TK the individual levels are not resolved, thus the laser excites what is effectively a continuum with a density of states ρ(E). In that case the ground state population simply decays exponentially at the rate given by first-order perturbation theory, Γ = (2π/~)d 2 E 2 ρ(E). At longer times t ≈ TK , the first exponential contributes, but the second does not, giving a simple Fourier series time dependence. In this regime the evolution of the wave function is just periodic motion of a wave packet around a Kepler orbit, as is illustrated in Fig. 73.6a. The coherent quantum wave packet behaves like a classical particle for many Kepler periods, gradually spreading out as the second exponential in the sum (73.28) begins to contribute. This spreading of the wave packet produces the intrinsic relaxation of the collective dipole moments from the various transitions. However, because the levels are discrete, this decay is not permanent, but is reversed and leads to a spontaneous “revival” of the original wave packet [73.6], without the need for a π-pulse to produce an “echo.” In its evolution toward the revival, the wave packet passes through a number of fractional revivals in which

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Part F 73.6

miniature replicas of the original wave packet are equally spaced around the orbit, each traveling at the velocity of a particle traveling in a corresponding classical Kepler orbit. This complex time evolution arises from the spreading of the wave packet all the way around the orbit so that the head and tail of the packet interfere with each other, producing interference fringes. The further evolution of this fringe pattern produces the various revival phenomena shown in Fig. 73.6. a)

73.6.2 Multiphoton Resonance and Two-Photon Bloch Equations Multiphoton transitions Fig. 73.7a introduce new coherent transient phenomena. If levels |g and |e have the same parity, two photons from the same laser field are sufficient to excite level |e. For simplicity we regard |e as a single state, but any number of intermediate levels | j of opposite parity may be present. Substituting the state vector  |Ψ(t) = ag (t)|g + b j (t) e−iωt | j j

+ ae (t) e−i2ωt |e ,

b)

(73.29)

into the Schrödinger equation yields dag 1 =− Ωg j b j , i dt 2

(73.30)

j

 db j 1 = ∆ jg b j − Ω jg ag + Ω je ae , dt 2 dae 1 = ∆eg ae − i Ωe j b j , dt 2

i

(73.31) (73.32)

j

c)

d)

where the ∆s are the detunings and the Ωs are the Rabi frequencies for the dipole-allowed transitions. For example, Ωg j = dg j E/~, and ∆ jg = (E j − E g )/~ − ω and ∆eg = (E e − E g )/~ − 2ω. If the states | j are not too close to resonance, the b j oscillate rapidly and to a first approximation average to zero. A better approximation is to retain the small nonzero solution for b j obtained by setting db j / dt = 0 in (73.31) to obtain
 Ω jg ag + Ω je ae , (73.33) bj = − 2∆ jg a)

Fig. 73.6a–d The free evolution of a Rydberg wave packet made up of a superposition of circular-orbit states centered about n = 360. (a) Initially the wave packet is to good approximation a minimum uncertainty wave packet in all three dimensions, but after 12 orbits (b) the packet has spread all the way around the orbit so that the head and tail of the wave packet overlap, producing interference fringes. (c) After 40 orbits, t = TR /3, the fringes have produced the one-third fractional revival in which three miniature replicas of the original wave packet are equally spaced around the orbit. (d) After 120 orbits, t = TR , the complete wave packet revival occurs

b) e ω ω

j

ωa j g

ωb

e

g

Fig. 73.7a,b Model two-photon resonances. Two photons

couple the ground state |g with an excited state |e. Many intermediate nonresonant levels | j are present. In (a) we have the cascade system, and in (b) the Λ pump–probe system

Coherent Transients

dU (2) = −∆(2) V (2) , dt dV (2) = ∆(2) U (2) + Ω (2) W (2) , dt dW (2) = −Ω (2) V (2) , dt

(73.34)

Here the superscript (2) indicates that the variables are identified with the two-photon |g → |e transition. The various coefficients are similarly generalized [73.11]. For example, the two-photon Rabi frequency is given by Ω (2) ≡

1  dg j d je 2 E . 2 ~2 ∆ jg

(73.35)

j

and the two-photon detuning ∆(2) incorporates the laserinduced level shifts ∆(2) ≡ ∆e j + ∆ jg 1  |de j |2 E 2 (t) 1  |d jg |2 E 2 (t) + + . 4 4 ~2 ∆e j ~2 ∆ jg j

j

(73.36)

The last two terms give the difference in the ac Stark shifts of the upper and lower levels produced by the laser field. W (2) is the inversion as before, but U (2) and V (2) are somewhat different. They cannot be directly tied to the expectation value of a dipole moment because levels |g and |e have the same parity. Thus, while the quantity U (2) − iV (2) is the two-photon analog of U − iV in the original OBE’s, it cannot serve as a source term in the Maxwell equation. In the case of cw applied fields, the solutions to the two-photon OBEs are formally identical to those for a two-level atom. In the case of pulsed fields, however, the detuning ∆(2) (t) is automatically “chirped” in frequency by the Stark shifts. This chirping may significantly modify the dynamics. Multiphoton generalizations of the Bloch equations can be made for other arrangements and numbers of levels. The two-photon

version applies as well to three-level Λ and V configurations as to the cascade system shown in Fig. 73.7a, for which they were derived.

73.6.3 Pump–Probe Resonance and Dark States Dark states or trapping states occur whenever a fielddependent linear combination of active levels is dynamically disconnected from the other levels. This occurs, e.g., in a pump-probe interaction, which fits the scenario of Fig. 73.7 if two lasers instead of one are used to excite level |e from the ground level via two-photon resonance. A strong steady laser a is applied for the |g → | j transitions and a weak tunable “probe” laser b for the | j → |e transitions. In the simplest format, ∆eg = 0, and all the | j levels can be combined into a single level labeled |2. The three-level state vector can be written in terms of field-free states |Ψ(t) = ag |g + b2 |2 + ae |e ,

(73.37)

or, in terms of field-dependent dressed states |Ψ(t) = AT |T  + b2 |2 + A S |S ,

(73.38)

where Ω|T  ≡ Ωa |e − Ωb |g, Ω|S ≡ Ωa |g + Ωb |e and AS (t) ≡ Ω −1 [Ωa ag + Ωb ae ] , AT (t) ≡ Ω −1 [Ωa ae − Ωb ag ] . (73.39)  The normalizing factor is Ω ≡ Ωa 2 + Ωb 2 . The state |T  is an eigenvector of the three-level RWA Hamiltonian, with eigenvalue zero, and the amplitude AT (t) is a constant of motion. Thus |T  is termed a trapping state, and population in |T  is inaccessible to the (possibly very strong) laser fields. At two-photon resonance this conclusion is robust, not depending strongly on the idealized conditions assumed here. In fact, AT , the trapping state amplitude, is an adiabatic invariant, remaining constant to first order even under slow changes in Ωa and Ωb . In a pump-probe experiment, this trapping state is observed as an abrupt drop in probe absorption as the probe frequency is tuned through two-photon resonance. Since only two-photon resonance is required (both transitions can be equally detuned) this coherent transient effect has no analog in two-level physics. The ideal method for exciting the trapping state from the ground state uses another coherent transient process called counter-intuitive pulse sequencing in which pulse b is turned on first. The trapping state |T  is essentially

1073

Part F 73.6

which can be used to eliminate b j from the equations for ag and ae . This is called adiabatic elimination of dipole coherence. In this approximation, levels |g and |e are directly coupled to each other and two-photon coherence arises. The coupling of levels |g and |e is similar to the two-level coupling described in Sect. 73.1 and two-photon Bloch equations analogous to (73.5) are the result:

73.6 Multi-Level Generalizations

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Part F 73.7

the ground state |g if Ωa = 0. Thus if Ωb is turned on first, and then Ωa turned on later, the ground state adiabatically becomes the trapping state and all initial probability flows smoothly with it. An essential point is the ease with which pumpprobe adiabaticity is maintained, particularly for strong fields on resonance, in contrast to one-photon adiabaticity, which is never achieved at strong-field resonance. In the pump-probe case one must only satisfy the inequality  

3/2  dΩa dΩb  2 2  Ω Ω − , (73.40) a  Ωa + Ωb  dt b dt which is automatically accomplished by counterintuitive pulse sequencing. The inequality allows one to tolerate rapid change of Ωb while Ωa = 0. Then after Ωb has reached a high value, Ωa can also be turned on very rapidly because the right side of (73.40) is already very large. This is “counter-intuitive” excitation because if the population is in level |g it is “natural” to turn on pulse Ωa first, not Ωb . It can be dramatically beneficial to use counter-intuitive excitation when it is important to avoid relaxation associated with level |2.

73.6.4 Three-Level Transparency The foregoing results for three-level excitation can be extended to resonant pulse propagation in three-level media. The equations governing simultaneous two-pulse evolution in the local-time coordinates cT ≡ ct − z and ζ ≡ z are ∂Ea ~ = i µa a∗g b2 , ∂ζ da ∂Eb ~ = i µb ae∗ b2 , (73.41) ∂ζ db where 4πda2 Nωa , etc. (73.42) µa = ~c Note that the bilinear combination 2a∗g b2 corresponds to U − iV in (73.16).

Soliton-like pulses can propagate in three-level media. Both pulses must compete for interaction with level |2. They depend only on a single variable Z ≡ ζ − uT where u is the pulse’s constant velocity in the moving frame. Soliton solutions are given (for µa = µb ) for Λ media by Ea = ~/da = A sech K Z , Eb = ~/db = B tanh K Z ,

(73.43)

and ag = − tanh K Z −2iKu b2 = sech K Z , A B ae = sech K Z , A

(73.44)

where the parameters A, B, Ku are nonlinearly related to the pulse length τ:  2 Ku ≡ 1/τ and 2/τ = A2 − B 2 . (73.45) The moving frame velocity is given by 1/u = 2µ/A2 and the expression for the lab frame velocity V is 1/V = 1/c + 2µ/A2 .   If B → 0, then Ea → 2~/τda sechK Z, which is the exact McCall–Hahn formula for the two-level one-pulse soliton amplitude [73.1]. No adiabatic condition was invoked in obtaining the soliton solutions. The physical measure of adiabaticity comes from the pulse duration τ. If τ is short, an appreciable population appears transiently in level |2, but if τ is long (an adiabatic pulse), the population skips level |2 and goes directly from |g to |e and back again during the pulse. Note that the sech and tanh functions are ideally counter-intuitive, with pulse b starting infinitely far ahead of pulse a. In practice, the infinite leading edge of the tanh function plays no role and can be truncated to several times τ without appreciable change in the character of the pulse pair.

73.7 Disentanglement and “Sudden Death” of Coherent Transients The existence of entanglement (non-separability) of states is the most prominent evidence that quantum mechanics is a truly nonlocal theory. This has consequences for coherent transients, allowing them to exhibit nonintuitive effects unlike any discussed up to this point. We will choose an example that directly illustrates this

point by showing that two entangled atoms whose inversion and coherence decay exponentially can have an entanglement that not only does not decay exponentially but which reaches its steady state long before the atoms reach their final states. In this sense the nonlocal transients of the pair of atoms are not at all

Coherent Transients

ρ AB (t) =

4  µ=1

K µ (t)ρ AB (0)K µ† (t),

(73.46)

where the so-called Kraus operators [73.14] K µ (t) are available in closed form in this case [73.15]. For illustration we will choose a partially coherent initial state, expressed by a two-atom density matrix with a single free parameter a:   a 0 0 0  1 0 1 1 0  (73.47) ρ AB (0) =  . 3 0 1 1 0  0 0 0 1−a Here the convention is to label the rows and columns in the order ++, +−, −+, −−. The transient decay of either atom’s excitation can be calculated separately from their reduced density matrices ρ A ≡ TrB {ρ AB }, etc. For example: ρ A =
+ B |ρ AB |+ B  +
− B |ρ AB |− B    1 1+a 0 , = 3 0 2−a

Atom A

Atom B

Cavity A

Cavity B

Fig. 73.8 Illustration of a set-up in which two partially

excited atoms A and B are located inside two spatially separated cavities that are possibly very remote from each other. The two atoms are assumed initially entangled, but they have no interaction

of Wootters [73.17], which has the convenient normalization 1 ≥ C(t) ≥ 0, where C = 1 represents completely entangled atoms (such as in a pure Bell state, for example) and C = 0 denotes the complete absence of entanglement. For the specific case a = 1 one finds for the state (73.47) the initial concurrence C(0) = 23 , indicating a state with partial two-party coherence (incomplete entanglement). At time t one then finds: C AB (t) =

  2 max 0, e−t/τ0 f(t) , 3

(73.49)

√ where f(t) = 1 − 2ω2 + ω4 , and ω ≡ 1 − e−t/τ0 . The strikingly non-intuitive consequence of this expression is that C(t) → 0 abruptly after a short time (disentanglement suffers a “sudden death”) if 2ω2 + ω4 ≥ 1. In fact, only a minor algebraic rearrangement shows that this strange condition must occur, and the finite sudden

C( ρ ) 1 2⁄

3

(73.48)

and the upper level excitation of atom A can A (t) = be found to behave exactly as expected: ρ++ a+1 −t/τ0 e , where τ is the usual spontaneous emission 0 3 lifetime. For nonlocal transients we will need a timedependent measure of entanglement, and there are several options [73.16], all related to the joint entropy of the two-atom system. We will use the concurrence C(t)

1⁄

3

0 0.5

0

1 0.5 2 2.5 t 3

1⁄

3

2⁄ a

1075

Part F 73.7

intuitively related to the local transients affecting the atoms separately. We imagine the situation sketched in Fig. 73.8, two two-level atoms A and B of the type discussed at length already, prepared in a partially excited entangled mixed state and located remotely from each other, without direct or indirect interaction. They each must eventually, because of spontaneous emission, come to their ground states, creating the final joint state |− A  ⊗ |− B , which is clearly in factored form (disentangled). The question is, what is the manner of evolution by which the quantum entanglement feature evolves toward zero. The standard Master Equation methods [73.12] for investigating spontaneous emission [73.13] can be applied to each atom separately since they are not interacting with each other. For any initial state ρ AB (0), the density operator at t can be expressed as

73.7 Disentanglement and “Sudden Death” of Coherent Transients

3

Fig. 73.9 Effect of spontaneous emission on concurrence

of two two-level atoms given the initially entangled mixed state (73.47) depending on the single parameter 1 ≥ a ≥ 0

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Part F 73

death time t0 is given by  √  t0 2+ 2 ≈ 0.53 . ≡ ln τ0 2

(73.50)

The non-local coherent transient behavior of entanglement for the entire range of allowed a values [73.15] is shown in Fig. 73.9. This shows that the concurrence undergoes familiar smooth and infinitely long decay only for a values in the limited range 13 > a ≥ 0. Otherwise

sudden death occurs sooner or later. In our example, non-local coherence becomes zero most abruptly after a finite time for a = 1, the case that was calculated in (73.49). Although not yet observed experimentally, it appears that these results are not exceptional. Sudden termination of entanglement has also been predicted for two-party continuum states as well as for qubit pairs experiencing only T2 decay, in contrast to the combined T1 and T2 decay appropriate to spontaneous emission as treated here.

References 73.1 73.2 73.3 73.4 73.5 73.6 73.7 73.8

L. Allen, J. H. Eberly: Optical Resonance and TwoLevel Atoms (Dover, New York 1987) B. W. Shore: Theory of Coherent Atomic Excitation, Vol. 1,2 (Wiley, New York 1990) P. Meystre, M. Sargent III: Elements of Quantum Optics (Springer, Berlin, Heidelberg 1990) C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg: Atom-Photon Interactions (Wiley, New York 1992) M. O. Scully, M. S. Zubairy: Quantum Optics (Cambridge Univ. Press, Cambridge 1997) W. P. Schleich: Quantum Optics in Phase Space (Wiley-VCH, New York 2001) C. C. Gerry, P. L. Knight: Introductory Quantum Optics (Cambridge Univ. Press, Cambridge 2005) P. W. Milonni, J. H. Eberly: Lasers (Wiley, New York 1988)

73.9 73.10 73.11 73.12 73.13 73.14

73.15 73.16 73.17

M. V. Fedorov: Atomic and Free Electrons in a Strong Light Field (World Scientific, Singapore 1997) J. D. Jackson: Classical Electrodynamics, 2nd edn. (Wiley, New York 1975) F. H. M. Faisal: Theory of Multiphoton Processes (Plenum, New York 1987) See Sect. 78.7 in this book See Sect. 78.12 in this book A complete discussion of Kraus operators appropriate to Bloch vector evolution is given in S. Daffer, ´dkiewicz, J. K. McIver: J. Mod. Optics 51, 1843 K. Wo (2004) Ting Yu, J. H. Eberly: Phys. Rev. Lett. 93, 140404 (2004) See Chapt. 81 in this book W. K. Wootters: Phys. Rev. Lett. 80, 2245 (1998)

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Multiphoton a 74. Multiphoton and Strong-Field Processes

74.1

74.1.4 74.1.5 74.1.6 74.1.7

Autoionization .......................... 1079 Coherence and Statistics ............ 1079 Effects of Field Fluctuations ........ 1079 Excitation with Multiple Laser Fields .............................. 1080

74.2 Strong-Field Multiphoton Processes ...... 1080 74.2.1 Nonperturbative Multiphoton Ionization ................................ 1081 74.2.2 Tunneling Ionization ................. 1081 74.2.3 Multiple Ionization.................... 1081 74.2.4 Above Threshold Ionization ........ 1081 74.2.5 High Harmonic Generation ......... 1082 74.2.6 Stabilization of Atoms in Intense Laser Fields ............... 1083 74.2.7 Molecules in Intense Laser Fields .............................. 1084 74.2.8 Microwave Ionization of Rydberg Atoms ...................... 1084 74.3 Strong-Field Calculational Techniques ... 1086 74.3.1 Floquet Theory .......................... 1086 74.3.2 Direct Integration of the TDSE ..... 1086 74.3.3 Volkov States ............................ 1086 74.3.4 Strong Field Approximations....... 1087 74.3.5 Phase Space Averaging Method ... 1087

Weak Field Multiphoton Processes ........ 1078 74.1.1 Perturbation Theory................... 1078 74.1.2 Resonant Enhanced Multiphoton Ionization .............. 1078 74.1.3 Multi-Electron Effects ................ 1079

References .................................................. 1088

The excitation of atoms by intense laser pulses can be divided into two broad regimes determined by the characteristics of the laser pulse relative to the atomic response. The first regime involves relatively weak optical laser fields of long duration (> 1 ns), and the second involves strong fields of short duration (< 10 ps). These will be referred to as the weak-long (WL) and strongshort (SS) cases respectively. In the case of atomic excitation by WL pulses, the intensity is presumed to be high enough for multiphoton transitions to occur. The resulting spectroscopy of absorption to excited states is potentially much richer than single-photon excitation because it is not limited by the single-photon selection rules for radiative transitions. However, the intensity is still low enough for a theoretical description based on perturbations of field-free atomic states to be valid, and the time depen-

dence of the field amplitude does not play an essential role. The SS case is fundamentally different in that the atomic electrons are strongly driven by fields too large to be treated by perturbation theory, and the time dependence of the pulse as it switches on and off must be taken into account. Atoms may absorb hundreds of photons, leading to the emission of one or more electrons, as well as photons of both lower and higher energy. Because the flux of incident photons is high, a classical description of the laser field is adequate, but the time dependent Schrödinger equation (TDSE) must be solved directly to obtain an accurate representation of the atom–field interaction. For SS pulses of optical wavelength, it is sufficient in most cases to consider only the electric dipole (E1) interaction term defined in Chapt. 68. The atom–field

Part F 74

The excitation of atoms by intense laser pulses can be divided into two broad regimes: the first regime involves relatively weak optical laser fields of long duration, and the second involves strong fields of short duration. In the first case, the intensity is presumed to be high enough for multiphoton transitions to occur. The resulting spectroscopy is not limited by the single-photon selection rules for radiative transitions. However, the intensity is still low enough for a theoretical description based on perturbations of field-free atomic states to be valid, and the time dependence of the field amplitude does not play an essential role. In the second case, the field intensities are too large to be treated by perturbation theory, and the time dependence of the pulse must be taken into account. A discussion on the generation of sub-femtosecond pulses is included.

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Part F 74.1

interaction can then be expressed in either the length gauge or the velocity gauge [74.1] (Chapt. 21). In the length gauge, the TDSE is  ∂Ψ(r, t)
i~ = H0 + er · E(t) Ψ(r, t) , (74.1) ∂t where H0 is the field-free atomic Hamiltonian, r the collective coordinate of the electrons, and E(t) the electric field of the laser given by 
E(t) = E(t) xˆ cos(ωt + ϕ) +  yˆ sin(ωt + ϕ) . (74.2)

Here ϕ is the phase and  defines the polarization: linear if  = 0 and circular if || = 1. In the velocity gauge, the TDSE is   e2 ie~ ∂ψ(r, t) 2 = H0 − A(t) · ∇ + A (t) i~ ∂t mc 2mc2 (74.3) × ψ(r, t) .  t    Here A(t) = −c E t dt is the vector potential of the laser field. The solutions of (74.1) and (74.3) are related

by the phase transformation   ie Ψ(r, t) = exp r · A(t) ψ(r, t) . ~c

(74.4)

Since lasers usually must be focussed to reach the strong field regime, measured electron and ion yields include contributions from a distribution of field strengths. The photoemission spectrum, on the other hand, contains a coherent component due to the macroscopic polarization of all the atoms and therefore is sensitive also to the laser phase variations within the focal volume. In this chapter methods for solving (74.1) and (74.3) are discussed along with details of the atomic emission processes. References [74.1–3] are three recent books which provide excellent introductions to this subject. Further developments are well described in the proceedings of the International Conferences on Multiphoton Physics [74.4–7] and the NATO workshop on SuperIntense Laser-Atom Physics [74.8, 9].

74.1 Weak Field Multiphoton Processes 74.1.1 Perturbation Theory Since atomic ionization energies are generally
10 eV, while optical photons have energies of only a few eV, several photons must be absorbed to produce ionization, or even electronic excitation in the case of the noble gases. For WL pulses, the electronic states are only weakly perturbed by the electromagnetic field. The rate of an n-photon transition can then be calculated using nth order perturbation theory for the atom–field interaction. For an incident photon number flux φ of frequency ω, the rate is  2παφω n

(n)

2 (n) Wi→ f = 2π

Ti→ f δ(ωi + nω − ω f ) , e2 (74.5)

where



(n)

Ti→ f = f d G[ωi + (n − 1)ω] × d G[ωi + (n − 2)ω] · · · d G[ωi + ω] d|i ,

(74.6)

|i is the ith eigenstate of the field-free atomic Hamiltonian, d = e
ˆ · r, with
ˆ the polarization direction and | j j|  . (74.7) G(ω) = ω − ω j + iΓ j /2 j

The sum over j includes an integration over the continuum for all sequences of E1 transitions allowed by angular momentum and parity selection rules. Methods for calculating cross sections and rates in the weak-field regime are described in [74.1, 10] and in Chapt. 24.

74.1.2 Resonant Enhanced Multiphoton Ionization For multiphoton ionization, ω can be continuously varied because the final state in (74.5) lies in the continuum. If ω is tuned so that ωi + mω  ω j for some contributing intermediate state | j in (74.7), then that state lies an integer m photons above the initial state, and the corresponding denominator vanishes (to within the level width Γ j ), producing a strongly peaked resonance. Since it takes k = n − m additional photons for ionization, the process is called m, k resonant enhanced multiphoton ionization (REMPI). Measurements of the photoelectron angular distribution are useful in characterizing the resonant intermediate state. Calculations using the semi-empirical multichannel quantum defect theory to provide the needed matrix elements have been very successful in describing experimental results. This technique is discussed in more detail in Chapt. 24.

Multiphoton and Strong-Field Processes

74.1.3 Multi-Electron Effects Multiply excited states can play a role in multiphoton excitation dynamics. These states are particularly important if their energies are below or not too far above the first ionization potential. Configuration expansions including these states have been used successfully in studies, for example, of the alkaline earth atoms, which have many low-lying doubly excited states. The presence of these states also can enhance the direct double ionization of an atom [74.12].

74.1.4 Autoionization The configuration interaction between a bound state and an adjacent continuum leads to an absorption profile in the single photon ionization spectrum with a Fano lineshape. The actual lineshape reflects the interference between the two pathways to the continuum. Autoionizing states can also be probed via multiphoton excitation [74.13, 14]. Because, in the strong field regime, coupling strengths and phases change with intensity, the lineshapes can be strongly distorted by changing the incident intensity. At particular intensities, the phases of the excited levels can be manipulated to prevent autoionization completely. Then a trapped population with energy above the ionization limit can be created [74.15].

74.1.5 Coherence and Statistics Real laser fields exhibit various kinds of fluctuations, and so are never perfectly coherent. The effects of such fluctuations on the complex electric field amplitude
 (74.8) E(t) = E(t) exp − iωt + ϕ(t ] can be modeled by a variety of stochastic processes [74.16], depending on the conditions [74.10, 17– 20], as follows. For cw lasers, a phase diffusion model (PDM) is often used for which E(t) = E = const. and √ ϕ(t) (74.9) ˙ = 2bF(t) ,

1079

where F(t) describes white noise by a real Gaussian function [74.16] characterized by the averaged values F(t) = 0, F(t)F(t  ) = 2bδ(t − t  ). The stochastic electric field then has an exponential autocorrelation function






 E(t)E ∗ (t  ) = E 2 exp −b t − t  − iω(t − t  ) , (74.10)

and a Lorentzian spectrum of width b. Far off resonance, such a Lorentzian spectrum often gives unrealistic results, and the model (74.9) is then replaced by an Ornstein–Uhlenbeck process,  ϕ(t) (74.11) ¨ = −β ϕ(t) ˙ + 2bβ F(t) , where the parameter β for β b plays the role of a cutoff of the Lorentzian spectrum. A multimode laser with a large number M of independent modes has a field of the form E(t) = M j=1 E j exp[−iω j t + iϕ j (t)], and according to the central limit theorem [74.16], can be described for large M as a complex Gaussian process defined to be a chaotic field,  

˙ = −(b + iω)E(t) + 2b |E(t)|2 F(t) , (74.12) E(t) where F(t) is now a complex white noise, and ω is the central frequency of the field. The field, (74.12) has an exponential autocorrelation function, and a Lorentzian spectrum of width b. Various other stochastic models have been discussed in the literature. These include Gaussian fluctuations of the real amplitude of the field E(t); Gaussian chaotic fields with non-Lorentzian spectra; non-Gaussian, nonlinear diffusion processes (that describe for instance a laser close to threshold [74.16]); multiplicative stochastic processes (that describe a laser with pump fluctuations [74.18]) and jump-like Markov processes [74.21–23]. Statistical properties of laser fields can sometimes be controlled experimentally to a great extent [74.19, 20].

74.1.6 Effects of Field Fluctuations Since the response of systems undergoing multiphoton processes is in general a nonlinear function of the field intensity (and, in particular, of the field amplitude), it depends in a complex manner on the statistics of the field. The enhancement of the nonresonant multiphoton ionization rate illustrates the point. According to the perturbation equation (74.5), the rate of an n-photon process is proportional to φn ; i. e., to I n , where I is the field

Part F 74.1

The perturbation equation (74.5) indicates that the rate for nonresonant multiphoton ionization scales as φn for an n-photon process [74.11]. However, this is not the case for REMPI since the resonant transition saturates and (74.5) no longer applies. Then the rate can be controlled either by the m-photon resonant excitation step, or by the number of photons k needed for the ionization step.

74.1 Weak Field Multiphoton Processes

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intensity. For fluctuating fields, the average response is thus 

 

2n  W (n) ∝ I n ∝ E(t) . (74.13) i→ f

Part F 74.2

Phase fluctuations (as described by PDM) do not affect the average. On the other hand, for complex chaotic fields, the average is I n   n!In ,

(74.14)

i. e., significant enhancement of the rate for n > 1. Field fluctuations lead to more complex effects in resonant processes. Two well-studied examples are the enhancement of the ac Stark shift in resonant multiphoton ionization [74.24], and the spectrum of double optical resonance – a process in which the ac Stark splitting of the resonant line is probed by a slightly detuned fluctuating laser field [74.18]. Double optical resonance is very sensitive not only to the bandwidth of the probing field, but also to the shape of its frequency spectrum.

74.1.7 Excitation with Multiple Laser Fields The simultaneous application of more than one laser field produces interesting and novel effects. If a laser and its second (2ω) or third (3ω) harmonic are combined and the relative phase between the fields controlled, product state distributions and yields can be altered dramatically. The effects include reducing the excitation or ionization

rates in the ω − 3ω case [74.25] or altering the photoelectron angular distributions and the harmonic emission parity selection rules using ω − 2ω [74.26]. A laser field can dress or strongly mix the fieldfree excited states, including the continuum, of an atom. This can produce a number of effects depending on how the dressed system is probed. By coupling a bound, excited state with the continuum, ionization strengths and dynamics are altered, resulting in new resonance-like structures where none existed before. This effect is called laser-induced continuum structure, or LICS [74.15,27]. This general idea has been exploited to design schemes for lasers without inversion [74.28] in which the dressed atom can have an inverted population, allowing gain even though in terms of the undressed states the lower level has the largest population. A laser can produce dramatic changes in the index of refraction of an atomic medium [74.29], creating, at specific frequencies, laser-induced transparency for a second, probe laser field.Multistep ionization, where each step is driven by a laser at its resonant frequency, has resulted in two useful applications. These are: efficient atomic isotope separation [74.23]; and the detection of small numbers of atoms in a sample, called single-atom detection. This technique is extremely sensitive because the use of exact resonance for each step yields very large cross sections for ionization, and the efficiency of collecting ions is high [74.30].

74.2 Strong-Field Multiphoton Processes Recently developed laser systems can produce very short pulses, some as short as a few to tens of femtoseconds, while at the same time maintaining the pulse energy so that the peak power becomes very high. Focused intensities up to 1019 W/cm2 have been achieved. Because the pulses are short, atoms survive to much higher intensities before ionizing, making possible studies of laser-atom interactions in an entirely new regime. A discussion of the status of short pulse laser development is given in Chapt. 71 and in [74.31]. With increasing intensity, higher-order corrections to (74.6) contribute to the transition rate. The next order correction comes from transitions involving two additional photons, one absorbed and one emitted, leading to the same final state. One effect of these terms is to shift the energies of the excited states in response to the oscillating field. This is called the dynamic or ac Stark shift. The ac Stark shift of the ground state tends to be small because of the large detuning from the ex-

cited states for long wavelength photons. On the other hand, in strong fields the shift of the higher states and the continuum can become appreciable. Electrons in highly excited states respond to the oscillating field in the same manner as a free electron. Their energies shift with the continuum by the amount Up =

(1 + )e2 E 2 , 4mω2

(74.15)

where Up is the cycle-averaged kinetic energy of a free electron in the field and  defines the polarization of the field in (74.2). Up is called the ponderomotive or quiver energy of the electron. For strong laser fields, Up can be several eV or more, meaning that during a pulse, many states shift through resonance as their energies change by an amount larger than the incident photon energy. The resulting intensity-induced resonances can dominate the ionization dynamics.

Multiphoton and Strong-Field Processes

which is less than unity when tunneling dominates and larger than unity when multiphoton ionization dominates.

74.2.1 Nonperturbative Multiphoton Ionization The breakdown of perturbation theory for nth-order multiphoton processes occurs when the higher-order correction terms become comparable to the nth-order term. Assuming that the dipole strength is ∝ ea0 , where a0 is the Bohr radius, and the detuning is δ ∝ ω, the ratio of an (n + 2)-order contribution to the nth-order term from (74.6) is [74.1]     2παφω  ea0 2 I ωa 2 Rn+2,n  = , 2 ω Iγ 2ω e (74.16)

where ~ωa  27.2114 eV is the atomic unit of energy e2 /a0 , and Iγ  3.509 45 × 1016 W/cm2 is the intensity corresponding to an atomic unit of field strength, 1/2 given by E a = αc m/a03  5.1422 × 109 V/cm. The atomic unit of intensity itself is defined by Ia = φa ~ωa = αcE a2 ,

attraction of the ion core combines with the laser electric field to form an oscillating barrier through which the electron can escape by tunneling, if the amplitude of the laser field √ is large enough. The dc rate for this process is eE/ 2m IP , where IP is the ionization potential of the electron. When this rate is comparable to the laser frequency, tunneling becomes the most probable ionization mechanism [74.34–36]. The ratio of the incident laser frequency to the tunneling rate is called the Keldysh parameter, and is given by  (74.18) γ = IP /2Up ,

(74.17)

6.436 414 (4) × 1015 W/cm2 .

which is Thus Iγ = Ia /(8πα). For photon energies of 1 eV, Rn+2,n becomes unity for I ∼ 1014 W/cm2 . Because of the large number of (n + 2)-order terms, perturbation theory actually fails for I > 1013 W/cm2 . Above this critical intensity, nonresonant n-photon ionization ceases to scale with the φn dependence predicted by perturbation theory.

74.2.2 Tunneling Ionization At sufficiently high intensity and low frequency, a tunneling mechanism changes the character of the ionization process. For lasers in the ir or optical range, a strongly bound electron can respond to the instantaneous laser field since the oscillating electric field varies slowly on the time scale of the electron. The Coulomb

74.2.3 Multiple Ionization Excitation and ionization dynamics are dominated by single electron transitions in the strong field regime. Although atoms can lose several electrons during a single pulse, the electrons are released sequentially. There is no convincing evidence of significant collective excitation in atoms in strong fields, even though it has been extensively sought. Once one electron is excited in an atom, the remaining electrons have much higher binding energies. As a result, the laser field is unable to affect them significantly until it reaches much higher intensity. By that time the first electron has been emitted. Simultaneous ejection of two electrons occurs as a minor channel (< 1%) in strong field multiple ionization. Although it is possible that doubly excited states of atoms could assist in the double ionization, in the helium and neon cases studied, these states are unlikely to be contributors [74.37].

74.2.4 Above Threshold Ionization In strong optical and ir laser fields, electrons can gain more than the minimum amount of energy required for ionization. Rather than forming a single peak, the emitted electron energy spectrum contains a series of peaks separated by the photon energy. This is called above threshold ionization, or ATI [74.38–40]. The peaks appear at the energies E s = (n + s)~ω − IP ,

(74.19)

where n is the minimum number of photons needed to exceed IP , and s = 0, 1, . . . is called the number of

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Part F 74.2

Electrons promoted into the continuum acquire the ponderomotive energy, oscillating in phase with the field. In a linearly polarized field, the amplitude of the quiver motion of a free electron, given by eE/4mω2 , can become many times larger than the bound state orbitals. If the initial velocity of an electron is small after ionization, it can be accelerated by the field back into the ion core. The subsequent rescattering changes the photoelectron energy and angular distributions, and allows the emission of high energy photons [74.32, 33]. This simple dynamical picture forms the basis of the current understanding of many strong-field multiphoton processes.

74.2 Strong-Field Multiphoton Processes

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excess photons or above threshold photons carried by the electron. Calculations in the perturbative regime for ATI are given for hydrogen in [74.11].

Part F 74.2

Peak Shifting As the intensity approaches the nonperturbative regime, the ac Stark shift of the atomic states begins to play a significant role in the structure of the ATI spectrum. The first effect is a shift of the ionization potential, given roughly by the ponderomotive energy Up . Additional photons may then be required in order to free the electron from the atom; i. e., enough to exceed IP + Up . If the emitted electron escapes from the focal volume while the laser is still on, it is accelerated by the gradient of the field. The quiver motion is converted into radial motion, increasing the kinetic energy by Up and exactly canceling the shift of the continuum. The electron energies are still given by (74.19). However, when Up exceeds the photon energy, the lowest ATI peaks disappear from the spectrum. In this long pulse limit, no electron is observed with energy less than Up . This is called peak shifting in that the dominant peak in the ATI spectrum moves to higher order as the intensity increases. ATI Resonance Substructure If the laser pulse is short enough (< 1 ps for the typical laser focus), the field turns off before the electron can escape from the focal volume. Then the quiver energy is returned to the field and the ATI spectrum becomes much more complicated. The observed electron energy corresponds to the energy

E s (shortpulse) = (n + s)~ω − (IP + Up ) .

(74.20)

relative to the shifted ionization potential. Electrons from different regions of the focal volume are thus emitted with different ponderomotive shifts, introducing substructure in the spectrum which can be directly associated with ac Stark-shifted resonances [74.38, 39]. ATI in Circular Polarization The above discussion is appropriate for the case of linear polarization where the excited states of the atom can play a significant role in the excitation. In a circularly polarized field, the orbital angular momentum L must increase one unit with each photon absorbed so that multiphoton ionization is allowed only to states which have high L, and hence a large centrifugal barrier. The lower energy scattering states then cannot penetrate into the vicinity of the initial state. Thus the ATI spectrum in circular polarization peaks at high energy and is very small near threshold.

74.2.5 High Harmonic Generation High-order harmonic generation (HG) in noble gases is a rapidly developing field of laser physics [74.41, 42]. When an SS pulse interacts with an atomic gas, the atoms respond in a nonlinear way, emitting coherent radiation at frequencies that are integral multiples of the laser frequency. Due to the inversion symmetry of the atom, only odd harmonics are  of the fundamental  emitted. In the high intensity > 1013 W/cm2 , low frequency regime, the harmonic strengths fall off for the first few orders, followed by a broad plateau of nearly constant conversion efficiency, and then a rather sharp cut-off [74.41, 42]. The plateaux extend to well beyond the hundredth order of the 800–1000 nm incident wavelengths, using the light noble gases as the active medium. There has also been experimental evidence of HG from ions. Harmonic generation provides a source of very bright, short-pulse, coherent XUV radiation which can have several advantages over the other known sources, such as the synchrotron. Plateau and Cut-off A recently developed two-step model [74.32,33], which combines quantum and classical aspects of laser-atom physics, accounts for many strong field phenomena. In this model, the electron first tunnels [74.43] from the ground state of the atom through the barrier formed by the Coulomb potential and the laser field. Its subsequent motion can be treated classically, and primarily consists of oscillatory motion in phase with the laser field. If the electron returns to the vicinity of the nucleus with kinetic energy T , it may recombine into the ground state with the emission of a photon of energy (2n + 1)~ω ≤ T + IP , where n is an integer. The maximum kinetic energy of the returning electron turns out to be T  3.2 Up , resulting in a cut-off in the harmonic spectrum at the harmonic of order

Nmax  (IP + 3.2Up )/~ω .

(74.21)

Theoretical Methods Calculation of harmonic strengths requires the evaluation of the time-dependent dipole moment of the atom, 

d(t) = Ψ(t)|er|Ψ(t) . (74.22)

The strength of harmonics emitted by a single atom are then related to Fourier components of d(t), or more ¨ precisely, its second time derivative, d(t). The induced dipole moment d(t) can be directly evaluated from the numerical [74.44] or Floquet [74.26]

Multiphoton and Strong-Field Processes

solutions of the TDSE. Good agreement with numerical and experimental data is also obtained using a strong field approximation discussed below and a Landau– Dyhne formula. This approach can be considered to be a quantum mechanical implementation of the two-step model [74.45].

(74.23)

where n M (r) is the refractive index of the medium (which depends on atomic, electronic and ionic dipole polarizabilities), while PM (r) is the polarization induced by the fundamental field only. It can be expressed as
 PM (r) ∝ N(r)dM M(r) exp − iM∆(r) , (74.24) where N(r) is the atomic density, dM (r) is the Mth Fourier component of the induced dipole moment, and ∆(r) is a phase shift coming from the phase dependence of the fundamental beam due to focusing. All of these quantities may have a slow time dependence, reflecting the temporal envelope of the laser pulse. Phase matching is most efficient in the forward direction. In general, the strength and spatial properties of an harmonic depend in a very complex way on the focal parameters, the medium length and the coherence length of a given harmonic. Propagation and phase matching effects can lead to a shift of the location of the cut-off in the harmonic spectrum [74.47]. Harmonic Generation by Elliptically Polarized Fields The two-step model implies that for harmonic emission it is necessary that the tunneling electrons return to the

nucleus and recombine into their initial state. According to classical mechanics, there are many trajectories in a linearly polarized field that involve one or more returns to the origin. However, there are practically no such trajectories in elliptically polarized fields. As a result, the two-step model predicts a strong decrease of the harmonic strengths as a function of the laser ellipticity. This prediction has been confirmed experimentally [74.48]. The Generation of Sub-Femtosecond XUV Pulses Manipulation of generated harmonics by allowing the temporal beating of superposed high-order harmonics can produce a train of very short intensity spikes, on the order of ∼ 100 attoseconds and shorter, where 1 as = 10−18 s [74.49]. The structural characteristics of the generated pulse-trains depend on the relative phases of the harmonics combined. Employing driving pulses that were themselves only a few femtoseconds long, experimental groups in Vienna [74.50] and Paris [74.51] reported the first observations and measurements of such sub-femtosecond UV/XUV light pulse-trains. The scientific importance of breaking the femtosecond barrier is obvious: the time-scale necessary for probing the motion of an electron in a typical bound, valence state is measured in attoseconds (atomic unit of time ≡ 24 as). Attosecond pulses will allow the study of the timedependent dynamics of correlated electron systems by freezing the electronic motion, in essence exploring the structure with ultra-fast snapshots. A crucial aspect for all attosecond pulse generation is the control of spectral phases. Measurements of the timing of the attosecond peaks relative to the absolute phase of the ir driving field have been accomplished [74.52]. This provides insight into the recollision, harmonic generation process. Also, the control of the group velocity phase relative to the envelope of the few cycle driving pulses allows the production of reproducible pulse-trains [74.53]. Thus, the highly non-perturbative, nonlinear multiphoton interactions of very short ir or visible light pulses with atoms or molecules is becoming a novel, powerful, and unique source for studies of very rapid quantum-electronic processes.

74.2.6 Stabilization of Atoms in Intense Laser Fields It has been argued [74.54] that in very intense laser fields of high frequency, atoms undergo dynamical stabilization and do not ionize. The stabilization effect can be explained by gauge transforming the TDSE (74.1)

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Part F 74.2

Propagation and Phase Matching Effects A single atom response is not sufficient to determine the macroscopic response of the atomic medium. Because different atoms interact with different parts of the focused laser beam, they feel different peak field intensities and phases (which actually undergo a rapid π shift close to the focus). The total harmonic signal results from coherently adding contributions from single atoms, accounting for propagation and interference effects. The latter effects can wipe out the signal completely a constructive phase matching takes place. The propagation and phase matching effects in the strong field regime [74.46] can be studied by solving the Maxwell’s equations for a given harmonic component of the electric field E M (r) (68.23),  1 Mω 2 2 ∇ EM (r) + n M (r)EM (r) = − PM (r) , 0 c

74.2 Strong-Field Multiphoton Processes

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Part F 74.2

to the Kramers–Henneberger (K–H) frame; i. e., a noninertial oscillating frame which follows the motion of the free electron in the laser field. The K–H transformation consists of replacing r → r + α(t), where, for the linearly polarized monochromatic  laser field,  α(t) = xˆ α0 cos(ωt − ϕ); α0 = eE/ mω2 is the excursion amplitude of a free electron, and xˆ is the polarization direction. The TDSE in the K–H frame is  2 2 
 ∂Ψ(r, t) ~ ∇ i~ = − + V r + α(t) Ψ(r, t) , ∂t 2m (74.25)

i. e., it describes the motion of the electron in an oscillatory potential. In the high frequency limit, this potential may be replaced by its time average ω VK–H (r) = 2π

2π/ω 


 dt V r + α(t) ,

(74.26)

0

and the remaining Fourier components of V [r + α(t)] treated as a perturbation. When α0 is large, the effective potential (74.26) has two minima close to r = ±xˆ α0 . The corresponding wave functions of the bound states are centered near these minima, thus exhibiting a dichotomy. The ionization rates from the K–H bound states are induced by the higher Fourier components of V(r + α(t)). For large enough α0 , the rates decrease if either the laser intensity increases or the frequency decreases. Numerical solutions of the TDSE [74.55, 56] show that stabilization indeed occurs for laser field strengths and frequencies of the order of one atomic unit. More importantly, stabilization is possible even when the laser excitation is not monochromatic, but rather, is produced by a short laser pulse. Physically, free electrons in a monochromatic laser field cannot absorb photons due to the constraints imposed by energy and momentum conservation. Absorption is possible only in the vicinity of a potential, such as the Coulombic attraction of the nucleus. In the case of strong excitation, i. e., when α0 is much larger than the Bohr radius, the electron spends most of the time very far from the nucleus, and therefore does not absorb energy from the laser beam. Thus stabilization, as viewed from the K–H frame, has a classical analog. Other mechanisms of stabilization based on the quantum mechanical effects of destructive interference between various ionization paths have also been proposed [74.57]. Due to classical scaling (Sect. 74.2.8) stabilization is predicted to occur for much lower laser frequencies if the atoms are initially prepared in highly excited states. If

additionally, the initial state has a large orbital angular momentum corresponding to classical trajectories that do not approach the nucleus, stabilization is even more easily accomplished. The stabilization of a 5g Rydberg state of neon has recently been reported [74.58].

74.2.7 Molecules in Intense Laser Fields Molecular systems are more complex than atoms because of the additional degrees of freedom resulting from nuclear motion. Even in the presence of a laser field, the electron and nuclear degrees of freedom can be separated by the Born–Oppenheimer approximation, and the dynamics of the system can be described in terms of motions on potential energy surfaces. In strong fields, the Born–Oppenheimer states become dressed, or mixed by the field, creating new molecular potentials. Because of avoided crossings between the dressed molecular states, the field induces new potential wells in which the molecules become trapped. These states, known as laserinduced bound states, are stable against dissociation, but exist only while the laser field is present [74.59]. Their existence affects the spectra of photoelectrons, photons, and the fragmentation dynamics. If the field is strong enough, many electrons can be ejected from a molecule before dissociation, producing highly charged, energetic fragments [74.60]. Such experiments are similar to beam-foil Coulomb explosion studies of molecular structure. However, because of changes from the fieldfree equilibrium geometries in laser dissociation, the energies of the fragments lie systematically below the corresponding values from Coulomb explosion studies.

74.2.8 Microwave Ionization of Rydberg Atoms Similar phenomena appear in the ionization of highly excited hydrogen-like (Rydberg) atoms by microwave fields [74.61,62], but the dynamical range of the parameters involved is different from the case of tightly bound electrons. Recent developments have greatly extended techniques for the preparation and detection of Rydberg states. Since, according to the equivalence principle, highly excited Rydberg states exhibit many classical properties, a classical perspective of ionization yields useful insights (Sect. 74.3.5). Classical Scaling The classical equations of motion for an electron in both a Coulomb field and a monochromatic laser field polarized along the z-axis are invariant with respect to the

Multiphoton and Strong-Field Processes

following scaling transformations: p ∝ n −1 ˜, 0 p

r ∝ n 20 r˜ ,

t ∝ n 30 t˜ ,

ϕ ∝ ϕ˜ ,

ω ∝ n −3 ˜, 0 ω

˜ E ∝ n −4 0 E .

(74.27)

  p˜2 1 − + ze H˜ = ˜ E˜ cos ω˜ t˜ + ϕ˜ , 2m r˜

(74.28)

i. e., it depends only on ω˜ and E˜ . In experiments, the principal quantum number n 0 of the prepared initial state typically ranges from 1 to 100. Classical scaling extends to the fields of other polarization and to pulsed excitation, provided that the number of cycles in the rise, top and fall of the pulse is kept fixed. This scaling does not hold for a quantum Hamiltonian, unless one also rescales Planck’s constant, ~˜ = ~/n 0 . In practice, increasing n 0 , keeping E˜ and ω˜ constant, corresponds to a decrease in the effective ~ toward the classical limit. In view of this classical scaling, experimental and theoretical results are usually analyzed in terms of the scaled variables. Since the classical dynamics generated by the Hamiltonian (74.28) exhibits chaotic behavior in some regimes, the dynamics of the corresponding quantum system is frequently referred to as an example of quantum chaos [74.63–65]. Regimes of Response By varying the initial n 0 , several regimes of the scaled parameters can be covered. The experimentally measured response of Rydberg atoms in microwave fields can be divided into six categories: The Tunneling Regime. For ω˜ ≤ 0.07, the response of the system is accurately represented as tunneling through the slowly oscillating potential barrier composed of the Coulomb and microwave potentials. The Low Frequency Regime. For 0.05 ≤ ω˜ ≤ 0.3, the ionization probability exhibits structure (bumps, steps, changes of slope) as a function of the field strength. The quantum probability curves might be lower or higher than the corresponding classical counterparts calculated with the aid of the phase averaging method (Sect. 74.3.5). The Semiclassical Regime. For 0.1 ≤ ω˜ ≤ 1.2, the ion-

ization probabilities agree well for mostfrequencies

with the results obtained from the classical theory. In particular, the onset of ionization and appearance intensities (i. e., the intensities at which a given degree of ionization is achieved) coincide with the onset of chaos in the classical dynamics. Resonances in the ionization probabilities appear that correspond to the classical trapping resonances [74.63–66]. The Transition Region. For 1 ≤ ω˜ ≤ 2, the differences

between the quantum and classical results are visible. Quantum ionization probabilities are frequently lower and appearance intensities higher than their classical counterparts. The High Frequency Regime. For ω˜ ≥ 2, quantum re-

sults for ionization probabilities are systematically lower and appearance intensities higher than their classical counterparts. This apparent stability of the quantum system has been attributed to three kinds of effects: quantum localization [74.66], quantum scars [74.67], and perhaps to the stabilization of atoms in intense laser fields (Sect. 74.2.6). The Photoeffect Regime. When the scaled frequency

becomes greater than the single photon ionization threshold, the system undergoes single photon ionization (the photoeffect). Quantum Localization The classical dynamics changes as the field increases. Chaotic trajectories start to fill phase space and, as the KAM tori (describing periodic orbits) [74.63–65] break down, the motion becomes stochastic, resembling a random walk. This process, in which the mean energy grows linearly in time, is termed diffusive ionization. In the quantum theory, diffusion corresponds to a random walk over a ladder of suitably chosen quantum levels. However, both diagonal and off-diagonal elements of the evolution operator, which describe quantum mechanical amplitudes for transitions between the levels, depend in a quasiperiodic manner on the quantum numbers of the levels involved. Such quasiperiodic behavior is quite analogous to a random one. Electronic wave packets that initially spread in accordance with the classical laws tend to remain localized for longer times due to destructive quantum interference effects. Quantum localization is an analog of the Anderson localization of electronic wave functions propagating in random media [74.66].

1085

Part F 74.2

In the scaled units, the Hamiltonian H˜ = n −2 0 H of the system becomes

74.2 Strong-Field Multiphoton Processes

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Quantum Optics

Part F 74.3

Quantum Scars Even in the fully chaotic regime, classical phase space contains periodic, though unstable, orbits. Nevertheless, quantum mechanical wave function amplitudes can become localized around these unstable orbits, resulting in what are called quantum scars. The increased stability of the hydrogen atom at ω˜  1.3 has been in fact attributed [74.67] to the effects of

quantum scars. These effects are very sensitive to fluctuations in the driving laser field. Control of the laser noise therefore provides a powerful spectroscopic tool to study such quantal phenomena [74.68]. Using this tool, it has recently become possible to demonstrate the effects of quantum scars in the intermediate regime of the scaled frequencies (less than but close to 1).

74.3 Strong-Field Calculational Techniques The SS pulse regime requires a nonperturbative solution of the TDSE. Two approaches have been developed: the explicit numerical solution of the TDSE and the Floquet expansion technique. In addition to these, several approximate methods have been proposed.

74.3.1 Floquet Theory The excitation and ionization dynamics of an atom in a strong laser field can be determined by turning the problem into a time-independent eigenvalue problem [74.26,69]. From Floquet’s theorem, the eigenfunctions for a perfectly periodic Hamiltonian of the form H = H0 + HN e−iNωt (74.29) N=0

can be expressed in the form e−iNωt ψ N . Ψ(t) = e−iXt/~

(74.30)

N

Putting this into the time-dependent Schrödinger equation results in an infinite set of coupled Floquet equations for the harmonic components ψ N . In the velocity gauge, the Floquet equations are (X + N ~ω − H0 )ψ N = V+ ψ N−1 + V− ψ N+1 , (74.31)

where, for a vector potential of amplitude A, e V+ = − A· p , 2mc †

(74.32)

and V− = V+ . The equations (74.31) have been solved, after truncation to a manageable number of terms, using many techniques to provide what are called the quasi-energy states of the laser-atom system. The eigenvalues X of these equations are complex, with Im(X) giving the decay or ionization rate for the system. The generated rates are found to be very accurate as long

as the pulse length of the laser field is not too short, at least hundreds of cycles. The eigenfunctions provide the amplitudes for the photoelectron energy spectra, and the time-dependent dipole of the state can be related to the photo-emission spectrum of the system. Yields for slowly varying pulses can be constructed by combining the results from the individual, fixed-intensity calculations [74.26]. The Floquet method can be applied for any periodic Hamiltonian. In strong enough fields of high frequency, the Floquet equations can be truncated to a very small set in the K–H frame [74.54].

74.3.2 Direct Integration of the TDSE Methods for the direct solution of the time-dependent Schrödinger equation are described in general in Chapt. 8 and in [74.70,71] for multiphoton processes. The wave functions are defined on spatial grids or in terms of an expansion in basis functions. The time evolution is obtained by either explicit or implicit time propagators. All these methods are capable of generating numerically exact results for an atom with a single electron in a short pulsed field for a wide range of pulse shapes, wavelengths and intensities. The solutions are time-dependent wave functions for the electrons which can be analyzed to obtain excitation and ionization rates, photoelectron energies, angular distributions, and photoemission yields. The ability to generate an explicit solution of the TDSE allows the study of arbitrary pulse shapes and provides insight into the excitation dynamics. For multi-electron atoms, one generally has to limit the calculations to that for a single electron in effective potentials which represent, as well as possible, the influences of the remaining atomic electrons. This approach is called the single active electron approximation, and it gives generally accurate results for systems with no low-lying doubly excited states, for example, the noble gases [74.70]. In these cases, the excitation dynamics

Multiphoton and Strong-Field Processes

are dominated by the sequential promotion of a single electron at a time.

74.3.3 Volkov States

(74.33) × ei[k−eA(t)/~c]·r , t where A(t) = −c E(t  ) dt  is the vector potential of the field. ΨV is called a Volkov state. In a linearly polarized field, the electron oscillates along the direction of polarization with an amplitude α0 = e~A/(mcω). In the strong field regime, this amplitude can greatly exceed the size of a bound state orbital. Volkov states provide a useful tool that can be applied in various strong field approximations discussed in the next Section.

1087

terms corresponding to matrix elements for free electrons plus corrections due to the potential [74.45]. In the latter version of SFA, the amplitude of the electronic wave function b( p) corresponding to outgoing momentum p is given by tF b( p) = i


 dt d p − eA(t)/c · E(t)

0


× exp − iS(tF , t)/~ .

(74.34)

Here d[ p − eA(t)/c] denotes the dipole matrix element for the transition from the initial bound state to the continuum state in which the electron has the kinetic momentum p − eA(t)/c, tF is the switch-off time of the laser pulse, and 
 2 tF p − eA(t  )/c  + IP (74.35) S(tF , t) = dt 2m t

is a quasiclassical action for an electron which is born in the continuum at t and propagates freely in the laser field. The form of the expression (74.34) is generic to the SFA.

74.3.4 Strong Field Approximations

74.3.5 Phase Space Averaging Method

There have been several attempts to solve the TDSE in the strong field limit using approximate, but analytic methods. Such strong field approximations (SFA) typically neglect all the bound states of the atom except for the initial state. In the tunneling regime (γ < 1), and a quasistatic limit (ω → 0), one can use a theory [74.43] in which the ionization occurs due to the tunneling through the Coulomb barrier distorted by the electric field of the laser. The wave function is constructed as a combination of a bare initial wave function of the electron (close to the nucleus) and a wave function describing a motion of the electron in a quasistatic electric field (far from the nucleus). In an second approach [74.34–36], the elements of the scattering matrix Sˆ are calculated assuming that initially the electronic wave function corresponds to a bare bound state. On the other hand, the final, continuum states of the electron are described by dressed wave functions that account for the free motion of the electron in the laser field. In the simplest case, such dressed states are Volkov states (74.33). ˆ Alternatively, the time-reversed S-Matrix is obtained by dressing the initial state and using field-free scattering states for the final state. Yet another method consists of expanding the electronic continuum–continuum dipole matrix elements in

The methods of classical mechanics are particularly useful in describing the microwave excitation of highly excited (Rydberg) atoms [74.61, 62] (Sect. 74.2.8), but have also been applied to describe high harmonic generation, stabilization of atoms in super intense fields and two electron ionization [74.72–74]. The classical phase space averaging method [74.75] solves Newton’s equations of motion r˙ = p/m , p˙ = −∇V(r) − eE(t) ,

(74.36) (74.37)

for the electron interacting with the ion core and the laser field. A distribution of initial conditions in phase space is chosen to mimic the initial quantum mechanical state of the system, and a sample of classical trajectories generated. Quantum mechanical averages of physical observables are then identified with ensemble averages of those observables over the initial distribution. Since the dynamics of multiphoton processes is very complex, the neglected phases in this approach generally cause negligible errors and the results can be in quite good agreement with quantum calculations. Additionally, an examination of the trajectories provides details of the excitation dynamics which are often difficult to extract from a complex time-dependent wave function.

Part F 74.3

A laser interacting with free a electron superimposes an oscillatory motion on its drift motion in response to the field. The wave function for an electron with drift velocity v = ~k/m is given by   t  2 e i   ~k − A(t ) dt  ΨV (r, t) = exp − 2m ~ c

74.3 Strong-Field Calculational Techniques

1088

Part F

Quantum Optics

References 74.1 74.2 74.3

Part F 74

74.4

74.5

74.6

74.7

74.8

74.9

74.10 74.11 74.12 74.13 74.14 74.15 74.16

74.17

74.18

74.19

74.20

F. H. M. Faisal: Theory of Multiphoton Processes (Plenum, New York 1987) M. Gavrila (Ed.): Atoms in Intense Laser Fields (Academic Press, San Diego 1992) M. H. Mittleman: Theory of Laser–Atom Interactions, 2nd edn. (Plenum, New York 1993) P. Lambropoulos, S. J. Smith (Eds.): Proceedings of the International Conference of Multiphoton Processes III, 1984, Vol. 2 (Springer, Berlin, Heidelberg 1984) S. J. Smith, P. L. Knight (Eds.): Proceedings of the International Conference on Multiphoton Processes IV, 1988, Vol. 8 (Cambridge Univ. Press, Cambridge 1988) G. Mainfray, P. Agostini (Eds.): Proceedings of the International Conference on Multiphoton Processes V, 1991 (Centre d’Etudes de Saclay, Saclay 1991) L. F. DiMauro, R. R. Freeman, K. C. Kulander (Eds.): Proceedings of the International Conference of Multiphoton Processes VIII, AIP Conference Proceedings, Vol. 525 (American Institute of Physics, Melville 2000) B. Piraux, A. L’Huillier, K . Rza ¸ z˙ ewski (Eds.): SuperIntense Laser–Atom Physics, Vol. 316 (Plenum, New York 1993) B. Piraux, K. Rza ¸ z˙ ewski (Eds.): Super-Intense LaserAtom Physics, NATO ASI Series Ii, Vol. 12 (Kluwer Academic, The Netherlands 2001) P. Lambropoulos: Adv. At. Mol. Phys. 12, 87–158 (1976) Y. Gontier, M. Trahin: J. Phys. B 13, 4383 (1980) X. Tang, P. Lambropoulos: Phys. Rev. Lett. 58, 108 (1987) J. H. Eberly: Phys. Rev. Lett. 47, 408 (1981) P. Lambropoulos, P. Zoller: Phys. Rev. A. 24, 379 (1981) P. L. Knight, M. A. Lauder, B. J. Dalton: Phys. Rep. 190, 1 (1990) H. Risken: The Fokker–Planck Equation: Methods of Solution and Applications, ed. by H. Haken (Springer, Berlin, Heidelberg 1984) J. H. Eberly: Laser Spectroscopy, ed. by H. Walther, K. W. Rothe (Springer, Berlin, Heidelberg 1979) p. 80 P. Zoller: Proceedings of the International Conference of Multiphoton Processes III, 1984, Vol. 2, ed. by P. Lambropoulos, S. J. Smith (Springer, Berlin, Heidelberg 1984) pp. 68– 75 D. S. Elliot: Proceedings of the International Conference of Multiphoton Processes III, 1984, Vol. 2, ed. by P. Lambropoulos, S. J. Smith (Springer, Berlin, Heidelberg 1984) pp. 76–81 D. S. Elliot et al.: Phys. Rev. A. 32, 887 (1985)

74.21 74.22 74.23 74.24 74.25 74.26

74.27 74.28 74.29 74.30 74.31 74.32

74.33 74.34 74.35 74.36 74.37 74.38

74.39

74.40 74.41

74.42 74.43 74.44 74.45 74.46 74.47

74.48 74.49

A. I. Burshtein et al.: Sov. Phys. JETP. 21, 597 (1965) A. I. Burshtein et al.: Sov. Phys. JETP. 22, 939 (1996) B. W. Shore: The Theory of Coherent Atomic Excitation (Wiley, New York 1990) L. A. Lompré, G. Mainfray, C. Manus, J. P. Marinier: J. Phys. B 14, 4307 (1981) C. Chen, D. S. Elliot: Phys. Rev. Lett. 65, 1737 (1990) R. M. Potvliege, R. Shakeshaft: Atoms in Intense Laser Fields, ed. by M. Gavrila (Academic Press, San Diego 1992) pp. 373–434 O. Faucher et al.: Phys. Rev. Lett. 70, 3004 (1993) J. E. Field, S. E. Harris: Phys. Rev. Lett. 66, 1154 (1991) S. E. Harris: Phys. Rev. Lett. 70, 552 (1993) G. S. Hurst, M. G. Payne, S. D. Kramer, J. P. Young: Rev. Mod. Phys. 52, 767 (1979) M. D. Perry, G. Mourou: Science 264, 917 (1994) K. C. Kulander, K. J. Schafer, J. L. Krause: SuperIntense Laser-Atom Physics, NATO ASI Series Ii, Vol. 12, ed. by B. Piraux, K. Rza ¸ z˙ ewski (Kluwer Academic, The Netherlands 2001) pp. 95–110. P. B. Corkum: Phys. Rev. Lett. 73, 1995 (1993) L. V. Keldysh: Sov. Phys. JETP 20, 1307 (1965) H. R Reiss: Phys. Rev. A 22, 1786 (1980) F. Faisal: J. Phys. B 6, 312 (1973) D. Fittinghoff, P. R. Bolton, B. Chang, K. C. Kulander: Phys. Rev. A 49, 2174 (1994) H. G. Muller, P. Agostini, G. Petite: Atoms in Intense Laser Fields, ed. by M. Gavrila (Academic Press, San Diego 1992) pp. 1–42 R. R. Freeman et al.: Atoms in Intense Laser Fields, ed. by M. Gavrila (Academic Press, San Diego 1992) pp. 43–65 J. H. Eberly, J. Javanainen, K. Rza ¸ z˙ ewski: Phys. Rep. 204, 331 (1991) A. L’Huillier, L.-A. Lompr´ e, G. Mainfray, C. Manus: Atoms in Intense Laser Fields, ed. by M. Gavrila (Academic Press, San Diego 1992) pp. 139–205 Y. Liang, M. V. Ammosov, S. L. Chin: J. Phys. B 27, 1269 (1994) M. V. Ammosov, N. B. Delone, V. P. Krainov: Sov. Phys. JETP 64, 1191 (1986) J. L. Krause, K. J. Schafer, K. C. Kulander: Phys. Rev. A 45, 4998 (1992) M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, P. Corkum: Phys. Rev. A 49, 2117 (1994) A. L’Huillier, K. J. Schafer, K. C. Kulander: J. Phys. B 24, 3315 (1991) A. L’Huillier, M. Lewenstein, P. Salières, Ph. Bal¨m: Phys. Rev. A 48, cou, J. Larsson, C. G. Wahlstro 4091 (1993) K. S. Budil, P. Salières, A. L’Huillier, T. Ditmire, M. D. Perry: Phys. Rev. A 48, 3437 (1993) S. E. Harris, J. L. Macklin, T. W. Hänsch: Opt. Comm. 100, 487 (1993)

Multiphoton and Strong-Field Processes

74.50 74.51 74.52 74.53 74.54

74.56 74.57

74.58

74.59 74.60

74.61

74.62

74.63 74.64 74.65 74.66 74.67 74.68

74.69 74.70

74.71 74.72

74.73

74.74

74.75

M. C. Gutzwiller: Chaos in Classical and Quantum Mechanics (Springer, Berlin, Heidelberg 1990) F. Haake: Quantum Signatures of Chaos (Springer, Berlin, Heidelberg 1991) G. Casati, B. Chirikov, D. L. Shepelyansky, I. Guarnieri: Phys. Rep. 154, 77 (1987) G. Casati, I. Guarneri, D. L. Shepelyansky: Physica A 163, 205 (1990) and references therein R. V. Jensen, M. M. Sanders, M. Saraceno, B. Sundaram: Phys. Rev. Lett. 63, 2771 (1989) L. Sirko, M. R. W. Bellermann, A. Haffmans, P. M. Koch, D. Richards: Phys. Rev. Lett. 71, 2895 (1993) S. I. Chu: Adv. Chem. Phys. 73, 739 (1989) K. C. Kulander, K. J. Schafer, J. L. Krause: Atoms in Intense Laser Fields, ed. by M. Gavrila (Academic Press, San Diego 1992) pp. 247–300 K. Burnett, V. C. Reed, P. L. Knight: J. Phys. B 26, 561 (1993) M. Lewenstein, K. Rza ¸ z˙ ewski, P. Sali` eres: SuperIntense Laser-Atom Physics, NATO ASI Series Ii, Vol. 12, ed. by B. Piraux, K. Rza ¸ z˙ ewski (Kluwer Academic, The Netherlands 2001) pp. 425–434 P. B. Lerner, K. LaGattuta, J. S. Cohen: SuperIntense Laser-Atom Physics, NATO ASI Series Ii, Vol. 12, ed. by B. Piraux, K. Rza ¸ z˙ ewski (Kluwer Academic, The Netherlands 2001) pp. 413–424 V. V` eniard, A. Maquet, T. M` enis: Super-Intense Laser-Atom Physics, NATO ASI Series Ii, Vol. 12, ed. by B. Piraux, K. Rza ¸ z˙ ewski (Kluwer Academic, The Netherlands 2001) pp. 225–232 J. G. Leopold, I. C. Percival: J. Phys. B 12, 709 (1979)

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Part F 74

74.55

M. Drescher et al.: Science 291, 1923 (2001) P. M. Paul et al.: Scienc 292, 1689 (2001) L. C. Dinu et al.: Phys. Rev. Lett. 91, 063901 (2003) A. Baltuˇska et al.: Nature 421, 611 (2003) M. Gavrila: Atoms in Intense Laser Fields, ed. by M. Gavrila (Academic Press, San Diego 1992) pp. 435–510 and references therein Q. Su, J. H. Eberly, J. Javanainen: Phys. Rev. Lett. 64, 862 (1990) K. C Kulander, K. J. Schafer, J. L. Krause: Phys. Rev. Lett. 66, 2601 (1991) M. V. Fedorov: Super-Intense Laser-Atom Physics, NATO ASI Series Ii, Vol. 12, ed. by B. Piraux, K. Rza ¸ z˙ ewski (Kluwer Academic, The Netherlands 2001) pp. 245–259 M. P. de Boer, J. H. Hoogenraad, R. B. Vrijen, L. D. Noordam, H. Muller: Phys. Rev. Lett. 71, 3263 (1993) A. D. Bandrauk (Ed.): Molecules in Laser Fields (Dekker, New York 1994) D. Normand, C. Cornaggia: Super-Intense LaserAtom Physics, NATO ASI Series Ii, Vol. 12, ed. by B. Piraux, K. Rza ¸ z˙ ewski (Kluwer Academic, The Netherlands 2001) pp. 351–362 P. M. Koch: Super-Intense Laser-Atom Physics, NATO ASI Series Ii, Vol. 12, ed. by B. Piraux, K. Rza ¸ z˙ ewski (Kluwer Academic, The Netherlands 2001) pp. 305–316 P. M. Koch: Proceedings of the Eigth South African Summer School in Physics, 1993 (Springer, Berlin, Heidelberg 1993)

References

1091

Cooling and Tr 75. Cooling and Trapping

75.1

Notation ............................................. 1091

75.2 Control of Atomic Motion by Light ......... 1092 75.2.1 General Theory ......................... 1092 75.2.2 Two-State Atoms....................... 1094 75.2.3 Multistate Atoms ....................... 1097 75.3 Magnetic Trap for Atoms ...................... 1099 75.4 Trapping and Cooling of Charged Particles ............................. 1099 75.4.1 Paul Trap ................................. 1099 75.4.2 Penning Trap ............................ 1101 75.4.3 Collective Effects in Ion Clouds .... 1102 75.5 Applications of Cooling and Trapping .... 1103 75.5.1 Neutral Atoms........................... 1103 75.5.2 Trapped Particles ...................... 1104 References .................................................. 1105 give a current snapshot. Additional references are occasionally listed here to accentuate specific points. These citations are to either particularly representative papers or to the most recent articles on the subject, and are intended as entries to the literature. No assignment of credit or priority is implied.

75.1 Notation In this Chapter, the lower and upper states of an optical transition are denoted by the respective labels g and e, for “ground” and “excited”. The notation Jg → Je stands for a transition in which the lower and upper levels have the angular momentum degeneracies 2Jg + 1 and 2Je + 1. The resonance frequency of the transition is ω0 . The detuning of the driving monochromatic light of frequency ω from the atomic resonance is ∆ = ω0 − ω. Γ is the spontaneous decay rate. Spontaneous emission is taken to be the sole mechanism of line broadening, so that the HWHM linewidth of the transition is γ = Γ/2. The Rabi frequency is Ω = DE/~, where D is the reduced dipole moment matrix element that would apply to a transition with unit Clebsch–Gordan coefficient, and E is the electric field amplitude of the laser. The

corresponding intensity scale is the saturation intensity 4π 2 ~cΓ , (75.1) 3λ3 defined in such a way that the light intensity I satisfies Is =

Ω = Γ ⇒ I = Is .

(75.2)

If multiple laser beams are explicitly mentioned, laser intensity and Rabi frequency are quoted for each of the equally intense beams. The momentum of a photon with the wave vector k is ~k. The recoil velocity vr =

~k m

(75.3)

Part F 75

Interactions of light with an atomic particle are accompanied by exchange of momentum between the electromagnetic field and the atom. Narrow-band resonance radiation from tunable lasers enhances the ensuing mechanical effects of light to the extent that it is literally possible to stop atoms emanating from a thermal gas, and to trap atoms with light. References [75.1] and [75.2] are two early sources on laser cooling and trapping of the traditional two-state model atom. A number of articles on cooling and trapping of atoms with the inclusion of angular momentum degeneracy are contained in [75.3]. While the development based on the atom-field dressed states is followed sparingly in the present Chapter, an authoritative survey of this approach is given in [75.4]. Reviews on traps for charged particles include [75.5] and [75.6]. These articles, as well as [75.2], also discuss cooling of trapped particles. Cooling and trapping of atomic particles are now standard technologies, but development and extension of the methods to new applications continues. The special issues [75.7] and [75.8]

1092

Part F

Quantum Optics

Table 75.1 Laser cooling parameters for the lowest S1/2 –P3/2 transition of hydrogen and most alkalis (the D2 line). Also

shown are the nuclear spin I and the ground state hyperfine splitting ∆νhfs . Γ is typically known to within a few per cent, so these values of Γ , TD and Is may not all be accurate to the full displayed precision

Part F 75.2

Parameter

1H

6 Li

7 Li

23 Na

39 K

40 K

85 Rb

87 Rb

133 Cs

m λ vr Γ TD εr Tr Is I ∆νhfs

1.67 121.6 326 98.9 2390 13 396 643 14 509 1/2 1420

9.99 670.8 9.89 5.92 142 73.7 3.54 5.13 1 228.2

11.7 670.8 8.48 5.92 142 63.2 3.03 5.13 3/2 803.5

38.2 589.0 2.95 9.90 238 25.0 1.20 12.7 3/2 1772

64.7 766.5 1.34 6.16 148 8.72 0.418 3.58 3/2 461.7

66.4 766.5 1.30 6.16 148 8.50 0.408 3.58 4 1286

141 780.0 0.602 5.89 141 3.86 0.185 3.24 5/2 3036

144 780.0 0.589 5.89 141 3.77 0.181 3.24 3/2 6835

221 852.1 0.352 5.22 125 2.07 0.0992 2.21 7/2 9193

equals the change of the velocity of an atom of mass m when it absorbs a photon with wave number k = 2π/λ. The kinetic energy of an atom with velocity vr and the corresponding frequency, R 1 R = mvr2 , εr = 2 ~

(75.4)

are referred to as recoil energy and recoil frequency. Two temperatures, the Doppler limit TD and the recoil

Units 10−27 kg nm cm/s 2π MHz µK 2πkHz µK mW/cm2 2πMHz

limit Tr are often cited in laser cooling. They are TD =

~γ , kB

Tr =

R , kB

(75.5)

where kB is the Boltzmann constant. Table 75.1 lists numerical values of pertinent parameters for laser cooling and trapping using the D2 line for most stable and long-lived alkali isotopes and hydrogen.

75.2 Control of Atomic Motion by Light frequencies νi (i = x, y, z), the cm Hamiltonian is

75.2.1 General Theory Hamiltonian The mechanical effects of light may be derived from the Hamiltonian

ˆ r) , Hˆ = Hˆ A + Hˆ cm + Hˆ F − dˆ · E(ˆ

(75.6)

where Hˆ A , Hˆ cm and Hˆ F are the Hamiltonians for the internal degrees of freedom of the atom, center-of-mass (cm) motion of the atom, and free electromagnetic field. ˆ r ), where rˆ is the cm The quantized electric field is E(ˆ position operator. The dipole operator dˆ acts on the inˆ r) ternal degrees of freedom of the atom. Since the dˆ · E(ˆ term couples all degrees of freedom, the possibility of influencing cm motion by light immediately follows. The inclusion of the quantized cm motion is the essential ingredient not contained in traditional theories of lightmatter interactions. For an atom with mass m trapped in a possibly anisotropic harmonic oscillator potential with

Hcm =


mν2 rˆ 2 pˆ 2 i i + , 2m 2

(75.7)

i=x,y,z

where pˆ is the cm momentum operator. For a free atom, νi = 0. Master Equation With the aid of Markov and Born approximations, the vacuum modes of the electromagnetic field may be eliminated as described in Sect. 78.7. This gives a master equation for the reduced density operator ρˆ that contains the internal and cm degrees of freedom of the atom. Relaxation terms proportional to Γ and γ are all that is left of the quantized fields. Consider as an example a two-state atom in a traveling wave of light with the electric field strength

1 E(r, t) = E ei(k·r−ωt) + c.c. . 2

(75.8)

Cooling and Trapping

Master equations are conveniently written using Wigner functions to represent the cm motion. Given the internalstate labels i and j = g or e, and the three-dimensional variables r, p, the Wigner functions are defined as  1 d3 u eiu· p/~ ρij (r, p) = (2π ~)3   1 1 × r − u|i|ρ| (75.9) ˆ j|r + u . 2 2

iΩ d ρˆ ge ( p) = − (γ − i∆)ρˆ ge ( p) − dt 2  1 × e−ik·r ρgg ( p − ~k) 2   1 ik·r − e ρee p + ~k , 2 iΩ d ρˆ eg ( p) = − (γ + i∆)ρeg ( p) + dt 2    1 × eik·r ρgg p − ~k 2   1 −ik·r ρee p + ~k . −e 2

sphere. Representative expressions for W(n) ˆ are W(n) ˆ =

1 , 4π

3 1 − (ˆe · n) ˆ 2 , 8π

3

1 + (ˆe · n) ˆ 2 . 16π (75.15)

These apply, respectively, for isotropic spontaneous emission, for spontaneous emission in a ∆m = 0 transition, and in ∆m = ±1 transitions; eˆ stands for the unit vector in the direction of the quantization axis for angular momentum. Only the p dependence has been denoted explicitly in the Wigner functions, as the recoil effects displayed on the right-hand sides of (75.10–75.13) take place at a fixed position r. Semiclassical Theory Suppose that vcm  vr and τ  τcm , where τ and τcm are the time scales for light-driven changes of the internal state and cm motion of the atom. Then the internal degrees of freedom may be eliminated adiabatically from the master equations in favor of the position-momentum distribution for the cm motion,
f(r, p, t) = ρii (r, p, t) , (75.16) i

where the sum runs over the internal states of the atom. Technically, the recoil velocity vr is treated as an asymptotically small expansion parameter. The result is the Fokker–Planck equation for the cm motion
∂2 ∂ d f =− · (F f ) + (Dij f ) . (75.17) dt ∂p ∂ pi ∂ p j i, j

(75.12)

(75.13)

Here the convective derivative that describes the motion of the atom in the absence of light is d ∂ p ∂
∂ = + · − mνi ri ; (75.14) dt ∂t m ∂r ∂ pi i

cf. Hcm in (75.7). W(n) ˆ is the angular distribution of spontaneous photons, and the integral runs over the unit

1093

In this semiclassical theory the cm motion of the atom is regarded as classical. The atom moves under the optical force F(r, p, t), which models the coarse-grained flow of momentum between the electromagnetic field and the atom. Dij (r, p, t), with i, j = x, y, z, is the diffusion tensor. Diffusion is an attempt to model quantum mechanics with a classical stochastic process, including discreteness of recoil kicks, random directions of spontaneous photons, and random timing of optical absorption and emission processes. A general prescription exists for calculating the force and the diffusion tensor for an arbitrary atomic level scheme and light field [75.9]. However, the study of diffusion amounts to an involved analysis of photon statistics of the scattered light, and here only the force is considered explicitly. Let Vˆ (r) be the dipole interaction operator coupling the driving field and the internal

Part F 75.2

The Wigner function is one of the quantum mechanical quasiprobability distributions, Sect. 78.5, with the special property that the marginal distribution of r obtained by integrating over p coincides with the correct quantum probability distribution for position, and vice versa with r and p interchanged. In the rotating wave approximation, Sect. 68.3.2, the master equations are    d 1 Ω ik·r ρee ( p) = − Γρee ( p) + i e ρˆ ge p − ~k dt 2 2   1 , (75.10) − e−ik·r ρˆ eg p − ~k 2  iΩ d ρgg ( p) = Γ d2 nW(n)ρ ˆ ee ( p + ~kn) ˆ − dt 2    1 × eik·r ρˆ ge p + ~k 2 −ik·r ρˆ eg ( p + 12~k) , (75.11) −e

75.2 Control of Atomic Motion by Light

1094

Part F

Quantum Optics

state for an atom at position r. By assumption, Vˆ (r) has been rendered slowly varying in time with the aid of a suitable rotating wave approximation. To compute the force, one takes an atom that travels along a hypothetical trajectory unperturbed by light in such a way that at time t it arrives at the phase space point (r, p), whereupon the density operator of the internal degrees of freedom is . ˆ The force is then  ∂ Vˆ . Fi (r, p, t) = −Tr ˆ (75.18) ∂ri

Part F 75.2

Quantum Theory When either vcm
vr or τ  τcm , the full quantum theory of cooling and trapping is needed. Master equations such as (75.10–75.13) must then be solved without the assumption that vr is small. Most practical calculations have been numerical case studies [75.10–12]. A truncated basis, e.g., of plane waves, is used to expand the cm state. Density matrix equations are solved numerically, either directly, or by resorting to quantum trajectory simulations, Sect. 78.11. Qualitative Origin of Laser Cooling Velocity dependent dissipative forces are needed for cooling. They arise because the evolution of the internal state of a moving atom has a finite response time τ. The atom conveys the memory of the field it has sampled over the length  = vτ on its past trajectory. If   λ, a nonequilibrium component proportional to  is present in the density operator of the internal state of the atom. Further interactions with light convert this component into a velocity dependent force of the form

F = −mβv ,

β ∝ Iτ .

(75.19)

If the damping constant β is positive, (75.19) describes exponential damping of the velocity on the time scale β −1 . In the contrary case   λ, when the atom travels many wavelengths during the memory time, linear dependence of force on velocity breaks down. The watershed is the critical velocity or velocity capture range λ vc ≈ . (75.20) τ One-Dimensional Considerations Most specific results cited here are one-dimensional. By default, the propagation direction of light and the direction of vector quantities other than light polarization is eˆ x . The relevant components of position, velocity and momentum are denoted by x, v, and p.

The general one-dimensional Fokker–Planck equation for a particle trapped in a harmonic oscillator potential with a cm oscillation frequency ν is   p ∂ ∂ ∂ + − mν2 x f ∂t m ∂x ∂p ∂2 ∂ = − (F f ) + 2 (D f ) . (75.21) ∂p ∂p For the force (75.19) with constant β = β0 and D(z, p) = D0 , the steady state of the Fokker–Planck equation is a thermal distribution of the form    β0 m p2 mν2 x 2 + , f(x, p) = K exp − D0 2m 2 (75.22)

where K is a normalization coefficient. Since Wigner functions give correct quantum mechanical marginal distributions for r and p, expectation values of kinetic and potential energy may be calculated from the distribution function (75.22) as if it were a classical phase space density. For a free atom with ν = 0, the temperature is directly proportional to the kinetic energy, D0 . (75.23) β0 mkB However, for a trapped particle with ν = 0 the Fokker– Planck equation may be valid all the way to the quantum mechanical zero-point energy. Then temperature and energy are no longer directly proportional to one another. For a trapped particle, the safe interpretation of (75.22) is that the total cm energy of the particle is T=

E=

D0 . β0 m

(75.24)

75.2.2 Two-State Atoms A two-state or two-level atom, discussed in detail in Sect. 68.3, stands for a closed (recycling) transition with one lower state and one excited state. In practice, a twostate system is often realized by driving a J → J + 1 transition with circularly polarized light. This leads to optical pumping to the states with maximal (or minimal) component of angular momentum along the quantization axis, say, to the transition m = J → J + 1. Two types of force are generally distinguished: light pressure, or scattering force, or spontaneous force, and dipole, or gradient, or induced force. However, the distinction is neither exclusive, nor exhaustive. Here the two types of force are approached by way of examples.

Cooling and Trapping

75.2 Control of Atomic Motion by Light

Traveling Waves Light Pressure. Consider a cycle of absorption and spon-

beams may be added when averaged over a wavelength. For velocities well below the critical velocity

taneous emission. In an absorption, the atom receives a photon recoil kick in the direction of the laser beam, while in spontaneous emission the recoil kick has a random direction and zero average. The atom is on the average left with a velocity change equal to vr . The corresponding force is along k, and is given by

Γ , (75.30) k the wavelength-averaged force is of the form of (75.19),

F = Fm

Ω 2 /2 γ 2 + ∆2 (v) + Ω 2 /2

.

(75.25)

1 Mvr Γ , 2

(75.26)

∆(v) = ∆ + kv

(75.27)

Fm = and

is the effective detuning, which includes the Doppler shift experienced by the moving atom. Diffusion. For a traveling wave, the diffusion coefficient accompanying light pressure is

D (1 + α)Ω 2

 = ~2 k2 Γ 4 ∆2 (v) + γ 2 + Ω 2 /2 

2 ∆ (v) − 3γ 2 Ω 4 − 3 , 4 ∆2 (v) + γ 2 + Ω 2 /2 where α=

(75.28)

vc,D =

4Ω 2 γ∆ ¯ , β¯ =  F = −m βv 2 εr . ∆2 + γ 2

(75.31)

When light is tuned below the atomic resonance (“red detuning” with ∆ > 0), exponential damping of the atomic velocity with the time constant β¯ −1 ensues. No matter which way the atom moves, it is always Doppler tuned toward resonance with the light wave that propagates opposite to its velocity, and away from resonance with the light wave that propagates along its velocity. Net momentum transfer therefore opposes the motion of the atom. This is known as Doppler cooling. Optical Molasses. For three pairs of counterpropagating

waves in three orthogonal directions, (75.31) is valid in all coordinate directions, and hence as a vector equation between the force F and velocity v. For ∆ > 0 an atom experiences an isotropic viscous damping force, as if it were moving in a thick liquid. Such a field configuration is dubbed optical molasses. Two counterpropagating beams make a one-dimensional optical molasses. Limit of Doppler Cooling. Under the conditions

 d nW(n)(ˆ ˆ ex · n) ˆ 2

2

(75.29)

depends on W(n), ˆ see (75.15). Representative values are α = 1/3 for isotropic spontaneous emission, and α = 2/5 (α = 3/10) for spontaneous emission with ∆m = 0 (∆m = ±1) with respect to a quantization axis that is perpendicular (parallel) to the direction eˆ x . Spontaneously emitted photons cover all of the 4π solid angle, and so do the directions of photon recoil kicks on the atom. Absorption from a light wave traveling in a particular direction leads to transverse diffusion also in the orthogonal directions, which is not accounted for by the one-dimensional (75.28). Phenomenology in Multimode Fields Doppler Cooling in Standing Waves. Next take an atom

in two counterpropagating plane waves of light. At low intensity, Ω  Γ , forces of the form (75.25) for the two

of (75.31), the diffusion coefficients for the two counterpropagating beams averaged over a wavelength may be added, and the v = 0 form suffices for slow atoms. This yields ¯ = 0) D(v (1 + α) Ω 2 .  = ~2 k2 Γ 2 ∆2 + γ 2

(75.32)

The random diffusive motion of the atom corresponds to diffusive heating that competes with Doppler cooling. In equilibrium, the temperature is   ¯ = 0) D(v ~γ ∆ γ + . (75.33) T= = (1 + α) ¯ B 4kB γ ∆ m βk Equation (75.33) also applies to three-dimensional Doppler cooled molasses, provided one uses α = 1 corresponding to added transverse diffusion. The minimum temperature is reached at Γ ∆=γ = . (75.34) 2

Part F 75.2

Here the maximum of light pressure force, a convenient scale for optical forces, is

1095

1096

Part F

Quantum Optics

For three-dimensional molasses, the Doppler limit TD of (75.5) is obtained. For Ω > Γ , the performance of Doppler cooling deteriorates. Qualitatively, power broadening increases the effective linewidth γ . Dipole Forces. Dipole forces are the resonant analog

Part F 75.2

of ponderomotive forces discussed in Sect. 74.2. They arise from successions of absorption and induced emission driven by photons with different momenta. Such processes occur only if there is more than one wave vector present in the field, i. e., if there is an intensity gradient. For a zero-velocity two-state atom, the gradient force in a monochromatic field with the local total intensity I(r) is Fg (r) =

4~∆γ 2 ∆2 + γ 2 [1 + 2I(r)/I

s]

∇ I(r) . Is

(75.35)

The dipole force may be derived from the potential energy   2γ 2 I(r)/Is . Vg (r) = −2~∆ ln 1 + (75.36) ∆2 + γ 2 The atoms are strong field seekers for ∆ > 0, and weak field seekers when ∆ < 0.

As explained in Sect. 68.3.4, one may diagonalize the Hamiltonian to obtain the dressed atom-field states. Because the light field depends on position, so do the energies of the dressed states and their decompositions into plain atomic states. In Fig. 75.1 drawn for ∆ < 0, the dressed state with a minimum (maximum) at the field nodes coincides with the bare ground state (excited state) at the nodes. At the antinodes the admixtures of ground and excited states are evened out to some extent. The energy of a dressed state acts as potential energy for the cm motion of an atom residing in that particular state. In fact, the gradient force is the force derived from these potential energies, averaged over the occupation probabilities of the dressed states. The occupation probability is larger for the dressed state with a larger ground state admixture. From Fig. 75.1 one therefore sees that the atom predominantly resides in the dressed state that has a minimum of energy at the nodes. The atom is a weak-field seeker, as it should for ∆ < 0. Spontaneous emission remains to be considered. It gives rise to transitions between the dressed states. These transitions may go both ways between the dressed states, because the states are in general superpositions of the bare ground state and the excited state. The rate of spontaneous transitions from one dressed state to another

Optical Trap and Optical Lattice. Dipole forces are utilized in the optical trap for atoms, and even molecules. A common configuration consists of a focused laser beam tuned below resonance. The focus becomes the trap. The detuning from resonance may be substantial; lasers such as CO2 and Nd–YAG have been used. A standing wave of light makes a periodic array of optical traps called an optical lattice. Optical lattices may be set up in 1D, 2D, and 3D configurations. Induced Diffusion. Random motion of atoms in velocity space owing to absorptions and induced emissions of photons with different momenta leads to induced diffusion. Contrary to diffusion in a traveling wave as in (75.28), induced diffusion does not saturate at high intensity. Instead, the diffusion coefficient continues to grow linearly with I. Induced diffusion is another reason why the lowest Doppler cooling temperatures are generally reached at low (I < Is ) light intensities. Sisyphus Effect. In a standing wave at high intensity and large detuning, another kind of optical force becomes important that cannot be categorized either as light pressure or gradient force.

Fig. 75.1 Qualitative origin of Sisyphus effect. The hatched

pattern represents a standing light wave. The energies of the two dressed states are drawn as a function of position, along with a few filled circles representing the admixture of the ground state in each dressed state at selected field positions. Larger circles correspond to larger ground state admixtures, and hence, larger equilibrium populations of the dressed state. This figure applies for tuning of the laser above the atomic resonance (“blue detuning”)

Cooling and Trapping

Exact Results for Standing Waves Force. The force on a two-state atom in a one-

dimensional standing light wave may be expanded analytically to first order in velocity. With the field phase chosen so that the antinode is at x = 0, the force is F(x, v) = Fg (x) − mβ(x)v ,

(75.37)

where the gradient force is

~k∆ Ω 2 sin 2kx , d and the damping coefficient is   β(x) = 8∆Ω 2 εr γ −1 d −3 1 − cos2 kx  × ∆2 γ 2 + γ 4 − 2γ 2 Ω 2 cos2 kx  − 2Ω 4 cos4 kx , Fg (x) = −

(75.38)

(75.39)

with d = ∆2 + γ 2 + 2Ω 2 cos2 kx. Diffusion. For a standing wave, the v = 0 form of the

diffusion coefficient is D(v = 0) ~2 k2 Γ  Ω 2 αγ 2 cos2 kx + γ 2 sin2 kx + 2Ω 2 sin2 kx cos2 kx = 2γ 2 d   2 4 2 2 ∆ Ω sin kx cos kx ∆2 + 5γ 2 + 4Ω 2 cos2 kx . − γ 2d3 (75.40)

Semiclassical versus Quantum Theory When γ  εr , the r.m.s. velocity of a cooled two-state atom is always  vr , and semiclassical theory is valid.

1097

Under the same condition γ  εr , the Doppler-limit r.m.s. velocity also is less than the critical velocity vc,D from (75.30). Velocity expansions such as in (75.37) and (75.40) are then justified. In the contrary case, γ
εr , the full quantum theory of trapping and cooling must be employed. The cooled velocity distribution is not thermal, and temperature is ill-defined. The lowest expectation value of kinetic energy for a two-state atom in a linearly polarized standing wave occurs at low I for ∆ = 4.4εr , and is equal to 0.53 R.

75.2.3 Multistate Atoms Energy levels of atomic systems usually have angular momentum degeneracy. In addition, the polarization of light in general depends on position. A combination of these aspects leads to phenomena beyond the two-state atomic model. Polarization Gradient Cooling As explained in connection with (75.19), a finite memory time of the internal atomic state may lead to damping of the cm motion. For a two-state atom, internal equilibration arises from spontaneous emission. The time scale is τD ∼ Γ −1 , and Doppler cooling ensues. However, an atom whose ground state has angular momentum degeneracy is also subject to optical pumping. If the polarization of light varies as a function of position, optical pumping is needed to reach local equilibrium. The pumping time scale τp ∝ I −1 then becomes relevant for a moving atom. The associated cooling is known as polarization gradient cooling. Its hallmark is that, for low I, the damping coefficient β ∝ Iτp is independent of intensity. Two detailed mechanisms of polarization gradient cooling have been described [75.13], although in three-dimensional light fields they are intertwined. The Sisyphus effect works like the Sisyphus effect for a twostate atom, except that it relies on light shifts and optical pumping within the ground state manifold. Induced orientation cooling is analogous to Doppler cooling. Velocity dependence of optical pumping in counterpropagating waves leads to pumping to a state for which the force due to the wave propagating opposite to the atom exceeds the force due to the wave propagating along with the atom. Lin ⊥ Lin Molasses. One-dimensional lin ⊥ lin molasses

consists of two counterpropagating waves with orthogonal linear polarizations. The net polarization varies

Part F 75.2

increases (decreases) with the excited (ground) state admixture of the initial state. In reference to Fig. 75.1, suppose that the atom is coming from the left in the upper dressed state. The probability that the atom makes a transition to the lower dressed state, as marked by the downward vertical arrow, is largest at the node. If this transition takes place, near the next antinode the most probable transition is as shown by the upward vertical arrow. In this manner the atom spends most of its time at an uphill climb against the potential, and is therefore slowed down. In reference to Greek mythology, this is called the Sisyphus effect. In the two-state model atom, cooling takes place when the laser frequency is higher than the atomic resonance frequency.

75.2 Control of Atomic Motion by Light

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Part F

Quantum Optics

from linear – via elliptical – to circular over a distance of λ/8. With the phases chosen such that light is linearly polarized at x = 0, in the limit of nonsaturating intensity, and at low velocity, the semiclassical force on an atom with a J = 1/2 → 3/2 transition is   2 1 −1 F = −~k∆ s sin 4kx − qvγ (1 + cos 4kx) . 3 3 (75.41)

Here the saturation parameter s is s=

Ω 2 /2

Part F 75.2

. (75.42) ∆2 + γ 2 In this configuration, only the Sisyphus effect contributes to cooling. Cooling takes place for ∆ > 0, and the resulting temperature obtained from the positionaveraged quantities is   135∆2 + 296γ 2 2IΓ TD . (75.43) T= 1080(∆2 + γ 2 ) Is |∆| σ+ –σ− Molasses. One-dimensional σ + –σ − molasses

consists of two counterpropagating waves with opposite circular polarizations. The net polarization is linear everywhere, but the direction of polarization rotates as the point of observation is displaced along the propagation axis; hence, the alternative name corkscrew molasses. At low intensity and low velocity, the force on an atom with a J = 1 → 2 transition is 60 ∆γ F=− ~k 2 v . (75.44) 2 17 5γ + ∆2 In this configuration only induced orientation cooling contributes. Cooling again takes place for ∆ > 0, and the resulting temperature is   29∆2 + 1045γ 2 2IΓ   TD . (75.45) T= Is |∆| 300 ∆2 + γ 2 Lin Lin Molasses. The designation lin lin denotes

a standing wave with the same linear polarization for both counterpropagating beams. In one dimension there is no polarization gradient, but three lin lin pairs in orthogonal directions (often with mutually orthogonal polarizations) make a three-dimensional optical molasses with potential polarization gradient cooling. Experimental Molasses. For IΓ/Is |∆| < 1 and |∆| > Γ , the temperatures (75.43) and (75.45) both reduce to the form 2IΓ ~Ω 2 =C TD . (75.46) T =C kB |∆| Is |∆|

Under these conditions the same scaling is approximately observed also in three-dimensional six-beam optical molasses operating with atoms that have a degenerate ground state. The constant C depends on the degeneracy of the transitions and on the polarizations of the molasses beams. Measured values are mostly in the range 0.25 < C < 0.5 [75.14]. Limit of Cooling. While the expressions (75.43), (75.45)

and (75.46) suggest that T goes all the way to zero as I → 0 or ∆ → ∞, there is a lower limit of T reached in polarization gradient cooling. T eventually starts to rise abruptly when |∆| is increased or I is decreased. The empirical rule of thumb is that T ∼ 10 Tr is the lowest temperature one can expect. Semiclassical versus Quantum Theory. According to the semiclassical theory, T ∝ I, so the r.m.s. velocity of cooled atoms is proportional to I 1/2 . Now, the critical velocity of polarization gradient cooling, estimated roughly as

vc,p =

λ λγΩ 2 ≈ 2 , τp ∆ +γ2

(75.47)

is proportional to I. At low enough I, vc,p is therefore smaller than the velocity width of the cooled atoms. Expansions of force and diffusion in velocity are no longer useful, and temperature predictions of the type (75.46) fail. This occurs, at the latest, when the r.m.s. velocity equals a few recoil velocities. Semiclassical theory does not lead to predictions that grossly violate its key premise that the ensuing velocity distribution is much broader than vr . However, reliable theoretical limits of temperature for polarization gradient cooling may only be obtained from the full quantum treatment. Magneto-Optical Trap Since spontaneous forces may be strong already at modest light intensities ∼ lmathrm S use of light pressure to trap neutral atoms appears desirable. However, the optical Earnshaw theorem states that (in the limit of low I) the spontaneous force on a two-state atom is sourceless. While confinement may be possible in some directions, escape routes for atoms remain open in others. Threedimensional trapping of a two-state atom with light pressure is not possible. A Magneto-optical trap (MOT) defeats the Earnshaw theorem by relying on angular momentum degeneracy. Consider an atom with a J = 0 → 1 transition in a magnetic field B that depends linearly on position around the

Cooling and Trapping

zero at x = 0. Suppose that the gradient of B is chosen in such a way that the m = 1 (m = −1) magnetic substate of the excited state has the higher (lower) energy for x > 0, and that the atom is illuminated by σ ± polarized beams propagating in the ±x-directions, tuned below resonance. When the atom is displaced from x = 0 in either direction, it is closer to resonance with the beam that pushes it back toward x = 0. This makes the restoring force responsible for trapping. A magneto-optical trap can be set up also in three dimensions. A quadrupole magnetic field of the form (75.48)

is produced by reversing the direction of current in one of the two Helmholtz coils. Three orthogonal pairs of light beams, each in the σ + –σ − configuration, complete the trap. The magnetic field is sourceless. To compensate for the ensuing signs of the field gradients, one of the σ + –σ − corkscrews has the opposite handedness from the other two.

The mechanism of the magneto-optical trap for the J = 0 → 1 configuration is the same as the mechanism for Doppler cooling, except that position dependent level shifts of the excited states take the place of velocity dependent Doppler shifts. The restoring force and the damping coefficient of Doppler cooling are closely related. For the coordinate directions u = x, y, z the relation is   mge µ B  ∂Bu  Fu = −κu u, κu = β , (75.49) ~k  ∂u  where ge is the Landé factor for the excited state. A magneto-optical trap may similarly be based on the induced orientation mechanism of polarization gradient cooling. In that case, the Landé factor of the ground state, gg , should probably be used in (75.49). This may be the true mechanism of most magnetooptical traps, but insufficient quantitative understanding precludes firm conclusions. Atoms in a well-aligned magneto-optical trap reside near the zero of B, so that the magnetic field has little effect on polarization gradient cooling. Trapping and cooling are achieved simultaneously.

75.3 Magnetic Trap for Atoms The magnitude B(r) of a magnetic field may have a minimum in free space, as in (75.48). A particle with a magnetic dipole moment µ then experiences a trapping potential U(r) = µB(r) if µ and B are antiparallel. µ remains locked antiparallel to B if the field seen by the moving dipole satisfies the adiabatic condition   1  dB  µB  (75.50) B  dt  ~ (Section 73.3.4). However, if B(r) = 0 at the minimum, the adiabatic condition is violated and the dipole may flip (Majorana transition). The particle may end up in a repulsive potential, and get expelled from the trap. This becomes a problem at low temperatures, when the

particles accumulate near the minimum of the potential. Trap configurations are therefore designed in which B(r) = 0 at the minimum. Evaporative Cooling. A magnetic trap is often combined with evaporative cooling. The most energetic atoms from the tail of the thermal distribution escape from the trap, whereupon the average energy of the remaining atoms decreases. Successful operation of evaporative cooling requires a high enough rate of elastic collisions so that the atoms thermalize in a time short compared with the lifetime of the sample. In order to sustain the rate of evaporation, the effective depth of the trap is lowered as the atoms cool.

75.4 Trapping and Cooling of Charged Particles Since the potential Φ(r) of a static electric field satisfies Laplace’s equation, Φ(r) cannot have an extremum in free space. A static electric field therefore cannot serve as an ion trap (Earnshaw’s theorem). Paul and Penning traps circumvent this limitation by making use of an alternating electric field and a magnetic field, respectively. Cooling is often employed to assist trapping.

1099

75.4.1 Paul Trap Trapping Configuration. Consider an ideal trap whose sur-

faces are hyperboloids of revolution; see Fig. 75.2. The two “endcaps” and the intervening “ring” are equipotential surfaces of the quasistatic electric

Part F 75.4

   ∂Bz  1 1 B(r)  x e yˆ e − − z e ˆ ˆ z x y ∂z r=0 2 2

75.4 Trapping and Cooling of Charged Particles

1100

Part F

Quantum Optics

equal to the cycle-averaged kinetic energy in the micromotion. Explicitly,   q 2 E 2 (r) q 2 V 2 x 2 + y2 + 4z 2 UP (r) = = , 4m ω˜ 2 4m ω˜ 2 40

U – Vcos ω˜ t

(75.56) z0

where E(r) is the ac field amplitude. This is an anisotropic harmonic oscillator potential characterized by the oscillation frequencies √ 2qV νz = 2νx,y = . (75.57) m ω ˜ 20

⬇

ρ0

Part F 75.4

Fig. 75.2 Electrode configuration and voltages of an ideal

hyperboloid Paul trap

potential Φ(x, y, z) =

  Φ0 (t) 2z 2 − x 2 − y2 2 20

,

(75.51)

where 0√is the distance from the center to the ring, z 0 = 0 / 2 is the distance to the endcaps, and Φ0 (t) is a voltage applied between the endcaps and the ring, Φ0 (t) = U − V cos ωt ˜ .

(75.52)

Motion of an Ion. In the ideal three-dimensional Paul trap, Newton’s equations of motion for the coordinates u = x, y or z may be recast as Mathieu’s equations,

d2 u + (au − 2qu cos 2τ)u = 0 , (75.53) dτ 2 where τ = ωt/2 ˜ is a dimensionless quantity proportional to time, the parameters are 8qU , m ω˜ 2 20 4qV qz = − 2qx,y = − , m ω˜ 2 20 az = − 2ax,y = −

(75.54) (75.55)

and m and q are the mass and charge of the particle. Stable trapping ensues when au and qu are such that the motion of the ion is stable in all directions. A Paul trap normally operates in the first stability region of (75.53). Stable motion may be qualitatively divided into forced micromotion at the frequency ω˜ of the external drive, and into slower secular motion of the center of the micromotion. If U = 0, the secular motion takes place in an effective ponderomotive potential UP , Sect. 74.2,

Quantization of C.M. Motion. The separation of micromotion and secular motion is excellent, and the trap is stable, when νx,y,z  ω. ˜ Ignoring the micromotion, the cm motion of the ions in the potential UP (r) may be quantized readily. The energy of a state with n x,y,z quanta in the coordinate directions x, y, z is  
1 . ~νi n i + (75.58) E= 2 i=x,y,z

Variations of Paul Trap. Little practical advantage usu-

ally arises from a realization of the ideal shape. Even a single electrode with an applied ac voltage may work as a Paul trap. A linear trap is basically a two-dimensional Paul trap with an added static longitudinal potential to prevent escape of the ions from the ends of the trap. A closed race track Paul trap is obtained by bending a linear trap into a ring. Cooling Laser Cooling in One Dimension. The secular motion

of an ion may be cooled using lasers. Consider the motion of the ion in one of the principal-axis directions x, y, z, with ν denoting the corresponding cm frequency. In the common case where γ  ν, Doppler cooling works basically as with a free atom. In the contrary case, ν  γ , cooling may be achieved by tuning the laser to ω = ω0 − ν. Resonant photoabsorption starting with n cm quanta decreases the quantum number from n to n − 1, and subsequent spontaneous emission on the average leaves the cm energy nearly untouched. The net effect is reduction of the cm energy by ~ν in such a Raman process. Since the oscillating ion sees a frequency-modulated laser with sidebands, this method of ion cooling is called sideband cooling. For one-dimensional motion of a two-state ion in a traveling light wave at low I, the velocity damping

Cooling and Trapping

rate is β=

75.4 Trapping and Cooling of Charged Particles

1101

Energy in Micromotion. Possibly with the aid of com-

2Ω 2 γ∆   εr , 2 (∆ + ν) + γ 2 (∆ − ν)2 + γ 2

(75.59)

and the expectation value of the cm energy is  ~ E= ∆2 + γ 2 + ν2 4∆

  (∆ − ν)2 + γ 2 (∆ + ν)2 + γ 2 +α , ∆2 + γ 2 (75.60)

(75.61)

For optimal sideband cooling, ν  γ and ∆ = ν, the result is     γ 2 1 γ 4 . n = (1 + 4α) +O (75.62) 4 ν ν In principle, by decreasing the linewidth γ , the ion may be put arbitrarily close to the ground state of the cm harmonic oscillator. Such a decrease is not practical in real two-state systems, but is routinely achieved by using the two-photon resonance in a three-state Λ configuration as an effective two-state system; see Sect. 73.6.2. Laser Cooling in Three Dimensions. Either by design

or chance, no two of the νi are precisely degenerate. If the damping rate β and the trap frequencies νi satisfy β
|νi − ν j |, i = j ,

(75.63)

the motion of the ion in each principal axis direction of the trap is cooled independently of the other directions. For γ  νx,y,z , a singlelaser propagat√ beam  ing approximately in the direction 1/ 3 (ˆex + eˆ y + eˆ z ) suffices to cool all components of the secular motion.

75.4.2 Penning Trap Trapping Configuration. In the Penning trap, a dc voltage U is

applied between the endcaps and the ring, and a constant magnetic field B in the direction of the trap axis z is added. The magnetic field forces an ion escaping toward the ring to turn back. Motion of an Ion. The motion of an ion is a super-

position of three periodic components. For the same ideal hyperboloid shape that was discussed with the Paul trap, (75.51) and Fig. 75.2, the three components are completely decoupled. Firstly, in the axial direction, the ion executes oscillations at the axial frequency  1/2 2qU νz = . (75.64) m 20 Secondly, the ion undergoes cyclotron motion in the plane perpendicular to the trap axis. As a result of the electric field, the frequency of the cyclotron motion   1 1 2 1 2 1/2  ν − ν (75.65) νc = νc + 2 4 c 2 z is displaced from the cyclotron frequency νc = qB/m of a free ion. Thirdly, the guiding center of cyclotron motion rotates about the trap axis at the magnetron frequency   1 1 2 1 2 1/2 νc − νz . (75.66) νm = νc − 2 4 2 The frequencies typically satisfy νm  νz  νc .

(75.67)

Magnetron motion has unusual properties. It takes up the majority of the electrostatic energy in the transverse directions, which in the absence of the magnetic field would lead to expulsion of the ion. Relative to a stationary ion at the trap center, the energy of the magnetron motion is bounded from above by zero. The radius, as well as velocity and kinetic energy of magnetron motion, decreases with increasing total energy. The energy for a state with n c , n z and n m quanta in the cyclotron,

Part F 75.4

where α characterizes the angular distribution of spontaneous emission, see (75.29). The result (75.60) is useful when either εr  γ or εr  ν. The limit ν  γ is for Doppler cooling; the temperature from (75.60) coincides with (75.33) for a free atom. The case with both ν  γ and εr  ν corresponds to sideband cooling in the Lamb–Dicke regime, in which the cooled ion is confined to a region much smaller than λ. In connection with sideband cooling, it is convenient to cite the expectation number of harmonic oscillator quanta n instead of energy or temperature; the latter are    −1 1 ~ν  ln 1 + n−1 E = ~ν n + ,T = . 2 kB

pensating static electric fields, one cooled ion may be confined near the zero of the trapping ac electric fields. Then the energy in the micromotion is comparable to the energy in the secular motion.

1102

Part F

Quantum Optics

axial and magnetron motions is therefore     1 1 + ~νz n z + E = ~νc n c + 2 2   1 . − ~νm n m + 2

(75.68)

Part F 75.4

Cooling Laser Cooling. For ions in a practical Penning trap, the frequencies νc , νz and νm are < γ . If k is not orthogonal to either the cyclotron motion or the axial motion, Doppler cooling proceeds essentially as for a free atom. However, energy should be added to the magnetron motion in order to reduce the magnetron radius and velocity. The solution is to aim a finite-size laser beam off the center of the trap in such a way that an ion experiences a higher (lower) intensity over the part of its magnetron orbit in which it travels in the direction of (opposite to) the laser beam. With a proper choice of the parameters, the ensuing addition of energy overcomes Doppler cooling of the magnetron motion. Other Cooling Methods. Precision measurements are

carried out in Penning traps with objects that do not have an internal level structure suitable for laser cooling; and thus, other cooling methods are used. For light particles such as electrons, characteristic times of radiative damping of the cyclotron motion are in the subsecond regime, and hence, so are the equilibration times with blackbody radiation. Cooling is accomplished by enclosing the trap in a low-temperature (e.g. liquid helium) environment. For protons and heavier particles, the equilibration times of the cyclotron motion with the environment are impracticably long, and the same applies to the axial and magnetron motions even for electrons. A workable cooling scheme for the axial motion is based on the charges that the oscillating particle induces on the endcaps. The charges generate currents in an external circuit connecting the endcaps. The endcaps are thus coupled to a cooled resonant circuit tuned to the axial frequency, and axial motion relaxes to thermal equilibrium with the resonant circuit. A variant of this resistive cooling , in which the ring is split into electrically insulated segments, is used to cool the cyclotron motion of protons and heavier ions. Magnetron motion of an electron or proton is cooled by sideband cooling. An electric field with components in both the z-direction and xy-plane, and tuned to ω = νz + νm , drives transitions which may either increase or decrease the number of quanta in each mode.

However, the matrix elements favor transitions with ∆n z = 1 and ∆n m = −1. Pumping of the axial motion is canceled by axial cooling, while an equilibrium with low kinetic energy ensues for the magnetron motion. Ideally, the ratio of kinetic energies becomes Tkin,m νm = . Tkin,z νz

(75.69)

75.4.3 Collective Effects in Ion Clouds As soon as there is more than one ion in the trap, Coulomb interactions between the ions profoundly shape the physics [75.15, 16]. Ion Crystal In the standard Paul trap radio frequency heating due (presumably) to transfer of energy from micromotion to secular motion limits the number of ions that can be cooled efficiently by a laser. Nevertheless, at a low temperature, the ions settle to equilibrium positions corresponding to a minimum of the joint trapping and Coulomb potentials, and form a “crystal”. Depending on the trap parameters, the ions may also execute quasiperiodic or chaotic collective motion, or move nearly independently of one another. Changes between crystalline and liquid forms of the ion cloud resembling phase transitions are observed. Strongly Coupled Plasma Cooling of a large number of ions is possible in a Penning trap. However, magnetron motion becomes uniform rotation of the entire cloud, and Coulomb interactions set a lower limit on the attainable radius of the cloud. This leads to a lower limit on the kinetic energy and second-order Doppler shift. In a co-rotating frame, the ions behave like a onecomponent plasma on a neutralizing background. The characteristic parameter for a one-component plasma with charge per particle q and density n is   q2 4πn 1/3 ΓP = , (75.70) 3 4π0 kB T

essentially the ratio of the Coulomb energy between two nearest-neighbor ions divided by the thermal kinetic energy. ΓP > 1 indicates a strongly coupled plasma; for ΓP > 2 and ΓP > 170 solid and liquid phases are expected in an infinite plasma. Experiments with a Penning trap have produced ΓP  300. Concentric shells of ions or various more or less crystalline arrangements are seen depending on the experimental conditions.

Cooling and Trapping

Sympathetic Cooling In a trap that holds two or more species of charged particles, cooling of the motion of one species is trans-

75.5 Applications of Cooling and Trapping

1103

ferred by Coulomb interactions to the other species. This sympathetic cooling broadens the scope of ion cooling methods.

75.5 Applications of Cooling and Trapping

75.5.1 Neutral Atoms Experimental Considerations Originally the experiments often started with a longitudinal deceleration and cooling of an atomic beam by a counterpropagating laser beam. To compensate for the change of the Doppler shift of the atoms while they slowed down, the position dependent magnetic field of a tapered solenoid shifted the transition frequency of the atoms to keep them near resonance while they moved down the solenoid. A magneto-optical trap then scooped some atoms, cooled them further, and captured them. Nowadays this Zeeman slower ist mostly supplanted by various schemes in which a MOT directly captures atoms from the low-velocity wing of the thermal distribution. Depths of neutral-atom traps are below 1 K. Storage times are typically of the order of 1 s, limited at high densities by exothermal binary collisions and at low densities by collisions with the atoms in the background gas. The temperature of cooled atoms may be measured by the time-of-flight method. All cooling and trapping fields are turned off, whereupon the atoms fall freely under gravity. The distribution of arrival times of atoms at a probe laser beam underneath the initial molasses is compared with a numerical model for the disintegration of the molasses. A fit gives the temperature. The focus is on Li, Rb and Cs, to a large extent because the required laser frequencies can be generated

using inexpensive diode lasers. Hyperfine structure of the ground state of the alkalis complicates experiments because the atoms may end up in an inert hyperfine level outside the active cooling/trapping transitions. To counteract this, a second appropriately tuned repump laser is added to return such atoms to circulation. A few experiments use lanthanide atoms or metastable states of rare-gas atoms, some isotopes of which do not have hyperfine structure. Cold Collisions In the molecular picture of a collision involving a laser, optical excitation takes place at the interatomic distance for which the difference between the potential curves of the incoming and excited states equals the energy of a laser photon [75.17]. The end products of an inelastic collision (fine or hyperfine structure changing collision, associative ionization, radiative escape, etc.) are normally determined at shorter interatomic distances, when the potential curve of the excited state has an anti-crossing with the potential curve of the product channel. The novel feature of ultracold collisions is the long duration due to the low velocity of the collision partners. Spontaneous emissions and other phenomena irrelevant at room temperature may take place during the collision. On the scale of typical resonance widths, at very low temperatures the collision partners are in effect in a single continuum state with zero energy. This facilitates photoassociation spectroscopy. Laser-induced transitions from the initial continuum state to bound vibrational states of the molecule are observed. Collisions are of practical importance in that they limit the achievable atom density in a trap: an inelastic collision may release more kinetic energy than the trap can contain, which results in a loss of an atom (or two atoms) from the trap. Collision rates are actually measured by monitoring the loss rate of atoms as a function of density. Precise energies of the molecular vibrational states from photoassociation spectroscopy are used as input to determine [75.18] s-wave scattering lengths for atoms and to measure molecular parameters to an accuracy that far exceeds the capabilities of ab-initio calculations.

Part F 75.5

Trapping and cooling offer increased interaction times between the atoms/ions and the light. This leads to reduced transit time broadening, and indeed to macroscopic (> 1 s) interaction times. Laser cooling in a magneto-optical trap routinely gives temperatures so low that the Doppler width is below the natural linewidth of the cooling transition. A homogeneously broadened atomic sample is thus prepared. Cooling also enables reduction of the second-order Doppler effect. Various frequency measurements are the prime beneficiary of cooling and trapping. Potential applications range from detection of the change of natural constants in time Chapt. 30 to such feats of technology as the Global Positioning System.

1104

Part F

Quantum Optics

Frequency Standards An atomic fountain starts with an optical molasses or a magneto-optical trap. The laser beams are then manipulated to give an upward push to the atoms. The atoms fly up against gravity for a few tens of centimeters, then turn back and, because of the initial transverse velocities, fan out to a “fountain”. In a fountain clock [75.19], the fountain erupts through a microwave cavity that drives a hyperfine transition in the atoms. The clock is in effect an accurate measurement of the transition frequency. The fountain is beneficial because the interrogation times ≈ 1 s are longer and the atomic velocities ≈ 1 m/s slower than in traditional beam clocks.

Part F 75.5

Bose–Einstein Condensate At present, the most prominent basic-physics applications of cooling and trapping of atoms undoubtedly are in studies of Bose–Einstein condensation and quantum degenerate Fermi gases in dilute atomic vapors. This topic is covered in Chapt. 76.

75.5.2 Trapped Particles Experimental Considerations Both Paul and Penning traps behave like a conservative potential, and scatter rather than confine a particle coming from the outside. One method to load a trap is to generate the ions in situ, e.g., by letting a beam of atoms and electrons collide inside the trap. Time dependent electric potentials are another loading method. The trapped species is injected thorough a hole in the endcap, and the opposing endcap is raised to an electric potential that makes the entering particles stop. The potential is then lowered before it ejects the particles. A single electron, positron, proton, antiproton or ion may be loaded. Typical depths of ion traps are ≈ 1 eV or ≈ 104 K. With the aid of cooling, the storage time may be made infinite for all practical purposes. Trap frequencies are measured by observing the resonance excited by added ac fields. For instance, an electric field near the axial cm resonance frequency may be coupled between the ring and one endcap. A resonance circuit coupled between the ring and the other endcap is used to detect the resonance. Alternatively, ejection of the driven ions is monitored. For an electron in a Penning trap, the cyclotron frequency is in the extreme microwave region. Detection of the resonance is achieved indirectly. The uniform magnetic field is perturbed with a piece of a ferromagnet to make a magnetic bottle. The axial motion and the cyclotron motion are then coupled. A resonant

microwave drive adds energy to the cyclotron motion, which detectably alters the axial frequency. The three trap frequencies satisfy 2 νc2 = ν2c + νm + νz2 .

(75.71)

This relation remains valid even if the magnetic field is misaligned with respect to the trap axis, and is also insensitive to small imperfections in the cylindrical symmetry of the electrodes. The bare cyclotron frequency may therefore be deduced accurately. For an ion with a dipole-allowed resonance transition, fluorescence of a single ion is readily detected. Even absorption of a single ion may be measurable. Various methods of finding the temperature have been devised. At temperatures of 1 K and higher, Doppler broadening of a dipole-allowed optical transition is observable. The size of the single-ion cloud is a measure of temperature. Finally, motional sidebands in the absorption of a narrow transition (γ  ν), not necessarily the same transition as the one used for cooling, may be measured to find n. In the Lamb–Dicke regime only the carrier absorption at ∆ = 0 and sidebands at ∆ = ±ν are significant, and the ratios of the peak absorptions are εr εr α− : α0 : α+ = n : 1 : (1 + n) . (75.72) ν ν In an ion crystal, the ions have collective vibration modes akin to phonons, instead of the three vibration modes along the principal axes of the trap of a single ion. Doppler cooling and sideband cooling work for such collective modes much like they work for the vibration modes of a single ion. Quantum Jumps Ion traps make it possible to isolate an individual atomic scale particle for studies for a practically indefinite time, which enables clean experiments on various fundamental aspects of quantum mechanics and quantum electrodynamics. Quantum jumps are a case in point. Suppose that, in addition to an optically driven twolevel system, a single ion has a third shelving state. The ion infrequently makes a transition to the shelving state, stays there for a long time compared with the time scale of spontaneous emission of the active system, and then returns to the two-level system. When the ion makes a transition to the shelving state, fluorescence from the two-level system suddenly ceases; and the fluorescence reappears, equally abruptly, when the ion returns to the two-level system. The jumps in light scattering are the quantum jumps [75.20]. They are a method to detect

Cooling and Trapping

a weak transition with an enormous amplification: a single transition to or from the shelving state may mean the difference between the presence or absence of billions of fluorescence photons. g − 2 Measurements For an electron, the cyclotron frequency νc and the spinflip frequency νs are related by

1 νs = gνc . 2

(75.73)

Measurements of Mass Ratios As the cyclotron frequency is inversely proportional to the mass of the ion (or electron, positron, proton, antiproton, etc.), an accurate measurement of the cyclotron frequencies of two species in the same Penning trap amounts to an accurate measurement of the ratio of

the masses [75.22]. A sufficient resolution to weigh molecular bonds is conceivable. Quantum System of Motional States A vibrational mode in a trapped ion and an effective twostate system for the internal degrees of freedom make a realization of the Jaynes–Cummings model (discussed in detail in Sect. 79.5.1). Moreover, sideband cooling enables an experimenter to put this mode cleanly in its lowest quantum state. These observations have inspired quantum-state engineering with the objective of generating an arbitrary state of the vibrational motion of the ion [75.23]. In many-ion crystals the collective vibration modes may be used to couple and entangle the internal degrees of freedom of two or more ions. As discussed in Chapt. 81, prototype quantum gates have been demonstrated in this manner. More generally, experiments have come to the point when it is possible to address joint quantum states for the internal and cm degrees of freedom almost at will. This facilitates new cooling schemes. Time evolution derived from a Hamiltonian can never lead to cooling; an irreversible mechanism such as spontaneous emission is always needed. The idea of many cooling schemes therefore is to pump atoms optically around the quantum states in such a way there is no pathway out of the target state, so that the atoms eventually accumulate there. Velocity selective coherent population trapping [75.24], Raman sideband cooling of an ion, and Raman cooling of an atom in an optical lattice [75.25] work in this way.

References 75.1 75.2 75.3 75.4

75.5 75.6 75.7 75.8

The Mechanical Effects of Light, J. Opt. Soc. Am. B 2(11) (1985), special issue S. Stenholm: Rev. Mod. Phys. 58, 699 (1986) Laser Cooling and Trapping of Atoms, J. Opt. Soc. Am. B 6(11) (1989), special issue C. Cohen-Tannoudji: Fundamental Systems in Quantum Optics, ed. by J. Dalibard, J. M. Raymond, J. Zinn-Justin (Elsevier, Amsterdam 1991) p. 1 D. J. Wineland, W. M. Itano, R. S. Van Dyck Jr.: Adv. At. Mol. Phys. 19, 135 (1984) L. S. Brown, G. Gabrielse: Rev. Mod. Phys. 58, 233 (1986) Physics of Trapped Ions, J. Opt. Soc. Am. B 20(5) (2003), special issue Laser Cooling of Matter, J. Phys. B 36(3) (2003), special issue

1105

75.9 75.10 75.11 75.12 75.13 75.14

75.15 75.16 75.17

J. Javanainen: Phys. Rev. A 46, 5819 (1992) Y. Castin, K. Mølmer: Phys. Rev. Lett. 74, 3772 (1995) M. R. Doery, E. J. D. Vredenbregt, T. Bergeman: Phys. Rev. A 51, 4881 (1995) A. C. Doherty, T. W. Lynn, C. J. Hood, H. J. Kimble: Phys. Rev. A 63, 013401 (2001) J. Dalibard, C. Cohen-Tannoudji: J. Opt. Soc. Am. B 6, 2023 (1989) C. Gerz, T. W. Hodapp, P. Jessen, K. M. Jones, C. I. Westbrook, K. Mølmer: Europhys. Lett. 21, 661 (1993) H. Walther: Adv. At. Mol. Opt. Phys. 31, 137 (1993) J. J. Bollinger, D. J. Wineland, D. H. E. Dubin: Phys. Plasmas 1, 1403 (1994) P. S. Julienne, A. M. Smith, K. Burnett: Adv. At. Mol. Opt. Phys. 30, 141 (1993)

Part F 75

Due to quantum electrodynamic corrections g = 2, and so the anomaly frequency νa = νc − νs ∼ 10−3 νc is nonzero. A magnetic field at the anomaly frequency causes a simultaneous flip of the spin and a loss or gain of one quantum of energy in the cyclotron motion. In a magnetic bottle, the change in the cyclotron motion causes a change in the axial resonance frequency, which is detected. The anomaly frequency can thus be measured accurately. Together with a measurement of the cyclotron frequency, this yields a measurement of the g-factor of the electron, or positron [75.21].

References

1106

Part F

Quantum Optics

75.18

75.19

75.20 75.21

E. G. M. van Kempen, S. J. J. M. F. Kokkelmans, D. J. Heinzen, B. J. Verhaar: Phys. Rev. Lett. 88, 093201 (2002) Y. Sortais, S. Bize, M. Abgrall, S. Zhang, C. Nicolas, C. Mandache, R. Lemonde, P. Laurent, G. Santarelli, P. Petit, A. Clairon, A. Mann, S. Chang, C. Salomon: Phys. Scr. T95, 50 (2001) R. Blatt, P. Zoller: Eur. J. Phys. 9, 250 (1988) R. S. Van Dyck Jr., P. B. Schwinberg, H. G. Dehmelt: Phys. Rev. Lett. 59, 26 (1987)

75.22 75.23 75.24

75.25

M. P. Bradley, J. V. Porto, S. Rainville, J. K. Thompson, D. E. Pritchard: Phys. Rev. Lett. 83, 4510 (1999) D. Leibfried, R. Blatt, C. Monroe, D. Wineland: Rev. Mod. Phys. 75, 281 (2003) J. Lawall, F. Bardou, B. Saubamea, K. Shimizu, M. Leduc, A. Aspect, C. Cohen-Tannoudji: Phys. Rev. Lett. 73, 1915 (1994) S. P. Hamann, D. L. Haycock, G. Klose, P. H. Pax, I. H. Deutsch, P. S. Jessen: Phys. Rev. Lett. 80, 4149 (1998)

Part F 75

1107

Quantum Dege 76. Quantum Degenerate Gases

Bose–Einstein condensation in dilute alkali metal vapors has realized a source of atoms with properties analogous to the properties of laser light, and more recently, ultralow-temperature Fermi gases have come under study. The field of quantum degenerate gases has become a main theme in AMO physics. Dilutevapor systems are weakly interacting, and subject to

76.1

Elements of Quantum Field Theory ........ 1107 76.1.1 Bosons..................................... 1108 76.1.2 Fermions.................................. 1109 76.1.3 Bosons versus Fermions ............. 1109

76.2 Basic Properties of Degenerate Gases .... 1110 76.2.1 Bosons..................................... 1110 76.2.2 Meaning of Macroscopic Wave Function .................................. 1114 76.2.3 Fermions.................................. 1115 76.3 Experimental ...................................... 1115 76.3.1 Preparing a BEC......................... 1115 76.3.2 Preparing a Degenerate Fermi Gas.......................................... 1117 76.3.3 Monitoring Degenerate Gases ..... 1117 76.4 BEC Superfluid ..................................... 1117 76.4.1 Vortices.................................... 1117 76.4.2 Superfluidity ............................ 1118 76.5 Current Active Topics ............................ 1119 76.5.1 Atom–Molecule Systems............. 1119 76.5.2 Optical Lattice with a BEC ........... 1121 References .................................................. 1123 references are meant to be entries to the literature only. Assignment of credit or priority is never implied.

a degree of experimental control not seen before in traditional low-temperature condensed matter systems. Ultralow-temperature gases have thereby also given a new lease on life to investigations of superfluid systems in condensed matter physics. The result is a broad interdisciplinary effort that is still expanding at the time of writing.

76.1 Elements of Quantum Field Theory A Bose–Einstein condensate and a degenerate Fermi gas are both consequences of particle statistics, exchange symmetries of the many-particle wave function. It is possible, in principle, to deal directly with the wave

functions, but in practice analyses of many-body systems are usually carried out using the methods of second quantization and field theory. In first quantization, the particles are labeled as if each one had a unique tag

Part F 76

The purpose of this Chapter is to summarize the basic physics of dilute quantum degenerate gases. Given the broad activity in the field, many choices have to be made regarding the topics to include and the style of the discussion. Emphasis is placed on AMO physics, as opposed to condensed matter physics. One related choice is that virtually nothing is said about temperature dependence. Inside AMO physics the approach is in the vein of quantum optics, as opposed to atomic/molecular structure and collisions. For the most part, the coverage is on elementary concepts and basic material. The exception to this is Sect. 76.5, where a few topical issues are addressed. The review article [76.1] has become the standard reference on the basic properties of a Bose–Einstein condensate (BEC), [76.2] is its contemporary with more of a quantum optics slant, [76.3] concentrates on conceptual issues, [76.4] makes connections between the present theories and traditional condensed matter physics, and [76.5] is particularly explicit about the structure and excitations of a BEC. Here, references are usually not given for topics that are discussed in these reviews, or where a full discussion is easily traced from them. Otherwise,

1108

Part F

Quantum Optics

on it, and the wave function for more than one indistinguishable particle must be symmetrized explicitly. In second quantization, the question is how many particles are in a given state without a distinction between identical particles. The exchange symmetries are then taken care of automatically. Here we briefly summarize [76.6] elementary features of quantum field theories for both bosons and fermions.

is the operator for the total number of particles in the system. The annihilation and the creation operators have the usual boson commutators,   † † † ak , ak = δkk . (76.5) [ak , ak ] = ak , ak = 0, The boson field operator is defined as  ˆ ψ(x) = u k (x)ak .

(76.6)

k

76.1.1 Bosons Particles with an integer value of the angular momentum obey the Bose–Einstein statistics. The characteristic property is that a one-particle state can accommodate an arbitrary number of bosons. State Space for Bosons. Specifically, first consider one

Part F 76.1

particle whose states are completely specified by a set of quantum numbers k. As a notational device for the purposes of the present Chapter, all of the quantum numbers are assumedly mapped in a one-to-one fashion to nonnegative integers, and correspondingly the quantum numbers are written k = 0, 1, 2, . . . . The quantum numbers written here always incorporate a description of the state of the c.m. motion of the particle. We therefore have an orthonormal basis of wave functions to represent any state of a particle, {u k (x)}k , where x stands for the c.m. coordinate. Given the one-particle states, the postulate is that the Fock states |n 0 , n 1 , . . . , n ∞  with n k = 0, 1, 2, . . . particles in the states k = 0, 1, 2, . . . form an orthonormal basis for the many-body system. Second-Quantized Operators for Bosons. The annihilation operator for the state k, ak , is defined by

ak |n 0 , n 1 , . . . , n k , . . . , n ∞  √ = n k |n 0 , n 1 , . . . , n k − 1, . . . , n ∞  .

(76.2)

It follows that †

† nˆ k = ak ak

(76.3)

and so is called the number operator for the state k. Correspondingly,  † Nˆ = ak ak (76.4) k

(76.7)

follows from boson commutators and the completeness of the wave functions {u k (x)}k . The orthogonality of the wave functions gives the expression  † ˆ ˆ N = d3 x ψˆ (x) ψ(x) (76.8) for the particle number operator. The positive operator †

ˆ n(x) = ψˆ (x) ψ(x) ˆ

(76.9)

evidently represents the density of the particles at the position x. The second-quantized operators introduced thus far can be used to express all observables acting on indistinguishable bosons. The most relevant here are the oneand two-particle operators. One-particle operators, such as the kinetic energy, act on one particle at a time, while two-particle operators, such as atom–atom interactions, refer to two particles. In first quantization, these are of the form  1 V(xn ), O2 = u(xn , xn  ) , O1 = 2  n n,n

(76.10) (76.1)

Its Hermitian conjugate, the creation operator, behaves as
† ak | . . . , n k , . . .  = n k + 1 | . . . , n k + 1, . . .  .

ak ak | . . . , n k , . . .  = n k | . . . , n k , . . .  ,

The commutator for the field operator,   † ˆ ψ(x), ψˆ (x ) = δ(x − x ) ,

where the sums run over the labels of the particles. The corresponding second-quantized operators are  † ˆ (76.11) Oˆ 1 = d3 x ψˆ (x)V(x)ψ(x),  1 † † d3 x d3 x  ψˆ (x)ψˆ (x ) Oˆ 2 = 2 ˆ  )ψ(x) ˆ . (76.12) × u(x, x )ψ(x When the particles have internal degrees of freedom in addition to the c.m. motion, such as hyperfine and Zeeman states, it is convenient for the present purposes to regard particles in each internal state as a separate species. Thus, if the quantum number breaks up into

Quantum Degenerate Gases

k ≡ { p, α}, where p stands for quantum numbers of the center of the mass and α for the quantum numbers of the internal state, it is expedient to define a quantum field for each species α as ψˆ α (x) =



u pα (x)a pα .

k−1

= n k (−1)

p=0 n p

|n 0 , n 1 , . . . , n k − 1, . . . , n ∞  ,

Bose–Einstein Condensate. The state of a boson sys-

tem of particular interest here is the BEC. In an ideal gas condensation entails a macroscopic fraction of the particles occupying the ground state of the c.m. motion. Condensation is a phase transition that occurs when either the density of the gas is increased or the temperature is lowered. In a homogeneous ideal Bose gas the governing parameter is the phase space density ζ defined as 3/2 (76.14)

where m is the mass of the condensing atoms, T is the temperature, and n is the density of the condensing species. For the purposes of quantum degeneracy, each internal state of an atom behaves as a separate species. Bose–Einstein condensation takes place when the phase space density satisfies ζ = 2.612. Depending on whether density or temperature is regarded as a constant, (76.14) may be regarded as an equation for the critical temperature Tc or the critical density n c for Bose–Einstein condensation.

76.1.2 Fermions Particles with a half-integer angular momentum obey the Fermi–Dirac statistics. Each Fock state may then only have the occupation numbers n k = 0, 1. The conventional definition of the annihilation operator contains

[A, B]+ ≡ AB + BA rather than the commutator. For instance,  † ak , ak + = δkk . [ak , ak ]+ = 0,

(76.16)

(76.17)

Except for the use of anticommutators in lieu of commutators, all formal expressions for field operators and one- and two-particle operators written down for bosons in Sect. 76.1.1 remain valid as stated. Degenerate Fermi Gas. A degenerate Fermi gas realized in a dilute atom vapor is the fermion counterpart of a BEC. The basic parameter of a free noninteracting Fermi gas is the Fermi energy, the chemical potential at zero temperature. It is given by



1/3 ~2 kF2 ; kF = 6π 2 n , (76.18) 2m where n once more is the density for the relevant fermion species. In the limit of zero temperature the Fermi gas makes a Fermi sea; the states below the Fermi energy are filled with one particle each, the states above the Fermi energy are empty. The gas begins to show substantial deviations from the Maxwell-Boltzmann statistics and may be regarded as degenerate when the temperature is below the Fermi temperature, T ≤ TF = F /kB . Except for a numerical factor, in terms of density and temperature the condition T ≤ TF is the same as the condition for Bose–Einstein condensation. F =

76.1.3 Bosons versus Fermions Isotopes of alkali metals with an odd mass number 7 Li, 23 Na, 39 K, 85 Rb, 87 Rb, 133 Cs make Bose–Einstein gases, while isotopes with an even mass number 6 Li,

40 K make Fermi–Dirac gases. Atoms are composite particles consisting of fermions, and how they may act as bosons is a legitimate question. Whether a satisfactory formal answer to this question exists may be debatable, but in practice atoms seem to obey the correct statistics in processes that do not expose their individual constituents.

Part F 76.1

wave function quantum mechanics to second quantization, the state of the system must be specified in second quantization. For instance, take the Hamiltonian Hˆ and ˆ According to statistical the particle number operator N. mechanics a system characterized by the temperature T and chemical potential µ is in the state with the density ˆ ˆ operator ρˆ = e−(H−µ N)/kB T/Z, where the grand partition ˆ ˆ −( function is Z = Tr e H−µ N)/kB T .

n,

ak |n 0 , n 1 , . . . , n k , . . . , n ∞ 

and fermion operators are governed by the anticommutator

States of Bosons. To complete the transformation from

2π ~2 mkB T

a phase factor,

(76.15)

Mechanisms that cause transitions between the internal states couple the fields ψˆ α (x) for different α.



1109

(76.13)

p

ζ=

76.1 Elements of Quantum Field Theory

1110

Part F

Quantum Optics

When two bosonic atoms with integer angular momenta combine into a molecule, the molecule has an integer angular momentum and behaves as a boson. On the other hand, two fermionic atoms also make a bosonic molecule. Models for this latter type of system are basic-

ally ad hoc since, at this point in time, no microscopic theory for such a reorganization of the statistics exists. Nonetheless, empirically, diatomic molecules formed by combining two fermionic atoms indeed appear to be bosons.

76.2 Basic Properties of Degenerate Gases Atoms Are Trapped. Quantum degenerate alkali va-

pors are typically prepared in an atom trap. Close to the bottom almost every trap is a three-dimensional harmonic oscillator potential completely characterized by the principal-axis directions and the corresponding trap frequencies ωi , the (angular) frequencies at which a single atom would oscillate back and forth in the given principal-axis direction. In the principal-axis coordinate system the trapping potential reads

Part F 76.2

V(x) =

1 2

3 

mωi2 xi2 .

(76.19)

i=1

It is convenient to introduce the characteristic harmonicoscillator frequency scale and the corresponding harmonic-oscillator length scale as  ~ . (76.20) ω¯ = (ω1 ω2 ω3 )1/3 , = m ω¯ Atom–Atom Interactions. At low temperatures/ener-

gies only s-wave collisions are significant. In the theory of quantum degenerate gases these are frequently represented by a pseudopotential tailored to give the correct s-wave phase shift. For two atoms the atom–atom interaction is 4π ~2 a δ(x1 − x2 ) , (76.21) u(x1 , x2 ) = m where a is the s-wave scattering length. Qualitatively, the scattering length is positive (negative) if the interaction is repulsive (attractive). Model Hamiltonian. Quantum field theory for a single-

component Bose gas usually starts with the Hamiltonian  ˆ ˆ , (76.22) H = d3 x H(x) where the Hamiltonian density is

~2 2 † ˆ ˆ ˆ ∇ + V(x) ψ(x) H(x) = ψ (x) − 2m 2π ~2 a † † ˆ ψ(x) ˆ ψˆ (x)ψˆ (x)ψ(x) . + m

(76.23)

ˆ As in Sect. 76.1.1, ψ(r) is the boson field operator, the kinetic energy −~2 ∇ 2 /2m and trapping potential V(x) are one-particle operators, and atom–atom interactions are governed by the two-body operator u of (76.21). Analogous models can be written down for multicomponent boson and fermion fields, for coupling between atoms and molecules, and so on.

76.2.1 Bosons Gross–Pitaevskii Equation Mean-Field approximation. Conventionally, the next

step for bosons is to go over to the corresponding classical field theory. The result is referred to as mean-field theory or semiclassical theory. Formally, one first writes down explicitly the Heisenˆ berg equation of motion for the boson field ψ,   ∂ ˆ ˆ t) = ψ(x, t), Hˆ , i~ ψ(x, (76.24) ∂t and then declares that in the equations of motion ψˆ → ψ is a classical field not a quantum field anymore. We call ψ the macroscopic wave function of the condensate. This approximation is precisely analogous to using the classical instead of the quantum description for the electric and magnetic fields of the light coming out of a laser. Time-Dependent Gross–Pitaevskii Equation. The

time-dependent Gross–Pitaevskii equation (GPE) is

∂ ~2 2 ∇ + V(r) ψ(r, t) i~ ψ(r, t) = − ∂t 2m 4π ~2 a |ψ(r, t)|2 ψ(r, t) . (76.25) + m This equation is nonlinear, and normalization of the macroscopic wave function ψ is important. Quantum mechanically, the particle number operator is given by (76.8), so that the normalization for a system with N particles naturally reads  d3 x |ψ(x, t)|2 = N . (76.26)

Quantum Degenerate Gases

Time evolution under (76.25) preserves the normalization. Obviously, and in accordance with (76.9), n(x) = |ψ(x)|2

(76.27)

is the local density of the gas. Time-Independent Gross–Pitaevskii Equation. So-

lutions to the time-dependent GPE of the form ψ(x, t) = φ(x) e−iµt/~ are stationary states with no time evolution in the physics. The analog of the energy of a stationary state is called the chemical potential µ. The corresponding wave function φ satisfies the timeindependent GPE   4π ~2 a 2 ~2 2 µφ = − ∇ +V φ+ |φ| φ . (76.28) 2m m

Sign of Scattering Length. The qualitative properties

of a condensate, as per the GPE, depend on the sign of the scattering length. For repulsive atom–atom interactions or no atom–atom interactions, a ≥ 0, both the time-dependent and the time-independent forms are mathematically well behaved. Unless otherwise noted, the scattering length is always assumed non-negative. In the case of a negative scattering length a BEC may, in principle, decrease its energy without a bound by collapsing to a point. Mechanisms such as three-body recombination or molecule formation would eventually set in as the density increases and the collapse would stop, but the condensate must then be presumed lost. In the absence of an external potential, a condensate with a negative scattering length is unconditionally unstable against collapse. For a bounded condensate the increase in the kinetic energy coming with the decreasing size may hold off the collapse, provided the number of atoms in the condensate is sufficiently small. Simple dimensional-analysis arguments give the condition of stability in a harmonic trap as N|a|
. Behavior attributed to a collapse has been observed in 7 Li for trapped states with a negative scattering length. By using a Feshbach resonance it is also possible to adjust the (apparent) scattering length (Sect. 76.5.1) which has led to further demonstrations of collapse-like physics.

1111

Healing Length. Consider

the time-independent GPE (76.28) without an external potential, and scale the various quantities as follows:

√ 4π ~2 na ¯ µ=µ , n φ, ¯ m 1 x = ξ x¯ ; ξ = √ . (76.29) 8πna Here n is the density scale for the gas, and the length scale is ξ. In terms of these new variables the timeindependent GPE reads  2 µ (76.30) ¯ φ¯ = −∇¯ 2 φ¯ + φ¯  φ¯ . φ=

There is a solution in all of space with µ ¯ = 1, φ¯ = 1. If for some reason, such as at an edge of the sample, the condensate wave function must vanish, the length scale over which the wave function grows back to one (in the scaled units) is of the order In fact, (76.30) has √ unity. of

¯ x) = tanh z/ the solution φ(¯ ¯ 2 in the half-space z¯ ≥ 0. The quantity ξ is the minimum length scale over which a condensate wave function can build up to the density n. It is called the healing length. Thomas–Fermi Approximation. Without atom–atom interactions, the ground state of the trapping potential V(x) would be the lowest-energy (lowest µ) solution to (76.28). However, experience has shown that even modest repulsive atom–atom interactions (a > 0) spread out the macroscopic wave function of the condensate a great deal. With increasing size comes decreasing kinetic energy, according to the Heisenberg uncertainty principle. This suggests the Thomas–Fermi approximation, in which the kinetic energy term in (76.28) is simply ignored. The density of the gas is then easily solved to be     m µ − V(x) , µ > V(x) 2 n(x) = (76.31)  4 π~ a 0, otherwise,

an inverted image of the trapping potential. The normalization (76.26) can be used to find the relation between chemical potential and particle number, and all of the unknown quantities may, in principle, be found. For a harmonic potential the Thomas–Fermi approximation can be worked out explicitly with the results     1 15Na 2/5 15Na 1/5 µ = ~ω¯ , R= , 2   1 15Na 2/5 . (76.32) n(0) = 8π 3 a

Part F 76.2

Both GPEs are nonlinear variants of the Schrödinger equation, and in other contexts they are often referred to as nonlinear Schrödinger equations. The nonlinear term approximates the interaction energy of an atom with the other atoms in an averaged way by relying on the local density of the atoms, hence the term mean-field theory.

76.2 Basic Properties of Degenerate Gases

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Part F

Quantum Optics

The quantity R represents the size of the condensate. In particular, it equals the radius of the spherical condensate if the trap is isotropic with ω1 = ω2 = ω3 . Finally, n(0) is the central, maximum, density of the atoms. The relevant dimensionless parameter is Na/ , which can easily be much larger than unity in the experiments. When the Thomas–Fermi approximation is accurate, the chemical potential exceeds the typical level spacing of the harmonic-oscillator trap, and the condensate is larger than the ground-state wave function of the harmonic oscillator would be. Ordinarily, the condensate is also much larger than the healing length. Small Excitations in a BEC Linearizing the GPE. The time-dependent GPE is

Part F 76.2

nonlinear, but may be linearized around a stationary solution. Consider the special case without a trapping potential, V(x) ≡ 0. The stationary solutions are plane waves, √ (76.33) φ(x) = n ei p·x . This corresponds to a flow of the gas at the velocity v = ~ p/m and with a momentum ~ p per atom. The chemical potential of such a mode is 1 µ =  p + 0 , ~ 2

(76.34)

where p =

~ p2 , 2m

0 =

mc2 , ~

c=

√ ~ 8πna ~ = m mξ (76.35)

are the dispersion relation of free atoms, a peculiar analog of the rest energy, and the speed of sound in the BEC. The ansatz for small deviations from the stationary solution is written as √ µ  ψ(x, t) = n ei( p·x− ~ t) 1 + u ei(q·x−νt) ∗  + v∗ e−i(q·x−ν t) , (76.36) where ~q and ~ν are the momentum and energy associated with the excitation relative to the momentum and energy of the original flow. The GPE mixes the field ψ and its complex conjugate ψ ∗ , so that two small amplitudes u and v are needed for the excitations. The ansatz (76.36) is a solution to the time-dependent GPE to the lowest nontrivial order in u and v if these amplitudes and the frequency of the excitation satisfy the

eigenvalue equation      u u q + 0 + q · v 0 . =ν −v v 0 q + 0 − q · v (76.37)

The remaining problem is that the eigenvalue equation has two solutions for each q, which gives twice as many small-excitation modes as there are degrees of freedom. The extra modes are the penalty one pays for the linearization of the GPE. The criterion |u|2 − |v|2 > 0 picks out the correct small-excitation modes. The corresponding dispersion relation for the excitations is  (76.38) ν(q) = q · v + q (q + 20 ) . For a stationary BEC with v = 0 and in the limit q → 0, (76.38) gives ν  cq. This confirms the identification of c as the speed of sound. In the BEC experiments the condensates are trapped, but in principle the same analysis of small-excitation modes may be carried out both numerically and in a myriad of analytical approximations. The generic result is that the trap frequencies lend their frequency scale to small excitations. At low enough temperatures, excitation frequencies calculated in this way agree well with the experiments. Within the mean-field approximation small excitations may be analyzed similarly in all boson systems, for instance, in a multi-component Bose–Einstein condensate or a joint atom–molecule condensate. The evolution frequencies may be complex, which signals a dynamical instability of the stationary configuration; there are small-excitation modes that grow exponentially. The instability of a free gas with a negative scattering length, which is apparent in (76.38) for 0 < 0, is a simple example. Bogoliubov Theory. Bogoliubov theory is the many-

body quantum version of the analysis of small excitations. The idea is to treat the condensate mode ψ0 , containing n 0 atoms, separately in the field operator √ ψˆ = n 0 ψ0 + δψˆ , (76.39) expand the Hamiltonian in the lowest nontrivial (second) ˆ and diagonalize. order in the remnant quantum field δψ, The result is small-excitation modes with the annihilation operators Ak , where k stands for the appropriate quantum numbers. It turns out that the core mathematics of Bogoliubov theory is the same as the mathematics of small excitations, but two features are added.

Quantum Degenerate Gases

First, Bogoliubov theory explicitly shows that the coefficients u and v in the analog of (76.37) need to satisfy |u|2 − |v|2 = 1 to ensure boson commutators for the operators Ak . Second, with quantum fluctuations, atom– atom interactions force atoms out of the condensate even at zero temperature. In a homogeneous (untrapped) condensate, in the limit na3 1, at T = 0, the fraction of noncondensate atoms  is 8 N − n0 = N 3

na3 . π

(76.40)

When the gas parameter na3 is much smaller than unity, at low enough temperatures most of the atoms are in the condensate. Mean-field theory and the GPE are expected to apply, and empirically, they do. Numerical Methods for GPE Mathematical Properties of the GPE. Let us momen-

Split-Step Fourier Method. The superposition principle does not hold for the solutions of the time-dependent GPE, and the excited states are usually not orthogonal to one another in any useful sense. Methods based on eigenstate expansions for solving the time-dependent GPE are cumbersome at best. Instead, one often simply integrates the GPE as a partial differential equation in time. A number of different methods are used, but here we only discuss an elementary split-step Fourier method [76.7]. This is an exceedingly popular algorithm for parabolic equations, easy to implement, and with minor modifications also solves the time-independent GPE in any number of dimensions. Thus, consider integration of (76.25) forward in time over a step from t to t + ∆t. For this purpose assume first that |ψ|2 in the GPE were a constant equal to its value at time t, then the evolution over the time step ∆t would be given by   ~∇ 2 + U(x) ψ(x, t + ∆t) = exp −i∆t − 2m × ψ(x, t) , (76.41)

where U(x) is a given function of position. In the algorithm the exponential is first split approximately, for instance, as   ~∇ 2 + U(x) exp −i ∆t − 2m   2   ∆t ~∇ exp − i ∆t U(x)  exp i 2 2m   ∆t ~∇ 2 × exp i 2 2m (76.42) ≡ T˜ U˜ T˜ . The exponential of the kinetic-energy operator is diagonal in the Fourier representation. Consequently, carrying out the Fourier transform F and its inverse with the aid of the Fast Fourier Transformation gives the split-step algorithm ˜ −1 T˜ F ψ(t) ψ(t + ∆t) = F −1 T˜ F UF

(76.43)

with obvious efficient implementations. The inaccurate constant |ψ|2 may be improved upon in a corrector step in which the average of the initial wave function ψ(t) and the ψ(t + ∆t) obtained in the first pass is used as |ψ|2 , and step (76.43) is taken again. This split-step algorithm preserves the normalization of the macroscopic wave function, and features a high-order approximation to the exponential operator of the kinetic energy.

1113

Part F 76.2

tarily assume that by separation of variables, or by some fiat, the problem of solving the time-independent GPE has been rendered one-dimensional. The Schrödinger equation is linear and any constant multiple of a solution is also a solution. One parameter, e.g., the logarithmic derivate of the wave function at a given point in space, determines a stationary state completely. This does not hold for the corresponding GPE, for which the values of the wave function and its derivative can be specified independently at (almost) every fixed point in space. As a result of the added flexibility, and unlike the Schrödinger equation, the GPE has bounded solutions for continuous ranges of the values of the chemical potential µ. However, the time-independent GPE (76.28) comes with the added normalization condition (76.26). Normalization quantizes the values of µ for the bound states. In practice one might, for instance, find a solution that satisfies the boundary conditions for a given µ with the shooting method, then adjust the value of µ until normalization holds. Techniques used in the first numerical analyses of the time-independent GPE in the context of atom vapor condensates were variations of this theme. Such schemes are not feasible in spatial dimensions greater than one. In general there is one solution to the timeindependent GPE that can be chosen to be positive everywhere, the ground state with the lowest chemical potential. Excited steady states exist, but only a few, such as the flowing states of (76.33) and vortices discussed in Sect. 76.4.1, have obvious physical meanings. As the GPE is nonlinear, excited states are not the same as small excitations.

76.2 Basic Properties of Degenerate Gases

1114

Part F

Quantum Optics

Integration in Imaginary Time. The split-step algorithm also provides a global method to find the ground state. To this end the time-dependent GPE is integrated in imaginary time, i. e., replacing ∆t → −i∆t, starting from a more or less random initial wave function and normalizing after every step. If the GPE were linear, this procedure would emphasize the lowest-energy component of the wave function until it is the only one that remains to within a prescribed accuracy. It is not clear that the same should apply to the nonlinear GPE, but often this is the case. In nonlinear problems, split-operator methods, in spite of their seeming simplicity, often exhibit spurious behavior. Successful applications of these techniques require skill and experience in the art of numerical methods.

Part F 76.2

Local-Density Approximation While an experimental BEC is usually trapped, it is often much easier to study the theory for a formally infinite homogeneous condensate. As long as the phenomena under investigation involve length scales much smaller than the size of the condensate and time scales much shorter than the inverse trap frequencies, trapping cannot affect the behavior of the gas locally. Under such conditions one may analyze the gas at each position x as if it were homogeneous, and at the end of the calculations average over the density distribution. The unit-normalized distribution of the density of the gas used in the averaging is    δ  − n(x) n(x) d3 x  P() = . (76.44) n(x) d3 x For instance,
15 n 0 −   P() = H()H(n 0 − ) (76.45) 5/2 4n 0 holds for the Thomas–Fermi approximation with the maximum density n 0 ≡ n(0). The Heaviside step functions H restrict the density to the correct range 0 ≤  ≤ n 0 . As an example, the average density in the Thomas–Fermi model is  4 P() d = n 0 . (76.46) 7

76.2.2 Meaning of Macroscopic Wave Function Here the macroscopic wave function ψ has been introduced by replacing a boson field theory with a classical field theory.

The intuitive interpretation is that, for interacting particles, the atoms condense not to the ground state of the confining potential, but to the one-body state whose wave function is the macroscopic wave function. This notion may be criticized on various grounds, but in practice it makes a useful picture. A precise formal meaning of the macroscopic wave function, and of Bose–Einstein condensation for interacting systems, is found by considering the one-particle density matrix  †  ˆ ) , ρ(x, x ) = ψˆ (x)ψ(x (76.47) which is sufficient to determine the expectation value of any one-particle operator. This is the position representation of a positive Hermitian operator with the trace equal to particle number (or its expectation value) N. In this way, with an orthonormal set of functions {ψk (x)}k and nonnegative eigenvalues n k , such that  nk = N , (76.48) k

an expansion of the form  n k ψk (x)ψk (x ) ρ(x, x ) =

(76.49)

k

exists. The system is a BEC if at least one eigenvalue n k is of the order of the number of particles and does not formally go to zero in the thermodynamic limit (if the limit exists and is sensible). The usual case is that only one eigenstate, call it k = 0, has such a large eigenvalue. The macroscopic wave function is the corresponding eigenfunction ψ ≡ ψ0 , and n 0 gives the number of condensate atoms. If there is more than one macroscopic eigenvalue, the condensate is called fragmented. Another interpretation of the macroscopic wave function comes from statistical mechanics. In a continuous (second-order) phase transition typically a symmetry of the system is spontaneously broken. For example, below the Curie temperature a single-domain ferromagnet magnetizes in some specific direction, and the state has a lower symmetry than the rotationally invariant Hamiltonian of an isotropic ferromagnet. Any quantity that appears in a continuous phase transition and characterizes the breaking of the symmetry may be called an order parameter. The macroscopic wave function can be viewed as the order parameter associated with spontaneous breaking of the global phase or “gauge” symmetry of quantum mechanics. Specifically, in quantum mechanics the state of the system is

Quantum Degenerate Gases

76.2.3 Fermions Static Fermi Gas Thomas–Fermi Approximation. Consider an ideal

single-species Fermi gas of trapped atoms. The original Thomas–Fermi approximation (see Chapt. 20) was formulated for fermions, namely, electrons, and in the present case it is modified as follows. For the atom density n(x) at position x, at a low temperature, the corresponding local internal chemical potential is approximated according to (76.18) as 2/3  ~2 6π 2 n(x) . F (x) = (76.50) 2m Given the trapping potential V(x), the density of the gas adjusts in such a way that the sum of the external potential energy and the local internal chemical potential, the

Fermi energy, is a constant across the gas, 2/3  ~2 6π 2 n(x) V(x) + = µ, (76.51) 2m the global chemical potential. One may solve the density for a given chemical potential as √ 3 3    2 m 2 µ − V(x) 2 , V(x) < µ; n(x) = 3 π 2 ~3   0, otherwise. (76.52)

Finally, the integral of the density over all space should equal the atom number, which gives an equation to determine the chemical potential µ. For a harmonic trap this program can be carried out in an exact, analytical manner with the result that R = 22/3 31/3 N 1/6 , µ = 61/3 N 1/3 ~ω, ¯ √ 2 N . (76.53) n(0) = √ 3π 2 3 The quantities ω, ¯ , and R have the same meaning as in the BEC case. The Thomas–Fermi approximation for fermions should be applicable whenever N 1. In a one-component Fermi gas at low temperature atom–atom interactions are typically negligible for a multitude of reasons. There is no s-wave scattering, and the presence of the Fermi sea tends to suppress repulsive interactions. However, in the case of attractive interactions between two species, the Fermi sea may be thermodynamically unstable; the energy may be lowered by pairing fermions into Cooper pairs. This is the mechanism behind the BCS theory of superconductivity [76.8]. Excitations in a Fermi Gas If the interactions do not render a fermion system into a superfluid, see Sect. 76.5.1, the elementary excitations of a degenerate Fermi gas with short-range interactions are basically atom–hole pairs. What happens in the contrary case for trapped and strongly interacting atoms is presently an active area of research.

76.3 Experimental 76.3.1 Preparing a BEC In a trapped gas the density n(x) is self-determined from the atom number N, and the condition for a BEC in an

1115

ideal gas is most readily expressed in terms of the total number of atoms as kB Tc = 0.94 ~ωN ¯ 1/3 .

(76.54)

Part F 76.3

unchanged if the wave function is multiplied by an arbitrary complex phase factor eiϕ . But to write the wave function as ψ(x) already implies a preferred phase, and likewise even if the wave function is adorned with random but, for any given condensate, fixed phase, as in eiϕ ψ(x). For one condensate the random phase is inconsequential. Suppose, however, that two BECs with the wave functions eiϕ1 ψ1 (x) and eiϕ2 ψ2 (x) are combined. If the macroscopic wave functions behave as wave functions should, the combination of two condensates displays the density n(x) = | eiϕ1 ψ1 (x) + eiϕ2 ψ2 (x)|2 . There should be an interference pattern between the condensates. Two BECs indeed produce an interference pattern when they are combined, although the randomness or the absence thereof of the phases ϕ1,2 is difficult to verify experimentally. From the quantum optics viewpoint, a condensate is a given number of atoms in a given one-particle state, a number state, and cannot possess any phase at all. This seems to contradict the observations of an interference pattern. The resolution is that the process of measurement in itself produces a phase difference between the condensates even if there initially is none.

76.3 Experimental

1116

Part F

Quantum Optics

In practice, at the bottom of the trap the conditions on temperature and density for a BEC are similar to the conditions for a BEC in a free gas. In the thermodynamic limit, such that ω¯ → 0, N → ∞ with ωN ¯ 1/3 held constant, below the critical temperature Tc the fraction of condensate atoms behaves as a function of temperature T as  3 n0 T = 1− . N Tc

(76.55)

Part F 76.3

The experimental realizations of alkali vapor condensates are based on techniques of laser cooling and trapping of atoms. The following discussion relies heavily on material from Chapt. 75. A BEC in a dilute atomic gas is usually prepared using a two-stage process. First, a magneto-optical trap is used to capture a sample of cold atoms and to cool it to a temperature of the order of a few tens of microkelvin. The atoms are then transferred to a magnetic trap for evaporative cooling that leads to condensation. A magnetic trap is based on a combination of two ideas. First, if an atom that starts out with its magnetic moment antiparallel to the magnetic field moves slowly enough in a position dependent magnetic field, its magnetic moment remains adiabatically locked antiparallel to the magnetic field. The energy of the atom is then a minimum where the magnetic field is a minimum. Second, the absolute value of the magnetic field may have a minimum in free space. The minimum is then a trap for atoms whose magnetic moments are suitably oriented. The downside is that only atoms in the right magnetic (Zeeman) states are trapped. While the atoms cool down, they accumulate at the center of the trap. The center should not be a zero of the magnetic field, because at zero field an atom would lose the lock between the directions of the magnetic moment and the magnetic field necessary for trapping. A time orbiting potential (TOP) trap starts with the same kind of magnetic field that is used in a magnetooptical trap. A time-dependent magnetic field is then added in such a way that the zero of the magnetic field orbits around the center of the trap. If the frequency at which the zero orbits is high enough so that the atoms cannot follow, they see an effective potential with a minimum at the center of the trap and do not sample the zero. Alternatively, it is possible to wind a coil in such a way that it makes a magnetic field whose absolute value has a minimum that is not zero. In this type of a Ioffe–Pritchard trap the winding of the wire resembles the seams on a US baseball.

The basic idea of evaporative cooling is that the most energetic atoms escape from the trap, then the remaining atoms thermalize to a lower temperature. Some atoms are lost in the process, but with the decreasing temperature the density at the trap center nonetheless tends to increase and the phase space density increases even more due to the cooling. The cooling is usually forced by an rf drive. The transition frequency between the Zeeman states depends on the magnetic field, and increases toward the edges of the trap. Atoms are removed where the rf frequency is on resonance and drives transitions to untrapped Zeeman states. Thus, while the atoms cool and concentrate at the center, the radio frequency is swept down in such a way that the “rf knife” removing the atoms slides in from the edge of the trap. At some radio frequency a condensate abruptly emerges. The temperature can be further lowered by continuing evaporative cooling, albeit at the expense of loss of atoms. As a rule of thumb, an atom needs to experience a hundred collisions before condensation occurs, and a typical time needed to prepare a condensate is a few seconds. In a good vacuum a condensate may live for tens of seconds. It is also possible to condense atoms trapped in a faroff resonant optical trap based on the dipole forces of light, instead of the magnetic trap [76.9]. For tuning below the resonance, atoms are strong-field seekers. A focused laser beam is a three-dimensional trap for atoms, as is an arrangement with two crossed beams focused to the same spot. Furthermore, with extreme off-resonant light from a CO2 or a Nd-YAG laser, absorption of photons and the associated photon recoil kicks and heating may be negligible. An optical trap may also be added after a BEC is prepared in a magnetic trap. The advantage is that an optical trap will hold the atoms regardless of their magnetic state, so that multicomponent “spinor” condensates may be studied. Moreover, while an adiabatic change of the strength of a trap cannot change the phase space density, the phase space density may be altered by changing the shape of the trap by adding a tight optical trap to the bottom of a much wider magnetic trap. Reversible condensation inside an added optical subtrap based on such an increase in the phase space density has been demonstrated. Methods to condense atoms that might be suited for future technological applications are being pursued. For instance, by lithographic techniques it is possible to put conducting wires on a substrate to make an atom chip. With currents flowing, the wires produce magnetic fields that guide the atoms. Condensation

Quantum Degenerate Gases

in such a configuration has been reported [76.10]. Two-dimensional condensation in what is known as a gravito-optical surface trap has also been achieved experimentally [76.11]. There is an analogy between a condensate and a beam of light from a laser that we rely on extensively. However, the analogy is only partial. By dropping a condensate under gravity one makes a pulsed atom laser, and by coupling a trapped Zeeman state to an untrapped state by rf excitation it is possible to make a condensate leak slowly out of the trap. Nonetheless, at this time a method to produce a continuous beam of condensate atoms, a continuous-wave atom laser, is yet to be demonstrated.

76.3.2 Preparing a Degenerate Fermi Gas

76.3.3 Monitoring Degenerate Gases Orders of Magnitude. As a rule of thumb, the trapping frequencies in a magnetic trap are ω¯ ≈ 2π × 10 Hz, while

the frequencies in an optical trap may reach into the kHz regime. A typical oscillator length is ≈ 1 µm. A usual number of atoms is N ≈ 106 . Scattering lengths are of the order of a ≈ 10 nm. The size of a degenerate gas is in the neighborhood of R ≈ 0.1 mm, the maximum density is about n 0 ≈ 1015 cm−3 , and the BEC transition temperature and the Fermi temperature are of the order of Tc ≈ TF ≈ 1 µK. However, much lower temperatures are readily reached in a BEC. Phase Contrast Imaging. It is possible to monitor condensate features substantially larger than the wavelength of the light used in the measurements nondestructively, in situ, by using phase contrast imaging. In this method the light is detuned far off resonance so that absorptions with the accompanying photon recoil kicks on the atoms are rare, but the phase of the light nonetheless changes upon propagation through the sample. The phase change may be detected by interfering the transmitted light with the original light, with the phase of the latter suitably shifted. Time-of-Flight Imaging. Usually, though, the obser-

vation of a degenerate gas at the end of an experiment is by time-of-flight imaging. The trap is suddenly removed, whereupon the gas expands freely. After the atom cloud has grown to a size large enough compared to the wavelength of the resonant light used to monitor the gas, an absorption image of the cloud is taken. This gives the projection of the density of the gas onto a plane perpendicular to the direction of propagation of the light. Except for the effects of atom–atom interactions, after a sufficiently long time of free flight the density reflects the initial momentum distribution of the atoms. Timeof-flight images bear the signs of both condensation in a Bose gas and quantum degeneracy in a Fermi gas. Nontrivially, other features of interest such as vortex cores are also preserved and can be detected after the free expansion. The downside is that the time-of-flight method is destructive. After each snapshot the sample will have to be prepared again.

76.4 BEC Superfluid 76.4.1 Vortices Flow Velocity in a Superfluid. By manipulating the

Heisenberg equations of motion for a Bose field under the Hamiltonian (76.22) it is easy to derive the equation

1117

of continuity for the atoms, ∂ nˆ + ∇ · Jˆ = 0 , ∂t

~ † ˆ nˆ = ψˆ ψ, ψ∇ψ † − ψ † ∇ψ , Jˆ = i 2m

(76.56) (76.57)

Part F 76.4

A single-species, very-low temperature Fermi gas is an uninteresting system, as the Fermi–Dirac statistics forbids s-wave interactions between the atoms and the gas is nearly ideal. In experiments the gas usually has two species, different states of the same atom. The interactions between the species are comparable in strength to the interactions between bosonic atoms. Evaporative cooling works in a two-species gas, and can be used to prepare a degenerate Fermi gas either in a magnetic trap [76.12] or in an optical trap [76.13]. Second, one can use a gas of bosons, and indeed a BEC, as a refrigerator [76.14]. At this writing the lowest temperatures are of the order of 0.1 TF . Reaching lower temperatures is complicated by various factors, such as the very low heat capacity of a BEC and collisions becoming inefficient at low temperature because the inert Fermi sea reduces the available phase space. Nonetheless, lower temperatures seem to be mainly a matter of advances in technology.

76.4 BEC Superfluid

1118

Part F

Quantum Optics

which identifies nˆ and Jˆ as the operators for atom density and atom current density. The corresponding mean-field quantities are obtained when again the boson fields are replaced with the corresponding classical fields. Writing the classical field in terms of the density n(x, t) and phase ϕ(x, t) in the form √ ψ = n eiϕ , (76.58) the local flow velocity v = j/n becomes ~ v = ∇ϕ . m The velocity field is irrotational.

(76.59)

Quantization of Circulation. Integration of the flow

velocity around an arbitrary loop gives  2π ~ ~ dl · v = ∆ϕ = p , m m p = 0, ±1, ±2, . . . ,

(76.60)

Part F 76.4

since the change of the phase ∆ϕ around a closed loop must be an integer multiple of 2π. Equation (76.60) expresses the quantization of circulation in a superfluid. A medium described by a macroscopic wave function, such as a BEC, cannot sustain arbitrary flow velocities. Vortices. As an example, in bulk rotation at an angular velocity Ω the line integral around a loop at the distance r from the axis of rotation would be 2πr 2 Ω, which is not permitted for an arbitrary r. Instead, upon an attempt to make a BEC rotate, the angular velocity will be carried by vortex lines. These are lines through the condensate, entering and exiting at the surface, such that each vortex carries one quantum of circulation. At the core of a vortex the flow velocity should be infinite to sustain a finite circulation, which is physically impossible. Nature solves this problem by making the vortex core normal (not BEC), so that the macroscopic wave function does not apply. The diameter of the vortex core is of the order of the healing length ξ, given by (76.29). When the trapping potential on the atoms is rotated, it is convenient to study the physics in the co-rotating frame. Given a frame rotating at the angular velocity Ω and the angular momentum operator L = x × p per particle, transformation to the rotating frame adds the one-particle term

Hr = −Ω · L

(76.61)

to the Hamiltonian. Any particular configuration of vortices is a thermodynamically stable equilibrium if it is the minimum of energy in the co-rotating frame. For

a trapped condensate, at zero rotation velocity the state without vortices is the energy minimum, and increasing the rotation speed makes states with an increasing number of vortices the stable configuration. However, a vortex configuration may be metastable and live for a long time even if it is not the minimum of energy. Conversely, even the energy-minimum configuration of vortices must first be nucleated. Since the circulation can only have quantized values, it cannot change in a continuous process. It takes a zero condensate density somewhere to create or destroy a vortex. These alternatives provide a large number of experimental scenarios involving rotation of the trap or stirring of the condensate, condensation of a rotating normal gas by taking it across the transition temperature, and so forth. For instance, when a trap containing a BEC is rotated, vortices are generated at the surface where they start their lives as dynamical instabilities. The vortices then drift in and form a regular vortex array [76.15]. When the rotation is halted, the vortices drift out to the surface and disappear.

76.4.2 Superfluidity A BEC also has the remarkable property that it may sustain persistent currents that are completely immune to viscosity. The qualitative reason may be seen from the dispersion relation of small excitations (76.38). As long as the flow speed |v| is less than the speed of sound c, all excitation energies are positive, so that the flowing state is the state of lowest energy and is thermodynamically stable. On the other hand, when the flow velocity exceeds the speed of sound, the system has excitations that lower the energy, ν(q) < 0 for some q. The flowing state is then not a minimum of energy. The flow is not thermodynamically stable, and it decays when it interacts with an environment by sending off small excitations. The speed of sound gives the Landau critical velocity for superfluidity. The critical velocity c tends to zero when the atom–atom interactions vanish with a → 0. While the condensate wave function may be written down whether the atoms interact or not, superfluidity and persistent flows rely on the interactions. The same applies to vortices, as in the limit of a noninteracting gas the healing length and the radius of the vortex core tend to infinity. The conventional picture is that superfluid flow in an inhomogeneous medium is unstable if the local flow velocity exceeds the local (density dependent) speed of sound. In practice, in liquid He experiments and numerical simulations of dilute condensates the current

Quantum Degenerate Gases

often dissipates by shedding vortices when it flows too fast past an obstacle. It is not clear if the conventional picture is, or should be expected to be, quantitatively

76.5 Current Active Topics

1119

accurate. In fact, at this time there are no experiments with alkali vapor condensates in toroidal geometries that would offer natural conduits for persistent currents.

76.5 Current Active Topics 76.5.1 Atom–Molecule Systems

Basic Atom–Molecule Model for Bosons. Consider

a minimal

model for conversion of bosonic atoms boson ˆ The Hamiltonian field φˆ into bosonic molecules (ψ). density reads     ~ 2 ˆ ˆ† ~ 2 Hˆ † ˆ ∇ φ +ψ − ∇ + δ ψˆ =φ − ~ 2m 4m † † † + g(ψˆ φˆφˆ + ψˆ φˆ φˆ ) .

(76.62)

Effective Scattering Length. Heisenberg equations of

motion for the atomic and molecular fields read d ~ 2ˆ † i φˆ = − ∇ φ + 2gφˆ ψˆ , (76.63) dt  2m  d ~ 2 ∇ + δ ψˆ + gφˆφˆ . i ψˆ = − (76.64) dt 4m Suppose now that the detuning from resonance, |δ|, is the largest frequency parameter in the problem, then one may solve the molecular field adiabatically from (76.64) ˆ Inserting this into (76.63) gives as ψˆ = −gφˆφ/δ.

~ 2 ˆ 2g2 ˆ † ˆ ˆ d φˆ = − ∇ φ− φ φφ , (76.65) dt 2m δ which is the Heisenberg equation of motion for the atomic field that ensues from an effective Hamiltonian density   ~2 2 ˆ 2π ~2 aE ˆ † ˆ † ˆ ˆ † ∇ φ+ φ φ φφ (76.66) Hˆ E = φˆ − 2m m i

with the effective scattering length mg2 . (76.67) 2π ~δ Experimentally, the modification of the scattering length is in addition to a constant “background” scattering length a0 , and the sum of the two scattering lengths aE = −

a = a0 + aE .

(76.68)

is usually reported. As long as one stays sufficiently far away from an atom–molecule resonance, tuning the resonance condition is tantamount to tuning the atom–atom scattering length. When the detuning is negative (positive), the energy of a molecule is lower (higher) than the energy of an on-threshold pair of atoms, the corresponding induced scattering length is positive (negative), and the

Part F 76.5

Diatomic molecules and conversion between atoms and molecules at temperatures low enough to render the system quantum degenerate are at present probably the most active frontier in the studies of ultracold gases. Experimental achievements include a condensate of molecules, and coherent transitions between two chemically different species. More broadly, unforeseen new angles open up into long-standing issues in superfluid systems in condensed matter physics, such as strongly interacting superfluids and BEC-BCS crossover. A snapshot of the field at this time is given in the present section. Two colliding asymptotically free atoms cannot in general combine into a diatomic molecule, as energy and momentum would not be conserved in the process. However, there are two mathematically equivalent methods to adjust energy conservation, photoassociation and Feshbach resonance [76.16], both of which may lead to molecule formation. The underlying idea is that two seemingly free atoms may be regarded as a dissociated state of a corresponding diatomic molecule. In photoassociation a laser drives transitions from a dissociated two-atom state to a bounded state of the molecule. Energy conservation is adjusted by tuning the laser. In a Feshbach resonance hyperfine interactions drive transitions from a two-atom state in a particular manifold of electronic states to a molecular state in another manifold of electronic states. The magnetic moments of the two-atom state and the bounded molecular state are different, so that the resonance may be tuned by varying the magnetic field applied on the atom–molecule gas.

This model is for free atoms. Atom–molecule coupling is described as a contact interaction characterized by the coupling coefficient g. The detuning δ may be adjusted in photoassociation by tuning the laser, and in a Feshbach resonance by varying the magnetic field.

1120

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Quantum Optics

induced atom–atom interactions are in effect repulsive (attractive). It should be noted, though, that the scattering length does not simply become very large. Vernacular of this style, and associated attempts to study the theory of an interacting Bose gas in the limit when the gas parameter naE3 is not small, are occasionally misguided and misleading. The physics rather is in resonant, energy-conserving conversion between atoms and molecules. Atom–Molecule Coupling Strength. Given the differ-

ence in magnetic moments ∆µ between a molecule and two atoms and the position of the Feshbach resonance B0 , the detuning is

∆µ(B − B0 ) . (76.69) ~ The variation of the scattering length is conventionally parametrized in terms of the magnetic field width of the Feshbach resonance ∆B as   ∆B (76.70) a = a0 1 − . B − B0 δ=

Part F 76.5

A combination of (76.67–76.70) gives the relation between the field width and the contact interaction parameter characterizing the Feshbach resonance,  2π|a0 ∆µ ∆B| g= . (76.71) m Unfortunately, the difference in magnetic moments ∆µ is not always publicized and one may be forced to estimate, say, ∆µ ≈ µB . Two-Mode Model. Consider next, in the mean-field approximation, the case when only uniform atomic and molecular condensates are present. The condensates are represented by complex √ amplitudes α and β such that √ φˆ → n α and ψˆ → n/2 β, where n now is the invariant density equal to atom density plus twice the density of the molecules. It follows from (76.63) and (76.64) that the probability amplitudes for atoms and molecules, normalized as |α|2 + |β|2 = 1, satisfy

Ω iα˙ = √ α∗ β, 2

Ω iβ˙ = δβ + √ α2 . 2

(76.72)

These are nonlinear variations of the usual two-level equations (see√Chapt. 73) of quantum optics, with the quantity Ω = n g playing the role of the Rabi frequency. This system displays analogs of coherent optical transients (see Chapt. 73), such as Rabi oscillations between atomic and molecular condensates, and adiabatic

following from an atomic condensate to a molecular condensate when the detuning is swept through the resonance [76.17]. The two-mode model is simplistic in that it ignores processes in which molecules dissociate into correlated pairs of noncondensate atoms [76.18, 19]. There are also secondary complications. The molecules created in a Feshbach resonance are highly vibrationally excited, and tend to get quenched in collisions with atoms and other molecules. Typical lifetimes are in the millisecond regime. Usual one-color photoassociation from two atoms to a molecule with the absorption of a photon, on the other hand, creates an electronically excited molecule, which decays spontaneously on a time scale far shorter than the typical photoassociation time scales. To mitigate spontaneous emission, one usually resorts to two-color photoassociation, in which a second laser takes the photoassociated molecules to another more stable level [76.20]. At present, the two-mode system with just an atomic and a molecular condensate has never been realized cleanly in an experiment. Nonetheless, experiments using a Feshbach resonance have demonstrated Ramsey fringes in transitions between atomic and molecular condensates [76.21], and formation of what probably is a (short-lived) molecular condensate from the bosonic isotope 23 Na [76.22]. Fermion Systems. Combining two fermionic atoms

gives a bosonic molecule, and Feshbach resonance in a Fermi gas is currently a popular topic. The main interest is in the BEC-BCS crossover. Basically, if the magnetic field is tuned so that the detuning is negative, molecules have a lower energy than atoms. Thermal equilibration then leads to molecules that will condense if the temperature is low enough. On the other hand, if the detuning is positive, atom–atom interactions are attractive. The atoms may undergo a phase transition into a fermion superfluid that is analogous to the BCS phase transition in a superconductor. What happens in between has been a question in theory for a while [76.23], and is finally accessible to experiments. Collisionless adiabatic transfer from atoms to molecules by sweeping the magnetic field across a Feshbach resonance works with fermions much like with bosons [76.24]. Moreover, molecules formed in the 834 G Feshbach resonance in the fermionic isotope 6 Li may live for seconds. It is now possible to study thermal equilibrium and long-lived excited states in the neighborhood of the resonance [76.25–27]. In particular, the observation of a vortex lattice over a wide range

Quantum Degenerate Gases

of magnetic fields on both sides of the resonance [76.24] indicates that close to the resonance the gas is a strongly interacting superfluid.

76.5.2 Optical Lattice with a BEC

Optical Lattice. Dipole forces of standing-wave fields of light generate a periodic potential, an optical lattice, on the atoms. If the light is detuned far enough from atomic resonances, absorption and spontaneous emission are negligible and the potential is conservative. In one dimension, the potential energy for the motion of the atoms is typically of the form V(x) = V0 sin2 kx , (76.73) where k is the wave number of the lattice light, and the depth of the lattice V0 can be inferred from the known parameters of the atoms and the light as explained in Sect. 75.2.2. Double-Well Potential. One can integrate the GPE numerically for an arbitrary potential, but many more insights have been gained from restricted models. The simplest one is a double-well potential in the two-state approximation, in which only the ground state of the atoms in both wells is taken into account. The Hamiltonian is

 † 2

2  ∆ † Hˆ . = − al ar + ar† al + 2κ al al + ar† ar ~ 2 (76.74)

Here ∆ is a parameter characterizing tunneling between the “left” and the “right” potential well, al,r are the annihilation operators for ground-state bosons in each well, and κ is a measure of atom–atom interactions. The one-particle states in a symmetric double-well potential with weak tunneling come in doublets, one even

1121

state φ+ and one odd state φ− , with respect to the center of the double-well trap. The choices of signs of the wave functions are assumed to work out √ in such a way that the left and right states are φl,r = 1/2 (φ+ ± φ− ). Equation (76.74) could be the version of the Hamiltonian (76.22) restricted to the basis of the two states φl,r . There are many forms of the Hamiltonian (76.74) that differ by a polynomial of the conserved particle † †

number Nˆ = al al + ar ar . Inasmuch as particle number is fixed, adding any function of the conserved quantity Nˆ to the Hamiltonian has no effect on the dynamics, and so such forms are functionally equivalent. Here polynomials of Nˆ are added to the Hamiltonian without further notice to produce the simplest-looking results. Semiclassical Approximation. The usual method of go-

ing to the classical field theory gives the equations for √ the semiclassical amplitudes defined by al,r / N → αl,r , 1 iα˙ l = − ∆ αr + χ|αl |2 αl , 2 1 iα˙ r = − ∆ αl + χ|αr |2 αr , (76.75) 2 with χ = 4κN. Without atom–atom interactions these are the equations of a resonant two-level system and describe Josephson oscillations of the atoms between the sides of the double-well potential. Interactions temper the oscillations, or stop them completely [76.28]. Phase Diffusion in the Double Well. The model (76.74)

is unusual in that one can easily go beyond the semiclassical approximation [76.29]. The entire state space for N atoms is spanned by the vectors |n l , n r  = |n l , N − n l  with n l = 0, . . . , N. The Hamiltonian can be diagonalized and the time dependence of the system solved numerically even for large N. Moreover, the common case in which the atom number fluctuations at the sites are at least of the order unity, but small in a relative sense, is amenable to a simple analytical approximation. The phase difference of the condensates between the two traps, ϕ, ˆ is a case in point. One can measure it by releasing the atoms from the trap and letting them interfere. Although there are serious in-principle problems with this interpretation, phase difference can be viewed qualitatively as the canonical conjugate of the difference between the number of atoms on the sides of the double well, nˆ ≡ nˆ l − nˆ r , with [n, ˆ ϕ] ˆ = −i. The minimum uncertainty product of phase difference and atom number difference is then 1/2. On this basis, results for various experiments can be qualitatively and quantitatively predicted.

Part F 76.5

A BEC confined to a periodic potential enjoys a longstanding popularity for reasons that have varied in time. In the early days of BEC the Josephson effect and phase behavior of a BEC were topical. The possibility of a quantum phase transition in an optical-lattice system was the next broad topic to emerge, and nowadays speculations about using condensates in a lattice either as supporting technology or as the active element in quantum information processing abound. The deceptively simple theoretical models of these systems add to their staying power. In this section, a brief discussion of an optical lattice holding a BEC is presented. The quantum information view is not pursued here, as so far no specific experimental progress has appeared in print.

76.5 Current Active Topics

1122

Part F

Quantum Optics

Part F 76.5

For instance, suppose the system is prepared in the ground state of the Hamiltonian (76.74). The ground state is an even split of the atoms between the two traps, but atom number fluctuations depend on the ratio between tunneling and atom–atom interactions. For |∆|/χ 1, atom–atom interactions dominate, and since moving an atom from one side of the trap to the other costs much interaction energy, the ground state is close to a number state with half of the atoms in each potential well. In the contrary case, the ground state is essentially the many-body state with all N atoms in the symmetric state φ+ , which breaks up into a Poissonian distribution of the atoms between the states φl and φr . As long as the standard deviation of the atom number difference is at least of the order of unity, it is given by 1/4  √ ∆ , (76.76) ∆n = N ∆ + 4Nκ and the phase fluctuations are 1 . (76.77) ∆ϕ = 2 ∆n As discussed above in Sect. 76.3.2, a measurement of the phase difference will produce a definite result. In the interaction-dominated case the result should in effect be random, while in the tunneling-dominated case the phase difference should come out √ the same every time, save for fluctuations of the order 1/ N. Similarly, if one were to start from the ground state in the case when tunneling dominates, and then suddenly turn off tunneling by adjusting the potential well, atom–atom interactions would lead to diffusion (or rather, dispersion) of the phase difference. The result of a phase measurement becomes increasingly random with time according to  1 1 + 16Nκ 2 t 2 . ∆ϕ(t) = (76.78) 2 N

One-particle eigenstates in a periodic lattice are organized in energy bands, but there is a well-known transformation in condensed matter physics that makes orthonormal Wannier states , more or less localized in the lattice sites, out of the states in each band. The most rigorous interpretation of the Hamiltonian (76.79) is that it is the representation of the Hamiltonian (76.22) in the Wannier states belonging to the lowest energy band of the lattice, ignoring tunneling between non-adjacent sites. Viewed in this way, the model is only valid in the limit when the energy per atom for atom–atom interactions is small compared to the energy spacing between the bands. Nonetheless, direct integrations of the GPE with atom–atom interactions also show energy bands, and for suitably picked parameters the Hamiltonian (76.79) should be generic for the case when interband transitions are negligible. Now, without atom–atom interactions the width of the energy band from the Bose–Hubbard model would be ~∆. On the other hand, the band structure for the potential energy (76.73) of an optical lattice comes from the Mathieu equation [76.30]. A comparison gives the estimate      8 E r V0 3/4 V0 ∆= √ exp −2 . (76.80) Er Er π ~

Bose–Hubbard Model. The corresponding multi-well

Phase Diffusion in the Bose–Hubbard Model. The

problem goes under the rubrics fo the Bose–Hubbard model and tight-binding approximation. The Hamiltonian is



2 ∆ † Hˆ  † − an+1 an + an−1 an + 2κ an† an . = ~ 2 n

dominant feature of the Bose–Hubbard model again is competition between tunneling and atom–atom interactions. Also, phase and atom number fluctuations may again be studied analytically under the assumptions that atom number fluctuations at each site are at least of the order of unity, but small in a relative sense. For the same numbers of atoms per site, the results are basically the same in the two- and multi-well cases. In fact, multi-well counterparts [76.31] of the experiments on the phase relations are well ahead of the experiments on two-well systems [76.32]. The multiwell experiments are in satisfactory agreement with the theory.

(76.79)

The sum runs over the sites of the optical lattice, which is taken to be one-dimensional in this example, and ∆ and κ again characterize tunneling and atom–atom interactions. At the ends of the lattice there are some boundary conditions, but for a long enough lattice they do not influence the physics.

This is an asymptotic expression for the limit when the recoil energy E r = ~2 k2 /2m and the lattice depth V0 satisfy V0 E r . Next suppose that one ascribes to each potential well the unit-normalized wave function φ(x), then an estimate for the atom–atom interaction parameter κ comes from (76.23) in the form  π ~a d3 x |φ(x)|4 . (76.81) κ= m

Quantum Degenerate Gases

Superfluid-Mott Insulator Transition. Because the size of the state space tends to grow as L L with the number of lattice sites L, direct numerical solutions to the multiwell problem are computationally intractable. This is somewhat unfortunate, as a long lattice also presents behaviors with no obvious analogs in the two-well case, and for which analytical approximations have proven hard to come by. When tunneling dominates, the system is in what is referred to as the superfluid phase. Fluctuations of atom number between the states are relatively large. On the other hand, if atom–atom interactions dominate, it becomes costly in energy to put anything but an exact number state of atoms at each lattice site. This is the Mott insulator phase. According to calculations carried out using the so-called Gutzwiller ansatz, the ground state of an optical lattice inserted in an atom trap consists of regions with the same integer number of atoms

References

at the lattice sites within each region [76.33]. When the parameters of the system are varied, in what is known as a quantum phase transition, the system should abruptly switch between these phases. The superfluid-Mott insulator transition has been observed experimentally [76.34]. Lattice parameters, especially tunneling, can be varied easily by changing the intensity of the lattice light. The observation of the transition is by means of phase coherence. In the superfluid state the system is characterized by a global macroscopic wave function. When the atoms are released from the lattice, atoms originating from different sites are capable of interference, and the interference pattern reflects the lattice structure. On the other hand, in the Mott insulator phase the lattice sites are in number states with little phase coherence between them, and there is no interference pattern.

76.2 76.3 76.4 76.5 76.6

76.7

76.8 76.9

76.10

76.11

76.12

76.13

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, S. Stringari: Rev. Mod. Phys. 71, 463 (1999) A. S. Parkins, D. F. Walls: Phys. Rep. C303, 1 (1998) A. J. Leggett: Rev. Mod. Phys. 73, 307 (2001) S. Stenholm: Phys. Rep. C363, 173 (2002) A. L. Fetter: J. Low. Temp. Phys. 129, 263 (2002) A. L. Fetter, J. D. Walecka: Quantum Theory of Many-Particle Systems (McGraw-Hill, New York 1971) M. D. Feit, J. A. Fleck Jr., A. Steiger: Solution of the Schrödinger equation by a spectral method, J. Comput. Phys. 47, 412 (1982) J. R. Schrieffer: Theory of Superconductivity (Addison-Wesley, Redwood City 1988) M. D. Barrett, J. A. Sauer, M. S. Chapman: Alloptical formation of an atomic Bose-Einstein condensate, Phys. Rev. Lett. 87, 010404 (2001) S. Schneider, A. Kasper, Ch. vom Hagen, M. Bartenstein, B. Engeser, T. Schumm, I. Bar-Joseph, R. Folman, L. Feenstra, J. Schmiedmayer: BoseEinstein condensation in a simple microtrap, Phys. Rev. A 67, 023612 (2003) D. Rychtarik, B. Engeser, H.-C. Nägerl, R. Grimm: Two-dimensional Bose–Einstein condensate in an optical surface trap, Phys. Rev. Lett. 92, 173003 (2004) B. DeMarco, D. S. Jin: Onset of Fermi degeneracy in a trapped atomic gas, Science 285, 1703 (1999) K. M. O’Hara, S. L. Hemmer, M. E. Gehm, S. R. Granade, J. E. Thomas: Observation of a strongly interacting degenerate Fermi gas of atoms, Science 298, 2179 (2002)

76.14

76.15

76.16

76.17

76.18

76.19

76.20

76.21

76.22

76.23

A. G. Truscott, K. E. Strecker, W. I. McAlexander, G. B. Partridge, R. G. Hulet: Observation of Fermi pressure in a gas of trapped atoms, Science 291, 2570 (2001) J. R. Abo-Shaeer, C. Raman, J. M. Vogels, W. Ketterle: Observation of Vortex Lattices in BoseEinstein Condensates, Science 292, 476 (2001) E. Timmermans, P. Tommasini, M. Hussein, A. Kerman: Feshbach resonances in atomic Bose-Einstein condensates, Phys. Rep. C315, 199 (1999) J. Javanainen, M. Mackie: Coherent photoassociation of a Bose-Einstein condensate, Phys. Rev. A 59, R3186 (1999) S. J. J. M. F. Kokkelmans, M. J. Holland: Ramsey fringes in a Bose–Einstein condensate between atoms and molecules, Phys. Rev. Lett. 89, 180401 (2002) M. Mackie, K.-A. Suominen, J. Javanainen: Meanfield theory of Feshbach-resonant interactions in 85 Rb condensates, Phys. Rev. Lett. 89, 180403 (2002) R. Wynar, R. S. Freeland, D. J. Han, C. Ryu, D. J. Heinzen: Molecules in a Bose-Einstein condensate, Science 287, 1016 (2000) E. A. Donley, N. R. Claussen, S. T. Thompson, C. E. Wieman: Atom-molecule coherence in a BoseEinstein condensate, Nature 417, 529 (2002) K. Xu, T. Mukaiyama, J. R. Abo-Shaeer, J. K. Chin, D. E. Miller, W. Ketterle: Formation of quantumdegenerate sodium molecules, Phys. Rev. Lett. 91, 210402 (2003) M. Holland, S. J. J. M. F. Kokkelmans, M. L. Chiofalo, R. Walser: Resonance superfluidity in a quantum

Part F 76

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degenerate Fermi gas, Phys. Rev. Lett. 87, 120406 (2001) C. A. Regal, C. Ticknor, J. L. Bohn, D. S. Jin: Creation of ultracold molecules from a Fermi gas of atoms, Nature 424, 47 (2003) S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl, C. Chin, J. Hecker Denschlag, R. Grimm: Bose–Einstein condensation of molecules, Science 302, 2101 (2003) M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, A. J. Kerman, W. Ketterle: Condensation of pairs of fermionic atoms near a Feshbach resonance, Phys. Rev. Lett. 92, 120403 (2004) M. W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, C. H. Schunck, W. Ketterle: Vortices and superfluidity in a strongly interacting Fermi gas, Nature 435, 1047 (2005) S. Raghavan, A. Smerzi, S. Fantoni, S. R. Shenoy: Coherent oscillations between two weakly coupled Bose-Einstein condensates: Josephson effects,

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76.30 76.31

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76.34

π oscillations, and macroscopic quantum selftrapping, Phys. Rev. A 59, 620 (1999) J. Javanainen, M. Yu. Ivanov: Splitting a trap containing a Bose-Einstein condensate: Atom number fluctuations, Phys. Rev. A 60, 2351 (1999) M. Abramowitz, I. A. Stegun: Handbook of Mathematical Functions (Dover, New York 1970) C. Orzel, A. K. Tuchman, M. L. Fenselau, M. Yasuda, M. A. Kasevich: Squeezed states in a Bose-Einstein condensate, Science 291, 2386 (2001) Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, A. E. Leanhardt: Atom interferometry with Bose-Einstein condensates in a doublewell potential, Phys. Rev. Lett. 92, 050405 (2004) D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, P. Zoller: Cold bosonic atoms in optical lattices, Phys. Rev. Lett. 81, 3108 (1998) M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, I. Bloch: Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms, Nature 415, 39 (2002)

Part F 76

1125

De Broglie Opt 77. De Broglie Optics

77.1

Overview............................................. 1125

77.2

Hamiltonian of de Broglie Optics .......... 1126 77.2.1 Gravitation and Rotation ........... 1127

77.2.2 77.2.3 77.2.4 77.2.5 77.3

Charged Particles ...................... 1127 Neutrons .................................. 1127 Spins ....................................... 1127 Atoms ...................................... 1127

Principles of de Broglie Optics .............. 1129 77.3.1 Light Optics Analogy .................. 1129 77.3.2 WKB Approximation................... 1130 77.3.3 Phase and Group Velocity........... 1130 77.3.4 Paraxial Approximation ............. 1130 77.3.5 Raman–Nath Approximation ...... 1131

77.4 Refraction and Reflection ..................... 1131 77.4.1 Atomic Mirrors .......................... 1131 77.4.2 Atomic Cavities ......................... 1132 77.4.3 Atomic Lenses ........................... 1132 77.4.4 Atomic Waveguides ................... 1132 77.5 Diffraction .......................................... 1133 77.5.1 Fraunhofer Diffraction ............... 1133 77.5.2 Fresnel Diffraction ..................... 1133 77.5.3 Near-Resonant Kapitza–Dirac Effect .................. 1133 77.5.4 Atom Beam Splitters .................. 1134 77.6 Interference ........................................ 1135 77.6.1 Interference Phase Shift ............. 1135 77.6.2 Internal State Interferometry ...... 1136 77.6.3 Manipulation of Cavity Fields by Atom Interferometry.............. 1137 77.7

Coherence of Scalar Matter Waves ......... 1137 77.7.1 Atomic Sources.......................... 1137 77.7.2 Atom Decoherence .................... 1138

References .................................................. 1139

77.1 Overview Any collection of particles of mass M and momentum p has a de Broglie wavelength λdB = 2π ~/ p. Originally meant to explain the orbits of a single Coulombbound electron, the ultimate wave character of matter has been confirmed for all fundamental particles, and also for composite particles such as ions, atoms and molecules.

In terms of the Bohr radius a0 , the fine-structure constant α, and the electron mass m e 2παa0  = 2π λdB =

v/c M/m e



Eh m e 2T M

1/2 a0 , (77.1)

Part F 77

De Broglie optics concerns the propagation of matter waves, their reflection, refraction, diffraction and interference. The main subfields of de Broglie optics are electron optics [77.1], neutron optics [77.2], and atom optics [77.3]. Well-established applications are found in electron diffraction and microscopy [77.4], electron holography [77.5], neutron diffraction and interferometry [77.6]. The subject of atom optics is relatively new, and applications are currently being developed in precision spectroscopy, precision measurement, atom lithography, atom microscopy, and atom interferometry [77.7–9]. This chapter concentrates on the principles of de Broglie optics. Illustrations of these principles will be presented mainly in the framework of atom optics. A typical de Broglie optical experiment involves a source, a beam of particles produced by that source, an array of optical elements, possibly a probe, other optical elements placed behind the probe, and finally a detector. Optical elements are collimators, apertures, lenses, mirrors, and beamsplitters. A probe could be a crystal, a biological sample, or just another unknown optical element whose properties are to be investigated. To avoid proliferation of notation, we refer to any object placed in the beam path simply as an “optical element”.

1126

Part F

Quantum Optics

where v = p/M is the velocity, T = 12 Mv2 is the kinetic energy, and E h is the atomic unit of energy (Table 1.4 of Sect. 1.2). For electrons, λdB ≈ 1.226 426 nm/(T /eV)1/2 . The thermal de Broglie wavelength is defined in (77.80). Regardless of the particle species, the theory of de Broglie optics divides into two distinct parts: the theory of dispersion and the theory of the optical phenomena. In the theory of dispersion an effective Hamiltonian is derived which describes the interaction of the particles with the optical elements. In the theory of optical phenomena, the ensuing Schrödinger equation is solved and put into the context of geometrical and Fourier optics (e.g., by relating distributions of intensity on the detector screen to properties of a given optical element).

Part F 77.2

Dispersion of de Broglie Waves The theory of dispersion is highly particle specific, since it depends on any internal degrees of freedom which may give rise to permanent magnetic moments or to induced electromagnetic moments either in static or in optical light fields. Interaction with the latter fields also may give rise to spontaneous emission. Spontaneous emission has important consequences for the coherence properties of atomic matter waves because of the random recoil associated with it (Chapt. 75). Dispersion of matter waves differs from that of light waves in a number of regards. First, the dispersion relation of free particles is quadratic in the wavenumber giving rise to spatial spreading of wavepackets even in one-dimensional configurations. Second, particles may be brought to rest, which is impossible for

photons. And finally, particles in beams may show selfinteraction giving rise to nonlinear optical phenomena even for freely traveling particle waves. Electrons and ions, for example, experience a particle density dependent Coulomb broadening in the focus of a lens (Boersch effect [77.5]). In atom optics, nonlinear phenomena occur due to atom–atom interactions in the ensemble, the nature of which may be significantly influenced by laser light [77.10]. In the presence of laser light, atom–atom interactions mainly result from photon exchange which, in most cases, leads to a repulsive interaction. Details of the microscopic basis of atom–atom interactions, in particular, those concerning cold collisions, are presented in Sect. 75.5.1. Characteristic effects of nonlinear atom optics like four-wave mixing [77.11] and parametric amplification [77.12] have been observed with the highly dense samples provided by Bose–Einstein degenerate gases (Chapt. 76). Optics of de Broglie Waves In contrast to the theory of dispersion, the theory of optical phenomena of matter waves is quite universal and bears strong resemblance to ordinary light optics. This resemblance is closest if the particles can be described by a scalar wave function of a structureless point particle. However, in the presence of resonant laser fields, electronic levels, their Zeeman sublevels and possible hyperfine structure may play an important role, in which case a multicomponent spinor wave function must be used to describe the atom optical phenomena properly.

77.2 Hamiltonian of de Broglie Optics A large class of phenomena of particle optics is welldescribed by an effectively one-particle Hamiltonian model in which the particles are not assumed to mutually interact. In such a model the ensemble of particles is described by a wave function ψ(x, t) = x|ψ(t) whose time evolution is governed by the Schrödinger equation ∂ |ψ(t) = H(t)|ψ(t) (77.2) ∂t with a one-particle Hamiltonian of the generic form  2 pˆ − A(ˆx, t) + U(ˆx, t) . (77.3) H(t) = 2M i~

Here, pˆ and xˆ denote the canonically conjugate operators of the center of mass momentum and position, respectively. The cartesian components of pˆ and xˆ obey the fundamental commutation relation 

 xˆi , pˆ j = i~δij .

(77.4)

The vector potential A(x, t) and scalar potential U(x, t) account for the interaction of the particle with all the optical elements, probes, and samples which are placed between the source and the detector, including the effects of gravitation and rotation.

De Broglie Optics

All particles are subject to the influence of gravitation and rotation, which both may be viewed as being special cases of an accelerated frame of reference. Effects of uniform gravitation are described by A(x) = 0 ,

(77.5)

where g describes the direction and magnitude of gravitational acceleration. Effects of uniform rotation are described by M U(x) = − (Ω × x)2 , 2

A(x) = M (Ω × x) , (77.6)

where the direction and magnitude of Ω refer to the orientation of the axis of rotation and the angular velocity, respectively. Here, U(x) is the potential of the centrifugal force while A(x) is the potential of the Coriolis force.

77.2.2 Charged Particles The interaction of charged particles (electrons, ions) with the electromagnetic field is described by A(x, t) = (q/c)A(x, t) , (77.7)

where q is the particle charge (q = −e for electrons), and A and Φ denote the gauge potentials of the electromagnetic fields E and B, respectively. The fields are given by 1∂ A(x, t) − ∇Φ(x, t) , c ∂t B(x, t) = ∇ × A(x, t) .

E(x, t) = −

(77.8) (77.9)

77.2.3 Neutrons For neutrons interacting with a spatially homogeneous gas, liquid, or amorphous solid contained in a volume V , a common model is  U0 inside V , (77.10) U(x) = 0 outside V , where U0 = 2π ~2 b/M,  being the number density of scatterers in the volume, and b the bound coherent scattering length. For neutrons interacting with perfect crystals, U(x) is a smooth periodic function with the same periodicity

77.2.4 Spins The interaction of the spin related magnetic moment µ with the electromagnetic field is described by ˆ U(x, t) = − µ ˆ · B(x, t) ,   ˆ A(x, t) = µ ˆ × E(x, t) /c

(77.11)

Here, the vector potential is due to the motional correction of the magnetic dipole interaction. Usually, it is neglected. However, it does play an important role for the Aharonov–Casher effect (Sect. 77.7). For fundamental particles with spin 12 , one has

~e σ, (77.12) 4Mc where σ is the vector of Pauli matrices, and g is the g-factor of the particle (Sect. 75.5.2). Here and in what follows, a hat on U and A indicates the matrix character of the hatted quantity, the matrix indices referring to the internal degrees of freedom, like spin. Similarly, Ψ(x, t) denotes a spinor-valued wave function. The wave function of a spin- 12 particle, for example, is displayed in the form  ψ↑ (x, t) , Ψ(x, t) = (77.13) ψ↓ (x, t) µ ˆ =g

where ψ↑ (ψ↓ ) is the scalar wave function of the σ3 = +1(−1) component of the state |Ψ(t).

77.2.5 Atoms Many optical elements in atom optics are based on the mechanical effects of the radiation interaction. In the electric dipole approximation, the interaction of a single atom with the electromagnetic field is described by ˆ U(x, t) = − dˆ · E(x, t) , ˆ ˆ , A(x, t) = B(x, t) × d/c

(77.14)

where dˆ is the operator of electric dipole transition. The vector potential Aˆ is due to the motional correction of the electric dipole interaction. Usually it is neglected; see however, the paragraph Electric Dipole Phase in Sect. 77.7. For a monochromatic field of frequency ω E(x, t) = E(+) (x) e−iωt + h.c. ,

(77.15)

Part F 77.2

U(x, t) = qΦ(x, t) ,

1127

as the lattice and an average value U0 within a single unit cell (see [77.2] for details).

77.2.1 Gravitation and Rotation

U(x) = −Mg · x ,

77.2 Hamiltonian of de Broglie Optics

1128

Part F

Quantum Optics

where E(+) (x) defines both polarization and spatial characteristics of the field. A standing wave laser field with optical axis in the x-direction and linear polarization , for example, is described by (+)

E

(x) =  E0 f(x, y, z) cos(kx) ,

(77.16)

where E0 is the electric field amplitude, k = ω/c is the wavenumber, and the slowly varying function f(x, y, z) accounts for the transverse profile of the laser field. The electric dipole operator dˆ acts in the Hilbert space of electronic states of the atom. In the particular case that the polarization of the laser field is spatially uniform and that spontaneous emission does not play a role, two states are generally sufficient and the atom may be modeled by a two-level atom with electronic levels |e and |g of energy E e and E g , respectively (E e > E g ). With the spinor representation   0 1 , , |g = |e = (77.17) 1 0 the electric dipole operator assumes the form  0 1 , dˆ = d  d 1 0


 Beff (x) = − Re[Ω1 (x)], Im[Ω1 (x)], ∆ . (77.22)

As it stands, the Hamiltonian (77.21) describes the precession and center of mass motion of a fictitious spin in an external “magnetic field” Beff (x). Spatial variations of this field give rise to the Stern–Gerlach effect, i. e., the splitting of the atomic center of mass wave function (see also Sect. 77.5.3). Adiabatic Approximation In the position representation, the Hamiltonian (77.19) acts on state vectors of the form of a bispinor  ψe (x, t) . (77.23) Ψ(x, t) = ψg (x, t)

Alternatively, this state vector may be expanded in terms of the local eigenvectors α± (x), β± (x) of the interaction matrix, also called dressed states,   α+ (x) α− (x) + ψ− (x, t) , Ψ(x, t) = ψ+ (x, t) β+ (x) β− (x) (77.24)

(77.18)

Part F 77.2

where

where d is the matrix element of the dipole transition and  d denotes its polarization. The laser field is assumed resonant with 
to be near the e ↔ g transition at ω0 = E e − E g /~, and we denote by ∆ ≡ ω0 − ω the atom-laser detuning. Using the rotating wave approximation (Sect. 68.3.2) in an interaction picture with respect to the laser frequency, the Hamiltonian describing the atomic dynamics – both internal and center-of-mass – is given by  pˆ 2 ~ −∆ Ω1 (ˆx) , H= − (77.19) 2M 2 Ω1 † (ˆx) ∆ where

  ~Ω(x) α± (x) α± (x) ˆ U(x) =± , 2 β± (x) β± (x)

(77.25)

with eigenvalues determined by the dressed Rabi frequency

Ω(x) = |Ω1 (x)|2 + ∆2 , (77.26) and coefficients (we suppress the x-dependence for notational clarity)   cos ϑ2 − e−iϕ sin ϑ2 α+ α− = iϕ ≡ Sˆ . β+ β− e sin ϑ2 cos ϑ2 (77.27)

Ω1 (x) = 2d  d · E(+) (x)/~

(77.20)

is the spatially dependent bare Rabi frequency. In the field (77.16), Ω1 (x) is cosinusoidal with peak value Ω0 = 2dE0 /~. Atom Optical Stern–Gerlach Effect The Hamiltonian (77.19) may be written in the form

H=

where

pˆ 2 ~ + σ · Beff (ˆx) , 2M 2

(77.21)

Here, the Stückelberg angle ϑ ≡ ϑ(x) and phase angle ϕ ≡ ϕ(x) are defined in terms of the polar representation of the effective magnetic field Beff = Ω (cos ϕ sin ϑ, sin ϕ sin ϑ, cos ϑ) .

(77.28)

In the dressed state basis, the transformed Hamiltonian assumes the form   ˆ x) 2 ~ pˆ − A(ˆ ˜ † ˆ ˆ ˆ ˆ H ≡ S HS = + Ω(ˆx)σ3 , (77.29) 2M 2

De Broglie Optics

with matrix-valued vector potential ˆ A(x) = i~ S† (x) [∇ S(x)] .

77.3 Principles of de Broglie Optics

atom in the dressed ground state experiences a potential which is approximately given by (77.30)

This matrix is not diagonal; its off-diagonal elements describe nonadiabatic transitions between the dressed states. If the detuning is sufficiently large, and the atom moves sufficiently slowly, these nonadiabatic transitions may be neglected in a first approximation. In such an approximation, which is akin to the Born–Oppenheimer approximation in molecular physics, the dynamics of the atom is described by two decoupled Hamiltonians of the generic form (77.3), with scalar potentials U given by

~|Ω1 (x)|2 . (77.32) 4∆ For red detuning we have ∆ > 0, in which case the atom is attracted towards regions of high intensity (high field seeker). For blue detuning, the atom is repelled from such regions (low field seeker). A potential similar to (77.32) also applies for complex particles like molecules whose transitions are far detuned from the light frequency. The potential is proportional to the dynamic polarizability α(ω) and the field intensity. U(x) = −

Atom Optical Nonlinearity In very dense atom ensembles at low temperatures, collisions between particles can be described by a contact interaction whose strength depends, in the simplest case, on a single parameter, the s-wave scattering length a (Sect. 75.5.1). In the mean field approximation, these interactions translate into a density-dependent potential

4π ~2 a(N − 1) |ψ(x)|2 , (77.33) M where N is the number of particles. This nonlinearity leads to the occurrence of atom solitons [77.13], four-wave mixing [77.11], and parametric amplification [77.12]. U(x) =

77.3 Principles of de Broglie Optics Since the Schrödinger equation is a linear partial differential equation, de Broglie optics shares most of its principles with principles of other wave phenomena, and in particular with the optical principles of electromagnetic waves.

77.3.1 Light Optics Analogy The analogy of de Broglie optics and light optics becomes particularly transparent for monoenergetic beams of scalar particles. Such beams are described by a time harmonic wave function ψ(x, t) = e−iEt/~ ψE (x), where ψE (x) obeys the stationary Schrödinger equation (77.34)

Setting A(x) = 0 in (77.3) for simplicity, this equation assumes the form    U(x) ∇ 2 + k˜ 02 1 − ψE (x) = 0 , (77.35) E

where the wavenumber k˜ 0 is related to the energy E via the dispersion relation E≡

~2 k˜ 02 . 2M

(77.36)

If U = 0 at the entrance to the interaction region, E is the kinetic energy of the freely traveling de Broglie wave and k˜ 0 is the related wavenumber. Comparing (77.35) with the scalar Helmholtz equation of electromagnetic theory, and identifying  1/2 n˜ E (x) ≡ 1 − U(x)/E (77.37) as an index of refraction for matter waves, one observes the complete analogy of scalar optics of stationary matter waves and monochromatic light waves. This analogy can be generalized for spinor valued wave functions which would correspond to vector wave optics in anisotropic index media. However, in contrast to light, spinor-valued wave functions do not obey a transversality condition.

Part F 77.3

~ U(x) = ± Ω(x) (77.31) 2 and vector potentials A(x) given by the diagonal elements of (77.30). The vector potential is usually neglected. If included, it describes the Berry phase of the mechanical effects of the radiation interaction of a two-level atom. The idea behind the adiabatic approximation is that the internal state of the atom, which is described by any of the locally varying dressed states, has enough time to adjust smoothly to the motion of the atom. For the important case of strong detuning |∆| |Ω1 (x)|, this assumption is usually well justified. In this case, an

HψE = EψE .

1129

1130

Part F

Quantum Optics

In (77.35–77.37), the parameter E describes the kinetic energy of the incoming beam. Thus, E is positive, and therefore n˜ E < 1 for positive values of the potential, while n˜ E > 1 for negative values of the potential. For neutrons, one generally has n˜ E < 1. This contrasts to the index of refraction for light waves which is generally larger than one. For electrons, ions, and atoms both n˜ E < 1 and n˜ E > 1 may be realized.

77.3.2 WKB Approximation Waves are described by amplitude and phase. Particles are described by position and momentum. The link between these concepts is provided by Hamilton’s ray optics. For scalar matter waves, a ray follows a classical trajectory. The optical signature of the ray is the phase associated with it. The quantum mechanical version of Hamilton’s ray optics is obtained in the WKB approximation of the stationary Schrödinger equation (77.34). Any solution of (77.34) may be written in the form ψE (x) = A(x) eiW(x)

(77.38)

with real-valued A(x) and W(x). In the WKB approximation A(x) ≈ 1, and

Part F 77.3

W(x) = k˜ 0

P n˜ E (s) ds + ~

−1

x

(77.39)

which is called eikonal in Hamilton’s ray-optics. In this expression, n˜ E (s) ≡ n˜ E (x(s)), where x(s) denotes the classical trajectory of energy E connecting the point Pi with the point P, xi and x are the coordinates of Pi and P, respectively, and ds ≡ | dx| is the element of arc length measured along the classical trajectory. Note that the second contribution in (77.39) is generally gauge-dependent. However, for closed loops which are frequently encountered in interferometry, the gauge-dependence disappears by virtue of Stokes theorem which transforms the path integral into an area integral over the rotor of A. The eikonal (77.39) may also be written in the form of a reduced action

x

x 1



p(x ) · dx = k˜ (x ) · dx , (77.40) W(x) = ~ xi

77.3.3 Phase and Group Velocity The velocity of a particle which traverses a region of negative potential increases so that p(x) > p0 , and the  phase advances: δW = [ p(x) − p0 ] · dx > 0. In quantum mechanics, the classical velocity corresponds to the group velocity, while the evolution of the phase is determined by the phase velocity. The phase and group velocities of de Broglie waves are given by  1 E E vp (x) ≡ = , (77.41) p(x) n˜ E (x) 2M  ∂E 2E = n˜ E (x) , vg (x) ≡ ∂ p(x) M

A(x ) · dx ,

xi

Pi

wave vector, and the integral is evaluated along the classical trajectory of the particle. Note that in the presence of a vector potential, p(x) and dx are no longer parallel as a result of the difference between canonical momentum p and kinetic momentum M( d/ dt) x ≡ p − A. The WKB approximation becomes invalid in the vicinity of caustics where neighboring rays intersect. There, connection formulae are used to find the proper phase factors picked up by the ray in traversing the caustics. Depending on the topology of the intersecting rays, different classes of diffraction integrals provide uniform approximations for the wave amplitude near caustics. For further details see [77.14] and [77.15].

xi

where p(x) is the local value of the canonical momentum of the particle, k˜ (x) ≡ p(x)/~ is the corresponding

(77.42)

respectively. Note that the product vp vg = E/M is independent of n˜ E (x).

77.3.4 Paraxial Approximation The paraxial approximation is useful in describing the evolution of wave-like properties and/or distortion of wavefronts in the immediate neighborhood of an optical ray. Let the z-axis be the central optical axis of symmetry along which the optical elements are aligned. Using the ˜ Ansatz ψE (x) = eik0 z φ(x, y; z), and dropping ∂ 2 φ ∂z 2 in a slowly varying envelope approximation, one obtains ∂ φ(x, y; z) ∂z ~2 2 ∇⊥ + U(x, y; z) φ(x, y; z) , (77.43) = − 2M    2 where ∇⊥ = ∂ 2 ∂x 2 + ∂ 2 ∂y2 , and v0 = ~k˜ 0 M is the longitudinal velocity of incoming particles. This equation has exactly the form of a time-dependent i~v0

De Broglie Optics

Schrödinger equation in two dimensions, with z/v0 playing the role of a fictitious time t. With this interpretation, the spatial evolution of phase fronts along z can be analyzed in dynamical terms of particles moving in the xy-plane.

φ(x, y; z) =  i exp − ~v0

77.4 Refraction and Reflection Consider a particle beam of energy E incident on a medium with constant index of refraction n˜ E . The boundary plane at z = 0 in Fig. 77.1 divides the vacuum from the medium. At the boundary, the beam is partially reflected and partially transmitted, with the angles determined by Snell’s law of refraction (77.45)

(77.46)

˜ and transmittivity The coefficients of reflectivity R, T˜ = 1 − R˜ are given by the Fresnel formula    cos α − n˜ E cos β 2  . (77.47) R˜ =  cos α + n˜ E cos β  For n˜ E > 1, the interface is “attractive” and R˜ 1, with R˜ → 1 only for glancing incidence α → π/2. For n˜ E < 1, the interface is “repulsive” and total reflection

β z

α

U⬎0

Fig. 77.1 Reflection geometry

 dz U(x, y; z ) φ(x, y; z i ).

zi

In terms of a classical particle moving under the influence of U, the approximation loses validity for 1 2 2 Mv⊥
U, which is just a quarter cycle for a harmonic oscillator.




 R˜ = 1 occurs for α ≥ α˜ c , where α˜ c = sin−1 n˜ E is the critical angle. For α > α˜ c , the de Broglie wave becomes ˜ inside the medium evanescent, with ψE (z > 0) ∼ e−κz (z > 0), where
1/2 κ˜ = k˜ 0 sin2 α − sin2 α˜ c . (77.48) For thermal neutrons, π/2 − α˜ c ∼ 3 × 10−3 radians. If E < U, then n˜ E is imaginary and total reflection occurs for all α. In neutron optics, this total mirror reflection requires ultracold neutrons (T ≈ 0.5 mK). It has important applications for storage of ultracold neutrons in material cavities, and neutron microscopy using spherical mirrors. For details see [77.2].

77.4.1 Atomic Mirrors Inelastic processes, such as diffuse scattering and absorption, inhibit coherent reflection of atoms from bare surfaces. The surface must therefore be coated either with material of low adsorptivity (noble gas, see [77.16]) or electromagnetic fields (evanescent light or magnetic fields, see below). Reflection of Atoms by Evanescent Laser Light Evanescent light fields are produced by total internal reflection of a light beam at a dielectric–vacuum interface [77.17]. In the vacuum, the field decays exponentially away from the interface on a characteristic length κ −1 where
1/2 κ = k n 2 sin2 θi − 1 . (77.49)

Here, n is the light index of refraction of the dielectric, k is the wave number of the light beam in vacuo, and θi is its angle of incidence.

Part F 77.4

and the law of reflection

α⬘

z

(77.44)

In the Raman–Nath approximation (RNA) (also called the short-time, thin-hologram, or thin-lens approxima2 term in (77.43) is neglected. The potential tion), the ∇⊥ U(x, y; z) then acts as a pure phase structure, and the

α = α .

1131

solution of (77.43) becomes

77.3.5 Raman–Nath Approximation

sin α = n˜ E sin β ,

77.4 Refraction and Reflection

1132

Part F

Quantum Optics

If the light is blue-detuned from the atomic resonance, an incident beam of ground state atoms experiences the repulsive potential  1/2 ~  2 −2κ|z| U(x) = + ∆2 − |∆| . Ω0 e 2

low blue-detuned laser beam has been demonstrated by Hammes et al. [77.25]. In the atomic Fabry–Perot resonator both mirrors are realized by laser light [77.26].

77.4.3 Atomic Lenses

(77.50)

For α > α˜ c , the evanescent field acts as a nearly perfect mirror, the imperfections being due to nonadiabatic transitions into the “wrong” dressed state, and possible spontaneous emission. Reflection of atoms by evanescent laser light was demonstrated by Balykin et al. [77.18] at grazing incidence and by Kasevich et al. [77.19] at normal incidence.

De Broglie waves may be focused by refraction from a parabolic potential or by diffraction, e.g., by a Fresnel zone plate (Sect. 77.5.2). Consider focusing by the parabolic potential  1 2 (x 2 + y2 ) , −w ≤ z ≤ 0 Mω f 2 U(x, y; z) = 0 , otherwise . (77.52)

Part F 77.4

Reflection of Atoms by Magnetic Near Fields Magnetic near fields are produced above substrates with a spatially modulated permanent magnetization or close to arrays of stationary currents. In the vacuum above the substrate, the field decays approximately exponentially over a length comparable to the scale of the magnetic modulation. The motion of atoms that cross such inhomogeneous magnetic fields sufficiently slowly is governed by the analog of the adiabatic potential described in Sect. 77.2.5: 
 U(x) = −µ m s /s  B(x) , (77.51)

where µ is the magnetic moment and m s is the (conserved) projection of the atomic spin s onto the local magnetic field direction. A repulsive mirror potential is achieved for spin states with µm s < 0; these weak field seekers are repelled from the strong fields close to the substrate. Experiments have used magnetic recording media like magnetic tapes or hard disks [77.20], arrays of current-carrying wires [77.21], or amorphous magnetic substrates [77.22].

77.4.2 Atomic Cavities Atomic reflections are used in the two kinds of cavities proposed so far: the trampoline cavity and the Fabry– Perot resonator. The trampoline cavity, also called the gravito-optical cavity, consists of a single evanescent mirror facing upwards, the second mirror being provided by gravitation. A stable cavity is realized with the evanescent laser field of a parabolically shaped dielectric–vacuum interface, see [77.23] and the experiment by Aminoff et al. [77.24]. A cavity with transverse confinement provided by a hol-

For ground state atoms, such a potential is realized in the vicinity of the node of a blue detuned standing wave laser field of transverse width w. In this case 1/2
ω f = Ω0 ωrec /|∆| , (77.53)  2 where ωrec = ~k 2M is the recoil frequency. Comparison with the Raman–Nath approximation (77.44) at z = 0, with the phase fronts of a spherical wave converging towards a point x f = (0, 0, f ), shows that U describes a lens of focal length f=

v02 wω2f

.

(77.54)

The Raman–Nath approximation is only valid for a thin lens w f , and breaks down for w > wRN = πv0 /2ω f . In the latter case, oscillations of the particles in the harmonic potential become relevant, a phenomenon sometimes called channeling. Channeling may be used to realize thick lenses with focal length f = wRN corresponding to a quarter oscillation period. Focusing of a metastable helium beam using the anti-node of a large period standing wave laser field has been demonstrated [77.27]. Such a field is produced by reflecting a laser beam from a mirror under glancing incidence. The standing wave forms normal to the mirror surface. Similar interference patterns provide arrays of thick lenses that have been exploited in atom lithography to focus an atomic beam onto a substrate [77.28].

77.4.4 Atomic Waveguides Atomic waveguides can be realized with potentials that confine atoms in one or two dimensions [77.29–31]. These devices are key elements for integrated atom

De Broglie Optics

optics, a field that has seen a rapid evolution recently [77.32, 33]. A planar waveguide is provided by the one-dimensional confinement in an optical standing wave [77.34] or an atomic mirror combined with gravity [77.31]. The discrete nature of the waveguide modes in that case could be demonstrated by lowering the amplitude of the confining potential. Linear waveguides can be modeled by the parabolic transverse potential (77.52) that now extends along the waveguide axis (the zaxis). Physical realizations include hollow, blue-detuned laser beams [77.30], hollow fibers whose inner wall is coated with blue-detuned evanescent light [77.35], elongated foci of red-detuned light created by cylindrical

77.5 Diffraction

1133

lenses [77.36], and magnetic field minima along currentcarrying wires, possibly combined with homogeneous bias fields [77.37]. With typical thermal atomic ensembles, these waveguides operate in a multimode regime, and coherent operation has been demonstrated only with Bose–Einstein condensates. A strong transverse confinement that facilitates monomode operation, can be achieved with miniaturized wire networks deposited on a solid substrate [77.38, 39]. This approach may lead to the fabrication of atom chips [77.33]. Even multimode waveguides, however, can yield robust atomic interferometers, as suggested theoretically in [77.40] and [77.41].

77.5 Diffraction

77.5.1 Fraunhofer Diffraction In the Fraunhofer limit, the field at position (x, y) on a screen at a distance L downstream from the diffracting object is given by

dξ dη eik0 L ψ(x, y; L) = k˜ 0 iL 2π ˜

˜

(77.56) × e−i(kx ξ+k y η) ψ(ξ, η; 0) , ˜ ˜ ˜ ˜ where k x = k0 x/L, k y = k0 y/L. The field at the observation screen is thus given by the Fourier transform of

the field in the object plane; i. e. the momentum representation of the diffracted state. Since most diffraction experiments in atom optics are performed in the Fraunhofer limit, most calculations are done in the momentum representation. Atomic diffractions from microfabricated transmission gratings [77.42] and double slits [77.43] have been observed. Recent experiments have extended de Broglie wave diffraction to heavier, complex particles like fullerence molecules (C60 ) [77.44].

77.5.2 Fresnel Diffraction Typical applications of Fresnel diffraction are Fresnel zone plates and the effects of Talbot and Lau. Fresnel zone plates are microfabricated concentric amplitude structures which act like lenses. They are frequently employed in optics of α-particles and neutrons. In atom optics, focusing with a Fresnel zone plate was first demonstrated by Carnal et al. [77.45]. The Talbot effect and the related Lau effect refer to the self-imaging of a grating of period d, which appears downstream at distances that are integral multiples of the Talbot length L = 2d 2 /λdB . For a discrete set of smaller distances than the Talbot length, images of the grating appear with smaller periods d/n, n = 2, 3 · · · . For applications in matter wave interferometry, [77.46]; for applications in atom lithography see [77.47].

77.5.3 Near-Resonant Kapitza–Dirac Effect The near resonant Kapitza–Dirac effect refers to the diffraction of two-level atoms from a standing wave laser

Part F 77.5

The diffraction of matter waves is described by the solution of the Schrödinger equation (77.34) subject to the boundary conditions imposed by the diffracting object. For a plane screen Σ made of opaque portions and apertures, the solution in the source-free region behind the screen is given by the Rayleigh–Sommerfeld formulation of the Huygens principle

˜ k˜ 0 dξ dη eik0 R ψE (x) = i 2π R  Σ i n· R ψ(ξ) , × 1+ (77.55) R k˜ 0 R where ξ = (ξ, η, ζ) denotes coordinates of points on Σ, n is an inwardly directed normal to Σ at a point ξ, and R = x − ξ. A diffraction pattern only becomes manifest in the diffraction limit r d, where r is the distance to the observation point, and d is the length scale of the diffracting system. The two diffraction regimes are then the Fraunhofer limit r d 2 /λdB and the Fresnel regime r ≈ d 2 /λdB , also called near-field optics.

1134

Part F

Quantum Optics

field with a spatially uniform polarization. The dynamics of the effect is described by the Hamiltonian (77.19) with the mode function of the laser field given by (77.16). Consider atoms traveling predominantly in the zdirection, i. e., orthogonal to the axis of the laser field, with energy 12 Mv02 ~Ω0 (Fig. 77.2). Kapitza–Dirac diffraction is then observed in transmission, which in the theory of diffraction is called Laue geometry. In the paraxial approximation (Sect. 77.3.4) for motion in the z-direction, and assuming that the laser profile is homogeneous in the y-direction, the description reduces to an effectively one-dimensional model for the quantum mechanical motion along the x-axis of the laser field. Due to the periodicity of the standing wave light field, the transverse momenta of the transmitted atom waves differ by multiples of ~k from the transverse momentum of the undiffracted wave. For the important case of strong detuning, and assuming that the incoming atoms are in their electronic ground state moving with transverse momentum px = 0, the px -distribution of the outgoing wave is given in the Raman–Nath approximation (Sect. 77.3.5) by   2  Ω0 2   τ  , Prob( px = 2n ~k) ∝  Jn (77.57)  8|∆| 

Part F 77.5

where Jn is a Bessel function of order n, τ = w/v0 is an effective interaction time, w being the width of the laser field, and v0 the longitudinal velocity of the atoms. The distribution (77.57) was observed by Gould et al. [77.48].
−1/2 For τ
τRN , where τRN = ωrec Ω0 2 /|∆| , the Raman–Nath approximation becomes invalid. As a result of Doppler related phase-mismatch, the momentum spread saturates and shows a sequence of collapse and revival as a function of τ [77.49]. If the detuning is too small to allow for a scalar description, the two-level character of the atoms must be taken into account. For the particular case of ∆ = 0, the ground state evolves into √ an
equal superposition of the two diabatic states 1 2 |e ± |g while entering the interaction region. Inside the interaction region, these states experience potentials which differ only by their sign. For atomic beams with a small spatial spread δx 2π/k, the diabatic states experience opposite forces, leading to a splitting of the atomic beam called the atom-optical Stern–Gerlach effect [77.50]. In the general case of arbitrary ∆, the Kapitza– Dirac Hamiltonian (77.21) is most conveniently analyzed using band theoretical methods of solid state theory [77.51].

w k0

x z

Fig. 77.2 Geometry of the Kapitza–Dirac effect

77.5.4 Atom Beam Splitters Beam splitters are optical devices which divide an incoming beam into two outgoing beams traveling in different directions. For thermal neutrons, beam splitters may be realized by diffraction from perfect crystals in Laue geometry. For atoms, they can be realized using diffraction from crystalline surfaces, microfabricated structures (see Sect. 77.5.1), or by using diffraction from an optical standing wave. The Kapitza–Dirac effect, for example, may be exploited to split an atomic beam coherently using Bragg reflection at the “lattice planes” provided by the periodic intensity variations of a standing wave laser field [77.52]. This process is resonant for an incoming atomic beam traveling with transverse momentum px = ~k because it is energetically degenerate with the diffracted beam traveling with transverse momentum p¯ x = −~k. This level degeneracy is lifted while the atoms enter the interaction region. In the Bragg regime, the lifting happens slowly enough that only the momentum states | ± ~k participate in the diffraction (two-beam resonance), and their populations show Pendellösung type oscillations as a function of the transit time. The frequency of the oscillations is given by δE/~, where δE is the energy splitting of the two beams inside the interaction region. For a transit time given by a quarter period of the Pendellösung, a 50% beam splitting is observed [77.52]. In principle, Bragg resonances may also be realized for higher diffraction order px = n ~k ↔ p¯ x = −n ~k. However, in this case, intermediate momentum states become populated (multibeam resonance), which makes the higher-order Bragg resonances less suitable for beam splitting purposes.

De Broglie Optics

More promising for the realization of an atomic beam splitter is the magneto-optical diffraction which refers to the diffraction of three-level atoms from a laser field with a periodic polarization gradient (lin ⊥ lin configuration) (Chapt. 75), and a magnetic field aligned parallel to the optical axis of the laser field. This configuration realizes an interaction potential in the form of a blazed grating, i. e., a phase grating with an approximately triangular variation of phase. In an experiment by Pfau et al. [77.53], transverse split-

77.6 Interference

1135

ting of a beam of metastable helium by 42~k was observed [77.49]. Diffraction from an evanescent standing wave involves Bragg reflection of atoms under glancing incidence from the periodic grating of a blue detuned evanescent standing wave laser field [77.54– 56]. Diffraction at normal incidence has been demonstrated with sufficiently slow atoms and can be described by a generalization of the RNA (Sect. 77.3.5) [77.57, 58].

77.6 Interference where χ = W1 − W2 is the relative phase of the two components. The sinusoidal variations of I± with varying χ are called interference fringes, and χ/2π is called the fringe order number. In the WKB approximation, the phases W1 and W2 are given by

Wi = W0 + k˜ (x) · dx, i = 1, 2 , (77.61) i

where the integrals extend over the classical paths 1 and 2, respectively, and dx is an element of displacement along the paths. Using (77.61), the relative phase is

χ = k˜ (x) · dx , (77.62) C

77.6.1 Interference Phase Shift From a fundamental point of view, any interferometer is a ring. At the entrance port of a three-grating interferometer displayed in Fig. 77.3, for example, the wave function is split into two coherent parts which spatially evolve along different paths and subsequently come together at the exit port where they are superimposed to produce two outgoing waves

where C is the closed interferometer loop. Note, that on path 2, the path element dx and k˜ are antiparallel. Usually, the absolute value of χ is not measured, but only variations, called phase shifts, which result from displacements of the diffraction gratings or placement of optical elements into the beam path. Phase shifts

1 ψ± = √ (ψ1 ± ψ2 ) , (77.58) 2 where the components from path 1 and 2 are given by ψ1 = A1 exp (iW1 ) , ψ2 = A2 exp (iW2 ) .

Ψ1

Ψ+

Ψ2

Ψ–

υ (77.59)

√ For simplicity, assume A1 = A2 = A0 2. The relative flux of the outgoing waves is then  1
I± = 1 ± cos χ , (77.60) 2

D

D

Fig. 77.3 Geometry of the three-grating interferometer

Part F 77.6

While the overall phase of a wave function ψ is not observable, interferometry makes detectable the relative phases of two components ψ1 , ψ2 in a superposition ψ = ψ1 + ψ2 . Two types of interferometers are most common: the Young double slit as a paradigm for interferometers based on division of wavefront, and the Mach–Zehnder interferometer as a paradigm for interferometers based on division of amplitude. In de Broglie optics, the latter type is realized in the form of a threegrating interferometer, since division of amplitude is achieved by diffraction at gratings rather than by semitransparent mirrors. Experiments with this geometry have been reported for atoms [77.59] and recently for more massive, complex molecules (fullerenes) [77.60].

1136

Part F

Quantum Optics

come in two categories: dispersive and geometric. If a phase shift χ depends on v0 , it is called dispersive, otherwise it is called geometric. Geometric phases depend only on the geometry of the interferometer loop. The Sagnac effect (see below), for example, may be geometric. A phase which depends neither on v0 nor on the geometry of the interferometer loop is called topological. The Aharonov–Bohm effect (see below) is topological. Using (77.39), χ becomes χ = χ0 + χ{U} + χ{A} ,

(77.63)

where for weak potentials U and A, Mv0 (L 1 − L 2 ) , ~  1 U[x(s)] ds , χ{U} = − ~v0 1 A(x) · dx , χ{A} = ~ χ0 =

(77.64) (77.65) (77.66)

and L i is the geometric length of the path i. For a constant potential U0 intersecting the interferometer on a length w, χ{U} is given by χ{U} = −

Part F 77.6

U0 w . ~v0

(77.67)

Using Stokes theorem, χ{A} may be written in the manifestly gauge-invariant form

1 χ{A} = (77.68) [∇ × A(x)] · da , ~ where the integral extends over the area enclosed by the interferometer, and da is an infinitesimal area element. Dispersive Phase Shifts. Atom interferometers have been able to measure phase shifts of the form (77.65) due to, for example, the atomic level shift in an electric field (Stark effect) [77.59] or to coherent forward scattering by background gas atoms, see (77.10) and [77.61]. Sagnac Effect. The Sagnac effect refers to χ{A} in a ro-

tating interferometer. Inserting (77.6) into (77.68), and assuming that the axis of rotation is oriented perpendicular to the plane of the interferometer, the Sagnac phase shift is given by χSa = 4MΩA/~ ,

(77.69)

where A is the geometric area enclosed by the interferometer loop. χSa may be dispersive or geometric depending on the type of interferometer. In a Young

double slit, A is independent of energy and χSa is geometric. In a three-grating interferometer, the area is A ≈ ϑD2 , where ϑ is the splitting angle; see Fig. 77.3. In this case, χSa is dispersive because of the velocity dependence of ϑ ≈ 2~k/Mv0 . The Sagnac effect for de Broglie waves was first observed by Werner et al. [77.62] using a neutron interferometer. Aharonov–Bohm Effect. The Aharonov–Bohm effect refers to the χ{A} of charged particles encircling a magnetic flux line [77.63]. Inserting A from (77.7) into (77.68), and assuming particles of charge q encircling a line of flux Φ once, one finds

χAB = qΦ/~ .

(77.70)

A characteristic feature of the Aharonov–Bohm effect is that the particles actually never “see” the magnetic field of the flux line which is confined to some region inaccessible to the particles. χAB is strictly topological, and only depends on the linking number of the interferometer loop and the flux line. Its appearance is characteristic for all gauge theories. For further details and a summary on its experimental verification see [77.5]. Aharonov–Casher Effect. The Aharonov–Casher ef-

fect refers to χ{A} of a magnetic spin encircling an electric line charge [77.64]. Inserting A from (77.11) into (77.68), one obtains for proper alignment of µ and E |µ| r0 χAC = 2π , (77.71) µB ξ

where r0 is the classical electron radius, µB is the Bohr magneton, and ξ = e/ρel , ρel being the electric line charge density. χAC is topological only if the spin is aligned parallel to the electric line charge and both are oriented perpendicular to the plane of the interferometer. χAC for atoms has been observed by Sangster et al. [77.65]. Electric Dipole Phase. Electric dipole phase refers to the

χ{A} of an electric dipole moment encircling a magnetic line charge [77.66]. Inserting A from (77.14) into (77.68), one obtains for proper alignment of d and B |d| a0 χdE = 2π , (77.72) ea0 ξ

where a0 is the Bohr radius, and ξ = Φ0 /mg , Φ0 being the magnetic flux unit, and mg being the magnetic line charge density. In analogy to the Aharonov–Casher

De Broglie Optics

effect, χdE is topological, provided that d is aligned parallel to the magnetic line charge, and both are oriented perpendicular to the interferometer plane.

77.7 Coherence of Scalar Matter Waves

1137

is well described by the potential (77.32) U(x) = −~

g2 f(x)2 a† a , ∆

(77.73)

77.6.2 Internal State Interferometry Manipulation of the internal state of atoms by means of electromagnetic fields makes it possible to realize interferometric setups which involve separation of paths in internal space rather than in real space. Examples of such interferometers are the Optical Ramsey interferometer [77.67], the stimulated Raman interferometer [77.68], and the interferometers using static electric and magnetic fields [77.69, 70].

77.6.3 Manipulation of Cavity Fields by Atom Interferometry The entanglement of atomic states and quantized field states opens novel possibilities for manipulating and/or measuring nonclassical field states in a cavity. In the adiabatic limit, for example, and assuming sufficient detuning between the atom and the cavity field, the interaction and c.m. motion of an atom traversing a cavity

where g is the vacuum Rabi frequency, f(x) is a cavity mode function, and a, a† denote cavity photon annihilation and creation operators, respectively. Because of the presence of the photon-number operator a† a in (77.73), the deflection and phase shift of an atom traversing the cavity is quantized, displaying essentially the photon number statistics in the cavity. The quantized deflection is sometimes called the inverse Stern–Gerlach effect. Due to the entanglement of atom and cavity states, and the position dependence of the interaction strength, the phase shift induced by U(x) in a standing wave cavity may be used to measure either the atomic position via homodyne detection of the cavity field [77.71, 72], or the photon statistics via atom interferometry [77.73, 74]. In a ring cavity, the entanglement of c.m. motion and cavity field may be used to measure the atomic momentum [77.75] via homodyne detection of the cavity field. For further details see [77.76] and Chapt. 78.

The general solution of the free Schrödinger equation (77.2) may be written in the form

˜ ˜ ψ(x, t) = d3 k˜ a(k˜ ) ei(k·x−ω(k)t) , (77.74) where ω(k˜ ) ≡ E/~ = ~k˜ 2 /2M. If the coefficients a(k˜ ) are known, the state represented by ψ(x, t) is called a pure state. Otherwise it is called a mixed state, and physical quantities are obtained by an ensemble average over the possible realizations of a(k˜ ). The degree of coherence of matter waves is described by the autocorrelation function of Ψ(x, t):

Γ(x, t; x , t



) ≡ Ψ(x , t )∗ Ψ(x, t) ,

(77.75)

where the overline (· · · ) denotes the ensemble average over the possible realizations of a(k˜ ). In light optics, Γ(x, t; x , t ) is called the mutual coherence function. In particular, for equal times, Γ(x, t; x , t) describes the spatial coherence, and for equal positions, Γ(x, t; x, t ) describes the temporal coherence. For a beam of particles, coherence may be either longitudinal (measured along the beam) or transverse (measured across the beam). In contrast to light optics,

there is no simple relation between longitudinal coherence and temporal coherence because the dispersion relation of matter waves is quadratic in the wavenumber. The spatial coherence function is intimately related to the quantum mechanical density operator of the particles (Chapt. 7) ρ(t) = |Ψ(t)Ψ(t)| .

(77.76)

In the position representation, one has x|ρ(t)|x  ≡ ρ(x, x ; t) = Γ(x, t; x , t) .

(77.77)

Longitudinal and temporal coherence of a particle beam is determined mainly by the source of the beam. The thermal fission reactors used in neutron optics and the ovens used in atom optics are analogous to blackbody sources in light optics. In contrast, the transverse coherence is mainly determined by the way the particles are extracted from the oven to form a beam.

77.7.1 Atomic Sources To describe thermal sources, consider a single particle in an oven of temperature T and volume V , assuming

Part F 77.7

77.7 Coherence of Scalar Matter Waves

1138

Part F

Quantum Optics

that a(k˜ ) and a(k˜ ) are statistically independent: a(k˜ )∗ a(k˜ ) = ρ(k˜ )δ(k˜ − k˜ ) ,

Supersonic Beams. Supersonic beams are produced by (77.78)

where

 2 k˜ 2 λ3th ~ exp − ρ(k˜ ) = V 2MkB T

(77.79)

accounts for the thermal distribution of wavenumbers, and 1/2  2π ~2 λth = (77.80) MkB T denotes the thermal de Broglie wavelength. Using (77.78)–(77.80) in (77.75), the mutual coherence function becomes Γ(x, t; x , t ) = 1 1 V {1 + [i(t − t )/τth ]} 32  
2 x − x

, × exp −π 2 λth (1 + [i(t − t )/τth ])

(77.81)

Pulsed Beams. Pulsed beams are produced by chopping any of the beams described above. Important applications for pulsed beams are the resolution of temporal coherence and the mapping of the relative phases of the a(k˜ ) in matter wave interferometry. Laser-like Source of Atoms. In these sources, many

atoms with integral spin (Bosons) occupy one and the same quantum state of motion [77.78]. Their operational principle is rooted in the quantum statistical effects of indistinguishability. It may be viewed in close analogy to the operational principle of an ordinary laser (Chapt. 70) and the mechanism underlying Bose–Einstein Condensation (Sect. 76.1.1). Laser-like sources have indeed been achieved by letting a small current of atoms leak out of a trapped Bose–Einstein condensate [77.79].

77.7.2 Atom Decoherence

where

Part F 77.7

τth =

supersonic expansion of a high pressure gas which is forced through an appropriately designed nozzle. The expansion produces a velocity distribution in the longitudinal direction which is approximately Gaussian with a velocity ratio v/δv ≈ 10–20.

~ , kB T

(77.82)

is the thermal coherence time. According to (77.81), the spatial coherence of a thermal state falls off in a Gaussian manner on a scale given by λth . The temporal coherence, in contrast, falls off algebraically on a time scale given by τth . Expressed in physical units, one has 1.74( 5) × 10−9 m λth = √ , (M/u)(T/K) 7.63 × 10−12 s , τth = (T/K)

(77.83)

where u is the atomic mass unit. Atomic Beams Effusive Beams. Effusive beams are produced from ther-

mal sources by a suitable set of collimators placed in front of the opening of the oven. This produces a Maxwell–Boltzmann distribution of atomic velocities in the longitudinal direction. The coherence properties in the transverse direction are described by the van Cittert– Zernike theorem [77.77]; for details see any textbook on classical optics.

In any interferometer, the contrast of the interference fringes quantitatively measures the coherence of the wave involved. Partially coherent beams show an output flux given by I± =


 1 1 ± Re C eiχ , 2

(77.84)

instead of (77.60), with a complex number C. One has |C| ≤ 1, with the maximum achieved for a pure state; the phase of C is measured by scanning the interferometer phase shift χ. In de Broglie interferometry, coherence can be lost when the interfering matter wave gets entangled with other systems. This happens for atoms, for example, due to the emission or scattering of photons, as soon as the detection of these photons permits, in principle, the resolution of spatially separated paths in the interferometer. In fact, the width of Γ(x, t; x , t) as a function of x − x , also called the spatial coherence length, is reduced to the photon wavelength after a single scattering event, see [77.80, 81]. Interference can be restored when the emitted photons are detected and correlated with the atom output [77.82,83]. Collisions with background gas atoms between the optical elements of a three-grating interferometer also

De Broglie Optics

reduce coherence, as has been shown with fullerene molecules [77.84]. Finally, coherence is lost when atoms interact with random electromagnetic fields. This has become relevant for atom reflection from evanescent light because of the roughness of the dielectric surface

References

1139

used [77.85] (see also [77.86]). The coherent operation of integrated atom optics near metallic surfaces is limited by thermally excited electromagnetic near fields as shown in experiments by Harber et al. [77.87] (see also [77.88]).

References 77.1 77.2

77.3 77.4

77.5

77.6 77.7

77.10 77.11 77.12 77.13 77.14

77.15

77.16 77.17

77.18 77.19 77.20 77.21 77.22

77.23 77.24 77.25 77.26 77.27 77.28 77.29 77.30

77.31 77.32 77.33

77.34 77.35 77.36 77.37 77.38 77.39 77.40 77.41 77.42 77.43 77.44 77.45 77.46 77.47 77.48 77.49 77.50

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77.72 77.73 77.74 77.75 77.76

77.77 77.78 77.79 77.80 77.81 77.82 77.83 77.84 77.85 77.86 77.87 77.88

P. Storey, M. Collett, D. Walls: Phys. Rev. Lett. 68, 472 (1992) P. Meystre, E. Schumacher, S. Stenholm: Opt. Commun. 73, 443 (1989) A. M. Herkommer, V. M. Akulin, W. P. Schleich: Phys. Rev. Lett. 69, 3298 (1992) T. Sleator, M. Wilkens: Phys. Rev. A 48, 3286 (1993) S. Haroche, J. M. Raimond: Manipulation of non classical field states by atom interferometry. In: Cavity Quantum Electrodynamics, Adv. At. Mol. Opt. Phys., suppl. 2, ed. by P. R. Berman (Academic Press, Boston 1994) pp. 123–170 B. Taylor, K. J. Schernthanner, G. Lenz, P. Meystre: Opt. Commun. 110, 569 (1994) R. J. C. Spreeuw, T. Pfau, U. Janicke, M. Wilkens: Europhys. Lett. 32, 469 (1995) M. Köhl, T. W. Hänsch, T. Esslinger: Phys. Rev. Lett. 87, 160404 (2001) T. Pfau et al.: Phys. Rev. Lett. 73, 1223 (1994) O. Steuernagel, H. Paul: Phys. Rev. A 52, 905 (1995) M. S. Chapman et al.: Phys. Rev. Lett. 75, 3783 (1995) C. Kurtsiefer et al.: Phys. Rev. A 55, R2539 (1996) L. Hackermüller et al.: Appl. Phys. B 77, 781 (2003) A. Landragin et al.: Opt. Lett. 21, 1581 (1996) C. Henkel et al.: Phys. Rev. A 55, 1160 (1997) D. M. Harber, J. M. McGuirk, J. M. Obrecht, E. A. Cornell: J. Low Temp. Phys. 133, 229 (2003) C. Henkel, M. Wilkens: Europhys. Lett. 47, 414 (1999)

1141

78. Quantized Field Effects

Quantized Fiel

78.1

Field Quantization ............................... 1142

78.2 Field States ......................................... 1142 78.2.1 Number States .......................... 1143 78.2.2 Coherent States......................... 1143 78.2.3 Squeezed States ........................ 1144 78.2.4 Phase States ............................. 1145 78.3 Quantum Coherence Theory .................. 1146 78.3.1 Correlation Functions................. 1146 78.3.2 Photon Correlations ................... 1146

78.3.3 Photon Bunching and Antibunching ..................... 1147 78.4 Photodetection Theory ......................... 1147 78.4.1 Homodyne and Heterodyne Detection.......... 1147 78.5 Quasi-Probability Distributions ............. 1148 78.5.1 s-Ordered Operators .................. 1148 78.5.2 The P Function.......................... 1149 78.5.3 The Wigner Function.................. 1149 78.5.4 The Q Function.......................... 1151 78.5.5 Relations Between Quasi-Probabilities ...... 1151 78.6 Reservoir Theory .................................. 1151 78.6.1 Thermal Reservoir ..................... 1152 78.6.2 Squeezed Reservoir ................... 1152 78.7 Master Equation .................................. 1152 78.7.1 Damped Harmonic Oscillator....... 1153 78.7.2 Damped Two-Level Atom ........... 1153 78.8 Solution of the Master Equation ........... 1154 78.8.1 Damped Harmonic Oscillator....... 1154 78.8.2 Damped Two-Level Atom ........... 1155 78.9 Quantum Regression Hypothesis ........... 1156 78.9.1 Two-Time Correlation Functions and Master Equation ................. 1156 78.9.2 Two-Time Correlation Functions and Expectation Values .............. 1156 78.10 Quantum Noise Operators ..................... 1157 78.10.1 Quantum Langevin Equations ..... 1157 78.10.2 Stochastic Differential Equations . 1158 78.11 Quantum Monte Carlo Formalism .......... 1159 78.12 Spontaneous Emission in Free Space ..... 1159 78.13 Resonance Fluorescence ....................... 1160 78.13.1 Equations of Motion .................. 1160 78.13.2 Intensity of Emitted Light ........... 1160 78.13.3 Spectrum of the Fluorescence Light ........... 1161 78.13.4 Photon Correlations ................... 1161 78.14 Recent Developments........................... 1162 78.14.1 Literature ................................. 1162 78.14.2 Field States .............................. 1162 78.14.3 Reservoir Theory ....................... 1162 References .................................................. 1163

Part F 78

The electromagnetic field appears almost everywhere in physics. Following the introduction of Maxwell’s equations in 1864, Max Planck initiated quantum theory when he discovered h = 2π~ in the laws of black-body radiation. In 1905 Albert Einstein explained the photoelectric effect on the hypothesis of a corpuscular nature of radiation and in 1917 this paradigm led to a description of the interaction between atoms and electromagnetic radiation. The study of quantized field effects requires an understanding of the quantization of the field which leads to the concept of a quantum of radiation, the photon. Specific nonclassical features arise when the field is prepared in particular quantum states, such as squeezed states. When the radiation field interacts with an atom, there is an important difference between a classical field and a quantized field. A classical field can have zero amplitude, in which case it does not interact with the atom. On the other hand a quantized field always interacts with the atom, even if all the field modes are in their ground states, due to vacuum fluctuations. These lead to various effects such as spontaneous emission and the Lamb shift. The interaction of an atom with the many modes of the radiation field can conveniently be described in an approximate manner by a master equation where the radiation field is treated as a reservoir. Such a treatment gives a microscopic and quantum mechanically consistent description of damping.

1142

Part F

Quantum Optics

78.1 Field Quantization This section provides the basis for the quantized field effects discussed in this Chapter [78.1]. We expand the field in a complete set of normal modes which reduces the problem of field quantization to the quantization of a one dimensional harmonic oscillator corresponding to each normal mode. The classical free electromagnetic field, i. e., the field in a region without charge and current densities, obeys the Maxwell equations ∇·B=0, (78.1) ∇·D=0, (78.2) ∂B =0, ∇ × E+ (78.3) ∂t ∂D =0, ∇ × H− (78.4) ∂t where B = µ0 H, D = 0 E. The magnetic permeability µ0 connects the magnetic induction B with the magnetic field H and the electric permittivity 0 of free space connects the displacement D with the electric field E. In the case of a free field, E and B may be obtained from

Part F 78.2

B=∇×A, (78.5) ∂A , E=− (78.6) ∂t where the vector potential A obeys the Coulomb gauge condition ∇ · A = 0 and satisfies a wave equation. In order to solve this wave equation we expand the vector potential 1/2 2 

~ A(x, t) = 2ωk0 V k σ=1   × αkσ
kσ ei(k·x−ωk t) + c.c. (78.7) in a set of normal modes V −1/2 exp(ik · x)
kσ which are orthonormal in the volume V. Due to the gauge condition ∇ · A = 0, we obtain two orthogonal polarization vectors
k1 and
k2 with
kσ · k = 0 for each wave vector k. The dispersion relation is ωk = c|k|. The Fourier amplitudes αkσ are complex numbers in the classical theory.

The field is quantized by replacing the classical amplitude αkσ by the mode annihilation operator akσ . The ∗ is replaced by the mode creation complex conjugate αkσ † operator akσ . They obey the commutation relation   † (78.8) akσ , ak σ  = δkk‘ δσσ  . The representation of the electric field operator
 ~ωk 1/2 E(x, t) = i 20 V k,σ   × akσ
kσ ei(k·x−ωk t) − h.c. ≡ E+ (x, t) + E− (x, t)

(78.9)

in terms of these operators follows from (78.6) and the operator for the vector potential. Note also the often used decomposition of the electric field operator into the positive and negative frequency parts E+ and E− respectively. A similar relation holds for the operator describing the magnetic induction B. Using the operators for the electric and magnetic field, one can transform the field energy 

1 dV 0 E2 + B2 /µ0 H= (78.10) 2 into the form


† H= ~ωk akσ akσ + 1/2 ,

(78.11)

k,σ

which is a sum of independent harmonic oscillator Hamiltonians corresponding to each mode (k, σ). The † number operator Nkσ = akσ akσ represents the number of photons in the mode (k, σ), while ~ωk /2 is the energy of the vacuum fluctuations. Hence each mode of the electromagnetic field is equivalent to a harmonic oscillator. In the next section we discuss specific states of a single mode. The general quantum state of the electromagnetic field consisting of many modes is given by a superposition of product states that are composed out of these single mode states.

78.2 Field States This section summarizes the properties of several important states of the electromagnetic field. From the independence of the normal modes, the discussion may be restricted to a single normal mode. With the mode

index (k, σ) suppressed, a single mode Hamiltonian is

~ω ~ω 2 p2 + x . H = ~ω a† a + 1/2 = 2 2

(78.12)

Quantized Field Effects

In the second step, the quadrature operators

1 (78.13) x = √ a + a† , 2

1 p = √ a − a† (78.14) i 2 are introduced, which are equivalent to scaled position and momentum operators of a massive particle in a harmonic potential. The quadratures of a quantized field are measurable with the help of homodyne detection as discussed in Sect. 78.4.1 We shall now describe several states of this quantized field mode: number states, coherent states, squeezed states, Schrödinger cats, and phase states. A quantized field in a coherent state shows the most classical behavior. A superposition of two coherent states, which is a Schrödinger cat, already shows nonclassical features. Number states and squeezed states are further typical examples of nonclassical states.

78.2.1 Number States The eigenstates of the Hamiltonian (78.12) are the eigenstates of the number operator N = a† a, N|n = n|n ,

(78.15)

where n = 0, 1, 2, . . . denotes the excitations or the number of photons in the mode. The vacuum state of the mode |0, is defined by a|0 = 0 .

(78.16)

on a Fock state |n. These number or Fock states form a complete and orthonormal set of states so that ∞


|nn| = 1 ,

n|k = δnk .

(78.19)

n=0

Their quadrature representations are √ −1/2 2 x|n = π2n n! Hn (x) e−x /2 , (78.20) √ n −1/2 n − p2 /2  p|n = π2 n! (−i) Hn ( p) e . (78.21)

The states |x and | p are eigenstates of the quadrature operators x and p, (78.13) and (78.14).

1143

Number states provide a frequently used representation of a pure quantum state |ψ =

∞


cn |n ,

(78.22)

n=0

or a mixed quantum state given by the density operator
ρnk |nk| (78.23) ρ= n,k

(Chapt. 7).

78.2.2 Coherent States The coherent state is a specific superposition of number states. In contrast to a number state, a coherent state does not possess a definite number of photons: the photon distribution is Poissonian. For a large average photon number, the electric and magnetic fields have rather well defined amplitudes and phases with vanishing relative quantum fluctuations. Hence the Poissonian photon distribution frequently serves as a borderline between classical and nonclassical field states. Nonclassical states show a sub-Poissonian behavior. An extreme example is a field prepared in a number state. A parameter which quantifies the deviations from Poissonian behavior is the Q parameter introduced by Mandel [78.2]. We define the coherent state |α as an eigenstate of the annihilation operator a|α = α|α

(78.24)

with the complex amplitude α = |α|eiθ . The coherent state can be represented by |α = eαa

† −α∗ a

|0 = D(α)|0 ,

(78.25)

that is, by the action of the displacement operator D(α) on the vacuum. The number state representation of |α reads |α = e−|α|

2 /2

∞
αn √ |n . n! n=0

(78.26)

A coherent state |α0  that evolves in time according to the free field Hamiltonian (78.12) stays coherent, i. e., |ψ(t) = exp (−iHt/~) |α0  = e−iωt/2 |α(t) , (78.27)

with amplitude α(t) = α0 exp[−iωt].

Part F 78.2

The ladder of excitations can be climbed up and down via the application of creation and annihilation operators √ (78.17) a† |n = n + 1|n + 1 , √ a|n = n|n − 1 , (78.18)

78.2 Field States

1144

Part F

Quantum Optics

Another important representation of a coherent state is the x representation 

x|α = π −1/4 exp −[Re(α)]2

√ (78.28) × exp −x 2 /2 + 2αx , where |x denotes again the x quadrature eigenstate. The photon distribution in a coherent state |α|2n e−|α| (78.29) n! is a Poisson distribution with average  photon number 2 N = |α|2 and variance ∆N = N 2 − N2 = |α|2 . Hence the relative fluctuations (∆N )/N = N−1/2 vanish for a large average photon number. The Mandel Q parameter 2

|n|α|2 =

(∆N )2 − N Q≡ N

(78.30)

vanishes for a field in a coherent state. A nonclassical field may show sub-Poissonian behavior with Q < 0. As an example, the Schrödinger cat state is a macroscopic superposition of two coherent states −1/2   2 2 |cat = 2 + 2 cos α2 sin φ e−2α sin (φ/2)     × α eiφ/2 + α e−iφ/2 , (78.31)

Part F 78.2

where α is assumed to be real. The Q parameter for this superposition state, shown in Fig. 78.1, takes on negative values for specific angles φ. The nonclassical behavior of such a |cat-state can be explained [78.3] as a result of quantum interference between the two coherent states present in (78.31). The incoherent superposition described by the density operator 1  iφ/2  iφ/2   −iφ/2  −iφ/2 

ρ= αe + αe αe αe 2

the commutation relation [x, p] = i. Their uncertainties    (∆x)2 ≡ x 2 − x2 and (∆ p)2 ≡ p2 −  p2 fulfill the Heisenberg inequality ∆x∆ p ≥ 1/2 .

The coherent state is a special minimum uncertainty state √ with equal uncertainties ∆x = ∆ p = 1/ 2. Squeezed states comprise a more general class of minimum uncertainty states with reduced uncertainty in one quadrature at the expense of increased uncertainty in the other. These states |α,  are obtained by applying the displacement operator D(α) and the unitary squeeze operator 1 ∗ 2 1 †2 a − 2 a

S() = e 2 

|α,  = D(α)S()|0 .

(78.35)

The squeeze operator S() transforms a and a† according to S† ()a S() = a cosh r − a† e−2iφ sinh r , †

†

†

S ()a S() = a cosh r − a e where 

= r e−2iφ .

2iφ

sinh r ,

(78.36) (78.37)

The rotated quadratures

X 1 = x cos φ − p sin φ , X 2 = p cos φ + x sin φ ,

(78.38) (78.39)

transform according to S† ()(X 1 + iX 2 )S() = X 1 e−r + i X 2 er , which yields the uncertainties √ ∆X 1 = e−r / 2 , √ ∆X 2 = er / 2 .

(78.40)

(78.41) (78.42)

Q 2

does not have this nonclassical character: its Q parameter vanishes. Coherent states have a direct physical significance: the quantum state of a stabilized laser operating well above threshold can be approximated by a coherent state.

1.5

Squeezed states [78.4–6] minimize the uncertainty product of the quadrature components of the electromagnetic field. The quadrature components x and p of the single mode field are defined in (78.13) and (78.14). They obey

(78.34)

to the vacuum

(78.32)

78.2.3 Squeezed States

(78.33)

1 0.5 0.1

0.2

0.3

0.4 φ

–0.5 –1

Fig. 78.1 The Q parameter for a Schrödinger cat state (78.31) with amplitude α = 4

Quantized Field Effects

In particular, for φ = 0 the squeezed state |α, r is a minimum uncertainty state for the√quadratures x and p with √ ∆x = e−r / 2 and ∆ p = er / 2. The degree of squeezing in the quadrature x is determined by the squeeze factor r. The average photon number N = |α|2 + sinh2 r

(78.43)

of a squeezed state and its photon number variance  2 (∆N )2 = α cosh r − α∗ e−2iφ sinh r  + 2 cosh2 r sinh2 r

(78.44)

contain the coherent contribution α as well as squeezing contributions expressed by r and φ. In particular, for φ = 0, the Q parameter becomes negative for a large enough amplitude α and r > 0. The photon number distribution Wn = |α, r|n|2 becomes narrower than the one for the corresponding coherent state with the same α. This sub-Poissonian behavior is one of the nonclassical features of a squeezed state. Furthermore, Wn shows oscillations [78.7] for larger squeezing. The two regimes with sub-Poissonian and oscillating photon statistics Wn are shown in Fig. 78.2. A second representation of squeezed states has been introduced by Yuen [78.8]. In his notation, a squeezed state is an eigenstate of the operator b = µa + νa† ,

(78.45)

with |µ|2 − |ν|2 = 1 and eigenvalue β. This eigenstate can be written in the form

larization. Heidmann et al. [78.11] have shown that the difference intensity of these twin beams may exhibit reduced quantum fluctuations. Pulsed twin beams also contain reduced noise in the difference of their intensities.

78.2.4 Phase States The problem of a correct quantum mechanical description of phase has a long history in quantum mechanics [78.12]. First attempts to define a quantum phase are due to London and Dirac. The London phase state is ∞ 1
inφ |φ = √ e |n , 2π n=0

(78.47)

which is an eigenstate of the exponential phase operator  eiφ =

∞


|nn + 1| .

(78.48)

n=0

Since this operator is not unitary, it does not define a Hermitian operator φˆ for the phase. Nevertheless many  treatments of the phase of a quantum state |ψ = cn |n are based on the London phase distribution 2   
 1   2 −inφ cn e (78.49) Pr(φ) = φ|ψ| =  .   2π  n

(78.46)

which connects the squeezing operator S(r e−2iφ ) with the parameters µ = cosh r and ν = e−2iφ sinh r. In contrast to the definition (78.35), the displacement operator D(β) and the squeezing operator S() are applied now in reversed order. Nevertheless, the two equations (78.35) and (78.46) define the same state if the relation α = βµ + β ∗ ν is fulfilled. Several experiments have demonstrated the generation of squeezed light. Slusher et al. [78.9] obtained squeezing in the sidemodes of a four-wave mixing process. An optical parametric oscillator below threshold has been used by Wu et al. [78.10] in order to generate squeezed light. Nonclassical features can also be found in a down conversion process. This second-order process creates so-called signal and idler photons from one pump photon. Signal and idler beam are distinguished by frequency or po-

1145

Part F 78.2

|, β = S()D(β)|0 ,

78.2 Field States

Wn 0.1 r 0

0.05

0 30

50 70 90

5 n

Fig. 78.2 The photon number distribution Wn of a squeezed state |α, r with the coherent amplitude α = 7. For a squeezing parameter r = 0, the Poisson distribution of a coherent state |α = 7 is just visible. When r increases the photon distribution first becomes sub-Poissonian and then oscillatory

1146

Part F

Quantum Optics

Later treatments [78.13] rely on the Hermitian operators

1  † φ = sin eiφ −  eiφ , (78.50) 2i

†
φ = 1  eiφ +  cos eiφ , (78.51) 2 for the sine and cosine function of the phase.

Recently [78.14] a Hermitian phase operator was constructed starting from the phase state (78.47), restricted to a finite Hilbert space. An operational phase description has been proposed [78.15] in which a classical phase measurement is translated to the quantum realm by using an eight-port homodyne detector.

78.3 Quantum Coherence Theory This section introduces the correlation functions of the electromagnetic field. Ideal photon correlation measurements can bring out the phenomenon of photon bunching and antibunching.

78.3.1 Correlation Functions Correlation functions were originally introduced to describe an ideal photodetection process. Glauber [78.16] has presented a treatment based on an absorption mechanism in the detector which is sensitive to the positive frequency part E + of the electric field evaluated at the detector’s space-time position x ≡ (x, t). This leads to an average field intensity   (78.52) I(x) = Tr ρE − (x)E + (x)

Part F 78.3

at point x. Here the density operator ρ describes the state of the field. The ordering of the operators, i. e., E − E + ∼ a† a, is known as normal ordering with all annihilation operators to the right of all creation operators. The expression (78.52) now immediately generalizes to the correlation function of first order   G (1) (x1 , x2 ) = Tr ρE − (x1 )E + (x2 ) , (78.53) with x1 = (x1 , t1 ) and x2 = (x2 , t2 ). The classical interference experiments, such as Young’s double slit experiment, can be described in terms of G (1) . Furthermore, the correlation function of first order is connected to the power spectrum S(ω) of a quantized field via the Wiener–Khintchine theorem [78.17]. Under the assumption of a stationary process, i. e., when the au tocorrelation function E − (t)E + (t  ) depends only on the time difference τ = t − t  , then ∞   1 dτ E − (τ)E + (0) e−iωτ + c.c . S(ω) = 2π 0

(78.54)

This relation between the spectrum and the first-order correlation function is known as Wiener–Khintchine theorem.

In order to analyze the Hanbury–Brown and Twiss experiment [78.18] it is necessary to define higher order correlation functions. The general nth order correlation function is defined by G (n) (x1 , . . . , xn , xn+1 , . . . , x2n )  = Tr ρE − (x1 ) · · · E − (xn )  × E + (xn+1 ) · · · E + (x2n ) ,

(78.55)

where the field operators are again normal ordered. These correlation functions fulfill a generalized Schwartz inequality  2 G (1) (x1 , x1 )G (1) (x2 , x2 ) ≥ G (1) (x1 , x2 ) , (78.56)

which becomes, for the nth order functions, G (n) (x1 , .., xn , xn , .., x1 ) × G (n) (xn+1 , .., x2n , x2n , .., xn+1 ) 2  ≥ G (n) (x1 , .., xn , xn+1 , .., x2n ) .

(78.57)

A field is said to be first-order coherent when its normalized correlation function g(1) (x1 , x2 ) = 

G (1) (x1 , x2 )

1/2 G (1) (x1 , x1 )G (1) (x2 , x2 ) (78.58)

  satisfies g(1) (x1 , x2 ) = 1. In a Young type experiment, this case gives maximum fringe visibility. A more general definition of first order coherence is the condition that G (1) (x1 , x2 ) factorizes G (1) (x1 , x2 ) = G∗ (x1 )G(x2 ) ,

(78.59)

where G denotes some complex function. This definition can be readily generalized to the nth order case. The nth order coherence applies when the relation G (n) (x1 , . . . , x2n ) = G∗ (x1 ) · · · G∗ (xn )G(xn+1 ) · · · G(x2n )

(78.60)

Quantized Field Effects

holds. A field in a coherent state possesses nth order coherence.

78.3.2 Photon Correlations The Young experiment demonstrates the appearance of first-order correlations. However, experiments that can distinguish between the classical and quantum domains have to be based on measurements of second-order correlations. These experiments are of the Hanbury–Brown and Twiss type, and determine the arrival of a photon at detector position x and time t and another photon at time t + τ. Following the theory of Glauber, the second-order correlation function   G (2) (τ) = E − (t)E − (t + τ)E + (t + τ)E + (t) (78.61)

is measured. In this formula we have omitted the variable for the position x. Usually the normalized correlation function g(2) (τ) = 

G (2) (τ) 2 G (1) (0)

(78.62)

is introduced. The function g(2) is always positive, which is true for classical as well as for quantum fields; but there exists a purely quantum domain given by 0 ≤ g(2) (0) < 1 .

(78.63)

78.4 Photodetection Theory

1147

For example, for a number state |n, (2) (0) = 1 − 1/n , g|n

(78.64)

with n ≥ 1. In contrast, a coherent state |α yields (2) (0) = 1. g|α

78.3.3 Photon Bunching and Antibunching In a realistic theory (but not in an oversimplified one-mode model), the correlation function G (2) (τ) always factorizes on a sufficiently long time scale, and g(2) (τ) → 1. The photons are then no longer correlated, and they arrive randomly as in the case of coherent light; see for example (78.195). If g(2) (0) > 1, the photons show a tendency to arrive in bunches, an effect known as photon bunching. This effect has been observed for chaotic light. The opposite situation with 0 ≤ g(2) (0) < 1 demonstrates the reverse effect, namely photon antibunching. As seen from (78.63), this is a regime only accessible to nonclassical light. An example is given by the resonance fluorescence of a two-level atom, treated in Sect. 78.13. Note that we can rewrite g(2) (0) with the help of Mandel’s Q parameter  † †  a a aa (2) = 1+ Q . (78.65) g (0) = a† a2 Hence, a field state with Q < 0 shows the effect of photon antibunching.

bution contains the integrated intensity operator t+T I =η dt  E − (t  )E + (t  ) (78.67)

So far we have used a very simple theory of photodetection: any absorbed photon leads to a photoelectric emission which can be observed. But in any real experiment, these photons are counted over some time interval T and the observed photoelectric emissions are dominated by two statistics: (i) the statistics of photoelectric emission which is also present for a classical field and (ii) the specific quantum statistics of a quantized field. A detailed discussion of the quantum theory of photoelectric detection has been given by Kelley and Kleiner [78.19]. A central result is the formula  In  p(n, t, T) = : exp(−I) : (78.66) n!

containing the quantum efficiency η of the detector. The notation : · · · : indicates a quantum average where the operators have to be normally ordered and time ordered. This operator ordering reflects the process on which a photodetector is based. It annihilates or absorbs photons, one after the other. A good treatment of photoelectric detection can be found in [78.20].

for the probability of counting n photoelectrons in the time interval from t to t + T . This photocounting distri-

These detection methods allow the extraction of specific quantum features of a single mode quantum field,

t

78.4.1 Homodyne and Heterodyne Detection

Part F 78.4

78.4 Photodetection Theory

1148

Part F

Quantum Optics

the signal field. Figure 78.3 summarizes the principle of optical homodyning. Two quantum fields described by the annihilation operators a and b are mixed at a 50/50 beam splitter BS. Both fields have the same frequency. The mode a represents the signal mode whose quantum state is given by the density operator ρ. Mode b serves as a reference field, the local oscillator. The coherent state |α = ||α| eiθ  determines the quantum state of the local oscillator. Two ideal photodetectors 1 and 2 measure the number of photons in the output modes of the beam splitter. For a highly excited coherent state, i. e., a classical local oscillator, the statistics of the photocurrent difference ∆I can be described by the moments of the signal mode operator 1  X θ = √ a e−iθ + a† eiθ . 2

(78.68)

– ⌬I

2

BS a

1

ρ b

ⱍα⬎

Fig. 78.3 The principle of optical homodyning

For example, the photocurrent difference ∆I ∼ X θ  = Tr(ρX θ )

(78.69)

is proportional to the expectation value of X θ . In particular, for θ = 0 and θ = π2 one is able to measure all the moments of the two quadratures x and p (78.13) and (78.14) of the signal mode. In general, the statistics of the photocurrent ∆I reveal the probability distribution Pr(X θ ) = X θ |ρ|X θ 

(78.70)

of the observable X θ when the signal mode is in the state ρ. The states   |X θ  = π −1/4 exp − X 2θ /2 ∞
1 (78.71) Hn (X θ ) einθ |n × √ n n! 2 n=0 are eigenstates of the operator X θ , and are known as rotated quadrature states. The heterodyne technique [78.21,22] relies on a similar mixing of a signal field with a local oscillator at a beam splitter, but this time the local oscillator frequency is offset by the intermediate frequency ∆ω with respect to the frequency ω0 of the signal mode. Filters select the beat frequency components in the photocurrent of the detectors. This photocurrent contains the quantum statistics of the two quadratures of the signal field [78.21, 22].

Part F 78.5

78.5 Quasi-Probability Distributions Quasi-probability distributions play an important role in quantum optics for three reasons. First, they are a complete representation of the density operator of a quantum field. Second, they allow one to calculate expectation values in the spirit of classical statistical physics. Third, they offer the possibility of converting a master equation for the density operator into an equivalent c-number partial differential equation. In this section, we relate a specific quasi-probability function to a specific operator ordering.

78.5.1 s-Ordered Operators † A normally  †ordered m n product of a and a is a product of the form a a : the annihilation operators a stand to the right of the creation operators a† . In an antinormally

 m ordered product like an a† , the order of a and a† has changed. A generalized s-ordered product can be defined as   n 

  ∂ m ∂ m † n − ∗ a a ≡ s ∂ξ ∂ξ   ∗ × D(ξ, ξ , s) ∗ (78.72) ξ=ξ =0

with the generalized displacement operator   1 ∗ ∗ † ∗ D(ξ, ξ , s) ≡ exp ξa − ξ a + sξξ . 2

(78.73)

For s = 1 we find again normal ordering. The values s = 0 and s = −1 produce symmetric and antinormal ordered products. As an example we note {a† a}s = a† a − (s − 1)/2.

Quantized Field Effects

Expectation values of those s-ordered products are easily derived from the characteristic function   (78.74) χ(ξ, ξ ∗ , s) = Tr ρD(ξ, ξ ∗ , s) via differentiation   n   ∂ m ∂  χ(ξ, ξ ∗ , s) ∗ − ∗ ξ=ξ =0 ∂ξ ∂ξ = {(a† )n am }s  .

(78.75)

The Fourier transform of χ yields the quasi-probability distribution of Cahill and Glauber [78.23]  1 ∗ ∗ d2 ξχ(ξ, ξ ∗ , s) eαξ −α ξ , (78.76) W(α, s) = 2 π where d2 ξ = d Re(ξ) d Im(ξ). With this distribution one is able to calculate the expectation value of any s-ordered operator product     n a† am = (α∗ )n αm W(α, s) d2 α . (78.77) s

We concentrate now on three important quasiprobability distributions, namely the cases s = 1, s = 0, and s = −1, corresponding to the Glauber–Sudarshan distribution, the Wigner function, and the Q function respectively. A detailed discussion of these three functions can be found in [78.24].

78.5.2 The P Function

in terms of coherent states |α, (78.26). It is related to the Cahill–Glauber function W(α, s) via P(α) = W(α, s = 1) .

(78.79)

The expectation value of any normally ordered operator product   † m n  a a = (α∗ )m αn P(α) d2 α (78.80) has a particularly simple form in terms of P(α). From (78.78) the P function of a coherent state |α0  becomes P(α) = δ

(2)



(α − α0 )

(78.81)

   = δ Re(α) − Re(α0 ) δ Im(α) − Im(α0 ) .

Quantized fields for which the P function is positive do not show nonclassical effects such as squeezing and antibunching. For nonclassical states, such as number states or squeezed states, the P function only exists in terms of

1149

generalized functions, such as delta functions and their derivatives, which have a highly singular character. The positive P-representation P(α, β) for a nondiagonal decomposition of a density operator is [78.27, 28]  |αβ ∗ | P(α, β) . (78.82) ρ= d2 α d2 β ∗ β |α The function P(α, β) is a direct generalization of the Glauber–Sudarshan function P(α), (78.78). P(α, β) exists for any physical density operator ρ [78.27] and is given by   1 1 P(α, β) = 2 exp − |α − β ∗ |2 4 4π       1 ∗  1 ∗ × (α + β ) ρ  (α + β ) . (78.83) 2 2 ∗ Here the state |1/2(α + β ) denotes a coherent state with the complex amplitude 1/2(α + β ∗ ). Note that P(α, β) is always positive.

78.5.3 The Wigner Function This quasi-probability was first introduced by Wigner [78.29] and may be defined as the distribution function for a symmetrically ordered operator product which is obtained in the case s = 0. The Wigner function plays an important role in other branches of physics, such as quantum chaology, and in particular in any semiclassical phenomenon when one considers the transition from quantum mechanics to classical mechanics. Consider the Wigner function of a quantum mechanical particle of position  ~  a + a† , x= (78.84) 2mω and momentum  m ~ω  † p=i a −a . (78.85) 2 The Wigner function may be written in terms of position and momentum variables ∞ 1 dy e−2i py/~ x + y|ρ|x − y , W(x, p) = π~ −∞

(78.86)

where |x ± y denotes position eigenstates. The s = 0 Cahill–Glauber definition and the above definition of the Wigner function are related by   1 mωx + i p W(x, p) = W α = √ , s = 0 . (78.87) 2 2m ~ω

Part F 78.5

The quasi-probability function P(α) was introduced [78.25, 26] as a diagonal representation of the density operator  (78.78) ρ = P(α)|αα| d2 α

78.5 Quasi-Probability Distributions

1150

Part F

Quantum Optics

The position and momentum distributions of a particle, or equivalently the quadruture distributions in the case of a quantized field mode, are ∞ Pr(x) =

W(x, p) d p ,

(78.88)

  p ∂ ∂V ∂ − + W(x, p) m ∂x ∂x ∂ p  
1 i~ r−1 ∂ r V ∂ r W + =0. r! 2 ∂x r ∂ pr

(78.95)

r=3,5,..

−∞ ∞

Pr( p) =

and

W(x, p) dx .

(78.89)

−∞

Furthermore, the scalar product  2 dx d p W|ψ1  (x, p) |ψ1 |ψ2 | = 2π ~ × W|ψ2  (x, p)

The Wigner function has negative parts for most quantum states. For example, the Wigner function of a Fock state |n, W|n (x, ¯ p) ¯ =

(−1)n −x¯2 − p¯2  2 e L n 2x¯ + 2 p¯2 , π

(78.96)

(78.90)

of two quantum states is expressed by the phase space overlap of the two corresponding Wigner functions. Consequently, any Wigner function W(x, p) has to obey the necessary condition  dx d pW(x, p)W|ψ (x, p) ≥ 0 (78.91) for all W|ψ representing a pure state. For a normalized state |ψ,  2  1 . (78.92) dx d p W|ψ (x, p) = 2π ~

Part F 78.5

Instead of solving the Schrödinger equation for the dynamics of a massive particle in a potential V(x), we can try to solve the equation
1  i~ r−1 ∂ r V ∂ r W p ∂W ∂W =− + ∂t m ∂x r! 2 ∂x r ∂ pr

clearly becomes negative due to the oscillating Laguerre polynomial L n as shown in Fig. 78.4. Note that √we have introduced the dimensionless position x = mω/~ x ¯ √ and momentum p¯ = 1/ mω~ p. On the other hand, the Wigner function

√  2 1 W|α,r (x, ¯ p) ¯ = exp − e2r x¯ − 2 Re(α) π √  2  − e−2r p¯ − 2 Im(α) (78.97) of a squeezed state (78.35) is always positive as shown in Fig. 78.5. It is a long thin ellipse in phase space (i. e. a Gaussian cigar). Concerning the negative parts of the Wigner function, the Hudson theorem [78.30] states that a necessary and sufficient condition for the Wigner function of a pure state |ψ to be nonnegative is that it can be described by a wave function of the

r=1,3,..

(78.93)

for its Wigner function W(x, p, t). Note that here only the odd derivatives of the potential V enter. This equation is the quantum analogue of the classical Liouville equation, to which it reduces in the limit of ~ → 0. However, the initial distribution W(x, p, t = 0) has to be a Wigner function in the sense of (78.86). Furthermore the Wigner function of an energy eigenfunction in the potential V(x) may be obtained from the equations   2 p ~2 ∂ 2 ~2 ∂ 2 V ∂ 2 W(x, p) + V(x) − − 2m 8m ∂x 2 8 ∂x 2 ∂ p2  
1 i~ r ∂ r V ∂ r W + = EW(x, p) , r! 2 ∂x r ∂ pr r=4,6,..

(78.94)

W(x, p)

0.2 0.1 0 p 2 –2

0 0 –2 2 x

Fig. 78.4 The Wigner function (78.96) of a Fock state

|n = 4. The negative parts can be seen clearly

Quantized Field Effects

78.6 Reservoir Theory

1151

78.5.4 The Q Function The Q function is defined by the diagonal matrix elements Q(α) = α|ρ|α/π W(x, p) 0.4 0.2

–2 0 p

0 1

2 0 4

(78.101)

of the density operator ρ, where |α denotes a coherent state. The Q(α) function is always a positive and bounded function, which exists for any density operator ρ. The Q function is also known as Husimi’s function. It allows one to calculate expectation values of antinormally ordered operator products of the form     m (78.102) an a† = d2 ααn (α∗ )m Q(α) .

–1 x

Fig. 78.5 The Wigner function of a squeezed state |α, r with coherent amplitude α = 1 and squeezing e2r = 0.25. For these values the phase space ellipse is oriented along the x-axis and squeezed in the p-direction. Note that this function is positive everywhere

Moreover, since the Q function corresponds to the case s = −1 of the Cahill–Glauber distribution, Q(α) = W(α, −1) .

(78.103)

78.5.5 Relations Between Quasi-Probabilities

form

  1 x|ψ = exp − (ax 2 + bx + c) . 2

In general, the relation (78.98)

Here a, b and c denote some constants with Re(a) > 0. Finally, the Kirkwood distribution function ∞

dy e−2i py/~ x|ρ|x − 2y (78.99)

−∞

is a phase space function that resembles the Wigner function. In the case of a pure state ρ = |ψψ| this function reduces to ˜ p) e−ix p/~ , K(x, p) = ψ(x)ψ(

(78.100)

˜ p) denotes the Fourier transform of ψ(x). where ψ(

holds between two Cahill–Glauber distributions with the parameters s > s. In particular, the non-negative Q function  2 d2 β exp[−2|α − β|2 ]W(β, s = 0) Q(α) = π (78.105)

turns out to be a smoothed Wigner function W(β, s = 0). It is this smoothing process that washes out possible negative parts in the Wigner function.

78.6 Reservoir Theory Reservoir theory treats the interaction of one system with a few degrees of freedom, called the system, with another system with many degrees of freedom, called the reservoir. A typical application of reservoir theory is a microscopic theory of damping: the system interacts with a reservoir, called the heat bath. The system

dissipates energy into the heat bath whereas the heat bath introduces additional fluctuations to the system. Since the present chapter focuses on quantized field effects, the reservoir consists of the many modes of the radiation field in free space. Such a reservoir is modeled by a large number of independent harmonic

Part F 78.6

1 K(x, p) = π~

   2|α − β|2 2 2 d β exp −  W(α, s) = π(s − s) s −s  (78.104) × W(α, s )

1152

Part F

Quantum Optics

oscillators  
1 † , ~ωi bi bi + Hr = 2

are (78.106)

i

† bi

where bi and are the annihilation and creation operators for the ith harmonic oscillator of the reservoir. For convenience the interaction with the system is frequently approximated by a Hamiltonian of the form


† gi A bi + gi∗ A† bi , (78.107) Hint = ~ i

where A is an operator of the small system and gi is the coupling strength of this system to the ith oscillator of the reservoir. For example, A may be an annihilation operator if the system is a harmonic oscillator or a Pauli spin matrix in the case of a two-level atom coupled to the free space radiation field. Reservoir theory has important applications, and a detailed discussion can be found in various books, for example [78.17, 20, 27, 28, 31–33].

78.6.1 Thermal Reservoir The most commonly used reservoir is the thermal reservoir or thermal heat bath. Its characteristic properties

 †  † † bi  = bi = bi b j  = bi b j = 0 ,  †  bi b j = n i δij .

(78.108) (78.109)

Here ni =

1 exp (~ωi /kB T ) − 1

(78.110)

is the average number of photons at frequency ωi , T is the temperature of the reservoir, and kB denotes the Boltzmann constant.

78.6.2 Squeezed Reservoir Another example of a reservoir is a squeezed vacuum or squeezed reservoir. If, for example, multiwave mixing is used to squeeze the radiation field, conjugate pairs of the reservoir operators b are correlated.  † † Therefore, the expectation values bi b j  and bi b j may be nonvanishing. Apart from the average number n i of photons at frequency ωi , which  † take  into account nonvanishing expectation values bi bi , additional complex squeezing parameters are needed to describe the reservoir [78.28, 33, 34]. The characterization of a squeezed reservoir based on noise operators is discussed in Sect. 78.10.

Part F 78.7

78.7 Master Equation In quantum mechanics, density operators are used to describe mixed states, and are discussed in Chapt. 7. Here we introduce the concept of the reduced density operator ρs = Trr (ρsr ) ,

(78.111)

which is the density operator ρsr of the complete system traced over the degrees of freedom of the reservoir. The equation of motion for ρs in the Schrödinger picture is i ρ˙ s (t) = − Trr {[Hsr , ρsr (t)]} . ~

(78.112)

In the Born–Markov approximation the trace over the reservoir can be evaluated and leads to an equation of motion for ρs which no longer contains reservoir operators. This equation of motion is usually called the master

equation. The Born–Markov approximation consists of two different parts: 1. Born approximation: The coupling to the reservoir is assumed to be sufficiently weak to allow a perturbative treatment of the interaction between the reservoir and the system. 2. Markov approximation: The correlations of the reservoir are assumed to decay very rapidly on a typical time scale of the system, or equivalently, the reservoir has a very broad spectrum. This approximation involves the assumption that the modes of the reservoir are spaced closely together, so that the frequency ωi is a smooth function of i. Since a general treatment is rather technical, we consider two typical examples. A more general discussion can be found in [78.17, 20, 27, 28, 31–33]

Quantized Field Effects

78.7.1 Damped Harmonic Oscillator The universally accepted Hamiltonian in nonrelativistic QED for a harmonic oscillator of frequency ω coupled to a reservoir consisting of a large number of harmonic oscillators is given by the total Hamiltonian [78.35, 36]  
  1 1 † † Hsr = ~ω a a + + ~ωi bi bi + 2 2 i

+ Hlc + Hsi ,

(78.113)

with the linear coupling term




† Hlc = ~ gi a + a† bi + bi ,

(78.114)

i

and the self-interaction term

2
~ g2 i a + a† . Hsi = ωi

(78.115)

i

The approach used in quantum optics is to drop the term Hsi and to make the rotating-wave approximation, that † is, to drop the terms a bi and a† bi , see also Chapt. 68. Then the approximate total Hamiltonian reads  
  1 1 + ~ωi bi† bi + Hsr = ~ω a† a + 2 2 i


† +~ gi a bi + a† bi . (78.116) i

Harmonic Oscillator in a Thermal Bath Within the Born–Markov approximation the master equation is

1 ρ˙ = γ (n + 1) 2aρa† − a† aρ − ρa† a 2

1 † (78.117) + γ n 2a ρa − a a† ρ − ρa a† , 2 where

ρ(t) = eiωa

† a(t−t

0)

ρs (t) e−iωa

† a(t−t

0)

(78.118)

is the reduced density operator in the interaction picture. The damping constant γ is given by γ = 2πD(ω)|g(ω)|2 ,

(78.119)

1153

where g(ω) denotes the coupling strength at frequency ω. The number of thermal photons at frequency ω is 1 . n= (78.120) exp (~ω/kB T ) − 1 Thus the Born–Markov approximation replaces the discrete reservoir modes by a continuum of modes with a density D(ω). Harmonic Oscillator in a Squeezed Bath Within the Born–Markov approximation, the reduced density operator (78.118) in the interaction picture satisfies the master equation

1 ρ˙ = γ (n + 1) 2aρa† − a† aρ − ρa† a 2

1 + γ n 2a† ρa − a a† ρ − ρa a† 2

1 − γ m 2a† ρa† − a† a† ρ − ρa† a† 2 1 (78.121) − γ m ∗ (2aρa − a aρ − ρa a) . 2 Here γ is again given by (78.119). The squeezed reservoir is characterized by a real number n and a complex number m. Physically n is the number of photons at frequency ω, i. e., similar to the thermal reservoir, it measures the average energy at frequency ω. The complex number m determines the amount of squeezing. In general, the positivity of the density operator requires

|m|2 ≤ n (n + 1) .

(78.122)

A more quantitatively definition of n and m in terms of noise operators is given in Sect. 78.10.

78.7.2 Damped Two-Level Atom The interaction of a two-level atom with a classical electromagnetic field is already discussed in Chapt. 68. For a quantum mechanical treatment of the field we only have to replace the classical field by its quantum mechanical counterpart (78.9). We then find in the rotating-wave approximation (Chapt. 68), that the dynamics of a two-level atom with a transition frequency ω0 coupled to a reservoir consisting of a large number of harmonic oscillators is approximately described by the total Hamiltonian  
1 1 Hsr = ~ω0 σz + ~ωi bi† bi + 2 2 i


† gi σ− bi + gi∗ σ+ bi , (78.123) +~ i

Part F 78.7

Despite the problems with this approximate Hamiltonian (see Sect. V.D of [78.35,36] for a discussion) we adopt it in the present context because it leads to the widely used master equation for the damped harmonic oscillator. We consider two reservoirs: a thermal bath and a squeezed bath.

78.7 Master Equation

1154

Part F

Quantum Optics

where

 0 σ+ = 0  1 σz = 0

   1 0 0 , σ− = , 0 1 0  0 . −1

where (78.124)

(78.125)

Again, two reservoirs are considered: a thermal bath and a squeezed bath. Two-Level Atom in a Thermal Bath Within the Born–Markov approximation, the master equation is 1 ρ˙ = γ (n + 1) (2σ− ρσ+ − σ+ σ− ρ − ρσ+ σ− ) 2 1 + γ n (2σ+ ρσ− − σ− σ+ ρ − ρσ− σ+ ) , 2 (78.126)

ρ(t) = eiω0 σz (t−t0 )/2 ρs (t) e−iω0 σz (t−t0 )/2

(78.127)

is the reduced density operator in the interaction picture, and γ and n are given by (78.119) and (78.120). Two-Level Atom in a Squeezed Bath Within the Born–Markov approximation, the reduced density operator in the interaction picture, (78.127), satisfies the master equation

1 ρ˙ = γ (n + 1) (2σ− ρσ+ − σ+ σ− ρ − ρσ+ σ− ) 2 1 + γ n (2σ+ ρσ− − σ− σ+ ρ − ρσ− σ+ ) 2 − γ mσ+ ρσ+ − γ m ∗ σ− ρσ− , (78.128) where γ , n and m have the same meaning as in (78.121).

78.8 Solution of the Master Equation 78.8.1 Damped Harmonic Oscillator We consider only a thermal reservoir and present the solution of the master equation (78.117). For n = 0 it can be solved in terms of coherent states, see (78.26). For n = 0 we give solutions in terms of quasi-probability distributions.

Part F 78.8

Coherent States For n = 0, which is a good approximation for optical frequencies, if the system is initially in a coherent state |α0  with a density operator

ρ(t0 ) = |α0 α0 | ,

(78.129)

then there exists a simple analytical solution of the master equation (78.117) ρ(t) = |α0 e−γ (t−t0 )/2 α0 e−γ (t−t0 )/2 | .

(78.130)

A coherent state thus remains a coherent state with an exponentially decaying amplitude α0 e−γ (t−t0 )/2 . According to (78.78) a general solution     ρ(t) = d2 α0 P(α0 )α0 e−γ (t−t0 )/2 α0 e−γ (t−t0 )/2  (78.131)

can be constructed for an initial density operator  (78.132) ρ(t0 ) = d2 α0 P(α0 )|α0 α0 | .

If the system is initially in a superposition
|ψ(t0 ) = ci |αi 

(78.133)

i

of coherent states, the time evolution is given by  


1 ∗ −γ(t−t0 ) 2 |αi − αk | ci ck exp − 1 − e ρ(t) = 2 i,k

   × exp i 1 − e−γ(t−t0 ) Im αi αk∗    (78.134) × αi e−γ (t−t0 )/2 αk e−γ (t−t0 )/2  . For γ(t − t0 )  1, the interference terms |αi αk |, i = k decay with an effective decay constant γ |αi − αk |2 /2. Thus the damping constant is modified by the separation of the two coherent states in phase space. Fokker–Planck Equation A widely used procedure for solving the master equation for a damped harmonic oscillator, (78.117), or for similar problems, is to derive an equation of motion for the quasi-probability distributions W(α, α∗ ; s) defined in (78.76) from the master equation. The operators a and a† are replaced by appropriate differential operators. The substitution rules can be derived

Quantized Field Effects

from (78.73), (78.74) and (78.76) and are   k s+1 ∂ k a † a  ρ → α∗ − 2 ∂α   s−1 ∂  × α− W, 2 ∂α∗   k s+1 ∂  †  ρ a a → α− 2 ∂α∗   s−1 ∂ k ∗ × α − W, 2 ∂α   s−1 ∂ aρa† → α − 2 ∂α∗   s−1 ∂ W, × α∗ − 2 ∂α   s+1 ∂ a† ρa → α∗ − 2 ∂α   s+1 ∂ W, × α− 2 ∂α∗   s+1 ∂ a† ρa† → α∗ − 2 ∂α   s −1 ∂ W, × α∗ − 2 ∂α   s−1 ∂ aρa → α − 2 ∂α∗   s+1 ∂ W. × α− 2 ∂α∗

1155

The time-dependent solution of this Fokker–Planck equation has the form  G(α, α∗ , t|α , α∗ , t  ; s) W(α, α∗ , t; s) = × W(α , α∗ , t  ; s) d2 α ,

(78.138)

where G(α, α∗ , t| α , α∗ , t  ; s)     α − α e−γ (t−t  )/2 2  exp −   n s 1 − e−γ(t−t )

=  πn s 1 − e−γ(t−t )

(78.139)

is the Green’s function of the Fokker–Planck equation (78.136). The steady-state solution is W(α, α∗ , t → ∞; s) =

1 −|α|2 /n s e , πn s

(78.140)

which is the distribution function of a harmonic oscillator in thermal equilibrium with a reservoir of temperature T .

78.8.2 Damped Two-Level Atom

(78.135)

(78.136)

where 1 1−s 1−s = + . 2 exp (~ω/kB T ) − 1 2 (78.137)

The density operator   ρee ρeg ρ= ρge ρgg for a two-level atom can be written as     1 1 + σz  σ−  2   . ρ= 1 σ+  2 1 − σz 

(78.141)

(78.142)

Thus, a two-level atom is completely described by the expectation values σz  = ρee − ρgg , σ+  = ρge , σ−  = ρeg .

(78.143)

Hence the master equation (78.128) can be cast into the equations of motions for these expectation values   d 1 σ+  = −γ n + σ+  − γ m ∗ σ−  , dt 2

Part F 78.8

In general, this procedure leads to equations of motion which involve higher derivatives of W as exemplified by the quantum mechanical Liouville equation (78.93) for the Wigner function. For simple Hamiltonians, however, this equation has the form of a Fokker–Planck equation which is well known in classical stochastic problems [78.27, 37] (Sect. 78.10.2). In particular, for a damped harmonic oscillator described by the master equation (78.117), one obtains   γ ∂ ∂  ∗ ∂W ∂2 W = , (αW) + ∗ α W + γ n s ∂t 2 ∂α ∂α ∂α∂α∗

ns = n +

78.8 Solution of the Master Equation

1156

Part F

Quantum Optics

  1 d σ−  = −γ n + σ−  − γ mσ+  , dt 2   d 1 σz  = −2γ n + σz  − γ , (78.144) dt 2

which can easily be solved for arbitrary initial conditions. In contrast to a thermal reservoir (m = 0), a squeezed reservoir results  in two different decay constants γ n + 12 + |m| and  transverse γ n + 12 − |m| [78.34].

78.9 Quantum Regression Hypothesis In the Schrödinger picture, time-dependent expectation values for system operators A j can be calculated from the reduced density operator ρs (t) via   A j  = Trs A j ρs (t) . (78.145) The reduced density operator, however, is not sufficient to calculate two-time correlation functions such as A j (t + τ)Ak (t). For a definition of two-time correlation functions, the Heisenberg picture is more appropriate. Here, expectation values follow from   A j  = Trsr Usr† (t, t0 )A j (t0 )Usr (t, t0 )ρsr (t0 ) , (78.146)

where Usr (t, t0 ) describes the unitary time evolution of the complete system and ρsr (t0 ) is the density operator in the Heisenberg picture. Similarly, two-time correlation functions such as A j (t + τ)Ak (t) can be defined as A j (t + τ)Ak (t)  = Trsr Usr† (t + τ, t0 )A j (t0 )Usr (t + τ, t0 )  × Usr† (t, t0 )Ak (t0 )Usr (t, t0 )ρsr (t0 ) .

(78.147)

Part F 78.9

The quantum regression hypothesis avoids the calculation of Usr (t, t0 ). Two equivalent formulations exist, one based on the master equation for ρs and another based on the equation of motion for the expectation values A j , see for example [78.20, 27, 28, 31, 33].

78.9.1 Two-Time Correlation Functions and Master Equation It follows from their definition (78.147) in the Heisenberg picture that two-time correlation functions A j (t + τ)Ak (t) for system operators A j and Ak can be calculated with the help of the operator  Rs (t + τ, t) = Trr Usr (t + τ, t)  (78.148) × Ak ρsr (t)Usr† (t + τ, t) , where Usr (t + τ, t) describes the unitary time evolution of the complete system between t and t + τ. We find   A j (t + τ)Ak (t) = Trs A j Rs (t + τ, t) . (78.149)

Note, that in (78.148) and (78.149) we interpret A j and Ak as operators in the Schrödinger picture and have omitted the argument t0 . Because the reduced density operator   (78.150) ρs (t) = Trr Usr (t, t0 )ρsr (t0 )Usr† (t, t0 ) satisfies the master equation, it is plausible to assume, that when the time derivative is taken with respect to τ, the operator Rs (t + τ, t) also satisfies the master equation for ρs , subject to the initial condition Rs (t, t) = Ak ρs (t). However, this requires the additional assumption that the approximations made in the derivation of the master equation for ρs (t) are also valid for Rs (t + τ, t).

78.9.2 Two-Time Correlation Functions and Expectation Values A second formulation of the quantum regression hypothesis asserts that two-time correlation functions A j (t + τ)Ak (t) obey ∂ A j (t + τ)Ak (t) ∂τ
G j (τ)A (t + τ)Ak (t) , (78.151) = 

provided that the expectation values of a set of system operators A j satisfy
∂ A j (t) = G j (t)A (t) . (78.152) ∂t 

This is the form of the quantum regression hypothesis that was first formulated by Lax [78.38]. The equivalence of the two formulations follows from the interpretation of Rs (t + τ, t) on the right side of (78.149) as a “density operator”. Then  Trs A j Rs (t + τ, t) is an “expectation value” for which we assume that (78.152) is valid; i. e.,   ∂ Trs A j Rs (t + τ, t) ∂τ
  G j (τ)Trs A Rs (t + τ, t) . (78.153) = 

According to (78.149), this is identical to (78.151).

Quantized Field Effects

78.10 Quantum Noise Operators

1157

78.10 Quantum Noise Operators The master equation is based on the Schrödinger picture in quantum mechanics: the state of the system described by a density operator is time-dependent, whereas operators corresponding to observables are time independent. If we use the Heisenberg picture instead and make similar approximations as in the derivation of the master equation, we arrive at equations of motion for the Heisenberg operators, see for example [78.17,28,31,32]. Due to the interaction with a reservoir these equations have additional noise terms and damping terms.

78.10.1 Quantum Langevin Equations Again consider a damped harmonic oscillator. The equation of motion for the annihilation operator a(t) ˜ = eiω(t−t0 ) a(t)

(78.154)

in the interaction picture follows from the Heisenberg † equations for the operators a, a† , bi and bi and reads t
da˜  2 =− |gi | e−i(ωi −ω)(t−t ) a(t ˜  ) dt  dt i t0
−i gi∗ e−i(ωi −ω)(t−t0 ) bi (t0 ) . (78.155) i

In general, the noise operator
gi∗ e−i(ωi −ω)(t−t0 ) bi (t0 ) F(t) = −i

(78.156)

i

da˜ γ = − a(t) ˜ + F(t) , dt 2

(78.157)

with a damping term −γ a(t)/2 and a noise term F(t). ˜ Note that a simple damping equation such as γ ˙˜ = − a(t) a(t) (78.158) ˜ 2

F(t)F † (t  ) = γ(n + 1)δ(t − t  ) ,

(78.159)

where the averages are taken over the reservoir. The damping constant γ and the number of thermal photons n are given in (78.119) and (78.120). For more general relations, see [78.35, 36], where it is shown explicitly that correlation functions involving the fluctuation force do not in fact depend on the oscillator frequency. The condition of a sufficiently small reservoir correlation time requires that τ c ≈ ~/(kB T ) is small compared with the time scales of the systems. The only time scale in (78.157) is γ −1 . The relevant condition is therefore τc  γ −1 . For typical applications in quantum optics, a is the annihilation operator and a† is the creation operator of a single-mode cavity field. Here one can have quality factors of the cavity on the order of Q = ω/γ ≈ 106 . In terms of the quality factor, the condition of sufficiently small reservoir correlation times  requires ~ω/(kB T )  Q. For optical frequencies ω ≈ 3 × 1015 Hz) and T ≈ 300 K one has ~ω/(kB T ) ≈ 75. In the microwave regime (ω ≈ 30 GHz) one can have temperatures as low as T ≈ 3 mK and still have ~ω/(kB T ) ≈ 75. Therefore the assumption of delta-correlated noise is a good approximation for typical applications in quantum optics. Similarly, for a squeezed reservoir one has   F(t) = F † (t) = 0 , F(t)F(t  ) = γ mδ(t − t  ) ,  †  F (t)F † (t  ) = γ m ∗ δ(t − t  ) ,   † F (t)F(t  ) = γ nδ(t − t  ) ,   (78.160) F(t)F † (t  ) = γ(n + 1)δ(t − t  ) , which gives a quantitative definition of the parameters n and m in the master equations (78.121) and (78.128). Again, a detailed discussion in [78.35, 36] shows that correlation functions involving the fluctuation forces do not depend on the oscillator frequency. The Langevin equation (78.157) is based on the use of the approximate Hamiltonian given in (78.116),

Part F 78.10

is not delta-correlated, and there are also memory effects in (78.155). The noise operator F(t) can be used to classify the reservoir: if it is delta-correlated, that is, if the reservoir has a very broad spectrum, one speaks of white noise, see below. If the correlation time is finite so that there are memory effects, one speaks of colored noise. If the spectrum of the noise is very broad (as in the derivation of the master equation for the reduced density operator), the operator a(t) ˜ satisfies the quantum Langevin equation

is unphysical since it does not preserve the commutation   relation a, ˜ a˜† = 1. It is the noise term which saves the commutation relation. For a thermal reservoir with a sufficiently small correlation time, the standard derivations [78.32] give       F(t) = F † (t) = F(t)F(t  ) = F † (t)F † (t  ) = 0 ,   † F (t)F(t  ) = γ nδ(t − t  ) ,

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i. e., it is based on the rotating-wave approximation and the neglect of self-interaction √terms. The correspond ing Langevin equation for x = ~/(2mω) a + a† may be calculated and, not unexpectedly, it disagrees with the Abraham–Lorentz equation which Ford and O’Connell [78.39] showed could be derived systematically using the exact Hamiltonian (78.113). In fact, Ford and O’Connell showed that an improved equation for the radiating electron (improved in the sense that it is second-order and is not subject to the analyticity problems and the problems with runaway solutions associated with the Abraham–Lorentz equation) may be obtained by generalizing the Hamiltonian (78.113) to include electron structure. The implications following from these different equations are presently under study.

78.10.2 Stochastic Differential Equations In Sect. 78.10.1 we discussed one of the simplest quantum systems with dissipation, the damped harmonic oscillator. For more complicated systems the noise term can also contain system operators. In such cases there are two different ways to interpret the Langevin equation. In order to give a feeling for the two possible interpretations, we discuss the one dimensional classical Langevin equation dx = g(x, t) + h(x, t)F(t) dt

(78.161)

Part F 78.10

for the stochastic variable x(t) with delta-correlated noise F(t)F(t  ) = δ(t − t  ). Due to the singular nature of delta-correlated noise, such a Langevin equation does not exist from a strictly mathematical point of view. A mathematically more rigorous treatment is based on stochastic differential equations [78.27, 28, 37]. The variable x(t) is said to obey a stochastic differential equation dx(t) = g(x, t) dt + h(x, t)F(t) dt = g(x, t) dt + h(x, t) dW(t) ,

(78.162)

if, for all times t and t0 , x(t) is given by t x(t) = x(t0 ) +

g(x(t  ), t  ) dt 

t0

t + t0

h[x(t  ), t  ] dW(t  ) .

(78.163)

Here the last term is a Riemann–Stieltjes integral defined by t

  h x(t  ), t  dW(t  )

t0

= lim

n→∞

n−1


  h[x(τi ), τi ] W(ti+1 ) − W(ti ) ,

i=0

(78.164)

where τi is in the interval (ti , ti+1 ). There are two different approaches to such problems: the Ito approach and the Stratonovich approach. They differ in the definition of stochastic integrals.  In the Stratonovich approach, one evaluates h x(τi ), τi at τi = (ti + ti+1 in the Ito   )/2, whereas approach one evaluates h (x(τi ), τi ) at τi = ti . This slightly different definition of τi leads to different results because, as a consequence of the delta-correlated noise term, x(t) is not a continuous path. However, there is a relation between the solution of a Stratonovich stochastic differential equation and an Ito stochastic differential equation. Suppose x(t) is a solution of the Stratonovich stochastic differential equation dx(t) = g(x, t) dt + h(x, t) dW(t) .

(78.165)

Then x(t) satisfies the Ito stochastic differential equation   ∂h(x, t) 1 dt dx(t) = g(x, t) + h(x, t) 2 ∂x + h(x, t) dW(t) . (78.166) Instead of dealing with stochastic differential equations, one can derive a Fokker–Planck equation for the conditional probability P(x, t|x0 , t0 ). For the Stratonovich stochastic differential equation (78.165), the Fokker–Planck equation is   ∂ 1 ∂h(x, t) ∂P =− g(x, t) + h(x, t) P ∂t ∂x 2 ∂x 1 ∂2 2 + h (x, t)P , (78.167) 2 ∂x 2 which takes the form ∂ 1 ∂2 2 ∂P = − g(x, t)P + h (x, t)P , (78.168) ∂t ∂x 2 ∂x 2 if (78.165) is interpreted as a stochastic differential equation in the Ito sense. The two approaches have the following properties: (i) in most of the models used in physics the Stratonovich

Quantized Field Effects

definition of a stochastic integral is needed to give correct results, (ii) rules from ordinary calculus are applicable only in the Stratonovich approach, and (iii) for

78.12 Spontaneous Emission in Free Space

1159

Langevin equations with h(x, t) = const, as in (78.157), the Stratonovich interpretation and the Ito interpretation of stochastic integrals are equivalent.

78.11 Quantum Monte Carlo Formalism The quantum Monte Carlo formalism was developed to solve numerically master equations of the Lindblad type [78.20, 40–44] i ρ˙ s = − [Hs , ρs ] ~

1
† † † 2C j ρs C j − C j C j ρs − ρs C j C j . + 2 j

(78.169)

Here C j are arbitrary system operators. As an illustrative example, consider

1 i ρ˙ s = − [Hs ,ρs ]+ 2Cρs C † −C † Cρs −ρs C † C , ~ 2 (78.170)

where C is an arbitrary system operator. Instead of solving the master equation, one defines quantum trajectories or stochastic wave functions as follows. Starting from |ψ(t), there are two possibilities for the time evolution during the interval dt: 1. The system evolves according to the non-Hermitian Hamiltonian Hs − i2~ C † C; i. e.    1−i dt Hs − i2~ C † C /~ |ψ(t) . |ψ(t+ dt) = 1 − ψ(t)|C † C|ψ(t) dt (78.171)

2. The system makes a jump; i. e. |ψ(t + dt) =

C|ψ(t) ψ(t)|C † C|ψ(t)

.

(78.172)

Since both possibilities describe a nonunitary time evolution, |ψ must be normalized after each step. For each time interval dt one of these two possibilities is randomly chosen according to the probability P(t)dt = ψ(t)|C † C|ψ(t) dt

(78.173)

to make a jump between t and t + dt. We can now define a density operator ρs (t) = |ψ(t)ψ(t)| (78.174) for a specific quantum trajectory |ψ(t). The density operator ρs (t) = |ψ(t)ψ(t)|

(78.175)

averaged over all trajectories (indicated by the bar) is then a solution of the master equation (78.170). This method can easily be generalized to master equations of the form (78.169).

Consider an atom which is initially in one of its excited states and which interacts with the quantized electromagnetic field of free space. Even if none of the modes of the electromagnetic field is excited, there are still the vacuum fluctuations which “interact” with the atom and give rise to important effects: 1. Spontaneous emission: the atom spontaneously emits a photon and decays from the excited state. 2. Natural linewidth: due to the finite lifetime of the atomic levels, the radiation from an atomic transition has a finite linewidth, called the natural linewidth. 3. Lamb shift: the energy levels of the atom are shifted. The standard theory of spontaneous emission is the Wigner–Weisskopf theory [78.17, 32]. Here an initially

excited atomic state | decays exponentially according to |c (t)|2 = e−Γ t , where the decay constant Γ is given by
ω3 |di |2 i Γ = , 3π0 ~c3

(78.176)

(78.177)

i

and the sum is over all atomic states with an energy E i lower than the energy E  of the state |. ωi = (E  − E i )/~ is the transition frequency for the transition | → |i, and di = e|r|i is the corresponding dipole moment. The same decay constant Γ is also observed as a linewidth in the spectrum of the radiation scattered

Part F 78.12

78.12 Spontaneous Emission in Free Space

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Quantum Optics

by an atom when the incoming photon excites the atom to the level |. The energy level shift is more troublesome and needs the concept of mass renormalization, a standard problem in quantum electrodynamics. The theory and results are discussed in Chapt. 27 and Chapt. 28. Recent calculations of Pachucki [78.45] based on fully relativistic quantum electrodynamics and includ-

ing two-loop corrections predict 1 057 838(6) kHz for the energy difference between the 2s 1/2 state and the 2p 1/2 -state which is in excellent agreement with the experimental result of 1 057 839(12) kHz [78.46]. For a discussion of energy levels and transition frequencies in hydrogen and deuterium atoms see also Sect. 28.3.

78.13 Resonance Fluorescence Consider a two-level atom driven by a continuous monochromatic wave which is treated classically. The excited state of the atom can decay by spontaneous emission into vacuum modes of the electromagnetic field. This emission is called resonance fluorescence. Of particular interest are the properties of the emitted light. For a detailed discussion of resonance fluorescence, see for example [78.20, 31–33]. The far field at position R emitted by an atom at the origin is proportional to its dipole moment and can be expressed in terms of the dipole operators σ+ and σ− according to the relation [78.20] ω2 (d × R) × R σ− (t − r/c) , E+ (R, t) = − 0 4π0 c2 R3 ω2 (d ∗ × R) × R σ+ (t − r/c) , E− (R, t) = − 0 4π0 c2 R3 (78.178)

where

Part F 78.13

d = eg|r|e

(78.179)

is the atomic dipole matrix element and the field operators E+ (R, t) and E− (R, t) as well as the dipole operators σ+ (t) and σ− (t) are in the Heisenberg picture. Knowledge of the operators σ+ (t) and σ− (t) is therefore sufficient to study the properties of the emitted light in the far field.

78.13.1 Equations of Motion The total Hamiltonian for the system reads  
1 1 ~ωi bi† bi + Hsr = ~ω0 σz + 2 2 i


† gi σ− bi + gi∗ σ+ bi +~

where a resonant driving term has been added to the Hamiltonian (78.123). Here d is the projection of the dipole matrix element eg|r|e onto the polarization vector of the driving field with an amplitude E. The corresponding master equation in the interaction picture is   1 ρ˙ = − i Ω1 σ+ + σ− , ρ 2 1 + γ (2σ− ρσ+ −σ+ σ− ρ−ρσ+ σ− ) , (78.181) 2 where Ω1 = −E d ∗ /~ is the Rabi frequency associated with the driving field. The vacuum modes of the field are described by a thermal reservoir at zero temperature. The equations of motion for the expectation values σ+ , σ− , and σz  are the optical Bloch equations with radiative damping (Chapt. 68) and are γ Ω1 d σ+  = − σ+  − i σz  , dt 2 2 γ Ω1 d σ−  = − σ−  + i σz  , dt 2 2 d σz  = −γ (σz  + 1) − iΩ1 (σ+  − σ− ) . dt (78.182)

These expectation values determine the density operator (78.142) of the two-level atom. Because (78.182) are a system of linear differential equations for σ+ , σ− , and σz , they can be solved analytically. Furthermore, the quantum regression hypothesis allows one to calculate two-time correlation functions as shown in Sect. 78.9.

78.13.2 Intensity of Emitted Light

i

1 ∗ d E σ− eiω0 t + d ∗ E σ+ e−iω0 t , − 2

(78.180)

According to (78.52)) and (78.178), the intensity of the fluorescence light at position R is given by (78.183) I = E − (R, t)E + (R, t) ∝ σ+ σ−  , and can be decomposed into two parts: the coherent intensity (78.184) Icoh ∝ σ+ σ− 

Quantized Field Effects

originating from the mean motion of the dipole, and the incoherent intensity Iinc ∝ σ+ σ−  − σ+ σ−  ,

(78.185)

which is due to fluctuations of the dipole motion around its average value. The steady state intensities are Ω12 γ 2 Icoh ∝  2 , γ 2 + 2Ω12

(78.186)

2Ω14 Iinc ∝  2 . γ 2 + 2Ω12

(78.187)

and

For weak laser intensities (Ω1 small) the intensity of the fluorescence light is dominated by the coherent part whereas for high intensities (Ω1 large) it is dominated by the incoherent part.

78.13.3 Spectrum of the Fluorescence Light The Wiener–Khintchine theorem (78.54) allows one to express the steady state spectrum of the fluorescence light as the Fourier transform of the correlation function σ+ (τ)σ− (0)ss in the form ∞ 1 e−iωτ E − (τ)E + (0)ss dτ + c.c. S(ω) = 2π 0

∝

1 2π

∞

e−i(ω−ω0 )τ σ+ (τ)σ− (0)ss dτ + c.c.

0

(78.188)

Ω12 γ 2 Scoh (ω) ∝  2 δ(ω − ω0 ) . γ 2 + 2Ω12

(78.189)

The incoherent part of the fluorescence light has two qualitatively different spectra. For Ω1 < γ/4, it has a single peak at ω0 , whereas it consists of three peaks for Ω1 > γ/4. For Ω1  γ/4 it is given by  1 (γ/2)2 Sinc (ω) ∝ 2πγ (ω − ω0 )2 + (γ/2)2 (3γ/4)2 1 + 3 (ω − ω0 + Ω1 )2 + (3γ/4)2  (3γ/4)2 1 . + 3 (ω − ω0 − Ω1 )2 + (3γ/4)2 (78.190)

1161

The central peak at ω = ω0 has a width of γ/2 whereas the width of the two side peaks at ω = ω0 ± Ω1 is 3γ/4. Their heights are one third of the height of the central peak. This spectrum was predicted by Burshtein [78.47] and Mollow [78.48] and experimentally confirmed by Schuda et al. [78.49], Wu et al. [78.50], and Hartig et al. [78.51]. This triplet can be explained in terms of the dressed states |1, n and |2, n introduced in Chapt. 68 (78.50). If the driving field is resonant with the atomic transition, these states have the energies   1 ω0 − ~ Rn , E 1,n = ~ n + 2   1 ω0 − ~ Rn , (78.191) E 2,n = ~ n + 2 (78.52). The energy differences between the allowed transitions are E 2,n − E 2,n−1 = ~ω0 + ~ Rn − ~ Rn−1 ≈ ~ω0 , E 2,n − E 1,n−1 = ~ω0 + ~ Rn + ~ Rn−1 ≈ ~ω0 + ~Ω1 , E 1,n − E 2,n−1 = ~ω0 − ~ Rn − ~ Rn−1 ≈ ~ω0 − ~Ω1 , E 1,n − E 1,n−1 = ~ω0 + ~ Rn − ~ Rn−1 ≈ ~ω0 , (78.192)

where we have made the approximations Rn − Rn−1 ≈ 0 , Rn + Rn+1 ≈ 2g n¯ + 1 ≈ Ω1 .

(78.193)

This is a good approximation for an intense driving field which can approximated by a highly excited coherent state with an average photon number n. ¯ Figure 78.6 shows these energy levels and the allowed transition. Obviously, the transitions correspond to frequencies ω0 , ω0 − Ω1 and ω0 + Ω1 . The dressed ⱍ2, n冭

⍀1 ⱍ1, n冭 ω0

ω0

ω0 – ⍀ 1

ω0 + ⍀ 1

ⱍ2, n – 1冭

⍀1 ⱍ1, n –1冭

Fig. 78.6 Energy level diagram of dressed states. The transition frequencies are ω0 , ω0 − Ω1 and ω0 + Ω1

Part F 78.13

Again it consists of two contributions: a coherent part Scoh (ω), and an incoherent part Sinc (ω). The coherent part is

78.13 Resonance Fluorescence

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Quantum Optics

state picture also explains the 2:1 ratio for the integrated intensities of the central peak and the side peak in (78.190).

g(2) (τ) =

G (2) ss (τ) 2 |G (1) ss (0)|

  3γ sin δτ , = 1 − e−3γτ/4 cos δτ + 4δ

78.13.4 Photon Correlations In addition to the spectrum which is based on the correlation function E − (τ)E + (0)ss in Sect. 78.13.3, the second-order correlation function − − + + G (2) ss (τ) = E (0)E (τ)E (τ)E (0)ss

∝ σ+ (0)σ+ (τ)σ− (τ)σ− (0)ss

reads

(78.194)

can be measured to gain more insight into the fluorescence light, see also Sect. 78.3.2. Experimentally this is done by measuring the joint probability for detecting a photon at time t = 0 and a subsequent photon at time t = τ. Again the result can be obtained from the quantum regression hypothesis and

(78.195)

where δ is given by ! δ = Ω12 − γ 2 /4 .

(78.196)

For τ = 0, g(2) (0) = 0, indicating a tendency of photons to be separated. This tendency is known as photon antibunching and was first predicted by Carmichael and Walls [78.52, 53] and experimentally verified by Kimble et al. [78.54, 55]. Photon antibunching of radiation emitted from a two-level atom has a simple explanation: After the atom has emitted a photon it is in the ground state and must first be excited again before it can emit another photon.

78.14 Recent Developments

Part F 78.14

This chapter has discussed the fundamentals of the quantized electromagnetic field and applications to the broad area of quantum optics. However, in the last eight years, quantum optics has blossomed in several new directions particularly in the key role it is playing in recent investigations of the fundamentals of quantum theory and related applications. In particular, the superposition principle (the bedrock of quantum mechanics), entanglement, the quantum-classical interface, and precision measurements have become very topical research areas, especially in respect to their relevance to quantum information processing.

78.14.1 Literature During the last eight years, several books on quantum optics [78.56–62] have been published. These books cover the topics of this chapter to some extent and take into account recent developments. For an introduction to the rapidly evolving fields of quantum information processing, we refer the reader to Chapt. 81 and [78.63– 65].

78.14.2 Field States Recently, number states of the radiation field were observed in a cavity-QED experiment [78.66].

78.14.3 Reservoir Theory New research topics, such as quantum information processing, rely on the superposition principle and entangled quantum states. Since these states are very sensitive to decoherence, reservoir theory has attracted a lot of interest in recent years. Furthermore, as discussed in [78.67–72], decoherence is the physical process by which the classical world emerges from its quantum underpinning. Many investigations in this area involve the presence of a reservoir (heat-bath/environment) and master equations are a ubiquitous tool. The familiar master equations of quantum optics are in Lindblad form [78.73], which guarantees that the density matrix is always positive definite during time evolution. In the derivation of this equation [78.74, 75], rapidly oscillating terms are omitted by the method of coarse-graining in time; the high frequencies correspond to the oscillator frequency ω0 and, in the usual weak coupling limit, ω0  γ , where γ is a typical decay constant. This is the rotating wave approximation (Sect. 66.3.2). We have referred to the equations obtained prior to coarse-graining in time as pre-master (or pre-Lindblad) equations [78.74,76], and such equations have been used extensively in other areas of physics [78.77, 78]; other authors have simply referred to them as master equations

Quantized Field Effects

particular by the desire to study decoherence and other short time phenomena, an exact master equation was derived [78.85] to study the non-Markovian dynamics of a two-level atom interacting with the electromagnetic field. In addition, motivated by the desire to study a driven oscillator, the usual two-level atom master equation was generalized to include the case of an external force field [78.86, 87]. This generalized equation was then used not only to obtain the familiar zero-temperature Burshtein–Mollow spectrum, but also the corresponding high temperature results. For strong resonant driving at high temperature, the same three-peaked structure was observed in the zero temperature case, but a much larger width was found. The analysis, following other investigations, used the Lax formula for calculating two-time correlation functions. This formula is not a “quantum regression theorem” as it is often designated (see also Sect. 78.9), but simply an approximation (which more resembles an Onsager classical regression theorem [78.88]) which works very well in the case of weak coupling and for frequencies near a resonant frequency, but not otherwise [78.86, 87]. In Sect. 78.6, we stressed the usefulness of quasiprobability distributions instead of the density matrix, with particular attention to the Wigner distribution. In particular, for simple Hamiltonians, we pointed out that the equation for the corresponding Wigner function has the form of a Fokker–Planck equation and we considered the explicit form describing the usual master equation. The more general equations associated with an exact master equation and their solution was the subject of [78.83] and interesting limits of that equation, including the pre-master equation for both momentum coupling and coordinate coupling were discussed at length in [78.89, 90]. In the case of two-level systems, it is not convenient to use quasi-probability distributions; instead, it is found that the preferred tool is the polarization vector [78.91]. Surprisingly, it has not been generally adopted by the quantum optics community although its usefulness in that context has been demonstrated recently in [78.86, 87].

References 78.1

78.2

C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg: Photons, and Atoms. An Introduction to Quantum Electrodynamics (Wiley, New York 1989) L. Mandel: Opt. Lett. 4, 205 (1979)

78.3 78.4

W. Schleich, M. Pernigo, Fam Le Kien: Phys. Rev. A 44, 2172 (1991) H. J. Kimble, D. F. Walls (Eds.): J. Opt. Soc. Am. B 4(10), 1453–1737 (1987)

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but, to avoid confusion, we reserve the latter term for equations in Lindblad form. Pre-master equations, like the master equations, describe an approach to the equilibrium state. This equilibrium state is the same in either case [78.76], but with pre-master (non-Lindblad) equations the approach can be through non-physical states of negative probability. However, as recently demonstrated, pre-master equations have other advantages vis a` vis master equations: (a) they lead to the exact expression for the mean value of x(t) (as obtained from the exact Langevin equation for the problem); (b) they lead, in the classical limit (~ → 0), to the familiar Fokker–Planck equation of classical probability; and (c) the exact master equation [78.79–83] is for long times of pre-master form. However, the general expectation (based on the time dependence of the coefficients) that the exact master equation preserves positivity for all times has not been realized since Ford and O’Connell have recently shown that, even in high temperature regime, the density matrix is not necessarily positive [78.84]. In traditional  quantum optics, the emphasis has been on long-time t  γ −1 phenomena, for which the use of either master or pre-master equations is justified. However, they are both inadequate for dealing with short-time  t  γ −1 phenomena (as can be shown most simply by calculating the mean-square displacement, a key ingredient in decoherence calculations), which are of much recent interest. Thus, it is desirable to use exact master equations. In that respect, the exact master equation of Hu et al. [78.79, 80] for an oscillator is an arbitrary dissipative environment has proved to be a popular and useful tool for which an exact solution has now been obtained [78.83]. However, it should also be mentioned that the solution of the initial value quantum Langevin equation gives all the same information as the exact master equation, and in fact, the solutions of the former were used to obtain the solutions of the latter [78.83]. The familiar two-level atom master equation is, of course, similar in form to the usual Lindblad-type master equation for the oscillator. However, motivated in

References

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Quantum Optics

78.5 78.6 78.7 78.8 78.9 78.10 78.11 78.12

78.13 78.14 78.15 78.16 78.17 78.18 78.19 78.20 78.21 78.22 78.23 78.24

Part F 78

78.25 78.26 78.27 78.28 78.29 78.30 78.31 78.32 78.33 78.34 78.35 78.36

P. Knight, R. London (Eds.): J. Mod. Opt. 34(6) (1987) E. Giacobino, C. Fabre (Eds.): Appl. Phys. B 55(3) (1992) W. Schleich, J. A. Wheeler: J. Opt. Soc. Am. B 4, 1715 (1987) H. P. Yuen: Phys. Rev. A 13, 2226 (1976) R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, J. F. Valley: Phys. Rev. Lett. 55, 2409 (1985) L. A. Wu, M. Xiao, H. J. Kimble: J. Opt. Soc. Am. B 4, 1465 (1987) A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre: Phys. Rev. Lett. 59, 2555 (1987) W. P. Schleich, S. M. Burnett (Eds.): Special issue on Quantum Phase and Phase Dependent Measurements, Phys. Scr.T. T48 (1993) P. Carruthers, M. M. Nieto: Rev. Mod. Phys. 40, 411 (1968) D. T. Pegg, S. M. Barnett: Phys. Rev. A 39, 1665 (1989) J. W. Noh, A. Fougères, L. Mandel: Phys. Rev. A 45, 424 (1992) R. J. Glauber: Phys. Rev. 130, 2529 (1963) W. H. Louisell: Quantum Statistical Properties of Radiation (Wiley, New York 1973) R. Hanbury Brown, R. Q. Twiss: Nature 177, 27 (1956) P. L. Kelley, W. H. Kleiner: Phys. Rev. 136, 316 (1964) H. Carmichael: An Open Systems Approach to Quantum Optics (Springer, Berlin, Heidelberg 1993) H. Yuen, H. P. Shapiro: IEEE Trans. Inf. Theory 26, 78 (1980) J. H. Shapiro, S. S. Wagner: IEEE J. Quantum Electron. 20, 803 (1984) K. E. Cahill, R. J. Glauber: Phys. Rev. A 177, 1882 (1969) M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner: Phys. Rep. 106, 121 (1984) R. J. Glauber: Phys. Rev. 131, 2766 (1963) E. C. G. Sudarshan: Phys. Rev. Lett. 10, 277 (1963) C. W. Gardiner: Handbook of Stochastic Methods (Springer, Berlin, Heidelberg 1985) C. W. Gardiner: Quantum Noise (Springer, Berlin, Heidelberg 1991) E. Wigner: Phys. Rev. 40, 749 (1932) R. L. Hudson: Rep. Math. Phys. 6, 249 (1974) P. Meystre, M. Sargent III: Elements of Quantum Optics (Springer, Berlin, Heidelberg 1991) C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg: Atom-Photon Interactions (Wiley, New York 1992) D. F. Walls, G. J. Milburn: Quantum Optics (Springer, Berlin, Heidelberg 1994) C. W. Gardiner: Phys. Rev. Lett. 56, 1917 (1986) G. W. Ford, J. T. Lewis, R. F. O’Connell: Phys. Rev. A 37, 4419 (1988) R. F. O’Connell: Dissipation in a squeezed-state environment, Proceedings of the Second International Workshop on Squeezed States and Uncertainty Relations, Moscow, Russia 1992, ed. by

78.37 78.38

78.39 78.40 78.41 78.42 78.43 78.44 78.45 78.46 78.47 78.48 78.49 78.50 78.51 78.52 78.53 78.54 78.55 78.56 78.57 78.58

78.59 78.60 78.61 78.62 78.63

D. Han, Y.-S. Kim, V. I. Man’ko (NASA, Maryland, USA 1993) H. Risken: The Fokker-Planck Equation (Springer, Berlin, Heidelberg 1989) M. Lax: Fluctuations and Coherence Phenomena in Classical and Quantum Physics. In: Brandeis University Summer Institute Lectures, Vol. 2, ed. by M. Chretin, E. P. Gross, S. Deser (Gordon and Breach, New York 1966) G. W. Ford, R. F. O’Connell: Phys. Lett. A 157, 217 (1991) J. Dalibard, Y. Castin, K. Mølmer: Phys. Rev. Lett. 68, 580 (1992) K. Mølmer, Y. Castin, J. Dalibard: J. Opt. Soc. Am B 10, 524 (1993) R. Dum, P. Zoller, H. Ritsch: Phys. Rev. A 45, 4879 (1992) C. W. Gardiner, A. S. Parkins, P. Zoller: Phys. Rev. A 46, 4363 (1992) R. Dum, A. S. Parkins, P. Zoller, C. W. Gardiner: Phys. Rev. A 46, 4382 (1992) K. Pachucki: Phys. Rev. Lett. 72, 3154 (1994) E. W. Hagley, F. M. Pipkin: Phys. Rev. Lett. 72, 1172 (1994) A. I. Burshtein: Sov. Phys. JETP 22, 939 (1966) B. R. Mollow: Phys. Rev. 188, 1969 (1969) F. Schuda, C. R. Straud, Jr., M. Hercher: J. Phys. B 7, L198 (1974) F. Y. Wu, R. E. Grove, S. Ezekiel: Phys. Rev. Lett. 35, 1426 (1975) W. Hartig, W. Rasmussen, R. Schieder, H. Walther: Z. Phys. A 278, 205 (1976) H. J. Carmichael, D. F. Walls: J. Phys. B 9, L43 (1976) H. J Carmichael, D. F. Walls: J. B. Phys 9, 1199 (1976) H. J. Kimble, M. Dagenais, L. Mandel: Phys. Rev. Lett. 39, 691 (1977) M. Dagenais, L. Mandel: Phys. Rev. A 18, 201 (1978) M. O. Scully, M. S. Zubairy: Quantum Optics (Cambridge Univ. Press, Cambridge 1996) H.-A. Bachor: A Guide to Experiments in Quantum Optics (Wiley-VCH, Weinheim 1998) H. J. Carmichael: Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer, Berlin, Heidelberg 1999) P. Meystre, M. Sargent III: Elements of Quantum Optics (Springer, Berlin, Heidelberg 1999) C. W. Gardiner, P. Zoller: Quantum Noise (Springer, Berlin, Heidelberg 2000) M. Orszag: Quantum Optics (Springer, Berlin, Heidelberg 2000) W. P. Schleich: Quantum Optics in Phase Space (VCH-Wiley, Weinheim 2001) D. Bouwmeester, A. Ekert, A. Zeilinger (Eds.): The Physics of Quantum Information: Quantum Cryptography, Quantum Teleportation, Quantum Information (Springer, Berlin, Heidelberg 2000)

Quantized Field Effects

78.64

78.65

78.66 78.67 78.68 78.69

78.70 78.71

78.72 78.73 78.74 78.75

M. A. Nielsen, I. L. Chuang: Quantum Computation and Quantum Information (Cambridge Univ. Press, Cambridge 2000) G. Alber, T. Beth, M. Horodecki, P. Horodecki, R. Horodecki, M. Rötteler, H. Weinfurter, R. Werner, A. Zeilinger: Quantum Information: An Introduction to Basic Theoretical Concepts and Experiments, Springer Tracts in Modern Physics, Vol. 173 (Springer, Berlin, Heidelberg 2001) B. T. H. Varcoe, S. Brattke, M. Weidinger, H. Walther: Nature 403, 743 (2000) H. D. Zeh: Found. Phys. 1, 69 (1970) A. J. Leggett: Suppl. Prog. Theor. Phys. 69, 80 (1980) E. P. Wigner: In: Quantum Optics, Experimental Gravity and Measurement Theory, ed. by P. Meystre, M. O. Scully (Plenum, New York 1983) p. 43 D. F. Walls, G. J. Milburn: Phys. Rev. A 31, 2403 (1985) E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, I.-O. Stamatescu: Decoherence and the Appearence of a Classical World in Quantum Theory, 2nd edn. (Springer, Berlin, Heidelberg 2003) W. H. Zurek: Rev. Mod. Phys. 75, 715 (2003) G. Lindblad: Commun. Math. Phys. 48, 119 (1976) G. W. Ford, J. T. Lewis, R. F. O’Connell: Ann. Phys. (NY) 252, 362 (1996) J. T. Lewis, R. F. O’Connell: Ann. Phys. (NY) 269, 51 (1998)

78.76 78.77 78.78 78.79 78.80 78.81 78.82 78.83 78.84

78.85 78.86

78.87 78.88 78.89 78.90 78.91

References

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G. W. Ford, R. F. O’Connell: Phys. Rev. Lett. 82, 3376 (1999) A. O. Caldeira, A. J. Leggett: Physica A 121, 587 (1983) R. Dekker: Phys. Rep. 80, 1 (1981) B. L. Hu, J. P. Paz, Y. Z. Zhang: Phys. Rev. D 45, 2843 (1992) J. J. Halliwell, T. Yu: Phys. Rev. D 53, 2012 (1996) F. Haake, R. Reibold: Phys. Rev. A 32, 2462 (1985) R. Karrlein, H. Grabert: Phys. Rev. E 55, 153 (1997) G. W. Ford, R. F. O’Connell: Phys. Rev. D 64, 105020 (2001) G. W. Ford, R. F. O’Connell: Limitations on the utility of exact master equations, Ann. Phys (NY) 319, in press C. Anastopoulos, B. L. Hu: Phys. Rev. A 62, 033821 (2000) R. F. O’Connell: Optics Comm. 179, 451 (2000) Reprinted in Ode to a Quantum Physicist edited by W. Schleich, H. Walther, W. E. Lamb (Elsevier,Amsterdam, 2000) R. F. O’Connell: Optics Comm. 179, 477 (2000) G. W. Ford, R. F. O’Connell: Phys. Rev. Lett. 77, 798 (1996) G. W. Ford, R. F. O’Connell: Acta Phys. Hung. B 20, 91 (2004) R. F. O’Connell: J. Optics B 5, S349 (2003) E. Merzbacher: Quantum Mechanics, 3rd edn. (Wiley, New York 1998) p. 394

Part F 78

1167

Entangled Ato 79. Entangled Atoms and Fields: Cavity QED

Although the concept of a “free atom” is of use as a first approximation, a full quantum description of the interaction of atoms with an omnipresent electromagnetic radiation field is necessary for a proper account of spontaneous emission and radiative level shifts such as the Lamb shift (Chapt. 27). This chapter is concerned with the changes in the atom-field interaction that take place when the radiation field is modified by the presence of a cavity. An atom in the vicinity of a plane perfect mirror serves as an example of cavity quantum electrodynamics [79.1–5]. The primary focus in this chapter is the two extreme cases of weak coupling and strong coupling, as exemplified by spontaneous emission.

79.1

Atoms and Fields ................................. 1167 79.1.1 Atoms ...................................... 1167 79.1.2 Electromagnetic Fields ............... 1168

79.2 Weak Coupling in Cavity QED ................. 1169 79.2.1 Radiating Atoms in Waveguides .. 1169 79.2.2 Trapped Radiating Atoms and Their Mirror Images ............. 1170 79.2.3 Radiating Atoms in Resonators ... 1170 79.2.4 Radiative Shifts and Forces......... 1171

In the strong coupling regime, the excited atom is strongly coupled to an isolated resonant cavity mode. In the absence of damping, an oscillatory exchange of energy between the atom and the field replaces exponential decay (Fig. 79.1b) with a coherent evolution in time. Experimental investigations of these effects began [79.8] with the development of suitable resonators and techniques for producing atoms with long lived excited states and strong dipole transition moments.

79.1 Atoms and Fields 79.1.1 Atoms The essential features of cavity QED are elucidated by the two-level model atom discussed in Chapts. 68,

69, 70, and 77 (see also [79.9]). A ground state |g
and an excited state |e
are coupled to the radiation field by a dipole interaction. Using the formal equivalence to a spin-1/2 system, the Pauli spin operators

Part F 79

In the weak coupling regime, the coupling of an excited atom to a broad continuum of radiation modes leads to exponential decay (Fig. 79.1a), as first described by Weisskopf and Wigner [79.6]. Spontaneous emission may be enhanced or suppressed in structures such as waveguides or “bad” cavities. Cavities also introduce van der Waals forces and the subtle Casimir level shifts [79.7].

79.2.5 Experiments on Weak Coupling ... 1172 79.2.6 Cavity QED and Dielectrics........... 1173 79.3 Strong Coupling in Cavity QED ............... 1173 79.4 Strong Coupling in Experiments ............ 1174 79.4.1 Rydberg Atoms and Microwave Cavities.............. 1174 79.4.2 Strong Coupling in Open Optical Cavities ............. 1174 79.5 Microscopic Masers and Lasers .............. 1175 79.5.1 The Jaynes–Cummings Model ..... 1175 79.5.2 Fock States, Coherent States and Thermal States.................... 1175 79.5.3 Vacuum Splitting ....................... 1177 79.6 Micromasers........................................ 1178 79.6.1 Maser Threshold........................ 1178 79.6.2 Nonclassical Features of the Field 1179 79.6.3 Trapping States ......................... 1179 79.6.4 Atom Counting Statistics............. 1180 79.7 Quantum Theory of Measurement ......... 1180 79.8 Applications of Cavity QED .................... 1181 79.8.1 Detecting and Trapping Atoms through Strong Coupling ............ 1181 79.8.2 Generation of Entanglement ...... 1181 79.8.3 Single Photon Sources................ 1182 References .................................................. 1182

1168

Part F

Quantum Optics

are σx = σ † + σ ,
 σ y = −i σ † − σ ,   σz = σ † σ − σσ † = σ † , σ ,

(79.1)

with σ † = |e
g| and σ = |g
e|. The quadratures (out of phase components) of the atomic polarization are given by σx and σ y , while σz is the occupation number difference. The free atom Hamiltonian is 1 Hatom = ~ω0 σz , (79.2) 2 where ~ω0 = E e − E g is the transition energy.

79.1.2 Electromagnetic Fields Classical Fields Classical electromagnetic fields have longitudinal and transverse components:

E(r, t) = El (r, t) + Et (r, t) .

(79.3)

In the Coulomb gauge, the longitudinal part is the instantaneous electric field. The transverse part is the radiation a) Pe

b) Pe

1

1

0.5

0.5

0

0

1

2

0

3 t/τ

c)

0

1

2

3
/t

d)


field which obeys the wave equation   1 ∂2 1 ∂ j(r, t) . ∇ 2 − 2 2 Et (r, t) = c ∂t 0 c2 ∂t

(79.4)

In empty space, the driving current density j(r, t) vanishes, and the field may be expanded in a set of orthogonal modes as  Et (r, t) = E µ (t) e−iωµ r uµ (r) + c.c. , (79.5) µ

with slowly varying amplitudes E µ (t). The spatial distributions uµ (r) obey the vector Helmholtz equation 
ω 2 µ 2 (79.6) uµ (r, t) = 0 , ∇ + c depending on geometric boundary conditions as imposed by conductive or dielectric mirrors, waveguides, and resonators. In free space, plane wave solutions uµ (r, t) = u eik·r have a continuous index µ = (k, ) with wave vector k and an index  for the two independent polarizations. The orthogonality relation

1 uµ · u∗ν d3r = δµν (79.7) V V

applies. For a closed cavity, V is the resonator volume. In waveguides and free space, an artificial boundary is introduced and then increased to infinity at the end of a calculation, such that the final results do not depend on V . Quantum Fields The quantum analog of the classical transverse field in (79.4) is obtained through a quantization of its harmonic modes leading to a number state expansion. Field operators obey standard commutation relations † [aµ , aν ] = δµ,ν , and for a single mode with index µ, the amplitude E µ in (79.5) is replaced by the corresponding operator

Part F 79.1

E µ (t) = Eµ aµ e−iωt ,

† † iωt Eµ (t) = Eµ ∗ aµ e .

(79.8)

ω0

(ω –ω0)

ω0

(ω – ω0)

Fig. 79.1a–d Upper row: Excitation probability of an excited atom. (a) Exponential decay in free space or bad cavities in the weak coupling limit. (b) Oscillatory evolution in good cavities or in the strong coupling case. Lower row: The spectral signature of exponential decay is a Lorentzian line shape (c) while the so-called vacuum Rabi splitting (d) is observed in the strong coupling case

The normalization factor Eµ is chosen such that the energy difference between number states |n
µ and |n + 1
µ in the volume V is ~ωµ , giving

~ωµ . 20 V The Hamiltonian of the free field is    1 . ~ωµ aµ† aµ + HField = 2 µ Eµ Eµ † =

(79.9)

(79.10)

Entangled Atoms and Fields: Cavity QED

In the Coulomb gauge, the vector potential A(r) is related to the electric field E = −∂ A/∂t by  Eµ
† iωt Aµ (r, t) = − e aµ e−iωt + aµ uµ (r) . ωµ (79.11)

The ground state |0
µ is called the vacuum state. While the expectation value n|E|n
= 0 for a number state, the variance is not zero, since n|EE∗ |n
> 0, giving rise to nonvanishing “fluctuations” of the free electromagnetic field. Dipole Coupling of Fields and Atoms The combined system of atoms and fields can be described by the product quantum states |a, n
of atom states |a
and field states |n
. The interaction Hamiltonian H I of the atom and the radiation field is given by (The A2 -term plays an important role in energy shifts and can only be neglected when radiative processes involving energy exchange are considered.)

q2 2 q p · A(r) + A (r) . (79.12) m 2m This interaction causes the atom to exchange energy with the radiation field. In the dipole approximation, the coupling strength is proportional to the component of the atomic dipole moment deg = qe|reg |g
along the electric field, with coupling constant HI = −

gµ (r) = |deg · uµ (r)Eµ |/~ .

(79.13)

In the rotating wave approximation (RWA) (Chapts. 68, 69, and 70),  
~ gµ σ † aµ + gµ∗ aµ† σ , (79.14) HRWA = µ

atomic level at the rate [79.6]

2π    2 Γeg = 2 gµ δ ωµ − ω0 . ~ µ ˜

(79.15)

k

Here we have separated the discrete (µ) ˜ and the continuous part (wave vector k) of the mode index µ. If gµ (79.13) does not vary much across a narrow resonance, then  2  Γeg  2π gµ (ω0 ) ρµ˜ (ω0 ) . (79.16) µ ˜

The density of states corresponding to the continuous mode index k of dimension ν can be evaluated on a νdimensional fictitious volume V (ν) as

∞  V (ν) ρµ˜ = δ(ωµ,k dν kδ(ωµ,k ˜ − ω) → ˜ − ω) , (2π)ν k

0

(79.17)

provided ω(k) is known, and by converting the sum (This is formally accomplished by taking the limit of ∆k = 2π/l for large l, where l is a linear dimension of an artificial resonator, and the resonator volume is V = l 3 . If the relation between mode spacing ∆k and geometric dimension is nonlinear in a more complex geometry, this analysis can be very complicated.) over plane wave vectors k into an integral. The Rate of Spontaneous Emission In free space [ω(k)2 = (ck)2 ], the sum in (79.16) contributes a factor of two, due to polarization, to the total density of states in free space, ρfree (ω) = Vω2 /π 2 c3 . When the vector coupling of atom and field (79.13) is replaced by its average in isotropic free space, that is, by 1/3, the result

Γeg = Aeg =

2 ω3 e2reg

(79.18)

3π0 ~c3 is obtained for the decay rate Aeg as measured by the natural linewidth Γeg .

79.2 Weak Coupling in Cavity QED The regime of weak cavity QED generally applies when an atom is coupled to a continuum of radiation modes. This is always the case with mirrors, waveguides, or bad cavities. The signatures of weak cavity QED are modifications of the rate of spontaneous emission, as well as the existence of van der Waals and Casimir forces. Formally, this regime is well described by perturbation theory.

1169

79.2.1 Radiating Atoms in Waveguides Within the continuous spectrum of a waveguide, radiative decay of an excited atomic level remains exponential, and Γeg may be determined as in the preceding section. We now consider the modifications of spontaneous decay in a parallel plate waveguide. Ac-

Part F 79.2

where we have used the atomic operators of (79.1). In a continuous electromagnetic spectrum, the atom interacts with a large number of modes having quantum numbers µ, yielding exponential decay of an excited

79.2 Weak Coupling in Cavity QED

1170

Part F

Quantum Optics

cording to (79.16), the theoretical problem is reduced to a geometric evaluation of mode densities. Between a pair of mirrors it is convenient to distinguish TEnk and TMnk modes, where n is the number of half waves across the gap of width d. The dispersion relation ω(k) reflects the discrete standing wave part (nπ/d) and a running wave part as in free space,

2 ω2n,k = c2 |k|2 + nπ/d

n = 0, 1, 2, . . . TM n = 1, 2, . . . TE . (79.19)

The average mode density [du = 1, (79.13)] is evaluated [(79.17), ν = 2] with an appropriate quantization volume containing the area of the plates, V = Ad, giving ωc [ω] ρfree (ωc ) , 2ω2c ωc [ω + 1] ρTM (ω) = ρfree (ωc ) , 2ω2c ρTE (ω) =

(79.20)

where [x] is the largest integer in x, and ωc = πc/d gives the waveguide cutoff frequency. Below ωc , the TE-mode density clearly vanishes and, with the pictorial notion of turning off the vacuum introduced by Kleppner [79.10], inhibition of radiative decay is obvious. Figure 79.2 shows the calculated mode density for a parallel plate waveguide. The decay rate can be calculated from (79.16), with the spatial variation of gµ included. This configuration was used for the first experiments which showed the suppression of spontaneous emission in both the microwave and the near optical frequency domain [79.11, 12] with atomic beams.

79.2.2 Trapped Radiating Atoms and Their Mirror Images Boundary conditions imposed by conductive surfaces may also be simulated by appropriately positioned image charges. Inspired by classical electrodynamics, this image charge model can be successfully used to determine the modifications of radiative properties in confined spaces. In the simplest case, an atom is interacting with its image produced by a plane mirror. Trapped atoms and ions allow one to control their relative position with respect to a mirror to distances below the wavelength of light. Hence they are ideal objects for studying the spatial dependence of the mirror induced modifications of their radiative properties. In an experiment with a single trapped ion (see Fig. 79.3), its radiation field was superposed onto its mirror image [79.13, 14], yielding a sinusoidal variation of both the spontaneous decay rate and the mirror induced level shift with excellent contrast.

79.2.3 Radiating Atoms in Resonators Resonators In a resonator, the electromagnetic spectrum is no longer continuous and the discrete mode structure can also be resolved experimentally. While a resonator is only weakly coupled to external electromagnetic fields, it still interacts with a large thermal reservoir through currents induced in its walls. The total damping rate is due to resistive losses in the walls (κwall ) and also due to transmission at the radiation ports, 1/τµ = κµ = κwall + κout . An empty resonator stores energy for times

τµ = Q/ωµ , ρ /ρ free(ω0) 15

Part F 79.2

ρfree

10

ρcav 5

0

0

1

2

3

4 (ω – ω0)

Fig. 79.2 Modification of the average vacuum spectral

density (ρTE + ρTM ) in a parallel plate cavity (thick line) compared with free space (thin line)

(79.21)

and the power transmission spectrum is a Lorentzian with width ∆ωµ = ωµ /Q µ . The index µ, for instance, represents the TElm and TMlm modes of a “pillbox” microwave cavity, or the TEMklm modes of a Fabry– Perot interferometer (Fig. 79.4). When cavity damping remains strong, Γµ  Γeg , the atomic radiation field is “immediately” absorbed and Weisskopf–Wigner perturbation theory remains valid. In this so-called bad cavity limit, resonator damping can be accounted for by an effective mode density of Lorentzian width ∆ωµ for a single isolated mode, ωµ /2Q µ 1 ρµ (ω) = . (79.22) π (ω − ωµ )2 + (ωµ /2Q µ )2 Bad and Good Cavities The modification of spontaneous decay is again calculated from (79.16). For an atomic dipole aligned

Entangled Atoms and Fields: Cavity QED

parallel to the mode polarization, and right at resonance, ωµ = ω0 , the enhancement of spontaneous emission is found to be proportional to the Q-value of a selected resonator mode: cav Γeg free Γeg

ρµ |u(r)| 3Qλ3 3Qλ3 2 |u(r)| = = = , ρfree 4π 2 V 4π 2 Veff (79.23)

where the effective mode volume is Veff = V/|u(r)|2 . The lowest possible value Veff  λ3 is obtained for ground modes of a closed resonator. For an atom located at the waist of an open Fabry–Perot cavity with length L, it is much larger. Special limiting conc cases for concentric and confocal cavities are Veff conf 2 2 = λ L(R/D) and Veff = λL /2π, respectively, where (R/D) gives the ratio of mirror radius to cavity diameter. At resonance, the atomic decay rate Γµ grows with Q µ , whereas the resonator damping time constant κµ is reduced. Eventually, the energy of the atomic radiation field is stored for such a long time that reabsorption becomes possible. Perturbative Weisskopf–Wigner the-

79.2 Weak Coupling in Cavity QED

1171

ory is no longer valid in this good cavity limit, which is separated from the regime of bad cavities by the more formal condition cav Γeg > κµ .

(79.24)

The strong coupling case is considered explicitly in Sect. 79.3. Antenna Patterns Since the reflected radiation field of an atomic radiator is perfectly coherent with the source field, the combined radiation pattern modifies the usual dipole distribution of a radiating atom. The new radiation pattern can be understood in terms of antenna arrays [79.15]. For a single atomic dipole in front of a reflecting mirror for example, one finds a quadrupole type pattern due to the superposition of a second, coherent image antenna. In some of the earliest experimental investigations on radiating molecules in cavities, modifications of the radiation pattern were observed [79.16].

79.2.4 Radiative Shifts and Forces When the radiation field of an atom is reflected back onto its source, an energy or radiative shift is caused by the corresponding self polarization energy. An atom in

493-nm photon counts in 0.2 s 1800 1600

a) Open resonator

1400 1200 1000 800 600 400 200 –200

Movable mirror

–100

0

Ion trap

100 200 Mirror shift (nm)

b) Closed resonator

Detector

Fig. 79.3 Sinusoidal variation of the λ = 493 nm spontaneous emission rate of a single trapped Ba ion caused by self-interference from a retroreflecting mirror. The experimental arrangement is sketched at the bottom [79.13]

Fig. 79.4a,b Two frequently used resonator types for cavity QED: (a) Open Fabry–Perot optical cavity. (b) Closed “pillbox” microwave cavity

Part F 79.2

0

1172

Part F

Quantum Optics

a)

The Retarded Limit: Casimir Forces At large separation, retardation becomes relevant, since the contributions of individual atomic oscilllation frequencies in (79.25) cancel by dephasing, thus reducing the ∆vdW . A residual Casimir–Polder [79.18] shift may be interpreted as the polarization energy  a slowly fluc of tuating field with squared amplitude E 2 = 3~c/640 z 4 originating from the vacuum field noise

b)

2 π

σ

π

π

0

σ

1 –0.5 σ 0

0

Γcav / Γfree 1

2 2z/λ

∆cav /Γfree 1

0

1

2 2z/λ

∆CP = −

1 3~cαst , 4π0 8πz 4

(79.26)

Fig. 79.5 (a) Normalized rate of modified spontaneous

emission in the vicinity of a perfectly reflecting wall for σ and π orientation of the radiating dipole. (b) Corresponding energy shift of the resonance frequency. Shaded area indicates contribution of static van der Waals interaction

the vicinity of a plane mirror (Fig. 79.5) again makes a simple model system. Since the energy shift depends on the atom wall separation z, it is equivalent to a dipole force Fdip whose details depend on the role of retardation. Here we distinguish between the two cases where no radiation energy is exchanged between the atom and the field (van der Waals, Casimir forces) and where the atomic radiation causes forces by self-interference.

Part F 79.2

The Unretarded Limit: van der Waals Forces When the radiative round trip time tr = 2z/c is short compared with the characteristic atomic revolution period 2π/ωeg , retardation is not important. In this quasistatic limit, van der Waals energy shifts for decaying atomic dipoles vary as z −3 with the atom–wall separation. Such a shift is also present for a nonradiating atom in its ground state. In perturbation theory, the van der Waals energy shift of an atomic level |a
is   a|q 2 (d 2 · xt )2 + 2(d 2 · z )2 |a
. ∆vdW = − 64π0 z 3 (79.25)

Since the van der Waals force is anisotropic for electronic components parallel ( z ) and perpendicular ( xt ) to the mirror normal, the degeneracy of magnetic sublevels in an atom is lifted near a surface. The total energy shift is ≈ 1 kHz for a ground state atom at 1 µm separation, and very difficult to detect. However, the energy shifts grow as n 4 since the transition dipole moment scales as n 2 . With Rydberg atoms, van der Waals energy shifts have been successfully observed in spectroscopic experiments [79.17].

where αst is the static electric polarizability. The vacuum field noise ∆CP replaces ∆vdW at distances larger than characteristic wavelengths, and is even smaller. Only indirect observations have been possible to date, relying on a deflection of polarizable atoms by this force [79.19, 20]. The Casimir–Polder force can also be regarded as an ultimate, cavity induced consequence of the mechanical action of light on atoms [79.21]. It is an example of the conservative and dispersive dipole force which is even capable of binding a polarizable atom to a cavity [79.22]. Radiative Self-Interference Forces Spontaneous emission of atoms in the vicinity of a reflecting wall also provides an example of cavity induced modification of the dissipative type of light forces, or radiation pressure. If the returning field is reabsorbed, the spontaneous emission rate is reduced and a recoil force directed away from the mirror is exerted. If the returning radiation field causes enhanced decay, a recoil towards the mirror occurs due to stimulated emission. If the photon is detected at some angle with respect to the normal vector connecting the atom with the mirror surface, two paths for the photon are possible: It can reach a detector directly, or following a reflection off the wall. At small atom–mirror separation these paths are indistinguishable, the atom is thus left in a superposition of two recoil momentum states.

79.2.5 Experiments on Weak Coupling Perhaps the most dramatic experiment in weak coupling cavity QED is the total suppression of spontaneous emission. For the experiments which have been carried out with Rydberg atoms and for a low-lying near infrared atomic transition [79.11, 12], it is essential to prepare atoms in a single decay channel. In addition, the atoms must be oriented in such a way that they are only coupled to a single decay mode (see the model waveguide

Entangled Atoms and Fields: Cavity QED

in Fig. 79.4). This may be interpreted as an anisotropy of the electromagnetic vacuum, or as a specific antenna pattern. An important problem in detecting the modification of radiative properties – changes in emission rates as well as radiative shifts – arises from their inhomogeneity due to the dependence on atom–wall separation. This difficulty has been overcome by controlling the atom–wall separation at microscopic distances through light forces [79.17], or by using well localized trapped ions [79.13, 14]. Furthermore, spectroscopic techniques that are only sensitive to a thin layer of surface atoms [79.23] have been used to clearly detect van der Waals shifts. An atom emitting a radiation field in the vicinity of a reflecting wall will experience an additional dipole optical force caused by its radiation field. This force has been observed as a modification of the trapping force holding an ion at a fixed position with respect to the reflector [79.24]. Conceptually most attractive and experimentally most difficult to detect is the elusive Casimir interaction. Only for atomic ground states is this effect observable, free from other much larger shifts. The influence of the corresponding Casimir force on atomic motion has been observed in a variant of a scattering experiment, confirming the existence of this force in neutral atoms [79.19, 20]. The success of this experiment shows that spectroscopic techniques involving the exchange of photons are not suitable for the Casimir problem. A notable exception could be Raman spectroscopy of the magnetic substructure in the vicinity of a surface. In general, scattering or atomic interferometry experiments are more promising methods. The experiment by Brune et al. [79.25] may be interpreted in this way.

79.3 Strong Coupling in Cavity QED

79.2.6 Cavity QED and Dielectrics There are two variants of dielectric materials employed to study light-matter interaction in confined space: Conventional materials such as glass or sapphire, and artificial materials called photonic materials or metamaterials. While dielectric materials are theoretically more difficult to treat than perfect mirrors, since the radiation at least partially enters the medium, they have a similar influence on radiative decay processes. One new aspect is, however, the coupling of atomic excitations to excitations of the medium, which was observed for the case of a surface-polariton in [79.26]. Cavities with dimensions comparable to the wavelength promise the most dramatic modification of radiative atomic properties, but micrometer sized cavities for optical frequencies with highly reflecting walls are difficult to manufacture. So-called whispering gallery modes of spherical microcavities [79.27] have been intensely studied, but no simple way of coupling atoms to these resonator modes has been found yet. On the other hand, dielectric materials with a periodic modulation of the index of refraction may exhibit photonic bandgaps in analogy with electronic bandgaps in periodic crystals [79.28,29]. Electronic phenomena of solid state physics can then be transferred to photons. For example, excited states of a crystal dopant or a quantum dot cannot radiate into a photonic bandgap, the radiation field cannot propagate, and the excitation energy remains localized. The bandgap behaves like an empty resonator, and if a resonator structure is integrated into the device, the regime of strong coupling [79.30,31] can be achieved with such photonic structures. An overview of suitable systems can be found in [79.32].

H = Hatom ⊗ Hfield ,

(79.27)

The interaction of a single cavity mode with an isolated atomic resonance is now characterized by the Rabi nutation frequency, which gives the exchange frequency of the energy between atom and field. For an amplitude E corresponding to n photons, √ (79.29) Ω(n) = gµ n + 1 .

(79.28)

This is the simplest possible situation of a strongly coupled atom–field system. The new energy eigenvectors are conveniently expressed in the dressed atom

which is spanned by the states |n; a
= |n
|a
.

Part F 79.3

79.3 Strong Coupling in Cavity QED Strong coupling of atoms and fields is realized in a good cavity when Γµ < Γeg (79.24). The Hilbert space of the combined system is then the product space of a single two-level atom and the countable set of Fock-states of the field,

1173

1174

Part F

Quantum Optics

model [79.33]: |+, n
= cos θ|g, n + 1
+ sin θ|e, n
, |−, n
= − sin θ|g, n + 1
+ cos θ|e, n
,

(79.30)

√ with tan 2θ = 2gµ n + 1/(ω0 − ωµ ). The separate energy structures of free atom and empty resonator are now replaced by the combined system of Fig. 79.6. At resonance, the new eigenstates are separated by 2~ΩR , where ΩR = gµ is the vacuum Rabi frequency.

79.4 Strong Coupling in Experiments In order to achieve strong coupling experimentally, it is necessary to use a high-Q resonator in combination with a small effective mode volume. This condition was first realized for ground modes of a closed microwave cavity [79.8], and later also for open cavity optical resonators (Fig. 79.6) [79.34]. It is interesting to control the interaction time of the atoms with the cavity field. In earlier experiments, this was typically achieved by selecting the passage time for an atom transiting the cavity. The advancement of atom trapping methods has also led to the observation of a truly one-atom laser at optical frequencies [79.35]. More recently, this situation has also been realized for artificial atoms including superconducting systems [79.36, 37] and quantum dots [79.30, 31].

79.4.1 Rydberg Atoms and Microwave Cavities

Part F 79.4

At microwave frequencies, very low loss superconducting niobium cavities are available with Q ≈ 1010 . Resonator frequencies are typically several tens of GHz and can be matched by atomic dipole transitions between two highly excited Rydberg states. By selective field ionization, the excitation level of Rydberg atoms can be detected, and hence it is possible to measure whether a transition between the levels involved has occurred. The efficiency of this method approaches unity, so that experiments can be performed at the single atom level. The interaction or transit time T is usually much shorter than the lifetime τRy of the Rydberg states involved. For this reason, circular Rydberg states with quantum numbers l = m = n − 1 are particularly suitable. Rydberg atoms [79.38] are prepared in an atomic beam, selectively excited to an upper level, and then sent through a microwave cavity where the upper and lower levels are coupled by the electromagnetic field. If the atom is detected in the lower of the coupled levels as it leaves the resonator, the excitation energy has been stored in the resonator field. Thus the evolution of the

resonator field is recorded as a function of the atomic interaction. A microwave cavity in interaction with a single or a few Rydberg atoms is called a micromaser (formerly a one atom maser) [79.8]. The experimental conditions may be summarized as gµ > 1/T > 1/τRy > κµ .

(79.31)

79.4.2 Strong Coupling in Open Optical Cavities At optical wavelengths, a cavity with small Veff in (79.23) is clearly more difficult to construct than at centimeter wavelengths. However, dielectric coatings are now available which allow very low damping rates ωµ /Q µ for optical cavities. Very high finesse F  107 (which is a more convenient measure for the damping rate of an optical Fabry–Perot interferometer) has been achieved. By reducing the volume of such a high-Q cavity mode, strong coupling of

l+, 1> le, 1>, lg, 2> l–, 1>

le, 0>, lg, 1>

l+, 0> l–, 0>

Fig. 79.6a,b Level diagram for the combined states of noninteracting atoms and fields (a) which are degenerate at resonance. Degeneracy is lifted by strong coupling of atoms and fields (b) yielding new “dressed” eigenstates

Entangled Atoms and Fields: Cavity QED

atoms and fields at optical frequencies has been demonstrated [79.34]. In open structures, the atoms can still decay into the continuum states with a rate γ . Therefore the condition

79.5 Microscopic Masers and Lasers

1175

for strong coupling in such systems is usually given as 2 gµ >1. (79.32) κµ γ

79.5 Microscopic Masers and Lasers In a microscopic laser, simple atoms are strongly coupled to a single mode of a resonant or near resonant radiation field. Collecting atomic and field operators from (79.2), (79.10), and (79.14), this situation is described by the Jaynes–Cummings model Hamiltonian [79.39, 40] HJC = Hatom + Hfield + HRWA   1 1 † = ~ω0 σz + ~ωµ aµ aµ + 2 2 † σ) . + ~gµ (σ † aµ + aµ

and (79.33)

The Jaynes–Cummings model (79.33) represents the most basic and, at the same time, the most informative model of strong coupling in quantum optics. It consists of a single two-level atom interacting with a single mode of the quantized cavity field. The time evolution of the system is determined by ∂ψ = Hψ . ∂t

(79.34)

This model can be solved exactly due to the existence of the additional constant of motion N = a † a + σz + 1 ,

(79.35)

2 ∞   n=0 j=1

j

Cn (t)|n, j
,

 δ sin[Ω(n + 1)t] 2Ω(n + 1) √  gµ n + 1 1 Cn+1 (0) sin[Ω(n + 1)t] −i Ω(n + 1)    1 t , (79.38) × exp −iωµ n + 2

(79.36)

with δ = ωµ − ω0 the detuning between the atom and 2 n)1/2 is the generalized cavity and Ω(n) = 12 (δ2 + 4gµ j Rabi frequency. The coefficients Cn (0) are determined by the initial preparation of atom and cavity mode. The result simplifies considerably for δ = 0 to |Ψ(t)
=

∞   1 Cm (0) e−iωµ (m−1/2)t m=0

 √ × cos gµ mt |m; 1
 √ − i sin gµ m t |m − 1; 2
2 + Cm (0) e−iωm u(m+1/2)t  √

× cos gµ m + 1 t |m; 2


 √ − i sin gµ m + 1t |m + 1; 1
. (79.39) j

The coefficients Cn (0) represent any initial state of the system, from uncorrelated product states to entangled states of atom and field. There exist numerous generalizations of this model which include more atomic levels and several coherent fields.

Part F 79.5

i. e., conservation of the “number of excitations”. Its eigenvalues are the integers N which are twofold degenerate except for N = 0. The simultaneous eigenstates of H and N are the pairs of dressed states defined in (79.30) which are not degenerate with respect to the energy H. The initial state problem corresponding to (79.34) is solved by elementary methods in terms of the expansion |Ψ(t)
=

  2 Cn (0) cos[Ω(n + 1)t]

Cn2 (t) =

+i

79.5.1 The Jaynes–Cummings Model

i~

where the expansion coefficients are    δ Cn1 (t) = Cn1 (0) cos[Ω(n)t] − i sin[Ω(n)t] 2Ω(n) √  ngµ 2 C (0) sin[Ω(n)t] −i Ω(n) n−1    1 t (79.37) × exp −iωµ n − 2

1176

Part F

Quantum Optics

79.5.2 Fock States, Coherent States and Thermal States We now illustrate the properties of the Jaynes– Cummings model by specifying the initial state. Assume that the atom and field are brought into contact at time t = 0 and that all correlations that might exist due to previous interactions are suppressed. Rabi Oscillations If the atom is initially in the excited state and the field contains precisely m quanta, then j

Cn (t = 0) = δn,m δ j,2 .

The occupation probabilities of the atomic states evolve in time according to √

n 2 (t) = Ψ(t)|2
2|Ψ(t)
= cos2 gµ m + 1 t , (79.42)

√

n 1 (t) = Ψ(t)|1
1|Ψ(t)
= sin2 gµ m + 1 t .

(79.43)

The photon number and its variance are √ 

 n(t)
= Ψ(t)a† aΨ(t) = m + sin2 gµ m + 1 t , 

(79.44)

  2  ∆ n
= Ψ(t) a† a − a† a Ψ(t) √

sin2 2gµ m + 1 t = . (79.45) 4 √ In the limit of large m, gµ m + 1 is proportional to the field amplitude and the classical Rabi oscillations in a resonant field are recovered. The nonclassical features of the states are characterized by Mandel’s parameter  2  ∆ n − n
≥ −1 . QM = (79.46) n


Part F 79.5

For the present example, √

1 sin2 2gµ m + 1 t Q M = −1 + √

. 4 m + sin2 gµ m + 1 t

(79.48)

while the atom starts from the excited state |α|n 2 j (79.49) Cn (0) = e−|α| /2 √ δ j,2 . n! In this case, the general solution specializes to |Ψ(t)
=

(79.40)

The solution of (79.34) assumes the form

 √ |Ψ(t)
= e−iωµ (m+1/2)t cos gµ m + 1 t |m; 2
√

 − i sin gµ m + 1 t |m + 1; 1
. (79.41)

2

The Coherent State Consider the case where the field is initially prepared in a coherent state ∞ 

αn 2 |α
= exp αa† − α∗ a |0
= e−|α| /2 √ |n
, n! n=0

(79.50)

and the occupation probability of the excited state is   ∞  √

|α|2n −|α|2 1 1+ n 2 (t) = cos 2gµ n + 1 t . e 2 n! n=0

(79.51)

From here, detailed quantitative results can only be obtained by numerical methods [79.41]. However, if the coherent state contains a large number of photons |α|2  1, the essential dynamics can be determined by elementary methods. Initially, the population oscillates with the Rabi frequency Ω1 ≈ gµ |α|, which is proportional to the average amplitude of the field, as expected from its classical counterpart. With increasing time, the coherent oscillations tend to cancel due to the destructive interference of the different Rabi frequencies in the sum:  1 2 n 2 (t) = 1 + cos(2gµ |α| t) e−(gt) /2 . (79.52) 2 However, strictly aperiodic relaxation of n 2 (t) is impossible since the exact expressions, (79.36) and (79.37), represent a quasiperiodic function which, given enough time, approaches its initial value with arbitrary accuracy. For short times, the oscillating terms in the sum cancel each other due to the slow evolution of their frequency with n. However, consecutive terms interfere constructively for larger times tr , such that the phases satisfy φn+1 (tr ) − φn (tr ) = 2π .

(79.47)

Q M ≥ 0 indicates the classical regime, while Q ≤ 0 can only be reached by a quantum process.

∞  αn 2 √ e−iω(n+1/2)t e−|α| /2 n! n=0  √

× cos gµ n + 1 t |n; 2
√

 − i sin gµ n + 1 t |n + 1; 1
,

For

|α|2

(79.53)

 1, the increment of the arguments is

φn+1 − φn = gµ tr /|α| ,

(79.54)

Entangled Atoms and Fields: Cavity QED

and therefore the first revival of the Rabi oscillations occurs approximately at tr = π|α|/gµ . A clear distinction of Rabi oscillation, collapses, and revivals requires a clear separation of the three time scales t1  t2  t3 ,

(79.55)

|α|)−1

−1 for where t1 ≈ (gµ for Rabi oscillation, t2 ≈ gµ collapse, and t3 ≈ |α|/gµ for revival. The typical features of the transient evolution starting from a coherent state are shown in Fig. 79.7. With time increasing even further, revivals of higher order occur which spread in time, and finally can no longer be separated order by order.

The Thermal State Consider a microwave resonator brought into thermal contact with a reservoir, inducing loss on a time scale κ −1 and thermal excitation. The dissipative time evolution is described by the master equation

ρ˙ = (L 0 + L)ρ

    ≡ i[H, ρ]/~ + κ(n th + 1) a, ρa† + aρ, a†     + κn th a† , ρa + a† ρ, a , (79.56)

where n th = [exp(β ~ω) − 1]−1 , at T = kB β −1 , is the equilibrium population of the cavity mode, L 0 symbolizes the unitary evolution according to the Jaynes– Cummings dynamics and L is a dissipation term. The solution of this model can be expressed in terms of an eigenoperator expansion of the equation Lρ = −λρ .

(79.57)

The eigenvalues λ that determine the relaxation rates, as well as the eigenoperators, are known in closed form for

1

Population n2(t), α = 4

0.5

0.25

0

0

20

40

60 gt

Fig. 79.7 Rabi oscillations, dephasing, and quantum revival

1177

the case of vanishing temperature [79.42]. Since energy is exchanged between the nondecaying atom and the decaying cavity mode, cavity damping is modified in a characteristic way due to the presence of the atom. The technical details can be found in [79.43].

79.5.3 Vacuum Splitting In the classical case, the eigenvalues of the interaction free Hamiltonian are degenerate at resonance. The atom–field interaction splits the eigenvalues and determines the Rabi frequency of oscillation between the two states. One consequence is the existence of side bands in the resonance fluorescence spectrum [79.44]. In the quantum case, the field itself is treated as a quantized dynamical variable determined from a self-consistent solution for the complete system of atom plus field. The vacuum Rabi frequency Ωvac = gµ remains finite, and accounts for the spontaneous emission of radiation from an excited atom placed in a vacuum. In the limiting case of a single atom interacting with the quantized field, the photon number n can only change by ±1, and the population oscillates with the frequency Ω(n) given by (79.29). For an ensemble of N atoms, n can in principle change by up to ±N. However, if the field and atoms are only weakly excited, the collective frequency of the ensemble is determined by the linearized Maxwell–Bloch equations. The eigenfrequencies are given by   1 2 N − (γ − κ)2 , i(γ⊥ + κ) ± 4gµ λ± = ⊥ 2 (79.58)

γ⊥−1

where is the phase relaxation time of the atom and κ −1 the decay time of the resonator. This is the polariton dispersion relation in the neighborhood of the polariton gap. The spectral transmission    κ[γ + i(ω0 − ω)] 2   (79.59) T(ω) = T0  (ω − λ+ )(ω − λ− )  of an optical cavity containing a resonant atomic ensemble of N atoms reveals the internal dynamics of the coupled system and a splitting of the resonance line occurs. T0 is the peak transmission of the empty cavity. The splitting increases either with the number of √ photons, approaching n + 1 in the presence of a single √ atom, or with the number of atoms, approaching N in the resonator when the field is weak. The latter case is demonstrated in Fig. 79.8 [79.34] for an optical resonator with 1–10 atoms interacting with a field that contains, on average, much less than a single photon.

Part F 79.5

1.75

79.5 Microscopic Masers and Lasers

1178

Part F

Quantum Optics

79.6 Micromasers Sustained oscillations of a cavity mode in a microwave resonator can be achieved by a weak beam of Rydberg atoms excited to the upper level of a resonant transition. For a cavity with a Q ≈ 1010 , much less than a single atom at a time, on average, suffices to balance the cavity losses. Operation of a single atom maser has been demonstrated [79.8]. The atoms enter the cavity at random times, according to the Poisson statistics of a thermal beam, and interact with the field only for a limited time. In order to restrict the fluctuations of the atomic transit time, the velocity spread is reduced. This is achieved either by Fizeau chopping techniques, or by making use of Doppler velocity selection in the initial laser excitation process. Since most of the time no atom is present, it is natural to separate the dynamics into two

0.05

parts [79.45]: 1. For the short time while an atom is present, the state evolves according to the Jaynes–Cummings dynamics, where H is defined in (79.33), ρ(t) ˙ = i[H, ρ]/~ ,

and damping can safely be neglected. The formal solution is abbreviated by ρ(t) = F(t − t0 )ρ(t0 ). 2. During the time interval between successive atoms, the cavity field relaxes freely toward the thermal equilibrium according to (79.56) with L 0 = 0: ρ(t) ˙ = Lρ ,

with the formal solution ρ(t) = exp[L(t − t0 )]ρ(t0 ).

N = 10.7 atoms

ρ(ti+1 ) = exp(Lt p )F(τ)ρ(ti ) ,

(79.62)

where t p = ti+1 − ti − τ, and τ is the transit time. If τ  ti+1 − ti on average, then t p ≈ ti+1 − ti . After averaging (79.62) over the Poisson distribution P(t) = R exp(−Rt p ) for t p , where R is the injection rate, the mean propagator from atom to atom is –10

0

10 20 Frequency
(MHz) N = 1.0 atoms

0.10 (i)

Part F 79.6

(i) (ii) –10

(79.64)

Due to the continuous injection of atoms, the field never becomes time independent, but may relax toward a stroboscopic state defined by

(ii)

0

R F(τ)ρ(ti )
. (79.63) R−L After excitation, the reduced density matrix of the field alone becomes diagonal after several relaxation times κ −1 : ρ(ti+1 )
=

n|Tratom (ρ)|m
= Pn δn,m .

0.05

0 –20

(79.61)

The time development of the micromaser therefore consists of an alternating sequence of unitary F(t) and dissipative e(Lt) evolutions. Atoms enter the cavity one by one at random times ti . Until the next atom enters at time ti+1 , the evolution ti is given by

n

0 –20 n

(79.60)

10 20 Frequency
(MHz)

Fig. 79.8 Intracavity photon number (measured from

a transmission experiment, [79.34]) as a function of probe frequency detuning, and for two values of N, the average number of atoms in the mode. Thin lines give theoretical fits to the data, including atomic number and position fluctuations. Curve (ii) in the lower graph is for a single intracavity atom with optimal coupling gµ

ρ(ti+1 )
= ρ(ti )
.

(79.65)

The state of the cavity field can be determined in closed form by iteration: Pn = N

n  n th κ + Ak , (n th + 1)κ

(79.66)

k=1

where N guarantees√normalization of the trace and Ak = (R/n) sin2 (gµ τ n), and exact resonance between

Entangled Atoms and Fields: Cavity QED

cavity mode and atom is assumed. Since all off-diagonal elements vanish in steady state, (79.66) provides a complete description for the photon statistics of the field.

79.6.1 Maser Threshold The steady state distribution determines the mean photon number of the resonator as a function of the operating conditions: ∞  n
= n Pn . (79.67)

1

79.6 Micromasers

1179

Average photon number nth = 0 nth = 0.1

0.8 0.6 0.4 0.2

n=0

A suitable dimensionless control parameter is  1 Θ = gµ τ R/κ . (79.68) 2 For Θ  1, the energy input is insufficient to counterbalance the loss of the cavity, effectively resulting in a negligible photon number. With increasing pump rate R, a threshold is reached at Θ  1, where n
increases rapidly with R. In contrast to the behavior of the usual laser, the single atom maser displays multiple thresholds with a sequence of minima and maxima of n
as a function of Θ [79.46]. This can be related to the rotation of the atomic Bloch vector. When the atom undergoes a rotation of about π during the transit time τ, a maximum of energy is transferred to the cavity and n
is maximized. The converse applies if the average rotation is a multiple of 2π. This behavior is shown in Fig. 79.9. The minima in n
are at Θ  2nπ.

79.6.2 Nonclassical Features of the Field

0

0

5

10 15 Normalized transittime Θ

Fig. 79.9 Average photon number as a function of the normalized transit time defined by (79.68)

4

Variance of photon number distribution nth = 0 nth = 0.1

3

2

1

0

0

5

10 15 Normalized transittime Θ

Fig. 79.10 Variance normalized on the average photon number σ 2 /σ
. Values below unity indicate regions of nonclassical behavior

is a measure of the randomness of the field intensity. Classical Poisson statistics require that σ 2 ≥ n
. A value below unity indicates quantum behavior, which has no classical analog. In Fig. 79.10, the variance is plotted as a function of Θ. Regions of enhanced fluctuations σ 2 > n
alternate with regions with sub-Poissonian character σ 2 < n
[79.47]. When n
approaches a local maximum it is accompanied by large fluctuations, while at points of minimum field strength the fluctuations are reduced below the classical limit. This feature is repeated with a period of Θ  2π, but finally washes out at large values of Θ. The large variance of n is caused by a splitting of the photon distribution Pn into two peaks, which gives

rise to bistability in the transient response [79.48]. The sub-Poissonian behavior of the field is reflected in an increased regularity of the atoms leaving the cavity in the ground state.

79.6.3 Trapping States If cavity losses are neglected, operating conditions exist which lead directly to nonclassical, i. e., Fock states. If the cavity contains precisely n q photons, an atom that enters the resonator in the excited state leaves it again in the same state provided the condition [79.49]  gµ τ n q + 1 = 2qπ (79.70)

Part F 79.6

Fluctuations can be of classical or of quantum origin. The variance of the photon number

σ 2 = n 2
− n
2 (79.69)

1180

Part F

Quantum Optics

is satisfied, i. e., the Bloch vector of the atom undergoes q complete rotations. If the maser happens to reach such a trapping state |n q
, the photon number n q can no longer increase irrespective of the flux of pump atoms. With the inclusion of cavity damping at zero temperature, n q still represents an upper barrier that cannot be overcome, since damping only causes downward transitions. Even in the presence of dissipation, generalized trapping states exist with a photon distribution that vanishes for n > n q and has a tail towards smaller photon numbers n ≤ n q . However, thermal fluctuations at finite temperatures destabilize the trapping states since they can momentarily increase the photon number and allow the distribution to jump over the barrier n = n q . Nevertheless, even for n th < 10−7 , remnants of the trapping behavior persist, and can be seen in the transient response of the micromaser (Sect. 79.6.4).

79.6.4 Atom Counting Statistics Direct measurements of the field in a single atom maser resonator are not possible because detector absorption would drastically degrade its quality. However, the field can be deduced from the statistical signature of the atoms leaving the resonator. The probability P(n) of finding n atoms in a beam during an observation interval t is given by the classical Poisson distribution P(n) = (Rt)n e−Rt /n! .

(79.71)

Information on the field inside is then revealed by the conditional probability W(n, |g
, m, |e
, T ) of finding n atoms in the ground state and m atoms in the excited state during a time t. Since there are only two states, it is sufficient to determine the probability ∞  W(n, |g
, m, |e
, t) (79.72) W(n, |g
, t) = m=0

Part F 79.7

for being in the ground state [79.50]. For n = 0, the probability of observing no atom in the ground state during the period t is W(0, |g
, t) =

Tr(ρstst ) exp{L + R[O|g
+(1 − η)O|e
− 1]t} , (79.73)

where O| j
=  j|F(τ)| j
(79.60) and ρstst is the steady state of the maser field. This probability is closely related to the waiting time statistic P2 (0, |g
, t) between two successive ground state atoms, a property which is easily determined in a start-stop experiment. For an atom detector with finite quantum efficiency η for state selective detection, the waiting time probability is P2 (0, |g
, t)  = Tr(ρstst )O|g


 × exp[L + R[O|g
+ (1 − η)O|e
− 1]T ]O|g
2  Tr(ρstst )O|g
(79.74)

How a specific field state is reflected in the atom counting statistics will be illustrated for two situations: the region of sub-Poisson statistics and the region where the trapping condition is satisfied. Increased regularity of the cavity field Q M ≤ 0 manifests itself in increased regularity of ground state atoms in the beam. The statistical behavior exhibits “anti-bunching”, i. e., P2 (0, |g
, t) has a maximum at finite t, indicating “repulsion” between successive atoms in comparison with a Poissonian beam. If the transit time τ is chosen in such a way that gτ  2π, the chance of observing an initially excited atom in the ground state is negligible. At some point, however, an unlikely thermal fluctuation occurs, adding a photon. The rotation√angle of the Bloch vector suddenly increases to 2π 2  3π, and the atoms tend to leave the cavity in the ground state. After a typical cavity lifetime, the field decays and the trapping condition is restored again. Under this operation condition, the statistics of ground state atoms is governed by two time constants: 1. a short interval, in which successive atoms leave the cavity in the ground state after a thermal fluctuation; 2. a long time interval, in which the trapping condition is maintained and all atoms leave the resonator in their excited state until the next fluctuation occurs. The probability P2 (0, |g
, t) is plotted in Fig. 79.11. The plot clearly shows the two time regimes that govern the imperfect trapping situation.

79.7 Quantum Theory of Measurement When the object of interest consists of only a few atoms and a few photons, the puzzling consequences of quantum mechanical measurement become visible. In the case of the micromaser, the information on the state of

the field is imprinted in a subtle way on the atomic beam. While photon counting is normally a destructive operation, the dispersive part of the photon-atom interaction may be used to determine the photon number inside

Entangled Atoms and Fields: Cavity QED

5

P2(0, lb>,η ;T)

10 104 103 102 101 100 10–1 10–2 10–3 10–4 10–5 –2

–1

0

1

2

3

4

5 6 log(T/Tcav)

Fig. 79.11 Waiting time probability for atoms in the ground state while cavity is operated at vacuum trapping-state condition

a resonator without altering it, on average. Dispersive effects shift the phase of an oscillating atomic dipole without changing its state. The phase shift due to the field in the resonator can be measured in a Ramsey-type experiment [79.51]. Consider an atom with two transitions |g
→ |e
and |e
→ |i
. The first is far from resonance with the cavity and the second is close to resonance, but with a detuning δie = ω − ωie large enough so as not to change the cavity photon number as the atom passes through. The dynamic Stark effect of the |g
→ |e
transition frequency due to state |i
is then  √ 2 (79.75) ∆ωeg = gie n + 1 /δie .

79.8 Applications of Cavity QED

1181

If the resonator is now placed between the two Ramsey cavities, which are tuned to ωR ≈ ωeg , such that the polarization of the |e
→ |g
transition is rotated by ≈ π/2, then the additional phase shift ∆ωeg τ, where τ is the transit time through the optical resonator, can be measured, and hence the photon number n. Since Rydberg states have a large coupling constant gµ , the phase shift due to a single atom is detectable [79.51]. A complete measurement of n requires a sequence of N atoms because a single Ramsey measurement only determines whether the atom is in state |e
or |g
, and hence ∆ωeg τ to within ±π/2. Since each measurement provides one binary bit of information, a sequence of N measurements can in principle distinguish 2 N possible Fock states for the photon field. However, with a monoenergetic beam, integral multiples of 2π remain undetermined. A distribution of velocities, and hence transit times, is therefore desirable. An entropy reduction strategy for selecting an optimal velocity distribution, based on the outcome of previous measurements, is described in [79.52]. As a consequence of the uncertainty principle, a measurement of the photon number destroys all information about the phase of the field. In the present case, the noise in the conjugate variable (the phase) is prevented from coupling back on the measured one, and hence the measurement is called a quantum nondemolition experiment. Many other aspects of phase diffusion, entangled states, and quantum measurements in the micromaser are discussed in [79.53].

79.8 Applications of Cavity QED 79.8.1 Detecting and Trapping Atoms through Strong Coupling

79.8.2 Generation of Entanglement In the middle of the 1990s, it was realized that fully controlled quantum systems could be used to implement a revolutionary type of information processing now called quantum computing [79.55]. From the beginning, cavity QED has conceptually played an important role for experimental realizations, since it offers a route to manipulate, in principle, all physical parameters of a coherently interacting system. With the well established microwave-cavity–Rydberg-atom system, it was proven

Part F 79.8

From Fig. 79.8 it is obvious that an atom travelling through the cavity will modify the transmission properties of this cavity. Strong coupling thus enables the experimenter to detect the presence of a single atom dispersively by monitoring cavity transmission or reflection. Laser cooled atoms have low velocities and spend sufficient time in the cavity even in free flight to generate the transmission signal shown in Fig. 79.12. The signals correspond to individual atom transits, and the shape depends on the detuning of the probe laser from the resonantly interacting cavity-atom system. If an atom absorbs a photon inside the cavity, a strong dipole force can be exerted due to the inhomogeneous

field distribution of the cavity mode. Trapping of atoms with a single photon was achieved [79.54], and from the time variation of the cavity transmission a reconstruction of atomic trajectories became possible.

1182

Part F

Quantum Optics

a) T(∆) 1

0.1

0.01

0

0.2

0.4

0.6

b) T(∆)

0.8 1 Time (ms)

2

1.5 1

79.8.3 Single Photon Sources

0.5 0

0

0.2

0.4

0.6

0.8 1 Time (ms)

0.2

0.4

0.6

0.8 1 Time (ms)

c) T(∆) 4 3 2 1 0

The first ‘application’ of cavity QED was the transfer of the strong coupling idea to the combined internal and motional quantum states of trapped ions [79.58]. Here the harmonic oscillation of the ion replaces the electric field of the conventional cavity-QED system. This quantum gate was realized with a system of two trapped ions coupled to each other by Coulomb forces [79.59]. Ideas about how to use the strong coupling of atoms and photons [79.60–62] for the generation of atom– photon, or atom–atom (by insertion of more than one atom) entanglement abound, but entanglement generation by means of cavity-QED with a controlled source of atoms or ions remains a challenge for the future.

0

Fig. 79.12 Transmission of a strongly coupled cavity for individual atom transits. Caesium atoms and cavity are in perfect resonance at λ = 852 nm while the probe laser is increasingly detuned to the red side of the resonance from top to bottom 79.57

that the generation of correlated and nonlocal, so-called ‘entangled’ quantum states, is possible [79.56].

Coherent laser fields are considered the ultimate source of classical radiation fields, and they are characterized by the random arrival time of photons. Nonclassical light sources with, for instance, a regularized stream of photons offer interesting properties for low-noise measurement applications. Cavity-QED systems offer an attractive light-matter process for the generation of such ‘photon-bit-streams’, or single photon sources [79.63]. In such devices, a single photon state can, for instance, be created by Raman processes involving a classical field, which serves as the control parameter for the process, and the vacuum field of the optical resonator. The Raman process leaves a single photon in the cavity, which only weakly interacts with the atom. If the resonator has suitable transmission properties, this photon will then escape with predetermined frequency, shape, and propagation direction. Deterministic single photon sources have been realized with quantum dots [79.64,65], single molecules [79.66], and also with slow [79.67] or trapped [79.68] cold atoms and ions [79.69] inside optical cavities.

Part F 79

References 79.1

79.2 79.3 79.4

Cavity QED is reviewed in detail in S. Haroche: Fundamental Systems in Quantum Optics, ed. by J. Dalibard et al. (Elsevier, Amsterdam 1992) D. Meschede: Phys. Rep. 211(5), 201–250 (1992) P. Berman (Ed.): Cavity Quantum Electrodynamics (Academic, Amsterdam 1994) An introduction into the more general framework of low energy Quantum Electrodynamics may be found in the recent textbook P. Milonni (Ed.): The Quantum Vacuum (Academic, Boston 1994)

79.5 79.6 79.7 79.8 79.9

79.10

P. Berman (Ed.): Cavity Quantum Electrodynmics (Academic, Boston 1994) V. Weisskopf, E. Wigner: Z. Phys. 63, 54 (1930) F. Levin, D. Micha (Eds.): Long-Range Casimir Forces (Plenum, New York 1993) D. Meschede, H. Walther, G. Müller: Phys. Rev. Lett. 54, 551 (1985) L. Allen, J. Eberly: Optical Resonance and TwoLevel-Atoms (Dover, New York 1987), reprint of the original 1975 edn. D. Kleppner: Phys. Rev. Lett. 47, 233 (1981)

Entangled Atoms and Fields: Cavity QED

79.11 79.12 79.13 79.14

79.15 79.16 79.17 79.18 79.19 79.20 79.21

79.22 79.23 79.24

79.25

79.26 79.27 79.28 79.29 79.30

79.32 79.33

79.34 79.35 79.36

79.37

79.38

79.39 79.40 79.41 79.42 79.43 79.44 79.45 79.46 79.47 79.48 79.49 79.50 79.51 79.52 79.53 79.54

79.55 79.56 79.57 79.58 79.59

79.60 79.61 79.62 79.63 79.64

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, R. J. Schoelkopf: Nature 431, 164 (2004) R. F. Stebbings, F. B. Dunning (Eds.): Rydberg States of Atoms and Molecules (Cambridge Univ. Press, Cambridge 1983) E. T. Jaynes, F. W. Cummings: Proc. IEEE 51, 89 (1963) H. Paul: Ann. Phys. 11, 411 (1963) J. H. Eberly, N. B. Narozhny, J. J. SanchezMondragon: Phys. Rev. Lett. 44, 1323 (1980) H. J. Briegel, B. G. Englert: Phys. Rev. A 47, 3311 (1993) C. Ginzel, H. J. Briegel, U. Martini, B. G. Englert, A. Schenzle: Phys. Rev. A 48, 732 (1993) B. R. Mollow: Phys. Rev. 188, 1969 (1969) P. Filipowicz, J. Javanainen, P. Meystre: Phys. Rev. A 34, 3077 (1986) G. Rempe, H. Walther: Phys. Rev. A 42, 1650 (1990) G. Rempe, F. Schmidt-Kaler, H. Walther: Phys. Rev. Lett. 64, 2783 (1990) U. Benson, G. Raithel, H. Walther: Phys. Rev. Lett. 72, 3506 (1994) P. Meystre, G. Rempe, H. Walther: Opt. Lett. 13, 1078 (1988) C. Wagner, A. Schenzle, H. Walther: Opt. Commun. 107, 318 (1994) M. Brune, S. Haroche, V. Lefevre, J. M. Raimond, N. Zagury: Phys. Rev. Lett. 65, 976 (1990) R. Schack, A. Breitenbach, A. Schenzle: Phys. Rev. A 45, 3260 (1992) C. Wagner, A. Schenzle, H. Walther: Phys. Rev. A 47, 5068 (1993) A. C. Doherty, T. W. Lynn, C. J. Hood, H. J. Kimble: Phys. Rev. A 63, 013401 (2001) and references therein D. P. DiVincenzo: Fortschr. Phys. 48, 771–783 (2000) J. M. Raimond, M. Brune, S. Haroche: Rev. Mod. Phys. 73, 565 (2001) C. J. Hood, M. S. Chapman, T. W. Lynn, H. J. Kimble: Phys. Rev. Lett. 80, 4157 (1998) J. I. Cirac, P. Zoller: Phys. Rev. Lett. 74, 4091 (1995) F. Schmidt-Kaler, H. Häffner, M. Riebe, S. Gulde, G. P. T. Lancaster, T. Deuschle, C. Becher, C. F. Roos, J. Eschner, R. Blatt: Nature 422, 408 (2003) T. Pellizari, S. A. Gardiner, J. I. Cirac, P. Zoller: Phys. Rev. Lett. 75, 3788 (1995) A. S. Soerensen, K. Moelmer: Phys. Rev. Lett. 91, 097905 (2003) L. You, X. X. Yi, X. H. Su: Phys. Rev. A 67, 032308 (2003) C. K. Law, H. J. Kimble: J. Mod. Opt. 44, 2067 (1997) P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, E. Hu. Lidong Zhang, A. Imamoglu: Science 290, 2282 (2000)

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79.31

R. G. Hulet, E. Hilfer, D. Kleppner: Phys. Rev. Lett. 55, 2137 (1985) W. Jhe, A. Anderson, E. Hinds, D. Meschede, L. Moi, S. Haroche: Phys. Rev. Lett. 58, 666 (1987) J. Eschner, Ch. Raab, F. Schmidt-Kaler, R. Blatt: Nature 413, 495 (2001) M. A. Wilson, P. Bushev, J. Eschner, F. SchmidtKaler, C. Becher, R. Blatt, U. Dorner: Phys. Rev. Lett. 91, 213602 (2003) J. Dowling: Quantum Measurements in Optics, ed. by P. Tombesi, D. Walls (Plenum, New York 1992) K. H. Drexhage: Progress in Optics, Vol. 12, ed. by E. Wolf (North-Holland, Amsterdam 1974) V. Sandoghar, C. Sukenik, E. Hinds, S. Haroche: Phys. Rev. Lett. 68, 3432 (1992) H. Casimir, D. Polder: Phys. Rev. 73, 360 (1948) C. Sukenik, M. Boshier, D. Cho, V. Sandoghar, E. Hinds: Phys. Rev. Lett. 70, 560 (1993) A. Shih, D. Raskin, P. Kusch: Phys. Rev. A 9, 652 (1974) C. Cohen-Tannoudji: Fundamental Systems in Quantum Optics, ed. by J. Dalibard et al. (Elsevier, Amsterdam 1992) S. Haroche, M. Brune, J. M. Raimond: Europhys. Lett. 14, 19 (1991) M. Chevrollier, M. Fichet, M. Oria, G. Rahmat, D. Bloch, M. Ducloy: J. Phys. (Paris) 2, 631 (1992) P. Bushev, A. Wilson, J. Eschner, C. Raab, F. Schmidt-Kaler, C. Becher, R. Blatt: Phys. Rev. Lett. 92, 223602 (2004) M. Brune, P. Nussenzveig, F. Schmidt-Kaler, R. Bernadot, A. Maali, J. M. Raimond, S. Haroche: Phys. Rev. Lett. 72, 3339 (1994) H. Failache, S. Saltiel, A. Fischer, D. Bloch, M. Ducloy: Phys. Rev. Lett. 88, 243603 (2002) W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, V. Lefèvre-Seguin: Opt. Lett. 26, 166 (2001) E. Yablonovitch: Phys. Rev. Lett. 58, 2059 (1987) K. Inoue, K. Ohtaka (Eds.): Photonic Crystals (Springer, Berlin, Heidelberg 2004) J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, A. Forchel: Nature 432, 197 (2004) T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, D. G. Deppe: Nature 432, 200 (2004) K. Vahala: Nature 424, 839 (2003) C. Cohen-Tannoudji: Frontiers in Laser Spectroscopy, ed. by R. Balian et al. (North-Holland, Amsterdam 1977) R. Thompson, G. Rempe, H. Kimble: Phys. Rev. Lett. 68, 1132 (1992) J. McKeever, A. Boca, A. D. Boozer, J. R. Buck, H. J. Kimble: Nature 425, 268 (2003) I. Chiorescu, P. Bertet, K. Semba, Y. Nakamura, C. J. P. M. Harmans, J. E. Mooij: Nature 431, 159 (2004)

References

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79.65 79.66 79.67

C. Santori, M. Pelton, G. Solomon, Y. Dale, Y. Yamamoto: Phys. Rev. Lett. 86, 1502 (2001) B. Lounis, W. E. Moerner: Nature 407, 491 (2000) A. Kuhn, M. Hennrich, G. Rempe: Phys. Rev. Lett. 89, 067901 (2002)

79.68

79.69

J. McKeever, A. Boca, A. D. Boozer, R. Miller, J. R. Buck, A. Kuzmich, H. J. Kimble: Science 303, 1992 (2004) M. Keller, B. Lange, K. Hayasaka, W. Lange, H. Walther: Nature 431, 1075 (2004)

Part F 79

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80. Quantum Optical Tests of the Foundations of Physics

Quantum Opti

Quantum mechanics began with the solution of the problem of blackbody radiation by Planck’s quantum hypothesis: in the interaction of light with matter, energy can only be exchanged between the light in a cavity and the atoms in the walls of the cavity by the discrete amount E = hν, where h is Planck’s constant and ν is the frequency of the light. Einstein, in his treatment of the photoelectric effect, reinterpreted this equation to mean that a beam of light consists of particles (“light quanta”) with energy hν. The Compton effect supported this particle viewpoint of light by demonstrating that photons carried momentum, as well as energy. In this way, the wave–particle duality of quanta made its first appearance in connection with the properties of light. It might seem that the introduction of the concept of the photon as a particle would necessarily also introduce the concept of locality into the quantum world. However, in view of observed violations of Bell’s inequalities, exactly the opposite seems to be true. Here we review some recent results in quantum optics which elucidate nonlocality and other fundamental issues in physics. In spite of the successes of quantum electrodynamics, and of the standard model in particle physics, there is still considerable resistance to the concept of the photon as a particle. Many papers have been written trying to explain all optical phenomena semiclassically, i. e., with the light viewed as a classical wave, and the atoms treated quantum mechanically [80.1–4]. We first present some quantum optics phenomena which exclude this semiclassical viewpoint.

80.1 The Photon Hypothesis ........................ 1186

80.4 Complementarity and Coherence........... 1191 80.4.1 Wave–Particle Duality ................ 1191 80.4.2 Quantum Eraser ........................ 1191 80.4.3 Vacuum-Induced Coherence ....... 1192 80.4.4 Suppression of Spontaneous Down-Conversion ..................... 1192 80.5 Measurements in Quantum Mechanics ... 1193 80.5.1 Quantum (Anti-)Zeno Effect ........ 1193 80.5.2 Quantum Nondemolition ........... 1193 80.5.3 Quantum Interrogation .............. 1194 80.5.4 Weak and “Protected” Measurements .......................... 1195 80.6 The EPR Paradox and Bell’s Inequalities 1195 80.6.1 Generalities.............................. 1195 80.6.2 Polarization-Based Tests ............ 1196 80.6.3 Nonpolarization Tests ................ 1196 80.6.4 Bell Inequality Loopholes ........... 1198 80.6.5 Nonlocality Without Inequalities . 1199 80.7 Quantum Information .......................... 1200 80.7.1 Information Content of a Quantum: (No) Cloning ........ 1200 80.7.2 Super-Dense Coding .................. 1200 80.7.3 Teleportation ............................ 1200 80.7.4 Quantum Cryptography .............. 1201 80.7.5 Issues in Causality ..................... 1202 80.8 The Single-Photon Tunneling Time ....... 1202 80.8.1 An Application of EPR Correlations to Time Measurements............... 1202 80.8.2 Superluminal Tunneling Times .... 1203 80.8.3 Tunneling Delay in a Multilayer Dielectric Mirror ... 1203 80.8.4 Interpretation of the Tunneling Time................ 1204 80.8.5 Other Fast and Slow Light Schemes ............. 1205 80.9 Gravity and Quantum Optics ................. 1206 References .................................................. 1207

Part F 80

80.2 Quantum Properties of Light................. 1186 80.2.1 Vacuum Fluctuations: Cavity QED ................................ 1186 80.2.2 The Down-Conversion Two-Photon Light Source ........... 1187 80.2.3 Squeezed States of Light ............ 1187

80.3 Nonclassical Interference ..................... 1188 80.3.1 Single-Photon and Matter–Wave Interference ... 1188 80.3.2 “Nonlocal” Interference Effects and Energy–Time Uncertainty ..... 1189 80.3.3 Two-Photon Interference ........... 1190

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80.1 The Photon Hypothesis In an early experiment, Taylor reduced the intensity of a thermal light source in Young’s two-slit experiment, until, on the average, there was only a single photon passing through the two slits at a time. He then observed a two-slit interference pattern which was identical to that for a more intense classical beam of light. In Dirac’s words, the apparent conclusion is that “each photon then interferes only with itself” [80.6]. However, a coherent state, no matter how strongly attenuated, always remains a coherent state (Sect. 78.2.2); since a thermal light source can be modeled as a statistical ensemble of coherent states, a stochastic classical wave model yields complete agreement with Taylor’s observations. The one-by-one darkening of grains of film can be explained by treating the matter alone quantum mechanically [80.2]; consequently, the concept of the photon need not be invoked, and the claim that this experiment demonstrates quantum interference of individual photons is unwarranted [80.7]. This weakness in Taylor’s experiment can be removed by the use of nonclassical light sources; as discussed by Glauber [80.8], classical predictions diverge from quantum ones only when one considers counting statistics, or photon correlations. In particular, two-photon light sources, combined with coincidence detection, allow the production of single-photon (n = 1 Fock) states with near certainty. In the first such experiment [80.9], two photons, produced in an atomic cascade within nanoseconds of each other, impinged on two beam splitters, and were then detected in coincidence by means of four photomultipliers placed at all possible exit ports. In a simplified version of this experiment [80.5], one of the beam splitters and its two detectors are replaced with a single detector D1 (Fig. 80.1). We define

2

1

3

Fig. 80.1 Triple-coincidence setup of Grangier et al. [80.5]

the anticorrelation parameter α ≡ N123 N1 /N12 N13 ,

(80.1)

where N123 is the rate of triple-coincidences between detectors D1 , D2 and D3 ; N1 is the singles rate at D1 ; and N12 and N13 are double-coincidence rates. Then from Schwarz’s inequality [80.5, 7, 10], α ≥ 1 for any classical wave. In essence, since the wave divides smoothly, the coincidence rate between D2 and D3 is never smaller than the “accidental” coincidence rate, even when measurements are conditioned on an event at D1 . (The Hanbury–Brown and Twiss experiment [80.11] can be explained classically, because the thermal fluctuations lead to “bunching,” or a mean coincidence rate which is greater than the mean accidental rate; Sect. 78.3.3.) By contrast, the indivisibility of the photon leads to strong anticorrelations between D2 and D3 , making α arbitrarily small. In agreement with this quantum mechanical picture, Grangier et al. observed a 13-standard-deviation violation of the inequality [80.5], corroborating the notion of the “collapse of the wave packet” as proposed by Heisenberg [80.12].

80.2 Quantum Properties of Light 80.2.1 Vacuum Fluctuations: Cavity QED

Part F 80.2

The above considerations necessitate the quantization of the electromagnetic field, which in turn leads to the concept of vacuum fluctuations [80.4] (Sect. 78.1). Difficulties with this idea, such as the implied infinite zero-point energy of the universe, have led some reseachers to attempt to dispense with this concept altogether, along with that of the photon, in every explanation of electromagnetic interactions with matter. Of course, it is impossible to explain all phenomena, such as

spontaneous emission and the Lamb shift, without some kind of fluctuating electromagnetic fields (Chapt. 78 and Sect. 79.2.4), but one can go a long way with an ad hoc ambient classical electromagnetic noise-field filling all of space, in conjunction with the radiation reaction [80.1, 4]. In particular, even the Casimir attraction between two conducting plates (Sect. 79.2.4), which has now been verified with high precision [80.13–19] can be explained semiclassically in terms of dipole forces between electrons in each plate as they un-

Quantum Optical Tests of the Foundations of Physics

dergo zero-point motion that induce image charges in the other plate. Nevertheless, the effects of cavity QED (Chapt. 79) [80.20–22], including the influence of cavity-induced boundary conditions on energy levels and spontaneous emission rates, are most easily unified via quantization of the electromagnetic field. By coupling highly excited Rydberg atoms to photons in a high-finesse superconducting microwave cavity, Haroche et al. have observed single-photon driven Rabi oscillations [80.23], and have used these to study decoherence effects [80.24], atom–photon entanglement [80.25], and quantum nondemolition measurements [80.26] (Sect. 80.5.2). Kimble et al. [80.27] and Rempe et al. [80.28] have performed similar experiments, coupling atoms to small optical cavities, and even trapping the atoms with light fields at the singlephoton level [80.29,30]. By monitoring the amplitude of the light transmitted through the cavity (which depends on the precise location of the atom inside the cavity volume), the trajectory of the atom can be determined with ultrahigh resolution, much smaller than an optical wavelength [80.31, 32].

80.2.2 The Down-Conversion Two-Photon Light Source

Deep red

1187

Red

UV pump

KDP crystal

Orange

Fig. 80.2 Conical emissions of down-conversion from a nonlinear crystal (for type-I phase-matching). Photon energy depends on the cone opening angle, and conjugate photons lie on opposite sides of the axis, e.g., the inner “circle” orange photon is conjugate to the outer “circle” deep-red photon, etc.

are within femtoseconds of each other, so that detection of one photon implies with near certainty that there is exactly one quantum present in the conjugate mode [80.43]. In type-I phase-matching the correlated photons share the same polarization, while in type-II phase-matching they have orthogonal polarizations . We will see below (Sect. 80.6.2) how both of these can enable the production of photons that are entangled in polarization, as well as in other degrees of freedom. This production technique allowed for the first reconstruction of the Wigner distribution for a single photon [80.44], which is manifestly non-classical in that the quasiprobability for both quadratures of the field to vanish is negative. In contrast to an earlier demonstration of a negative Wigner function [80.45] (using atoms, not photons), this measurement was possible using essentially classical measurement techniques with no quantum assumptions, and is in this sense a direct demonstration of the non-classical nature of the electromagnetic field. Later work [80.46] used these techniques to demonstrate that a single photon forced to choose between two output ports of a beam splitter exhibits quantum correlations.

80.2.3 Squeezed States of Light The creation of correlated photon pairs is closely related to the process of quadrature-squeezed light production (Sect. 78.2.2, and the review in [80.47]). For example, when the gain arising from parametric amplification in a down-conversion crystal becomes large, there is a transition from spontaneous to stimulated emission of pairs. This gain is dependent on the phase of amplified light relative to the phase of

Part F 80.2

The quantum aspects of electromagnetism are made more striking with a two-photon light source, in which two highly correlated photons are produced in spontaneous parametric down-conversion, or parametric fluorescence [80.33–36]. In this process, an ultraviolet “pump” photon produced in a laser spontaneously decays inside a crystal with a χ (2) nonlinearity into two highly correlated red photons, conventionally called the “signal” and the “idler” (Sect. 72.3.4). (The quantum state of the light is more correctly written as |ψ ∝ |vacuum + |1s |1i + 2 |2s |2i + . . . , but since the amplitude process itself is 
of the down-conversion very weak  is of order 10−6 , one often neglects the terms containing 2 or more pairs. However, recent experiments have begun to exploit these higher-order terms, e.g., to investigate 3-, 4-, 5-photon quantum effects [80.37–40]. Very recently, a stimulated downconversion process [80.41] has indicated the presence of 12-photon entanglement [80.42].) As shown in Fig. 80.2, a rainbow of colored cones is produced around an axis defined by the direction of the uv beam (for the case of type-I phase-matching), with the correlated downconversion photons always emitted on opposite sides of the UV beam (Sect. 72.2.2). Their emission times

80.2 Quantum Properties of Light

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Quantum Optics

the pump light. As a result, the vacuum fluctuations are reduced (“squeezed”) below the standard quantum limit (SQL) in one quadrature, but increased in the other, in such a way as to preserve the minimum uncertainty-principle product [80.48]. This periodicity of the fluctuations at 2ω is a direct consequence of the fact that the light is a superposition of states differing in energy by 2~ω – the quadrature-squeezed vacuum state   1 ∗ 1 † † |ξ = exp ξ aa − ξa a |0 (80.2) 2 2 represents a vacuum state transformed by the creation (a† a† ) and destruction (aa) of photons two at a time. Essentially any optical processes operating on photon pairs (e.g., four-wave mixing [80.49, 50]) can also produce such squeezing. Amplitude squeezing involves preparation of states with well-defined photon number, i. e., states lacking the Poisson fluctuations of the coherent state. The possibility of producing such states (e.g., via a constant-currentdriven semiconductor laser [80.51]) demonstrates that “shot noise” in photodetectionshould not be thought of as merely the result of the probabilistic (à la Fermi’s

Golden Rule, Sect. 69.4) excitation of quantum mechanical atoms in a classical field, but as representing real properties of the electromagnetic field, accessible to experimental control. The highest level of number-squeezing reported ( − 5.7 ± 0.1) dB, corresponding to a noise reduction 73% below the SQL) employed an asymmetric fiber loop to squeeze solitons [80.52]. The highest level of quadrature-squeezing reported has been ( − 7.0 ± 0.2) dB (80% below the SQL), using a χ (2) crystal optical parametric oscillator [80.53]. This was a continuous-wave vacuum-squeezed beam – the same technique was used to produce − 5.0 dB of bright continuous-wave squeezing locked for several hours. Squeezed light has begun to have impact in metrology. A few examples suffice: the generation of audio-band squeezed light [80.54] and the demonstration of a squeezing-enhanced power-recycled Michelson interferometer [80.55], both suitable for gravity-wave interferometry (Sect. 80.9); the use of squeezing to measure displacement of a light beam below the standard quantum limit [80.56], suitable for atomic force microscopy; and demonstrations of squeezed light spectroscopy [80.57–59].

80.3 Nonclassical Interference 80.3.1 Single-Photon and Matter–Wave Interference

Part F 80.3

The first truly one-photon interference experiment [80.5] used the cascade source discussed in Sect. 80.1. One of the photons was directed to a “trigger” detector, while the other, thus prepared in an n = 1 Fock state, was sent through a Mach–Zehnder interferometer. The output photon, detected in coincidence with the trigger photon, showed fringes with a visibility > 98%. Dirac’s statement that a single photon interferes with itself is thus verified. Of course, matter can also display interference, determined by the deBroglie wavelength (Chapt. 77). There have been significant recent advances in atomic matter–wave interferometry and its applications [80.60], ever since the early experiments of Pritchard, which used standing-light gratings and nanofabricated diffraction gratings to construct Mach–Zehnder-type interferometers for sodium atoms [80.61] and molecules [80.62] from a supersonic source. Chu and Kasevich introduced the use of STIRAP (Stimulated Raman Adiabatic fast

Passage; Sect. 69.7) to produce coherent beam splitters for cold atomic beams, also in a Mach–Zehnder-type interferometer, but whose source were cesium atoms cooled in and launched from a magneto-optical trap (MOT) [80.63]. Matter–wave interferometry has now been applied to precision measurements of the acceleration g due to Earth’s gravity [80.64], gravity gradiometry [80.65], and Sagnac matter–wave gyroscopes [80.66, 67]. To date some of the largest systems to display quantum interference are large molecules like carbon 60 (“Buckyballs”) and carbon 70 [80.68]. These are significant in that the average deBroglie wavelength of the molecules, emitted from an oven, was 2.8 pm, actually about 350 times smaller than the molecule itself. Arndt et al. have also demonstrated multislit diffraction with the biological molecule porphyrin, and with fluorofullerenes (C60 F48 ) [80.69]. With a mass of 1632 amu, the latter are currently the largest single objects to display interference. Looking ahead, others have suggested
that it may be possible to put a micron-scale mirror with 14 ≈ 10 atoms into a superposition of resolvable spatial

Quantum Optical Tests of the Foundations of Physics

1189

80.3.2 “Nonlocal” Interference Effects and Energy–Time Uncertainty The energy–time uncertainty principle, ∆E∆t ≥ ~/2 has been tested in a down-conversion interference experiment [80.81]. The down-conversion process conserves energy and momentum:

~ω0 = ~ω1 + ~ω2 , ~k0 ≈ ~k1 + ~k2 ,

(80.3) (80.4)

where ~ω0 (~k0 ) is the energy (momentum) of the parent photon, and ~ω1 (~k1 ) and ~ω2 (~k2 ) are the energies (momenta) of the daughter photons; k1 and k2 sum to k0 to within an uncertainty given by the reciprocal of the crystal length [80.82]. Since there are many ways of partitioning the parent photon’s energy, each daughter photon may have a broad spectrum, and hence a wave packet narrow in time. However, ω1 + ω2 = ω0 is extremely well-defined, so that the difference in the daughter photons’ arrival times, and the sum of their energies can be simultaneously known to high precision. Thus, the daughter photons of a parent photon of sharp energy E 0 are in an energy-“entangled” state, a nonfactorizable sum of product states [80.83]: E0 |Ψ  =

dE A(E) |E |E 0 − E ,

(80.5)

0

where A(E) is the probability amplitude for the production of two photons of energies E and E 0 − E. A measurement of the energy of one of the photons to be E 1 can be interpreted as causing an instantaneous “collapse” of the system to the state |E 1 |E 0 − E 1 , implying an instantaneous increase of the width of the other photon’s wave packet. (Of course, the notion of collapse need not be invoked to explain such results. One can view the detection of the trigger photon as conditionally selecting a particular subensemble of the pairs. However, as discussed in Sect. 80.6.3, it is not correct to interpret the down-conversion photons as possessing a well-defined energy prior to measurement.) In the experiment, one photon was used as a trigger, while the other was sent into an adjustable Michelson interferometer (Fig. 80.3), used to measure its coherence length. (The same apparatus was also used to demonstrate that Berry’s phase in optics has a quantum origin [80.84].) If the trigger photon passed through an interference filter F1 of narrow width ∆E and was detected, then the conjugate photon occupied a broad wavepacket of duration ∆t ≈ ~/∆E, and displayed interference. When

Part F 80.3

locations [80.70] – the mirror, part of a high-finesse optical cavity forming one arm of a Michelson interferometer, could be mounted on a high-quality mechanical oscillator, whereby the interaction with a single photon would change the frequency of the oscillator. Two other systems, demonstrating Bose–Einstein condensation (BEC) (see Chapt. 76), have also produced evidence of macroscopic quantum coherence. In the experiments of Ketterle et al., atoms from two different atomic vapor BEC clouds were allowed to fall onto the same detection region, and display interference fringes [80.71] (more recently interference from an array of 30 independent BECs has been observed [80.72]). In some ways this is the matter–wave equivalent of the famous Pfleegor– Mandel experiment [80.73], in which light from two separate lasers displays interference, even when attenuated to the single-photon level. The explanation in terms of the indistinguishability of the underlying processes is that one cannot ascertain from which laser source a given photon originated. However, this explanation must be applied carefully to the situation of the two atomic BECs: Unlike the lasers, the BEC clouds can – at least in principle – be prepared with a definite number of atoms, and it would therefore seem that one could in principle determine which cloud emitted a given detected atom. However, this determinacy is rapidly lost after a few atoms are detected [80.74]. Once the number becomes uncertain, a well-defined relative phase of the two BECs is established, according to the number-phase uncertainty relation ∆(N2 − N1 )∆(φ2 − φ1 )
1/2. In fact, discussions have recently arisen over whether or not lasers should not also be viewed as incoherent number-state combinations, instead of the usual coherent state |α (the issue is that in principle there is nothing in a laser to break the symmetry and select a particular phase) [80.75–77]. Finally, quantum coherence (though not explicitly spatial interference as in the previous examples), has been detected in the operation of a Josephson-junction linked superconducting loop – the group of Mooij was able to prepare a superposition of clockwise and counterclockwise circulating electrical currents [80.78]. Since the ≈ 0.5 µA currents corresponded to the motion of millions of Cooper pairs, this is arguably the largest system thus far to have displayed quantum coherence. This superconducting system also holds promise for quantum computing (see Chapt. 81), as Rabi oscillations between the different flux states have been observed [80.79, 80].

80.3 Nonclassical Interference

1190

Part F

Quantum Optics

F1 UV laser

D1

Idler KDP Beam dump Signal F2 D2 Fixed QWP F3

Rotatable QWP D3

Fig. 80.3 The energy–time uncertainty relation and wave function collapse were studied by investigating the effect of various filters before the detectors in a single-photon interference experiment [80.5, 81]

there was no trigger, no fringes were observed, implying a much shorter wave packet. This is a nonlocal effect in that the photons could in principle be arbitrarily far away from each other when the collapse occurs.

80.3.3 Two-Photon Interference

Part F 80.3

In the above experiments, interference occurs between two paths taken by a single photon. An early experiment to demonstrate two-photon interference using the down-conversion light source was performed by Ghosh and Mandel [80.85]. They looked at the counting rate of a detector illuminated by both of the twin beams. No interference was observed at the detector, because although the sum of the phases of the two beams emitted in parametric fluorescence is well defined (by the phase of the pump), their difference is not, due to the number-phase uncertainty principle. However, the rate of coincidence detections between two such detectors whose separation was varied did display high-visibility interference fringes. Whereas in the standard two-slit experiment, interference occurs between the two paths a single photon could have taken to reach a given point on a screen, in this case it occurs between the possibility that the signal photon reached detector 1 and the idler photon detector 2, and the possibility that the reverse happened. This experiment provides a manifestation of quantum nonlocality; interference occurs between alternate global histories of a system, not between local

fields. At a null of the coincidence fringes, the detection of one photon at detector 1 excludes the possibility of finding the conjugate photon at detector 2. Such interference becomes clearer in the related interferometer of Hong et al. [80.86] (Fig. 80.4). The identically polarized conjugate photons from a downconversion crystal are directed to opposite sides of a 50–50 beam splitter, such that the transmitted and reflected modes overlap. If the difference in the path lengths ∆L prior to the beam splitter is larger than the two-photon correlation length (of the order of the coherence length of the down-converted light), the photons behave independently at the beam splitter, and coincidence counts between detectors in the two output ports are observed half of the time – the other half of the time both photons travel to the same detector. However, when ∆L ≈ 0, such that the photon wave packets overlap at the beam splitter, the probability of coincidences is reduced, in principle to zero if ∆L = 0. One can explain the coincidence null at zero path-length difference using the Feynman rules for calculating probabilities: add the probability amplitudes of indistinguishable processes which lead to the same final outcome, and then take the absolute square. The two indistinguishable processes here are both photons being reflected at the beam splitter (with Feynman amplitude r · r) and both photons being transmitted (with Feynman amplitude t · t). The probability of a coincidence detection is then    i i 1 1 2 Pc = |r · r + t · t|2 =  √ · √ + √ · √  = 0 , 2 2 2 2 (80.6)

assuming a real transmission amplitude, and where the factors of i come from the phase shift upon reflection at a beam splitter [80.87, 88]. The possibility of a perfect null at the center of the dip is indicative of a nonclassical effect. Indeed, classical field predictions allow a maximum coincidence-fringe visibility of only 50% [80.89]. The tendency of the photons to travel off together at the beam splitter can be thought of as a manifestation of the Bose–Einstein statistics for the photons [80.90]. In practice, the bandwidth of the photons, and hence the width of the null, is determined by filters and/or irises before the detectors [80.82]. Widths as small as 5 µm have been observed, corresponding to time delays of only 15 fs [80.91]. Consequently, one application is the determination of single-photon propagation times with extremely high time resolution (Sect. 80.8).

Quantum Optical Tests of the Foundations of Physics

80.4 Complementarity and Coherence

1191

80.4 Complementarity and Coherence 80.4.1 Wave–Particle Duality The complementary nature of wave-like and particlelike behavior is frequently interpreted as follows: due to the uncertainty principle, any attempt to measure the position (particle aspect) of a quantum leads to an uncontrollable, irreversible disturbance in its momentum, thereby washing out any interference pattern (wave aspect) [80.93, 94]. This picture is incomplete though; no “state reduction,” or “collapse,” is necessary to destroy interference, and measurements which do not involve reduction can be reversible. One must view the loss of coherence as arising from an entanglement of the system wave function with that of the measuring apparatus (MA) [80.95]. Previously interfering paths can thereby become distinguishable, such that no interference is observed. Consider the simplest experiment, a Mach–Zehnder interferometer with a 90◦ polarization rotator in arm 1. If horizontally polarized light is input, the state before √ the recombining beam splitter is
|1|V  + |2|H)/ 2 , where |1 and |2 indicate the path of the photon. Because the polarization – playing the role of the MA – labels the path, no interference is observed at the output. Englert [80.96] has introduced a generalized relation quantifying the interplay between the wave-like attributes of a system (as measured by the fringe visibility V) and the particle-like character (as measured by the distinguishability D of the underlying quantum processes): V 2 + D2 ≤ 1 .

of the MA is maintained, then interference may be recovered. The first demonstration of a quantum eraser was based on the interferometer in Fig. 80.4 [80.92]. A half waveplate inserted into one of the paths before the beam splitter serves to rotate the polarization of light in that path. In the extreme case, the polarization is made orthogonal to that in the other arm, and the r·r and t·t processes become distinguishable; hence, the destructive interference which led to a coincidence null does not occur. The distinguishability can be erased, however, by using polarizers just before the detectors. In particular, if the initial polarization of the photons is horizontal, and the waveplate rotates one of the photon polarizations to vertical, then polarizers at 45◦ before both detectors restore the original interference dip. If one polarizer is at 45◦ and the other at −45◦ , interference is once again seen, but now in the form of a peak instead of a dip (Fig. 80.5). There are four basic measurements possible on the MA (here the polarization) – two of which yield which-path information, one of which recovers the initial interference fringes (here the coincidence dip), and one of which yields interference anti-fringes (the peak instead of the dip). In some implementations, the decision to measure wave-like or particle-like behavior may even be delayed until after detection of the original quantum, an irreversible process [80.103–105]. (This is an extension of the original delayed-choice discussion by

(80.7)

The equality holds for pure input states. This relation has now been well verified in optical systems like that described above [80.97, 98], as well as in atom interferometry (Sect. 77.6) [80.99]. In the latter, the role of the polarization was played by internal energy states of an atom diffracted off a standing light wave.

D1 A UV

Nonlinear crystal

D2

Fig. 80.4 Simplified setup for a Hong–Ou–Mandel (HOM)

80.4.2 Quantum Eraser

Part F 80.4

The interference lost to entanglement may be regained if one manages to “erase” the distinguishing information. This is the physical content of quantum erasure [80.100, 101]. The primary lesson is that one must consider the total physical state, including any MA with which the interfering quantum has become entangled, even if that MA does not allow accessible which-path information [80.102]. If the coherence

interferometer [80.86]. Coincidences may result from both photons being reflected, or both being transmitted. When the path lengths to the beam splitter are equal, these processes destructively interfere, causing a null in the coincidence rate. In a modified scheme, a half waveplate in one arm of the interferometer (at “A”) serves to distinguish these otherwise interfering processes, so that no null in coincidences is observed. Using polarizers before the detectors, one can “erase” the distinguishability, thereby restoring interference [80.92] (Sects. 80.4.2 and 80.6.2)

1192

Part F

Quantum Optics

Coincidence rate (s–1) 45°, 45° Theory 400 45°, 22° 350 Theory 45°, – 45° 300 Theory 250

Ds

s1

450

BS

NL1 i1

s2

A NL2

UV

200

i1, i2

B

Di

Fig. 80.6 Schematic of setup used in [80.109]. The idler

150

photons from the two crystals are indistinguishable; consequently, interference fringes may be observed in the signal singles rate at detector Ds . Additional elements at A and B can be used to make a quantum eraser

100 50 0 –1110

–1090

–1070

–1050

–1030 –1010 –990 Trombone prism position (µm)

Fig. 80.5 Experimental data and scaled theoretical curves (adjusted

to fit observed visibility of 91%) with polarizer 1 at 45◦ and polarizer 2 at various angles. Far from the dip, there is no interference and the angle is irrelevant [80.92]

Wheeler [80.106], and the experiments by Hellmuth et al. and Alley et al. [80.107,108], in which the decision to display wave-like or particle-like aspects in a light beam may be delayed until after the beam has been split by the appropriate optics.) But in all cases, one must correlate the results of measurements on the MA with the detection of the originally interfering system. This requirement precludes any possibility for superluminal signaling. In a related atom-optics experiment, researchers observed contrast loss in an atom interferometer when single photons were scattered off the atoms (yielding which-path information) [80.110]. They further demonstrated that the lost coherence could be recovered by observing only atoms that were correlated with photons emitted into a limited angular range, in essence realizing a quantum eraser.

80.4.3 Vacuum-Induced Coherence

Part F 80.4

A somewhat different demonstration [80.109, 111] of complementarity involves two down-conversion crystals, NL1 and NL2, aligned such that the trajectories of the idler photons from each crystal overlap (Fig. 80.6). A beam splitter acts to mix the signal modes. If the path lengths are adjusted correctly, and the idler beams overlap precisely, there is no way to tell, even in principle, from which crystal a photon detected at Ds originated. Interference appears in the signal singles rate at Ds , as any of the path lengths is varied. If the

idler beam from crystal NL1 is prevented from entering crystal NL2, then the interference vanishes, because the presence or absence of an idler photon at Di then “labels” the parent crystal. One explanation for the effect of blocking this path is that coherence is established by the idler-mode vacuum field seen by both crystals. Experiments have also been performed in which a time-dependent gate is introduced in the idler arm between the two crystals [80.112]. As one expects, the presence or absence of interference depends on the earlier state of the gate, at the time when the idler photon amplitude was passing through it.

80.4.4 Suppression of Spontaneous Down-Conversion A modification [80.113] of this two-crystal experiment uses only a single nonlinear crystal (Fig. 80.7). A given pump photon may down-convert in its initial right-ward passage through the crystal, or in its left-going return trip (or not at all, the most likely outcome). As in the previous experiment, the idler modes from these two processes are made to overlap; moreover, the signal modes are also aligned to overlap. Thus, the left-going and right-going production processes are indistinguishable and interfere. The result is that fringes are observed in all of the counting rates (i. e., the coincidence rate and both singles rates) as any of the mirrors is translated. A different interpretation is as a change in the spontaneous emission of the down-converted photons, akin to the suppression of spontaneous emission in cavity QED demonstrations, discussed in Sects 80.2.1 and 79.2. Subsequent theoretical and experimental work has shown that, in a sense, there are always photons between the down-conversion crystal and the mirrors, even

Quantum Optical Tests of the Foundations of Physics

in the case of complete suppression of the spontaneous emission process [80.114, 115]. This same conclusion should also apply in the atom case [80.116], though in contrast to that system, for the down-conversion experiment the distances to the mirrors are much longer than the coherence lengths of the spontaneously emitted photons. One recent application of this phenomenon is to study effective nonlinearities at the single-photon level, which has enabled the construction of a 2-photon switch [80.117]. Finally, with the inclusion of waveplates to label the photons’ paths, and polarizers to erase this information, an improved quantum eraser experiment was also completed, in which the which-path information for one photon was carried by the other photon [80.104].

80.5 Measurements in Quantum Mechanics

1193

Signal

Pump LiIO3 Idler

Fig. 80.7 Schematic of the experiment to demonstrate enhancement and suppression of spontaneous downconversion [80.113]

80.5 Measurements in Quantum Mechanics 80.5.1 Quantum (Anti-)Zeno Effect

80.5.2 Quantum Nondemolition The uncertainty principle between the number of quanta N and phase φ of a beam of light, ∆N∆φ ≥ 1/2 ,

(80.8)

implies that to know the number of photons exactly, one must give up all knowledge of the phase of the wave. In theory, a quantum nondemolition (QND) process is possible [80.124]: without annihilating any of the light quanta, one can count them. It might seem that this would make possible successive measurements on noncommuting observables of a single photon, in violation of the uncertainty principle; it is the unavoidable introduction of phase uncertainty by any number measurement which prevents this. QND schemes [80.125] often employ the intensitydependent index of refraction arising from the optical

Part F 80.5

A strong measurement of a quantum system will project it into one of its eigenstates [80.95]. If the system evolves slowly out of its initial state: |Ψ(t = 0) = |Ψ0  → (1 − t 2 /τ 2 )1/2 |Ψ0  + t/τ|Ψ1 , then repeated measurements with an interval much less than τ can inhibit this evolution. If there are N total measurements within τ, then the probability
for the system to still be in the initial state is P(τ) = 1 − (τ/N )2 /τ 2 N → 1 as N → ∞. This phenomenon, known as the quantum Zeno effect [80.118], has been experimentally observed using 3 levels in 9 Be+ ions [80.119]. The ions were prepared in state |i, and weakly coupled to state | f  via RF radiation that induced a slow Rabi oscillation between the two states. Thus, in the absence of any intervening measurements, the ions evolved sinusoidally into state | f . When rapid measurements were made (by a laser strongly coupling state | f  to readout state |r, hence leading to strong fluorescence only if the atom was in state | f ), the effect was to inhibit the |i → | f  transition. Note that here it was the absence of fluorescent photons which projected the state at each measurement back into the state |i (Sect. 80.5.3). Also, although we have explained the effect in terms of a repeated “collapse” of the wave function back into its initial state, equally valid explanations without such reductions are also possible [80.120, 121]. Koffman and Kurizki have pointed out that the above inhibition phenomenon depends on there being a bounded number of final states (the ion example had

only one). If instead the measurement process actually increases the number of accessible final states | f , then one obtains the “anti-Zeno” effect, in which the |i → | f  rate is enhanced rather than suppressed by frequent measurements [80.122]. For example, this would be the case in the ion example if the |i → | f  transition were spontaneous (allowing all frequencies) instead of driven (proceeding only at the driving Rabi frequency). The anti-Zeno effect has been observed by monitoring the survival time (against tunneling escape) of atoms trapped in an accelerating far-detuned standing wave of light [80.123].

1194

Part F

Quantum Optics

Kerr effect (Sect. 72.4.2) – the change in the index due to the intensity of the “signal” beam changes the optical phase shift on a “probe” beam [80.126–128]. Other proposals include using the Aharonov–Bohm effect to sense photons via the phase shift their fields induce in passing electrons [80.129, 130]. To date the closest experimental realization [80.26] of a QND measurement – of the photon number in a microwave cavity – was performed by passing Rydberg atoms [80.131, 132] in a superposition of the ground and excited states through the cavity. The interaction with the cavity photon is adjusted to be equivalent to a 2 π-pulse (Sect. 79.5). The result is that in the absence of any photon, the quantum state of the atoms after the cavity was unchanged; with a photon in the cavity, the ground state acquired an extra relative phase of π, which was then detected by measuring the atom’s quantum state. An efficiency approaching 50% was achieved. Recently, it was suggested that optical QND measurements could enable scalable quantum computing [80.133].

80.5.3 Quantum Interrogation

Part F 80.5

The previous section described techniques to measure the presence of a photon without absorbing it. Now we discuss a method – quantum interrogation – to optically detect the presence of an object without absorbing or scattering a photon. The possibility that the absence of a detection event – a “negative-result” measurement – can lead to wavepacket reduction was first discussed by Renninger [80.134] and later by Dicke [80.135]. Here we consider the Gedankenexperiment proposed by Elitzur and Vaidman (EV), a simple single-particle interferometer, with particles injected one at a time [80.136]. The path lengths are adjusted so that all the particles leave a given output port (A), and never the other (B). Now suppose that a nontransmitting object is inserted into one of the interferometer’s two arms – to emphasize the result, EV considered an infinitely sensitive “bomb”, such that interaction with even a single photon would cause it to explode. By classical intuition, any attempt to check for the presence of the bomb involves interacting with it in some way, and, by hypothesis, inevitably setting it off. Quantum mechanics, however, allows one to be certain some fraction of the time that the bomb is in place, without setting it off. After the first beam splitter of the interferometer, a photon has a 50% chance of heading towards the bomb, and thus exploding it. On the other hand, if the photon takes the path without the bomb, there is no more interference, since the nonex-

plosion of the bomb provides welcher Weg (“which way”) information (see Sect. 80.4.1). Thus the photon reaches the final beam splitter and chooses randomly between the two exit ports. Some of the time (25%), it leaves by output port B, something which never happened in the absence of the bomb. This immediately implies that the bomb (or some nontransmitting object) is in place – even though (since the bomb is unexploded) it has not interacted with any photon; EV termed this an “interaction-free measurement”. (We prefer the more general description “quantum interrogation”, which then includes cases – e.g., detecting a semi-transparent or quantum object – where it may not be possible to logically exclude the possibility of an interaction.) It is the mere possibility that the bomb could have interacted with a photon which destroys interference. An initial experimental implementation of these ideas [80.137] used down-conversion to prepare the single photon states (Sect. 80.1), and a single-photon detector as the “bomb”. Subsequently the technique was implemented incorporating focusing lenses, which would enable the image (more correctly, the silhouette) of an object to be determined with less than one photon per “pixel” being absorbed [80.138]. By adjusting the beamsplitter reflectivities in the above example, one can achieve at most a 50% fraction of measurements that are interaction-free. An improved method, relying on the quantum Zeno effect [80.118] (Sect. 80.5.1), was discovered with which one can in principle make this fraction arbitrarily close to 1 [80.137]. For example, consider a photon initially in cavity #1 of two identical cavities coupled by a lossless beam splitter whose reflectivity R = cos2 (π/2N ). If the photon’s coherence length is shorter than the cavity length, after N cycles the photon will with certainty be located in cavity #2, due to an interference effect (the equivalent of a π-pulse interaction). However, if cavity #2 instead contains an absorbing object (e.g., the ultra-sensitive bomb), at each cycle there is only a small chance (= 1 − R) that the photon will be absorbed; otherwise, the non absorption projects the photon wave packet entirely back onto cavity #1. After all N cycles, the total probability for the photon to be absorbed by the object is 1 − R N , which goes to 0 as N becomes large. (In practice, unavoidable losses in the system limit the maximum number of cycles and hence the achievable performance [80.139].) The photon effectively becomes trapped in cavity #1, thus indicating umambiguously the presence of the object in cavity #2. This quantum-Zeno version of interrogation was first implemented using the inhibited rotation of a pho-

Quantum Optical Tests of the Foundations of Physics

ton’s polarization (the object to be detected blocked one arm of a polarizing interferometer through which the photon was repeatedly cycled), achieving an efficiency of 75% [80.139]. A cavity-based implementation, in which the presence of the absorbing object inside a high-finesse cavity vastly increased the reflection off the cavity [80.140], detected the presence of the object with only 0.15 photons on average being absorbed or scattered [80.141].

80.5.4 Weak and “Protected” Measurements Aharonov, Albert, and Vaidman extended quantum measurement theory by introducing “weak” measurement, a procedure that determines a physical property of a quantum system belonging to an ensemble that is both preselected and postselected [80.142, 143]. In the standard theory, a quantum system is measured by entangling its eigenstates with distinguishable pointer states of a measurement device, completely resolving the observable eigenvalue spectrum. The measurement can be weakened by increasing the overlap of the pointer states, consequently reducing the resolution of the eigenstates. When performed between two measurements this can give surprising results – in contrast to ordinary expectation values, the pointer can lie outside the range of the eigenvalue spectrum of the measured observable O. The “weak value” Ow ≡ Ψini |O|Ψfin / Ψfin |Ψini  between preselected (|Ψini ) and postselected (|Ψfin ) states completely characterizes the outcome of the weak measurement. Weak measurements do not disturb each other, so that the weak values of non-commuting observables can be measured simultaneously [80.144, 145]. Furthermore, they have proved to yield meaningful results in many different circumstances, e.g., Hardy’s paradox of Sect. 80.6.5 [80.146], measurement of negative kinetic energies [80.147], and the “observation” of a single

80.6 The EPR Paradox and Bell’s Inequalities

1195

particle in two locations [80.148]. One other potentially powerful application of weak measurements is the amplification of weak signals, which was first demonstrated by amplifying the birefringence-induced small displacement of optical fields [80.149, 150]. It was recently shown that weak measurements of this kind in fact arise naturally in fiber optics telecom networks, due to polarization-mode dispersion and polarizationdependent losses [80.151]. One recent proposal [80.152] suggests that the controversy over whether or not “welcher Weg” information may be obtained (and interference consequently destroyed) without disturbing a particle’s momentum (Sect. 80.4.2) may be resolved by making weak measurements of momentum inside an interferometer. The above results push us to reexamine our interpretation of wave functions. We customarily use the wave function only as a calculational tool, but we have also learned that it is in some sense physical, and should not be regarded merely as some distribution from classical statistics. One proposal [80.153, 154] suggests that the wave function of a single particle should be regarded as a real entity. When a state is “protected” from change, e.g., by an energy gap, and measurements are performed sufficiently “gently”, one should be able to determine not just the expectation value of position, but the wave function at many different positions, without altering the state of the particle. (This idea of measuring the entire wave function of a single particle should not be conflated with Raymer et al.’s fascinating work on the reconstruction of the quantum state of a light field by repeated sampling of a large ensemble; see [80.155] and Sect. 78.4.) No violation of the uncertainty principle or the no-cloning theorem (Sect. 80.7.1) arises from this, as the ability to “protect” a state relies on some preexisting knowledge about the state; but it assigns a deeper significance to the wave function, one Aharonov terms “ontological,” as opposed to merely epistemological (but see also [80.156]).

80.6 The EPR Paradox and Bell’s Inequalities 80.6.1 Generalities

√ |ψ −  = (|H1 , V2  − |V1 , H2 ) / 2 ,

(80.9)

where the letters denote horizontal (H) or vertical (V) polarization, and the subscripts denote photon propagation direction. This state, analogous to the singlet state of a pair of spin-1/2 particles, is isotropic – it has the same form regardless of what basis is used to describe

Part F 80.6

Nowhere is the nonlocal character of the quantum mechanical entangled state as evident as in the “paradox” of Einstein, Podolsky, and Rosen (EPR) [80.157], the version of Bohm [80.158], and the related inequalities by Bell [80.159, 160]. Consider two photons traveling off back-to-back, described by the entangled

state

1196

Part F

Quantum Optics

it. Measurement of any polarization component for one of the particles will yield a count with 50% probability; individually, each particle is unpolarized. Nevertheless, if one measures the polarization component of particle 1 in any basis, one can predict with certainty the polarization of particle 2 in the same basis, seemingly without disturbing it, since it may be arbitrarily remote. Therefore, according to EPR, to avoid any nonlocal influences one should ascribe an “element of reality” to every component of polarization. A quantum mechanical state cannot specify that much information, and is consequently an incomplete description, according to the EPR argument. The intuitive explanation implied by EPR is that the particles leave the source with definite, correlated properties, determined by some local “hidden variables” not present in quantum mechanics (QM). For two entangled particles, a local hidden variable (LHV) theory can be made which correctly describes perfect correlations or anti-correlations (i. e., measurements made in the same polarization basis). The choice of an LHV theory versus QM is then a philosophical decision, not a physical one. However, in 1964 John Bell [80.159] discovered that QM gives different statistical predictions than does any LHV theory, for situations of nonperfect correlations (i. e., analyzers at intermediate angles). Bell’s inequality (BI) constrains various joint probabilities given by any local realistic theory, and was later generalized to include any model incorporating locality [80.161, 162], and also extended to apply to real experimental situations [80.163, 164]. With the caveat of supplementary assumptions (Sect. 80.6.4), Bell’s inequalities have now been tested many times, and the vast majority of experiments have violated them, in support of QM. One general interpretation is that the predictions of QM cannot be reproduced by any completely local theory. It must be that the results of measurements on one of the particles depend on the results for the other, and these correlations are not merely due to a common cause at their creation [80.165, 166].

80.6.2 Polarization-Based Tests

Part F 80.6

The first BI tests were performed with pairs of photons produced via an atomic cascade, and a later version incorporated rapid (albeit periodic) switching of the analyzers [80.167, 168]. Unfortunately, the angular correlation of the cascade photons is not very strong. In contrast, the strong correlations of the down-converted photons make them ideal for such tests, the first of which were performed using setups essentially identical to that

already discussed in connection with quantum erasure (Fig. 80.4). Orthogonally polarized (e.g., horizontal and vertical) but otherwise identical photons are combined on a nonpolarizing 50–50 beam splitter. If one considers only the events with a single photon in each output (i. e., ignoring the cases for which both photons use the same beam splitter output port), one obtains the (postselected) entangled state (80.9) [80.169, 170]. (In fact, this technique is now used as a method for characterizing the indistinguishability of photons from independent sources, e.g., quantum dots [80.171] or independent down-conversion crystals [80.172, 173].) Down-conversion schemes have also been developed to produce entangled states without the need to postselect out half of the photons. For example, consider a type-I phase-matched crystal (Sect. 72.2.2) that downconverts H-polarized pump photons into V-polarized pairs; and an adjacent, identical crystal that is rotated by 90◦ , thus down-converting V-polarized pump photons into H-polarized pairs. By coherently pumping the two √ crystals with light polarized at |45 ≡ (|V  + |H)/ √2, one obtains the entangled state (|HH + |VV / 2. (More generally, pumping α|V  + eiϕ β|H produces arbitrary nonmaximally entangled states of the form α|HH + eiϕ β|VV [80.174].) Such a source has produced the largest and fastest violations of Bell inequalities to date (over 200-σ violation in less than 1 second) [80.175, 176]. Using type-II phase-matching one also can produce polarization entanglement from a single crystal [80.177]. One member of each down-conversion pair is emitted along an ordinary polarized cone while the other is emitted along an extraordinary polarized cone. If the photons happen to be emitted along the intersection of the two cones, neither photon will have a definite √ polarization – they will be in the state (|HV  + |VH/ 2. This entanglement source has now been used in a variety of quantum investigations, including Bell inequality tests [80.177, 178], quantum cryptography [80.179, 180] (Sect. 80.7.4 and Chapt. 81) and teleportation [80.181, 182], and as a resource for studying entanglement of more than 2 photons [80.37–42].

80.6.3 Nonpolarization Tests The advent of parametric down-conversion has also led to the appearance of several nonpolarization-based BI tests, using, for example, an entanglement of the photon momenta (Fig. 80.8) [80.183]. By use of small irises (labeled ‘A’ in the figure), Rarity and Tapster examined four down-conversion modes: 1s, 1i, 2s, and 2i. Beams

Quantum Optical Tests of the Foundations of Physics

A KD*P

1s

Ds Di

2i

UV Ps 1i A

2s

Di
Pi

Ds


Fig. 80.8 Outline of Rarity and Tapster apparatus used

to demonstrate a violation of a Bell’s inequality based on momentum entanglement [80.183]

1197

Several groups [80.191, 192] have violated a BI based on energy–time entanglement of the photons [80.193]. In the method due to Franson, one member of each down-converted pair is directed into an unbalanced Mach–Zehnder-like interferometer, allowing both a short and long path to the final beam splitter; the other photon is directed into a separate but similar interferometer. There arises interference between the indistinguishable processes (“short–short” and “long– long”) which could lead to coincidence detection. Using fast detectors to select out only these processes, the reduced state (80.9) is  1 |ψ = |S1 , S2  − eiφ |L 1 , L 2  , (80.10) 2 where the letters indicate the short or long path, and the phase is the sum of the relative phases in each interferometer. Although no fringes are seen in any of the singles count rates, the high-visibility coincidence fringes (Fig. 80.9) lead to a violation of an appropriate BI. One conclusion is that it is incorrect to ascribe to the photons a definite time of emission from the crystal, or even a definite energy, unless these observables are explicitly measured. This same sort of arrangement, modified to work with a pulsed pump, has been used to demonstrate the longest violation of local realism, in which Gisin’s group has observed a 16-σ BI violation (modulo the detection and timing loopholes discussed in Sect. 80.6.4) with photons separated by 10.9 km [80.194]. In a related experiment, they have used a similar system to place limits on the “speed of collapse” of the 2-photon wave function, i. e., how fast a nonlocal “signal” would need to propagate from one side of the experiment to the other to account for the measured nonclassical correlations. Depending on some assumptions about the detection process and which inertial frame of reference is considered, the nonlocal-influence speed was constrained to be at least 104 c to 107 c [80.195]. In one interesting variant, the researchers arranged to have moving detectors, such that in the local reference frame of each detector, it was the other detector which initiated the collapse. (Due to the experimental difficulty of accelerating actual detectors to high velocities, a rapidly rotating absorbing disk was placed close to one output port of a polarizing beamsplitter; following ideas discussed in Sect. 80.5.3, the non-absorbance of the photon by the absorber was deemed sufficient to cause a reduction of the wave function.) As expected, the measured correlations were in no way reduced, but this experiment did rule out one alternative theory of nonlocal collapse [80.196,197]. Finally,

Part F 80.6

1s and 1i correspond to one pair of conjugate photons; beams 2s and 2i correspond to a different pair. Photons in beams 1s and 2s have the same wavelength, as do photons in beams 1i and 2i. With proper alignment, after the beam splitters there is no way to tell whether a pair of photons came from the 1s–1i or the 2s–2i paths. Consequently, the coincidence rates display interference, although the singles rates at the four detectors indicated in Fig. 80.8 remain constant. This interference depends on the difference of phase shifts induced by rotatable glass plates Pi and Ps in paths 1i and 2s, respectively, and is formally equivalent to the polarization case considered above, in which it is the difference of polarization-analyzer angles that is relevant. By measuring the coincidence rates for two values for each of the phase shifters – a total of four combinations – the experimenters were able to violate an appropriate BI. One interpretation is that the emission directions of a given pair of photons are not elements of reality. Momentum conservation in the down-conversion process (80.4) also leads to entanglement directly in the spatial modes in the correlated photons. For example, Zeilinger et al. [80.184, 185] and White et al. [80.186] have demonstrated entanglement between the orbital angular momentum of the photons, of the form (| + 1, −1 + |0, 0 + | − 1, +1, where 0 and ±1 respectively denote modes with no orbital angular momentum (gaussian spatial profiles) and ±~ (Laguerre–Gauss-Vortex modes). Note that this enables one to investigate correlations for degrees of freedom that reside in larger Hilbert spaces than do the 2-level systems (e.g., polarization) discussed above. The nonlocal spatial correlations of the down-conversion photons have also given rise to many interesting experiments in the area of quantum imaging [80.187–190], where one is able to obtain √ spatial resolution beyond that predicted by the usual N shot-noise limitations.

80.6 The EPR Paradox and Bell’s Inequalities

1198

Part F

Quantum Optics

Coincidence counts N12 (per 24 s) 700  2 = 0°  2 = 90° Visib. = 92.8  0.9 % Visib. = 90.3  0.7 % 600 500 400 300 200 100 0

Interferometer 1 phase shift,  1

Fig. 80.9 High-visibility coincidence fringes in a Franson

dual-interferometer experiment [80.192] for two values of the phase in interferometer 2 as the phase in interferometer 1 is slowly varied. The curves are sinusoidal fits

by using more than two possible creation times, e.g., with a mode-locked pulsed laser, Gisin’s group demonstrated entanglement for a two-photon state in a Hilbert space of at least dimension 11 [80.198]. EPR-like correlations have also been observed directly between two correlated field modes via homodyne tomography [80.199], though the joint Wigner function is positive-definite here. In contrast, a homodyne measurement on the state of a single photon split between two paths (Sect. 80.4.4) took advantage of the nonclassical nature of the initial state to violate a Bell inequality [80.46]. Like all existing Bell-inequality experiments, this work suffered from several loopholes (Sect. 80.6.4), but a new proposal suggests that the high efficiency of homodyne detection may provide a unique opportunity for a loophole-free test of Bell’s inequalities [80.200].

80.6.4 Bell Inequality Loopholes

Part F 80.6

In fact, to date no single experiment has unambiguously violated a Bell inequality, due to the existence of two experimental challenges, the detection and locality “loopholes”. All of the experiments discussed thus far have required supplementary assumptions, e.g., the “fair-sampling” assumption that the fraction of pairs detected is representative of the entire ensemble emitted by the source. [In fact, for the entangled photons emitted in the atomic cascade experiments, this assumption is manifestly false, because the strong polarization correlations only exist for those photons emitted nearly in opposite directions. If one were to collect all of the emitted photon pairs, they would not lead to a violation (see [80.201]

for a fuller discussion).] To close the locality (or “timing”) loophole requires that the analyzer settings be switched rapidly and randomly, in order to guarantee that no (sub)luminal information transfer could account for the observed correlations. The necessary conditions have been met only in the down-conversion experiment by Zeilinger et al., which separated the photons by 400 m and used ultrafast random number generation and electronic polarization-analysis choice to ensure space-like separated observers [80.178]. However, in that experiment the detection efficiency was less than 5%. In order to understand the detection loophole, consider the Clauser–Horne (CH) form of the Bell inequality [80.164], which relates the directly observable singles rates S1 and S2 and the coincidence rate C12 , rather than “inferred” probabilities, by C12 (a, b) + C12 (a, b ) + C12 (a , b) − C12 (a , b ) ≤ S1 (a) + S2 (b) , (80.11) where a and a (b and b ) are any pair of analyzer (e.g., polarizer) settings at detector 1 (2). For certain choices of a, a , b, and b , quantum mechanics predicts the left hand side of the CH inequality can exceed the right hand side. However, in practice this is very difficult to observe, since the coincidence rates fall as η2 (η is the detection efficiency), compared to the singles rates on the right hand side, which fall only as η. In order to close the detection loophole, one requires η ≥ 83% [80.201] (for maximally entangled photons. (Eberhard has shown that the required detection efficiency may be reduced to 67% by using nonmaximally entangled quantum systems [80.202, 203]. The idea is that one can choose the analysis settings a and b to reduce the value of the RHS of the CH inequality.) In essence, one requires high detection efficiencies to ensure that the contributions from undetected events are not sufficient to cause the total ensemble to satisfy the Bell inequality even while the detected events violate it. To date, only the entangled-ion experiment of Wineland et al. [80.204] has had sufficiently high efficiencies to close the detection loophole. In this experiment the entangled variables were the hyperfine energy levels of 9 Be+ ions. By employing a cycling transition that leads to the emission of many photons if the atom is in one of the states, a detection efficiency in excess of 98% was achieved, allowing an 8-σ Bell inequality violation. However, because the ions were separated by only 3 µm in the same linear Paul trap, and in fact were measured

Quantum Optical Tests of the Foundations of Physics

using the same laser pulse, there was no possibility of closing the locality loophole. More recently, Monroe et al. have demonstrated the entanglement of a trapped ion and a photon [80.205], and have used this to violate a Bell inequality [80.206]. Similar experiments have also enabled the production of up to four entangled ions [80.207]. Though neither of the experimental loopholes was closed in these experiments, they are noteworthy as the first controlled demonstrations of entanglement in massive particles. Efforts are now underway to attempt a direct violation of (80.11) with no auxiliary assumptions using down-conversion photons and high-efficiency (> 85%) single-photon detectors [80.208, 209], in addition to atomic schemes [80.210] and homodyne schemes [80.200] (Sect. 80.6.3), which enjoy even higher intrinsic efficiency.

In the above experiments for testing nonlocality, the disagreement between quantum predictions and Bell’s constraints on local realistic theories are only statistical. Greenberger, Horne, and Zeilinger (GHZ) pointed out that in some systems involving three or more entangled particles, a contradiction could arise even at the level of perfect correlations [80.211, 212]. A schematic of one version of the GHZ Gedankenexperiment is shown in Fig. 80.10. The source at the center is posited to emit

α

a

a


Φa

γ

Φc

c

γ


b

trios of correlated particles. Just as the Rarity–Tapster experiment selected two pairs of photons (Fig. 80.8), the GHZ source selects two trios of photons; these are denoted by abc and a b c . Hence, the state coming from the source may be written  √
(80.12) |ψ = |abc + |a b c  / 2 . After passing through a variable phase shifter (e.g., φa ), each primed beam is recombined with the corresponding unprimed beam at a 50–50 beam splitter. Detectors (denoted by Greek letters) at the output ports signal the occurrence of triple coincidences. The following simplified argument conveys the spirit of the GHZ result. Given the state (80.12), one can calculate from standard QM the probability of a triple coincidence as a function of the three phase shifts: 1 P(φ1 , φ2 , φ3 ) = [1 ± sin(φa + φb + φc )] , 8

β


b
β

Fig. 80.10 A three-particle Gedankenexperiment to demon-

strate the inconsistency of quantum mechanics and any local realistic theory. All beam splitters are 50–50 [80.111]

where the plus sign applies for coincidences between all unprimed detectors, and the minus sign for coincidences between all primed detectors. For the case in which all phases are 0, it will occasionally happen (1/8th of the time) that there will be a triple coincidence of all primed detectors. Using a “contrafactual” approach, we ask what would have happened if φa had been π/2 instead. By the locality assumption, this would not change the state from the source, nor the fact that detectors β and γ went off. But from (80.13) the probability of a triple coincidence for primed detectors is zero in this case; therefore, we can conclude that detector α would have “clicked” if φa had been π/2. Similarly, if φb or φc had been π/2, then detectors β or γ would have clicked. Consequently, if all the phases had been equal to π/2, we would have seen a triple coincidence between unprimed detectors. But according to (80.13) this is impossible: the probability of triple coincidences between unprimed detectors when all three phases are equal to π/2 is strictly zero! Hence, if one believes the quantum mechanical predictions for these cases of perfect correlations, it is not possible to have a consistent local realistic model. Down-conversion experiments have enabled the production of 3- and 4-photon GHZ states, with results in good agreement with theory (the all-or-nothing arguments given above become inequalities in any real experiment) [80.37–39]. By similar arguments, Hardy has shown the inconsistency of quantum mechanics and local realism in a Gedankenexperiment with just two particles [80.213, 214]. When the arguments are suitably modified to

Part F 80.6

c


Φb

1199

(80.13)

80.6.5 Nonlocality Without Inequalities

α


80.6 The EPR Paradox and Bell’s Inequalities

1200

Part F

Quantum Optics

deal with real experiments, inequalities once again result; these have also been experimentally violated, using non-maximally entangled states from

down-conversion [80.215, 216], further underscoring the inconsistency between quantum theory and locality.

80.7 Quantum Information 80.7.1 Information Content of a Quantum: (No) Cloning The inherent nonlocality of particles in an entangled state cannot be used to transmit superluminal messages. For example, if A and B receive a polarization-entangled pair of photons, which A then collapses in a certain basis by performing a polarization measurement, B can only extract one bit of information from a measurement on his photon – this bit corresponds not to A’s choice of basis, but to the (random) outcome of A’s measurement. However, instantaneous communication would be possible if one could make copies (“clones”) of a single photon in an unknown polarization state: by performing measurements on n copies of his photon, B could determine its polarization to a resolution of n bits, thereby accurately determining A’s choice of basis. Taking into account quantum fluctuations [80.217], one finds that no physically allowed amplifier can make a sufficiently faithful copy for such a scheme to work – it is impossible to clone an unknown quantum state [80.218]. A simple proof is as follows. Consider an ideal cloner, initially in the state |ic , which would take |0|ic → |0|0c and |1|ic → |1|1c . For an accurate copier, one would expect (|0 + |1)|ic → (|0 + |1)(|0c + |1c ), but the linearity of quantum transformations instead yields |0|0c + |1|1c , i. e., an entangled state. Although perfect cloning is impossible, it is nevertheless possible to create copies which are “pretty good”. Specifically, using an optimal cloning strategy, one can in principle create a copy with a fidelity of 5/6 with the original state [80.219, 220]. Such a cloning procedure has been experimentally realized by several groups, e.g., relying on stimulated emission, or sending the photon to be cloned through a low noise optical amplifier, with results matching the theoretical predictions [80.221–223].

Part F 80.7

80.7.2 Super-Dense Coding The previous considerations make the work by Bennett et al. on “quantum teleportation” and related effects all the more remarkable. In the quantum dense-coding protocol, a single photon can be used to transfer two bits of information, when it is part of an en-

tangled EPR pair [80.224]. Again consider A and B, each possessing one photon of such a pair. By manipulating only her photon (via a polarization rotator and a phase shifter), A can convert the initial joint state |ψ −  (80.9) into any of the four two-particle √ “Bell states” [|ψ ± B  ± |V A , H B ) / 2, AB  = (|H A , V√ |φ± AB  = (|H A , H B  ± |V A , V B ) / 2], and then send her photon to B. By making a suitable measurement on both photons, B can then in principle determine which of the four states A produced [80.225]: A’s single photon carried two bits. This protocol has been experimentally realized using down-conversion photons [80.226], though only two of the four Bell states could be reliably distinguished. (Standard polarization Bell state analysis is implemented by combining the two photons on a nonpolarizing 50–50 beam splitter. For any of the triplet states |ψ + , |φ± , the photons will both travel to the same output port due to the Hong–Ou–Mandel interference discussed in Sect. 80.3.3; only for the state |ψ −  will the photons travel to different outputs, resulting in a coincidence detection.)

80.7.3 Teleportation In the even more striking quantum teleportation effect [80.227], an unknown polarization state f (with its in-principle infinite amount of information) can be “teleported” from A to B, if each already possesses one photon of an EPR pair (e.g., in the singlet state |ψ − AB ). First, A jointly measures her EPR photon and the photon F (whose state f is to be teleported) in the basis defined by the Bell states of these two photons. Via a mere two bits of classical information, A then informs B which of the four (equally probable) Bell states she actually measured. With this information, B can transform the state of his EPR particle into f . For ex− ample, if A found the singlet state |ψBF , then the polarization of her EPR photon must have been orthogonal to that of F (because the polarizations of particles in a singlet state are always perfectly anticorrelated, regardless of the quantization basis). But because the two EPR photons were initially also in a singlet state, their polarizations must also be orthogonal, so B’s EPR pho− ton is already in state f . If instead A found |φBF ,

Quantum Optical Tests of the Foundations of Physics

1201

interconverted into “out” qubits, thus obeying timereversal symmetry. To resolve the paradox of the apparent loss of information of matter falling into a black hole, Lloyd and Ng propose that pairs of entangled photons can materialize at the event horizon of a black hole. One member of the photon pair flies outward to become the Hawking blackbody radiation; the other falls into the black hole and hits the singularity together with the matter that formed the hole. The annihilation of the infalling photon acts as a measurement on the infalling matter in a quantum teleportation-like process, transporting the information contained in the infalling matter to the outgoing Hawking radiation, using the Horowitz–Maldacena mechanism [80.236].

80.7.4 Quantum Cryptography Although EPR schemes cannot send signals superluminally, they have other potential applications in cryptography. In the “one-time pad” of classical cryptography [80.237], two collaborators share a secret “key” (a random string of binary digits) in order to encode and decode a message. Such a key may provide an absolutely unbreakable code, provided that it is unknown to an eavesdropper. The problem arises in key distribution: any classical distribution scheme is subject to noninvasive eavesdropping, e.g., using a fiber-coupler to tap the line, without disturbing the transmitted classical signal. In quantum cryptography proposals, security is guaranteed by using single-photon states [80.238, 239], some of the schemes employing particles prepared in an EPRentangled state. Each collaborator receives one member of each correlated pair, and measures the polarization (or whatever degree of freedom is carrying the information) in a random basis. After repeating the process many times, the two then discuss publicly which bases were used for each measurement, but not the actual measurement results. The cases where different bases were chosen are not used for conveying the key, and may be discarded, along with instances where one party detected no photon. In cases where the same bases were used, however, the participants will now have correlated information, from which a random, shared key can be generated. As long as single photons are used, any attempt at eavesdropping, even one relying on QND, will necessarily introduce errors due to the uncertainty principle. If the eavesdropper uses the wrong basis to study a photon before sending it on to the real recipient, the very act of measuring will disturb the original state.

Part F 80.7

for example, then B simply makes the same transformation (with a polarization rotator) that would have − changed |ψ − AB  into |φ AB , again leaving his photon in the state f . Thus, although one may only extract one bit of (normally useless) information from an EPR particle, the perfect correlations may be used to transfer an infinite amount of information, i. e., precise specification of a point in the state space of the particle (the Poincaré sphere for a photon or the Bloch sphere for an electron). The “no-cloning” theorem (Sect. 80.7.1) is not violated, since A irrevocably alters the state of F by the measurement she performs, leaving only one particle in f . A number of experiments have now experimentally realized quantum teleportation. The first of these used polarization-entangled down-conversion photons [80.181], but was limited by the impossibility to resolve all four Bell states using only linear optics. Teleportation of continous-variable states has also been observed [80.228], using twin-beam squeezed states (Sect. 80.2.3). Recently, the first teleportation in a matter system has been achieved: the groups of Blatt et al. [80.229] and Wineland et al. [80.230] have successfully teleported the (energy) quantum state of an ion to a separate ion. Although the overall distance was less than 1 mm, these challenging experiments are significant because they incorporate most of the techniques necessary for scalable quantum information processing in an ion-trap system (see Sect. 81.7.2). In an interesting extension of the original teleportation protocol, one can ask what happens if the photon to be teleported is itself entangled to a 4th photon G. In this case, known as “entanglement swapping”, a successful teleportation will lead to the entanglement of photons G and B, even though these have never directly interacted. Entangled down-conversion photons have been used to demonstrate such “entanglement swapping” achieving a violation of Bell’s inequalities between the two noninteracting photons [80.231]; similar results have been achieved with continous variables as well [80.232]. Such procedures may one day enable construction of a quantum “repeater” which could enable the transmission of quantum information over long distances [80.233, 234]. A fundamental connection between quantum information and black holes has recently been suggested by Lloyd and Ng [80.235] and others. The basic idea is that every physical object, including a black hole, can be thought of as a quantum computer that unitarily transforms input states to output states; i. e., “in” quantum bits (“qubits”) can always be reversibly

80.7 Quantum Information

1202

Part F

Quantum Optics

Although one can perform quantum cryptography with single-photon states or even weak coherent states (Sect. 81.2.1), there are a number of advantages to using entanglement: there is inherent randomness; photons from different pairs show no correlations (unlike multiple photons in an attenuated coherent state, which may be used by an eavesdropper to gain information), and any information leaked to other degrees of freedom automatically shows up as increased error rate [80.240]. This last feature means that the sources are “self-checking” in a sense, and one could even let an adversary have control over the source, secure in the knowledge that any tampering would become evident in the nonlocal correlations. A number of quantum cryptography implementations using down-conversion photon pairs have been realized, either using polarization entanglement [80.179, 180, 241, 242], or energy–time entanglement [80.243, 244]. One main experimental challenge will be to increase the rates of entangled pair production (the typical source emission rate for these cryptography experiments is only 10 000 /s), to

make them competitive with current weak coherent pulse schemes (which can easily operate at over 10 MHz).

80.7.5 Issues in Causality Outstanding causal paradoxes in optics include the paradox of Barton and Scharnhorst [80.245], closely related to the Casimir effect (Sect. 79.2.4). In this case, the amplitude for light-by-light scattering is modified by the presence of closely-spaced parallel conducting plates. It appears that this can lead to propagation of light in vacuum (albeit a vacuum “colored” by the presence of the Casimir plates) faster than c. Hegerfeldt has pointed out similar paradoxes in connection with localization of any particle in a quantum field theory [80.246] and with the Glauber theory of photodetection (Sect. 78.4) [80.247]. However, at least for the simplest example – the interaction between two widely separated atoms – as long as one considers only probabilities that depend on the separation r, the second atom cannot be excited by light from the first until after a time r/c [80.248].

80.8 The Single-Photon Tunneling Time 80.8.1 An Application of EPR Correlations to Time Measurements

Part F 80.8

In this section we discuss experiments involving the quantum propagation of light in matter. Due to the sharp time correlations of the paired photons from spontaneous down-conversion, one can use the HOM interferometer (Sect. 80.3.3, and Fig. 80.4) to measure very short relative propagation delay times for the signal and idler photons. One early application was therefore to confirm that single photons in glass travel at the group velocity [80.91]. At least until recently, the only quantum theory of light in dispersive media was an ad hoc one [80.249–256]. The shift of the interference dip resulting from a medium introduced into one of the interferometer arms can be accurately measured by determining how much the path lengths must be changed to compensate the shift and recover the dip. This result suggests that when looking for a microscopic description of dielectrics, it is unnecessary to consider the medium as being polarized by an essentially classical electric field due to the collective action of all photons present, and reradiating accordingly. Linear dielectric response is not a collective effect in this sense – each photon interferes only with itself (as per Dirac’s dictum) as it is par-

tially scattered from the atoms in the medium. The single-photon group velocity thus demonstrates “wave– particle unity.” The standard limitation for measurements of shorttime phenomena is that to have high time-resolution, one needs short pulses (or at least short coherence lengths), but these in turn require broad bandwidths and are therefore very susceptible to dispersive broadening. It is a remarkable consequence of the EPR energy correlations of the down-conversion photons (Sect. 80.6.3) that time measurements made with the HOM interferometer are essentially immune to such broadening [80.91, 257, 258]. In effect, the measurement is sensitive to the difference in emission times while the broadening is sensitive to the sum of the frequencies. While frequency and time cannot both be specified for a given pulse, the crucial feature of EPR correlations such as those exhibited by down-converted photons is that this difference and this sum correspond to commuting observables, and both may be arbitrarily well defined. The photon which reaches detector 1 could either have traversed the dispersive medium and been transmitted, or traversed the empty path and been reflected, leaving its twin to traverse the medium. The medium thus samples both of the (anticorrelated) frequencies, leading to an automatic cancellation of

Quantum Optical Tests of the Foundations of Physics

any first-order (and in fact, all odd-order) dispersive broadening. Measurements can be more than 5 orders of magnitude more precise than would be possible via electronic timing of direct detection events, and in principle better than those performed with nonlinear autocorrelators (which rely on the same nonlinear physics as down-conversion, but do not benefit from a cancellation of dispersive broadening).

80.8.2 Superluminal Tunneling Times Another well-known problem in the theory of quantum propagation is the delay experienced by a particle as it tunnels. There are difficulties associated with calculating the “duration” of the tunneling process, since evanescent waves do not accumulate any phase [80.259–261]. First, the kinetic energy in the barrier region is negative, so the momentum is imaginary. Second, the transit time of a wavepacket peak through the barrier, defined in the stationary phase approximation by

(80.14) τ (φ) ≡ ∂ arg t(ω) eikd /∂ω , tends to a constant as the barrier thickness diverges, in seeming violation of relativistic causality. (Actually, it is shown in [80.88] that such saturation of the delay time is a natural consequence of time-reversal symmetry, and in [80.262] that one can deduce from the principle of causality itself that every system possesses a superluminal group delay, at least at the frequency where its transmission is a minimum.) For example, the transmission function for a rectangular barrier, t(k, κ) =

e−ikd −k cosh κd + i κ 2kκ sinh κd 2

2

,

(80.15)

1203

imply that it is a better measure of the duration of the interaction than is the group delay [80.259]. The Larmor time [80.264] is one of the early efforts to attach a “clock” to a tunneling particle, in the form of a spin aligned perpendicular to a small magnetic field confined to the barrier region. The basic idea is that the amount of Larmor precession experienced by a transmitted particle is a measure of the time spent by that particle in the barrier. This clock turns out to contain components corresponding both to the distance-independent “dwell time” and the linear-in-distance semiclassical time. Curiously, the most common theories for tunneling times become superluminal in certain cases anyway, whether or not they deal with the motion of wave packets. Here, we will restrict ourselves to discussing the time of appearance of a peak of a single-photon wave packet. While other tunneling-time experiments have been performed in the past [80.265], optical tests offer certain unique advantages [80.266], including the ease of construction of a barrier with no dissipation, very little energy-dependence, and a superluminal group delay. The transmitted wave packets suffer little distortion, and are essentially indistinguishable from the incident wave packets. At a theoretical level, the fact that photons are described by Maxwell’s (fully relativistic) equations is an important argument against interpreting superluminal tunneling predictions as a mere artifact of the nonrelativistic Schrödinger equation. Also, one is denied the recourse suggested by some workers [80.267] of interpreting the superluminal appearance of transmitted peaks to mean that only the high-energy components (which, for matter waves, traveled faster even before reaching the barrier [80.268]) were transmitted.

80.8.3 Tunneling Delay in a Multilayer Dielectric Mirror A suitable optical tunnel barrier can be a standard multilayer dielectic mirror. The alternating layers of low and high index material, each one quarter-wave thick at the design frequency of the mirror, lead to a photonic bandgap [80.269] analogous to that in the Kronig– Penney model of solid state physics (Sect. 79.2.6). The gap represents a forbidden range of energies, in which the multiple reflections will interfere constructively so as to exponentially damp any incident wave. The analogy with tunneling in nonrelativistic quantum mechanics arises because of the exponential decay of the field envelope within the periodic structure, i. e., the imaginary value of the quasimomentum. The same qualitative features arise for the transmission time: for thick barriers,

Part F 80.8

leads in the opaque limit (κd  1) to a traversal time of 2m/~kκ, independent of the barrier width d. The same result applies to photons undergoing frustrated total internal reflection [80.88], when the mass m is replaced by n 2 ~ω/c2 , and similar results apply to other forms of tunneling. Some researchers have therefore searched for some more meaningful “interaction time” for tunneling, which might accord better with relativistic intuitions and perhaps have implications for the ultimate speed of devices relying on tunneling [80.263]. The “semiclassical time” corresponds to treating the magnitude of the (imaginary) momentum as a real momentum. This time is of interest mainly because it also arises in Büttiker and Landauer’s calculation of the critical timescale in problems involving oscillating barriers, which they take to

80.8 The Single-Photon Tunneling Time

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it should saturate at a constant value, as was verified in a recent experiment employing short classical pulses [80.270]. (A more direct analogy, that of waveguides beyond cutoff, yielded similar results in a classical microwave experiment [80.271], while another paper has reported superluminal effects related to the penetration of diffracted or “leaky” microwaves into a shadow region [80.272]. All these experiments involve very small detection probabilities, just as in Chu and Wong’s pioneering experiment on propagation within an absorption line [80.273]. However, it has been predicted that superluminal propagation could occur without high loss or reflection [80.274–276], by operating outside the resonance line of an inverted medium (Sect. 70.1). One can understand the effect as off-resonance “virtual amplification” of the leading edge of a pulse.) The phenomenon was investigated at the singlephoton level by using the high time-resolution techniques discussed in Sect. 80.8.1 to measure the relative delay experienced by down-conversion photons [80.277] when such a tunnel barrier (consisting of 11 layers) was introduced into one arm of a HOM interferometer. The transmissivity of the barrier was relatively flat throughout the bandgap (extending from 600 nm to 800 nm; Fig. 80.11), with a value of 1% at the gap center (700 nm), where the experiment was performed. The HOM coincidence dip was measured both with the barrier (and its substrate) and with the substrate Transmission probability (%) 100

Time (fs)

20

90

18

80

16

70

14

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12

50

10

40

8

30

6

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4

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2

Part F 80.8

0 450

550

650

750

850

0 950 1050 Wavelength (nm)

Fig. 80.11 Transmission probability for the tunnel barrier used

in [80.277] (heavy dotted curve); heavy black, dashed brown, and solid grey curves show group delay, Larmor time, and semiclassical time. Also shown for comparison is the “causality limit” d/c = 3.6 fs (horizontal line)

Coincidence counts (normalized)

Without barrier

With barrier

∆ t ≈ –2 fs Time

Fig. 80.12 Coincidence profiles with and without the tunnel barrier map out the single-photon wave packets. The lower profile shows the coincidences with the barrier; this profile is shifted by ≈ 2 fs to negative times relative to the one with no barrier (upper curve): the average particle which tunnels arrives earlier than the one which travels the same distance in air

alone (Fig. 80.12). Each dip was subsequently fitted to a Gaussian, and the difference between their centers was calculated. When several such runs were combined, it was found that the tunneling peak arrived 1.47 ± 0.21 fs earlier than the one traveling through air, in reasonable agreement with the theoretical prediction of approximately 1.9 fs. Taking into account the 1.1 µm thickness of the barrier, this implies an effective photon tunneling velocity of 1.7 c. The results exclude the “semiclassical” time but are consistent with the group delay. An investigation of the energy-dependence of the tunneling time was also performed, by angling the dielectric mirror, thus shifting its bandgap [80.278]. The data confirm the group delay in this limit as well, and rule out identification of Büttiker’s Larmor time with a peak propagation time, at least in this optical system.

80.8.4 Interpretation of the Tunneling Time Even though a wave packet peak may appear on the far side of a barrier sooner than it would under allowed propagation, it is important to stress that no information is transmitted faster than c, nor on average is any energy. These effects occur in the limit of low transmission, where the transmitted wavepacket can be considered as a “reshaped” version of the leading edge of the incident pulse [80.273, 279, 280]. At a physical level, the

Quantum Optical Tests of the Foundations of Physics

1205

question of whether time scales as defined by the Bohm model have any physical meaning. The “weak measurement” approach of Aharonov et al. [80.142, 143] or equivalently, complex conditional probabilitiy amplitudes obeying Bayes’s theorem [80.288, 289], can be used to address the question of tunneling interaction times in an experimentally unambiguous way. The real part of the resulting complex times determine the effect a tunneling particle would have on a “clock” to which it coupled, while the imaginary part indicates the clock’s back-action on the particle. They unify various approaches such as the Larmor and Büttiker–Landauer times, as well as Feynman-path methods. In addition, they allow one to discuss separately the histories of particles which have been transmitted or reflected by a barrier, rather than discussing only the wave function as a whole. Interestingly, these calculations do not support the assertion that transmitted particles originate in the leading ramp-up of a wave packet.

80.8.5 Other Fast and Slow Light Schemes In addition to the case of tunneling described already, apparently superluminal propagation was observed in a Bessel-beam geometry [80.290], and in the case of an inverted medium [80.291], as described in [80.274– 276]. While the former case may be explained geometrically, the latter – in which superluminal group velocities occur without significant gain, loss, or distortion – raises difficult questions about the speeds of propagation of both energy and information. For two contrasting perspectives, see [80.292] and [80.293]. Much more work has followed, including theoretical treatments of the role of quantum noise in preventing superluminal information transfer [80.294], and attempts to experimentally compare the velocity of information transfer with the group velocity [80.295]. The latter work seemed to verify the claim, previously tested only in an electronic analog [80.296], that even in the regime of superluminal propagation, new information was limited to causal speeds. Some dispute has persisted [80.297], and it seems clear that a more rigorous definition of information velocity is required; somewhere between the idealized extremes of infinitely sharp signal fronts and strictly finite signal bandwidths lies the real world, and neither the front velocity nor the group velocity should be expected to completely describe the behavior of actual information-carrying pulses. At the same time, the definition of the energy

Part F 80.8

reflection from a multilayer dielectric is due to destructive interference among coherent multiple reflections between the different layers. At times before the field inside the structure reaches its steady-state value, there is little interference, and a non-negligible fraction of the wave is transmitted. This preferential treatment of the leading ramp-up engenders a sort of “optical illusion,” shifting the transmitted peak earlier in time. A signal, such as a front with a sharp onset, relies on arbitrarily high-frequency components, which would not benefit from this illusion, but instead travel arbitrarily close to the vacuum speed of light c. Even for a smooth wave packet, no energy travels faster than light; most is simply reflected by the barrier. Only if one considers the Copenhagen interpretation of quantum mechanics, with its instantaneous collapse, does one find superluminal propagation of those particles which happen to be transmitted. This leads to the question of whether it is possible to ask which part of a wave packet a given particle comes from. One paper argued that transmitted particles do in fact stem only from the leading ramp-up of the wave packet [80.281]. While it is true that the transmission only depends on causally connected portions of the incident wave packet, further analysis revealed that simultaneous discussion of such particle-like questions and the wave nature of tunneling ran afoul of the complementarity principle [80.282]. In essence, labeling the initial positions of a tunneling particle destroys the careful interference by which the reshaping occurs (as in the quantum eraser, Sect. 80.4.1). However, one picture in which the transmitted particles really do originate earlier is the Bohm–de Broglie model of quantum mechanics [80.283, 284]. This theory considers Ψ to be a real field (residing however in configuration space, thus incorporating nonlocality) which guides pointlike particles in a deterministic manner. It reproduces all the predictions of quantum mechanics without incorporating any randomness; the probabilistic predictions of QM arise from a range of initial conditions. Bohm’s equation of motion has the form of a fluid-flow equation, v(x) = ~∇ arg Ψ(x)/m, implying that particle trajectories may never cross, as velocity is a single-valued function of position. Consequently, all transmitted particles originate earlier in the ensemble than all reflected particles [80.282, 285]. This approach yields trajectories with well-defined (and generally subluminal) dwell times in the barrier region. However, the fact that the mean tunneling delay of Bohm particles diverges as the incident bandwidth becomes small, along with other interpretational issues [80.286, 287], leaves open the

80.8 The Single-Photon Tunneling Time

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velocity in active media cannot be resolved without reviving long-standing conundra about how to apportion energy between the propagating field and energy stored in the medium [80.298]. Since 1999, there has actually been much more excitement over so-called “slow light” than over “fast light” [80.299]. Building on the concepts of electromagnetically-induced transparency [80.300] (Sect. 69.7), two groups succeeded in that year in utilizing the steep dispersion curves which can be generated by extremely narrow holes in absorption lines to slow light by a remarkable factor, with group velocities as low as 17 m/s [80.301, 302]. Two years later, two experiments succeeded in bringing light to a standstill [80.303, 304]. This can be un-

derstood in terms of a “dark-state polariton” model, in which the propagating photon is adiabatically converted into a (stationary) metastable atomic excitation. In addition to the obvious possibilities for storage of optical pulses, particularly tantalizing at the quantum level [80.305], there have been several proposals for generating extremely strong optical nonlinearities in this system, perhaps directly applicable to quantum information processing [80.306, 307]. Both slow light and fast light have now been observed in solid-state systems [80.308, 309], as well. In both cases, the extension to the single-photon level remains a major goal, for both practical and theoretical reasons. For a recent review on anomalous optical propagation velocities, see [80.310].

80.9 Gravity and Quantum Optics

Part F 80.9

According to general relativity, gravitational radiation can be produced and detected by moving mass distributions [80.311–313]. However, gravitational radiation is coupled only to time-varying mass quadrupole moments in lowest order, since the mass dipole moment is  j m j r j = M Rcm and Rcm for a closed system can only exhibit uniform rectilinear motion. Current efforts focus on detecting gravitational waves (typically at 100 Hz to 1 kHz) from astrophysical sources, such as supernovae or collapsing binary stellar systems. For example, it is expected that in the nearby Virgo cluster of galaxies, several such events should occur per year, each yielding a fractional strain (∆L/L) of 10−21 on Earth. However, there are large uncertainties in this estimate. Two main efforts have been pursued toward gravitational wave detection. The first type of detector, the resonant-mass detector (sometimes known as a “Weber bar,” after its inventor), utilizes a large cylindrical mass whose fundamental mode of acoustical oscillation is resonantly excited by time-varying tidal forces produced by the passage of a gravitational wave. The induced motions are typically detected by piezoelectric crystals, or by SQUIDs (superconducting quantum interference devices) [80.314], yielding strain sensitivi√ ties better than 10−18 / Hz. Such detectors were first constructed in the 1960’s [80.313], and are still in use (e.g., such as the 2.3-ton bar at Louisiana State University, ALLEGRO [80.315]), but to date no incontrovertible detections have been reported. (Attempts to improve the signal-to-noise ratio in resonant-mass GW detectors led to the consideration of back-actionevading sensors (a special case of QND measurements,

Sect. 80.5.2) to circumvent the standard quantum noise limit [80.316].) More recently, a large amount of research has been devoted to using optical interferometry to detect gravitational radiation. A passing gravitational wave alters the relative path length in the arms of a Michelson interferometer, thereby slightly shifting the output fringes. Although the effective gravitational mass of the light is much smaller than that of the Weber bar, very long interferometer arms (2–4 km, with a Fabry–Perot cavity in each arm to increase the effective length) more than make up for this disadvantage. The signal-tonoise ratio for the detection of a fringe shift depends on the power of the light. The US initiative, called LIGO (Laser Interferometer Gravitational-Wave Observatory [80.317]) uses 10 W from a Nd:YAG laser, and an additional external mirror to recirculate the unmeasured light, thus increasing the stored light power up to 10 kW. There are three LIGO interferometers, located respectively in Hanford, Washington (with both a 2 km and a 4 km version) and Livingston, Louisiana (4 km version). The registration of coincident events at the separated interferometers allows one to rule out terrestial artifacts, but many problems involving seismic and thermal isolation, absorption and heating, intrinsic thermal noise and optical quality had to be addressed. The LIGO interferometers have had several preliminary science runs since Fall 2002; the first true “search run” is scheduled for 2005 [80.318]. The√present sensitivities, which range from 10−21 to 10−22 / Hz (at ≈ 200–300 Hz) √ are approaching the initial design goal of 3 × 10−23 / Hz.

Quantum Optical Tests of the Foundations of Physics

Other similar detectors nearing, or currently in, operation are VIRGO (in Pisa, Italy), GEO 600 (near Hannover, Germany), and TAMA (in Tokyo, Japan); planning for an instrument (ACIGA) in Australia is underway, as is planning for the NASA/ESA collaborative Laser Interferometer Space Antenna (LISA), currently aiming for launch in 2012 [80.319]); this space-based interferometer, with arms up to 5 × 106 km long, would probe frequencies from 10−4 to 10−1 Hz, inaccessible to terrestial experiments due to seismic and atmospheric disturbances. Because the standard quantum noise limit of these detectors is ultimately determined by the vacuum fluctuations incident on the unused input ports of the interferometers, it is in principle possible to achieve reduced noise levels by using squeezed vacuum instead [80.313, 320] (Sect. 80.2.3).

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This has been demonstrated experimentally in tabletop experiments [80.55, 321], though there are no plans to incorporate it into the current version of LIGO. It has also been suggested that matter waves which interact with gravity waves inside a matter–wave interferometer (Sect. 80.3.1) could lead to a sensitive method to detect gravitation radiation [80.322, 323]. Such a “Matter–wave Interferometric Gravitationalwave Observatory” (MIGO) may allow the detection of primordial gigahertz gravity waves arising from the Big Bang [80.324]. Moreover, quantum mechanical detectors based on the use of macroscopically coherent entangled states may enable quantum transducers which can interconvert between electromagnetic and gravitational radiations, based on time-reversal symmetry [80.325, 326].

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80.286 B.-G. Englert, M.O. Scully, G. Süssmann, H. Walther: Z. Naturforsch. A 47, 1175 (1992) 80.287 C. Dewdney, L. Hardy, E. J. Squires: Phys. Rev. A 184, 6 (1993) 80.288 A. M. Steinberg: Phys. Rev. A 52, 32 (1995) 80.289 A. M. Steinberg: Phys. Rev. Lett. 74, 2405 (1995) 80.290 D. Mugnai, A. Ranfagni, R. Ruggeri: Phys. Rev. Lett. 84, 4830 (2000) 80.291 L. J. Wang, A. Kuzmich, A. Dogariu: Nature 406, 277 (2000) 80.292 G. Nimtz, W. Heitmann: Prog. Quant. Electr. 21, 81 (1997) 80.293 R. Y. Chiao, A. M. Steinberg: Prog. Opt. 37, 345 (1997) 80.294 A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, R. Y. Chiao: Phys. Rev. Lett. 86, 3925 (2001) 80.295 M. D. Stenner, D. J. Gauthier, M. A. Neifeld: Nature 425, 695 (2003) 80.296 M. W. Mitchell, R. Y. Chiao: Phys. Rev. A 230, 133 (1997) 80.297 G. Nimtz: Nature 429, 695–698 (2004), Brief Communications Arising (6 May 2004) 80.298 G. Diener: Ann. Phys. (Leipzig) 7, 639 (1998) 80.299 P. W. Milonni: Fast Light, Slow Light, and LeftHanded Light (Institute of Physics, Bristol 2004) 80.300 J. P. Marangos: J. Mod. Opt 45, 471 (1998) 80.301 L. V. Hau, S. E. Harris, Z. Dutton, C. H. Behroozi: Nature 397, 594 (1999) 80.302 M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, M. O. Scully: Phys. Rev. Lett. 82, 5229 (1999) 80.303 C. Liu, Z. Dutton, C. H. Behroozi, L. V. Hau: Nature 409, 490 (2001) 80.304 D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, M. D. Lukin: Phys. Rev. Lett 86, 783 (2001) 80.305 C. H. van der Wal, M. D. Eisaman, A. André, R. L. Walsworth, D. F. Phillips, A. S. Zibrov, M. D. Lukin: Science 301, 196 (2003) 80.306 S. E. Harris, L. V. Hau: Phys. Rev. Lett. 82, 4611 (1999) 80.307 M. D. Lukin, M. Fleischhauer, R. Cote: Phys. Rev. Lett. 87, 037901 (2001) 80.308 A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar: Phys. Rev. Lett. 88, 023602 (2002) 80.309 N. S. Bigelow, N. N. Lepeshkin, R. W. Boyd: Science 301, 200 (2003) 80.310 R. W. Boyd, D. J. Gauthier: Prog. Opt. 43, 497 (2002) 80.311 S. Weinberg: Gravitation and Cosmology (Wiley, New York 1972) 80.312 C. W. Misner, K. S. Thorne, J. A. Wheeler: Gravitation (Freeman, San Francisco 1987) 80.313 K. S. Thorne: Gravitational radiation. In: Three Hundred Years of Gravitation, ed. by S. W. Hawking, W. Israel (Cambridge Univ. Press, Cambridge 1987) p. 400 80.314 P. F. Michelson, J. C. Price, R. C. Taber: Science 237, 150 (1987) 80.315 http://sam.phys.lsu.edu/

Quantum Optical Tests of the Foundations of Physics

80.316 C. M. Caves, K. S. Thorne, W. P. Drever, V. D. Sandberg, M. Zimmerman: Rev. Mod. Phys. 52, 341 (1980) 80.317 B. C. Barish, R. Weiss: Phys. Today 52, 44 (1999) 80.318 http://www.ligo.caltech.edu/ 80.319 http://lisa.jpl.nasa.gov/; http://sci.esa.int/science-e/www/area/index.cfm? fareaid=27 80.320 C. M. Caves: Phys. Rev. D 23, 1693 (1981) 80.321 M. Xiao, L.-A. Wu, H. J. Kimble: Phys. Rev. Lett. 59, 278 (1987) 80.322 R. Y. Chiao, A. D. Speliotopoulos: J. Mod. Opt. 51, 861 (2004)

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80.323 A. D. Speliotopoulos, R. Y. Chiao: Phys. Rev. D. 69, 084013 (2004) 80.324 S. Foffa, A. Gasparini, M. Papucci and R. Sturani: http://arxiv.org/abs/gr-qc/0407039 (2004) 80.325 R. Y. Chiao, J. Boyce, J. C. Garrison: Ann. N. Y. Acad. of Sci. 755, 400 (1995) 80.326 R. Y. Chiao: Conceptual tensions between quantum mechanics and general relativity: Are there experimental consequences? In: Science and Ultimate Reality, ed. by J. D. Barrow, P. C. W. Davies, C. L. Harper Jr. (Cambridge Univ. Press, Cambridge 2004) p. 254

Part F 80

1215

81. Quantum Information

For many years atomic physicists had used quantum mechanics very successfully to calculate energy levels, cross sections and other practical quantities, and for the most part, left the philosophical issues of interpretation to others. But after the work of Bell in the 1960’s showed that the peculiarly nonlocal nature of quantum correlations could be tested in the lab, a number of atomic physicists turned to the experimental study of entanglement and quantum measurement. A second phase began at the start of the 1990’s when it was realized that correlations and quantum superpositions could be exploited in quantum information processing and secure communication. This has led to an explosive growth of the subject over the past 10 years, fuelled by the long-term prospects of quantum computing and the nearer goal of quantum cryptography. We review some of these developments in this chapter.

81.1

81.2 Simple Quantum Protocols.................... 1218 81.2.1 Quantum Key Distribution .......... 1219 81.2.2 Quantum Teleportation .............. 1219 81.2.3 Dense Coding............................ 1220 81.3 Unitary Transformations....................... 1221 81.3.1 Single-Qubit Operations............. 1221 81.3.2 Two-Qubit Operations................ 1221 81.3.3 Multi-Qubit Gates and Networks . 1222 81.4 Quantum Algorithms............................ 1222 81.4.1 Deutsch–Jozsa Algorithm ........... 1222 81.4.2 Grover’s Search Algorithm .......... 1223 81.5 Error Correction ................................... 1223 81.6 The DiVincenzo Checklist ...................... 1224 81.6.1 Qubit Characterization, Scalability1224 81.6.2 Initialization ............................ 1224 81.6.3 Long Decoherence Times ............ 1224 81.6.4 Universal Set of Quantum Gates .. 1225 81.6.5 Qubit-Specific Measurement....... 1225 81.7

Physical Implementations .................... 1225 81.7.1 Linear Optics............................. 1225 81.7.2 Trapped Ions ............................ 1226 81.7.3 Cavity QED ................................ 1226 81.7.4 Optical Lattices, Mott Insulator.... 1227

Quantifying Information ...................... 1216 81.1.1 Separability Criterion ................. 1216 81.1.2 Entanglement Measures ............. 1217

81.8 Outlook .............................................. 1227

Quantum information theory is regarded as a mainly mathematics-based subject area which straddles the fields of theoretical physics (quantum mechanics and statistics), mathematics, and theoretical computer science. Its success stems from the introduction of novel methods into both physics and mathematics. The fundamental quantity and resource in many applications in quantum information processing is quantum-mechanical entanglement between spatially separated subsystems. Entanglement is a purely quantum-mechanical effect and has led to numerous speculations about the validity of quantum mechanics itself for its apparent paradoxical implications. Most, if not all, of these difficulties have been resolved and can be mostly attributed to the simple

fact that paradoxical behaviour is incompatible with common sense or everyday experience. This initial upsetting seems to be common to all revolutionary theories and has occurred most notably in Einstein’s theory of relativity [81.1]. These quantum-mechanical correlations have numerous applications in quantum cryptography [or rather quantum key distribution (QKD)], quantum communication, dense coding, and act as the main resource in quantum computing. We will briefly touch upon some mathematical issues concerning separability, quantification of entanglement and channel capacities before describing how quantum key distribution, teleportation and dense coding work. After that, a brief discussion of single-qubit and two-qubit quantum gates

References .................................................. 1228

Part F 81

Quantum Info

1216

Part F

Quantum Optics

Part F 81.1

follows before we describe the simplest quantum algorithms. The issues of error correction and fault tolerant computation as well as DiVincenzo’s checklist (which any realization should satisfy) provide the background for the discussion of some physical implementations. We are acutely aware of the fact that we can give only a brief introduction into what has become a major field of investigation over the last decade. There are al-

ready a number of review articles and textbooks on the market that cover the vast literature on this emerging subject. Amongst those are the first quantum computing compendium by Gruska [81.2] and the quantum information textbooks by Nielsen and Chuang [81.3] and Stolze and Suter [81.4]. A regularly updated annotated bibliography on this subject, compiled by Cabello, forms an invaluable resource for those interested in the subject of this chapter [81.5, 6].

81.1 Quantifying Information As already noted, entanglement comes about if a quantum-mechanical system can be divided into several parts. As an example, consider a two-photon emission process from a spin-zero particle by which two photons escape in opposite directions. Given that the photons are spin-one particles, their spin projections onto some axis must be mutually opposite. As there is no prior information about the actual orientation of the spin, the part of the photon wave function associated with the spin degree is therefore 1 |ψ
= √ (| ↑↓
− | ↓↑
) . (81.1) 2 The striking feature of this type of quantum state is that it describes correlations of two spatially separated particles. If the polarization state of one photon is measured, the state of the other particle, which can be far away, is then instantly predetermined. These (nonlocal) correlations that exist between the particles are of purely quantum origin and are called entanglement. Note, however, that no information can be transmitted faster than the speed of light with this type of set up because the (classical) information concerning the measurement result on one particle needs to be transmitted via a necessarily causal classical channel. The issue of nonlocality has been seen as a vital part in understanding the foundations of quantum mechanics itself (Chapt. 80). In 1935, Einstein, Podolsky, and Rosen argued on the basis of entangled states that quantum mechanics is incomplete [81.7]. They were most concerned about the existence of elements of reality within strongly correlated quantum systems and initiated the debate on quantum nonlocality. The non-existence of so-called local hidden variable theories for the description of states like (81.1) was finally demonstrated by Bell [81.8, 9]. He showed that maximally entangled states violate certain inequalities (now called Bell’s in-

equalities) which local hidden variable models would have obeyed. Later experiments showed the correctness of Bell’s demonstration [81.10–14]. In classical information theory, the unit of information is called a bit, which can be defined as the amount of information contained in a yes–no question. As a matter of fact, ‘bit’ is the abbreviation for ‘binary digit’ and refers to Boolean algebra in which the allowed states of a system are the logical 0 and the logical 1. Therefore, by abuse of language, one bit (as a unit) is the information carried by one bit (as a binary digit) [81.15]. In quantum mechanics, however, due to its inherent linearity, two ‘quantum bits’ (qubits for short) can be in superpositions of the logical states |01
and |10
, or | ↑↓
and | ↓↑
, as in the example above. This typical example of an entangled state shows that quantifying the amount of information contained in a quantum state is different from what is known in classical information theory because of the superposition property. The very same linearity prohibits us from copying an arbitrary quantum state. This effect is known as the no-cloning theorem [81.16]. However, universal copying machines can be constructed within the constraints of quantum mechanics [81.17].

81.1.1 Separability Criterion From the above it is clear that entangled states play a major role in defining the differences between classical and quantum information. Let us begin by asking under which circumstances a particular given quantum state is entangled or not. For this, we need to give a criterion which allows one to decide this crucial question. Consider a bipartite quantum state, i. e., a state which is decomposed into two distinct, albeit possibly correlated, subsystems A and B. Note that these subsystems themselves might consist of ensembles of particles, in which

Quantum Information

i

i

The range of summation in (81.2) is limited by a theorem due to Caratheodory which states that every point in a convex set can be reached by suitable convex combinations of its extreme points. All states that cannot be written in the form of (81.2) are said to be entangled. Note that the set of separable states form a convex subset of the convex set of all possible states. We will now give a simple criterion which decides whether a given state is actually separable or not. For this, one notes that by transposing the part of the density operator associated with the subsystem B, an operation which is called partial transposition, the resulting operator will not necessarily stay positive. However, if the density operator is separable, then its partial transpose is again a positive operator, and hence is a valid density operator. This condition of a state possessing a positive partial transpose is a necessary separability criterion [81.18] but sufficient only in the case of density matrices having Hilbert space dimensions 2 × 2 or 2 × 3 [81.19]. In higher-dimensional Hilbert spaces there exist states with positive partial transposes (PPT) which are nevertheless inseparable. This phenomenon is called bound entanglement [81.20, 21].

Because of the convexity of the set of separable ˆ states, one can construct an operator (a hyperplane) W that separates an entangled state from the disentangled states,   ˆ ˆ AB < 0 ⇔ ˆ AB inseparable , tr W   ˆ ˆ AB ≥ 0 ⇔ ˆ AB separable . tr W (81.3) Such an operator is called an entanglement witness [81.22, 23], and its existence is ensured by a consequence of the Hahn–Banach theorem [81.24]. A similar separability criterion can be found for a particularly interesting class of quantum states in infinite-dimensional Hilbert spaces, the Gaussian states. Gaussian states are most frequently encountered in quantum optics. They comprise all coherent, squeezed and thermal states, and combinations of them. Although being infinite-dimensional, these states permit a complete description in terms of their first and second moments. The characteristic function of a single-mode Gaussian state with λT = (x, p) is given by [81.25]   1 χ(λ) = exp imT λ − λT V λ , (81.4) 4 where m is a vector containing the first moments and V is the covariance matrix containing the second moments. A necessary and sufficient criterion for separability of a bipartite Gaussian state is that the partially transposed covariance matrix still possesses positive symplectic eigenvalues [81.26, 27].

81.1.2 Entanglement Measures ␳ˆAB E(␳ˆAB ) ˆ W Set of all density matrices

Separable states

Once one has checked for inseparability, the obvious question to ask concerns the amount of entanglement, hence the amount of nonclassical correlations in the given state. For bipartite pure states the answer is unique and given by the von Neumann entropy of one subsystem, viz., E(|ψ AB
) = S A (ˆ B ) = S B (ˆ A ) ,

(81.5)

with S A (ˆ B ) = − tr ˆ B ln ˆ B where ˆ A(B) = tr |ψ AB
ψ AB | . B(A)

Fig. 81.1 Convex set of bipartite density matrices; the inner convex set represents the separable states. The witness opˆ forms a hyperplane that separates ˆ AB from the erator W set of separable states

The second equality in (81.5) follows from the left-hand side of the Araki–Lieb inequality [81.28] |S A − S B | ≤ S AB ≤ S A + S B

(81.6)

1217

Part F 81.1

case we are looking at a bipartite cut through the whole system. Then we say that a bipartite state is not entangled and hence separable if its density operator can be written as a convex combination of tensor product states, viz.

ˆ = pi ˆ iA ⊗ ˆ iB , pi = 1 . (81.2)

81.1 Quantifying Information

1218

Part F

Quantum Optics

Part F 81.2

when noting that the entropy of a pure state vanishes. Obviously, since S AB = 0, no information can be extracted from the total state, all information is contained in the correlations between the subsystems A and B which are revealed by performing a measurement on one of the subsystems. For bipartite quantum systems prepared in mixed states the answer is not so obvious. However, some insight can already be gained by looking at the Schmidt decomposition of the state (which, for bipartite states, always exists) [81.29, 30], in particular, the number of elements in the decomposition, named the Schmidt number [81.31]. For a more precise definition of mixed bipartite entanglement, something more is needed. Recall that the set of separable density matrices forms a convex subset of all feasible density matrices. It therefore makes sense to look for a distance-type measure between the given state and the convex hull of product states. Note that the possibility of defining such a measure is provided by the convexity of the separable states and a consequence of the Hahn–Banach theorem [81.24]. Generally, agreement has been reached on what properties any feasible entanglement measure must fulfil [81.32–34]. Let E(ˆ AB ) be a real-valued functional over the tensor-product Hilbert space of bipartite density matrices. If in addition E(ˆ AB ) has the following properties: 1. E(ˆ AB ) = 0 for all separable states; 2. E(ˆ AB ) is invariant trans unitary  under local † †  formations, viz., E Uˆ A ⊗ Uˆ B ˆ AB Uˆ A ⊗ Uˆ B = E(ˆ AB ); 3. E(ˆ AB ) is non-increasing under general local operations by classical  communication, viz.,  assisted ˆ i † ≤ E(ˆ AB ); ˆ i ˆ AB Vˆ i † W E i VˆAi ⊗ W B A B 4. E(ˆ AB ) reduces to the reduced von Neumann entropy for pure states, then E(ˆ AB ) is called an entanglement measure. Important examples of widely used entanglement measures are the entanglement of formation [81.35]
E F ( AB ) = pi E(|ψi
) , (81.7) min ˆ AB =

i

pi |ψi
ψi |

i

and the relative entropy of entanglement [81.32, 33]   tr ˆ AB (ln ˆ AB − ln σ) E R (ˆ AB ) =  min ˆ . σ= ˆ

i

pi σˆ iA ⊗σˆ iB

(81.8)

In general, both of these measures are hard to evaluate. Analytical formulas are known only in special cases. For qubits, the entanglement of formation is also a monotonic function of the concurrence [81.36,37]. The definition of the entanglement of formation, (81.7), can also be extended to cover Gaussian states [81.38]. The number of singlets, i. e., states of the form (81.1), that can be distilled from an ensemble of non-maximally entangled states is called the entanglement of distillation [81.39]. The entanglement of formation and the entanglement of distillation differ by the amount of bound entanglement (Sect. 81.1.1). In some instances, when it is not necessary to comply with all of the above properties of entanglement measures, other quantities can be used to assess the entanglement content of a bipartite state. Particularly useful is the logarithmic negativity [81.40, 41] E N (ˆ AB ) = log2 ˆ P.T. (81.9) AB 1 , where · 1 denotes the trace norm and ˆ P.T. AB the partial transpose of ˆ AB . This measure is often used in connection with Gaussian states. In close analogy to classical information theory, the amount of nonclassical correlations is measured in ebits when one computes entropies with the dual logarithm (log2 ). For example, a pure state with state vector 1 (81.10) |ψ
= √ (|01
+ |10
) 2 in an abstract two-particle Hilbert space spanned by the basis states {|00
, |01
, |10
, |11
} contains 1 ebit of entanglement. It is also a maximally entangled state associated with this Hilbert space since the von Neumann entropy of any state in a Hilbert space of dimension N is bounded from above by log2 dim N. We have concentrated here on bipartite entanglement. The extension to multipartite systems is by no means trivial and much remains to be done on this subject [81.42–44].

81.2 Simple Quantum Protocols In this section we describe the historically first and simplest quantum protocols – quantum key distribution, quantum teleportation, and super-dense coding –

that make use of inherently ‘quantum’ properties of quantum-mechanical systems. These are either entanglement or, in the case of the simplest version of quantum

Quantum Information

81.2.1 Quantum Key Distribution Historically, the earliest protocol that used quantummechanical features in order to realize some specific task that could not have been performed classically was a protocol for secure distribution of a key in cryptography, known as the BB84-protocol after its inventors Bennett and Brassard and the year of its invention [81.48]. Although it is commonly referred to as the first example of quantum information processing, it does not make use of entanglement which was only done some years later, by Ekert [81.49]. The BB84 protocol works in the following way. The sender A prepares a random sequence (or string) of single photons in a polarization state which is chosen out of a set of four basis states, horizontally and vertically (H and V ) polarized, and 45◦ and 135◦ (L and R) polarized. In each of the two basis sets {H, V }, {L, R} one of the states is used to encode the logical value 0 (say in H and L) and the other states encode the logical value 1 (V and R). The random sequence is sent to the receiver B who performs measurements on the sequence of signals by randomly choosing analogous basis states. The result will be another string of 0’s and 1’s that generically does not coincide completely with the original string. To rectify this problem, sender and receiver communicate over a classical public channel where the sender announces the sequence of basis sets in which the photon states were prepared. The receiver compares its sequence of randomly chosen basis states with the announced string and keeps all measureTable 81.1 BB84 protocol for secret key distribution. The sender A sends information encoded in either of two basis sets. The receiver B randomly chooses a measurement basis which is publicly communicated. For those cases when sender and receiver chose the same basis, the receiver’s measurement yields a secure bit Sender A Receiver B Key

→ ↑

↑



1

→ → ↑ 1



0

↑



→ → ↑ 1

ment results for which the choice of basis had been the same. In that way a common secret key is established (Table 81.1). The security against eavesdropping of this simple protocol comes from the fact that even by knowing the measurement basis (say {H, V }) no information has been revealed about the choice of the actual bit value (H or V ). Hence, it is the quantum-mechanical measurement process itself that provides security of the protocol. The first quantum key distribution experiments were reported in [81.50–52]. However, imperfections in the generation and detection of photons, transmission losses and polarization drift causes an actual experimental realization to be far from ideal. In practise, encodings other than polarization may be used (for example a time-binned interferometric basis [81.53]). Despite these error sources, unconditionally secure quantum key distribution can be [81.54–57] and has been achieved [81.58]. Some fiber-based systems have reached distances of more than 100 km [81.59, 60], but discussions of their security continue. For a review of theoretical and experimental aspects of quantum cryptography, see [81.61].

81.2.2 Quantum Teleportation An important utilization of entanglement as a necessary resource can be found in what is commonly known as quantum teleportation. The task of teleportation is to transmit the complete information of an arbitrary unknown quantum state to a spatially different location with the aim of re-creating it. The simplest and obvious way to perform this task would be to take the quantum object which is prepared in the original state and physically transport it to a different location. But sometimes this is not possible because for example an ion needs to be stored in a trap and cannot be moved. The next obvious thing to do would be to measure the quantum state and to re-create it at a different position using the classical information obtained during the measurement. However, single measurements on a quantum system yield only partial information and multiple measurements on many identically prepared copies would have to be performed. The protocol, which was originally proposed in [81.62] for qubits and later generalized to states in infinite-dimensional Hilbert spaces in [81.63], makes use of the existence of maximally entangled states. Let the unknown quantum state which is to be teleported be a qubit superposition state of the form |ψ
= α|0
+ β|1
,

|α|2 + |β|2 = 1 .

(81.11)

1219

Part F 81.2

key distribution, properties of the quantum-mechanical measurement process. We should mention here the pioneering work of Holevo [81.45] who showed that there are fundamental limits on the amount of information that can be extracted by measurements. The application of his ideas to channel capacity and communication [81.46,47] are well described in [81.3] and space limitations prevent us from elaborating on it in this chapter.

81.2 Simple Quantum Protocols

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Part F

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Part F 81.2

Output state 2 bit classical information

Bell measurement

Input state

Pauli rotation

Entangled resource

Fig. 81.2 Schematic outline of an ideal teleportation

protocol

Then one prepares a maximally entangled state of the form (81.10) which is one of the four so-called Bell states defined by 1 |Ψ ±
= √ (|01
± |10
) , 2 1 ± (81.12) |Φ
= √ (|00
± |11
) . 2 We then form the tensor product state |ψ
|Ψ +
as α + |ψ A
|Ψ BC
= √ (|0 A 0 B 1C
+ |0 A 1 B 0C
) 2 β + √ (|1 A 0 B 1C
+ |1 A 1 B 0C
) 2  + 1

= (α|0C
+ β|1C
)Ψ AB 2  − + (α|0C
− β|1C
)Ψ AB  + (α|1C
+ β|0C
)Φ + AB  + (α|1C
− β|1C
)Φ − (81.13) AB , where we have explicitly indexed the relevant subsystems. After performing a joint measurement on subsystems A and B in the Bell basis (this is called a Bell-state measurement [81.64, 65]) one obtains one of four possible results. If the measurement result was |Ψ +
, then the subsystem C is indeed prepared in the original unknown quantum state |ψ
, hence the state has been ‘teleported’ from subsystem A to C. For all other measurement results the outcome is not exactly the same quantum state as intended, but the difference is just a unitary transformation which is uniquely determined by the outcome of the Bell measurement. For example, measuring |Ψ −
means one has to perform a σˆ z -operation that flips the sign of the state |1
, whereas on obtaining |Φ +
or |Φ −
the operations to be applied have to be σˆ x or σˆ z σˆ x , respectively.

Note that this quantum teleportation protocol works with perfect fidelity only if a maximally entangled state has been used, i. e., a state containing 1 ebit of quantum information. In the course of the Bell measurement, the quantum information is used up, and two classical bits of information (the measurement result) have to be communicated to C in order to restore the original quantum state. In this sense, entanglement can be regarded as a resource or ‘fuel’ for certain tasks in quantum information processing. The first experimental demonstrations of teleportation of qubits were performed in [81.66–68] and of continuous variables in [81.69–71]. Recently, a teleportation experiment over 2 km standard telecommunication fibre has been reported [81.72]. A generalization of teleportation is entanglement swapping, in which EPR correlations are established between previously uncorrelated particles by Bell-state measurements [81.73, 74].

81.2.3 Dense Coding The complementary protocol to teleportation is characterized by the name of (super) dense coding [81.75]. The idea here is to transmit two classical bits of information at the expense of consuming 1 ebit of quantum information. The similarity to teleportation is best seen by noting that if the experimental apparatus of sender and receiver are interchanged and the protocol reversed (Fig. 81.3), then one reduces to the other. The mathematical equivalence of the teleportation and dense coding schemes has been beautifully shown in [81.76]. As in teleportation, sender and receiver initially share a two-particle maximally entangled state, i. e., one of the Bell states defined in (81.12). By acting with one of the four operations Iˆ, σˆ x , σˆ z , or σˆz σˆ x on the qubit on the sender’s side, the total 2 bit classical output 1 qubit transmission

Pauli rotation

2 bit classical input

Bell measurement

Entangled resource

Fig. 81.3 Schematic outline of an ideal superdense coding

protocol

Quantum Information

taries) can be transmitted using only a single qubit at a time. An experiment using entangled photon pairs was reported in [81.77], which demonstrated dense coding in practise.

81.3 Unitary Transformations As in classical information theory, one has to define a certain set of allowed operations or maps between states of an information-carrying system. Classical information processing allows operations such as the NAND (Not-AND), which is defined by the Boolean operation X 1 ∧ X 2 ∧ · · · ∧ X n on the Boolean variables X 1 , X 2 , . . . , X n . This operation is not reversible in the sense that, given the outcome of the operation, there is no unique way of determining what the input was. Hence, such types of classical gates destroy information during the course of their operation. Loss of information or irreversibility of an operation is accompanied by an increase in entropy of the state that has been operated on. For an initially pure quantum state having zero entropy, this means mixing the state and destroying its superposition nature and hence its quantum-mechanical entanglement. Therefore, the advantages of parallelism inherent in the superposition is lost. Thus, valid quantum operations in this sense can only be those that preserve the purity of states, hence unitary operations and partial projective measurements. Of course, in order to reset a quantum register, information has to be erased [81.78, 79]. This erasure procedure is described by a completely positive map (see, e.g. [81.3]). Note that any allowed quantum operation is completely positive; unitary operations are a special class of these. An example of how this constrains operations in quantum rather than classical information theory is the absence of a NOT operation in the former, as a NOT operation cannot be described in terms of completely positive maps [81.80].

81.3.1 Single-Qubit Operations It is instructive to give an example of how to classify all possible unitary operations that can act on a single qubit. A unitary operation acting upon the basis states {|0
, |1
} can be represented by a unitary (2 × 2)-matrix, hence a matrix that represents an element of the unitary group U(2). This group has four generators, the identity

matrix and the three Pauli matrices. Hence, all unitary (2 × 2)-matrices are linear combinations of those four matrices. Given the way they act upon basis states, they can be written as ˆI = |0
0| + |1
1| ,

(81.14)

σˆ x = |0
1| + |1
0| ,

(81.15)

σˆ z = |0
0| − |1
1| ,

(81.16)

and, by virtue of the commutation rules for U(2)generators, σˆ y = iσˆ x σˆ z . Sometimes, the short-hand notation X ≡ σˆ x , etc., is used. A particularly useful single-qubit gate which is not just one of the Pauli operators is the Hadamard gate H. In terms of Pauli operators it is defined as H = (X + Z)/ √ 2. Its purpose is to transform each basis state into equal √ superpositions of basis√ states, i. e., |0
→ (|0
+ |1
)/ 2 and |1
→ (|0
− |1
)/ 2. The Hadamard gate is used to initialize an equal superposition of all possible N-qubit basis states from the state |0
⊗N . Hence, N  i=1

1
Hi |0
⊗N = √ |xk
, N! k

(81.17)

where the |xk
are all N! possible words of length N containing 0’s and 1’s.

81.3.2 Two-Qubit Operations Similarly to the single-qubit case, one can write down all possible unitary operations on two qubits by noting that they constitute a representation of U(4). We will not give an exhaustive list of all 16 generators of this group since they can be found in the literature. Instead, we give examples of particularly useful two-qubit gates. Trivially, the group U(4) contains an 8-parameter subgroup U(2)×U(2) which consists of operations such as X 1 ⊗ X 2 , etc. Particular examples of nontrivial two-qubit gates are the controlled-NOT and the controlled-phase gate,

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two-qubit state is again in one of the four Bell states (81.12). Since they are mutually orthogonal to each other, the receiver can tell them apart by measuring in the Bell basis. In that way, two classical bits of information (the information about the single-qubit uni-

81.3 Unitary Transformations

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Part F 81.4

81.3.3 Multi-Qubit Gates and Networks Control Target

00典

00典

01典

01典

10典

11典

11典

10典

Fig. 81.4 Symbol and truth table of the controlled-NOT gate. The target qubit is flipped depending on the state of the control qubit

defined in terms of Pauli operators as C 12 = |01
01 | ⊗ I2 + |11
11 | ⊗ X 2 , Π12 = |01
01 | ⊗ I2 + |11
11 | ⊗ Z 2 ,

(81.18) (81.19)

where in both cases qubit 1 acts as the ‘control’ and qubit 2 acts as the ‘target’ (Fig. 81.4). The net effect of the controlled-NOT gate is to interchange the states |10
↔ |11
, whereas the controlled-phase gate changes the phase of the basis state |11
by π and leaves all other states unchanged. The controlled-NOT gate has an interpretation as a sum gate in that it performs a mapping |x, y
→ |x, x ⊕ y
where the addition has to be taken modulo 2. Moreover, it acts as an entangling gate when acting on tensor products of superpositions.

To realize a unitary operation on many qubits for a particular algorithm one would need a network of single-particle and multi-particle quantum gates. Quantum networks enable a prepared input state to be transformed by the appropriate unitary operator to a final state which is then measured. Deutsch’s model of quantum networks enables us to decompose the network into component gates in diagrammatic form [81.81]. The task is then to optimize the sequence of gates. One can treat quantum gates on N qubits as being elements of  acting  the group U N 2 which has N 4 generators. This, however, is not a particularly transparent or useful way of looking at these gates. Much more useful, and of immense practical importance, is a result essentially from linear algebra which states that every N-qubit gate can be decomposed into a network of single-qubit and twoqubit operations [81.82]. As a matter of fact, there is an even deeper result which says that every N-qubit gate can be generated by a network that consists only of very few elementary building blocks, the so-called universal set of quantum gates [81.83–85]. This set contains all possible single-qubit rotations and one nontrivial twoqubit gate, such as the above-mentioned controlled-NOT or controlled-phase gate.

81.4 Quantum Algorithms The search for algorithms that would run faster on a quantum computer than on any classical computer is a formidable task. When we say faster, we actually mean that the temporal complexity in performing a given task should be drastically reduced. The hope is that eventually one will find algorithms that provably run exponentially faster on a quantum computer compared to a classical computer.

81.4.1 Deutsch–Jozsa Algorithm Let us give a particularly instructive example known as the Deutsch–Jozsa algorithm [81.86]. Let us suppose one is given a string of N bits and a Boolean N-bit function f(x) such   that |x
|y
→ |x
|y ⊕ f(x)
for x ∈ 0, . . . , 2 N − 1 . From f(x) is known that it either returns always 0 or 1 (in which case one calls it ‘constant’) or returns 0 and 1 with equal probability (in which case it is ‘balanced’). The task is to find out whether f(x) is constant or balanced. Classically, one needs at least 2 N−1 + 1 strings to find the answer.

Quantum-mechanically, one prepares the trial input in a superposition of all possible computational basis states using the Hadamard gate from (81.17) and uses one function evaluation on all basis states simultaneously (Fig. 81.5). A measurement outcome other than 0 on any of the N query qubits then tells that the function f(x) is balanced. If it were constant, the measurement outcome would be 0 in all query qubits. This quantum parallelism is at the heart of the increase in speed that occurs in quantum computation. Versions of the Deutsch–Jozsa

0典

N

H

N

x

x

H

N

Uf 1典

H

y

y

f (x)

Fig. 81.5 Gate network for implementing the Deutsch–

Jozsa algorithm

Quantum Information

81.4.2 Grover’s Search Algorithm In contrast to the preceding example, which always gives the desired answer after exactly one trial, the quantum search algorithm by Grover [81.90] uses a procedure that amplifies the sought after result by a method called ‘inversion about average’. The goal of Grover’s algorithm is to search an unsorted database with 2 N entries out of which only one fulfils a given criterion. As in the Deutsch–Jozsa algorithm described above, the query is simultaneously run on all 2 N possible N-qubit basis states, being prepared in an equal superposition. It is assumed that the state that satisfies the search criterion will acquire a phase shift of π. After this step, the inversion about average is carried out. It is represented by a diffusion operator Dˆ = 2 Pˆ − Iˆ ,

(81.20)

where Iˆ is the identity operator and Pˆ a projection operator that averages each input vector with respect to its components. Compared to the previous average value of probability amplitudes, after each of these steps the   magnitude of the desired state increases by O 2−N/2 .  This procedure is repeated, and after only O 2 N/2 steps a projective measurement yields the desired result with

probability of O(1), or more precisely, of more than a half. This is a quadratic increase in speedcompared to  classical search algorithms, which need O 2 N steps. We have seen in these two examples that the use of quantum-mechanical superpositions can lead to a speed increase compared to the best classical algorithms. The most prominent example of such increases in speed is found in Shor’s algorithm for factoring large numbers [81.91,92]. Its core element is essentially a quantum Fourier transform to find the period of a Boolean string. This algorithm provides an exponential speed increase over any known classical algorithm. It should be noted, however, that the fastest known classical algorithm has not yet been proven to be optimal. Implementations in nuclear magnetic resonance systems with a few qubits have been reported for the quantum Fourier transform [81.93] as well as Shor’s factoring algorithm [81.94]. It turns out that there exists a whole class of algorithmic problems, the hidden subgroup problems [81.95– 97], whose quantum-mechanical analogues can lead to exponential increases in speed over their classical counterparts, a particular example of which is Shor’s factoring algorithm. Another instance of quantummechanically exponentially faster processes is found in quantum random walks on hypercubes where hitting times, i. e., the traversal time of an excitation across the cube’s diagonal, can be exponentially faster than for classical random walks [81.98].

81.5 Error Correction As we have discussed, the essence of quantum information processing is the use of quantum superpositions, interference, and entanglement. But quantum interference is fragile. It appears in practice that it is very difficult to maintain a superposition of states of many particles in which each particle is physically separated from all the others. Entanglement turns out to be incredibly delicate. The reason for this is that all systems, quantum or classical, are not isolated; they interact with everything around them: local fluctuating electromagnetic fields, the presence of impurity ions, coupling to unobserved degrees of freedom of the system containing the qubit, etc. These fluctuations destroy quantum interference. A simple analogy is the interference of optical waves in Young’s double slit experiment. In that apparatus waves from two spatially separated portions of a beam are brought together. If the two parts of the beam have the same phase, then the fringe pattern re-

mains stable. But if the phase of one part of the beam is drifting with respect to the other, then the fringe pattern will be washed out. And the more slits there are in the screen, the lower the visibility for the same amount of phase randomization per pair of slits. The sensitivity of an N-qubit register to decoherence is even worse, as a maximally entangled N-particle state decoheres at a rate N times faster than a single particle [81.99], one of the reasons why the world around us appears so classical. A single bit of information lost to an unobserved degree of freedom will result in the reduction of the quantum superposition to a mixed state. Yet correcting errors due to environmental interactions is essential if a quantum computer is to be constructed: to do ‘fault tolerant’ computing we need to be able to execute many gate operations coherently within the decoherence time if we are to have a chance of building a scalable quantum register [81.100].

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Part F 81.5

algorithm involving a few qubits have been implemented in nuclear magnetic resonance systems [81.87, 88] as well as ion traps [81.89].

81.5 Error Correction

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Part F 81.6

It might appear that the problem of stabilizing a register of qubits is hopeless, like trying to balance several pencils on their tips while on the deck of a ship in a storm. But, amazingly, quantum mechanics provides a way to solve this problem, through the use of even higher levels of entanglement. Shor and coworkers, and independently Steane showed in the mid-1990’s that encoding information in entangled sets of qubits offered the opportunity to execute quantum error correction [81.101, 102]. That one can do this is a remarkable consequence of entanglement. In classical information processing, inevitable environmental noise is dealt with by error correction. In its simplest form, this involves repeating the message transmission or calculation until a majority result is obtained. But there are more efficient ways, for example, use of a parity check on a block of bits. It turns out that a similar notion can be applied to a quantum register. However, the application is not straight forward because the contents of a register cannot be measured without destroying the superposition state encoded in it. The problem then is to determine what errors might be present in a quantum register without looking at the qubits. The elegant solution is to entangle the qubits in question with those in an ancillary register and measure the ancillary register. Because the two registers are correlated, the results of the measurement of the ancilla reveal any errors present in the processing register

without destroying any coherent superpositions in the processing register itself. The first experimental demonstration of quantum error correction used NMR techniques [81.103–105], but given the inherently mixed nature of NMR quantum computing [81.106], this has had limited impact on quantum information processing (QIP). However, very recently, Wineland’s group in Boulder has succeeded in implementing quantum error correction using laser-cooled trapped ions [81.107]. Another way to prevent the register coherence from falling apart is to know a little about the sort of noise that is acting on it. If the noise has some very slow components (or those with very long wavelengths), then it is sometimes possible to find certain combinations of qubit states for which the noise on one qubit exactly cancels the noise on another. These qubit states live in a ‘decoherence-free subspace’ (DFS) [81.108–111]. A computer will then be immune to environmental perturbations if all the computational states lie in this DFS. The connection between DFS and quantum error correction codes has been shown in [81.112]. Kwiat [81.113], using photonic qubits, and Wineland’s group [81.114], using trapped ions, have demonstrated the use of DFS experimentally. Although these results are encouraging, we are still a long way from the figure of merit for gate time to decoherence time needed for fault tolerance.

81.6 The DiVincenzo Checklist DiVincenzo gave a list of requirements that a physical implementation must fulfil in order to qualify as a sensible candidate for an implementation of quantum information processing [81.115].

the system can be scaled up to contain potentially many qubits.

81.6.1 Qubit Characterization, Scalability

Once the qubits have been specified, each quantum information processing or quantum computation task needs to be able to start from a well-defined state. This can be basically any quantum state of the many-qubit system as long as it is a product state and can be prepared errorfree. Commonly, this state is then called the ground state and denoted by |0
⊗N .

Each physical implementation must be tested upon how qubits should be encoded. For a qubit being essentially a two-level system, this task is generally not too difficult. Several candidates, such as electronic or nuclear spin, photon polarization, choice of path in an interferometer, degenerate ground states of an atom or ion, charge or flux states in superconducting quantum interference devices (SQUIDs) or exciton population, have all been recently explored. Much more challenging will be the question as to whether there are fundamental or technological limitations of having many of those qubits being operated upon seperately, hence whether

81.6.2 Initialization

81.6.3 Long Decoherence Times In order to ensure error-free computation without loss of purity of quantum superpositions, the decoherence times that are relevant for the quantum operation should be much longer than the gate operation time itself. In most

Quantum Information

81.6.4 Universal Set of Quantum Gates A necessary prerequisite for quantum information processing and quantum computing is the ability to generate

a set of quantum gates that can be considered universal. With such a set it will then be possible to generate all other quantum gates by concatenating them to form suitably arranged networks. The choice of which set out of the many possible is taken, depending on the physical implementation itself. Basically, it is determined by the operations that are intrinsically simple for the given interaction Hamiltonian. In some applications, such as the ion trap experiments, the controlled-NOT gate is preferred as the nontrivial two-qubit gate, whereas in linear optical networks one rather works with the controlledphase gate.

81.6.5 Qubit-Specific Measurement The last requirement is to be able to read out the result of the computation. That is, there has to be a way of providing a selective projective measurement. This proves to be a major challenge in most proposals for implementing quantum computing. Examples of the challenges involved are the necessity to provide photon-number resolving photodetectors, single-electron charge measurement devices, or singlespin measurements.

81.7 Physical Implementations Quantum information theory regards relevant objects as abstract quantities in a Hilbert space of a certain dimension. The different strands in its development can be roughly divided into generalized spin systems (qubits, qudits, the d-dimensional generalization of qubits) living in finite-dimensional Hilbert spaces, and harmonic oscillator systems which naturally live in infinite-dimensional Hilbert space. The latter are hence called continuous-variable (cv) systems. Examples for generalized spin systems are polarization states of a photon, magnetic sublevels of atomic hyperfine states or, to a good approximation, electronic levels of atoms and ions. Harmonic oscillator systems can be found in atomic populations in optical lattices, electromagneticfield modes or indeed any excitation of a bosonic quantized field. Both strands have their own virtues and disadvantages. Harmonic oscillator systems are naturally abundant and, in their materialization as photons, rather easily accessible and manipulable. However, due to their Hilbert space dimensionality, the nonlinear operations that are required for quantum gates are generally hard to achieve. In contrast, spin-like systems (apart from

photon polarization) require more experimental effort in preparing them but, unlike harmonic oscillator systems, they can show effective nonlinearities due to the finite dimensionality of their Hilbert space (the nonlinearities appear when coupled to another physical system which can then be traced out).

81.7.1 Linear Optics The use of photons as carriers of quantum information seems to be a straightforward matter for several reasons. First, they are easy to produce and to manipulate, and second, they both show spin-like behaviour (polarization) and can be treated as continuous-variable systems (Gaussian states). There exists, however, yet another possibility to store and manipulate quantum information in photons, namely when encoding information in number states or Fock states. But as noted before, for photons being bosonic systems, there are no natural nonlinearities (at least none which is strong enough) on the level of single or few photons. The trick here is to use conditional measurements or measurement-induced nonlinearities. The idea was first

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Part F 81.7

situations, decoherence limits the number of qubits that can be worked on simultaneously, thus affecting the scalability of the system. Typical examples of decoherence processes are heating mechanisms in ultracold systems, such as ion trap or atom chip experiments, spin relaxation in NMR-type experiments, or absorption in linear optical elements. Generally, decoherence is unavoidable due to the basic principles upon which quantum information processing is supposed to work. Avoiding decoherence means isolating the system from the outside world, the environment. But controlling the interaction between subsystems always has the negative effect of bringing the system in contact with the environment and therefore necessarily introduces decoherence. Once one has accepted that decoherence is unavoidable, ways have to be found to guard against it. Several error-correction schemes have been proposed that can correct for certain small amounts of decoherence as described in Sect. 81.5.

81.7 Physical Implementations

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Part F 81.7

Ancilla photon

PBS

Control qubit

Target qubit PBS 45

introduced to describe the properties in 3D optical lattices [81.126] (Sect. 81.7.4). In the cluster-state model the computational process is not described by a succession of elementary gates that act upon an (in principal arbitrary) input state, but by a well-defined sequence of single-qubit measurements performed on a maximally entangled ‘cluster’ of qubits. It has been realized that the cluster-state model represents an alternative model for quantum computing, the so-called ‘one-way quantum computer’ [81.127]. It was later found that there also exists a linear optical realization of the cluster-based approach [81.128, 129] which has been experimentally verified [81.130].

81.7.2 Trapped Ions Fig. 81.6 Schematic setup of an all-optical controlled-NOT

gate. Control and target qubits are encoded in the polarization of single photons. These are fed into polarizing beam splitters (PBS), one of which is rotated by 45 degrees

put forward in [81.116, 117] and realized experimentally in [81.118,119], where a polarization-encoding was used. Figure 81.6 shows the schematic setup of a simplified version of a controlled-NOT gate with one single ancilla photon (after [81.120]). Measurement-induced nonlinearities make use of the fact that unitary transformations in a larger Hilbert space, e.g., with added auxiliary photon modes combined with photodetection, can yield effective nonlinearities [81.121]. The drawback is, however, that the wanted nonlinearity is conditioned on the appearance of a certain measurement pattern which means that these schemes work only with a certain probability. Bounds for certain classes of gates have been reported in [81.122–125]. The set of quantum gates that can be considered fundamental differs slightly from most other physical implementations. Within the qubit encoding in photonnumber states the gates that can actually be implemented efficiently are those that act within Fock layers (subspaces of fixed total photon number), such as Z, the controlled-phase gate, or the swap gate. Other gates that do not fall into this class require excessively more resources unless other types of qubit encodings are used simultaneously. Gate operation times can be very fast and are only limited by the gating times of the photodetectors. However, a major experimental challenge is mode-matching in larger networks and interferometric set-ups. A complementary approach to the gate model is based on so-called cluster states which were originally

So far, the most advanced method in terms of the number of qubits and the number of gates that have been generated is by using ultracold ions stored in linear Paul traps (Chapt. 75) in which radio-frequency fields are used to generate confining potentials. The ions are trapped in the radial direction by electric quadrupole fields and in the axial direction by a static repulsive Coulomb force [81.131, 132]. The ions are cooled into their motional ground states by Doppler cooling [81.133, 134] and further cooled by resolved sideband cooling [81.135]. The qubits are encoded into two metastable electronic states. Various groups have used either transitions to metastable states or Raman coupling to avoid the decoherence that an upper-state lifetime would generate. The coupling between qubits is provided by the common vibrational motion in the Paul trap [81.136,137]. For reviews of the dynamics of lasercooled ions [81.138–141]. To date, a few qubits have been entangled and coherently manipulated. Simple quantum algorithms have been demonstrated [81.142] and teleportation achieved [81.143, 144].

81.7.3 Cavity QED Cavity QED provided the very first examples of atomfield entanglement. Single atoms interacting with single cavity-field modes are well described by the Jaynes– Cummings model [81.145]. In this model, excitation is transferred periodically between atoms and field provided the Q-factor of the cavity is high enough (Chapt. 79). The Rabi flopping can be used to generate controlled superpositions, and the cavity field used as a catalyst to entangle atoms [81.146]. Although it is possible to coherently manipulate single or few qubits in cavity QED, scaling to large numbers of qubits would

Quantum Information

odic spatially varying trapping potential through the AC Stark shift

seem very difficult. Nevertheless, trapped atoms within cavity QED environments offer great potential as local processors linked by quantum communication channels [81.147]. Progress towards this has been reported by several groups [81.148–151].

81.7.4 Optical Lattices, Mott Insulator Another possible way of implementing quantum computation is with cold atom technology. This includes the application of optical lattices in a sufficiently cold cloud of atoms showing Bose–Einstein condensation (BEC). In recent years it has been realized that Bose– Einstein condensates can undergo a phase transition if loaded into a three-dimensional periodic potential which, for example, can be realized by standing-wave optical fields [81.152, 153]. That is, one starts off with a BEC in its superfluid phase, in which the relative phases (or rather correlations) between the atoms are well-defined such that the whole ensemble of atoms can be described by a single macroscopic wave function (in first approximation). By loading this condensate into the optical lattice (Fig. 81.7) the number of atoms per lattice site is undetermined and can vary widely. However, when increasing the strength of the potential by increasing the power of the laser beams that create the standing-wave potential, eventually there will be a phase-transition to a state of the condensate in which each lattice site is occupied by a fixed and well-defined number of atoms (ideally we would like to have exactly one atom per site). In this so-called Mott-insulator phase, the relative phases (or correlations) between neighboring lattice sites are undetermined. Experimental evidence

of this phase-transition has been obtained in the beautiful experiments described in [81.154–156]. Although a Bose–Einstein condensate really exists only in three dimensions (since only there one finds a phase transition from a thermal cloud to a condensate), there are analogous systems, such as the quasi-condensate [81.157] and the Tonks–Girardeau gas [81.158] (for recent experiments, see [81.159, 160]), in one dimension that have similar properties. Such a system is well described by the Bose– Hubbard Hamiltonian [81.152, 153], in which the collisional interaction between atoms at the same lattice site provides the necessary nonlinearity. Atoms trapped in a one-dimensional optical lattice could serve as an atomic register that promises well controlled singlequbit and two-qubit manipulability. A universal set of quantum gates can be realised by manipulations of the lattice potential with additional laser fields [81.161]. The different types of quantum gates could, for example, be realized if the atoms possess two degenerate ground states that are used for the qubit encoding (as sketched in Fig. 81.7). A Raman transition between the ground states would result in single-qubit operations, whereas controlled collisions between atoms in neighboring lattice sites would produce two-qubit gates. Recently, threequbit gates [81.162] and global adressing of strings of qubits have been proposed [81.163]. The estimated gate evolution times in the adiabatic regime are roughly O(100 ms), which is just one to two orders of magnitude below the trapping lifetime measured in recent experiments [81.164–166]. The dominant loss effect is thereby a thermally induced spin flip mechanism that causes the atoms to leave the trapping region [81.167–169]. The gate evolution times could be reduced by several orders of magnitude in non-adiabatic regimes. The price to pay is that the temporal evolution of the laser pulse envelope has to be controlled much more precisely. Although quantum information processing using cold atom technology is still in its infancy, it promises to provide relatively long decoherence times. Moreover, scalability seems possible as rather long one-dimensional strings of atoms could be formed. Experimental evidence for this has of course yet to be shown.

81.8 Outlook We have discussed the very basic ideas behind quantum information and described a few possible applications. However, the immense wealth of ideas and possible

routes have barely been touched upon. Quantum key distribution is already at a stage where private companies are selling component parts to set up commercial

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Fig. 81.7 Counter-propagating laser beams induce a peri-

81.8 Outlook

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Part F 81

QKD systems. Only a few qubits have been maximally entangled and manipulated experimentally, so far. But a great number of qubits has been partially entangled within optical lattices [81.170] or in atomic vapours [81.171]. The dynamics of qubit interactions in such systems is closely related to systems studied in phase-transition theory, pointing to yet another application of the subject. The simulation of many-particle quantum systems is of course intrinsically difficult and could well require a quantum computer for its analysis [81.172–175]. If and when a quantum computer

can be built remains shrouded in mist. However, the ideas and methods that have already come out of quantum information theory provide useful tools for tackling other, seemingly unrelated, problems. One direction of current research regards many-body problems in condensed matter systems and quantum field theory (see, for example, [81.176]). We have touched upon only a small part of a rapidly developing subject – one in which quantum effects are the enablers of new technology [81.177]. We are confident that much more remains to be discovered.

References 81.1 81.2 81.3

81.4

81.5 81.6 81.7 81.8 81.9

81.10 81.11 81.12 81.13 81.14 81.15 81.16 81.17 81.18 81.19 81.20 81.21 81.22 81.23

B. Russell: The ABC of Relativity (Allen Unwin, London 1969) J. Gruska: Quantum Computing (McGraw-Hill, London 1999) M. A. Nielsen, I. L. Chuang: Quantum Computation and Quantum Information (Cambridge Univ. Press, Cambridge 2000) J. Stolze, D. Suter: Quantum Computing: A Short Course from Theory to Experiment (Wiley-VCH, Weinheim 2004) A. Cabello: quant-ph/0012089 A. Cabello: URL www.adancabello.com A. Einstein, B. Podolsky, N. Rosen: Phys. Rev. 47, 777 (1935) J. S. Bell: Physics 1, 195 (1964) J. S. Bell: Speakable and unspeakable in quantum mechanics (Cambridge Univ. Press, Cambridge 1987) S. J. Freedman, J. F. Clauser: Phys. Rev. Lett. 28, 938 (1972) A. Aspect, P. Grangier, G. Roger: Phys. Rev. Lett. 47, 460 (1981) A. Aspect, J. Dalibard, G. Roger: Phys. Rev. Lett. 49, 1804 (1982) J. G. Rarity, P. R. Tapster: Phys. Rev. Lett. 64, 2495 (1990) W. Tittel, J. Brendel, H. Zbinden, N. Gisin: Phys. Rev. Lett. 81, 3563 (1998) T. M. Cover, J. A. Thomas: Elements of Information Theory (Wiley, New York 1991) W. H. Zurek, W. K. Wootters: Nature (London) 299, 802 (1981) M. Hillery: Phys. Rev. A 54, 1844 (1996) A. Peres: Phys. Rev. Lett. 77, 1413 (1996) M. Horodecki, P. Horodecki, R. Horodecki: Phys. Lett. A 223, 1 (1996) P. Horodecki: Phys. Lett. A 232, 333 (1997) M. Horodecki, P. Horodecki, R. Horodecki: Phys. Rev. Lett. 80, 5239 (1998) B. Terhal: Phys. Lett. A 271, 391 (2000) M. Lewenstein, B. Kraus, J. I. Cirac, P. Horodecki: Phys. Rev. A 62, 052310 (2000)

81.24

81.25 81.26 81.27 81.28 81.29 81.30 81.31 81.32 81.33 81.34 81.35 81.36 81.37 81.38 81.39 81.40 81.41 81.42 81.43 81.44 81.45

81.46 81.47

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Quantum Information

81.49 81.50 81.51 81.52 81.53 81.54 81.55 81.56 81.57 81.58

81.59 81.60 81.61 81.62

81.63 81.64 81.65 81.66

81.67 81.68

81.69 81.70

81.71

81.72 81.73 81.74 81.75 81.76

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81.95

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Quantum Information

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References

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Part G

Application Part G Applications

82 Applications of Atomic and Molecular Physics to Astrophysics Alexander Dalgarno, Cambridge, USA Stephen Lepp, Las Vegas, USA 83 Comets Paul D. Feldman, Baltimore, USA 84 Aeronomy Jane L. Fox, Dayton, USA 85 Applications of Atomic and Molecular Physics to Global Change Kate P. Kirby, Cambridge, USA Kelly Chance, Cambridge, USA 86 Atoms in Dense Plasmas Jon C. Weisheit, Pullman, USA Michael S. Murillo, Los Alamos, USA

87 Conduction of Electricity in Gases Alan Garscadden, Wright Patterson Air Force Base, USA 88 Applications to Combustion David R. Crosley, Menlo Park, USA 89 Surface Physics Erik T. Jensen, Prince George, Canada 90 Interface with Nuclear Physics John D. Morgan III, Newark, USA James S. Cohen, Los Alamos, USA 91 Charged-Particle–Matter Interactions Hans Bichsel, Seattle, USA 92 Radiation Physics Mitio Inokuti, Argonne, USA

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82. Applications of Atomic and Molecular Physics to Astrophysics

Applications o

Almost all our information about the Universe reaches us in the form of photons. Observational astronomy is based on measurements of the distribution in frequency and intensity of the photons that are emitted by astronomical objects and detected by instrumentation on ground-based and space-borne telescopes. Information about the earliest stages in the evolution of the Universe before galaxies and stars had formed is carried to us by blackbody background photons that attended the beginning of the Universe. The photons that are the signatures of astronomical phenomena are the result of many processes of nuclear physics, plasma physics and atomic, molecular and optical physics. The processes that modify the photons on their journey from distant origins through intergalactic and interstellar space to the Earth belong mostly to the domain of atomic, molecular, and optical physics, as do the instruments that detect and measure the arriving photons and their spectral distribution. The spectra are used to classify galaxies and stars and to identify the astronomical entities and phenomena such as quasars, active galactic nuclei, grav-

82.1 Photoionized Gas ................................ 1235 82.2 Collisionally Ionized Gas ....................... 1237 82.3 Diffuse Molecular Clouds ...................... 1238 82.4 Dark Molecular Clouds .......................... 1239 82.5 Circumstellar Shells and Stellar Atmospheres ...................... 1241 82.6 Supernova Ejecta ................................. 1242 82.7 Shocked Gas........................................ 1243 82.8 The Early Universe ............................... 1244 82.9 Recent Developments........................... 1244 82.10 Other Reading ..................................... 1245 References .................................................. 1245

itational lensing, jets and outflows, pulsars, supernovae, novae, supernova remnants, nebulae, masers, protostars, shocks, molecular clouds, circumstellar shells, accretion disks and black holes. Quantitative analyses of the spectra of astronomical sources of photons and of the atomic, molecular, and optical processes that populate the atomic and molecular energy levels and give rise to the observed absorption and emission require accurate data on transition frequencies and wavelengths, oscillator strengths, cross sections for electron impact, rate coefficients for radiative, dielectronic and dissociative recombination, and cross sections for heavy particle collisions involving charge transfer, excitation, ionization, dissociation, fine structure, and hyperfine structure transitions, collision-induced absorption and line broadening. Data on radiative association and ion–molecule and neutral particle reaction rate coefficients are central to the interpretation of measurements of chemical composition in molecular clouds, circumstellar shells and supernova ejecta.

82.1 Photoionized Gas The Universe contains copious sources of energetic photons most often in the form of hot stars, and much of the material of the Universe exists as photoionized gas.

Photoionized gas produces the visible emission from emission nebulae, planetary nebulae, nova shells, starburst galaxies and probably active galactic nuclei [82.1].

Part G 82

The range of physical conditions of density, temperature, and radiation fields encountered in astrophysical environments is extreme and can rarely be reproduced in a laboratory setting. It is not only reliable data on known processes that are needed but also a deep understanding so that the relevant processes can be identified and the influence of the conditions in which they occur fully taken into account. We present here a summary of the processes that take place in photoionized gas, collisionally ionized gas, the diffuse interstellar medium, molecular clouds, circumstellar shells, supernova ejecta, shocked regions and the early Universe.

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Applications

Part G 82.1

Emission nebulae are extended regions of luminosity in the sky. They arise from the absorption of stellar radiation by the gas surrounding one or more hot stars.The gas is ionized by the photons and excited and heated by the electrons released in the photoionizing events. A succession of ionization zones is created in which highly ionized regions give way to less ionized gas with increasing distance from the central star as the photon flux is diminished by geometrical dilution and by absorption. The outer edge of a nebula is a front of ionization pushing out into the neutral interstellar gas. The densities are typically between 100 and 10 000 cm−3 and the temperatures between 5000 and 15 000 K. Nebulae are also called Hii regions. At low densities, the luminosity is low, but the ionized regions can still be detected by radio observations. Planetary nebulae are smaller in extent and more dense. They have a passing similarity in appearance to planets. Planetary nebulae are produced by the photoionization of shells of gas that have been ejected from the parent star as it evolved to its final white dwarf stage. Because the core of the parent star is very hot the irradiated gas is more highly ionized than are emisssion nebulae and has a distinctive spectrum. Photoionized gas is also found around novae. Novae are stars that have undergone spasmodic outbursts and they are surrounded by faint shells of ejected gas, photoionized and excited by the stellar radiation. Some supernova remnants, which are what remains after Counts / 320 s 17

15

Fe XVII 2p6–2p53d 15

10

X (m−1)+ + hν → X m+ + e−

(82.1)

and radiative X m+ + e− → X (m−1)+ + hν

(82.2)

and dielectronic


∗ X m+ + e− → X (m−1)+ → X (m−1)+ + hν (82.3)

recombination, and in plasmas with a significant population of neutral hydrogen and helium, by charge transfer recombination X m+ + H → X (m−1)+ + H+ X

m+

+ He → X

(m−1)+

(82.4)

+

+ He .

(82.5)

Wavelength (Å) 16

20

O Fe XVII 2p6–2p53s

a massive star has exploded, have spectra that also appear to be emanating from photoionized gas. The source of ionization may be synchrotron radiation. Figure 82.1 shows the X-ray emission spectrum of a supernova remnant. The nuclei of starburst galaxies have spectra like those of emission nebulae. They result from gas photoionized by radiation from hot stars created in a period of rapid star formation. Active galactic nuclei, such as quasars, have a different spectrum characterized by broad lines indicating a large range of velocities. Photoionized gas is the most likely interpretation.The ionizing source may be an accretion disk around a compact object such as a black hole. The ionization structure in a photoionized gas is determined by a balance of photoionization

Ly β

14

13 Ne IX 1s2 – 1s 2p, s R F I

12 Fe XVII 2p6–2p5 4d Ne X Ly α

VIII Ly γ

Ne IX 1s2 – 1s 3p

Ly δ

5

0 700

800

900

1000

1100 Energy (eV)

Fig. 82.1 X-ray spectrum of the supernova remnant Puppis A as observed by the Einstein satellite. Note the high level of ionization with hydrogen-like ions of oxygen and neon, suggesting a high temperature. After [82.2]

Applications of Atomic and Molecular Physics to Astrophysics

of the rate coefficients [82.14]. The resulting cooling rates increase exponentially with temperature and keep the temperature of the gas between narrow limits. Some contribution to cooling occurs from recombination and from free-free emission by electrons moving in the field of the positive ions. The luminosity of the photoionized gas comes from the photons emitted in the cooling processes and from radiative and dielectronic recombination. The radiative recombination spectrum of hydrogen extends from the Ly α line at 121.6 nm to radio lines at meter wavelengths. The recombination spectrum can be predicted to high accuracy, and calculations for a wide range of temperature, density, and radiation environments have been carried out for diagnostic purposes [82.15–18]. Electron impact and proton impact induced transitions are important in determining the energy level populations and the resulting spectrum. Stimulated emission often affects the intensities of the radio lines, especially those from extragalactic sources. Comparisons of the predicted intensities in the visible and infrared with theoretical predictions yield information on interstellar extinction in the nebula and along the line of sight. The relative intensities of the lines emitted by different metastable levels depend exponentially on the temperature. The relative intensities of the lines at 500.7 nm and 436.3 nm originating in the 1D 2 and 1S 0 levels of O++ vary as exp(33170/T ), and are commonly used to derive the temperature T . The electron density can be inferred from the lines emitted from neighboring levels with different radiative lifetimes for which there occurs a competition between spontaneous emission and quenching by electron impact. There are many possible combinations of lines. The lines at 372.89 nm and 372.62 nm emitted by the 2D 2 + 3/2 and D 5/2 levels of N are readily observable and their relative intensity yields the electron density. Radiative and dielectronic recombination lines are often seen in the spectra, as are a few lines due to charge transfer recombination. Fluorescence of starlight and resonance fluorescence of lines emitted in the nebula (called Bowen fluorescence by astronomers) also contribute to the spectra of photoionized gases. Many data are needed to adequately interpret the observations.

82.2 Collisionally Ionized Gas Hot gas is found in the coronae of stars and particularly the Sun, and in young supernova remnants, in the hot phase of the interstellar medium, and in inter-

galactic space. In a hot gas the ionization is produced by the impact ionization of the fast thermal electrons and recombination is radiative and dielectronic [82.19].

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Part G 82.2

Many detailed calculations of the ionization structure of photoionized regions have been carried out [82.1]. The ionizing source spectra of hot stars can be obtained from calculations of stellar atmospheres Sect. 82.5. Approximate values of cross sections for photoionization for a wide range of atomic and ionic systems in many stages of ionization are available [82.3–6]. Calculations of higher precision and reliability that incorporate the contributions from autoionizing resonance structures exist for specific systems [82.7]. They are undergoing continual improvements as increasingly powerful computational techniques are brought to bear on the calculations. The cross sections for radiative recombination are obtained by summing the cross sections for capture into the ground and excited states of the recombining system. Because of the contribution from highly excited states which are nearly hydrogenic, the rate coefficients are similar for different ions of the same excess nuclear charge. They vary slowly with temperature. In contrast, dielectronic recombination is a specific process whose efficiency depends on the energy level positions of the resonant states. For nebular temperatures, the rate coefficients vary exponentially with temperature. Explicit calculations have been carried out for many ionic systems [82.8–10]. Because the photoionization cross sections of the major cosmic gases hydrogen and helium diminish rapidly at high frequencies, multiply charged ions and neutral gas coexist in cosmic plasmas produced by energetic photons and charge transfer recombination may control the ionization structure. For multiply charged ions with excess charge greater than two, charge transfer is rapid. For doubly charged and singly charged ions, the cross sections are sensitive to the details of the potential energy curves of the quasimolecule formed in the approach of the ion and the neutral particle. Few reliable data exist. Some recent calculations may be found in the papers [82.11–13]. Photoionized gas is heated by collisions of the energetic photoelectrons and cooled by electron impact excitation of metastable levels, principally of O+ and O++ , N+ and N++ , and S+ and S++ , followed by emission of photons which escape from the nebula. Considerable attention has been given to the determination

82.2 Collisionally Ionized Gas

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Part G

Applications

Part G 82.3

The rate coefficients for electron impact ionization and for recombination for any given ionization stage are functions only of temperature, and hence so is the resulting ionization distribution. When ionization and recombination balance, coronal equilibrium is attained in which the ionization structure is specified by the temperature. Recombination at high temperatures is dominated by dielectronic recombination. At high temperatures, dielectronic recombination is stabilized by transitions in which the core electrons are the active electrons. The associated emission lines lie close in frequency to that of the resonant transition of the parent ion. They are called satellite lines. Together with lines generated by electron impact excitation, they provide a powerful diagnostic probe of density and temperature. In many circumstances such as in supernova remnants, coronal equilibrium does not hold, and the ionization and recombination must be followed as functions of time. The temperature also evolves as the hot plasma is cooled by electron impact excitation and ionization. The recombining gas produces X-rays and extreme UV radiation which modify the ionization structure. There is a particular need for more reliable data on high energy photoionization cross sections, on collision cross sections for electrons and positive ions, and on

–log P / ne nH (erg × cm3/ s) 21

H

C

O

He

22

N

Fe

Si Mg

23

Ne

(OII) 4.0

Si

5.0

6.0

7.0 log T (K)

Fig. 82.2 Total emissivity and emissivity by element as

a function of temperature in coronal equilibrium. The heavy solid curve is total emissivity and the lighter lines are contributions from individual elements. After [82.20]

the energy levels and transition probabilities of highly stripped complex ions. Figure 82.2 shows the emissivity of coronal gas.

82.3 Diffuse Molecular Clouds Diffuse molecular clouds are intermediate between the hot phase of the galaxy and the giant molecular clouds where much of the gas resides. They are called diffuse because they have optical depths of order unity, so photons can penetrate from outside the cloud and affect the chemical composition. The atoms and molecules are observed in absorption against background stars. Translucent clouds with optical depths between about 2 and 5 are intermediate between diffuse and dark clouds where photons from the outside still affect the chemistry. They can be observed both in absorption against a background source or in emission in the radio. The temperature is 100–200 K at the edges of a diffuse cloud with a density of about 100 cm−3 . In a typical diffuse cloud the temperature decreases to about 30 K at the center while the density increases to about 300–800 cm−3 . The chemistry is driven by ionization from interstellar UV photons and from cosmic rays. Interstellar UV photons ionize species which have ionization potentials less then that of atomic hydrogen.

Atomic hydrogen is so pervasive in the galaxy that UV photons with energies higher than 13.6 eV are absorbed near the source. The UV flux is a very important parameter in determining the composition of a diffuse cloud. Photodissociation provides destruction which limits the buildup of more complex species and so diffuse clouds are dominated by simpler diatomic species. Species with ionization potentials greater then hydrogen are mainly ionized by cosmic rays. Cosmic rays are high energy nuclei which stream through the galaxy. The cosmic ray ionization rate, the number of cosmic ray ionizations per second per particle, is an important parameter in interstellar chemistry. A lower limit to the cosmic ray ionization rate may be set by measured high energy cosmic rays reaching earth, giving an ionization rate of ≈ 10−17 s−1 . More realistic estimates of the cosmic ray ionization rate from looking at recombination lines suggest values of a few ×10−17 s−1 . The hydroxyl radical OH is produced in a manner similar to that discussed below in Sect. 82.5, and removed by photodissociation. Thus in diffuse clouds,

Applications of Atomic and Molecular Physics to Astrophysics

(82.16)

CH may also be removed by reactions with oxygen or nitrogen atoms to form CO and CN respectively. One of the outstanding problems in diffuse clouds is to understand the large abundance of CH+ relative to CH. The problem is producing the CH+ without producing additional CH. Since most reaction paths go through reaction (82.7), this is the most likely candidate. What is needed is some extra energy to overcome the endothermicity. This energy must come from either hot C+ or from hot or vibrationally excited H2 . The most popular model is gas heated by a shock, possibly a magnetic shock in which ions stream relative to the neutrals, giving a high effective energy. Unfortunately, though these shock models can reproduce the CH+ abundances, they also predict relative velocities between the CH+ and CH which are not often observed. Recently there have been suggestions that turbulence in the cloud could account for the CH+ abundance. The most comprehensive models of diffuse clouds are by van Dishoeck and Black [82.21]. A collection of photodissociation rates and photoionization rates is given in Roberge et al. [82.22]. The UV flux is predominantly from stars and may be as much as 105 times larger near an Hii region Sect. 82.1 than it is in the general interstellar medium. Regions in which the chemistry is dominated by photons are referred to as photon dominated regions or photodissociation regions (PDR’s). In the presence of high UV flux the cloud is much warmer than in a typical diffuse cloud. Temperatures may reach 1000 K near the edge of the cloud and 100 K far into the cloud. The chemistry differs from traditional diffuse cloud chemistry in that the high temperatures allow endothermic reactions to proceed. Sternberg and Dalgarno [82.23] have published a comprehensive model of photodominated regions.

Much of the mass of the galaxy is in the form of dark molecular clouds. The molecular clouds are sites of forming new stars. They are composed primarily of hydrogen, with about 10% helium and trace amounts of heavier elements. They have densities of approximately 103 or 104 cm−3 and temperatures between 10 and 20 K, and often contain denser clumps. The clouds are optically thick and so photons from the outside are absorbed on the surface of the clouds. The interiors are heated and ionized by cosmic rays which penetrate deep into the cloud.

The temperatures are too low to sustain much neutral chemical activity in the clouds, and cosmic ray ionization is important in driving the chemistry. In dense clouds, the cosmic rays both initiate the chemistry and limit it through the production of He+ and through cosmic ray induced photons. Table 82.1 is a list of molecules that have been observed in the interstellar medium, many of which have also been found in other galaxies. It is likely that all but H2 are formed in the gas phase by ion–molecule re-

C + hν → C+ + e− .

(82.6)

The carbon ion cannot react directly with H2 by C+ + H2 → CH+ + H ,

(82.7)

as this reaction is exothermic by 0.4 eV. Instead, the chemistry proceeds by the slow radiative association process C+ + H2 → CH+ 2 + hν .

(82.8)

The CH+ 2 ion may either dissociatively + CH2 + e− → CH + H ,

recombine (82.9)

or react with molecular hydrogen + CH+ 2 + H2 → CH3 + H .

The CH+ 3 then undergoes dissociative + CH3 + e → C + H2 + H

(82.10)

recombination

→ CH + H + H → CH2 + H → CH + H2 ,

(82.11) (82.12) (82.13) (82.14)

where the products are listed in order of decreasing likelihood. The CH is removed by photodissociation CH + hν → C + H

(82.15)

and by photoionization CH + hν → CH+ + e− .

82.4 Dark Molecular Clouds

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Part G 82.4

OH may be used to measure the cosmic ray ionization rate, subject to uncertainties in the OH photodissociation rate and the H+ 3 recombination rate. The OH abundances give rates of several ×10−17 s−1 for many diffuse clouds. The carbon chemistry begins with the ionization of C by UV photons:

82.4 Dark Molecular Clouds

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Applications

Table 82.1 Molecules observed in interstellar clouds

Part G 82.4

H2 CH+ C2 CO NH CS SO NS PN SiN H2 O HCN HCO N2 H+ HNO SO2 C2 O N2 O H2 CO NH3 HNCO C3 H C3 S C2 H2 HCNH+ CH4 HCOOH HC3 N HCCNC NH2 CH CH2 NH H2 CCCC C5 H CH3 OH NH2 CHO CH2 CHCN C6 H CH3 CHO CH3 C3 N CH32 O HC7 N HC9 N

Hydrogen Methylidyne ion Carbon Carbon monoxide Amidogen Carbon monosulphide Sulphur monoxide Nitrogen sulphide Phosphorus nitride Silicon nitride Water Hydrogen cyanide Formyl Protonated nitrogen Nitroxyl Sulphur dioxide Carbon suboxide Nitrous oxide Formaldehyde Ammonia Isocyanic acid Propynylidyne Tricarbon sulphide Acetylene Protonated hydrogen cyanide Methane Formic acid Cyanoacetylene Ethynyl isocyanide Cyanamide Methanimine Butatrienylidene Pentynylidyne Methyl alcohol Formamide Vinyl cyanide Hexatrinyl Acetaldehyde Methyl cyanoacetylene Dimethyl ether Cyanohexatriyne Cyano-octatetra-yne

action sequences initiated by cosmic ray ionization. The fact that isomers such as HCN and HNC are seen in approximately equal abundances suggests a low density gas phase environment. Reactions on surfaces and

CH OH CN CO+ NO SiO SO+ SiS HCl NH2 C2 H HNC HCO+ H2 S OCS HCS+ C2 S H2 CN H2 CS HCNS HOCO+ C3 N C3 O H3 O+ C3 H2 H2 CCC CH2 CO HNCCC C4 H CH2 CN CH3 CH CH3 SH HCC2 HO HC3 NH+ CH3 C2 H HC5 N CH3 NH2 HCOOCH3 CH3 C4 H CH3 CH2 CN CH3 CH2 OH HC1 1N

Methylidyne Hydroxyl Cyanogen Carbon monoxide ion Nitric oxide Silicon monoxide Sulphur monoxide ion Silicon sulphide Hydrogen chloride Amino radical Ethynyl Hydrogen isocyanide Formyl ion Hydrogen sulphide Carbonyl sulphide Thioformyl ion Dicarbon sulphide Methylene amidogen Thioformaldehyde Isothiocyanic acid Protonated carbon dioxide Cyanoethynyl Tricarbon monoxide Hydronium ion Cyclopropenylidene Propadienylidene Ketene Cyanoacetylene isomer Butadinyl Cyanomethyl radical Methyl cyanide Methyl mercaptan Propynal Protonated cyanoacetylene Methyl acetylene Cyanodiacetylene Methylamine Methyl formate Methyl diacetylene Ethyl cyanide Ethyl alcohol Cyano-decapenta-yne

the formation of grains are not well understood, but are surely important. The chemistry of molecular clouds is dominated by ion–molecule reactions driven by cosmic ray ionization.

Applications of Atomic and Molecular Physics to Astrophysics

The cosmic rays primarily ionize H2 : − (82.17) H2 → H+ 2 +e , + producing both H2 and fast electrons. The fast electrons produce additional ionizations. The H+ 2 quickly reacts with H2 to form H+ 3

(82.18)

The H+ 3 reacts with other species by proton transfer, which then drives much of the interstellar chemistry. As an example of the production of more complex molecules in interstellar chemistry, we examine the reaction networks leading to the production of water H2 O and the hydroxl radical OH. The H+ 3 ions formed by cosmic ray ionization react with atomic oxygen to form OH+ + O + H+ 3 → OH + H2 ,

which quickly reacts with H2 to form H3 abstraction sequence OH+ + H2 → H2 O+ + H , +

+

H2 O + H2 → H3 O + H .

(82.19)

O+

in an (82.20) (82.21)

O+

The H3 then undergoes dissociative recombination to form water and OH H3 O+ + e− → H2 O + H ,

(82.22)

→ OH + H2 .

(82.23)

The water is removed by reactions with neutral or ionized carbon, which eventually lead to the production of CO. OH is primarily removed by reactions with

atomic oxygen leading to O2 . The CO and O2 are removed by reactions with He+ . The He+ , generated by cosmic ray ionization of helium, does not react with H2 and so is available to remove species by reactions such as CO + He+ → C+ + O + He .

(82.24)

Water and OH are also removed by UV photons generated within the cloud. The clouds are too thick for external UV photons to penetrate, but cosmic rays excite H2 into electronically excited states which decay through emission of UV photons. These internally generated photons play an important role in determining the composition of the cloud. Gredel et al. [82.24] have compiled a list of the photodissociation and photoionization rates for cosmic ray induced photons. Modern chemical networks for molecular clouds include several hundred species and several thousand reactions. A standard set of reaction rates is provided by the UMIST (University of Manchester Institute of Science and Technology) dataset [82.25, 26]. The dataset may be obtained from the UMIST Astrophysics Group homepage (http://saturn.ma.umist.ac.uk:8000/). The clouds also contain dust particles as evidenced by the extinction curves for clouds and the observed depletions of heavier elements. The importance of surface chemistry on these dust particles to interstellar clouds is still uncertain. Dust particles are the best candidate to be the site of formation of molecular hydrogen, because known gas phase reactions fail to produce H2 in the quantities observed.

82.5 Circumstellar Shells and Stellar Atmospheres The continuum emission from a star is very nearly that of a blackbody. This emission is then absorbed and redistributed by the atmosphere of the star. The spectrum of the star is thus determined by its atmosphere. In the hottest stars, most of the material is ionized and the absorption lines are predominantly those of ions, while in the coldest stars, molecular lines are prominent. Kurucz has calculated models with continuum spectra and the inclusion of a large number of absorption lines [82.27]. There are two major projects for calculating the required atomic data. In 1984 an international collaboration named the Opacity Project was set up to calculate accurate atomic data needed for opacity calculations [82.28, 29]. The other earlier project is called

1241

OPAL. The two sources of data are compared in [82.30, 31]. Low and intermediate mass stars eject circumstellar envelopes in their red giant phase near the end of their evolution. Circumstellar envelopes are an important part of astronomy and they are a likely location for dust formation. They provide an interesting environment for studying molecules because they represent a transition between very high density stellar atmosphere environments to low density interstellar environments. These objects evolve to become planetary nebulae Sect. 82.1. We are fortunate in having one example, IRC 10216, which is very close to the Sun. The brightest 10 µm source beyond the solar system, IRC 10216 is a carbon-

Part G 82.5

+ H+ 2 + H2 → H3 + H .

82.5 Circumstellar Shells and Stellar Atmospheres

1242

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Applications

Part G 82.6

rich star surrounded by dust and gas it ejected in a strong stellar wind. The central star is so shielded that it is almost undetectable at optical wavelengths, and was not discovered until the 2 µm survey. IRC 10216 is where most of the circumstellar molecules are detected, and has greatly increased our understanding of circumstellar envelopes. The envelopes are ejected by the red giant in its final phase of evolution. The mass loss rates increase to ∼ 10−4 solar masses per year and temperatures in the envelope are of order 1000 K. Close to the star, the density is high and the chemistry is characteristic of thermal equilibrium. The situation is quite unlike any interstellar environments. For example in IRC 10216, the HNC is over one hundred times less abundant than HCN, whereas in molecular clouds, they have about the same abundance. If the star is oxygen-rich, large amounts of H2 O are formed and if it is carbon-rich, C2 H2 . This high temperature environment forms both molecules and grains.

The high obscuration of the central source indicates that grains are formed in these envelopes. Polycyclic aromatic hydrocarbons (PAH’s) or some similar species are observed in carbon rich planetary nebulae. These large molecules must have been produced when the object was a carbon rich circumstellar envelope. As the material flows out from the star the density and temperature decrease. As the density becomes lower, three body reactions become less important and at some point the products of these reactions are frozen out in a similar manner to the evolution of molecules in the early universe (Sect. 82.10). In the outermost portions of the circumstellar envelope, molecules are dissociated by interstellar UV photons. The penetrating UV radiation is shielded by dust, H2 , and CO. The relative abundances can vary rapidly with radius, and observations provide abundance and radial distribution, a wealth of data for modelers. Circumstellar chemistry is reviewed by Omont [82.32] and recent chemical models are given in [82.33–35].

82.6 Supernova Ejecta A supernova, the explosion of a massive star following core collapse, is one of the most spectacular displays in the Universe. The explosion occurs when the iron core of a massive star collapses to form a neutron star and the rebound shock and neutrino flux eject the outer portion of the star. The ejected portions of the star are rich in heavy elements produced in the interior of the progenitor star. We are fortunate to have had in our lifetime a supernova which was close and in an unobscured line of sight. Supernova 1987A, the first supernova observed in 1987, went off in the Large Magellanic Cloud, a small satellite galaxy to our own. It was the first supernova visible to the naked eye in nearly 400 years (since the Kepler supernova in 1604). Using the full range of modern astronomical instruments has allowed us to get detailed spectra of the evolving ejecta which has greatly increased our knowledge of supernovae. We will use SN1987A as an example of supernova ejecta. Initially the temperature of the ejecta of SN1987A was high, ≈ 106 K, but it quickly cooled through adiabatic expansion and radiation from the photosphere. The temperature leveled off at several thousand degrees because of heating by radioactive nuclei, first 56 Ni and then 56 Co, formed in the explosion. The dynamics is homologous free expansion: the velocity scales linearly

with the radius r(t) = vt where v the velocity and t the time since the explosion. The ejecta at first were optically thick and the spectrum resembled that of a hot star continuum with absorption lines from the surface. After a few days, the temperature dropped, but the ejecta remained optically thick and continued to show strong continuum emission. As the ejecta expand, the temperature drops, the ejecta become optically thin, and the spectrum is dominated by strong emission lines, superficially resembling an emission nebula Sect. 82.2. The emission is dominated by neutral atoms and singly ionized species. The gas is heated and ionized by the gamma rays from radioactive decay. The gamma rays Compton scatter, producing X-rays and fast electrons. The X-rays further ionize the gas and produce multiply charged ions through the Auger process. These multiply charged ions recombine through charge transfer with neutral atoms. Further charge transfer determines the relative ionization of different species, with the lowest ionization potential species more ionized than the higher ionization potential species. The development of the infrared and optical spectrum of Supernova 1987A has been recently reviewed by McCray [82.36]. One of the great surprises in the spectrum of Supernova 1987A was the discovery of molecules in the

Applications of Atomic and Molecular Physics to Astrophysics

infrared region. CO, SiO and possibly H+ 3 have been identified. In the absence of grains, molecules must be formed through either three-body or radiative processes. In the supernova ejecta, the densities are too low for three-body processes to be effective and molecules

82.7 Shocked Gas

are formed through radiative association reactions. The molecules are removed by reactions with He+ and the molecular abundances put a constraint on how much helium can be mixed back into the region with carbon and oxygen [82.37].

ates a hot dilute cavity in the interstellar medium with a temperature of millions of degrees. The density is low and the gas cools and recombines slowly. Overlapping supernova-induced cavities may be responsible for the hot gas that occupies a considerable volume of the interstellar medium in the Galaxy and in some external galaxies. The conditions are far from coronal equilibrium as the gas cools more rapidly than it recombines. The cooling radiation appears as soft X-rays and UV emission lines with a characteristic spectrum. As the gas cools below 10 000 K, molecular formation occurs. Molecular hydrogen is formed on the surfaces of grains as in molecular clouds and by the negative ion sequence that is effective in the early Universe Sect. 82.8. With the formation of H2 in a still warm gas, the chemistry is driven by exothermic and endothermic reactions with H2 . Thus OH is produced by the reaction of O atoms, and H2 O by the further reaction of OH with H2 . Enhanced abundances of other neutral and ionic molecules are the products of subsequent reactions with OH. The reactions of S+ and S with OH lead to SO+ and SO, and their simultaneous presence may be an indicator of a dissociative shock. There are in addition physical indicators of shocks, such as asymmetric line profiles indicating high velocities. In a nondissociative shock in a molecular gas, reactions with warm H2 dominate the chemistry as it does in the cooling zone of a dissociative shock. The composition is controlled by the post shock temperature and the H/H2 ratio. The warm H2 changes the ionic composition by converting C+ into CH+ . Evidence for a nondissociative shock is the infrared emission from H2 . The thermal emission from collisionally excited vibrational levels in shock-heated gas is readily distinguished from that discussed in Sect. 82.3 arising from UV pumping in a PDR. Emission from H2 has been detected in numerous objects in the Galaxy and in many distant external galaxies. In external galaxies, X-rays may contribute to the H2 infrared sprectrum through heating the gas and through excitation by photoelectron pumping to excited states followed by a downward cascade [82.39, 40].

Part G 82.7

82.7 Shocked Gas Shock waves occur in compressible fluids when the pressure gradients are large enough to generate supersonic motion, or when a disturbance is propagating through the fluid at supersonic velocities. Because information about the disturbance cannot propagate upstream in the fluid faster than the speed of sound, the fluid cannot respond dynamically until the shock arrives. The shock then compresses, heats, and accelerates the fluid. The boundary separating the hot compressed gas and the upstream gas is the shock front in which the energy of directed motion of the shock is converted to random thermal energy. Shocks are ubiquitous in the interstellar medium where they are driven by the ionization fronts of expanding Hii regions or nebulae, by outflowing gas accompanying stellar birth and evolution, and by supernova explosions. If the shock velocity is above 50 km/s, the shock gas is excited, dissociated, and ionized. The subsequent recombination and cooling radiation produces photons that may ionize and dissociate the gas components ahead of and behind the shock. This precursor radiation modifies the effects of the shock and influences its dynamical and thermal evolution. Fast shocks destroy all molecules by dissociating H2 by collisions with H, H2 and He and with electrons. Exchange reactions with H atoms destroy the other molecular species. At low densities, radiative stabilization occurs and dissociation is less efficient. Molecules reform in the cooling postshock gas. Slower shocks do not cause ionization or dissociation, but the chemical composition and the ion composition are modified by reactions taking place in the warm gas. The response of the interstellar gas to slow shocks is significantly affected by the presence of a magnetic field. In some ionization conditions, a magnetic precursor may occur in which a magnetosonic wave carries information about the shock, and the ionized and neutral components of the gas react differenly to the shock. Many different kinds of shock have been identified [82.38]. A very fast shock with a velocity of hundreds of km/s such as are driven by supernova explosions, cre-

1243

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Applications

82.8 The Early Universe

Part G 82.9

Molecules appeared first in the Universe after the adiabatic expansion had reduced the matter and radiation temperature to a few thousand degrees and recombination occurred, creating a nearly neutral Universe. The small fractional ionization that remained was essential to the formation of molecules. Molecular hydrogen formed through the sequences H+ + H → H+ (82.25) 2 + hν + H+ + H → H + H (82.26) 2 2 and (82.27) H + e− → H− + hν −

−

H + H → H2 + e , (82.28) the protons and electrons acting as catalysts. Many other atomic and molecularprocesses occurred Fig. 82.3, some involving excited hydrogen atoms. Thus − (82.29) H∗ + H2 → H+ 3 +e + was a source of H3 .

The Universe contained trace amounts of deuterium and 7 Li nuclei with which heteronuclear molecules could be made. Molecules with dipole moments may leave an imprint on the cosmic blackbody background radiation that occupies the Universe. The deuterated molecules HD form from D+ + H2 → HD + H+ , and H2 D+ from + D + H+ 3 → H2 D + H , + HD + H2 → H2 D+ + H , + HD + H+ 2 → H2 D + H .

ϑ

H

Li + H → LiH + hν

(82.34)

Li + H− → LiH + e−

(82.35)

−

LiH + H → Li + H2 e

H(n = 2)

ϑ H(n = 2)

e

H2

ϑ

H–

e

H(n = 2)

e, ϑ

e

H H+ H

ϑ

(82.33)

(82.36)

There are many destruction processes, of which

H–

H

(82.32)

Lithium hydride is formed through

Li + H → LiH + e . H+, ϑ

(82.31)

and

−

e

(82.30)

H+2

H+3 H2

Fig. 82.3 Diagram showing the important reactions in the

production of hydrogen molecules in the early Universe

(82.37)

may be the most severe, though its rate coefficient is uncertain. The chemistry of the early Universe is summarized in [82.41]. The formation of molecules was a crucial step in the fragmentation of the first gravitationally collapsing objects which separated out of the cosmic flow. Threebody recombination H + H + H → H2 + H Li + H + H → LiH + H

(82.38) (82.39)

may be a major source of molecules as the density increases.

82.9 Recent Developments While the core atomic and molecular process outlined are still unchanged, our understanding of the astrophysical environment has been greatly enhanced by a number of recent satellites. The Wilkinson Microwave Anisotropy Probe (WMAP) has given us the best map of the universe at the time of recombination and given us general confirmation of the Big Bang model. Perhaps the most surprising result is that stars seem to have

formed much sooner than would have been expected, about 180 million years after the big bang [82.42]. This makes it even more difficult to understand how the universe goes from the relative uniformity at the time of recombination to the collapse and formation of the first objects so quickly, a problem which is certainly controlled by atomic and molecular processes. The relevant atomic and molecular processes have been recently re-

Applications of Atomic and Molecular Physics to Astrophysics

for temperatures below 106 K, but still too slow to significantly reduce the helium ion abundance in supernova ejecta. The most likely explanation remains that mixing is not complete in supernova ejecta, and the molecules survive in regions of relatively low helium abundance. The state of modeling photoionized clouds has been recently reviewed by Ferland [82.50]. He also highlights the great need for atomic and molecular data for analyzing these clouds. New satellite data along with continued ground observations continually raise new astrophysical puzzles, puzzles which are controlled and probed by atomic and molecular processes. The astrophysical community owes a great debt to both atomic and molecular laboratory measurements and theoretical models of energy levels, reaction rates, and transition probabilities. In order to continue to progress in our understanding of the universe we will need to continue to fund the understanding of the atomic and molecular processes which control it.

82.10 Other Reading Astronomy is one of the oldest sciences and one of the fastest evolving. Advances in technology are rapidly increasing the sensitivity and resolution of our instruments and so new observations and more sophisticated models lead to an ever greater understanding of the Universe. This means that books will often be somewhat dated when they appear. However, the series Annual Review of Astronomy

and Astrophysics is a good source of recent review articles. In addition, good introductions or overviews of a particular field are given in [82.1, 51–56]. Many sources of atomic and molecular data are listed and discussed in [82.7]. For details on atomic spectroscopy, see [82.57, 58]. For details on molecular spectroscopy, see [82.59, 60].

References 82.1

82.2

82.3 82.4 82.5 82.6 82.7 82.8

D. E. Osterbrock: Astrophysics of Gaseous Nebulae and Active Galactic Nuclei (Univ. Science Books, Mill Valley 1989) P. F. Winkler, G. W. Clark, T. H. Markert, K. Kalata, H. W. Schnopper, C. R. Canizares: Astrophys. J. Lett. 246, 27L (1981) R. F. Reilman, S. T. Manson: Astrophys. J. Suppl. Ser. 40, 815 (1979) B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukoro, B. K. Fujikawa: Atom. Nucl. Data Tables 27, 1 (1982) M. Balucinska-Church, D. McCammon: 400, 699 (1992) D. A. Verner, D. G. Yakovlav, J. M. Band, A. B. Trzhaskavskaya: Atom. Nucl. Data Tables 55, 233 (1993) Special issue of Revista Mexicana de Astronomia y Astrofisica, March 23 (1992) H. Nussbaumer, P. J. Storey: Astron. Astrophys. 178, 324 (1978)

82.9 82.10 82.11 82.12 82.13 82.14 82.15 82.16 82.17 82.18

H. R. Ramadan, Y. Hahn: Phys. Rev. A 39, 3350 (1989) N. R. Badnell: Phys. Scr. T 28, 33 (1989) M. C. Bacchus-Montabonel, K. Amezian: Z. Phys. D 25, 323 (1993) P. Honvault, M. C. Bacchus-Montabonel, R. McCarroll: J. Phys. B 27, 3115 (1994) B. Herrero, I. L. Cooper, A. S. Dickinson, D. R. Flower: J. Phys. B 28, 711 (1995) C. Mendoza: Planetary Nebulae IAU Symp. 103, ed. by D. R. Flower (Reidel, Dordrecht 1983) p. 143 M. Brocklehurst, M. Salem: Comp. Phys. Commun. 13, 39 (1977) M. Salem, M. Brocklehurst: Astrophys. J. Suppl. Ser. 39, 633 (1979) P. G. Martin: Astrophys. J. Suppl. Ser. 66, 125 (1988) P. J. Storey, D. G. Hummer: Mon. Not. R. Astron. Soc. 272, 41 (1995)

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viewed by Lepp, Stancil and Dalgarno [82.43]. Two recent X-ray satellites, Chandra and XMM-Newton, both launched in 1999, have greatly increased our ability to detect hot ionized gas in stars, supernova remnants, active galaxies and other regions [82.44]. In particular, Chandra has allowed us for the first time to directly observe the hot gas between galaxies [82.45]. The Infrared Space Observatory (ISO) has provided a tremendous amount of data on cold regions in our own galaxy and allowed us to directly observe the icy mantles of dust grains [82.46]. Since the detection of molecules in SN 1987A, there have been many more observations of CO molecules in Type II supernova and they may even occur in every Type II [82.47]. CO has also been observed in a Type Ic supernova [82.48]. It remains a puzzle as to why the molecules are not rapidly removed by helium ions. A recent calculation of the O + He+ system [82.49] finds that radiative charge transfer is much faster then direct charge transfer

References

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Part G

Applications

82.19 82.20 82.21 82.22

Part G 82

82.23 82.24 82.25

82.26 82.27 82.28 82.29 82.30 82.31 82.32

82.33 82.34 82.35 82.36 82.37 82.38 82.39

M. Arnaud, R. Rothenflug: Astron. Astrophys. Suppl. 60, 425 (1985) T. J. Gaetz, E. E. Salpeter: Astrophys. J. Suppl. Ser. 52, 155 (1983) E. van Dishoeck, J. Black: Astrophys. J. Suppl. Ser. 62, 109 (1987) W. G. Roberge, D. Jones, S. Lepp, A. Dalgarno: Astrophys. J. Suppl. Ser. 77, 287 (1991) A. Sternberg, A. Dalgarno: Astrophys. J. Suppl. Ser. 99, 565 (1995) R. Gredel, S. Lepp, A. Dalgarno, E. Herbst: Astrophys. J. 347, 289 (1989) T. J. Millar, A. Bennett, J. M. C. Rawlings, P. D. Brown, S. B. Charnley: Astron. Astrophys. Suppl. 87, 585 (1991) P. R. A. Farquhar, T. J. Millar: CCP7 Newsletter 18, 6 (1993) R. L. Kurucz: Astrophys. J. Suppl. Ser. 40, 1 (1979) M. J. Seaton: J. Phys. B 20, 6363 (1987) A. E. Lynas-Gray, M. J. Seaton, P. J. Storey: J. Phys. B 28, 2817 (1995) M. J. Seaton, Y. Yan, B. Mihalas, A. K. Pradhan: Mon. Not. R. Astron. Soc. 266, 805 (1994) C. A. Iglesias, F. J. Rogers: Astrophys. J. 443, 460 (1995) A. Omont: Circumstellar Chemistry. In: Chemistry in Space, ed. by J. M. Greenberg, V. Pirronello (Kluwer Academic, Dordrecht 1991) p. 171 G. A. Mamon, A. E. Glassgold, A. Omont: Astrophys. J. 323, 306 (1987) L. A. M. Nejad, T. J. Millar: Astron. Astrophys. 183, 279 (1987) T. J. Millar, E. Herbst: Astron. Astrophys. 288, 561 (1994) R. McCray: Ann. Rev. Astron. Astrophys. 31, 175 (1993) W. Liu, S. Lepp, A. Dalgarno: Astrophys. J. 396, 679 (1992) B. T. Draine, C. F. McKee: Ann. Rev. Astron. Astrophys. 31, 373 (1993) S. Lepp, R. McCray: Astrophys. J. 269, 560 (1983)

82.40 82.41

82.42 82.43 82.44 82.45 82.46

82.47 82.48 82.49 82.50 82.51 82.52 82.53 82.54 82.55

82.56 82.57 82.58 82.59 82.60

R. Gredel, A. Dalgarno: Astrophys. J. 446, 852 (1995) A. Dalgarno, J. Fox: Ion Chemistry in Atmospheric and Astrophysical Plasmas. In: Unimolecular and Bimolecular Reaction Dynamics, ed. by C.-Y. Ng, T. Baer, I. Powis (Wiley, New York 1994) C. L. Bennett et al.: Astrophys. J. Suppl. 148, 1 (2003) S. Lepp, P. Stancil, A. Dalgarno: J. Phys. B 35, R57 (2002) F. Paerels, S. Kahn: Ann. Rev. Ast. Appl. 41, 291 (2003) F. Nicasto et al.: Nature 433, 495 (2005) C. Cesarsky, A. Salama: ISO Science Legacy: A Compact Review of ISO Major Achievements (Springer, 2005) J. Spyrimilo, B. Leibundgut, R. Gilmozzi: Ast. Ap. 376, 188 (2001) C. Gerardy et al.: PASJ 54, 905 (2002) L. B. Zhao et al.: Astrophys. J. 615, 1063 (2004) G. Ferland: Ann. Rev. Ast. Appl. 41, 517 (2003) C. W. Allen: Astrophysical Quantities (Athlone, London 1973) K. R. Lang: Astrophysical Formula (Springer, Berlin, Heidelberg 1980) W. W. Duley, D. A. Williams: Interstellar Chemistry (Academic, London 1984) L. Spitzer: Physical Processes in the Interstellar Medium (Wiley, New York 1978) A. Dalgarno, D. R. Layzer: Spectroscopy of Astrophysical Plasmas (Cambridge Univ. Press, Cambridge 1987) T. Hartquist: Molecular Astrophysics (Cambridge Univ. Press, Cambridge 1990) R. D. Cowan: The Theory of Atomic Structure and Spectra (Univ. California Press, Berkeley 1981) I. I. Sobelman: Atomic Spectra and Radiative Transitions (Springer, Berlin, Heidelberg 1979) G. Herzberg: Molecular Spectra and Molecular Structure (Prentice-Hall, New York 1939) P. F. Bernath: Spectra of Atoms and Molecules (Oxford Univ. Press, Oxford 1995)

1247

Comets

Comets are small bodies of the solar system believed to be remnants of the primordial solar disk. Formed near the orbits of Uranus and Neptune and subsequently ejected into an “Oort cloud” of some 40 000 AU in extent, these objects likely preserve a record of the volatile composition of the early outer solar system, and so are of great interest for the physical and chemical modeling of solar system formation. The comets arrive in the inner solar system as a result of galactic perturbations. The cometary volatiles are vaporized as their

83.1 Observations ....................................... 1247 83.2 Excitation Mechanisms......................... 1250 83.2.1 Basic Phenomenology................ 1250 83.2.2 Fluorescence Equilibrium ........... 1250 83.2.3 Swings and Greenstein Effects .... 1251 83.2.4 Bowen Fluorescence .................. 1252 83.2.5 Electron Impact Excitation .......... 1253 83.2.6 Prompt Emission ....................... 1253 83.2.7 OH Level Inversion..................... 1254 83.3 Cometary Models ................................. 1254 83.3.1 Photolytic Processes .................. 1254 83.3.2 Density Models ......................... 1255 83.3.3 Radiative Transfer Effects ........... 1256 83.4 Summary ............................................ 1256 References .................................................. 1257 analytical review of Festou et al. [83.2, 3] or in the compendia of Halley results [83.4]. The former also contains a comprehensive bibliography. Other sources concentrating largely on the physics and chemistry of comets include the volumes edited by Wilkening [83.5] and Huebner [83.6] and the pre-Halley review of Mendis et al. [83.7].

orbits bring them closer to the sun and it is solar radiation that initiates all of the processes that lead to the extended coma. Gas vaporization also leads to the release of dust into the coma, and the scattering of sunlight by dust is the major source of the visible coma and dust tail of comets. Somewhat fainter, and much more extended, is the plasma tail, resulting from photoionization by solar extreme UV radiation of the neutral volatiles and their subsequent interaction with the solar wind.

83.1 Observations In a rized sions trum

review in 1965, Arpigny [83.8] summathe known molecular and atomic emisdetected in the visible region of the spec(here defined as 3000 to 11 000 Å) as

follows: radicals: OH, NH, CN, CH, C3 , C2 , and NH2 + + ions: OH+ , CH+ , CO+ 2 , CO , and N2 ; metals: Na, Fe .

Part G 83

With the exception of the in situ measurements made by the Giotto and Vega spacecraft at comet 1P/Halley (the P/ signifies a periodic comet) during March 1986, all determinations of the volatile composition of the coma are derived from spectroscopic analyses. Detailed modeling is then used to infer the volatile composition of the cometary nucleus. This chapter focuses on the principal atomic and molecular processes that lead to the observed spectrum as well as the needs for basic atomic and molecular data in the interpretation of these spectra. The largely collisionless and low density coma, with no gravity or magnetic field, is a unique spectroscopic laboratory, as evidenced by the discovery of C3 before its identification in terrestrial laboratories [83.1]. Many key discrepancies remain to be resolved concerning the basic molecular composition and the elemental abundances of both the volatile and refractory components of the cometary nucleus, as well as the comet-to-comet variation (particularly between “new” and evolved periodic comets) of these quantities. These issues (and many others) are discussed in the recent

83. Comets

1248

Part G

Applications

Part G 83.1

The only known atomic feature was the O i forbidden red doublet at 6300 and 6364 Å. From the radicals and ions one could infer the presence of their progenitor “parent” molecules such as H2 O, NH3 , HCN, CO and CO2 , directly vaporizing from the comet’s nucleus. The metals, seen only in comets passing close to the sun, were assumed to come from the vaporization of refractory grains. The inventory of metals was soon expanded to include K, Ca+ , Ca, V, Cr, Mn, Ni and Cu, from observations of the sun-grazing comet IkeyaSeki (C/1965 S1) [83.9, 10] and H2 O+ was identified in comet Kohoutek (C/1973 E1). This latter comet was also the first to be extensively studied at wavelengths both shortward and longward of the visible spectral range. The first parent molecule to be directly identified was fluoresces in the Fourth Positive system 
1 CO, which A Π u − X 1Σ + in the VUV [83.11], although the in situ neutral mass spectrometer measurements made of Halley disclosed the presence of an extended, dominant source of CO [83.12] whose origin is still being debated [83.13]. Ideally, the molecular species should be detectable through their radio and sub-mm rotational transitions or through the detection of vibrational bands or individual ro-vibrational lines in the near IR. Water was first directly detected through ro-vibrational lines near 2.7 µm in comet Halley and again in comet Wilson (C/1986 P1) [83.14, 15]. However, due to the low column densities of the other expected species, typically ≈1 or less than that of H2 O, the direct detection of species such as H2 CO, H2 S and CH3 OH has only recently been made possible by the development of more sensitive instrumental techniques together with the fortuitous apparitions of two bright comets, C/1996 B2 (Hyakutake) and C/1995 O1 (Hale-Bopp) in 1996 and 1997. To date, more than two dozen parent molecules have been identified [83.16]. Isotope ratios, particularly the D/H ratio, in molecules such as HDO have been determined from sub-millimeter observations [83.17]. The ultimate result of solar photolysis (and to a lesser degree, the interaction with the solar wind) is the reduction of all of the cometary volatiles to their atomic constituents. The atomic inventory is somewhat easier to derive as the resonance transitions of the cosmically abundant elements H, O, C, N and S all lie in the VUV and, in principle, the total content of these species in the coma can be determined by an instrument with a suitably large field of view. Of course, a fraction of the atomic species of each element will be produced directly in ionic form, and will not be counted using this approach. In addition, another fraction exists in the coma in the solid grains, and this component will also not be included,

except for a small amount volatilized by evaporation or sputtering by energetic particles. The composition of the grains, though not the absolute abundance, has been determined from in situ measurements made by the Halley encounter spacecraft [83.18], and can be inferred, though not unambiguously, from reflection spectroscopy of cometary dust in the 3–5 µm range. The advent of space-borne platforms for observations in the VUV has produced a wealth of new information about the volatile constituents of the coma. The A 2 Σ + − X 2 Π (0, 0) band of the OH radical at ≈ 3085 Å was well known from ground-based spectroscopic observations, but as this wavelength lies very close to the edge of the atmospheric transparency window, the strength of this feature (relative to that of other species) was not appreciated until 1970 when comet Bennett (C/1969 Y1) was observed from space by the Orbiting Astronomical Observatory (OAO-2). The OAO-2 spectrum also showed a very strong, broadened H i Ly-α emission from H, the other principal dissociation product of H2 O. The broad shape of Ly-α seen in the OAO-2 spectrum is due to the large spatial extent of the atomic H envelope, the result of a high velocity acquired in the photodissociation process and a long lifetime against ionization. Later, at the apparition of comet Kohoutek (C/1973 E1), atomic O and C were identified in the spectra and direct UV images of the H coma, as well as of the O i and C i emissions, were obtained from sounding rocket experiments. These experiments were repeated for comet West (C/1975 V1) and led to the first detection of CO [83.11]. Between 1978 and 1996, over 50 comets were observed spectroscopically over the wavelength range 1200–3400 Å by the International Ultraviolet Explorer (IUE) satellite observatory [83.19, 20]. Most of the spectra were obtained at moderate resolution (∆λ = 6–10 Å), although high dispersion echelle spectra (∆λ = 0.2–0.3 Å) are useful for some studies, particularly those of fluorescence equilibrium (Sect. 83.2.2). For Halley alone, over 200 UV spectra were obtained from September 1985 to July 1986. The launch of the Hubble Space Telescope (HST) in 1990, together with subsequent enhancements to the spectroscopic instrumentation that were made on-orbit, marked another advance in sensitivity as well as the ability to observe in a small field-of-view very close to the nucleus. This yielded the first detection of CO Cameron band emission, a direct measure of CO2 being vaporized from the nucleus [83.21]. For an overview of a cometary spectrum, a composite spectrum of 103P/Hartley 2 spanning the region from H i Ly-α to 7000 Å taken with

Comets

10 000.00

Brightness (Rayleighs / Å) H

1000.00

strong absorption by water vapor in the terrestrial atmosphere. Prior to 1996, X-rays had not been detected in comets and the conventional wisdom was that they were unlikely to be produced in the cold, rather thin cometary atmosphere. The discovery of soft X-ray emission (E 30 km/s, the Doppler shift reduces the solar flux at the center of the absorption line to a very small value, so that the O i λ1302 line is expected to appear weakly, if at all, in the observed spectrum. Thus, it was a surprise that this line appeared fairly strongly in two comets, Kohoutek (C/1973 E1) and West (C/1975 V1), whose values of r˙ were both > 45 km/s at the times of observation. The explanation invoked the accidental coincidence of the solar H i Ly-β line at 1025.72 Å with the O i 3D − 3P transition at 1025.76 Å, cascading through the intermediate 3P state as shown in the simplified energy level diagram of Fig. 83.4 [83.35]. This mechanism, well known in the study of planetary nebulae, is referred to as Bowen fluorescence [83.38]. The g-factor due to Ly-β pumping is an order of magnitude smaller than that for resonance scattering, as shown in Fig. 83.3, but sufficient to explain the observations and to confirm that H2 O is the dominant source of the observed oxygen in the coma. Ly-β is also coincident with
the P1 line of the  (6,0) 1Σ + leadband of the H2 Lyman system B 1Σ + − X u g ing to fluorescence in the same line of several (6,v ) bands, the strongest being that of the (6,13) band at 1608 Å [83.39]. This line is, however, difficult to ob-

Comets

(3s⬘)3D

(3d)3D 7990

11287 (3p)3P

8447 (3s)3S

989

1027

1304

5577

2972 (2p4)⬘D

OI

6364 6300

(2p4)3P

Fig. 83.4 Simplified O i energy level diagram showing

transitions of interest in cometary spectra

serve because of the nearby strong CO Fourth Positive bands. Recently, the shorter wavelength (6,1), (6,2), and (6,3) bands have been detected in three comets using FUSE and the derived H2 column abundance was found to be consistent with a water photodissociation source [83.22, 23]. Another interesting example occurs for Ne i, where the second resonance transition at 629.74 Å coincides with the strong solar O v line at 629.73 Å. This line was used to set a sensitive upper limit on the Ne abundance in the coma of comet Hale-Bopp (C/1995 O1) [83.40].

83.2.5 Electron Impact Excitation The photoionization of the parent molecules and their dissociation products leads to the formation of a cometary ionosphere whose characteristics are only poorly known. Planetary ionospheres serve only as a poor model since the cometary atmosphere is gravitationally unbound and there is no constraining magnetic field. The in situ measurements of Halley provided order of magnitude confirmation of the theoretical modeling. In principle, electron impact excitation, which is often

the dominant source of airglow in the atmospheres of the terrestrial planets Chapt. 84, also contributes to the observed emissions, particularly in the UV, and so must be accounted for in deriving column densities from the observed emission brightnesses. However, one can use a very simple argument, based on the known energy distribution of solar UV photons, to demonstrate that electron impact excitation is only a minor source for the principal emissions. Since the photoionization rate of water (and of the important minor species such as CO and CO2 ) is ≈ 10−6 s−1 at 1 AU, and the efficiency for converting the excess electron energy into excitation of a single emission is of the order of a few percent, the effective excitation rate for any emission will be ≈ 10−8 s−1 or less at 1 AU [83.41]. Since the efficiencies for resonance scattering or fluorescence for almost all the known cometary emissions are much larger, electron impact may be safely neglected except in a few specific cases. The cases of interest are those of forbidden transitions, where the oscillator strength, and consequently the g-factor, is very small. Examples include the O i 5S − 3P 2 2,1 doublet at 1356 Å, which was observed in comets West and Halley by rocket-borne spectrographs and more recently in comets Hyakutake and Hale-Bopp, the O i 1D − 3P red lines at 6300 and 6364 Å, observed in many comets, and the CO Cameron bands [83.21]. However, the excitation of these latter two is dominated by prompt emission in the inner coma, the same region of the coma where electron excitation is important, as described in the next section.

83.2.6 Prompt Emission In cases where the dissociation or ionization of a molecule leaves the product atom or molecule in an excited state, the decay of this state with the prompt emission of a photon provides a useful means for tracing the spatial distribution of the parent molecule in the inner coma. The products of interest for the water molecule are described in Sect. 83.3.1. Prompt emission includes both allowed radiative decays (such as from the A 2 Σ + state of OH) as well as those from metastable states such as O(1D ), since the latter will move ∼150 km (for a comet at a geocentric distance of 1 AU, 1 arcsecond corresponds to a projected distance of 725 km) in its lifetime. The O i 1D – 3P transition at 6300 and 6364 Å Fig. 83.4 has been used extensively as a groundbased monitor of the water production rate with the caveat that other species such as OH, CO and CO2 may also contribute to the observed red line emission.

1253

Part G 83.2

(2p4)⬘S

83.2 Excitation Mechanisms

1254

Part G

Applications

Part G 83.3

In addition, when the density of H2 O is sufficient to produce observable red line emission, it is also sufficient to produce collisional quenching of the 1D state, and this must also be considered in the interpretation of the observations. The analogous 1D − 3P transitions in carbon occur at 9823 and 9849 Å and provide similar information about the production rate of CO. Carbon atoms in the 1D state, whose lifetime is ≈4000 s, are known to be present from the observation of the resonantly scattered 1P o − 1D transition at 1931 Å [83.42], and the 9849 Å line has been detected in comet HaleBopp [83.43]. The OH A 2 Σ + − X 2 Π prompt emission competes with that produced by the resonance fluorescence of OH and is difficult to detect, except close to the nucleus (inside 100 km) where the density of water molecules exceeds that of OH by a few orders of magnitude [83.44]. Again, the reason is that at the wavelengths below the threshold for simultaneous dissociation and excitation, the sun has much less flux than at the resonance wavelength. On the other hand, only a few rotational lines are excited in fluorescence equilibrium [83.34], while the prompt emission is characterized by a very “hot” rotational distribution, so in principle the two components

may be separated although observations at very high spatial and spectral resolution are required. OH prompt emission has also recently been detected in the infrared at 3.28 µm [83.45].

83.2.7 OH Level Inversion An important consequence of fluorescence equilibrium in the OH radical is the UV pumping of the hyperfine and Λ-doublet levels of the X 2 Π3/2 (J = 3/2) ground state, which results in a deviation of the population from statistical equilibrium [83.34]. Depending on the heliocentric velocity, this departure may be either “inverted” or “antiinverted” giving rise to either stimulated emission or absorption against the galactic background at 18 cm wavelength. This technique has been used extensively since 1974 to monitor the OH production rate in comets, even of those that appear close to the sun [83.46]. The resulting radio emissions are easily quenched by collisions with molecules and ions, the latter giving rise to a fairly large Rc that must be accounted for in interpreting the derived OH column density. Nevertheless, the radio and UV measurements give reasonably consistent results [83.46].

83.3 Cometary Models 83.3.1 Photolytic Processes As an example of the photolytic destruction processes occurring in the coma, consider the dominant molecular species, water. Water vapor is assumed to leave the surface of the nucleus with some initial velocity v0 and flow radially outward, expanding into the vacuum, and increasing its velocity according to thermodynamics [83.7]. Even though collisions are important at distances typically up to 104 km (depending on the density and consequently, on the total gas production rate), the net flow of H2 O molecules is radially outward, such that the density varies as R−2 near the nucleus, where R is the cometocentric distance. This is the basis for spherically symmetric coma models (the number of particles flowing through a spherical surface is conserved), which assume isotropic gas production, but appears to hold equally well for the case of Halley, which clearly was not outgassing uniformly over its surface [83.4]. The photolysis of H2 O can proceed by: a a

H2 O + hν→OH + H 2424.6 Å →OH(A 2 Σ+ ) + H 1357.1 Å

b b b c d e f

H2 O + hν→H2 + O(1D ) 1770 Å →H2 + O(1S ) 1450 Å →H + H + O(3P ) 1304 Å 984 Å →H2 O+ + e− →H + OH+ + e− 684.4 Å →H2 + O+ + e− 664.4 Å →OH + H+ + e− 662.3 Å

The right-hand column gives the energy threshold for each reaction, in wavelength units. The products are subsequently removed by: g h i j k l m

OH + hν→O + H →OH+ + e− H2 + hν→H + H − →H+ 2 +e + →H + H + e− O + hν→O+ + e− H + hν→H+ + e−

2823.0 Å 928 Å 844.79 Å 803.67 Å 685.8 Å 910.44 Å 911.75 Å

Reactions l and m can also occur by resonant charge exchange with solar wind protons. Reactions a , b and b correspond to the production of prompt emission, as

Comets

discussed in Sect. 83.2.6. The determination of column densities of H, O and OH simultaneously was convincing evidence that the dominant volatile species in the cometary nucleus was H2 O, long before the direct infrared detection of this species in the coma. Detailed cross sections for the absorption of UV photons by each of the reactants, including proper identification of the final product states, is necessary for the evaluation of the photodestruction rate in the solar radiation field of each of the above reactions. These rates are evaluated at 1 AU using whole disk measurements of the solar flux by integrating the cross section Jd =

πF σ d dλ .

(83.10)

0

Another quantity of interest in coma modeling is the excess velocity (or energy) of the dissociation or ionization products, and this requires knowledge of the partitioning of energy between internal and translational modes for each reaction [83.47]. Qualitatively, the photodissociation and photoionization rates can be estimated from the threshold energies given in the table above, since the solar flux is decreasing very rapidly to shorter wavelengths, as can be seen in Fig. 83.1. It is customary to specify the lifetime against photodestruction τi , of species i, which is just  (J d ) j , (83.11) (τi )−1 = j

where the sum is over all possible reaction channels, as well as the lifetimes into specific channels. Processes with thresholds near 3000 Å have lifetimes ≈104 s, those with thresholds near 2000 Å an order of magnitude longer, while those with thresholds below Ly-α, such as most photoionization channels, have lifetimes ≈106 s, all at 1 AU. In addition to uncertainties in the details of the absorption cross sections, further uncertainty is introduced into the calculation of J d by the lack of knowledge of the solar flux at the time of a given observation due to the variability of the solar radiation below 2000 Å, and most importantly, below Ly-α, where there have not been continuous space observations for more than a decade. The solar UV flux is known to vary considerably both with the 27-day solar rotation period and with the 11-year solar activity cycle. Also, at any given point in its orbit, a comet sees a different hemisphere of the sun than what is seen from Earth. Huebner et al. [83.48] have compiled an extensive list of useful

1255

photodestruction rates using mean solar fluxes to represent the extreme conditions of solar minimum and solar maximum. They also include the excess energies of the dissociation products. A detailed analysis of the rates for H2 O and OH, using surrogate solar indices such as the 10.7 cm solar radio flux, or the equivalent width of the He i line at 1.083 µm, is in good agreement with observations [83.49]. Similar analyses still remain to be carried out for other important species, such as CO and NH3 .

83.3.2 Density Models For parent molecules produced directly by sublimation from the surface of the comet, a spherically symmetric radial outflow model is often adopted. Such a model assumes a steady-state gas production rate Q i and a constant outflow velocity v, and gives rise to a density distribution as a function of cometocentric distance R given by n i (R) =

Qi e−R/βi , 4πvR2

(83.12)

where βi = vτi is the scale length of species i. The basic validity of this model was demonstrated by the Giotto neutral mass spectrometer measurements of H2 O and CO2 [83.4], although detailed analysis revealed that the velocity of the water molecules increased from 0.8 km/s at about 1000 km from the nucleus to 1.1 km/s at a radial distance of 10 000 km. The dependence of outflow velocity on heliocentric distance remains uncertain, although Delsemme [83.50] has suggested an r −1/2 dependence based on thermodynamic arguments. Sub-millimeter observations of H2 O have sufficient spectral resolution to permit the mapping of outflow velocities along various lines-of-sight to the comet [83.28]. For the dissociation products, the modeling is more complex. The simplest model assumes continued radial outflow, although at a different velocity, such as to maintain a constant flux of the initial particle across an arbitrary spherical surface surrounding the nucleus [83.51]. This model, which is valid only at distances equal to a few βi , is widely used as the densities can be easily expressed in analytical form. However, as surface brightness measurements are often made with small fields of view close to the nucleus, this model can lead to a factor of two error from the neglect of the dissociation kinematics. Since the solar photodissociation often leaves the product fragments with 1–2 eV of kinetic energy [83.47, 48], the resultant motion (which is assumed to be isotropic in the parent molecule’s rest frame), will contain a large nonradial

Part G 83.3

λth

83.3 Cometary Models

1256

Part G

Applications

Part G 83.4

component. Several approaches have been developed to account properly for the kinematics; notably, the vectorial model of Festou [83.52], and the average random walk model (a Monte Carlo method) of Combi and Delsemme [83.53]. The latter model has been extended to include time-dependent gas kinetics so as to properly account for regions of the coma where the gas is not in local thermodynamic equilibrium [83.54]. In addition to the photodestruction chains, chemical reactions, particularly ion-molecule reactions, can alter the composition within the collision zone defined in Sect. 83.2.2. While such reactions may produce numerous minor species, they do not erase the signatures of the original parent molecules. In fact, detailed chemical models have clearly demonstrated the need for complex molecules to serve as the parents of the observed C2 and C3 radicals in the coma, strengthening the connection between comet formation and molecular cloud abundances. Thus, the photochemical chains provide a valid means of relating the coma composition to that of the nucleus.

83.3.3 Radiative Transfer Effects The results of a model calculation for the density of a species must be integrated over the line of sight to obtain the column density at a given projected distance from the nucleus, and then integrated over the instrumental field of view for comparison with the observed average surface brightness or derived average column density Ni . This assumes that the coma is optically thin, and that all atoms or molecules have an equal probability of absorbing a solar photon. In practice, this is true for all molecular emissions except perhaps within 1000 km of the nucleus (i. e., for observations made at better than 1 resolution). Since the cross sections at

line center can be very large for an atomic resonance transition, the optical depth for the abundant species can exceed unity and radiative transfer along both the line of sight to the sun and that to the Earth must be considered. This is not a trivial problem as the velocity distribution of the atoms, particularly the component due to the excess energy of the photodestruction process, must be well known, as must be the shape of the exciting solar line. The most thoroughly studied case to date is that of H i Ly-α, whose angular extent, in direct images, can exceed several degrees on the sky [83.55]. An interesting case arises for resonance transitions between an excited 3 S1 state and the ground 3 P2,1,0 state, as for O and S, particularly the latter, as its concentration near the nucleus can be quite large due to the rapid decay of one of its parents, CS2 . For S i the three lines at 1807, 1820 and 1826 Å are not observed to have their statistical intensity ratio of 5:3:1, except at large distances from the nucleus. This is explained by noting that fine structure transitions will lead to all of the S atoms reaching the J = 2 ground state in a time short compared with that for absorbing a solar photon, and that the emitted 1807 Å photons will be re-absorbed and can then branch into the other two lines. The detailed solution to this problem has led to the conclusion that H2 S was the primary source of sulfur rather than CS2 , whose other product, CS, was simultaneously observed in the UV [83.56]. Millimeter and sub-millimeter observations of comet Hale-Bopp (C/1995 O1) subsequently showed that SO, SO2 , and OCS were also minor sources of atomic sulfur, comparable in abundance to CS2 [83.57]. Another minor source is S2 , initially observed in only one comet, IRAS-Araki-Alcock (C/1983 H1) [83.58], but recently seen in three additional comets by HST. The origin of S2 in the cometary nucleus remains a puzzle.

83.4 Summary This brief chapter can only hint at the wealth of observational data spanning the entire electromagnetic spectrum now routinely acquired at almost every comet apparition allowing for a statistically significant assessment of comet diversity and formation scenarios. Reference [83.59] will bring the interested reader up to date on all aspects of comet science. The next few years will see several spacecraft missions to comets, Stardust, Deep Impact and Rosetta, whose primary objective is the study of the cometary nucleus whose properties can only be inferred from remote

observations. Nevertheless, Earth-based observations of comets will continue to play an important role in understanding the physical and chemical environments of these objects left over from the formation of the Solar System. There are still significant challenges in understanding the atomic and molecular physics of the cometary atmosphere, an example being the identification of the large number of unidentified lines seen in high resolution spectra in both the far UV [83.22, 23] and visible [83.60] regions of the spectrum.

Comets

References

1257

References 83.1

83.2 83.3 83.4

83.5

83.7 83.8 83.9 83.10 83.11 83.12

83.13 83.14 83.15 83.16

83.17

83.18

83.19

83.20

83.21

83.22

83.23

83.24

83.25

83.26

83.27 83.28 83.29 83.30 83.31 83.32 83.33 83.34 83.35

83.36 83.37 83.38 83.39 83.40

83.41 83.42 83.43 83.44 83.45 83.46 83.47 83.48

83.49

H. A. Weaver, P. D. Feldman, M. R. Combi, V. Krasnopolsky, C. M. Lisse, D. E. Shemansky: Astrophys. J. Lett. 576, L95 (2002) H. A. Weaver, P. D. Feldman: Science with the Hubble Space Telescope, ed. by P. Benvenuti, E. Schreier (ESO, Garching 1992) p. 475 J. T. T. Mäkinen, J.-L. Bertaux, T. I. Pulkkinen, W. Schmidt, E. Kyrölä, T. Summanen, E. Qu´ emerais, R. Lallement: Astron. Astrophys. 368, 292 (2001) M. S. Povich, J. C. Raymond, G. H. Jones, M. Uzzo, Y.-K. Ko, P. D. Feldman, P. L. Smith, B. G. Marsden, T. N. Woods: Science 302, 1949 (2003) D. A. Neufeld et al.: Astrophys. J. Lett. 539, L151 (2000) A. Lecacheux et al.: Astron. Astrophys. 402, L55 (2003) C. M. Lisse, and 11 colleagues: Science 274, 205 (1996) T. E. Cravens: Science 296, 1042 (2002) J. B. Greenwood, I. D. Williams, S. J. Smith, A. Chutjian: Phys. Rev. A 63, 62707 (2001) V. Kharchenko, A. Dalgarno: Astrophys. J. Lett. 554, L99 (2001) P. Swings: Lick. Obs. Bull. XIX, 131 (1941) D. G. Schleicher, M. F. A’Hearn: Astrophys. J. 331, 1058 (1988) P. D. Feldman, C. B. Opal, R. R. Meier, K. R. Nicolas: The Study of Comets, NASA SP-393, ed. by B. Donn, M. Mumma, W. Jackson, M. A’Hearn, R. Harrington (NASA, Washington 1976) p. 773 J. L. Greenstein: Astrophys. J. 128, 106 (1958) K. F. Dymond, P. D. Feldman, T. N. Woods: Astrophys. J. 338, 1115 (1989) I. S. Bowen: Publ. Astron. Soc. Pacific 59, 196 (1947) P. D. Feldman, W. G. Fastie: Astrophys. J. Lett. 185, L101 (1973) V. A. Krasnopolsky, M. J. Mumma, M. Abbott, B. C. Flynn, K. J. Meech, D. K. Yeomans, P. D. Feldman, C. B. Cosmovici: Science 277, 1488 (1997) T. E. Cravens, A. E. S. Green: Icarus 33, 612 (1978) G. P. Tozzi, P. D. Feldman, M. C. Festou: Astron. Astrophys. 330, 753 (1998) R. J. Oliversen, N. Doane, F. Scherb, W. M. Harris, J. P. Morgenthaler: Astrophys. J. 581, 770 (2002) J.-L. Bertaux: Astron. Astrophys. 160, L7 (1986) M. J. Mumma, and 18 colleagues: Astrophys. J. 546, 1183 (2001) E. Gérard: Astron. Astrophys. 230, 489 (1990) H. Okabe: The Photochemistry of Small Molecules (Wiley, New York 1978) W. F. Huebner, J. J. Keady, S. P. Lyon: Solar Photo Rates for Planetary Atmospheres and Atmospheric Pollutants (Springer, New York 1992) S. A. Budzien, M. C. Festou, P. D. Feldman: Icarus 107, 164 (1994)

Part G 83

83.6

G. Herzberg: The Spectra and Structures of Simple Free Radicals: An Introduction to Molecular Spectroscopy (Dover, New York 1971) M. C. Festou, H. Rickman, R. M. West: Astron. Astrophys. Rev. 4, 363 (1993) M. C. Festou, H. Rickman, R. M. West: Astron. Astrophys. Rev. 5, 37 (1993) D. Krankowsky: Comets in the Post-Halley Era, ed. by R. L. Newburn, Jr. M. Neugebauer, J. Rahe (Kluwer, Dordrecht 1989) p. 855 L. Wilkening (Ed.) L. Comets (Univ. Arizona Press, Tucson 1982) M. F. A Hearn, M. C. Festou: Physics and Chemistry of Comets, ed. by W. F. Huebner (Springer, New York 1990) p. 69 D. A. Mendis, H. L. F. Houpis, M. L. Marconi: Fund. Cosmic Phys. 10, 1 (1985) C. Arpigny: Ann. Rev. Astron. Astrophys. 3, 351 (1965) G. W. Preston: Astrophys. J. 147, 718 (1967) C. D. Slaughter: Astron. J. 74, 929 (1969) P. D. Feldman, W. H. Brune: Astrophys. J. Lett. 209, L45 (1976) P. Eberhardt, D. Krankowsky, W. Schulte, U. Dolder, P. Lämmerzahl, J. J. Berthelier, J. Woweries, U. Stubbemann, R. R. Hodges, J. H. Hoffman, J. M. Illiano: Astron. Astrophys. 187, 484 (1987) M. A. DiSanti, M. J. Mumma, N. D. Russo, K. MageeSauer: Icarus 153, 361 (2001) M. J. Mumma, H. A. Weaver, H. P. Larson, D. S. Davis, M. Williams: Science 232, 1523 (1986) H. P. Larson, H. A. Weaver, M. J. Mumma, S. Drapatz: Astrophys. J. 338, 1106 (1989) D. Bockelée-Morvan, J. Crovisier, M. J. Mumma, H. A. Weaver: Comets II, ed. by M. C. Festou, H. A. Weaver, H. U. Keller (Univ. Arizona Press, Tucson 2004) R. Meier, T. C. Owen, H. E. Matthews, D. C. Jewitt, D. Bockelée-Morvan, N. Biver, J. Crovisier, D. Gautier: Science 279, 842 (1998) E. K. Jessberger, J. Kissel: Comets in the Post-Halley Era, ed. by R. L. Newburn, Jr. M. Neugebauer, J. Rahe (Kluwer, Dordrecht 1989) p. 1075 M. C. Festou, P. D. Feldman: Exploring the Universe with the IUE Satellite, ed. by Y. Kondo (Reidel, Dordrecht 1987) p. 101 M. C. Festou: International Ultraviolet Explorer – Uniform Low Dispersion Data Archive Access Guide: Comets, ESA SP-1134 (ESA, Noordwijk 1990) H. A. Weaver, P. D. Feldman, J. B. McPhate, M. F. A’Hearn, C. Arpigny, T. E. Smith: Astrophys. J. 422, 374 (1994) P. D. Feldman, H. A. Weaver, E. B. Burgh: Astrophys. J. Lett. 576, L91 (2002)

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83.50 83.51 83.52 83.53 83.54 83.55 83.56

A. H. Delsemme: Comets, ed. by L. L. Wilkening (Univ. Arizona Press, Tucson 1982) p. 85 L. Haser: Bull. Acad. R. Sci. Liège 43, 740 (1957) M. C. Festou: Astron. Astrophys. 95, 69 (1981) M. R. Combi, A. H. Delsemme: Astrophys. J. 237, 633 (1980) M. R. Combi: Icarus 123, 207 (1996) K. Richter, M. R. Combi, H. U. Keller, R. R. Meier: Astrophys. J. 531, 599 (2000) R. Meier, M. F. A’Hearn: Icarus 125, 164 (1997)

83.57

83.58 83.59 83.60

D. Bockelée-Morvan, D. C. Lis, J. E. Wink, D. Despois, J. Crovisier, R. Bachiller, D. J. Benford, N. Biver, P. Colom, J. K. Davies, E. Gérard, B. Germain, M. Houde, D. Mehringer, R. Moreno, G. Paubert, T. G. Phillips, H. Rauer: Astron. Astrophys. 353, 1101 (2000) M. F. A’Hearn, D. G. Schleicher, P. D. Feldman: Astrophys. J. Lett. 274, L99 (1983) M. C. Festou, H. A. Weaver, H. U. Keller (Eds.): Comets II (Univ. Arizona Press, Tucson 2004) A. L. Cochran, W. D. Cochran: Icarus 157, 297 (2002)

Part G 83

1259

Aeronomy

84. Aeronomy

84.1 Basic Structure of Atmospheres ............. 1259 84.1.1 Introduction ............................. 1259 84.1.2 Atmospheric Regions ................. 1260 84.2 Density Distributions of Neutral Species . 1264 84.2.1 The Continuity Equation ............. 1264 84.2.2 Diffusion Coefficients ................. 1265 84.3 Interaction of Solar Radiation with the Atmosphere ........................... 1265 84.3.1 Introduction ............................. 1265 84.3.2 The Interaction of Solar Photons with Atmospheric Gases ............. 1266 84.3.3 Interaction of Energetic Electrons with Atmospheric Gases ............. 1268 84.4 Ionospheres ........................................ 1271 84.4.1 Ionospheric Regions .................. 1271 84.4.2 Sources of Ionization ................. 1271 84.4.3 Nightside Ionospheres ............... 1277 84.4.4 Ionospheric Density Profiles........ 1277 84.4.5 Ion Diffusion ............................ 1279 84.5 Neutral, Ion and Electron Temperatures 1281 84.6 Luminosity .......................................... 1284 84.7 Planetary Escape ................................. 1287 References .................................................. 1290

84.1 Basic Structure of Atmospheres 84.1.1 Introduction In a stationary atmosphere, the force of gravity is balanced by the plasma pressure gradient force in the vertical direction, and the variation of pressure P(z) with altitude above the surface z is governed by the hydrostatic relation dP(z) = −ρ(z)g(z) , (84.1) dz where ρ(z) = n(z)m(z) is the mass density, n(z) is the number density, and m(z) is the average mass of the atmospheric constituents. In general, variables such as P, ρ, g, n and even m are functions of altitude, although it will often not be shown explicitly in the equations that follow for the sake of compactness. The acceleration of gravity g is usually taken to be the vector sum of the gravitational attraction per unit mass and the centrifugal

acceleration due to the rotation of the planet: g(r) = G M/r 2 − ω2r cos2 φ ,

(84.2)

where r = r0 + z is the distance from the center of the planet, r0 is the planetary radius, M is the planetary mass, G = 6.670 × 10−8 dyn cm2 g−2 is the gravitational constant, φ is the latitude, and ω is the angular velocity of the planet. When the hydrostatic relation (84.1) is combined with the ideal gas law in the form P = nkB T ,

(84.3)

where kB is Boltzmann’s constant and T is the temperature, and integrated, the the barometric formula   z 1 P(z) = P0 exp − dz
 , (84.4) H(z
) z0

Part G 84

We describe here the neutral and ionic structures of atmospheres, including the processes that determine the atmospheric layers, the distribution of the species, and the temperature profiles. We focus on the upper atmosphere, which comprises the thermosphere and the ionosphere, two regions which overlay and interact with each other. We describe the interaction of near and extreme ultraviolet solar photons and energetic electrons with the atmosphere and their role in ionization and dissociation of atmospheric species. We also review the production and loss processes that are important in the formation of the different layers of the dayside and nightside ionospheres, including ion and neutral diffusion. The processes that determine the neutral, ion and electron temperatures are discussed. We review the processes that are important in production of the luminosity of the upper atmospheres, including dayglow, nightglow and auroras. Finally, we describe atmospheric escape processes, including thermal and non-thermal mechanisms.

1260

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for the pressure P(z) above a reference level (denoted by the subscript 0) as a function of altitude results. The pressure scale height H(z) is defined as H(z) =

kB T . mg

(84.5)

In the lower and middle atmosphere, the mass m in (84.5) is the weighted average mass of the atmospheric constituents. When the ideal gas law (84.3) is substituted into the barometric formula (84.4), the altitude distribution   z T0 1
exp − dz  , (84.6) n(z) = n 0 T(z) H(z
) z0

Part G 84.1

for the number density n(z) above a reference altitude is obtained. Integration of (84.1) or (84.6) from z to infinity shows that the column density above that altitude is approximately N(z) = n(z)H(z). Thus the scale height can be thought of as the effective thickness of the atmosphere. In its lower and middle regions, the homosphere, the atmosphere is well-mixed by convection and/or turbulence. The upper boundary of this region is called

the homopause (or turbopause), and above this level, the major transport process is diffusion. The homopause is defined as the level at which the time constants for mixing and diffusion are equal, and usually occurs at n(z) ∼ 1011 –1013 cm−3 , depending on the strength of vertical mixing for a given planet. Since molecular diffusion coefficients vary from one species to another, the exact altitude of the homopause is species-dependent, with smaller species having lower homopause altitudes. In the terrestrial atmosphere, the homopause is near 100 km at n(z) ∼ 1013 cm−3 . Below the homopause, the mixing ratios (or fractions by number) of the constituent gases, apart from those minor or trace species whose density profiles are determined by photochemistry or physical loss processes, are fairly constant with altitude. Throughout the atmosphere, gravity exerts a force on each particle that is proportional to its mass. Below the homopause, however, the tendency of the species to separate out under the force of gravity is overpowered by large scale mixing processes, such as convection and turbulence. Above the homopause, each species is distributed according to its own scale height. Characteristics of the homopauses of the planets are presented in Table 84.1.

Table 84.1 Homopause characteristics of planets and satellites Planet

Altitude (km)

K (cm2 /s)

T (K)

nt (cm−3 )

P (µbar)

Venusb Earthc

135 100

4(7)a 1(6)

199 185

1.4(11) 1.3(13)

4.7(-3) 0.3

Marsd

120–125

5(7)

154

1.3(11)

2.8(-3)

Jupitere

500f

2(6)

600

1.4(13)

0.4

Saturng Uranush

1100f 300f

1.3(8) 1(4)

200 130

1.2(11) 1(15)

3(-3) 20

Neptunei Titanj

750f 1100

2(7) 1(9)

280 155

1.2(12) 3(9)

3.9(-2) 6.4(-5)

30

3(3)

40

3.3(14)

1.8

Tritonk

Read as 4 × 107 K from von Zahn et al. [84.1] and model atmosphere from Hedin et al. [84.3], for 1500 h, 15◦ N latitude, F10.7 = 150 c From The US Standard Atmosphere [84.6] d Viking model from [84.8] e From [84.13] and photochemical model of Kim [84.14].

Composition (Fraction by number) CO2 (76 ), N2 (7.6 ), O(9.3 ), CO(6.7 ), N(0.16 ), C(0.01 ) N2 (77%), O2 (18%), O(3 –4%), Ar(0.7%), He(9.5 ppm), H(1.3 ppm) CO2 (95%), N2 (2.5%), Ar(1.5%), O(0.9%), CO(0.42%) O2 (0.12%), NO(0.007%) H2 (95%), He(4.1%), H(0.055%), CH4 (200 ppb), C2 H2 (1.2 ppb),C2 H4 (2.5 ppb), C2 H6 (0.12 ppb) H2 (94%), He(6%), CH4 (178 ppm) H2 (85%), He(15%), CH4 (20 ppm), C2 H2 (10 ppb), C2 H6 (0.1 ppb), C4 H2 (0.05 ppt) H2 (83%), He(16%), CH4 (3 ppm) N2 (96%), CH4 (2.4%), C2 H2 (0.07%), C2 H4 (0.7%), C2 H6 (404 ppm), C4 H2 (105 ppm) N2 (99.9%), H2 (190 ppm), CH4 (37 ppm), H(3.1 ppm), N(4.1 ppb)

a

f

b

g

Altitude above the 1 bar level. From [84.2] h From [84.4, 5] i From [84.7] j From [84.9–12] k From [84.15, 16]

Aeronomy

84.1.2 Atmospheric Regions 600

Table 84.2 Molecular weights and fractional composition of dry air in the terrestrial atmospherea Species

Molecular Weight (g/mole)

Fraction by volume

N2 O2 Ar CO2 Ne He Kr Xe CH4 H2

28.0134 31.9988 39.948 44.009 95 20.183 4.0026 83.80 131.30 16.043 03 2.015 94

0.780 84 0.209 476 0.009 34 0.000 3756b 0.000 018 18 0.000 005 24 0.000 001 14 0.000 000 087 0.000 002 0.000 0005

a

Taken from The US Standard Atmosphere [84.6], except as noted b 2003 annual average value. The CO mixing ratio is increas2 ing at an annual rate of about 0.45%. Value is from [84.18]

1261

Altitude (km)

F10.7 = 75

250

140

200

500

400

300

200

100

Part G 84.1

The division of atmospheres into regions is based on the temperature structure of the terrestrial atmosphere, which is shown in Fig. 84.1. In the troposphere of a planet, above the boundary layer, T decreases at close to the adiabatic lapse rate (Γ ) for the constituent gases from the surface to the tropopause. For an atmosphere that is a mixture of ideal gases, Γ = g/cp , where cp is the specific heat of the gas mixture at constant pressure. The presence of a condensible constituent, such as water vapor in the terrestrial troposphere, and ammonia or methane in the atmospheres of the outer planets and satellites, decreases Γ because upward motion leads to cooling and condensation, which releases latent heat. On Earth, the dry adiabatic lapse rate is about 10 K/km and the moist adiabatic lapse rate is about 4–6 K/km in the lower to middle troposphere. The average lapse rate is about 6.5 K/km, and the altitude of the tropopause varies from about 9 to 16 km from the poles to the equator. The composition of the lower atmosphere of the Earth is given in Table 84.2. Above the terrestrial tropopause lies the stratosphere, a region of increasing T that is terminated at the stratopause, near 50 km. This increase in T is caused by absorption of solar near UV radiation by ozone in the Hartley bands and continuum (200–310 nm). In the terrestrial mesosphere, which lies above the stratosphere, T decreases again to an absolute minimum at the mesopause, where t ≈ 180 K and n(z) ≈ 1014 cm−3 . Above the mesopause, in the thermosphere, T increases

84.1 Basic Structure of Atmospheres

Mesopause Stratopause Tropopause

0

0

500

1000

1500 Temperature (K)

Fig. 84.1 Vertical distribution of temperature in the ter-

restrial atmosphere. The altitudes of the tropopause, stratopause and mesopause are indicated. The thermospheric temperatures depend on solar activity and profiles are shown for four values of the F10.7 index, from 75 (low solar activity) to 250 (high solar activity). The solid and dashed curves are for noon and midnight, respectively. After the MSIS model of Hedin [84.17]

rapidly to a constant value, the exospheric temperature, T∞ . The value of T∞ in the terrestrial atmosphere depends on solar activity and is usually between about 700 and 1500 K. Fig. 84.1 also shows altitude profiles of the noon and midnight thermospheric temperature for four values of the F10.7 index, (the 2800 MHz flux in units of 10−22 Wm−2 Hz−1 at 1 AU), which represent different levels of solar activity. The exosphere is a nearly collisionless region of the thermosphere that is bounded from below by the exobase. A particle traveling upward at or above the exobase will, with high probability, escape from the gravitational field of the planet. The exobase on Earth is located at about 450–500 km, depending upon solar activity. The surface P and T on Mars are about 6 mbar

1262

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Part G 84.1

and 230 K, respectively. Due to the effect of dust storms, the extent of the Martian troposphere is highly variable, with a lapse rate that is 2–3 K/km compared with the adiabatic lapse rate of 4.5 K/km and a variable thickness of 20–50 km [84.19]. The atmosphere of Mars, like many planetary atmospheres, does not have a stratosphere. A roughly isothermal mesosphere extends from the tropopause to the base of the thermosphere at about 90 km. The thermospheric T is sensitive to solar activity and, since Mars has a very eccentric orbit, to heliocentric distance; T∞ varies from about 180 to 350 K. Near the surface of Venus, T
700 K and P ∼ 95 bar. T decreases with a mean lapse rate of 7.7 K/km, compared with the adiabatic lapse rate of 8.9 K/km, from the surface to about 50 km. The region from 50 to 60 km contains the major cloud layer, and the tropopause is usually considered to be at about 60 km. In the mesosphere, between about 60 and 85 km, T decreases slowly from about 250 K to 180 K, and is nearly constant from 85 km to the mesopause at 100 km. The daytime exospheric temperature is only weakly dependent on solar activity, varying from about 230 to 300 K from low to high solar activity. The slow retrograde rotation of the planet, which results in a period of darkness that lasts 58 days, leads to the relative isolation of the nightside thermosphere, where T is found to decrease above the mesopause to an exospheric T ≈ 100 K. Because of this, the nightside Venus thermosphere has been called the “cryosphere”. The compositions of the lower atmospheres of Mars and Venus are given in Table 84.3. The giant planets, Jupiter, Saturn, Uranus and Neptune do not have solid surfaces, so their atmospheric regions are defined either in terms of pressure, altitude above the 1 bar level, or altitude above the cloud tops. The temperature structures of all but Uranus are influenced by internal heat sources that the terrestrial planets do not possess. The temperature structures near the tropopause can be determined from IR observations and radio occultation data, and at thermospheric altitudes from ultraviolet solar and stellar occultations performed by the Voyager spacecraft. In between these regions, there is a substantial gap in which only average temperatures can be inferred. Thus the location and temperature of the mesopauses are largely unknown. Below 300 mbar on Jupiter, the lapse rate is close to adiabatic (1.9 K/km). T at 1 bar is about 165 K, and the tropopause occurs near 140 mbar, where T ≈ 110 K. At 1 mbar, T again reaches 160–170 K. Temperature inversions have been reported in the stratosphere, and are probably due to absorption of solar radiation by dust or aerosols. Temperatures derived from the Voyager UV

stellar and solar occultations show that T increases from about 200 K near 1 µbar to an exospheric value of about 1100 K [84.2]. For P > 500 mbar on Saturn, the lapse rate approaches the adiabatic value of 0.9 K/km, and the tropopause, near the 100 mbar level, is characterized by T ≈ 80 K. Above the tropopause the temperature increases to about 140 K near a P ≈ 1 mbar, and above that altitude there are no measurements of T up to a pressure of about 10−8 bar, about 1000 km above the 1 bar level, where T is again about 140 K. Application of the hydrostatic equation to the altitude range 300–1000 km yields an average temperature near 140 K for the region. Above 1000 km, T increases to a T∞ ≈ 800 K [84.2]. The mixing ratios of the species in the lower atmospheres of Jupiter and Saturn are given in Table 84.4. The tropopauses on both Uranus and Neptune occur near 100 mbar, where T ≈ 50 K. The lapse rates in the troposphere are 0.7 and 0.85 K/km for Uranus and Neptune, respectively. The temperatures in the Uranus thermosphere range from 500 K near 10−7 bar (about 1000 km above the 1 bar level) to an exospheric value of about 800 K. At 300 km on Neptune, T attains a nearly constant value in the range 150 to 180 K. At 600 km where P ≈ 1 µbar, T increases again to a value that is Table 84.3 Composition of the lower atmospheres of Mars

and Venusa Species

Mixing Ratio Mars

Venus

CO2 N2 40 Ar O2 CO H2 O He Ne Kr Xe SO2 H2 S H2

0.953 0.027 0.016 0.0013 0.0008 0.0003c 4 ppme 2.5 ppm 0.3 ppm 0.08 ppm – – 15 ppmf

0.96 0.04 50 –120 ppmb 20 –40 ppm 20 –30 ppm 30 ppmd 10 ppm 5 –13 ppm 0.02– 0.4 ppm – 150 ppm 1 –3 ppm 0.1 ppmg

a

From [84.20], except as noted. Includes all isotopes of Ar. c Variable d From [84.21]. e From Krasnopolsky and Gladstone. [84.22] f Krasnopolsky and Feldman [84.23] g Yung and DeMore [84.24] b

Aeronomy

Table 84.4 Composition of the lower atmospheres of

Jupiter and Saturn Species H2 He CH4 NH3 H2 O C2 H6 PH3 C2 H2 20 Ne 36 Ar 84 Kr 132 Xe a b c

Mixing Ratio Jupiter 0.864b 0.136b 0.001 81b < 0.002b 520 ppmb 5 ppma 0.6 ppma 0.02 ppma ≤ 26 ppmb ≤ 9 ppmb ≤ 3.2 ppbb ≤ 0.38 ppbb

Saturn 0.94a 0.06a 0.0045a (0.5–0.2 ppmc ) 7.0 ppmc 1.4 ppmc 0.3 ppmc

probably about 600 K. Because the Voyager data have not been fully analyzed, the value of T∞ is uncertain [84.12]. The compositions of the lower atmospheres of Uranus and Neptune are given in Table 84.5. Titan, which is a satellite of Saturn, has an N2 /CH4 atmosphere of intermediate oxidation state. The mixing ratios of components of the lower atmosphere are given in Table 84.6. The surface P and T are 1.496 bar and 94 K, respectively. T decreases above the surface to about 71 K at the tropopause, which occurs at an altitude of 42 km and a pressure of 128 mbar. A reTable 84.5 Composition of the lower atmospheres of

Uranus and Neptunea Species H2 He CH4 HD CH3 D C2 H6 C2 H2 CO NH3 H2 O a b

Table 84.6 Composition of the lower atmosphere of Titana Species

Mixing Ratio

N2 CH4 H2 CO C2 H6 C3 H8 C2 H2 C2 H4 HCN

0.90–0.98 0.01–0.03b 2.0 × 10−3 60–150 ppm 20 ppm 4 ppm 2 ppm 0.4 ppm 0.2 ppm

Titan’s atmosphere may also contain up to 14% Ar [84.10] a From [84.29], except as noted b From [84.9]

analysis of the Voyager 1 solar occultation experiment showed that, above the tropopause, the temperature increases to a peak value of about 176 K at an altitude of about 300 km. The temperature then decreases to a T∞ of 153–158 K [84.9] Triton is a satellite of Neptune. It also has an N2 atmosphere with small amounts of methane, CO, H2 , and other species. The mixing ratios at 10 km are given in Table 84.7. The surface P is about 14–19 µbar. Methane in the troposphere is in equilibrium with a surface methane frost at about 38–50 K. The tropopause temperature is about 36 K, and occurs in the 8 to 12 km region. The middle atmosphere is isothermal with a temperture of about 52 K from 25 to 50 km, increasing to 78 K near 150 km [84.30]. T rises to an T∞ of about 100 K. Io and Europa are satellites of Jupiter. Both have transient atmospheres, with mean lifetimes of 2–3 days. The radius of Io is about 1821 km, and its atmosphere is Table 84.7 Composition of the atmosphere of Tritona

Mixing Ratio Uranus

Neptune

≈ 0.825 ≈ 0.152 ≈ 0.023 ≈ 148 ppm ≈ 8.3 ppm ≈ 1–20 ppb ≈ 10 ppb < 40 ppb < 100 ppb 5–12 ppb

≈ 0.80 ≈ 0.19 ≈ 0.01–0.02 ≈ 192 ppm ≈ 12 ppm ≈ 1.5 ppm ≈ 60 ppb 2.7 ± 1.8 ppmb < 600 ppb 1.5–3.5 ppb

After Lodders and Fegley [84.27], except as noted Courtin et al. [84.28]

1263

Species

Mixing Ratio

Comments

N2 CO CH4 H2 N N H H C2 H4 C2 H4

0.99 ± 0.01 0.0001– 0.01 113 ppm 75 ppm 3.8 × 10−5 ppm 290 ppm 0.092 ppm 1 ppm 3.9 × 10−4 ppm 2.9 × 10−2 ppm

Below ≈ 200 km Uncertain

a

100 km (near peak) 30 km (near peak) 26 km (near peak)

From [84.15]. Values are at 10 km, except as noted

Part G 84.1

After Strobel [84.25] After Niemann et al. [84.26] After Lodders and Fegley [84.27]

84.1 Basic Structure of Atmospheres

1264

Part G

Applications

Part G 84.2

mostly SO2 , which is produced by volcanic plumes. One model predicts that the average column density of SO2 is about 1016 cm−3 , and is larger at the equator than at the poles [84.31]. The atmospheric temperatures range from 100 to 2000 K, and the exospheric temperature is about 1800 K. The altitude of the exobase is about 1400 km. A plot of the number density and temperature as a function of altitude is shown in Figure 80–9 k. Europa is characterized by a radius of 1596 km. The atmosphere is mostly O2 with column density of 5 × 1014 cm−2 and a scale height of 145 km. The O2 is produced by sputtering of the ice-covered surface, and is removed in sputtering by torus thermal ions [84.32]. The ionosphere is produced by impact of electrons in Jupiter’s magnetosphere, and the maximum density of electrons is about 4 × 104 cm−3 [84.33]. Mercury does not have a troposphere, mesosphere, or stratosphere; The pressure at the surface is on the order of a picobar; thus the surface of the planet is the exobase. Nevertheless, several atomic species have been identified in fluorescence. They are listed in Table 84.8. Among the possible sources of atmospheric species are evaporation, ion sputtering, meteoroid bombardment, and photon-stimulated desorption. Ions produced by photoionization of neutrals may be picked up by the solar wind and lost from the atmosphere. Pluto and its satellite Charon form what is sometimes referred to as a double planet system. The radius of

Table 84.8 Number densities of species at the surface of

Mercurya Species

Number density (cm−3 )

H hot H He O Nab Kb

230 23 6 × 103 < 4.4 × 104 (1.7–3.8) × 104 5 × 102

a b

from Hunten et al. [84.34] Variable spatially and temporally

Pluto is 1150–1200 km, and that of Charon is about 600 km. The atmosphere of Pluto is mostly N2 , with small amounts of methane, CO, H2 , and H. Only upper limits are available for the mixing ratio of CO. The surface pressure and temperature are in the ranges 1 to 10 µbar and 35 to 57 K, respectively. Although the pressure at the surface is approximately the same as that of the base of the thermosphere on most planets, the thermal structure of the atmosphere is influenced by the large thermal escape flux at the top of the atmosphere and by adiabatic cooling. T maximizes near 1200–1260 km radius at about 100 K due to absorption of solar UV radiation. Above that radius, T decreases asymptotically to a value of about 80 K [84.35, 36].

84.2 Density Distributions of Neutral Species and the eddy diffusion velocity wKj :

84.2.1 The Continuity Equation The density distribution of a minor neutral species j in an atmosphere is determined by the continuity equation: ∂n j + ∇ · Φ j = Pj − L j , ∂t

(84.7)

where Φ j is the flux of species j, and P j and L j are the chemical production and loss rates, respectively. If only the vertical direction is considered, the divergence of the flux becomes ∂Φ j /∂z, and Φ j = n j w j , where w j is the vertical velocity of the species and n j is its number density. In one-dimensional models, transport due to turbulence and other macroscopic motions of air masses is often parametrized like molecular diffusion, using an eddy diffusion coefficient K in place of the molecular diffusion coefficient D j . The total transport velocity w j is then the sum of the diffusion velocity wDj

w j = wDj + wKj .

(84.8)

If there are no net flows of major constituents, wDj and wKj satisfy the equations   (1 + αTj ) dT 1 1 dn j D wj = − Dj + + , n j dz Hj T dz  wKj = − K

1 1 dT 1 dn j + + n j dz Havg T dz

(84.9)

.

(84.10)

In these expressions, αTj is the thermal diffusion factor (the ratio of the thermal diffusion coefficient to the molecular diffusion coefficient), and the pressure scale height Havg for a mixed atmosphere is given by (84.5) with m = m avg , the average molecular mass.

Aeronomy

For a stationary atmosphere, if molecular diffusion greatly exceeds eddy diffusion and if photochemistry can be neglected, then wDj = 0. The resulting number density distribution is called diffusive equilibrium, and is given by  

  1+αT z
j dz T0 . n j (z) = n j (z 0 ) exp − T Hi z0

(84.11)

When mixing processes dominate and wKj = 0, the distribution is given by (84.6), with H = Havg .

(84.12)

k= j

where f k is the mixing ratio of species k. The binary diffusion coefficient can be expressed as 3kB T , 16n t µ jk Ω jk

where µ jk is the reduced mass µ jk =

m j mk m j + mk

(84.14)

and n t = n j + n k is the total number density. The collision integral Ω jk is given by  5/2 µ 1 Ω jk = 2π 1/2 2kB T ∞

 × Q D (v)v5 exp −µv2 /2kB T dv 0

(84.13)

where v is the relative velocity of the particles, Q D (v) is the diffusion or momentum transfer cross section π Q (v) = 2π

σ el jk (θ, v)(1 − cos θ) sin θ dθ ,

D

0

(84.16)

and σ el jk (θ, v) is the differential cross section for elastic scattering of species j and k through angle θ. In practice, D jk is often expressed as b jk /n t where n t is the total number density and b jk is the binary collision parameter, which is usually given in tabulations in the semi-empirical form b = AT s . Here A and s (0.5 ≤ s ≤ 1.0) are parameters that are fit to the data. The binary collision parameter appears, for example, in the expression for the diffusion limited flux of a light species to the exobase of a planet (Sect. 84.7).

84.3 Interaction of Solar Radiation with the Atmosphere 84.3.1 Introduction The source for all atmospheric processes is ultimately the interaction of solar radiation, either photons or particles, with atmospheric gases. Since visible photons arise from the photosphere of the sun, which is characterized by T ≈ 6000 K, the solar spectrum in the visible and IR is similar to that of a black body at 6000 K. At longer (radio) and shorter (UV and X-ray) wavelengths, the photons arise from parts of the chromosphere and corona where the temperatures are higher (104 to 106 K). Thus the photon fluxes differ substantially from those which would be predicted for a 6000 K black body. Photons in the extreme and far UV regions of the spectrum are absorbed in the terrestrial thermosphere and X-rays in

the lower thermosphere and mesosphere. The solar Lyman α line at 1216 Å penetrates through a window in the O2 absorption cross sections to about 75 km. Near UV photons are absorbed by ozone in the stratosphere, and visible radiation is not appreciably attenuated by the atmosphere. The wavelength ranges that are most important for aeronomy are the UV and X-ray regions. A solar spectrum in the UV and soft X-ray regions at low solar activity is presented in Fig. 84.2a, and the ratio of a high solar activity photon fluxes to those at low solar activity is shown in Fig. 84.2b. The ratio is near unity at wavelengths longward of 2000 Å, but increases to factors that range between 2 and 3 over much of the extreme UV. At wavelengths between about 100 and

Part G 84.3

In the thermosphere of a planet, above the homopause, the major transport mechanism is diffusion, or transport by random molecular motions. The characteristic time τD for molecular diffusion is approximately H 2j /D j . The diffusion coefficient for a species j in a multicomponent mixture is usually taken as a weighted mean of binary diffusion coefficients D jk

D jk =

1265

(84.15)

84.2.2 Diffusion Coefficients

fk 1 = , Dj D jk

84.3 Interaction of Solar Radiation with the Atmosphere

1266

Part G

Applications

550 Å, the ratio of high to low solar activity fluxes reaches values as high as 100. The fluxes at X-ray wavelengths arise principally from solar flares and can increase by orders of magnitude from low to high solar activity. The sun also emits a stream of charged particles, the solar wind, which flows radially outward in all directions, and consists mostly of protons, electrons, and alpha particles. The average number density of solar wind protons is about 5 cm−3 , and the average speed is about 400–450 km/s at Earth orbit (1 AU). The interaction of these particles with the magnetic field (either induced or intrinsic) of a planet, and ultimately with the atmosphere, is the source of auroral activity. Terrestrial auroras arise mostly from precipitation of electrons with energies in the kilovolt range, although measured spectra vary widely. An example of a pria) log photon flux (106 cm –2Å–1)

Part G 84.3

6

4

2

mary electron auroral spectrum is shown in Fig. 84.3. Terrestrial auroral emissions maximize in the midnight sector, but dayside cusp auroras are produced by lower energy electrons, and diffuse proton auroras are also observed. Since charged particles are constrained to move along magnetic field lines, for planets with intrinsic magnetic fields, auroras usually occur in an oval near the magnetic poles, where the dipole field lines enter the atmosphere. For Venus, which has no intrinsic magnetic field, auroras are seen as diffuse and variable emissions on the nightside of the planet. On Earth, low latitude auroras, which arise from heavy particle precipitation, have also been observed. The primary particles that are responsible for Jovian aurora may be heavy ions originating from its satellite Io, protons, or electrons. Due to charge transfer, heavy particles spend part of their lifetime as neutral species, and their paths may then diverge from magnetic field lines. In any case, a large fraction of the effects of auroral precipitation is due to secondary electrons, regardless of the identity of the primary particles. In addition to producing emissions of atmospheric species in the visible, UV and IR portions of the spectrum, auroral particles ionize and dissociate atmospheric species and contribute to heating the neutrals, ions and electrons.

0

Flux (eV –1cm–2 s–1)

Downward flux

107 –2

0

500

1000

1500

2000

b) log Fhi/Flow

106

2 105

1.5

1

104

0.5 103 1 10 0

0

500

1000

1500 2000 Wavelength (Å)

Fig. 84.2 (a) Solar spectrum at 1 AU for 18–2000 Å. (b) Ratio of the flux at high solar activity to low solar

activity. Plotted with data from Tobiska [84.37]

102

103

104

105 Energy (eV)

Fig. 84.3 Downward electron flux as a function of energy measured by electron spectrometers on board a rocket traversing an auroral arc near Poker Flat, Alaska. After [84.38] with kind permission from Elsevier Science Ltd., UK

Aeronomy

84.3.2 The Interaction of Solar Photons with Atmospheric Gases The number flux of solar photons in a small wavelength interval around λ at an altitude z can, for the most part, be computed from the Beer–Lambert absorption law Fλ (z) = Fλ∞ exp[−τ(λ, z)] ,

(84.17)

Fλ∞

where is the solar photon flux outside the atmosphere, and τ(λ, z) is the optical depth which, in the plane parallel approximation, is given by 

∞

τ(λ, z) =

j

n j (z
)σ aj (λ) sec χ dz
.

(84.18)

z



∞

τ(λ, z) =

j

z

n j (z
)σ aj (λ) 

ro + z × 1− ro + z


−0.5

2

dz
.

sin χ 2

In a one-species atmosphere, the rate of absorption of solar photons of wavelength λ is q a (λ) = Fλ σ a (λ)n .

(84.21)

For an isothermal atmosphere in which H(z) ≈ const., the absorption maximizes where τ(λ, z) = 1. This is a fairly good approximation even for regions of the atmosphere where the H(z) is not constant. The altitude of unit optical depth is shown for wavelengths from X-rays to the near UV for overhead sun in the terrestrial atmosphere in Fig. 84.4a. Similar plots for Venus, Mars and Jupiter are shown in Fig. 84.4b, Fig. 84.4c, and Fig. 84.4d, respectively. N2 does not absorb longward of about 100 nm, so in the terrestrial atmosphere, O2 and O3 are the primary absorbers between 100 and 220 nm, while ozone dominates the absorption for wavelengths in the range 220–320 nm. On Venus and Mars, CO2 is the main absorber of FUV and EUV radiation, although at wavelengths less than about 100 nm, N2 , CO, and O also contribute. On Titan, methane is the primary absorber of UV radiation between 1400 Å and the absorption threshold of N2 , near 1000 Å. The interaction of UV photons with atmospheric gases produces ions and photoelectrons through photoionization, which may be represented as X + hν → X + + e− ,

(84.22)

and photodissociative ionization (84.19)

For χ larger than 90◦ , the optical depth is given by   τ(λ, z) = 2 n j (z
)σ aj (λ) ∞

j

zs



ro + z s × 1− ro + z
∞ −

2

◦

sin 90

dz


z

× 1−



ro + z ro + z


(84.23)

E pe = hν − I X − E ex

(84.24)

and in reaction (84.23) is

n j (z
)σ aj (λ)

AB + hν → A+ + B + e− .

In these equations, X represents any atmospheric species; A is either an atom or a molecular fragment and AB a molecule. The energy of the photoelectron in reaction (84.22) is given by

−0.5

2

−0.5

2 sin2 χ



dz
, (84.20)

where z s is the tangent altitude, the point at which the solar zenith angle is 90◦ for the path of solar radiation through the atmosphere.

1267

E pe = hν − E d − I A − E ex ,

(84.25)

where I j is the ionization potential of species j, E d is the dissociation energy of molecule AB, and E ex is the internal excitation energy of the products. Neutral fragments, which may be reactive radicals, are also produced in photodissociation AB + hν → A + B .

(84.26)

Part G 84.3

Here, σ aj (λ) is the absorption cross section of species j at wavelength λ, and the solar zenith angle χ is the angle of the sun with respect to the local vertical. For χ greater than about 75◦ , the variation of the solar zenith angle along the path of the radiation cannot be neglected; the optical depth must be computed by numerical integration along this path in spherical geometry. For χ ≤ 90◦ the optical depth is

84.3 Interaction of Solar Radiation with the Atmosphere

1268

Part G

Applications

a) Altitude (km)

b)

200 N2O 150

Altitude (km)

160

AIR

O2

O2

+ N+ O 2

150

NO+

140 O3

100 N+2

130

+

O

120 Lyman α

50

110 0

0

50

100

150

c) Altitude (km)

200

250 300 Wavelength (nm)

100

d) 1000

0

500

1000

1500 2000 Wavelength (Å)

1000

1500 2000 Wavelength (Å)

Altitude (km)

Mars Lo

160

800

Part G 84.3

140 600 120

400

100

80

200

0

500

1000

1500 2000 Wavelength (Å)

0

0

500

Fig. 84.4a–d The altitude where τ = 1 versus wavelength (a) Earth [84.39] (b) Venus (c) Mars [84.40] (d) Jupiter (Y. H.

Kim, unpublished)

The rate of ionization of a species j by a photon of wavelength λ at an altitude z is given by q ij (λ, z) = Fλ (z)σ ij (λ)n j (z) ,

(84.27)

where σ ij (λ) is the photoionization cross section. The rate for photodissociation is given by a similar expression, with the photoionization cross section replaced by the photodissociation cross section. The expression above must be integrated over the solar spectrum to give the total rate. In addition, it is often necessary to take into account ionization and/or dissociation to different final internal states of the products, so the partial cross sections or yields are needed. In the atmospheres of magnetic planets, photoelectrons may travel upward along the magnetic field lines to the conjugate point, where the field line re-enters the atmosphere. In order to model this effect, the differential (with respect to angle) cross sections for photoionization

σ ij (λ, θ) are necessary. The differential cross section is sometimes expressed as σ ij (λ, θ) =

σ ij (λ) 4π

1 [1 − β(λ)P2 (cos θ)] , 2

(84.28)

where θ is the angle between the incident photon beam and the ejected electron, P2 is a Legendre polynomial, and β is an asymmetry parameter.

84.3.3 Interaction of Energetic Electrons with Atmospheric Gases Suprathermal electrons, which are denoted here e∗− , and include both photoelectrons and auroral primary electrons, can also ionize species through electron-impact ionization X + e∗− → X + + e− + e


−

(84.29)

Aeronomy

and electron-impact dissociative ionization −

AB + e∗− → A+ + B + e− + e
.

(84.30)

In these reactions, e− represents the energy degraded photoelectron or primary electron, and e
− the secondary electron. The energy of the secondary electron E e
in an electron-impact ionization process (84.29) is given by E e
= E e* − I X − E ex − E e ,

(84.31)

where E e* is the energy of the primary or photoelectron, E e is the energy of the degraded primary or photoelectron, and E ex is the internal excitation energy of the product ions and/or neutral fragments. For the dissociative ionization process (84.30) the dissociation energy of the molecule must also be subtracted as well. Energetic electrons can also dissociate atmospheric species. In this process ∗−

→ A+ B+e

E e = E e* − D AB − E ex ,

(84.33)

where D AB is the dissociation energy of molecule AB. Collisions with suprathermal electrons can also promote species to excited electronic, vibrational or rotational states: AB + e∗− → AB † + e− ,

In general, the excitation rate q kj (z) of a species j to an excited level k with a threshold energy E k at an altitude z by electron impact is given by: ∞ q kj (z) = n j (z)

σ kj (E)

dF(z, E ) dE , dE

(84.34)

where the dagger denotes internal excitation. The energy lost by the electron is thus the excitation energy of the species. In determining the rate of ionization, dissociation and excitation by photoelectrons, the local energy loss approximation, that is, the assumption that the electrons lose their energy at the same altitude where they are produced, is fairly good near the altitude of peak photoelectron production. The mean free path of an electron near 150 km is about 30 m. Substantially above the altitude of peak production of photoelectrons, transport of electrons from below is important, and use of the local energy loss assumption causes the excitation, ionization, and dissociation rates to be underestimated. For keV auroral electrons, the computation of the energy deposition of the electrons must consider their transport through the atmosphere. Thus the elastic total and differential cross sections for electrons colliding with neutral species must be employed, as well as the inelastic cross sections, and the angles through which the electrons scatter must be taken into account.

(84.35)

Ek

σ kj (E )

where is the excitation cross section at electron energy E, and dF(z, E )/ dE is the differential flux of electrons (between energies E and E + dE ). The ionization rate q ij (z) of a species with ionization potential I j due to electron impact is given by q ij (z) = n j (z)

∞ (E−I  j )/2 dσ i (E ) j

dWs

0

Ij

dF(z, E ) dWs dE , dE (84.36)

(84.32)

the energy of the degraded electron is

1269

where dσ ij (E )/ dWs is the differential cross section for production of a secondary electron with energy Ws by a primary electron with energy E. The integral over secondary energies Ws terminates at (E − I j )/2 because the secondary electron is by convention considered to be the one with the smaller energy. Since the average energy of photoelectrons is less than 20 eV, the error incurred in cutting off the integrals in equations (84.35) and (84.36) at 200 eV or so, rather than (E − I j )/2 is not serious, although for high energy auroral electrons a larger upper limit may be required. An estimate of the number of ionizations in a gas produced by a primary electron with energy E p is E p /Wip , where Wip is the energy loss per ion pair produced, which approaches a constant value as the energy of the electron increases. Empirical values are available for Wip for many gases, and usually fall in the range 30–40 eV [84.41]. The total loss function or stopping cross section for an electron with incident energy E in a gas j is given by the expression σ kj (E )W kj L j (E ) = k (E−I  j )/2

+

(I j + Ws ) 0

dσ i j (E ) dWs , dWs (84.37)

where W kj is the energy loss associated with excitation of species j to excited state k. The differential cross

Part G 84.3

AB + e

−

84.3 Interaction of Solar Radiation with the Atmosphere

1270

Part G

Applications

section is usually adopted from an empirical formula that is normalized so that (E−I  j )/2

σ ij (E ) =

dσ ij (E ) dWs

0

dWs ,

(84.38)

where σ ij (E ) is the total ionization cross section at primary electron energy E. One formula in common use is that employed by Opal et al. [84.42] to fit to their data: dσ ij (E ) dWs

=



A(E )

1 + Ws /W

2.1 ,

(84.39)

Part G 84.3

where A(E ) is a normalization factor and W is an empirically determined constant, which has been found to be equal to within a factor of about 50% to the ionization potential for a number of species. For energy loss due to elastic scattering by thermal electrons, an analytic form of the loss function such as that proposed by Swartz et al. [84.43] may be used:  2.36 E − kB Te 3.37 × 10−12 L e (E ) = , E − 0.53kB Te E 0.94 n 0.03 e (84.40)

where Te is the electron temperature and n e is the number density of ambient thermal electrons. For high energy auroral electrons, the rate of energy loss per electron per unit distance over the path s of the electrons in the atmosphere can be estimated using the continuous slowing down approximation (CSDA) as dE − = n j (z)L j (E ) sec θ + n e (z)L e (E ) sec θ , ds j

(84.41)

where θ is the angle between the path of the primary electron s and the local vertical. In the CSDA, all the electrons of a given energy are assumed to lose their energy continuously and at the same rate. The rate of energy loss (− dE/ ds) is integrated numerically over the path of the electron, which degrades in energy until it is thermalized. In this approximation, inelastic processes are assumed always to scatter the electrons forward, so cross sections that are differential in angle are not required. Because electrons actually lose energy at different rates, however, and because elastic and inelastic scattering processes do change the direction of the electrons, the CSDA gives an estimate for the rates of electron energy loss processes that is increasingly inaccurate as the energy of the electron decreases. In practice, discrete energy loss of electrons can be easily treated numerically if the local energy loss

approximation is valid. The spectrum of electrons is divided into energy bins that are smaller than the energy losses for the processes, and the integrals in (84.35, 36) are replaced by sums over energy bins. Since elastic scattering of electrons by neutrals changes mostly the direction of the incident electron, and not its energy, only inelastic processes need be considered. In order to compute excitation and dissociation rates, only integral cross sections are required; the scattering angle is unimportant. For ionization, of course, the energy distribution of the secondary electrons must be considered, but not the scattering angles of either the primary or secondary electrons. Below the lowest thresholds for excitations, energetic electrons lose their energy in elastic collisions with thermal electrons. The process of energy loss to thermal electrons is often approximated as continuous, rather than discrete. The collision frequency νkj for a discrete electronimpact excitation process k of a species j is given by νkj (E ) = n j (z)ve (E )σ kj (E ) .

(84.42)

For energy loss due to elastic scattering from thermal electrons, a pseudo-collision frequency νe may be defined as  dE 1 − , (84.43) νe (E ) = ∆E dt where ∆E is the grid spacing in the calculation, and the energy loss rate is dE = ve (E )n e L e (E ) , (84.44) dt where L e is taken from (84.40). Since the energy bins should be smaller than the typical energy loss in order to obtain accurate rates for the excitation processes, it is often convenient to treat rotational excitation also as a continuous process, with a pseudo-collision frequency similar to that for elastic scattering from ambient electrons (84.43) with −

dE = ve (E )n j L rot (84.45) j (E ) , dt where the loss function for rotational excitation is given by J,J

L rot η Jj σ j (E )W jJ,J . (84.46) j (E ) = −

J

J
J ηj

In this expression, is the fraction of molecules jmeasured or computed cross section for electronimpact excitation of species j
from rotational state J to rotational state J
, and W jJ,J is the associated energy loss.

Aeronomy

The slowing down of high energy auroral primary electrons or photoelectrons arises from both elastic and inelastic scattering processes, and cannot be treated using the local energy loss approximation. In solving the equations for electron transport, theangle through which the primary electron is scattered, as well as the change in energy of the primary electron and the production of any secondaries, must be taken into account. Thus differential cross sections for the elastic and inelastic scattering of electrons by neutral species are required. The detailed equations for electron transport have been presented by, for example, Rees [84.44]. Several methods for approximating the energy deposition of auroral electrons are currently in use. The

84.4 Ionospheres

1271

CSDA has already been discussed, but it provides only a rough approximation to the depth of penetration of the electrons, and the rates of excitation, ion production, and other energy loss processes. In the two-stream approximation, the electrons are assumed to be scattered in either the forward or backward direction [84.45]. Implementation of this method requires only the backscattering probabilities, rather than complete differential cross sections. The method has been generalized to multi-stream models, in which the solid angle range of the electrons is divided into 20 or more intervals, so more or less complete differential cross sections are required [84.46,47]. Monte Carlo methods have also been used to model auroral precipitation [84.48].

84.4 Ionospheres 84.4.1 Ionospheric Regions

Part G 84.4

The division of the ionosphere into regions is based on the structure of the terrestrial ionosphere, which consists of overlapping layers of ions. These layers are the result of changes both in the composition of the thermosphere and in the sources of the ionization, and are shown schematically in Fig. 84.5. The major molecular ion layer is the F1 layer, which is produced by absorption of EUV (100–1000 Å) photons by the major thermospheric species, and occurs where the ion production maximizes. The E layer is below the F1 layer and is produced by shorter and longer wavelength photons that are absorbed deeper in the atmosphere: soft X-rays and Lyman β, which can ionize O2 and NO (Fig. 84.4a). In the D region, the densities of negative ions ecome appreciable and large densities of positive cluster ions appear. These ions are produced by harder X-rays, with λ  10 Å, and Lyman α, which penetrates to about 75 km, where it ionizes NO. The highest altitude peak in the terrestrial ionosphere is the F2 peak, which occurs near or slightly below 300 km, where the major ion is O+ . The peak density occurs where the chemical lifetime of the ion is equal to the characteristic time for transport by diffusion (∼ H 2 /D).

Altitude (km) Topside ionosphere

600

500

X+

400

300

F2 XY+

200

100

Cosmic rays X-Rays < 10Å

EUV

Lyα 1216 Å

E

XY – 102

F1

D

UV Lyβ 1026 Å X-Rays (10 – 100Å)

103

104

Bottomside ionosphere

105 106 Electron density (cm–3)

84.4.2 Sources of Ionization

Fig. 84.5 Ionospheric regions and primary ionization sources. After Bauer [84.49]

As discussed in Sect. 84.3, ionization can be produced either by solar photons and photoelectrons during the daytime or by energetic particles and secondary electrons during auroral events. Photoelectrons have suf-

ficient energy to carry out further ionization if they are produced by photons with λ  500 Å. These photons penetrate further and exhibit larger solar activity variations than longer-wavelength ionizing photons. Thus

1272

Part G

Applications

Part G 84.4

the ionization rate due to photoelectrons peaks below the main photoionization peak. Primary flux spectra of photoelectrons produced near the F1 peak (172 km) and below the ion peak (100 km) are shown in Fig. 84.6. The primary spectrum at the ion peak consists mostly of low energy electrons, whereas at 100 km, the low energy primaries are depleted, and there are relatively larger fluxes of electrons with E
50 eV. Figure 84.7 shows the primary and steady-state photoelectron spectra near the ion peaks on Venus and Titan. The major ions produced in the ionospheres of the earth and planets are usually those from the major + + + thermospheric species: N+ 2 , O2 , and O on Earth; CO2 , + + + + + O , N2 , and CO on Venus and Mars; and H2 , H , and + + He+ on the outer planets; N+ 2 , N , and CH4 in the iono+ + + sphere of Titan, and N2 , N , and C in the ionosphere of Triton. In the presence of sufficient neutral densities, however, ion–molecule reactions transform ions whose parent neutrals have high ionization potentials to ions whose parent neutrals have low ionization potentials. This is a rigorous rule only for charge transfer reactions, but it applies more often than not in other ion–molecule reactions as well. Because of transformations by ion–molecule reactions, the major ions in the F1 regions of the ionospheres + of Earth, Venus and Mars are O+ 2 and NO , in spite of the large differences in composition between the thermosphere of the earth and the thermospheres of Venus and Mars. A diagram illustrating the ion chemistry in the ionospheres of the terrestrial planets is shown in Fig. 84.8. The vertical positions of the ions in this figure represent the relative ionization potentials of the parent neutrals. In regions where there are sufficient neutral densities the ionization flows downward. Table 84.9 shows ionization potentials (IP ) for several major and minor species present in planetary thermospheres. Major atmospheric species generally have IP
12–13 eV (λ < 900–1000 Å). Only a few species can be ionized by the strong solar Lyman alpha line (1216 Å, 10.2 eV), including NO, and a few small hydrocarbons and radicals, such as CH3 and C2 H5 . Metal atoms, which are produced in the lower thermospheres and mesospheres of planets from ablation of meteors, have very low ionization potentials, and some can be ionized by photons with wavelengths longer than 2000 Å. Fig. 84.7 Computed primary and steady-state spectra for photoelectrons near the F1 peak on Venus at 1 eV resolution (top), and Titan at 0.5 eV resolution (bottom). The steadystate spectra are averaged over three intervals in both plots

log electron flux (cm–2s–1eV–1sr–1) 10

Earth 172 km

8

6 100 km 4

2

0

20

40

60

80

100 120 140 160 180 200 Energy (eV)

Fig. 84.6 Primary photoelectron spectrum for the terres-

trial atmosphere at 172 km (near the F1 peak) and at 100 km. The spectrum at 100 km is significantly harder than that at 172 km

log electron flux (cm–2s–1eV–1sr–1) Venus 140 km

10 Initial

8

6 Steady-state 4

2

8

0

50

100

150

log electron flux (cm–2s–1eV–1sr–1)

200 Energy (eV)

Titan 1000 km 6

Initial

4 Steady-state

2

0

0

50

100

150

200 Energy (eV)

Aeronomy

He

N2

e, hv

e,hv e, hv

O2

He+ 24.5

CO

N2

e,hv e,hv e,hv

e, hv e,hv

N2+ NO 15.6

+

N 14.02

N2,N,NO CO2 O2, NO

CO CO, CO2

e, hv

N (2D)

CO2

O

e, hv

O+(2D) 16.0

CO2

O2

e, hv

e, hv

e, hv

O2, CO2

CO2

N

O

CO2+ O 13.76

H

O+(4S) 13.62

O

O

H+ 13.60

O2

O, O2

O2

NO

C C NO

NO

N2, NO

NO N(2D) N, NO, N(2D)

Fig. 84.8 Diagram illustrating the ion chemistry in the ionospheres of the terrestrial planets. The numbers under the names of the ions indicate the ionization potentials of the parent neutral. In the presence of sufficient neutral densities, the ionization flows downward, the importance of dissociative recombination for the molecular ions increases as the ionization potentials of the parent neutrals decrease Table 84.9 Ionization potentials (IP ) of common atmospheric speciesa High IP Species

IP (eV)

Medium IP Species

IP (eV)

Ionized by Ly α Species

IP (eV)

He Ne Ar N2 H2 N CO CO2 O H HCN OH

24.59 21.56 15.76 15.58 15.43 14.53 14.01 13.77 13.62 13.60 13.60 13.00

CH4 CH4 O2 O2 C2 H6 C2 H2 C C3 H8 CH C2 H4 CH2 S

12.61 12.51 12.32 12.07 11.52 11.40 11.26 10.95 10.64 10.51 10.40 10.35

C4 H2 CH3 C3 H6 NO C2 H5 HCO C3 H7 Mg trans-HCNH cis-HCNH Ca Na

10.18 9.84 9.73 9.264 8.13 8.10 8.09 7.65 7.0b 6.8b 6.11 5.14

Computed with data taken from [84.50], except as noted;

b

From [84.51]

Part G 84.4

C+ 11.20 NO+ 9.76

O2+ 12.1

CO2, O2

O2

CO, CO2

a

1273

NO

CO2, O2

CO2

e, hv

O,O2

CO+ 14.01

H

O

N2 N

CO2

CO2

84.4 Ionospheres

1274

Part G

Applications

In ionospheres where hydrogen is abundant and sufficient neutral densities are present, ionization flows to species formed by protonation of neutrals that have large proton affinities. There are no in situ measurements of the ion composition of the outer planets, but models predict that H+ 3 and hydrocarbon ions dominate the lower ionospheres. In regions where meteor ablation occurs, metal ions may also be found. Many ion–molecule reactions proceed at or near gas kinetic (or collision) rates. The interaction of an ion with a nonpolar molecule is dominated by the ion-induceddipole interaction, for which the interaction potential is − 12 αd q 2 /r 4 , where αd is the polarizability of the neutral, q is the charge on the ion, and r is the distance between the particles. The Langevin rate coefficient is then given by kL = 2πq(α d /µ)1/2 ,

(84.47)

Part G 84.4

where µ is the reduced mass of the two species. For a singly charged ion, with αd in Å3 and µ in atomic mass units, this formula reduces to 2.34 × 10−9 (αd /µ)1/2 . The rate coefficient for an ion with a polar molecule is

 1/2  2 2πq 1/2 (84.48) , kd = 1/2 αd + cµd πkB T µ where µd is the dipole moment and c is a constant that is unity in the locked dipole approximation and is about 0.1 in the average dipole orientation (ADO) theory. Theories for ion–quadrupole interactions have also been developed, and the resulting formulas can be found in, for example, the review by Su and Bowers [84.59]. Measured rate coefficients for ion–molecule reactions have been compiled by Anicich et al. [84.60] and Ikezoe et al. [84.61]. Loss of ionization in planetary atmospheres proceeds mainly by dissociative recombination of molecular ions, which may be represented by AB + + e− → A + B .

(84.49)

Dissociative recombination coefficients are characteristically large, about 10−7 cm3 /s, at the electron temperatures Te typical of planetary ionospheres, which are usually within a factor of two or so of the neutral T ≈ 200–2000 K near the molecular ion density peak. Daytime peak electron densities are usually in the range 104 –106 cm−3 , and fractional ionizations are small, about 10−5 , near the F1 peak. The relative importance of ion–molecule reactions and dissociative recombination in the destruction of a particular ion is determined by the relative densities of electrons and neutrals with which the ions can react. In general, molecular ions whose parent neutrals have high IP are transformed by ion–molecule reactions preferentially to loss by dissociative recombination, and their peak densities occur higher in the atmosphere. Ions for which dissociative recombination is an important loss mechanism near the ion peaks of the terrestrial planets include NO+ and O+ 2 , and in the atmospheres of the outer planets, H+ 3 and hydrocarbon ions. For ions with + very high IP , such as N+ 2 and H2 , dissociative recombination is rarely important as a loss process, except at very high altitudes. It may, however, be important as a source of vibrationally or electronically excited fragments or hot atoms. Atomic ions may be destroyed by radiative recombination: X + + e− → X + hν ,

(84.50)

but the rate coefficients are small, about 10−12 cm3 /s at the typical Te of planetary ionospheres [84.62]. Atomic ions may dominate at high altitudes, where neutral densities are low, but in such regions, loss by downward diffusion is more important than chemical recombination. The ions diffuse downward to altitudes where the neutral densities are higher and are then destroyed in ion–molecule reactions. The major ions in the topside ionospheres of the planets tend to be atomic ions: O+ in the ionospheres of Earth and Venus, and H+ in the iono-

Fig. 84.9a–k Model thermospheres for the Earth and planets. The curves are number density profiles and are labeled by the species they represent. (a) Earth, based on the MSIS model of [84.17] for a latitude of 45◦ , a local time of noon for low (top) and high (bottom) solar activities; (b) High solar activity model of Venus, based on the model of [84.3] for 15◦ N latitude 15 h local time; (c) Low solar activity model of Mars, based on Viking 1 measurements [84.8]. Adapted after [84.52]; (d) Jupiter, After [84.53]. (e) Saturn, After [84.54]. (f) Uranus, after [84.55]; (g) Neptune [84.56]; (h) Pluto [84.57]. The abscissa is given in units of Pluto radii, and the ordinate in the top plot is number density in cm−3 ; the bottom is temperature in K. (i) Titan, After [84.10, 12]; and (j) Triton model, after [84.15]; (k) Io model,

temperature (upper scale), and number density (lower scale). The major constituent is SO2 , and transient with a lifetime of 2–3 days. From Strobel and Wolven [84.58]. The short dashed curves are for solar ionization only, and the solid and long-dashed curves are for different assumptions about the interaction of Triton’s thermosphere with electrons from Neptune’s magnetosphere. The solid curves are the recommended model (j) I

Aeronomy

1000

c)

Altitude (km)

400

Mars Low solar activity

F10.7 = 75 N

800

600

H H2

He CO He

N2 O2

300 O

Ar

400

H

0

200 5

10

NO

15 20 log density (cm–3)

C

Altitude (km)

N2

O2

N2

CO2

N

F10.7 = 200

He

800

100

N

104

105

106

107

108

109

1010

1011 1012 Density (cm–3)

d)

600

Pressure (mbar) Temperature (K) 0 100 200 300 400 500 600 700 800 900 1000

O

400

10 –7 10 –6 10 –5 10 –4 10 –3 10 –2 10 –1 100 101 102 103 104

Ar

200

H

0

5

10

15 20 log density (cm–3)

Altitude (km) 400 Venus Hi standard

He H

T

Gladstone NEB Model Atmosphere H H2 C2H2

105

e)

300

1010

Altitude (km)

1015

150 100 50

C2H6 CH4

1000 800 600 500 400 350 300 250 200

He

0 – 50

1020 Density (cm–3) Pressure (mbar)

O 10 –10 3000 CO

H 2000

200

NO

C O2

H2

N

CO2 Ar

N2

100 103

104

105

106

107

108

109 1010 1011 1012 1013 Density (cm–3)

1000

H2

10 –8

He CH4

10 –6 C2H2 10 –4

H2 O

1 CH3C2H C2H6 0 104 105 106 107 108 109 1010 1011 1012 1013 1014 1015 1016 Number density (cm–3)

Part G 84.4

O2

b)

O

Ar

200

1000

1275

Altitude (km)

Altitude (km)

a)

84.4 Ionospheres

1276

Part G

Applications

f) Altitude (km)

Temperature (K) 400 600

200

log10 PH2 (µbar) i) Altitude (km) 3000 800

–6

6000

2500

T

5000

–5

H2

2000

4000

H

–4

3000

1500

–3 1000

2000

–2

He 1000

Part G 84.4

3000

–1 0 2

HC 8 6 Altitude (km)

C2H2

C2H6

H2

g)

Titan

H

7000

10

12

14

16 log10 n (cm–3)

C4H2

N2

CO C2H4

CH4

4

6

8

10 12 log densities (cm–3)

j) Altitude (km) 20

1000

40

60

80

Temperature (K) 100 120

Neptune 800

2500

N

2000 H

600

H2

1500

H2

H 400

He 1000

C2H4 C2H6

C2H2

500 6

8

h) Radial distance (km) 20 1600

40

60

200

CH4 10

12 14 log densities (cm–3)

Temperature (K) 80 100 120

Pressure (µbar) 140 160 10 –2

Hybrid model (troposphere extension) High density model

1500

T

N2

0 3

4

5

6

7

8

9

10 11 12 13 14 15 16 log density (cm–3)

k) Height (km) 0

500

1000

Temperature (K) 1500

1000 n(z)

800 10 –1

1400

N(z)

T(z)

600

T(z) 1300 1

Γd(N2) = –0.5 K/km

1200

1208 1164 1140

1100 106

Tmin = 35 K

τv = 0.15

200

Tsurf = 57 K

10

Tsurf = 37 K

108

400

0 1010

1012 1014 Number density (cm–3)

106

107

108

109 1010 Number density (cm–3)

Aeronomy

spheres of the outer planets. On Mars, however, the O+ peak density does not exceed that of O+ 2 even at high altitudes. Model thermospheres for Earth and selected planets and satellites are shown in Fig. 84.9a–k. Measured or computed ion density profiles for the Earth and selected planets and satellites are shown in Fig. 84.10a–h.

84.4.3 Nightside Ionospheres

84.4.4 Ionospheric Density Profiles Density profiles of molecular ions can often be approximated as idealized Chapman layers. A Chapman layer of ions is one in which the ions are produced by photoionization and lost locally by dissociative recombination. The ionization rate q i in a one-species Chapman layer for monochromatic radiation is given by q i = Fσ i n ,

(84.51)

1277

where σ i is the ionization cross section, and F = F ∞ exp[−τ(z)] is the local solar flux. For an isothermal atmosphere, the scale height is approximately constant and therefore n = n 0 exp(−z/H). Sometimes an ionization efficiency ηi is defined such that σ i = ηi σ a .

(84.52)

Near threshold, the ionization efficiency for molecules is usually about 0.3–0.7 but it increases rapidly to 1.0 at shorter wavelengths. Since the maximum ionization rate in an isothermal atmosphere occurs where the optical depth (τ = n Hσ a sec χ) is unity and therefore n = 1/(σ a H sec χ), the maximum ionization rate in a Chapman layer is i qmax,χ =

i qmax,0 σi F∞ = . e σ a H sec χ sec χ

(84.53)

If the altitude of maximum ionization for overhead sun is defined as z = 0, then n 0 = (σ a H)−1 , and, expressing i , the ionization rate is F ∞ in terms of qmax,0 

z i q i (z) = qmax,0 exp 1 − − sec χ e−z/H . H (84.54)

It is apparent that at high altitudes (z → ∞) the ionization profile follows that of the neutral density, and below the peak (z → −∞), the ionization rate rapidly approaches zero. As the solar zenith angle increases, the peak rises and the magnitude of the density maximum decreases. Figure 84.12 shows a production profile for an idealized Chapman layer on both linear and semilog plots. The asymmetry with respect to the maximum is more obvious for the semilog plot. If photochemical equilibrium prevails, the production rate of the major ion is equal to the loss rate due to dissociative recombination q i (z) = αdr n i n e = αdr n 2i ,

(84.55)

where αdr is the dissociative recombination coefficient, n i is the ion density, and n e is the electron density. Therefore the density of an ion in a Chapman layer (in the photochemical equilibrium region) is given by  i 1/2 q (z) n i (z) = αdr  i  qmax,0 1/2 1 z 1 − sec χ e−z/H . = exp − αdr 2 2H 2 (84.56)

Actual ionization profiles differ from the idealized Chapman profile for several reasons. First, ionization is

Part G 84.4

Nightside ionospheres can result from several sources including remnant ionization from dayside, like O+ in the terrestrial ionosphere. While the lower molecular ion layers recombine, the F2 peak persists through the night, although it rises and the peak density is reduced by a factor of 10. Electron density profiles for day and night at high and low solar activities are shown in Fig. 84.11. In the auroral regions of the Earth, the precipitating electrons may also produce significant ionization, which maximizes in the midnight sector of the auroral oval. The nightside ionosphere of Venus is highly variable, but has been shown to contain the same ions as the dayside ionosphere. The densities are, however, lower by factors of 10 or more than those of the dayside ionosphere, and the average peak in the electron density profile is about (1 to 2) × 104 cm−3 . It is produced by a combination of precipitation of suprathermal electrons that have been observed at high altitudes in the umbra, and transport of atomic ions (mostly O+ ) at high altitudes from the dayside. For Mars, only a narrow range of solar zenith angles near the terminator at low solar activity has been measured by the radio occultation experiments on the Viking spacecraft [84.63], the Mariner 9 spacecraft, and more recently by the Mars Global Surveyor (MGS) spacecraft radio sciences (RS) experiment [84.64]. The electron densities are apparently low, and no composition information is available. At this time, there is no information available about the nightside ionospheres of the other planets.

84.4 Ionospheres

1278

Part G

Applications

d)

a) Altitude (km) N+

1000

4000

–

10–10

H3+

3000

He+ H+

10–8

2000

300 250

+

He

NO+

200

10–6

N2+

1000

O+

100 102

103

Ne– O2+

NO+

O2+ 104

105

C3H5+ 0

106 Number (cm3)

Altitude (km)

e)

400

Part G 84.4

H+

C+

CO2+

10

1

1000 104 105 Number density (cm–3)

100

Pressure (mbar)

b. 10 O+

N+

1

10–4 Voy2 36° N Voy2 31° S

Venus High

He+

CO+

300

e–

O2+

N+

150

b)

H+

Ne O+

500

Pressure (mbar)

Altitude (km)

–10

Electron (observed)

10–9 H3+

N2+

10–8 e

200 NO+ + O2

Electron (model)

H+

10–7 10–6

CH5+

–5

10 100 1

10

100

1000

c) Altitude (km)

104

1 × 101

105 106 Density (cm–3)

f)

1 × 102

1 × 103

1 × 104

1 × 105 1 × 106 Density (cm–3)

Altitude (km)

3000

400 Mars Lo v 1.24

300

Neptune

2500 H3+

2000 O+

O++ He+ H+ 200 N+

C+

ne

N2+ CO+

NO+

CO2+ O2+

1500

1000 Mg+

500

100 1

10

H+

100

1000

104 105 Density (cm–3)

–1

0

1

2

3 4 5 log ion densities (cm–3)

Aeronomy

g) Altitude (km)

h) Altitude (km)

3000

800

Pressure (µbar)

H+

N2H+

N2H+

10–6

N2+

600 CxHyNz+

10–5 CH5+

2000 H3+ N2

1500

400

C2H5+

+

C4H3+

H2CN+

N+

–1

0

1

C+

10–4

e

10–3 200

CxHy+

1000

–2

2 3 4 log densities (cm–3)

Egress Ingress 0

0

1

2

3

4 5 log10 density (cm–3)

10–2 10–1 100 101

Altitude (km)

700 Solar min

600

Solar max

500 400 F2

300

Day Night

produced by photons over a range of wavelengths, which do not all reach unit optical depth at the same altitude. Second, thermospheres are often not isothermal near the altitude of peak ion production. Third, photoionization is supplemented by photoelectron-impact ionization, which peaks lower in the atmosphere; and finally, the major ion produced is often transformed by ion–molecule reactions before it can recombine dissociatively. Nonetheless, the idealized concept of the Chapman profile is useful in understanding the general shape of ion profiles and their behavior as the solar zenith angle changes. In addition, ion layers produced by auroral precipitation may take on a similar appearance to a Chapman-type layer, although energetic electrons are not always extinguished, as are photons, in ion production.

200 F1 D 0

84.4.5 Ion Diffusion

E

100

108

109

1010

1011

1012 1013 Electrons (m–3)

Fig. 84.11 Typical midlatitude ionospheric electron density profiles for sunspot maximum and minimum, day and night. After Richmond [84.69]

Above the photochemical equilibrium layer of the ionosphere, upward and downward transport of ions must be considered. The motions of ions, neutrals, and electrons are coupled, and the momentum equation, which determines the fluxes or velocities of the ions must take into account these interactions. The interaction of an ion, denoted by a subscript i, with a neutral species de-

Part G 84.4

Fig. 84.10a–h Ionospheric density profiles. (a) Measured profiles for Earth [84.65]; Computed profiles for (b) Venus, for high solar activity and a solar zenith angle of 45◦ ; (c) Mars, for a low solar activity model for Viking 1 conditions and a solar zenith angle of 45◦ ; the dashed curves are measured profiles from Viking [84.66]; (d) Jupiter, After [84.67]; (e) Saturn, After [84.54]; (f) Neptune [84.56]. See also [84.68] (g) Titan, After Fox and Yelle (unpublished, 1995); and (h) Triton [84.16]

800

1279

Ne

Titan 2500

84.4 Ionospheres

1280

Part G

Applications

8

coefficient is

Reduced altitude (Z/H)

ct Din =

6

3(π/2)1/2



8n n Q ct in

kB Ti mi

1/2

2 75°

60°

0

30°

0°

–2

8

, (84.59)

4

–4

1 (1 + Tn /Ti )1/2

0

0.2

0.4

0.6 0.8 1 Production rate (Q/Qmax)

Reduced altitude (Z/H)

(84.60)

where es is the charge on species s, ln Λ is related to the Debye shielding length, and Tis = (m s Ti + m i Ts )/(m i + m s ) is a reduced temperature. Numerically, ln Λ is about 15, and the collision frequency is approximately [84.71]

6 4

Part G 84.4

2 75° 0

60°

30° 0°

–2 –4 –3

–2

–1 0 log [production rate–1(Q/Qmax)]

Fig. 84.12 Chapman production profile as a function of altitude on linear (top) and semilog (bottom) plots. The production rate is divided by the maximum production rate and the altitude by the scale height H, which is assumed to be constant. The origin on the altitude scale corresponds to the point of maximum absorption for overhead sun

noted by a subscript n, is through the ion-induced-dipole attraction or, for the diffusion of an ion through its parent neutral, by resonant charge transfer. For the former process, the ion–neutral diffusion coefficient is given by Din =

kB T , m i νin

(84.57)

where m i is the mass of the ion, and the ion–neutral momentum transfer collision frequency nnmn νin = 2.21π m i+m n



where Q ct in is the average charge transfer cross section [84.72]. The momentum transfer collision frequency for Coulomb interactions between an ion i and another ion or electron denoted by the subscript s is  3/2 2 2 µis 16π 1/2 n s m s ei es νis = ln Λ , 3 m i + m s 2kB Tis µ2is

αn e2 µin

1/2 ,

1/2

νis = 1.27

Z i2 Z s2 µis n s 3/2

m i Tis

s−1 ,

(84.61)

where Z is the species charge number, µ and m are in amu, and the number density is in units of cm−3 . The ion densities can be computed by solving the ion continuity equation, which is similar to (84.7) for neutral species, and in one dimension is ∂n i ∂Φi + = Pi − L i , ∂t ∂z

(84.62)

where the ion flux is given by Φi = n i wi . In general it is impossible to solve the momentum equation for the ion diffusion velocity wi in closed form, except for the special cases of a single major ion and of a minor ion moving through a dominant ion species. If motion of the ions only parallel to magnetic field lines is considered, the vertical velocity of a dominant ion (for which n i ≈ n e ) moving through a stationary neutral atmosphere is wi = −Da sin2 I  mi g 1 d(Te + Ti ) 1 dn i + + , × n i dz k(Te + Ti ) Te + Ti dz (84.63)

(84.58)

where αn is the polarizability of the neutral species (Dalgarno et al. [84.70]; Schunk and Nagy [84.71]). For the resonant charge transfer interaction, the diffusion

where I is the magnetic dip angle and the ambipolar diffusion coefficient defined as Da =

kB (Te + Ti ) . m i νin

(84.64)

Aeronomy

For a minor ion i diffusing through a major ion species j, its velocity is given by  kB Ti /m i 1 dn i Te /Ti dn e + wi = − (84.65) νi j + ν in n i ds n e ds νi j w j mi g 1 d(Te + Ti ) − + + . kB Ti Ti ds νi j + νin

84.5 Neutral, Ion and Electron Temperatures

1281

For regions in which there are large gradients in the ion or electron temperatures, thermal diffusion may also be important in determining the ion density profiles, especially those of light ions such as H+ and He+ . Equations for ion distributions in which thermal diffusion is included have been presented by, for example, Schunk and Nagy [84.71] and references therein.

84.5 Neutral, Ion and Electron Temperatures The temperature distribution in planetary thermospheres/ionospheres can be modeled by solving the equation for conservation of energy, which, in simplified form in the vertical direction is  ∂ N ∂Tm ∂Tm − n m kB κm = Q m − L m (84.66) 2 ∂t ∂z ∂z

Part G 84.5

where N is the number of degrees of freedom (3 for an atom, and 5 for a diatomic molecule), the subscript m refers to the neutrals, ions or electrons, κm is the thermal conductivity, Q m is the volume heating rate, and L m is the volume cooling rate. If horizontal variations are considered, the model becomes multidimensional and advective terms must be added to the equations. The Tn are also affected by compression or expansion due to subsidence or upwelling, respectively. Viscous heating may be a factor where there are local regions of intense energy input, such as in auroral arcs. These terms are not shown in the energy equation above, but may be found in standard aeronomy texts, such as Banks and Kockarts [84.72] or Rees [84.44] and Schunk and Nagy [84.73]. For planets with intrinsic magnetic fields, the electrons and ions are constrained to move along magnetic field lines, and the second term on the left-hand side of (84.66) must be multiplied by a factor sin2 I. The neutral thermospheres of planets are mostly heated by absorption of solar radiation in the 10 to 2000 Å range, although on planets with powerful auroras, electron precipitation may be an important source  of heat. Absorption of EUV radiation 100–1000 Å largely results in ionization of the major thermospheric species, in which most of the excess energy is carried away by the photoelectron. The photoelectron, however, may produce further dissociation or excitation of neutral species along the path to thermalization, and these processes may result in neutral heating. Photons near and longward of ionization thresholds in the FUV may lose their energy in photodissociation, in which the excess energy of the photon appears as kinetic or internal energy of the fragments.

Chemical reactions that follow ionization or dissociation release much of the absorbed solar energy as heat. Although the partitioning of kinetic energy released between the product species can be determined easily by conservation of energy and momentum, the fraction of energy that appears as internal or kinetic energy must be determined by measurements or theoretical calculations. If vibrationally or electronically excited states are produced in these interactions, however, the energy may be radiated to space, thus producing cooling. This may occur promptly if the radiative lifetime is short, or subsequent to an energy transfer process from a long-lived metastable species to a species for which radiation to a lower state is allowed. If the metastable species is quenched, however, its energy can also appear as heat. Thus the energy partitioning in chemical reactions and in the interactions of photons and photoelectrons with atmospheric species is important in understanding the temperature structure of thermospheres. A heating efficiency  is often defined as the fraction of energy absorbed at a given altitude that appears locally as heat. The heating efficiencies are in the range 30–40% in the terrestrial lower thermosphere. Above 200 km, the heating efficiency decreases because the energy of the important metastable species O(1 D) is lost as radiation rather than by quenching [84.74]. The heating efficiencies in the thermospheres of Venus and Mars are about 20% from 100 to 200 km [84.75, 76], and on Titan, they range from 20 to 30% from 800 to 2000 km. A column averaged heating efficiency for the Jovian thermosphere has been computed as 53% [84.77]. On Venus and Mars, CO2 is the major absorber of far UV radiation, whereas on the Earth, O2 plays that role. In the F1 regions of the ionospheres of the terrestrial planets, dissociative recombination of molecular ions tends to be the major source of heating. Below the F1 peak, photodissociation and neutral-neutral reactions, including quenching of metastable species, dominate. Since CH4 is a very strong absorber, the major heating

1282

Part G

Applications

Part G 84.5

mechanisms in the thermosphere of Titan are photodissociation and neutral-neutral reactions, both above and below the F1 peak. The few data that exist suggest that electron-impact dissociation is unimportant as a source of neutral heating, although further measurements would certainly be of benefit. Profiles of the heat sources in the terrestrial thermosphere and that of Mars are also shown in Fig. 84.13. Important cooling processes in planetary thermospheres include downward transport of heat by molecular and eddy conduction and infrared cooling from rotational and vibrational excitation of IR active species such as NO, CO and CO2 . Excitation of the fine structure levels of atomic oxygen and subsequent emission at 63 and 147 µm also plays a role in cooling the neutral species in the thermospheres of the terrestrial planets. In the outer planets and their satellites, hydrocarbon molecules such as CH4 and C2 H2 are the primary thermospheric IR radiators. The global circulation may play a role in redistributing the heat that is deposited in the dayside or auroral thermosphere [84.78, 79]. a) Altitude (km) 250

In order to model heating rates, cross sections for processes in which solar photons or photoelectrons interact with neutral species, and rate coefficients and product yields for chemical reactions of ions and neutral atmospheric species are necessary. In addition, it is necessary to know, for example, how much of the energy released appears as internal energy of the products in chemical reactions and how much appears as kinetic energy of the products. Knowledge of energy transfer processes, including vibration–vibration (V –V ) and translation– vibration (T –V ) transfer between atmospheric species is also important. For example, a particularly important cooling process for the thermospheres of the terrestrial planets is excitation of the CO2 15 µm bending mode in collisions with energetic O, and subsequent radiation [84.80]. The de-excitation rate is several percent of gas kinetic, which is anomalously large for a V –T process [84.81]. In the lower ionosphere, the electrons and ions are in thermal equilibrium with the neutral species, but at higher altitudes the plasma temperatures deviate from the neutral temperatures. Near the F1 peak, Te is usub) ZP 5

Chemical Rxns

J e– i

4 3 2

O2+ DR

200

SRB

O('D)

1

nc

0 Electron impact

O

100

0

1

2

3 4 5 log10 heating rate (eV/cm2 s)

SRC

–5 –6 –7

150 120

–4 Photodissociation

400 350 300 250 200

Qn

–2 –3

150

A

O3

–1 Quenching

500

iC

100 0

1.0

2.0

3.0 4.0 5.0 6.0 log heating rate (erg g m–1 s–1)

Fig. 84.13a,b Heating rates for the thermospheres of (a) Mars and (b) Earth. In (b), the curve labeled Qn is the total heating rate; e–i is the heating rate due to collisions between the neutrals and electron and ions, iC and nc are the heating rates due to exothermic ion–neutral and neutral–neutral chemical reactions, respectively; J is that  due  to Joule heating −1 ; A is that from auroral particle precipitation; O 1 D is the heating due for a superimposed electric field of 3.6 mV m  to quenching of O 1 D ; SRC and SRB are the heating rates due to absorption in the Schumann Runge continuum and bands, respectively; O is the heating from recombination of atomic oxygen; O3 is the heating rate due to absorption of photons by O3 in the Hartley bands. After [84.74]. In (a) the curve labeled “O+ 2 DR” is the heating rate due to ; that labeled “photodissociation” is heating due to the production of energetic dissociative recombination of O+ 2  neutrals  in photodissociation; the curve labeled “quenching” is that due to quenching of metastable species, such as O 1 D ; and “chemical reactions” denotes heating due to exothermic chemical reactions other than quenching of metastable species. After [84.76]

Aeronomy

a)

Ti in the ionosphere is elevated above Tn at high altitudes principally because of Coulomb collisions with energetic electrons. Another potentially important source of heat input to the ions near the ion peak is exothermic ion–neutral reactions, including quenching  of metastable ions, such as O+ 2 D , by neutrals. In the presence of electric fields, joule heating may be important and can cause Ti to exceed Te . The ions cool in elastic collisions and resonant charge transfer with neutral species, which are characterized by lower temperatures than the ions. The cooling rate for elastic collisions is mi 3 L in = −2n i νin kB (Ti − Tn ) . (84.67) mi + mn 2 Collisions between ions and neutrals (other than their parents) are dominated by the ion-induced-dipole interaction. The momentum transfer collision frequency is thus given by (84.58). Resonant charge transfer between an ion and its parent neutral, such as O+∗ + O → O∗ + O+ ,

1400

800

c) Altitude (km)

2500

1200

700

1000

600

800

500

1500

600

400

1000

400

300

200

200

0

0

1000

2000 3000 4000 Temperature (K)

(84.68)

leads to very effective ion cooling, which dominates at sufficiently high temperatures. Examples of Ti and Te profiles are shown in Fig. 84.14a–c for Earth, Venus and Titan. The International Reference Ionosphere (IRI) temperatures profiles

b) Altitude (km)

100 2 10

Altitude (km) Neutral

2000 Ion

Electron

500 103

1283

104 105 Temperature (K)

0

0

200 400 600 800 10001200 1400 1600

Temperature (K)

Fig. 84.14a–c Ion and electron temperature profiles. (a) Neutral (dot-dashedcurve), ion (dashed curve) and electron (solid curve) temperatures for the terrestrial ionosphere from the International Reference Ionosphere for equinox, noon and low and high solar activities. The electron temperature is found not to vary substantially with solar activity. After Bilitza and Hoegy [84.82] with kind permission from Elsevier Science Ltd., Kidlington UK. (b) Smoothed median ion (dashed curve) and electron (solid curve) temperatures in the Venus ionosphere as measured by the PV retarding potential analyzer and the Langmuir probe, respectively. The electron temperature profile is essentially constant with solar zenith angle. The ion temperature profile applies to solar zenith angles between 0 and 90◦ . After [84.83]. (c) Computed electron and ion temperatures for the ionosphere of Titan, including only solar photoionization as the source of electron heating. After [84.84]

Part G 84.5

ally larger than Ti , but Ti begin to diverge from Te at slightly higher altitudes. The energy source for the electrons on the dayside is largely photoionization, which, as discussed above, produces electrons with average energies in the 15 to 20 eV range. In slowing down, these energetic electrons lose their energy in inelastic processes with neutrals until E ≈ 1–2 eV. At this point, elastic scattering by the thermal electron population becomes the dominant energy loss process for the suprathermal electrons and the major source of heat for the thermal electrons. Other electron heating mechanisms include deactivation of electronically or vibrationally excited species, and, for the terrestrial planets, quenching of the fine structure levels of O. As for the neutrals, heat in the electron gas is redistributed by conduction at a rate that depends on the electron thermal conductivity. This quantity is inversely proportional to the sum of the momentum transfer collision frequencies of electrons with ions, neutrals, and ambient electrons. Cooling mechanisms for thermal electrons include Coulomb collisions with ions, rotational excitation of molecules, and, for the terrestrial planets, excitation of the fine structure levels of O. Because of the large mass difference, elastic collisions between neutrals and electrons are not effective in transferring kinetic energy.

84.5 Neutral, Ion and Electron Temperatures

1284

Part G

Applications

for the terrestrial thermosphere show the close coupling between the electrons, ions and neutrals at low altitudes and the ions and electrons at high altitudes. Ti increases with increasing solar activity at low altitudes and approaches Te at high altitudes. The values of Te and Ti are about 3000 K at 1000 km [84.82]. Electron and ion temperatures in the terrestrial ionosphere are discussed in Rees [84.44], Banks and Kockarts [84.72], and Whitten and Popoff [84.85]. Te and Ti in the Venus atmosphere were measured by instruments on the Pioneer Venus spacecraft. Ti , which approaches values of 2000–2500 K at high altitudes, has been found to be insensitive to solar zenith angle, except near the antisolar point, where it increases to values of 5000–6000 K [84.83]. T e also does not vary appreciably with solar zenith angle; the high altitude values are in the range 4000–6000 K [84.86].

The retarding potential analyzer (RPA) on the Viking spacecraft found that Ti on Mars decouples from the Tn near 180 km, and approach values of about 3000 K at high altitudes [84.66]. Te is predicted to diverge from the Tn in the lower ionosphere, and to approach values of 3000–4000 K at high altitudes [84.87]. There are no measurements of plasma temperatures in the ionospheres of the outer planets. The plasma temperatures on Titan have been predicted by a model [84.84]. The computed Ti are grater than Tn near the n e maximum near 1000 km, but approach values of about 300 K at high altitude. For the solar source only, Te increases rapidly to a constant value of about 800 K near 1200 km. Electrons from Saturn’s magnetosphere may interact with the Titan ionosphere during the part of its orbit that is within the magnetosphere, and in this case Te up to about 5000 K near 2000 km are predicted.

Part G 84.6

84.6 Luminosity The luminosity that originates in the atmospheres of the planets is generally classified as dayglow, nightglow, or aurora. Dayglow is the luminosity of the dayside atmosphere that occurs as a more or less direct result of the interaction of solar radiation with atmospheric gases. Among the sources of dayglow are photodissociative excitation and simultaneous photoionization and excitation. Dayglow may also include scattering of solar radiation by processes that are selective, such as resonance scattering by atoms and fluorescent scattering by molecules, but the term generally excludes nonselective scattering processes, such as Rayleigh scattering. In resonance scattering, the absorption of a photon by an atom in the ground state, causes a (usually dipole allowed) transition to a higher electronic state: A + hν → A∗ ,

(84.69)

followed by the emission of a photon as the state decays back to the ground state: A∗ → A + hν .

(84.70)

The wavelength of the emitted radiation is very nearly the same as the wavelength of the radiation absorbed. The cross section for absorption of a line in the solar spectrum is 2 c2 A21 φ(ν) , (84.71) 1 8πν2 where the subscript 1 indicates the lower state and 2 the upper state,  is the statistical weight of the state and a σ12 (ν) =

ν = c/λ is the frequency of the transition. A21 is the Einstein A coefficient for the transition, and φ(ν) is the lineshape function, which in this equation is normalized so that the integral over all frequencies is unity. If the linewidth is determined by the spread of velocities of the species, the lineshape φ(ν) is a Doppler (Gaussian) profile

  c ν − ν0 2 c2 φD (ν) = , (84.72) √ exp − ν0 u2 uν0 π where ν0 is the frequency at line center. The variable u = (2kB T/m)1/2 is the modal velocity of a gas in thermal equilibrium at temperature T . The width of the line at half maximum, ∆νD is 2ν0 u (ln 2)1/2 . (84.73) c If the linewidth is determined by the natural lifetime, the profile is a Lorentzian ∆νD =

∆νL /2π φL (ν) =  2  2 , ν − ν0 + ∆νL /2

(84.74)

where ∆νL = ΓR /2π is the line width at half maximum, and ΓR = Γ2 + Γ1 ,

(84.75)

where Γ2 and Γ1 are the inverse radiative lifetimes of the levels 2 and 1, respectively. Collisional broadening also results in a Lorentzian lineshape. If both Doppler

Aeronomy

and natural broadening mechanisms are important, the lineshape is a convolution of the two profiles, called a Voigt profile: ∞

φD (ν
)φL (ν − ν
) dν
.

(84.76)

−∞ a integrated over all The absorption cross section σ12 frequencies is proportional to the absorption oscillator strength f 12 :

∞ a σ12 (ν) dν = 0

πe2 f 12 , mec

(84.77)

where m e is the mass of the electron. A21 is related to the oscillator strength through A21 =

(84.78)

The excitation rate q2 of an upper level 2 by resonance scattering is given by q2 = F(ν)

πe2 2 λ4 f 12 = F(λ) A21 , mec 1 8πc

(84.79)

where F(ν) is the solar flux in units of photons cm−2 s−1 Hz−1 and F(λ) is the flux in units of photons cm−2 s−1 per unit wavelength interval. It should be noted that for radiative transfer purposes the photon flux that we have called F(ν) is sometimes denoted πF(ν). It is customary in aeronomy to define a “g-factor,” which is the probability per atom that a photon will be resonantly scattered in a particular transition: g21 = q2 A21 / A2i , (84.80) i

where the sum in the denominator is over all the lower states i that are accessible from the upper state 2. The g-factor for unattenuated solar radiation is often quoted at the mean sun-earth distance or at a particular planet. The volume emission rate ε21 (z) for resonance scattering of a solar photon is then given by ε21 (z) = g21 n 1 (z) ,

(84.81)

where n 1 (z) is the number density of atoms in level 1. In fluorescent scattering, a photon is absorbed by a molecule in a vibrational state v producing an excited electronic state with a vibrational quantum number v
AB(v) + hν → AB ∗ (v
) .

(84.82)

1285

This is followed by emission, at wavelengths that are usually the same as or longer than that of the absorbed photon, to a range of vibrational levels v

of a lower state AB ∗ (v
) → AB(v

) + hν
.

(84.83)

The volume emission rate of a transition from a level v
of the upper electronic state to a vibrational level v

of a lower electronic state at an altitude z is given by Av
v

εv
v

(z) = n(z)gv
v

= n(z)qv
 , (84.84) v

Av
v

where Av
v

is the transition probability, n(z) is the number density of the molecular species at altitude z, and qv
is the excitation rate of vibrational level v
of the upper electronic state from a range of lower states v. The latter quantity is πe2 2 ηv F(λ) λ f vv
, (84.85) qv
= m e c2 v where ηv is the fraction of molecules in the v vibrational level. Dayglow also includes emissions that are the result of the interaction of atmospheric species with the photoelectrons produced in solar photoionization, either by direct excitation A + e∗− → A∗ + e− ,

(84.86)

or by simultaneous dissociation and excitation AB + e∗− → A∗ + B + e− ,

(84.87)

or ionization and excitation X + e∗− → X +∗ + e− .

(84.88)

Electron impact processes are particularly important in producing excited states that are connected to the ground state by dipole forbidden transitions, whereas resonance and fluorescent scattering are largely limited to transitions that are dipole allowed. Dayglow emissions may also result from prompt chemiluminescent reactions, which occur when fragments or ions produced by dissociation or ionization recombine with the emission of a photon. As an example, the dayglow spectrum of the earth from 1200 to 9000 Å is shown in Fig. 84.15. An ultraviolet spectrum of Mars as measured by the Mariner 9 spectrometer is shown in Fig. 84.16a, and the UV dayglow of Saturn and Uranus, which were measured by the Voyager spacecraft, are compared in Fig. 84.16b. Nightglow arises from chemiluminescent reactions of species whose origin can be traced to species produced during the daytime or which have been transported

Part G 84.6

1 8π 2 e2 ν2 1 8π 2 e2 f = f 12 . 12 2 m e c3 2 m e cλ2

84.6 Luminosity

1286

Part G

Applications

Rayleighs/Å N2, Vegard–Kaplan Dayglow

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.10 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 3.4 3.5 3.6 +

NOγ

500 400

1.0

N2 , First negative

0.0 1.1

0.1

0.2 0.3 0.4 0.5 1.3 1.4 1.5 1.6

1.0 2.1 3.2

2.0 3.1

N2, Second positive 0.0

Lα(1216)

3.0

N2, LBH

Nll(2143)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

300

0.6 1.7

2.0 3.1

Oll(2470) Mgll(2795)

1.0 2.1 3.2

1.1 2.2 3.3

0.11

0.12 1.12

0.13 1.13

2.13

0.14 1.14

2.14

0.0 1.1 2.2 3.3

0.1 1.2 2.3 3.4

1.15 2.15

0.2

0.3

1.3

1.4

2.4

2.5

3.5

3.6

0.1 0.2 0.3 0.4 0.5 1.2 1.3 1.4 1.5 1.6 2.3 2.4 2.5 2.6 2.7 3.4 3.5 3.6 3.7 3.8 3.9

1.7 2.8

Call(3934)

Ol(2972)

1.0 1.11.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Call(3970)

200

[Nl](5200)

Lα(1216)

Ol (1304) Ol (1356)

Oll(4351) Oll(4368) Oll(4415)

Nl(3466)

100

1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000 4200 4400 4600 4800 5000 5200 Wavelength (Å) Rayleighs/Å N2, First positive

Part G 84.6

8.4 7.3 6.2

1.5

2.5

6.3 5.2 4.1 8.5 7.4

N2, Atmospheric

0.4

0.3 1.4

500

5.1 4.0

2.0

2.6

3.6 4.0

5.0

N+2 Meinel

3.0 8.6

7.5

6.4

5.3

4.2

3.0

2.0 7.6

8.7 0.0 1.1

1.0 2.1

5.1

3.1

4.3

5.4

5.2

2.1

3.2

0.1

2.2 2.0

4.1

6.5

3.1

4.2

1.0

1.2

5.3

[Ol](6300)

400

[Ol](6364)

[Oll](7320)

Ol(8446)

300 200 100

[Nl](5200)

Ol(7774)

[Ol](5577)

[Nl](5755)

5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000 7200 7400 7600 7800 8000 8200 8400 8600 8800 9000 Wavelength (Å)

Fig. 84.15 Terrestrial dayglow spectrum measured in a single 32-second exposure by the Arizona Imager/Spectrograph on board the Space Shuttle [84.88]

from the dayside. For example, on Venus, O and N produced on the dayside are transported by the subsolar to antisolar circulation to the nightside, where they subside and radiatively associate: N + O → NO + hν ,

(84.89)

producing emission in the δ and γ bands of NO (e.g., Stewart et al. [84.89]). The Venus UV nightglow spectrum is shown in Fig. 84.17. Similar phenomena have recently been observed by the ultraviolet spectrometer on the Mars Express spacecraft [84.90]. Auroral emissions are defined here as those produced by impact of particles other than photoelectrons. Although aurorae are usually thought of as confined to the

polar regions of the Earth and the outer planets, Venus, which does not have an intrinsic magnetic field, exhibits UV emissions on the nightside that are highly variable and cannot be explained as nightglow. It has been proposed that the emissions are produced by precipitation of soft electrons into the nightside thermosphere. Mars has recently been observed to exhibit auroral emissions, which are concentrated over magnetic field anomalies in the martian crust [84.91]. The intensities of airglow and aurora are usually measured in units of brightness called Rayleighs. One Rayleigh is an apparent column emission rate at the source of 106 photons cm−2 s−1 integrated over all angles, or 106 /4π photons cm−2 s−1 sr−1 . A comparison

Aeronomy

Fig. 84.16a,b Dayglow spectra of selected planets. (a) Martian airglow spectrum recorded by Mariner 9 at 15 Å resolution. After [84.92]. (b) Comparison of dayglow spec-

tra from Uranus (heavy line) and Saturn (thin line) recorded by Voyager 2 [84.4]I Total counts PV / OUVS night airglow spectrum of Venus 25 CO Cameron NO Gamma

20

NO Delta

84.7 Planetary Escape

a) Intensity k Rayleigh / 15 Å 30

~

24 2.0 1.0 0.0 0.1 0.2 0.3 1.0 0.0

18

0.1

1.5

0.2

OI 2972 Å

1.6

1.4

1.5

~

~

CO2+ A2Π –X2Π 4.0 3.0

CO+ B2 Σ+–X2Σ+

12

~

CO2+ B2 Σ+ –X2Π

CO σ3Π –X1Σ+

2.0

1.0

6 0

15

2000

10

Counts / 3840S 20

2500

40

b)

3000

60

80

Wavelength (Å)

100

Channel 120

Raman Lyα Solar Reflection CI

450 2000

2200

2400

H2Ly

2600 2800 Wavelength (Å)

H2Wr

Fig. 84.17 Far ultraviolet nightglow spectrum of Venus

obtained with the Pioneer Venus orbiter ultraviolet spectrometer. The predicted responses for three different band systems are also shown. After [84.93]

of auroral and dayglow emissions as measured by the Cassini UVIS as the spacecraft flew by Jupiter is shown in Fig. 84.18. (A. I. F. Stewart, private communication, 2004). A discussion of terrestrial airglow and auroral emissions can be found in Rees [84.44]. Meier [84.94] has reviewed spectroscopy and remote sensing of the terrestrial ultraviolet emissions. The airglows of Mars and Venus have been discussed by Barth [84.19], by Fox [84.95] and by Paxton and Anderson [84.40]. Airglow and auroral emissions on the outer planets have been reviewed by Atreya et al. [84.96], Atreya [84.2], and

300 H2 a–b

150 HI Rydberg

0

600

800

1000

1200

1400 1600 Wavelength (Å)

Strobel [84.4]. Airglow in the atmospheres of the planets has been reviewed by by Slanger and Wolven [84.97].

84.7 Planetary Escape Escape of species from atmospheres can occur by thermal and nonthermal mechanisms. Thermal processes include Jeans escape and hydrodynamic escape. Jeans escape is essentially evaporation of the energetic tail of the Maxwell–Boltzmann distribution, while hydrodynamic escape is a large-scale “blow-off” of the

atmosphere that occurs when the average molecular velocity is near or above the escape velocity. Although Jeans escape still occurs for light species in the thermospheres of small planets, hydrodynamic escape is thought to have occurred only in the early history of the terrestrial planets when the solar flux in the UV was

Part G 84.7

5

0 1800

1287

1288

Part G

Applications

Cassini uvis – Jupiter flyby 30 Dec 00

k Rayleigh/Å

1000

Atomic hydrogen Lyman series beta alpha

Solar lines CIV

Molecular hydrogen

HeII CI

SiII

Lyman and Werner bands

0.100

0.010

Part G 84.7

0.001

Heavy: Northern aurora Light: Equatorial airglow 1000

1200

1400

1600

1800 Wavelength (Å)

Fig. 84.18 Spectra of the northern aurora (heavy line) and the equatorial airglow (light line) taken by the UVIS spectrometer the Cassini spacecraft on closest approach on 30 December 2000. In both spectra, the Lyman and Werner bands of molecular hydrogen are prominent. They are excited by photoelectron impact in the airglow, and by primary and secondary auroral electrons. Locations of some of the brighter bands are indicated by dotted lines. Also shown are the Lyman-alpha and -beta lines of atomic hydrogen. At wavelengths longer than 1550 Å, the airglow spectrum is swamped by reflected UV sunlight; some solar lines are indicated. At shorter wavelengths, UV sunlight is absorbed by methane and acetylene, rather than reflected. The solar signal is much weaker in the auroral spectrum than in the equatorial, due to the higher angles of incidence and emission in the former [84.99]

higher. Nonthermal escape mechanisms, which dominate for heavy species on smaller bodies such as Mars and Titan, and for all species on Venus and Earth, include both photochemical and mechanical processes. A pedagogical discussion of escape processes can be found in Chamberlain and Hunten [84.98] Because of the exponential rate of change of density with altitude in atmospheres, escape is sometimes assumed to occur only at and above the exobase. The exobase is mathematically defined as the altitude where the mean free path l = (nσ c )−1 (where σ c is the collision cross section), is equal to the atmospheric scale height. The probability that a particle, moving upward from the exobase with sufficient velocity will actually escape without suffering another collision is 1/ e. The condition l = H therefore reduces to n Hσ = 1 or, equivalently, to N = (σ c )−1 , where N is the column density. Since a typical collision cross section is about 3 × 10−15 cm2 ,

the exobase is located near the altitude above which the column density is about 3.3 × 1014 cm−3 , although the collision cross section and thus the location of the exobase is different for different escaping species. Whether the trajectory of a particle moving upward at the exobase is ballistic (bound) or escaping (free) is determined by its total energy E, which is the sum of its kinetic and potential energies: rc 1 2 mG M E = mvc + dr , (84.90) 2 r2 ∞

where the symbols have the same meaning as in equation (84.2), and the subscript c refers to the critical level or exobase. If E < 0, the particle is bound. Expression (84.90) reduces to 1 (84.91) E = mvc2 − mgcrc 2

Aeronomy

where gc is the gravitational acceleration at the exobase. The escape velocity at the exobase, vesc , is then defined by the condition vesc = (2gcrc )1/2 . Particles with velocities greater than the escape velocity are assumed to escape if their velocity vector is oriented in the upward hemisphere and if they undergo no further collisions. The radius, gravitational acceleration, escape velocities and scale height at the equatorial exobases of the planets are given in Table 84.10. In the Jeans process, escape occurs when particles in the high energy tail of the Maxwellian distribution attain the escape velocity. The escape flux, ΦJ is given by ncu (84.92) ΦJ = √ (1 + λc ) exp(−λc ) , 2 π 1/2 where u = (2kB T/m) is the modal velocity and λ is the gravitational potential energy in units of kB T mgr r G Mm = = . λ= (84.93) rkB T kB T H

Planet

rc (km)

gca (cm s−2 )

a vesc,c (km s−1 )

a,b Havg (km)

Mercuryc Venusd Earthe Marsf Jupiterg Saturnh Uranusi Neptunej Titank Tritonl

2439 6250 6878 3593 73 000 67 000 31 800 27 300 4175 2222

378 831 842 333 2236 731 561 919 51.4 28.9

4.29 10.2 10.8 4.89 57.2 31.3 18.9 22.4 2.07 1.13

– 17 71 17 250 910 66 250 116 140

a

Values given are those at the equatorial exobase, and assume that the thermosphere co-rotates with the planet. b Average value computed from H avg = kB T/m avg g, but does not represent the local pressure scale height, except for cases where there is one major constituent. c Exobase is at the surface. d Model from Hedin et al. [84.3] for F ◦ 10.7 =150, 45 solar zenith angle. e MSIS model for F ◦ 10.7 =150, equator, 45 Solar Zenith Angle [84.17]. f Model from Nier and McElroy [84.8], and pertains to low solar activity conditions. g Model from [84.2, 14]. h Model from Atreya [84.2]. i Model from [84.55]. j Model from [84.56]. k Model from Strobel et al. [84.10]. l Model from Krasnopolsky et al. [84.15]

1289

Sometimes a correction factor is applied to the expression for the escape flux to account for the suppression of the tail of the distribution due to the escape of the energetic particles [84.100, 101]. Photochemical processes that produce energetic fragments include photodissociation and photodissociative ionization, photoelectron impact dissociation and dissociative ionization, as well as exothermic chemical reactions. The most important example of the latter are dissociative recombination reactions, which are very exothermic and tend to produce neutral fragments with large kinetic energies. Charge transfer processes such as H+∗ + O → O+ + H∗

(84.94)

can produce fast neutrals if the ion temperature is larger than the neutral temperature, as is usually the case near the exobase of a planet. In modeling these processes, the kinetic energy distribution of the product species is important, as well as the cross sections or reaction rates. Physical or collisional escape mechanisms include sputtering and “knock-on.” Sputtering can occur when a heavy ion picked up by the solar wind collides with an atmospheric species near or above the exobase, and in the process produces a “back-splash” in which the accelerated neutral may be ejected from the atmosphere. In knock-on, hot atmospheric neutral species, such as O atoms produced in exothermic chemical reactions near the exobase can collide with a lighter species, such as H, imparting sufficient kinetic energy to allow it to escape. Modeling these processes requires knowledge of the ion–neutral or neutral–neutral collision cross sections. The escape rate of a light species from a planetary atmosphere may be controlled by diffusion of the species from the lower atmosphere to the exobase, rather than by the escape process itself [84.102]. The limiting upward flux, φl of a species i with mixing ratio f i can be estimated as φl ≈ bi f i /Ha ,

(84.95)

where Ha is the average scale height of the atmosphere and bi is the binary collision parameter introduced in Sect. 84.2.2. Equation (84.95) above is usually evaluated at the homopause, with the mixing ratio taken from a suitable altitude in the middle atmosphere, but above the cold trap (where the species condenses), if one exists. The limiting flux obtains if and only if the mixing ratio is constant with altitude. The effect of photochemistry can be accounted for if all chemical forms of the species are considered in the calculation of f i .

Part G 84.7

Table 84.10 Exobase properties of the planets

84.7 Planetary Escape

1290

Part G

5000

Applications

Fig. 84.19 Spectrum of the oxygen green line taken on

Relative intensity Venus dark – Venus bright HIRES UT 20 Nov 99 15:59 – 16:08 500s Airmass = 1.7 Velocity = 12.45 km/s

4000 3000

Venus A = 4191 Γ = 0.103 Å T = 5577.574 Å

Terrestrial A = 3094 Γ = 0.103 Å T = 5577.344 Å

2000 1000 0

5576

5576.5

5577

5577.6

5578 5577.5 Wavelength (Å)

the nightside of Venus taken by the Keck/HIRES on 20 November 1999. The individual terrestrial and Dopplershifted Venusian components are shown by the dashed lines (from Slanger et al. 84.103)

Even if energetic particles released at the exobase of a planet do not have enough energy to escape, they may travel to great heights along ballistic orbits before falling back to the atmosphere. These particles are said to form a hot atom “corona”. Hot H and O coronas have been found to surround the Earth and Venus, and have been predicted for Mars (Fig. 84.19). Reviews of the H and O coronas of Venus have been presented by Fox and Bougher [84.79] and by Nagy et al. [84.104]. See Chamberlain and Hunten [84.98] for a detailed discussion of planetary coronal population processes.

Part G 84

References 84.1

84.2

84.3 84.4

84.5

84.6

84.7 84.8 84.9 84.10 84.11 84.12

U. von Zahn, S. Kumar, J. Niemann, R. Prinn: Composition of the Venus atmosphere. In: Venus, ed. by D. M. Hunten, L. Colin, T. M. Donahue, V. I. Moroz (Univ. Arizona Press, Tucson 1983) S. K. Atreya: Atmospheres and Ionospheres of the Outer Planets and Their Satellites (Springer, New York 1986) A. E. Hedin, H. B. Neimann, W. T. Kasprzak, A. Seiff: J. Geophys. Res. 88, 73 (1983) D. F. Strobel, R. V. Yelle, D. E. Shemansky, S. K. Atreya: The upper atmosphere of Uranus. In: Uranus, ed. by J. T. Bergstralh, E. D. Miner, M. S. Matthews (Univ. Arizona Press, Tucson 1991) S. K. Atreya, B. R. Sandel, P. N. Romani: Photochemical and vertical mixing. In: Uranus, ed. by J. T. Bergstralh, E. D. Miner, M. S. Matthews (Univ. Arizona Press, Tucson 1991) The U. S. Standard Atmosphere (National Oceanic and Atmospheric Administration, U.S. Government Printing Office 1976) R. V. Yelle, F. Herbert, B. R. Sandel, R. J. Vervack, T. M. Wentzel: Icarus 104, 38 (1993) A. O. Nier, M. B. McElroy: J. Geophys. Res. 82, 4341 (1977) R. J. Vervack Jr., B. R. Sandel, D. F. Strobel: Icarus 170, 91 (2004) D. F. Strobel, M. E. Summers, X. Zhu: Icarus 100, 512 (1992) Y. L. Yung, M. Allen, J. P. Pinto: Astrophys. J. Suppl. 55, 465 (1984) R. V. Yelle: private communication

84.13

84.14

84.15 84.16

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R. V. Yelle, L. A. Young, R. J. Vervack, R. Young, L. Pfister, B. R. Sandel: J. Geophys. Res. 101, 2149 (1996) Y. H. Kim: The Jovian Ionosphere. Ph.D. Thesis (State University of New York at Stony Brook, New York 1991) V. A. Krasnopolsky, B. R. Sandel, F. Herbert, R. J. Vervack: J. Geophys. Res. 98, 3065 (1993) D. F. Strobel, M. E. Summers: Triton’s upper atmosphere and ionosphere. In: Neptune, ed. by D. Cruikshank, M. S. Matthews (Univ. Arizona Press, Tucson 1996) A. E. Hedin: J. Geophys. Res. 96, 1159–1172 (1991) C. D. Keeling, T. P. Whorf: Atmospheric CO2 records from sites in the SIO sampling network. In: Trends: A Compendium of Data on Global Change. Carbon Dioxide Information Analysis Center, ed. by T. A. Boden, D. P. Kaiser, R. J. Sepanski, W. F. Stoss (Oak Ridge National Laboratory, U.S. Department of Energy, Oak Ridge, Tenn., U.S.A. 2004) C. A. Barth, A. I. F. Stewart, S. W. Bougher, D. M. Hunten, S. J. Bauer, A. F. Nagy: Aeronomy of the current Martian atmosphere. In: Mars, ed. by H. Kiefer, B. M. Jakosky, C. W. Snyder, M. S. Matthews (Univ. Arizona Press, Tucson 1992) J. S. Lewis, R. G. Prinn: Planets and their atmospheres: Origin and Evolution (Academic Press, Orlando 1984) T. M. Donahue, R. R. Hodges Jr.: Geophys. Res. Lett. 20, 591 (1993) V. A. Krasnopolsky, G. R. Gladstone: J. Geophys. Res. 101, 15,765 (1996)

Aeronomy

84.23 84.24 84.25

84.26 84.27 84.28 84.29

84.30 84.31 84.32

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84.35 84.36 84.37 84.38 84.39

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84.41 84.42 84.43 84.44 84.45 84.46

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84.56 84.57 84.58 84.59

84.60 84.61

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D. J. Strickland, R. E. Daniell, B. Basu, J. R. Jasperse: J. Geophys. Res. 98, 21533 (1993) S. C. Solomon: Geophys. Res. Lett. 20, 185 (1993) S. J. Bauer: Physics of Planetary Ionospheres (Springer, New York 1973) S. G. Lias, J. E. Bartmess, J. F. Liebman, J. L. Holmes, R. D. Levin, W. G. Mallard: Gas-phase ion, and neutral thermochemistry, J. Phys. Chem. Ref. Data 17, Suppl. 1 (1988) F. L. Nesbitt, G. Marston, L. J. Stief, M. A. Wickramaaratchi, W. Tao, R. B. Klemm: J. Phys. Chem. 95, 7613 (1991) J. L. Fox, A. Dalgarno: J. Geophys. Res. 84, 7315 (1979) G. R. Gladstone: private communication (2004) J. I. Moses, S. F. Bass: J. Geophys. Res. 105, 7013 (2000) F. Herbert, B. R. Sandel, R. V. Yelle, J. B. Holberg, A. L. Broadfoot, D. E. Shemansky: J. Geophys. Res. 92, 15093 (1987) J. R. Lyons, private communication M. Summers, D. F. Strobel: private communication (2005) D. F. Strobel, B. C. Wolven: Astrophys. Space Sci. 277, 271 (2001) T. Su, M. T. Bowers: Classical ion-molecule collision theory. In: Gas Phase Ion Chemistry, Vol. 1, ed. by M. T Bowers (Academic, New York 1979) V. G. Anicich: Astrophys. J. Suppl. Ser. 84, 215–315 (1993) Y. Ikezoe, S. Matsuoka, M. Takabe, A. Viggiano: Gas Phase Ion-Molecule Rate Constants Through 1986 (Maruzen Co., Tokyo 1986) D. R. Bates, A. Dalgarno: Electronic recombination. In: Atomic and Molecular Processes, ed. by D. R. Bates (Academic, New York 1962) pp. 245– 271 M. H. G. Zhang, J. G. Luhmann, A. J. Kliore: J. Geophys. Res. 95, 17095 (1990) D. P. Hinson and the Mars Radio Science team: Public access to MGS RS Standard electron density profiles, http://nova.stanford.edu/projects/mgs/ eds-public.html (2003) C. Y. Johnson: J. Geophys. Res. 71, 330 (1966) W. B. Hanson, S. Sanatani, D. R. Zuccaro: J. Geophys. Res. 82, 4351 (1977) A. N. Maurelis, T. E. Craveus: Icarus 154, 350 (2001) J. R. Lyons: Nature 267, 648 (1995) A. D. Richmond: The ionosphere. In: The Solar Wind and the Earth, ed. by S.-I. Akasofu, Y. Kamide (Terra Scientific Publishing Co., Tokyo 1987) A. Dalgarno, M. R. C. McDowell, A. Williams: Phil. Trans. Roy. Soc. London, Ser. A 250, 411 (1958) R. W. Schunk, A. F. Nagy: Rev. Geophys. Space Phys. 18, 813 (1980) P. M. Banks, G. Kockarts: Aeronomy (Academic, New York 1973)

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V. A. Krasnopolsky, P. D. Feldman: Science 294, 1914 (2001) W. B. DeMore: Icarus 51, 199 (1982) D. F. Strobel: The Photochemistry of Atmospheres: Earth, the Other Planets, and Comets, ed. by J. S. Levine (Academic, New York 1985) H. B. Niemann: J. Geophys. Res. 103, 22,831 (1998) K. Lodders, B. Fegley: The Planetary Scientist’s Companion (Oxford, New York 1998) R. Courtin, D. Gautier, D. Strobel: Icarus 123, 37 (1996) D. M. Hunten, M. G. Tomasko, F. M. Flasar, J. R. E. Samuelson, D. F. Strobel, D. J. Stevenson: Titan. In: Saturn, ed. by T. Gehrels, M. S. Matthews (Univ. Arizona Press, Tuscon 1984) J. L. Elliot, M. J. Person, S. Qu: The Astronomical Journal 126, 1041 (2003) D. F. Strobel, B. C. Wolven: Astrophysics and Space Science 277, 271 (2001) J. Saur, D. F. Strobel, F. M. Neubauer: J. Geophys. Res. 103, 19,947 (1998) A. J. Kliore, D. P. Hinson. F. M. Flasar, A. F. Nagy, T. E. Cravens: Science 277, 355 (1997) D. M. Hunten, T. H. Morgan, D. E. Shemansky: The Mercury atmosphere. In: Mercury, ed. by F. Vilas, C. R. Chapman, M. S. Matthews (Univ. Arizona Press, Tucson 1988) X. Zhu, M. E. Summers, M. H. Stevens: Icarus 120, 266 (1996) L. M. Lara, W. H. Ip, R. Rodrigo: Icarus 130, 16 (1997) W. K. Tobiska: J. Atmos. Terr. Phys. 53, 1005 (1991) D. Lummerzheim, M. H. Rees, H. R. Anderson: Planet. Space Sci. 37, 109 (1989) L. Herzberg: Solar optical radiation and its role in upper atmospheric processes. In: Physics of the Earth’s upper atmosphere, ed. by C. Hines, I. Paghis, T. R. Hartz, J. A. Fejer (Prentice-Hall, Englewood Cliffs, NJ 1965) L. J. Paxton, D. E. Anderson: Far ultraviolet remote sensing of Venus and Mars. In: Venus and Mars: Atmospheres, Ionospheres and Solar Wind Interactions, Geophysical Monograph, Vol. 66, ed. by J. G. Luhmann, M. Tatrallay, R. O. Pepin (American Geophysical Union, Washington, D.C. 1992) pp. 113– 190 J. M. Valentine, S. C. Curran: Rep. Prog. Phys. 21, 1 (1958) C. B. Opal, W. K. Peterson, E. C. Beaty: J. Chem. Phys. 55, 4100 (1971) W. E. Swartz, J. S. Nisbet, A. E. S. Green: J. Geophys. Res. 74, 6415 (1971) M. H. Rees: Physics and Chemistry of the Upper Atmosphere (Cambridge Univ. Press, Cambridge 1989) A. F. Nagy, P. M. Banks: J. Geophys. Res. 75, 6260 (1970) H. S. Porter, F. Varosi, H. G. Mayr: J. Geophys. Res. 92, 5933 (1987)

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R. W. Schunk, A. F. Nagy: Ionospheres: Physics, Plasma Physics and Chemistry (Cambridge Univ. Press, Cambridge 2000) R. G. Roble, E. C. Ridley, R. E. Dickinson: J. Geophys. Res. 92, 8745–8758 (1987) J. L. Fox: Planet. Space Sci. 37, 36 (1988) J. L. Fox, P. Zhou, S. W. Bougher: Adv. Space Res. 17, (11)203–(11)218 (1995) J. H. Waite, T. E. Cravens, J. Kozyra, A. F. Nagy, S. K. Atreya, R. H. Chen: J. Geophys. Res. 88, 6143 (1983) R. G. Roble: Rev. Geophys. 21, 217–233 (1983) J. L. Fox, S. W. Bougher: Space Sci. Rev. 55, 357 (1991) S. W. Bougher, D. M. Hunten, R. G. Roble: J. Geophys. Res. 99, 14609 (1994) R. P. Wintersteiner, R. H. Picard, R. D. Sharma, J. R. Winick, R. A. Joseph: J. Geophys. Res. 97, 18083 (1992) D. Bilitza, W. R. Hoegy: Adv. Space Res. 10, (8)81– (8)90 (1990) K. L. Miller, W. C. Knudsen, K. Spenner, R. C. Whitten, V. Novak: J. Geophys. Res. 85, 7759 (1980) A. Roboz, A. F. Nagy: J. Geophys. Res. 99, 2087 (1994) R. C. Whitten, I. G. Poppoff: Fundamentals of Aeronomy (Wiley, New York 1971) R. F. Theis, L. H. Brace, H. G. Mayr: J. Geophys. Res. 85, 7787 (1980) W. B. Hanson, G. P. Mantas: J. Geophys. Res. 93, 7538 (1988) A. L. Broadfoot, B. R. Sandel, D. Knecht, R. Viereck, E. Murad: Appl. Opt. 31, 3083 (1992) A. I. F. Stewart, J.-C. Gerard, D. W. Rusch, S. W. Bougher: J. Geophys. Res. 85, 7861 (1980) J. L. Bertaux, F. Leblanc, S. V. Perrier, E. Quemerais, O. Korablev, E. Dimarellis, A. Reberac, F. Forget,

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P. C. Simon, B. Sandel: Nightglow in the upper atmosphere of Mars and implication for atmospheric transport, Science 307(5709), 566–569 (2005) J. L. Bertaux, F. Leblanc, O. Witasse, E. Quemerais, J. Lilenstein, S. A. Stern, B. Sandel, O. Korablev: Nature, 435(7043), 790–794 (2005) C. A. Barth, C. W. Hord, A. I. Stewart, A. L. Lane: Science 175, 309 (1972) A. I. F. Stewart, C. A. Barth: Science 205, 59 (1979) R. R. Meier: Space Sci. Rev. 58, 1–187 (1991) J. L. Fox: Airglow and aurora in the atmospheres of Venus and Mars. In: Venus and Mars: Atmospheres, Ionospheres and Solar Wind Interactions, Geophysical Monograph, Vol. 66, ed. by J. G. Luhmann, M. Tatrallay, R. O. Pepin (American Geophysical Union, Washington, D.C. 1992) pp. 191–222 S. K. Atreya, J. H. Waite Jr., T. M. Donahue, A. F. Nagy, J. C. McConnell: Theory, measurements, and models of the upper atmosphere and ionosphere of Saturn. In: Saturn, ed. by T. Gehrels, M. S. Matthews (Univ. Arizona Press, Tucson 1984) T. G. Slanger, B. C. Wolven: Airglow processes in planetary atmospheres. In: Atmospheres in the Solar System: Comparative Aeronomy, Geophysical Monograph 130 (AGU, Washington, D.C. 2002) J. W. Chamberlain, D. M. Hunten: Theory of Planetary Atmospheres (Academic, New York 1987) A. I. F. Stewart: private communication (2004) J. W. Chamberlain: Planet. Space Sci. 11, 901 (1963) D. M. Hunten: Planet. Space Sci. 30, 773 (1982) D. M. Hunten: J. Atmos. Sci. 30, 1481 (1973) T. G. Slanger, D. L. Huestis, P. C. Cosby, N. Chanover: The Venus nightglow: Ground based observations and chemical mechanisms, Icarus (2004) in press A. F. Nagy, J. Kim, T. E. Cravens: Ann. Geophys. 8, 251 (1990)

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85. Applications of Atomic and Molecular Physics to Global Change

Applications o

While there has been a general understanding and appreciation of the science involved in both global warming and stratospheric ozone depletion by atmospheric scientists for some time, detailed understanding and rigorous proof has often been lacking. Over the last ten years, there have been many advances made in filling in the details and there will continue to be rapid advances in the future. This means that any article or book discussing this topic becomes out of date as soon as it is written. Nevertheless several recent references on these topics are recommended [85.1–3]. Atomic and molecular structure and spectroscopy, as well as collision processes involving atoms, molecules, ions and electrons, are important to the study of all planetary atmospheres. For additional information on this topic, see Chapt. 84 in this volume.

85.2 Atmospheric Models and Data Needs ................................... 1294 85.2.1 Modeling the Thermosphere and Ionosphere ........................ 1294 85.2.2 Heating and Cooling Processes .... 1295 85.2.3 Atomic and Molecular Data Needs1295 85.3 Tropospheric Warming/ Upper Atmosphere Cooling ................... 1295 85.3.1 Incoming and Outgoing Energy Fluxes ........................... 1295 85.3.2 Tropospheric “Global” Warming .. 1296 85.3.3 Upper Atmosphere Cooling ......... 1297 85.4 Stratospheric Ozone ............................. 1298 85.4.1 Production and Destruction ........ 1298 85.4.2 The Antarctic Ozone Hole ............ 1299 85.4.3 Arctic Ozone Loss ....................... 1300 85.4.4 Global Ozone Depletion.............. 1300 85.5 Atmospheric Measurements .................. 1300 References .................................................. 1301

85.1 Overview 85.1.1 Global Change Issues Over the last several decades there has been increasing concern about the global environment and the effect of human perturbations on it. This whole area, which involves a wide range of scientific disciplines, has become known as Global Change. Knowledge of processes taking place in the atmosphere, oceans, land masses, and plant and animal populations, as well as the interactions between these various earth-system components is essential to an overall understanding of global change – both natural and human-induced. The processes of atomic and molecular physics find greatest application in the area of atmospheric global change. The two major issues which have received significant attention in both the media and the scientific literature are: (1) global warming, due to the buildup of infrared-active gases; and (2) stratospheric ozone de-

pletion due to an enhancement of destructive catalytic cycles. Although both of these problems are thought to be caused by atmospheric pollutants due to industrialized human society, the general problem of air pollution and its direct effects on plant and animal populations will not be addressed here.

85.1.2 Structure of the Earth’s Atmosphere The vertical temperature structure of the earth’s atmosphere shown in Fig. 85.1 provides an important nomenclature that is widely used [85.4]. The atmosphere is divided into regions called “-spheres”, in which the sign of the temperature gradient with respect to altitude, dT/ dz, is constant. The regions in which the temperature gradient changes sign are called “-pauses”. The precise altitude pertaining to each of these regions can vary depending upon latitude and the time of year.

Part G 85

85.1 Overview............................................. 1293 85.1.1 Global Change Issues ................. 1293

85.1.2 Structure of the Earth’s Atmosphere ............................. 1293

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140

Altitude (km) Ionosphere

120

Thermosphere

100 Mesopause 80 60

Mesosphere Stratopause

40 Stratosphere 20 Troposphere 0 100

150

200

Equatorial tropopause Polar tropopause 250

300

350 400 Temperature (K)

Fig. 85.1 Vertical temperature profile of the atmosphere

Part G 85.2

In the troposphere, covering the range from 0 to ≈ 15 km above the earth, the temperature steadily decreases with altitude. This is the most complex region of the atmosphere, as it interacts directly with plant and animal life, land masses and the oceans. It is the region in which weather occurs. The change in sign of the temperature gradient at the tropopause to a positive dT/ dz in the stratosphere is due to heating by absorption of solar ultraviolet radiation which photodissociates O2 and O3 . The stratosphere, extending from ≈ 15 to 50 km above the earth, contains the ozone layer which shields the earth’s surface from harmful ultraviolet radiation in the range of 280–320 nm. At the stratopause, the heating processes have become too weak to compete with the cooling pro-

cesses, and throughout the mesosphere, approximately 50–85 km, the temperature again decreases with increasing altitude. Cooling processes, which will be discussed in Sect. 85.2.2, involve collisional excitation of molecular vibrational modes which decay by radiating to space. The coldest temperatures in the atmosphere are found at the mesopause, where the temperature gradient once again becomes positive. From approximately 70 km upward, a very diffuse plasma called the ionosphere exists due to photoionization of atoms and molecules by short wavelength (UV and EUV) solar radiation. Throughout the thermosphere, which extends from approximately 90 km upward, heating occurs because the atmosphere has become so thin that there are very few collisions and thus inefficient equilibration of the highly translationally excited atoms and ions with the molecular species which can radiate in the infrared. This “bottleneck” for energy loss causes increased heating. In the thermosphere and ionosphere, the thermal inertia is very small and there are huge temperature variations, both diurnally, and with respect to solar activity. The densities are low enough in the thermosphere, ionosphere, and mesosphere, that the primary processes determining the chemical and physical characteristics of these regions are two-body processes and “halfcollision events” discussed elsewhere in this volume: dissociative recombination, photoionization, photodissociation, charge transfer, and collisional excitation of molecular rotation and vibration. As the altitude decreases, the density increases. Then three-body interactions, interactions on surfaces (of aerosols and ices), and complex chemical cycles together with dynamical effects such as winds determine the chemical and physical characteristics of the stratosphere and troposphere.

85.2 Atmospheric Models and Data Needs While models are absolutely essential to the study of any system as complex as the earth’s atmosphere, they play a particularly fundamental role in exploring global change issues. Models not only provide predictions of future changes, but also allow exploration of sensitivities to particular parameters. Comparing the results of a model with observations ultimately tests and challenges scientific understanding. Of critical importance is the atomic and molecular data which goes into the models.

Generally, atmospheric models become increasingly complex as altitude decreases. General Circulation Models (GCMs), incorporating thousands of chemical reactions, global wind patterns, and abundances of large numbers of trace species, require supercomputers in order to model aspects of the troposphere and stratosphere. Tropospheric chemistry and transport models, such as GEOS-CHEM [85.5], model the sources, evolution, transport, and sinks of pollution, as well as the oxidative capacity of the troposphere.

Applications of Atomic and Molecular Physics to Global Change

85.2.1 Modeling the Thermosphere and Ionosphere

4 4 e + N+ 2 → N( S) + N( S) + 6 eV .

85.2.2 Heating and Cooling Processes In the upper atmosphere, heating occurs through absorption of short wavelength solar radiation to produce ionization and dissociation, and is mediated by collisions between electrons, ions, and neutrals. Ions and electrons are created during the daytime and to a great extent disappear during the night with the absence of solar radiation. Processes such as dissociative recombination, the primary electron loss mechanism, heat the gas: (85.1)

(85.2)

Cooling takes place when the kinetic energy of the gas is transformed through collisions into internal energy which can then be radiated away. The primary coolant above ≈ 200 km is the fine structure transition hν of atomic oxygen, O(3 P1 ) → O(3 P2 ), which is excited by thermal collisions and radiates at 63 µm. From approximately 120 km to 200 km, the fundamental band of NO, v = 1 → v = 0, which is excited by collisions with atomic oxygen and radiates at 5.3 µm, dominates the cooling. Below 120 km and throughout the mesosphere and stratosphere, the primary coolant is the ν2 band of CO2 radiating at 15 µm. This transition is excited by collisions of CO2 and atomic oxygen. Cooling throughout most of the atmosphere is accomplished through trace species because the major molecular species, N2 and O2 , are not infrared active.

85.2.3 Atomic and Molecular Data Needs Knowledge of rate coefficients for ion-neutral and neutral-neutral reactions as a function of vibrational and rotational excitation of the reactants is becoming increasingly important, as there is recent evidence of more internal excitation of molecular species than had previously been thought [85.6]. Accurate photoabsorption, photodissociation and photoionization cross sections as a function of wavelength for all the relevant species are important parameters determining the reliability and ultimate accuracy of an atmospheric model. Compilations of data, such as that by Conway [85.7] and Kirby et al. [85.8] are very useful, but can become rapidly outdated. The Smithsonian Astrophysical Observatory maintains the world standard database, HITRAN, for molecular line parameters and absorption cross sections from the microwave through the ultraviolet for analysis of atmospheric spectra [85.9]. Discussions of the needs for atomic and molecular data in the context of space astronomy, but including applications to atmospheric physics, can be found in a book edited by Smith and Wiese [85.10].

85.3 Tropospheric Warming/Upper Atmosphere Cooling 85.3.1 Incoming and Outgoing Energy Fluxes The overall temperature of a planet is determined by a balance between incoming and outgoing energy fluxes.

1295

In a steady state, the planet must radiate as much energy as it absorbs from the sun. The Earth, radiating as a black-body at an effective temperature TE , obeys the Stefan–Boltzmann law in which the energy emitted is expressed as σTE4 4πRE2 , with σ the Stefan–Boltzmann

Part G 85.3

The data necessary for modeling of the thermosphere and ionosphere are described here. The primary components of such a model include: a solar spectrum of photon fluxes as a function of wavelength, concentrations of neutral species, photoabsorption, photodissociation and photoionization cross sections as a function of wavelength. The computer code brings all these elements together, calculating opacity as the solar radiation propagates downward through the atmosphere, keeping track of ion production and electron production. Additional steps are required to calculate the abundances of trace species such as NO+ , necessitating the inclusion of all relevant ion-neutral reactions. Electron energy degradation can be tracked by including inelastic collisions of electrons with ions, atoms, and molecules. The primary neutral species are N2 , O2 , O, and He. Below about 100 km, the atmosphere is fully mixed by turbulence in the ratio 78% N2 , 21% O2 and 1% trace species such as O3 and CO2 . Above 100 km, turbulence dies out and the atmospheric species are in diffusive equilibrium, distributed by their molecular weight, with atomic oxygen dominating above ≈ 150 km and He + dominating much higher. The major ions are N+ 2 , O2 , + + O and NO .

3 3 e + O+ 2 → O( P) + O( P) + 7 eV ,

85.3 Tropospheric Warming/Upper Atmosphere Cooling

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constant, and RE the radius of the earth. An equation expressing the equality of energy absorbed and energy emitted can be written [85.11] as Fs πRE2 (1 − A) = σTE4 4πRE2 ,

(85.3)

Part G 85.3

where A is the albedo of the earth (the fraction of solar radiation reflected from, rather than absorbed by, the Earth), Fs is the solar flux at the edge of the earth’s atmosphere, and πRE2 is the Earth’s area normal to the solar flux. Solving this equation for TE , one obtains TE u 255 K (−18 ◦ C). The sun, which has a surface temperature of approximately 6000 K, emits most of its radiation in the 0.2–4.0 µm region of the spectrum (200–4000 nm). The upper atmosphere of the Earth (thermosphere, ionosphere, mesosphere, and stratosphere) absorbs all the solar radiation shortward of 320 nm. The atmosphere of the earth absorbs only weakly in the visible region of the spectrum where the solar flux peaks. The Earth, with an effective radiating temperature of 255 K, emits mainly long-wavelength radiation in the 4–100 µm region. Molecules naturally present in the atmosphere in trace amounts, such as carbon dioxide, water and methane, absorb strongly in this wavelength region [85.12]. Radiation coming from the Earth is thus absorbed, reradiated back to the surface, and thermalized through collisions with the ambient gas. This trapping of the radiation produces an additional warming of 33 K. Thus the mean surface temperature of the Earth is 288 K, not 255 K as found for TE above. This effect of the Earth’s atmosphere is known as the greenhouse effect. The greenhouse effect is what makes Earth habitable for life as we know it. Gases, both natural and man-made, which absorb strongly in the 4–100 µm region, are known collectively as greenhouse gases.

85.3.2 Tropospheric “Global” Warming According to a 2000 National Research Council Report, “the global-mean temperature at the earth’s surface is estimated to have risen by 0.25 to 0.4 ◦ C during the past 20 years” [85.13]. The Intergovernmental Panel on Climate Change (IPCC) has also concluded that global surface temperatures have increased and that “there is new and stronger evidence that most of the warming over the last 50 years is attributable to human activities” [85.14]. The Arctic region has warmed by an estimated 1 ◦ C in the past two decades, leading to substantial changes in the cryosphere [85.15]. Antarctic sea ice was stable from

1840 to 1950, but has since declined sharply. Sea ice extent shows a 20% decline since about 1950 [85.16,17]. From air bubbles trapped at different depths in polar ice, it is possible to determine carbon dioxide and methane concentrations several thousand years ago. Over the last two hundred years, CO2 levels have increased by 20%, from 280 to 330 ppm. Over the next century the total amount of CO2 in the atmosphere since 1900 is expected to double to as much as 600 ppm [85.18]. This increase is due primarily to the burning of fossil fuels. Although methane is present at levels several orders of magnitude less than CO2 , it is increasing much more rapidly. Methane concentrations have more than doubled over the last two hundred years due to industrial processes, fuels, and agriculture [85.18]. The man-made chlorofluorocarbons (CFCs), which have been widely used as refrigerants and in industry, have been increasing in the atmosphere at a rate of over 5% per year since the 1970s. Only recently has there been an indication that this trend is slowing down [85.19]. Ozone, which is a primary component of chemical smog, is a pollutant when it occurs in the troposphere and an effective greenhouse gas. It has been increasing worldwide also. This buildup of CO2 , CH4 , CFCs and tropospheric O3 causes a problem. In much of the spectral region from 5–100 µm, there is 100% absorption of radiation by the atmosphere – due mainly to naturally occurring water vapor. There is, however, a region of rather weak absorption, from ≈ 7–15 µm, known as the “atmospheric window”. Increased concentrations of the greenhouse gases strengthen the absorption in this region, tending to “close” this window, thus increasing the infrared opacity of the atmosphere. The increased opacity causes an immediate decrease in the thermal radiation from the planet-atmosphere system, forcing the temperature to rise until the energy balance is restored [85.20]. It is difficult to prove that the buildup of greenhouse gases is the cause of the observed temperature rise. Other possible causes include slight changes in solar activity and irradiance, and changes in ocean currents, which may have a profound effect on global temperature and climate. These are areas of active research. Given the increase in concentrations of greenhouse gases that has occurred and is predicted to continue, the change in radiative heating of the troposphere can be calculated. Models generally predict an increase in tropospheric temperatures ranging from 1.5 to 4.5 ◦ C, upon doubling the CO2 concentrations over the next cen-

Applications of Atomic and Molecular Physics to Global Change

tury. The 3 ◦ C range in temperature is due to the ways that different models incorporate climate feedbacks. Climate feedbacks include water vapor, snow and sea ice, and clouds. Rising temperatures increase the concentration of water vapor, which is itself a greenhouse gas, producing further warming. Rising temperatures reduce the extent of reflective snow and ice, thus reducing the Earth’s albedo. This leads to increased absorption of solar radiation, further increasing temperatures. Clouds both contribute to the albedo, thereby reducing the solar flux reaching the Earth, and absorb infrared radiation causing temperatures to rise. The modeling of clouds and their radiative properties is very difficult, and is one of the largest sources of uncertainty in the climate models. Understanding the role that the ocean, with its giant heat capacity, plays in global warming, and identifying and quantifying the various interactions occurring at the ocean-atmosphere interface, are vital areas of research which will affect the size of the predicted temperature increase. At present, there are few obvious opportunities for traditional atomic and molecular physics to play a significant part in global-warming research.

85.3.3 Upper Atmosphere Cooling

the lower boundary of the thermosphere. Using sophisticated atmospheric general circulation models, they predict that the stratosphere, mesosphere and thermosphere will show significant cooling — the largest cooling of 40–50 ◦ C occurring in the thermosphere. The extent of this cooling very much depends on the rate coefficient for the O + CO2 excitation of the ν2 bending mode. Rishbeth and Roble [85.22] assumed a value for this rate coefficient of 1 × 10−12 cm3 /s, intermediate between the value of Sharma and Wintersteiner [85.24] (6 × 10−12 cm3 /s) and an earlier value of 2 × 10−13 cm3 /s used by Dickenson [85.25]. The Sharma and Wintersteiner value, based on observations of 15 µm emission in the atmosphere around 100–150 km, was recently confirmed by Rodgers et al. [85.26], but Pollock et al. [85.27] obtain a value of 1.2 × 10−12 cm3 /s in laboratory experiments. Using the larger rate coefficient would result in even greater cooling [85.28]. The overall consequences of such a large temperature decrease in the upper atmosphere have not been fully explored — particularly the question as to how the dynamics of the atmosphere will be affected. Since many chemical reactions depend on temperature, there may be considerable readjustments in the vertical distribution of minor species in the atmosphere. Cooler temperatures cause the atmosphere to contract, reducing densities and, consequently, satellite drag. Cooler temperatures may also increase the occurrence of polar stratospheric clouds, thereby affecting ozone depletion (Sect. 85.4). Most significantly, tropospheric warming and upper atmosphere cooling both result from a buildup of CO2 . The size of the predicted cooling is greater by an order of magnitude than the amount of the predicted heating. Thus it may be possible to monitor the global warming trend by observing the predicted cooling in the upper atmosphere. There is evidence in the mesosphere that this cooling has already begun. Temperatures appear to have decreased by 3–4 ◦ C over the last decade [85.29, 30]. Gadsden [85.31] has also found that the frequency of occurrence of noctilucent clouds, the highest-lying clouds in the atmosphere, has more than doubled over the last twenty-five years. He has calculated that this change could result from a decrease in the mean temperature at the mesopause of 6.4 ◦ C during this time period. However, increased concentrations of water produced by oxidation of increased amounts of methane may be responsible for the more frequent appearance of the clouds. This is an ongoing area of research.

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The buildup of CO2 has an even greater effect on the temperature in the upper atmosphere than on that in the troposphere [85.21]. As discussed in the Sect. 85.2.2, CO2 is a coolant in the stratosphere, mesosphere and thermosphere, but as a greenhouse gas is involved in heating the troposphere. The explanation for this revolves around the collision physics issue of quenching versus radiating. In the troposphere, CO2 absorbs infrared radiation coming from the Earth, exciting the ν2 vibrational bending mode at 15 µm. The excited molecule can either reradiate or collisionally de-excite. In the lower atmosphere where densities are large, the lifetime against collisions is very short and the excited molecule is rapidly quenched. This transfer of energy from radiation through collisions into the kinetic energies of the colliding partners results in a net heating. In the stratosphere and above, atomic oxygen collisions with CO2 excite this same bending mode. But at these higher altitudes, densities are lower and quenching is greatly reduced. The excited molecule radiates and the radiation escapes to space. A net cooling results because the opacity is low at these altitudes. Roble and co-workers [85.22, 23] have investigated the doubling of CO2 and CH4 concentrations (as predicted for the next century) in the mesosphere and at

85.3 Tropospheric Warming/Upper Atmosphere Cooling

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85.4 Stratospheric Ozone 85.4.1 Production and Destruction Ozone production takes place continually in the stratosphere during daylight hours, as molecular oxygen is photodissociated and the resulting oxygen atoms undergo three-body recombination with O2 : O2 + hν → 2O , O + O2 + M → O3 + M .

(85.4) (85.5)

Ozone can be destroyed through photodissociation: O3 + hν → O2 + O ,

(85.6)

Part G 85.4

but because an oxygen atom is produced which immediately recombines with another O2 to form O3 , no net loss of O3 results. The photodissociation of O2 and O3 are important heating processes in the stratosphere. The amount of ozone in the stratosphere is quite variable, changing significantly with the seasons and with latitude. In the lower stratosphere, much of the ozone is created over the equatorial regions and then transported toward the poles. Besides being photodissociated, O3 is destroyed by reactions with radicals that are involved in catalytic cycles. The short-hand notation for the major cycles, NOx , HOx , ClOx (BrOx ) refers to the catalytically active forms involved in the cycles. Our knowledge about the relative importance of these catalytic cycles in ozone destruction has increased dramatically over the last decade. A number of these cycles are given below, with the ozone-destroying step listed first, and the rate-limiting step closing the catalytic cycle and regenerating the ozone-destroying radical, listed last. The net effect in each of these cases is to convert ozone and atomic oxygen (otherwise known as odd-oxygen) into molecular oxygen: NO + O3 → NO2 + O2 NO2 + O → NO + O2 NET: O3 + O → 2O2 ,

(85.7)

NO + O3 → NO2 + O2 NO2 + O3 → NO3 + O2 NO3 + hν → NO + O2 NET: 2O3 → 3O2 .

(85.8)

and

OH + O3 → HO2 + O2 HO2 + O3 → OH + 2O2 NET: 2O3 → 3O2 ,

(85.9)

and the halogen cycle, in which Z = Cl or Br: Z + O3 → ZO + O2 ZO + O → Z + O2 NET: O3 + O → 2O2 .

(85.10)

The following series of reactions couples the HOx and halogen cycles: HO2 + ZO → HOZ + O2 HOZ + hν → OH + Z Z + O3 → ZO + O2 OH + O3 → HO2 + O2 NET: 2O3 → 3O2 .

(85.11)

Finally the reaction set BrO + ClO → Br + Cl + O2 Br + O3 → BrO + O2 Cl + O3 → ClO + O2 NET: 2O3 → 3O2

(85.12)

is also important in the halogen destruction cycle. The coupling between these different cycles by reactions such as HO2 + NO → OH + NO2

(85.13)

turns out to be very important in understanding the details of ozone destruction, such as how much each mechanism contributes to the destruction as a function of altitude and in the presence of aerosols. Wennberg et al. [85.32] have recently shown that catalytic destruction by NO2 , which for two decades was considered to be the predominant loss process, accounted for less than 20% of the O3 removal in the lower stratosphere during May 1993. They further show that the cycle involving the hydroxyl radical accounted for nearly 50% of the total O3 removal and the halogen-radical chemistry was responsible for the remaining 33%. The NOx and HOx cycles are naturally occurring, whereas the ClOx and BrOx cycles are due mainly to man-made chemicals – the CFCs and halons. The amplification that takes place through a catalytic cycle is the

Applications of Atomic and Molecular Physics to Global Change

85.4.2 The Antarctic Ozone Hole The ozone depletion problem was largely theoretical until the discovery of the ozone hole over Antarctica. Following the 1985 announcement by Farman et al. [85.33] of ground-based observations of significant decline in O3 concentrations during springtime in the Southern Hemisphere, it was possible to map this event using archived satellite data beginning in 1979. The data depict a worsening event throughout the early 1980s. In 1987, 70% of the total O3 column over Antarctica was lost during the month of September and early October, and the areal extent of the hole was ≈ 10% of the Southern Hemisphere. The ozone hole has continued to grow in depth and width [85.34]. Recent data shows that this phenomenon continues, with the 2003 ozone

hole the second largest observed to date (the largest yet observed was on September 10, 2000) [85.35]. The causal link between the release and buildup of man-made CFCs and the ozone hole over Antarctica has been quite convincingly established by Anderson et al. [85.36] through in situ observations from high altitude aircraft flights into the polar vortex during the end of polar night and the beginning of Antarctic spring in 1987. The polar vortex is a stream of air circling Antarctica in the winter, creating an isolated region which becomes very cold during the polar night. Flights into the vortex were able to document a heightened, increasing level of ClO and a monotonically decreasing O3 concentration over a 3–4 week time period during late September and early October. The mechanism which appears to be repartitioning the chlorine from its reservoir form into its catalytically active form is a heterogeneous process occurring on the surfaces of polar stratospheric clouds. At the cold temperatures during the polar night, polar stratospheric clouds form, consisting of ice and nitric acid trihydrate. Gaseous ClONO2 collides with HCl that has been adsorbed onto the surface of the cloud crystals. Chlorine gas is liberated and the nitric acid formed in the reaction remains in the ice [85.36]: HCl + ClONO2 → Cl2 (g) + HNO3 .

(85.14)

As solar radiation starts to penetrate the region at the beginning of spring, the Cl2 molecules are rapidly photodissociated, producing Cl atoms which initiate the catalytic destruction of O3 . As there are no oxygen atoms around to complete the catalytic cycles, several mechanisms for regenerating the Cl and Br radicals have been proposed which involve only the ClO and BrO molecules themselves. Mechanism I [85.37] ClO + ClO + M → (ClO)2 + M (ClO)2 + hν → Cl + ClOO ClOO + M → Cl + O2 + M 2 × (Cl + O3 → ClO + O2 ) NET: 2O3 → 3O2 ;

(85.15)

Mechanism II [85.38] ClO + BrO → Cl + Br + O2 Cl + O3 → ClO + O2 Br + O3 → BrO + O2 NET: 2O3 → 3O2 .

(85.16)

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Part G 85.4

reason that these chemicals, which are only present at the level of parts per trillion, can have such a destructive effect. It is useful to think in terms of a total chemical budget for a radical such as Cl which enters into a cycle. Chlorine is put into the stratosphere when chemicals such as CF2 Cl2 are released into the atmosphere. Such compounds are chemically inert and insoluble in water, and therefore are not easily cleansed out of the lower atmosphere. In the stratosphere, however, the CF2 Cl2 is subjected to solar UV radiation and is photodissociated, producing the Cl radical. Chlorine exists in the upper atmosphere in catalytically active forms, Cl and ClO, as well as in stable, reservoir species, HCl and ClONO2 . The total chlorine budget consists of both the catalytically active plus reservoir species. Reactions which reduce the formation of reservoir species, or convert reservoir species to catalytically active forms, contribute to the ozone destruction. Photolysis of stable reservoir species, such as ClONO2 , can produce catalytically active forms. Bromine has an identical cycle to that of chlorine, but is 50 to 100 times more destructive than Cl because it does not react readily to go into its reservoir form, HBr. A knowledge of the photodestruction rates of all such species is important to an understanding of the overall ozone photochemical depletion problem. Studies of the Antarctic ozone hole show that gas phase photochemical cycles, as given above, are not the whole story with respect to ozone depletion. Heterogeneous chemistries taking place on the surfaces of ice crystals and sulfate aerosols play an important role also. These are discussed briefly in Sect. 85.4.2 and Sect. 85.4.4.

85.4 Stratospheric Ozone

1300

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While Mechanism I accounts for 75% of the observed ozone loss, the sum of I and II yields a destruction rate in harmony with the observed O3 loss rates [85.36].

85.4.3 Arctic Ozone Loss The region around the North Pole does not appear to exhibit an ozone hole as severe as that found in the Antarctic. Several factors lessen the probability of a significant ozone hole developing in the Arctic. First, a stable polar vortex does not get well established due to increased atmospheric turbulence from the greater land surface area in the Northern Hemisphere. Second, temperatures during the Arctic winter do not get as cold as during the Antarctic winter, so that polar stratospheric clouds (PSCs) do not form as easily. As seen in the preceding section, the surfaces of PSCs play an essential role in the O3 destruction mechanisms in the Antarctic. However, ozone levels are showing 20–25% reductions during February and March [85.39] over a much larger area around the North Pole than in the South. Thus ozone destruction is taking place during the transition from polar winter to spring in the Arctic but the phenomenon is more widespread, diffuse, and not as well-contained as in the Antarctic.

Part G 85.5

85.4.4 Global Ozone Depletion Over the last twenty-five years, satellite instruments have measured the total ozone column in the atmosphere. During this time ozone levels have been steadily decreasing globally, especially at mid- to high-latitudes. Recent analysis indicates the first evidence of recovery of stratospheric ozone levels, with diminished rates of ozone loss at altitudes of 35–45 km, coupled with a slowdown in the increase in stratospheric loading of chlorine [85.40]. Heterogeneous reactions on aerosol surfaces, as well as the homogeneous gas phase chemical cycles mentioned earlier, must be invoked to explain the global decline in ozone levels. A particularly important reaction appears to be the hydrolysis of N2 O5 on sulfate aerosols. This occurs very rapidly, converting reactive nitrogen, NO2 , into its reservoir species HNO3 : sulfate aerosol

N2 O5 + H2 O −−−−−−−−−→ 2HNO3 .

(85.17)

The N2 O5 is formed at night by reaction of NO2 and NO3 . Following the hydrolysis of N2 O5 , there is less reactive NO2 around to convert ClO into its reservoir species, ClONO2 , and less NO2 around to convert OH into the reservoir species, HNO3 . A heightened sensitivity of the ozone to increasing levels of CFCs develops [85.41]. It has been shown that certain regions of significantly depleted ozone also show high concentrations of sulfate aerosols. In addition, measurements of the ratio of catalytically active nitrogen to total nitrogen can be reproduced using the above heterogeneous reaction, and not by using gas phase processes alone. Study of further mechanisms at varying altitudes and latitudes is an active area of research. Record low global ozone measurements, 2% to 3% lower than any previous year, were reported beginning in 1992 [85.42] and continuing well into 1993. The increase in naturally occurring aerosols due to the eruption of Mount Pinatubo in June 1991 appears to explain this decline. During the winter of 1993–1994, total ozone levels returned to levels slightly above normal [85.43], presumably because the excess aerosols had been removed from the stratosphere by natural sedimentation processes. The continuing buildup of CO2 is predicted to contribute to increased cooling of the stratosphere. Declining temperatures in the stratosphere may increase the frequency of formation of polar stratospheric clouds which drive the destructive heterogeneous chemistry creating the Antarctic ozone hole. An increased occurrence of these clouds outside of the polar regions could affect ozone levels globally. There are also indications that certain ozone depletion chemistries taking place on the surface of sulfate aerosols may also be enhanced by lower temperatures [85.41]. Ozone itself is the dominant heat source in the lower stratosphere. Decreasing the amount of ozone drives temperatures still lower [85.44]. It is unfortunate that the two most significant atmospheric global change effects — the buildup of CO2 and the enhanced ozone destruction due to man-made CFCs — both cause decreasing temperatures in the stratosphere which may further enhance the destructiveness of the ozone photochemical cycles.

85.5 Atmospheric Measurements Ground-based observations, as well as measurements made by instruments carried aloft in satellites, bal-

loons, and high-flying aircraft, allow one to explore the atmosphere.

Applications of Atomic and Molecular Physics to Global Change

rate knowledge of the emission spectroscopy of species such as OH, HO2 , H2 O2 , H2 O, O3 , HNO3 , NO2 , N2 O, N2 O5 , HNO3 , ClNO3 , BrO, HCl, HOCl, and ClO is essential. Such measurements provide a rigorous test of atmospheric models. The recently-launched NASA EOS Aura satellite carries instruments that will make global measurements of a number of these species [85.47]. In order to analyze the data and deconvolve some of the line profiles to give information on concentrations as a function of altitude, molecular data such as line strengths and pressure broadening coefficients are needed [85.9]. Until recently, it has been impossible for remotesensing experiments to distinguish between ozone occurring in the stratosphere (where it is formed naturally) and ozone occurring in the troposphere (where it is a pollutant). Satellite instruments such as the ESA Global Ozone Monitoring Experiment (GOME), the Scanning Imaging Absorption Spectrometer for Atmospheric Chartography (SCIAMACHY) and the Ozone Monitoring Instrument (OMI) have broad enough spectral coverage and high enough resolution that the temperature dependence of the ozone absorption features from 300–340 nm, known as the Huggins bands, can be used to separate out the ozone concentrations in the middle and lower atmospheres [85.48, 49].

References 85.1 85.2

85.3 85.4 85.5

85.6

85.7 85.8

85.9

T. E. Graedel, P. J. Crutzen: Atmospheric Change: An Earth System Perspective (Freeman, New York 1993) J. Houghton: Global Warming: The Complete Briefing, 2nd edn. (Cambridge Univ. Press, Cambridge 1997) J. Staehelin, N. R. P. Harris, C. Appenzeller, J. Eberhard: Rev. Geophys. 39, 231 (2001) R. G. Roble: Encyclopedia of Applied Physics, Vol. 2 (VCH, Weinheim 1991) pp. 201–224 I. Bey, D. J. Jacob, R. M. Yantosca, J. A. Logan, B. D. Field, A. M. Fiore, Q. Li, H. Y. Liu, L. J. Mickley, J. Loretta, M.G. Schultz: J. Geophys. Res. 106(D19), 23073 (2001) P. S. Armstrong, J. J. Lipson, J. A. Dodd, J. R. Lowell, W. A. M. Blumberg, R. M. Nadile: Geophys. Res. Lett. 21, 2425 (1994) R. R. Conway: NRL Memorandum Report, MR-6155, 79 pp. (1988) K. Kirby, E. R. Constantinides, S. Babeu, M. Oppenheimer, G. A. Victor: At. Data Nucl. Data Tables 23, 63 (1979) L. S. Rothman, A. Barbe, D. C. Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Cler-

85.10

85.11 85.12 85.13

85.14 85.15

baux, V. Dana, V. M. Devi, A. Fayt J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, K. Yoshino: The HITRAN Molecular Spectroscopic Database: Edition of 2000 including Updates Through 2001, J. Quant. Spectrosc. Radiat. Transfer 82, 5–44 (2003) P. L. Smith, W. L. Wiese: Atomic and Molecular Data for Space Astronomy (Springer, Berlin, Heidelberg 1992) R. P. Wayne: Chemistry of Atmospheres (Oxford Univ. Press, New York 1991) p. 41 J. F. B. Mitchell: Rev. Geophys. 27, 115 (1989) National Research Council: Reconciling Observations of Global Temperature Change (National Academy Press, Washington, DC 2000) IPCC Third Assessment Report Climate Change (The Intergovernmental Panel on Climate Change 2001) J. C. Comiso, C. L. Parkinson: Phys. Today 57, 38 (2004)

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Measurements may be made either in situ or by remote-sensing techniques. The region of the atmosphere from ≈ 60 km to 120 km, encompassing the mesosphere and lower thermosphere and ionosphere, cannot be studied in situ as it is too high for balloons and aircraft, and too low (i. e., too much drag) for satellites. For this region, remote sensing experiments are essential and a comprehensive book on the subject is recommended [85.45]. Most of the instruments used to make atmospheric measurements have been developed in molecular physics and spectroscopy laboratories. Even experiments utilizing sophisticated techniques, such as laser induced fluorescence, and instruments, such as Fourier transform spectrometers, are being flown on payloads. An excellent compendium of ozone-measuring instruments for stratospheric research has been assembled by Grant [85.46]. A combination of good laboratory experiments, theoretical calculations, and ingenuity are necessary to extract accurate information from measurements made in the atmosphere. For instance, in order to understand the complicated interactions of the different photochemical cycles involved in ozone chemistry, spectroscopic emissions and absorptions of the many trace species are used to measure concentration profiles. An accu-

References

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85.16 85.17 85.18 85.19

85.20 85.21 85.22 85.23 85.24 85.25 85.26

85.27 85.28 85.29 85.30

Part G 85

85.31 85.32

85.33 85.34 85.35 85.36

E. W. Wolff: Science 302, 1164 (2003) M. A. J. Curran, T. D. van Ommen, V. I. Morgan, K. L. Phillips, A. S. Palmer: Science 302, 1203 (2003) J. Firor: The Changing Atmosphere (Yale University Press, New Haven 1990) J. W. Elkins, T. M. Thompson, T. H. Swanson, J. H. Butler, B. D. Hall, S. O. Cummings, D. A. Fishers, A. G. Raffo: Nature 364, 780–783 (1993) J. Hansen, D. Johnson, A. Lacis, S. Lebedeff, P. Lee, D. Rind, G. Russell: Science 213, 957 (1981) R. J. Cicerone: Nature 344, 104 (1990) H. Rishbeth, R. G. Roble: Planet. Space Sci. 40, 1011 (1992) R. G. Roble, R. E. Dickinson: Geophys. Res. Lett. 16, 1441 (1989) R. D. Sharma, P. P. Wintersteiner: Geophys. Res. Lett. 17, 2201 (1990) R. E. Dickenson: J. Atmos. Terr. Phys. 46, 995 (1984) C. D. Rodgers, F. W. Taylor, A. H. Muggeridge, M. Lopez-Puertas, M. A. Lopez-Valverde: Geophys. Res. Lett. 19, 589 (1992) D. S. Pollock, G. B. I. Scott, L. F. Phillips: Geophys. Res. Lett. 20, 727 (1993) S. W. Bougher, D. M. Hunten, R. G. Roble: J. Geophys. Res. 99, No. E7, 14,609 (1994) A. C. Aiken, M. L. Charin, J. Nash, D. J. Kendig: Geophys. Res. Lett. 18, 416 (1991) A. Hauchecorne, M.-L. Chanin, R. Keckhut: J. Geophys. Res. 96(D8), 15297 (1991) M. Gadsden: J. Atm. Terr. Phys. 52, 247 (1990) P. O. Wennberg, R. C. Cohen, R. M. Stimpfle, J. P. Koplow, J. G. Anderson, R. J. Salawitch, D. W. Fahey, E. L. Woodbridge, E. R. Keim, R. S. Gao, C. R. Webster, R. D. May, D. W. Toohey, L. M. Avallone, M. H. Proffitt, M. Loewenstein, J. R. Podolske, K. R. Chan, S. C. Wofsy: Science 266, 398–404 (1994) J. C. Farman, B. G. Gardiner, J. D. Shankin: Nature 315, 207 (1985) R. A. Kerr: Science 266, 217 (1994) http://jwocky.gsfc.nasa.gov/multi/multi.html J. G. Anderson, D. H. Toohey, W. H. Brune: Science 251, 39 (1991)

85.37 85.38 85.39

85.40

85.41

85.42

85.43

85.44

85.45

85.46

85.47 85.48 85.49

L. T. Molina, M. J. Molina: J. Phys. Chem. 91, 433 (1987) M. B. McElroy, R. J. Salawitch, S. C. Wofsy, J. A. Logan: Nature 321, 759 (1986) G. L. Manney, L. Froidevaux, J. W. Waters, R. W. Zurek, W. G. Read, L. S. Elson, J. B. Kumer, J. L. Mergenthaler, A. E. Roche, A. O’Neill, R. S. Harwood, I. MacKenzie, R. Swinbank: Nature 370, 429–434 (1994) M. J. Newchurch, E.-S. Yang, D. M. Cunnold, G. C. Reinsel, J. M. Zawodny, J. M. Russell III: J. Geophys. Res. 108(D16), 4507 (2003) D. W. Fahey, S. R. Kawa, E. L. Woodbridge, P. Tin, J. C. Wilson, H. H. Jonsson, J. E. Dye, D. Baumgardner, S. Borrmann, D. W. Toohey: Nature 363, 509–514 (1993) J. F. Gleason, P. K. Bhartia, J. R. Herman, R. McPeters, P. Newman, R. S. Stolarski, L. Flynn, G. Labow, D. Larko, C. Seftor, C. Wellemeyer, W. D. Komhyr, A. J. Miller, W. Planet: Science 260, 523–526 (1993) D. J. Hofmann, S. J. Oltmans, J. M. Harris, J. A. Lathrop, G. L. Koenig, W. D. Komhyr, R. D. Evans, D. M. Quincy, T. Deshler, B. J. Johnson: Geophys. Res. Lett. 21, 1779–1782 (1994) V. Ramaswamy, M.-L. Chanin, J. Angell, J. Barnett, D. Gaffen, M. Gelman, P. Keckhut, Y. Koshelkov, K. Labitzke, J.-J. R. Lin, A. O’Neill, J. Nash, W. Randel, R. Rood, K. Shine, M. Shiotani, R. Swinbank: Stratospheric Temperature Trends: Observations and Model Simulations, Rev. Geophys. 39, 71 (2001) J. T. Houghton, F. W. Taylor, C. D. Rodgers: Remote Sounding of Atmospheres (Cambridge Univ. Press, Cambridge 1984) W. B. Grant (Ed.): Ozone Measuring Instruments for the Stratosphere (Optical Society of America, Washington, DC 1989) Further information is available at http://aura.gsfc.nasa.gov/ K. V. Chance, J. P. Burrows, D. Perner, W. Schneider: J. Quant. Spectrosc. Radiat. Transfer 57, 467 (1997) R. Munro, R. Siddans, W. J. Reburn, B. Kerridge: Nature 392, 168 (1998)

1303

Atoms in Dens 86. Atoms in Dense Plasmas

When plasma densities are high enough that interparticle separations are comparable to atomic dimensions, there are important “environmental” consequences for atomic structure and atomic processes. Such conditions are found not only within stars and giant planets but, nowadays, also in the laboratory – especially in experiments related to the quest for inertial confinement fusion. After introducing important plasma concepts, we examine these consequences in regard to several issues: modification of atomic bound states, ionization balance, equation of state, and radiative and collisional processes that regulate transport coefficients and the spectral emission of non-equilibrium plasmas. Finally, we describe modern simulation methods that are being used to tackle various manybody problems in this subject. For nearly every issue we raise there is a need for better

understanding and for more, and more precise, data.

The focus of the present chapter is partially ionized matter in which important atomic phenomena are influenced by a dense plasma environment. As Fig. 86.1 (which is discussed in detail below) reveals, the densities in many laboratory and astrophysical plasmas are high enough to invalidate the presumption of isolated systems. The interaction of intense laser or particle beams with solid matter produces rapidly evolving, hot, and dense plasmas [86.3] that mimic some of the most extreme conditions in nature, including the thermonuclear environment of stellar interiors; these plasmas, as well as some produced in z-pinch implosions [86.4], are the basis of world-wide inertial confinement fusion (ICF) efforts. Dense plasmas also can be transient gain media for amplified spontaneous emission at X-ray wavelengths [86.5]. Additional impetus for the study of atoms in dense plasmas now comes from experiments involving irradiation of solids by ultra-short (sub-picosecond) laser pulses [86.6]. The moderate temperatures (tens of eV) but high (near-solid) densities typical of this so-called warm dense matter (WDM) regime [86.7] produce severely perturbed bound ionic states.

Part G 86

Ionized gases, or plasmas, are the predominant form of matter throughout the universe, and physical conditions in laboratory and cosmic plasmas vary greatly. No single experimental methodology or theoretical construct suffices to explore all aspects of the plasma state. Systematic study of plasmas began early in the 20th century, but until recently the physics of atoms in plasmas has been largely synonymous with the physics of isolated ions. This perspective is valid as long as the interparticle spacing is very much larger than the relevant atomic dimensions, typically a few to a few tens of Bohr radii. For example, ions are isolated in this sense in interstellar space where the electron density n e ≈ 1 cm−3 , or even in a tokamak, where n e ≈ 1014 cm−3 . For neutral and moderately charged atoms, data such as energy levels, oscillator strengths, and collision cross sections have long been obtained from traditional kinds of experiments and quantal calculations, as discussed elsewhere in this book. Progress in these areas continues to be made, with the X-ray spectrum of highly-charged Fe [86.1] and the Lamb shift in U+91 [86.2] being noteworthy examples of the kinds of accurate measurements that can now be made using electron beam ion traps and storage rings.

86.1 The Dense Plasma Environment ............ 1305 86.1.1 Plasma Parameters .................... 1305 86.1.2 Quasi-Static Fields in Plasmas..... 1305 86.1.3 Coulomb Logarithms and Collision Frequencies........... 1307 86.2 Atomic Models and Ionization Balance .. 1308 86.2.1 Dilute Plasma Models ................ 1308 86.2.2 Dense Plasma “Chemical” Models1309 86.2.3 Dense Plasma “Physical” Models . 1310 86.3 Elementary Processes ........................... 1311 86.3.1 Radiative Transitions and Opacity 1311 86.3.2 Collisional Transitions ................ 1312 86.4 Simulations......................................... 1313 86.4.1 Monte Carlo .............................. 1313 86.4.2 Molecular Dynamics................... 1313 86.4.3 The Deuterium EOS Problem........ 1315 References .................................................. 1316

1304

Part G

Applications

log[Te(K)] 2

1

0

–1

9 NIF main fuel

log [Re(a0)] –2 5 log[Θe (eV)]

Υ= 1

4

8 Sun 7

White dwarf 3

P = 1 Mbar

2

6 Γee = 1

0

4 18

1

Warm dense matter

5

Jupiter 21

24

27

30 log [ne(cm–3)]

Fig. 86.1 Plasma conditions discussed in the text are identified on this temperature-density plane. Plasmas below the line Γee = 1 are strongly coupled, and those below the line Υ = 1 contain degenerate electrons. The HEDP regime, which lies above and to the right of the line marked P = 1 Mbar, includes some conditions characteristic of warm dense matter. Also plotted are tracks representing conditions (as a function of radius) within the Sun, Jupiter and a typical white dwarf star; time-dependent conditions are also shown – from early compression through ignition – within the main (DT) fuel of a prototype target capsule for the National Ignition Facility

Part G 86

Dense cosmic plasmas – specifically, the interiors of stars and giant planets – are very large and have very long lifetimes. Their thermodynamic variables change only slowly with position or time and, hence, macroscopic regions can be considered as statistical systems evolving through a succession of states in local thermodynamic equilibrium (LTE). The equation of state (EOS), and the radiation and heat transport coefficients, viz. the opacity and thermal conductivity, are key to understanding the behavior of LTE plasmas. A recent monograph [86.8], plus comprehensive articles by More et al. [86.9], by Rogers and Iglesias [86.10], and by Saumon et al. [86.11] discuss many of the high density consequences for the opacity and for EOS. In contrast, the short lifetimes of dense plasmas created by intense beam irradiation or by explosive pinch devices often preclude the establishment of a thermal distribution of atomic level populations; in extreme cases, there is not even enough time to establish a Maxwellian distribution of particle velocities. Populations in highly nonequilibrium (non-LTE) laboratory plasmas, which must be found by solving rate equations [86.12, 13], are essential information for using X-ray line emission to diagnose conditions in ICF targets, or for identifying likely gain media for X-ray lasing. And, as we discuss in Sect. 86.3, the dense

plasma environment modifies transition rates themselves, further complicating the interplay of numerous collisional and radiative processes in such atomic kinetics calculations. The topics addressed here are needed for understanding non-LTE situations, as well as LTE ones. After characterizing the perturbing plasma environment in Sect. 86.1, we summarize well-known prescriptions for atomic structure and ionization balance in Sect. 86.2, and then discuss modified transition rates in Sect. 86.3 for ions in dense plasmas. Finally, we review in Sect. 86.4 how simulations are now being used to address a wide array of issues needed to accurately describe atoms in dense plasmas. There are several periodic meetings devoted to various aspects of this subject, and especially relevant ones include: Atomic Processes in Plasmas; Radiative Properties of Hot, Dense Matter; Spectral Line Shapes; and Strongly Coupled Coulomb Systems. Printed proceedings of these conferences are an excellent guide to recent developments in the topics discussed here, as well as numerous other, related ones. Additionally, three recent textbooks [86.14–16] provide detailed treatments of many of the subjects surveyed here. The present topic is an important part of what is now being termed “high energy-density physics”

Atoms in Dense Plasmas

(HEDP). Conventionally this interdisciplinary subject, which involves collective and/or non-linear phenomena in many-body systems, is defined as the study of matter in regimes where the total (matter plus electromagnetic

86.1 The Dense Plasma Environment

1305

field) pressure exceeds one megabar; this boundary is also marked in Fig. 86.1. Reference [86.17] is a recent National Research Council report on key issues and opportunities in HEDP.

86.1 The Dense Plasma Environment Most plasmas are charge neutral, so the mean number densities n i and n e of constituent ions (charge Z i e, mass m i ) and electrons (charge −e, mass m e ) satisfy
Z a na = 0 , (86.1) a

where the sum ranges over all particle species. Moreover, plasma conditions usually change slowly enough that each of the species is able to establish a thermal distribution of velocities, fixed by its temperature Θa = kB Ta (in energy units). Here, we assume these conditions hold.

86.1.1 Plasma Parameters A few key quantities characterize the plasmas under consideration. Derivations, and discussions of the roles of these and other auxiliary quantities can be found in standard plasma physics texts [86.18, 19], as well as a recent tutorial article [86.20]. 1. The plasma frequency  1/2
2 ωp = 4πn a (Z a e) /m a a

1/2 ωa2

(86.2)

  for the particle oscilladefines a timescale ∼ ω−1 e tions in response to a non-equilibrium charge density in the plasma. 2. The Debye length −1/2 
2 4πn a (Z a e) /Θa λD = a

a

=




−1/2 Da−2

(86.3)

a

is the distance beyond which plasma particles effectively screen any localized charge imbalance. 3. The ion-sphere (or electron-sphere) radius 1/3  3 (86.4) Ra = 4πn a

Z a Z b e2 (86.6) Rab Θab give the average ratio of potential to kinetic energies between species a and b. The reduced ion-sphere radius and temperature are Rab = 12 (Ra + Rb ) and Θab = (m a Θb + m b Θa )/(m a + m b ), respectively. When Γab is greater (less) than 1, that species is said to be strongly (weakly) coupled. And, when the number of particles of radius Da

a in a sphere  (a Debye sphere), 4πn a Da3 /3 = 1/(3Γaa )3/2 , is small, discreteness of the charge density can be important in describing certain plasma phenomena. Γab =

Figure 86.1 shows that dense plasmas can be strongly or weakly coupled; further, some of these plasma conditions involve degenerate electrons while others do not. And, in WDM, one encounters the situation where Υ ∼ 1 and Γee ∼ 1, which is particularly difficult to treat theoretically because several effects are competing amongst each other. In this figure, we plot the run of (n e , Θe ) values for the sun, for Jupiter, and for a typical white dwarf star (0.6 solar mass, pure C/O core, H/He outer layers). Also plotted is the track of DT fuel conditions in an imploding ICF capsule designed for the National Ignition Facility. Note that all of these systems sample wide portions of parameter space, and therefore an accurate description of each requires some very different plasma models.

Part G 86.1

=




defines a spherical volume associated with a single particle and is a measure of interparticle spacing (among particles of species “a”). 4. The Fermi energy  2/3 ~2 3π 2 n e ΘF = (86.5) 2m e characterizes the highest occupied energy level in a zero temperature system of electrons. A dimensionless measure of degeneracy is Υ = ΘF /Θe . Velocity distributions are either Maxwellian or Fermi–Dirac in the limits Υ  1 or Υ  1, respectively. 5. The Coulomb coupling parameters

1306

Part G

Applications

86.1.2 Quasi-Static Fields in Plasmas A simple, yet useful description of the plasma environment is given by the one-component plasma (OCP) model [86.22], in which particles of a single kind (Z a ,m a ) move against a smooth background of matter having on average the opposite charge density, ρ(r) = −Z a en a . Most plasma phenomena can be described within the context of either this or the twocomponent model (electrons and their parent ions, with n i = n e /Z i ). In the more realistic, two-component picture, electron screening of the (slower moving) ions is −1 established on a timescale ω−1 e  ωi , so it makes sense to speak of screened, quasi-static ionic fields. When there are many electrons in each Debye sphere, the electrostatic potential Φ(r) near a test charge Ze placed in an otherwise uniform, neutral plasma is exponentially reduced from the Coulomb expression, viz., Ze exp (−r/De ) . (86.7) r (A more elaborate version of this formula, for partially ionized atoms, has recently been proposed [86.23].) This “Debye screening” obtains only in weakly coupled plasmas and applies only to a test charge at rest. The faster the charge Ze moves, the less effective is the plasma at screening it, since only plasma particles with higher velocities can form the shielding cloud [86.18, 24]. In the opposite limit of large Γ -values, quasi-static screening is better described by the ion-sphere (IS) picture [86.19], in which each stationary ion of charge Z i e is surrounded by Z i electrons, uniformly distributed throughout a sphere of radius Ri , to produce the potential 

 (86.8) ΦIS (r) = Z i e 1/r − (1/2Ri ) 3 − r 2 /Ri2 ΦD (r) =

Part G 86.1

inside the sphere, and zero potential outside. Consideration of a plasma’s electric microfield illustrates these concepts. Moreover, microfields are a key ingredient in calculations of spectral line broadening in plasmas – an important subject discussed in Chapts. 59, 19 and in [86.14–16]. Local fluctuations in the density of any species about its mean value n a create a microscopic electric field E˜ a (r, t). There is a probability distribution P( E˜ a ) that characterizes the strengths of these microfields, which are quasi-static within time intervals short compared with the fluctuation timescale, 1/ωi . Holtsmark first calculated this distribution at an arbitrary position in an infinite, isotropic gas of noninteracting particles, and Chandrasekhar [86.25] gives a thorough account of this

famous stochastic problem. Holtsmark’s formula is ∞

 PH (ε) = (2ε/π) x sin(εx) exp −x 3/2 dx , (86.9) 0



where ε = E˜ a / (8π/25)1/3 E a is the scaled field, and E a = |Z a |e/Ra2 . The mean Holtsmark field is ε  2.99, and for ε  1, PH (ε) is well approximated by the distribution of fields due to a single nearest-neighbor in the gas,   Pnn (ε)  3/ 2ε5/2 . (86.10) Both of these distributions ignore the interactions among charged particles that become increasingly important as Γii grows, because particle positions then tend to be correlated. Quasi-static ion microfields – at the position of an ion – therefore become weaker, on average, as the coupling increases. Figure 86.2 illustrates this point and shows distributions computed with the P(ε ) 2.0 Γ = 2.0 1.5

1.0

Γ = 0.2 0.5

0.0

H

0

1

2

3

4 ε

Fig. 86.2 The probability distribution P(ε) of scaled microfield strengths ε for different plasma conditions. The curve marked H represents the Holtsmark distribution, which applies to an idealized case of non-interacting particles (i. e., Γ = 0). The other two curves, with Γ = 0.2 and Γ = 2.0, represent distributions determined by the APEX model for interacting ions (charge Z = 1) that are Debye-screened by plasma electrons. In these latter two cases, the ion density and temperature are, respectively, 1.0 × 1018 cm−3 and 1.15 eV, and 2.6 × 1024 cm−3 and 16 eV. After [86.21]

Atoms in Dense Plasmas

APEX method [86.26, 27], which uses a parameterized, two-particle distribution function “tuned” to yield the  exact second moment ε2 for the distribution function of field strengths experienced by any one ion in a plasma whose ions are all Debye-screened by electrons only. The potentials and microfields discussed above are based on classical statistical mechanics, even though dense plasmas are inherently quantum many-body systems. Reference [86.28] provides a careful discussion of the merits and limitations of this approach.

86.1.3 Coulomb Logarithms and Collision Frequencies

φ max

σm (E) = 2π

dφ [sin φ (1 − cos φ) σR (φ)] φmin

φmax , (86.11) ≈ πa ln φmin   where a =  Z a Z b e2  /E is a characteristic length. The familiar result (a/2b) = tan(φ/2) ≈ (φ/2) gives σm in terms of minimum and maximum impact parameters, bmin = a (from φmax = 1) and bmax = λD . Finally, if the actual collision energy in the argument of the logarithm, (λD /a), is replaced by its mean value, E = 32 Θ, the momentum transfer cross section takes the 2



1307

simple form  σm (E) = π

Z a Z b e2 E

2 ln Λ ;

(86.12)

for a two-component, electron-ion plasma, the argument of this Coulomb logarithm is 3/2

Λ ≈ 1/2ΓeZ ≈ (# particles in Debye sphere) . (86.13)

Spitzer’s result for σm yields the simplest expression for the frequency νeZ of electron-ion collisions in a twocomponent plasma, defined [86.18] as the mean value of the reciprocal of the time between collisions, n Z Z 2 e4  π 3/2 νeZ = √ ln Λ . (86.14) me 2Θ Equation (86.12) is commonly used to determine transport coefficients in weakly coupled plasmas, where ln Λ  1. In the dense plasma regime, however, the Coulomb logarithm can be small or even negative at high enough density, which yields meaningless results for the cross section. Physically, a small ln Λ arises from a small λD (high density and/or low temperature), which means that collisions can only occur at very small separations where the Coulomb potential is largest. Strong collisions can be included in the above analysis simply by not making a small-angle approximation in the evaluation of the cross section; the result is [86.30]  2  π Z a Z b e2 σm (E) = ln 1 + Λ2 . (86.15) 2 E In obtaining this result – which no longer yields a negative cross section – no assumption need be made about the value of φmax . One must still choose, however, bmax , which will not generally be given by λD , since Debye screening is invalid in the dense plasma regime. Strong collisions at small separations also bring in the effects of quantal scattering. These issues are best circumvented by obtaining the cross section directly from a quantal calculation involving the chosen screened potential [86.30, 31]. (Note that the formulae in [86.31] actually describe the scattering of one unscreened charge by another charge that is screened, so they are most relevant to the scattering of fast electrons by slow ions.) Eventually, even this formulation will fail when plasma kinetic processes affect the collision. For example, hard collisions can alter the velocity distribution function and collective modes can modify

Part G 86.1

It is well known that the total cross section for elastic scattering of two charged particles, Z a e and Z b e, diverges at any collision energy E , as a consequence of the infinite range of the Coulomb interaction. Plasma transport coefficients (e.g., electrical conductivity and thermal conductivity), however, which depend ultimately upon momentum transfer in this elementary process, are finite. This comes about because scattering at small center-of-mass angles φ, or, correspondingly, at large impact parameters b, is diminished by plasma screening, as in (86.7) and (86.8). Following Spitzer [86.29], we argue that there is some minimum effective scattering angle φmin . And, since the analysis will be based on classical formulae, there is some maximum scattering angle φmax beyond which quantum effects are important. Between these limits the Coulomb interaction is taken to be unscreened and Rutherford’s differential cross section σR (φ) is applicable. By assuming that φmax also is small, it follows that the momentum transfer cross section can be approximated as

86.1 The Dense Plasma Environment

1308

Part G

Applications

the static screened interaction potential. Furthermore, strong coupling introduces ionic structure that correlates the collisions. Accurate calculations must consider the collision process in the context of an appropriate kinetic equation [86.30, 32–34]. When such a process is carried out, the result can be inverted to yield “effective” or “generalized” Coulomb logarithms that typically are process dependent [86.18]. Li and Petrasso [86.32] obtained high-order correction terms for the Fokker–Planck equation, for example, and Boercker et al. [86.33] generalized the collision term in the Lenard–Balescu quantum ki-

netic equation to include strong coupling effects. Additionally, Berkovsky and Kurilenkov [86.34] extended the strong coupling description to also include “strong” collisions (those poorly treated within the Born approximation). The physics of collisions is directly measured by resistivity experiments, which employ some method to heat a solid and attempt to tamp the high-pressure plasma that is formed. The resistivity is then measured either by the reflectivity or directly through current and voltage probes. A review of these methods has recently been given by Benage [86.35].

86.2 Atomic Models and Ionization Balance A pervasive issue in the study of both laboratory and astrophysical plasmas is ionization balance: What is the distribution of charge states Z i of atomic ions in a particular plasma? Answers impact subjects as diverse as cosmic abundances deduced from astrophysical spectra, and the temporal behavior of laser-heated foils. Table 86.1 lists the charge-state dependence   of several plasma quantities [86.36]. Here, Z and Z 2 denote the mean and mean square ionic charge, respectively, viz.,
 n
n Z = Z i ni/ ni (86.16) ions

ions

Part G 86.2

For a dense plasma, experimental determination of actual charge-state distributions, or even Z, has proven difficult. Traditional spectroscopic methods (as described in the next section) require large atomic data bases and sophisticated kinetics models, which typically are run several times to find the best match to measured line shapes and intensities. Recently, however, an X-ray scattering method (based on the Compton effect) has been developed to determine Z in rapidly evolving plasmas [86.37]. Instead of detailed atomic data, this method requires accurate knowledge of the plasma’s dynamic structure factor [86.18], which in general must Table 86.1 Some plasma quantities that depend on its ion-

ization balance Quantity

Z-scaling (fixed nucleon density)

(Ideal) gas pressure Electrical resistivity Thermal conductivity

∼ (Z + 1)   ∼ Z 2 / Z   ∼ Z / Z 2  2 2 ∼ 1/ Z   ∼ Z Z 2

Ionic viscosity Bremsstrahlung

be obtained from a molecular dynamics simulation (as discussed in Sect. 86.4).

86.2.1 Dilute Plasma Models Consider a nondegenerate plasma in thermal equilibrium at a temperature Θ (for instance, some region of a star’s interior). The time independent ionization balance for each element is given by the Saha–Boltzmann formula [86.38]   1 2G Z+1 n Z+1 m e Θ 3/2 = exp(−I Z /Θ) , nZ ne GZ 2π ~2 (86.17)

for the density ratio of successive charge states, where G Z and I Z are, respectively, the partition function and ionization potential for the Z-times ionized atom. The solution of (86.17) is shown in Fig. 86.3 (top panel) for the case of solid density aluminum over a wide range of temperatures. The partition functions were determined from atomic states of the ground configuration only. The aluminum plasma is predominantly neutral at temperatures in the few electron volt range and ionizes stage by stage until it is nearly fully ionized just above one kilovolt. Of course, real aluminum is not an insulator at solid density and low temperatures, as Fig. 86.3 would suggest. Major corrections to (86.17) are evidently needed to incorporate the physics of WDM (Γ ∼ 1, Υ ∼ 1), especially corrections for partial electron degeneracy. We will return to this problem in later subsections. When conditions change too rapidly for LTE to be established, the plasma may evolve through a succession of “steady states” in which the relative abundances of dif-

Atoms in Dense Plasmas

ferent ion stages are determined by a balance of certain ionization and recombination rates. Then, in order to answer the straightforward question of what are the relative abundances of atomic ionization stages, one needs a vast data base of atomic energy levels, plus collisional and radiative rates. A set of rate equations must be solved to determine the populations n Z (α) for each quantum state α of each ion stage Z. Each equation involves transitions to and from all other states [86.12, 13]. The balance of photoionizations and dielectronic plus radiative recombinations in low-density, steadystate plasmas is termed nebular equilibrium, because these are the conditions appropriate to astrophysical nebulae – regions of ionized gas surrounding hot stars [86.39]. The balance of collisional ionizations and dielectronic plus radiative recombinations in low-density, steady-state plasmas is termed coronal equilibrium, because these are the conditions appropri-

1.0

Charge state fraction Ideal Saha

0.8

+11

0.6 +2

+4

+6

0.4 0.2 0.0

10

12

100 1000 Temperature (eV)

Saha + CL

10 8 6 Thomas-Fermi

4 2 0

Ideal Saha 1

10

100 1000 Temperature (eV)

Fig. 86.3 The top panel shows fractional abundances of

different charge states of aluminum in thermal equilibrium at solid density over a range of temperatures, as computed with the ideal Saha–Boltzmann equation (86.17). The corresponding mean charge Z is shown in the lower panel, in addition to the result from Thomas–Fermi theory, as given by (86.18). Also shown is a modified Saha formulation that includes continuum lowering shifts

1309

ate to the solar corona. Most tokamak plasmas also are in coronal equilibrium. Under coronal conditions, essentially all ions are in their respective ground states and the ionization balance is a function only of the plasma temperature [86.40]. As the plasma density increases, three-body collisions become important, and the resulting steady-state ionization balance, which depends on density as well as temperature, is termed collisionalradiative equilibrium [86.12]. This situation exists in most ICF experiments. Finally, conditions in sub-picosecond laser-plasma experiments can change so rapidly that none of the above simplifications apply. Ionization is strongly timedependent, and may involve multiphoton processes. As we describe below, a dense plasma environment vastly complicates the determination of ionization balance. In LTE cases, energy levels are changed and partition functions are truncated by the phenomenon of continuum lowering. In non-LTE cases these effects still occur, but, in addition, radiative and collisional rates themselves are altered.

86.2.2 Dense Plasma “Chemical” Models There are two distinct strategies taken to extend the results of the previous section. One strategy, known as the “chemical picture”, formulates the Saha–Boltzmann equation in terms of a free energy F(T, V, {n a }), where the species populations {n a } are to be determined, and various corrections due to couplings and degeneracy can be added to yield a thermodynamically consistent equation of state that includes atomic physics [86.41]. Because plasma screening attenuates the Coulomb interaction at long range, atoms and ions no longer have an infinite number of bound (Rydberg) states, and atomic partition functions are truncated in a natural way. The simplest chemical picture is that the onset of continuum energies has been “lowered” by some amount ∆I (relative to the atom’s ground state). When ∆I is a fixed quantity, continuum lowering eliminates bound states whose (unscreened) ionization potentials were less than ∆I, and moves all remaining states closer to the continuum by this same amount. Thus, levels get shifted but spectral lines do not. Schemes that use an effective, single-particle potential to determine a spectrum of modified eigenstates produce distinct plasma shifts for different levels and, hence, spectral line shifts. Experiments show, however, that almost all such predictions have been inaccurate: actual plasma-induced shifts are very small and, for most applications, ignorable ([86.14, Sect. 4.10], [86.15, Sect. 3.5]).

Part G 86.2

1 14

86.2 Atomic Models and Ionization Balance

1310

Part G

Applications

A variety of arguments has been put forth to quantify continuum lowering, including in particular: 1. determine ∆I from the last distinct spectral line near a series limit (the Inglis–Teller formula [86.42]); 2. determine ∆I from the atom’s dipolar interaction with the plasma’s microfield [86.41]; 3. determine ∆I from the binding energy of the ground state in some specified, screened Coulomb potential [86.43, 44]; 4. determine ∆I from a rigorous, statistical mechanical treatment of the atomic partition functions [86.45].

Part G 86.2

Figure 86.3 (lower panel) illustrates the effect of continuum lowering on the average ionization state Z; here, we solve (86.17) for solid density aluminum with the ionization potentials shifted by an amount determined by electron screening in the Debye approximation, ∆I = Z e2 /De . Although we do not plot Z for this case when the number of particles in a Debye sphere is less than ten, it is obvious that a somewhat higher degree of ionization exists when continuum lowering is accounted for. A recent experiment [86.46] suggests that the Inglis– Teller prescription for line merging accurately describes the disappearance of the uppermost members of a spectral series. But simply truncating the number of bound states and, hence, the internal partition function, does not yield a self-consistent thermodynamic description of the plasma [86.41, 47]. In this regard, the true situation in dense plasmas is far more complicated for two reasons, and both give rise to a gradual disappearance of high-lying bound states. First, excited ionic states can be strongly perturbed by one or more nearby ions, which means that (as the density increases) bound, quasi-molecular states form and eventually evolve to a conduction band. Models with names such as “incipient Rydberg states” [86.48], “quasi-localized states” [86.45], “cluster states” [86.49], “negative-energy continuum states” [86.50], and “collectivized states” [86.51] have been developed to capture the complicated physics of this intermediate regime. Second, space- and timedependent density fluctuations give rise to different perturbing configurations, which means that the plasma is more accurately described by the average of an ensemble of perturbed ionic states than by the individual states of an ion experiencing the mean (usually spherical) perturbation. In the chemical picture, the most common approach [86.41, 51, 52] reduces the effective statistical weight of each (unperturbed) ionic state by a factor representing the probability that the plas-

ma’s microfield is sufficiently strong to Stark ionize it. The actual inclusion of dynamical plasma screening effects on ionic bound states requires a much more elaborate model [86.53] that, as yet, has not been incorporated into computer codes simulating high energy-density plasma experiments. Other computational studies, involving simple continuum lowering prescriptions [86.54, 55], indicate that an accurate treatment of this phenomenon is essential for understanding non-LTE, as well as LTE, situations.

86.2.3 Dense Plasma “Physical” Models An alternative strategy abandons the distinction between atomic and plasma electrons; this is known as the “physical picture” [86.47]. The simplest model that accomplishes this is that of a nucleus centered in a charge-neutral, spherical cell of radius Rs . An electronic structure calculation for the total electron density n e (r) at temperature Θ, subject to the boundary condition dn e (Rs )/ dr = 0, is carried out and, once the density is known, various physical quantities can be obtained. The advantage of this approach is that effects such as continuum lowering and degeneracy are naturally and self-consistently incorporated. Models of this kind are referred to as either “statistical” models or as “average atom” (AA) models depending on the manner in which the electronic structure is determined. The accuracy of the approach depends on both the sophistication with which the density is computed and the validity of the spherical cell boundary condition. The simplest way to obtain the electronic density is with a statistical model, such as the finitetemperature Thomas–Fermi approximation and its various extensions to include exchange (“Thomas– Fermi–Dirac”) and gradient corrections (“Thomas– Fermi–Dirac–Weizsacker”); these models are covered in detail in Chapt. 20 for free atoms at Θ = 0. Briefly, the Thomas–Fermi (TF) model describes atomic charge densities by treating all electrons as a partially degenerate Fermi gas subject to a spherical, self-consistent electrostatic potential ΦTF (r) resulting from the nuclear charge Z n e and the electrons themselves. Given the simplicity of the TF model, agreement with experiment (for binding energies) is surprisingly good, usually well within a factor of two for the thousands of ions in the periodic table. Feynman and coworkers [86.56] were the first to use such models to describe hot, compressed atoms and their thermodynamic properties. Quantities such as the

Atoms in Dense Plasmas

internal energy, free energy, and pressure are readily computable. Extended Thomas–Fermi models are useful for describing properties of matter in dense, cold stars [86.57], for example. As we have emphasized, a quantity of particular interest is the average ionization state Z of the plasma, which can be determined from the electronic density that extends from cell to cell, viz., 4π 3 Z = R n e (Rs ) . (86.18) 3 s The definition (86.18) is not unique, however, and some authors prefer to define bound electrons as all those in negative energy states [86.58, 59]. It is interesting to note that this intuitive definition of Z, (86.18), is considerably different from that used in the Saha formulation. This Z generally will have a nonzero value even at Θ = 0, a phenomenon known as “pressure ionization,” because the ionization occurs solely due to the finite value of Rs . More [86.59] published a convenient prescription for finding ionization potentials and total energies, as predicted by the TF model, of ions with net charge Z i e between 0.1 Zn e and 0.9 Zn e. In the lower panel of Fig. 86.3 we show Z based on the More/TF result. There is general agreement with Saha at high temperatures, where ionic bound states are much smaller than the interparticle spacing, but important differences occur at low temperatures. Average atom models extend the statistical models by directly employing the Schrödinger equation

86.3 Elementary Processes

for the electron structure. Typically, a self-consistent electronic structure calculation is carried out such that the single-particle levels are thermally populated according to a Fermi–Dirac distribution. These models describe atomic shell structure, which is absent in the statistical models. Modern versions of AA are detailed quantum mechanical calculations based on, usually, finite-temperature density functional theory (DFT), with some approximation for the exchange-correlation potential. A good review of the finite-temperature DFT approach has been given by Gupta and Rajagopol [86.60], and [86.61] contains several numerical comparisons. This approach was pioneered by Rozsnyai [86.62, 63], who employed a TF approximation for the free electrons, and by Liberman [86.64] who constructed an AA based on a self-consistent field model with a thermal population of Dirac orbitals for all states. There are two major weaknesses of the AA method. First, the spherical cell neglects asymmetrical ionic configurations in the plasma and assumes that no ion can penetrate within the radius Rs . And, the AA does not straightfowardly yield the distribution of ionic stages, which is important for opacity and transport calculations. Ying and Kalman [86.65] have introduced a model that addresses the Z issue while also incorporating strong ionic correlations from neighboring ions. A DFT-based model that describes both strong coupling and the distribution of ionic stages also has been published [86.66].

centers in a dense plasma. Presently, analysis of any of these many-body problems requires considerable simplification.

86.3.1 Radiative Transitions and Opacity For a radiative transition between atomic states α and β, the absorption and emission rates are proportional to quantities of the form (∆E)n |α|d|β|2 summed over degenerate substates, where d is the electric dipole operator Chapt. 10, and n = 1 or 3 for the Einstein B and A coefficients, respectively. In a dense plasma, changes in these radiative quantities are due primarily to changes in the atomic wave functions. Theory predicts that plasma screening reduces line strengths, and that the reduction factor increases toward the series limit [86.67, 68].

Part G 86.3

86.3 Elementary Processes In a truly equilibrium plasma, atomic transitions do not modify the plasma’s physical state. However, the evolution of LTE and nonequilibrium plasmas is regulated by the time rate of change of quantities such as Θ and n e , and these in turn depend on transport coefficients such as the radiative opacity and the thermal conductivity. The processes controlling these coefficients are induced by various radiative and collisional interactions. Indeed, so many processes can occur that a major task is the identification of those which are most important in a particular situation. The plasma environment may also alter rates applicable to isolated atoms, through the perturbation of the atomic states involved and/or the screening of long-range Coulomb forces. Further complicating the usual two-body collision picture is the close proximity of many scattering

1311

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Part G 86.3

Also, the cross section for photoejection of an electron bound by any screened Coulomb potential must vanish at threshold – in marked contrast to the nonzero photoionization cross sections of isolated atoms and ions. Since the oscillator strength sum rule still holds, any diminution of the total bound-bound oscillator strength must be offset by an increase in the bound-free contribution; continuum lowering partly accounts for this latter increase. Spectroscopic observations of the reduction of decay rates by plasma screening have been reported [86.69]. But, the plasma densities evidently were too low for this effect to clearly manifest itself, and alternative explanations have since been given [86.70, 71]. In addition, there is the more fundamental question of whether static screening models are even appropriate for the description of radiative processes. As discussed below, this issue also arises in connection with inelastic collision processes in dense plasmas. Several large-scale computer codes are in wide use to calculate the opacity of hot, dense matter (in LTE). Among these, we note the code HOPE [86.62], which is based on the average atom model; the code LEDCOP [86.72], which uses accurate (Hartree–Fock) atomic term data; and the code OPAL [86.73], which uses detailed configuration accounting and parametric, (static) screened potentials to compute wave functions and energy levels. Also, there are some published results from the new code IDEFIX [86.74], which is based on a non-spherical (di-center) screened potential arising from the radiating ion and its nearest neighbor. When making comparisons among these models, it should be realized that the codes use quite different line-broadening and continuum lowering prescriptions.

86.3.2 Collisional Transitions Various screened Coulomb interactions also can be used to study plasma effects on inelastic scattering. References [86.31, 75] and citations therein use either Debye or ion-sphere potentials, and Born, distortedwave, or close coupling approximations, to investigate excitation processes in plasmas; however, bound states were left unperturbed. More elaborate static potentials and perturbed bound states were treated by Davis and Blaha [86.76, 77], but they did not self-consistently screen the interaction between projectile and target. For excitations involving a small transition energy ∆E, Kitamura [86.78] has recently published a self-consistent treatment of both (1) the quasi-static perturbations of

the target ion by the microfield, and (2) the dynamically screened electron-target interaction. The use of static screening models is invalid when ∆E  ~ωe because the collision duration is too short for any average description of the plasma’s screening to apply. In such cases, ionization being a particular example, one must consider the response of the target to electrodynamic disturbances [86.79]. Reference [86.80] gives a thorough discussion of this issue, and presents numerical examples of the effects of projectile and target screening in ionizing collisions. Bremsstrahlung is another important plasma collision process for which static screening models have been extensively used [86.81–83]. Unfortunately, most bremsstrahlung radiation emanating from hot plasmas represents free–free transitions in which ∆E  ~ωe (lower frequency emission being attenuated), and in these situations static screening models are suspect. In contrast, the formation of laser plasmas occurs mainly through inverse bremsstrahlung (free–free absorption) under conditions such that ∆E = ~ωlaser
~ωe , making static screening models relevant here. More sophisticated treatments of bremsstrahlung in dense plasmas [86.84–86] include one or more of the following: strong coupling effects among the plasma ions (introduced via radial distribution functions [86.22]), dynamic screening effects involving the electrons (introduced via frequency-dependent dielectric response functions [86.18]), possible degeneracy effects (introduced via Fermi–Dirac distribution functions for occupation probabilities of initial and final states), and only partial screening of the nuclear charge by the target ion’s bound electrons (introduced via a form factor for the target (Chapt. 56)). Calculations for plasmas with moderate coupling parameters (Γ ≤ few) reveal that the first three of these effects tend to reduce free-free emission and absorption rates, while the last effect tends to enhance the rates. At larger Γ -values, strong ion-ion coupling tends to drive these rates back up [86.87]. Advances in simulation capability (which we discuss next) are yielding ever more realistic descriptions of the dense plasma environment, but what is proving difficult to improve upon is the ubiquitous use of the Born approximation to treat all electron-ion scattering events (see, however, Berkovsky and Kurilenkov [86.34]). Strong collisions, i. e., those in which the photon energy ~ω is comparable with the relative kinetic energy of the collision, are particulary important for radiative losses from plasmas, but these also are just the collisions most likely to be poorly described by the Born approximation.

Atoms in Dense Plasmas

86.4 Simulations

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86.4 Simulations Most of the challenges pertaining to atomic phenomena in dense plasmas arise from the many-body nature of the atom-plasma interaction. Simple models are therefore subject to inaccuracies that may arise from inconsistencies or severe approximations which, in turn, degrade our understanding of experimental data. Because of this complexity, simulations are playing an increasingly important role in this field. Historically, simulations of dense plasmas have used simplified plasma models to address issues ranging from equations of state [86.88, 89] to plasma microfields [86.27, 28, 90]. Atomic physics is typically ignored within the simulation by assuming that an average charge Z, somehow known, can be assigned to each ion. For these simulation methods, there are many excellent textbooks that introduce the basic ideas [86.91–93]. Here, we focus instead on simulations that attempt to describe both the dense plasma and the atomic physics self-consistently. These simulation methods are categorized in terms of the underlying algorithm used and fall either into Monte Carlo or molecular dynamics methods; we discuss each in turn.

86.4.1 Monte Carlo

(86.19)

and, in principle, we could sample the multidimensional integral randomly to obtain a good estimate of the average value of O given a many-body Hamiltonian H. In practice, however, there are large portions of phase space that give very little contribution to the average and some method of “importance sampling” must be carried out. This problem was originally solved by Metropolis and coworkers [86.94] who introduced the Metropolis method, which uses a Markov chain of states in phase space that preferentially migrates towards states of higher probability [86.92]. The computation of properties of atomic systems in dense plasmas requires a quantum Monte Carlo (QMC) method because atomic systems are inherently quantum systems and the plasma itself can be degenerate. Al-

This expression can be used recursively to obtain matrix elements evaluated at higher and higher temperatures; this allows a high-temperature approximation to be made, albeit at the expense of having many more matrix elements to evaluate. It can be shown that in the simplest approximation each quantum particle can be replaced by a polymer of M classical particles linked by springs; this picture is referred to as the “classical isomorphism” [86.96]. All electrons in the system (bound and free) are treated on an equal footing. Although the PIMC method is, in principle, simple to implement and can be quite accurate, there are several issues that arise when it is applied to dense plasmas. There are difficulties with the deep attractive Coulomb well that is crucial for describing atomic physics; this leads to the need for enormous numbers of fictitious classical particles. A partially analytical or numerical solution can greatly mitigate this problem [86.93, 97]. PIMC also suffers from difficulty when treating fermion systems because of the so-called fermion sign problem in which many terms of opposite sign arise from the antisymmetric form of the N-electron wave function. Progress has been made in this direction as well [86.95]. Finally, the long-ranged nature of the Coulomb potential causes additional difficulties for describing bulk systems with periodic boundary conditions [86.98]. When these additional considerations are taken into account, good agreement with other methods is found, and results have led to important conclusions about experiments [86.99], which are detailed below.

86.4.2 Molecular Dynamics Molecular dynamics is a simulation method based on the time evolution of a many-body system [86.91–93].

Part G 86.4

Monte Carlo is a method to evaluate average quantities in thermodynamic equilibrium using random numbers. For example, to obtain a property of a classical system we might write  3N 3N   d r d p O r 3N , p3N exp (−βH)  O = , d3N r d3N p exp (−βH)

though there are a variety of QMC methods [86.93, 95], the most important for our purposes is the path-integral Monte Carlo (PIMC) method, which is formulated at finite temperatures and, in fact, exploits this condition by constructing an equivalent system of many more particles at a higher effective temperature. This is achieved by writing spatial matrix elements of the quantal version of the Boltzmann factor of (86.19) as    r| exp −β Hˆ r     r  ˆ = d3r  r| exp −β H/2      r  . ˆ (86.20) × r   exp −β H/2

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Part G 86.4

The vast majority of MD simulations are based on the solution of Newton’s equations for N classical particles in a main cell with periodic boundary conditions. Simulations can be carried out in various ensembles and one assumes that long simulations in the canonical ensemble will correctly sample canonical averages in the same manner as (86.19). Atomic properties in dense plasmas are usually obtained in the Born–Oppenheimer approximation, which freezes the ionic dynamics between time-steps and performs an electronic structure calculation to obtain the electronic density for that particular ionic configuration. The ions are then advanced using the forces from the resulting electronic density, and the procedure is repeated. Many methods are available, including basic TF [86.100], Hartree–Fock [86.101], and tight-binding [86.102] to compute the electronic structure. A commonly-used DFT code for warm matter is the Vienna Ab-Initio Simulation Package (VASP), which is widely distributed [86.103]. In contrast to most PIMC implementations, the deep Coulomb potential is often treated in VASP by softening the electron-ion interaction with some form of pseudopotential [86.103,104]. These methods trade between accuracy and range of validity; for example, TF theory is less accurate than DFT-MD, but is useful at higher temperatures (Θ > 50 eV) and for very dense plasmas. There are three advantages to the MD method. First, the MD approach greatly extends the AA model by including many nuclear centers in the main simulation cell, and these centers may arise from different elements. (This can also be done with PIMC.) Futhermore, DFT-MD, which can be very accurate for cold systems ([86.105], p. 117), has the additional advantage that single-particle (Kohn–Sham) orbitals can be used in formulae for linear, frequency-dependent response properties, such as the electrical conductivity. (It may seem paradoxical that electron dynamical information can be obtained from a static calculation. Strictly speaking this is not possible, although reasonable results can be obtained for some quantities [86.105, p. 49]). And finally, MD has the additional advantage that dynamical ionic properties can be obtained from the time evolution that is simulated. For example, DFT-MD simulations for the computation of the self-diffusion coefficient of dense hydrogen [86.106] have recently been performed; in principle, a wide range of other dynamical ionic properties are available, such as viscosity, thermal conductivity, and collective behavior. Quantum simulations beyond the Born–Oppenheimer approximation that treat electrons and ions on

an equal, dynamical footing are much more difficult since they involve a numerical solution of the N-body Schrödinger equation. Such simulations are necessary, however, for obtaining dynamical electron properties, especially under nonequilibrium conditions – those with time-dependent temperatures or nonthermal momentum distributions. Furthermore, these simulations can describe electronic properties, such as atomic physics, strong scattering, and degeneracy. Very simple models have been developed by Deutsch and coworkers, who constructed effective interactions between the electrons that yield some known property, such as the high-temperature pair correlation function [86.107]. These interactions can then be used directly in a simulation, but with modified equations of motion – an approach pioneered for dense plasmas by Hansen and coworkers [86.108]. Diffractive and symmetry effects can be accounted for in the high-temperature limit. For example, one model of the diffractive potential for the electron-ion interaction is vei (r) = −

 Ze2

1 − exp (−r/λ) , r

(86.21)

where λ is on the order of the electron deBroglie wavelength. This potential is finite at the origin, which prevents the classical collapse of a neutral system during the simulation. Unfortunately, such a method suffers from several weaknesses, including being limited to high temperatures, only describing Pauli exclusion by pair-interactions, and having incorrect atomic binding energies. There have been several attempts to improve upon simple potentials of the form (86.21). Since one of the main features of quantum mechanics is that conjugate space- and momentum-dependent quantities do not commute, it is natural to construct potentials v(r, p) that depend on both r and p. Such momentum-dependent potentials have been formulated in the context of nuclear physics and have recently been applied to dense plasmas by Ebeling and coworkers [86.109]. Although quite good atomic properties can be obtained, these potentials suffer from the fact that they are ad hoc, and one does not know how to choose adjustable parameters for unexplored conditions. A direct approach is to solve the time-dependent, many-particle Schrödinger equation, albeit approximately. This can be done by reducing the (infinite) degrees of freedom to a smaller, more manageable set. For example, Heller [86.110] first suggested using a Gaussian wavepacket to describe electron semiclas-

Atoms in Dense Plasmas

sical dynamics. This has been applied to ionization in a dense plasma by Ebeling [86.111]. In general, a many-body, antisymmetric wave function can be parametrized in terms of a few parameters for each particle and a time-dependent variational principle can be used to obtain the equations of motion of the parameters. For example, each particle in a Slater determinant can be chosen to be a Gaussian with a width w(t) with its conjugate momentum pw (t). This method is referred to as “Fermion molecular dynamics” or “wavepacket molecular dynamics” (WPMD). A review of this approach has recently been given by Feldmeir and Schnank [86.112]. Wavepacket shapes other than Gaussians, which can better reproduce atomic properties, have been proposed by Murillo and Timmermans [86.113].

86.4.3 The Deuterium EOS Problem

1315

Pressure (Mbar)

PIMC WPMD 1 LLNL/Nova

SNL/Z

0.1 0.5

0.6

0.7

0.8

0.9

1.0 1.1 Density (g/cm3)

Fig. 86.4 The equation of state of shocked deuterium; here

pressure (in Mbars) versus density (in g/cm3 ) is plotted along the Hugoniot. Results from two experiments are shown: Sandia (triangles) and Livermore (circles). Also shown are the results from two simulations: path integral monte carlo (diamonds) and wavepacket molecular dynamics (squares)

showed that details of antisymmetric wave functions become important below 2 Mbar. These results are shown in the figure, and better agreement with the Sandia result is found. Also shown in the figure are results from a WPMD calculation [86.118], which tend to agree with the Livermore data. The WPMD calculations, however, did not include full antisymmetrization of the electronelectron interaction. The temperatures predicted by the simulations tend to be in the fraction of an eV range at the lower part of the figure and up to tens of eV toward the top of the figure; thus the experiments and the simulations are probing the very interesting WDM regime in which molecular and atomic species are heated into a cool plasma state. Together with experiments on resistivity and Z, a more complete picture of the physics of atoms in dense plasmas is emerging. But, more theoretical and experimental developments are needed before we can tackle with confidence the wide range of dense plasmas that occur in the HEDP regime.

Part G 86.4

Experiments can be notoriously difficult to perform in the dense plasma regime because of the enormous pressures produced. Recently, experiments on compressed deuterium have been performed that pass from the molecular fluid phase into the dense plasma phase and therefore probe the physics of atomic and molecular states in a dense environment. The first of these was conducted at Livermore [86.114] using the Nova laser to shock compress liquid deuterium to 2 Mbar. The experiments indicated a higher compressibility compared with commonly used equation of state properties. These interesting results led to new experiments, again at Livermore [86.115], with pressures exceeding 3.0 Mbar, and at Sandia using a magnetically driven flyer plate on the Z machine [86.116] to achieve 0.7 Mbar. The Sandia results did not show the unexpected higher compressibility. Later, additional laser-based experiments were carried out at the Naval Research Laboratory [86.117] to 6 Mbar, which agreed with the orignial laser-based experiments but had significant error bars. The Livermore and Sandia results are shown in Fig. 86.4. (Error bars are not shown.) Interpreting these experiments has, in turn, led to increased activity in the use and development of various simulation methods. Also shown in Fig. 86.4 are results from PIMC and WPMD simulations. Early PIMC results, which treated the Fermion sign problem using properties of free particles, did not agree well with either experiment. A later calculation [86.99] improved upon that treatment and

10

86.4 Simulations

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References 86.1 86.2 86.3

86.4

86.5 86.6

86.7

86.8

86.9 86.10 86.11 86.12 86.13

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86.18 86.19 86.20 86.21 86.22 86.23

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86.74

86.75 86.76 86.77 86.78 86.79 86.80 86.81 86.82

86.83 86.84 86.85 86.86 86.87 86.88 86.89 86.90 86.91 86.92

86.93 86.94 86.95 86.96 86.97 86.98 86.99 86.100 86.101 86.102 86.103 86.104 86.105

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86.73

J. Al-Kuzee, T. A. Hall, H.-D. Frey: Phys. Rev. E 57, 7060 (1998) J. Seidel, S. Arndt, W. D. Kraeft: Phys. Rev. E 52, 5387 (1995) S. R. Stone, J. C. Weisheit: J. Quant. Spectrosc. Radiat. Transfer 35, 67 (1986) C. A. Iglesias, R. W. Lee: J. Quant. Spectrosc. Radiat. Transfer 58, 637 (1997) R. P. Feynman, N. Metropolis, E. Teller: Phys. Rev. 73, 1561 (1949) D. Lai, A. M. Abrahams, A. L. Shapiro: Astrophys. J. 377, 612 (1991) W. Zakowicz, I. J. Feng, R. H. Pratt: J. Quant. Spectrosc. Radiat. Transfer 27, 329 (1982) R. M. More: Adv. Am. Mol. Phys. 21, 305 (1985) U. Gupta, A. K. Rajagopal: Phys. Rep. 87, 259 (1982) P. Fromy, C. Deutsch, G. Maynard: Phys. Plasmas 3, 714 (1996) B. F. Rozsnyai: Phys. Rev. A 5, 1137 (1972) B. F. Rozsnyai: Phys. Rev. E 55, 7507 (1997) D. A. Liberman: J. Quant. Spectrosc. Radiat. Transfer 27, 335 (1982) R. Ying, G. Kalman: Phys. Rev. A 40, 3927 (1989) F. Perrot, M. W. C. Dharma-Wardana: Phys. Rev. E 52, 5352 (1995) J. C. Weisheit, B. W. Shore: Astrophys. J. 194, 519 (1974) L. G. D’yachkov, G. A. Kobzev, P. M. Pankratov: J. Quant. Spectrosc. Radiat. Transfer 44, 123 (1990) Y. Chung, H. Hirose, S. Suckewer: Phys. Rev. A 40, 7142 (1989) Y.-C. Chen, J. L. Libowitz: Phys. Rev. A 41, 2127 (1990) F. Aumayr, W. Lee, C. H. Skinner, S. Suckewer: J. Phys. B 24, 4489 (1991) N. H. Magee, A. L. Merts, J. J. Keady, D. P. Kilcrease: Los Alamos Report LA-UR-97-1038 (1997) C. A. Iglesias, F. J. Rogers: Astrophys. J. 464, 943 (1996) E. Minguez, P. Sauvan, J. M. Gil, R. Rodriguez, J. G. Rubiano, R. Florido, P. Martel, P. Angelo, R. Schott, F. Philippe, R. Leboucher-Dalimier, R. Mancini: J. Quant. Spectrosc. Radiat. Transfer 81, 301 (2003) B. L. Whitten, N. F. Lane, J. C. Weisheit: Phys. Rev. A 29, 945 (1984) J. Davis, M. Blaha: J. Quant. Spectrosc. Radiat. Transfer 27, 307 (1982) J. Davis, M. Blaha: Physics of Electronic and Atomic Collisions (North-Holland, Amsterdam 1982) H. Kitamura: Phys. Plasmas 11, 771 (2004) J. C. Weisheit: Adv. At. Mol. Phys. 25, 101 (1988) M. S. Murillo, J. C. Weisheit: Physics Reports 302, 1 (1998) I. P. Grant: Mon. Not. Roy. Astron. Soc. 118, 241 (1958) B. F. Rozsnyai, J. Quant: Spectrosc. Radiat. Transfer 22, 337 (1979)

References

1318

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86.113 M. S. Murillo, E. Timmermans: Contr. Plasma Phys. 43, 333 (2003) 86.114 L. B. Da Silva, P. Celliers, G. W. Collins, K. S. Bundil, N. C. Holmes, T. W. Barbee Jr., B. A. Hammel, J. D. Kilkenny, R. J. Wallace, M. Ross, R. Cauble, A. Ng, G. Chiu: Phys. Rev. Lett. 78, 483 (1997) 86.115 G. W. Collins, L. B. Da Silva, P. Celliers, D. M. Gold, M. E. Foord, R. J. Wallace, A. Ng, S. V. Weber, K. S. Bundil, R. Cauble: Science 281, 1178 (1998)

86.116 M. D. Knudsen, D. L. Hansen, J. E. Bailey, C. A. Hall, J. R. Asay, W. W. Anderson: Phys. Rev. Lett. 87, 225501 (2001) 86.117 A. N. Mostovych, Y. Chan, T. Lehecha, A. Schmitt, J. D. Sethian: Phys. Rev. Lett. 85, 3870 (2000) 86.118 M. Knaup, P.-G. Reinhard, C. Toepffer, G. Zwicknagel: J. Phys. A 36, 6165 (2003)

Part G 86

1319

87. Conduction of Electricity in Gases

Conduction of

The physics and chemistry of the conduction of electricity in ionized gases involves the interactions of the electrons and ions in the gas among themselves, with ground-state and excited-state gas atoms or molecules, with any surfaces that may be present, and with any electric or magnetic fields that are externally applied or generated by movement of the charged species. The electron collision-induced excitation and dissociation can result in new compounds being formed at rates which are

87.1

Electron Scattering and Transport Phenomena.............................................. 1320 87.1.1 Electron Scattering Experiments .. 1320 87.1.2 Electron Transport Phenomena ... 1321 87.1.3 The Boltzmann Equation ............ 1321 87.1.4 Electron-Atom Elastic Collisions .. 1322 87.1.5 The Electron Drift Current ........... 1322 87.1.6 Cross Sections Derived from Swarm Data........... 1326

87.2 Glow Discharge Phenomena ................. 1327 87.2.1 Cold Cathode Discharges ............ 1327 87.2.2 Hot Cathode Discharges.............. 1327 87.3 Atomic and Molecular Processes ............ 1328 87.3.1 Ionization ................................ 1328 87.3.2 Electron Attachment .................. 1329 87.3.3 Recombination ......................... 1330 87.4 Electrical Discharge in Gases: Applications .................................................. 1330 87.4.1 High Frequency Breakdown ........ 1331 87.4.2 Parallel Plate Reactors and RF Discharges ..................... 1331 87.5 Conclusions ......................................... 1333 References .................................................. 1333 ionization by electron collision, electron attachment, ion mobility, ion–ion and electron–ion recombination, and other important processes that affect the conduction of electricity in gases. Section 87.4 illustrates the importance of gaseous electronics with several important phenomena and technical applications.

orders of magnitude larger than those without a plasma present. For a gas pressure of 1 torr, the electron mean free path λmfp between collisions is ∼1 mm (∼ 109 collisions/second). The assumption that λmfp is small relative to the plasma physical dimensions is usually valid, however it is often comparable to, or larger than the space charge sheaths that form near electrodes and surfaces. Since the electron–atom interactions, in most cases of interest, have ranges shorter than the average gas-atom

Part G 87

The conduction of electricity through gases has played ubiquitous roles in science and technology. It was responsible for many of the fundamental discoveries in atomic and molecular physics; gas discharge lighting is essential to every night operations; gas discharge lasers are still important in research and manufacturing; and all of advanced microelectronics depends on plasma enhanced processing. To a large extent, the efficiencies of the above cited applications of gaseous electronics depend on the maintenance of the distinct non-equilibrium between the electrons and the gas or vapor. This nonequilibrium can be achieved by operating at low pressures or under pulsed excitation, where the duration of the energy input is less than the energy equilibration time between the electrons and the heavy particles. The term gas discharge originally described a transient or spark condition, but has been extended to mean the continuous conduction of electricity through gases. Section 87.1 treats the electron-velocity distribution and its effect on various measurements involving a swarm or distribution of electron velocities. In this section, low fractional ionization (< 10−6 ) is assumed; electron–ion collisions are negligible relative to electron–atom (and electron–molecule) collisions in so far as they affect electron mobility, diffusion or energy loss. Section 87.2 introduces the glow discharge and considers the cold cathode and hot cathode discharge phenomena. Section 87.3 discusses

1320

Part G

Applications

separation, they may be treated as binary collisions and the effects of neighboring gas atoms on a given electron– atom encounter can be neglected. Quantities involving such interactions in a partially ionized gas, where the electrons are distributed both in velocity and position, can be directly related to the corresponding quantities in

beam experiments, where the collisions are binary and the electrons have a defined velocity. The measurements and calculations of the electron velocity distributions, or of the electron energy distribution functions (EEDF), are therefore of fundamental importance in the description of partially ionized gases.

87.1 Electron Scattering and Transport Phenomena 87.1.1 Electron Scattering Experiments In 1903, Lenard [87.1] determined the attenuation of a beam of mono-energetic electrons by several gases. He measured the fraction of an electron beam that was transmitted without scattering through a field-free region containing the gas. Let an electron beam of density n e electrons/cm3 , traveling with velocity v, pass a distance dx through the gas. Let N be the number of atoms/cm3 . The loss of electrons by scattering per unit time due to collisions with atoms is then given by dn e = −Nvσn e , (87.1) dt where σ is the collision cross section for electrons of velocity v. Writing v dt = dx, we obtain n e = n e0 e−Nσx .

(87.2)

In relation to the conduction of electricity in gases, a more useful concept than σ is the momentum transfer

cross section σm . In general, σm is not equal to σ. Experimental determinations of σm , generally from electron swarm experiments, are summarized in [87.2]. Results for the noble gases are presented in Fig. 87.1 and for representative diatomic gases in Fig. 87.2. Above energies of about 10 eV, σm decreases as the electron energy increases. Towards higher electron energies in the monatomic gases, σm is inversely proportional to the ionization potential and directly proportional to the gas polarizability. The fact that σm has extremely low values in certain gases at low energies is known as the Ramsauer–Townsend (RT) effect. This can be explained by partial wave scattering theory as the point where the radial wave function has exactly one more oscillation inside the interaction potential well than does the corresponding free function. Outside the well the two functions are indistinguishable and there is no  = 0 (s-wave) scattering [87.3] (Sect. 45.2.7). For the scattering potentials of the heavier noble gases, this occurs at σm(10–16cm2)

100

σm(10–16cm2)

10

Part G 87.1

10

1 1 Helium Neon Argon Krypton Xenon

0.1 0.001

0.01

0.1

1

10

ε (eV)

Fig. 87.1 Momentum transfer cross section for noble gases

N2 O2 H2 CO 0.1 0.01

0.1

1

10

ε (eV)

Fig. 87.2 Momentum transfer cross section for some diatomic gases

Conduction of Electricity in Gases

low energies where the partial cross sections for the higher angular momenta ( = 1, 2, . . . ) are still negligible. Methane and silane (Fig. 87.3, [87.4, 5]) and possibly other nonpolar polyatomic gases, also display Ramsauer–Townsend (RT) minima. There is great similarity between the σm curves for atoms and molecules with similar external electron arrangements. The σm curves for H2 , O2 , N2 and CO (Fig. 87.2) show that these cross sections are large over the range 0.1 to 10 eV. Around 1.5 eV, N2 and CO display resonances which are associated with the formation of temporary negative ions followed by efficient vibrational excitation of these molecules [87.6].

87.1 Electron Scattering and Transport Phenomena

1321

The electron drift velocity wd is related to Γe and n e by wd = Γe /n e .

(87.5)

The electron drift velocity is usually small compared with the electron random velocity. The other limiting case is important in connection with ionic mobilities. It can also occur in electron transport in Ramsauer gases or gas mixtures that are used in fast particle counters [87.7]. The theoretical analysis of electron motion in gases uses primarily the Boltzmann transport equation approach [87.8]. Mean free path methods [87.9] and Monte-Carlo methods [87.10] have also been applied.

87.1.3 The Boltzmann Equation 87.1.2 Electron Transport Phenomena Electron transport phenomena in an ionized gas, such as diffusion under the influence of a density gradient and drift under the influence of an electric field, are directly related to the electron current density Γe which is caused by these influences. In order to calculate Γe , it is necessary to know the electron spatial and velocity density distribution f(v, r, t). The electron density n e and current density Γe are given by
f(v, r, t) d3 v , (87.3) n e (r, t) =
Γe (r, t) = v f(v, r, t) d3 v . (87.4)

100

σm(10–16cm2)

10

CH4 SiH4 0.1 0.01

0.1

1

10

ε (eV)

Fig. 87.3 Momentum transfer cross section for CH4 and

SiH4

Part G 87.1

1

The electron density distribution f(v, r, t) of a given kind of particle in phase space (configuration and velocity space) is determined by the combined effect of all the interactions to which the particle is subjected. If there are no sources or sinks, the number of particles in a volume element of the six-dimensional space which moves with the particles does not change with time. In configuration space, the continuity equation for f(v, r, t) is ∂f + ∇r · v f = 0 ; (87.6) ∂t in phase space, it has the form ∂f + ∇r · v f + ∇v · a f = 0 , (87.7) ∂t where the subscripts on the divergence operators denote the independent variable, v is the particle flow velocity in configuration space, and a is the particle acceleration (i. e., the flow velocity velocity space). This may be written as q(E + v × B)/m for a particle of charge q and mass m in the presence of electric and magnetic fields, E and B respectively. If there are sources or sinks of particles, then terms representing these appear on the right-hand side of (87.7). In principle, the effect of elastic and inelastic collisions may be included in the expressions for E and B, but they are usually treated as source terms which transport the colliding particles instantaneously from one volume element of velocity space to another. This is a good approximation, for example, in the case of electron– atom collisions, but it is not valid for the longer range electron–ion and electron–electron interactions in the presence of high-frequency electric fields. Other collisions, such as ionizing collisions, act to change the total n e as well as transporting electrons from one element of velocity space to another.

1322

Part G

Applications

The electron–electron collisions will produce a Maxwellian velocity distribution when the fractional ionization is sufficiently large. For molecular gases, this generally requires n e /N0 > 10−5 . Since the electron– electron Coulomb collision frequency varies as v−3 , these collisions are especially effective in the low energy elastic regime where there are no inelastic energy loss processes, and they may have an influence for n e /N0 as low as 10−7 for the noble gases [87.11]. Spitzer [87.12], Dreicer [87.13], and Butler and Buckingham [87.14] treat cases where the electron–electron term is included explicitly. A full treatment of this subject is given by Shkarofsky [87.15]. Using ∇v · a = 0, and including collisions as a source term, (87.7) becomes   ∂f ∂f + v · ∇r f + a · ∇v f = , (87.8) ∂t ∂t collisions where the subscript ‘collisions’ identifies the source term due to all types of collisions.

87.1.4 Electron-Atom Elastic Collisions

Part G 87.1

For electrons in a weakly ionized gas, where only electron–atom elastic collisions need be explicitly considered, the collision term is given by the Boltzmann collision integral [87.16]. This integral describes the rate at which electrons are brought into and removed from an element of volume in velocity space in terms of the differential elastic scattering cross section . The Boltzmann equation may be solved by a perturbation method [87.8] in which E, B, and the density gradients are the perturbations, starting from v f(v, r, t) ≈ f 0 (v, r, t) + · f1 (v, r, t) , (87.9) v where f 0 is the unperturbed isotropic term and f1 is the anisotropy due to E and B. Substituting into (87.8) and equating terms of the same angular type then yields  ∂ f0 v e ∂  2 + ∇r · f1 − v E · f 1 2 ∂t 3  3mv ∂v  kB Tg ∂ f 0 g ∂ 3 v νm f 0 + = 2 (87.10) mv ∂v 2v ∂v and eE ∂ f 0 ∂ f1 + v∇r f 0 − + ωB × f1 = −νm f1 , ∂t m ∂v

(87.11)

where νm = eE/(wd m) is the momentum transfer collision frequency from the Boltzmann collisions integral,

ωB = eB/m is a vector whose magnitude is the cyclotron frequency, and g = 2m/(M + m) controls the partition of kinetic energy in an elastic collision between particles with masses M and m. The right-hand term of (87.10) is the first-order correction due to elastic kinetic energy exchange between electrons and atoms. The E · f1 term arises from work done by E on the electrons, and the ∇r · f1 term accounts for particle loss from the volume element in phase space due to electric field drift and density gradients. Similarly, in (87.11), the E term accounts for loss due to electric field acceleration, and the v∇r f 0 term accounts for changes due to electron drift across a density gradient. The −νm f1 term results from randomization of the electron velocity direction in electron–atom collisions. The neglect of higher order terms in (87.9) is justified if δTe /Te  1, where Te is the gain in electron kinetic energy between collisions, and δ f 0 / f 0  1, where δ f 0 is the change in f 0 over λmfp . For the case of static E and B with E ⊥ B, the orbital time 1/ωB replaces the time between collisions if it is shorter, making the above conditions less stringent. However, for an ac field with the resonant frequency ωB , the original conditions apply. The off-resonant case in between. See [87.8, 17] for a further discussion of validity.

87.1.5 The Electron Drift Current Transport properties of a partially ionized gas are calculated from the moments of f(v, r, t). To obtain Γe , (87.11) must be solved for f1 . Assume that ωB = 0 and that E is constant in time. Then, since f 0 usually changes slowly compared with νm , ∂ f1 /∂t may be neglected to a first approximation, and    eE ∂ f v0 1 v∇r n e f v0 − ne , f1  − (87.12) νm m ∂v where n e (r, t) f v0 (v, r, t) = f 0 (v, r, t). Use of (87.3) gives Γe = −∇r (De n e ) − n e µe E ,

(87.13)

which defines the electron free diffusion coefficient

2
v v2 0 De = f v 4πv2 dv = , (87.14) 3νm 3νm and the static electron mobility
v ∂ f v0 e 4πv2 dv . µe = − (87.15) m 3νm ∂v In (87.13), ∇r should be interpreted as acting only on that part of De whose spatial dependence enters through an

Conduction of Electricity in Gases

energy change, such as a variation of Te with position (as in the theory of striations [87.18], but not on variations due to pressure gradients which would affect νm [87.8]). Electron Free Diffusion When the ionization density is low enough, the electrons diffuse freely to the walls, being unaffected by the space charge field caused by the heavier ions, which diffuse more slowly than the electrons. Setting E = 0 in (87.13), the electron current due to diffusion is

Γe = −∇r (De n e ) .

(87.16)

Since the electron heating and elastic recoil terms in (87.10) do not contribute to a net gain or loss of electrons, integration of (87.10) over v gives ∂n e ∂n e + ∇r · Γe = − ∇r2 (De n e ) = 0 , (87.17) ∂t ∂t for a decaying electron density. This equation is identical in form to the thermal conductivity equation, which can be solved analytically for certain simple geometries. For the case of a long cylindrical container of radius R, the equation becomes   ∂n e 1 ∂ ∂n e = De r , (87.18) ∂t r ∂r ∂r if D0 is constant and n e depends only on r. The general solution for n e satisfying n e (R, t) = 0 is    am r  t J0 , bm exp − (87.19) ne = tm R m=0

1323

For the case where a magnetic field is applied to the ionized gas, parallel to the cylinder axis so as to impede the electron diffusion to the walls, the free-diffusion coefficient in equation (87.18) is replaced by


νm v2 f v0 νm v2 2 4πv dv =  .  DeB = 2 + w2 2 + w2 3 νm 3 νm B B (87.22)

In this case, the electrons cannot diffuse to the walls unless they make collisions that interrupt their spiraling motion. Electron DC Mobility in Static Fields From (87.13), the electron particle current induced by an electric field E is Γe = −n e µe E, and the electron drift velocity is

wd =

Γe = −µe E . ne

(87.23)

Measurement of wd by a time-of-flight measurement can in principle, through the use of (87.14), provide some information about νm , provided that f v0 is known. Since a static electric field may perturb considerably the distribution function, and since n e is low in such measurements, f v0 should be determined using (87.10) after having eliminated f1 using (87.11). In the case where νm is independent of v, (87.15) after integration by parts, reduces to
e e µe = , (87.24) f v0 4πv2 dv = mνm mνm which is independent of f v0 . In general, wd exhibits a complex dependence on E/N, as shown for some example gases in Fig. 87.4 [87.19]. The high mobility of electrons in methane is of technical interest because of potential applications in plasma switches used for current interruption [87.20]. The local maximum of wd in methane for E/N = 3 Td (1 Townsend = 10−17 Vcm2 ) and its decrease with further increase of E/N arise because of the onset of inelastic collisions due to vibrational excitation at an energy close to the energy of the RT minimum [87.21]. As shown in Fig. 87.4, the drift velocities are generally in the range of 106 –107 cm/s. A detailed summary of electron drift velocities measured in many pure gases and gas mixtures of technical interest is given in [87.22, 23], and the techniques are elaborated in [87.24, 25]. The classic article on the drift velocity of ions in electric fields measured by a transit time technique is that

Part G 87.1

where tm = [De (am /R)2 ]−1 , and J0 is the zeroth-order Bessel function with roots J0 (am ) = 0, am+1 > am . The bm are determined by the radial dependence of n e at t = 0. (If the coaxial wall of radius r0 < R confines the plasma to an annular region r0 < r < R, the general solution must also include the zeroth-order Bessel function of the second kind, Y0 , to satisfy the boundary conditions). The time constants tm decrease rapidly with increasing m so that for sufficiently long t, the density distribution relaxes (e.g., after the microwave or static excitation is switched off) to    t a0 r  J0 . n e ≈ b0 exp − (87.20) t0 R This is called the fundamental diffusion mode. The above treatment gives a good approximation to n e (r) provided that λmfp  R. When νm is velocity-independent and f 0 is the Maxwellian of temperature Te , (87.14) becomes kB Te . (87.21) De = mνm

87.1 Electron Scattering and Transport Phenomena

1324

Part G

Applications

wd(cm/s)

(eV)

107

10

1 106 0.1 105

0.01 SiF4 CH4 SiH4

104 10–19

–18

10

–17

10

–16

10

0.001 0.001

H2
ε εk 0.01

0.1

1

–15

10 E/N (Vcm2)

10

100 1000 E/N (Td)

Fig. 87.5 Calculated u av and De /µe coefficients in H2

Fig. 87.4 Electron drift velocities in polyatomic gases

the generalization of (87.17) is of Hornbeck [87.26]. He showed that in some instances earlier workers had identified molecular ions as atomic ions. Contemporary measurements often use the guided ion beam (GIB) technique to measure such quantities as the symmetric charge transfer cross sections [87.27]. A review of measurement data has been given [87.28]. The Ratio of De /µe for Electrons From (87.14) and (87.15),  ∞ 4 v /νm f v0 dv De m = −  ∞0 3 . µe e 0 v /νm ∂ f v0 /∂v dv

(87.25)

The above ratio of the free-diffusion coefficient to the mobility is termed the characteristic energy and it is a measure of the average electron energy in certain cases. When f v0 is Maxwellian, then De /µe = kB Te /e

(87.26)

Part G 87.1

becomes a direct measure of the electron energy, independent of νm . Since the average electron energy u av = 32 kB Te , (87.26) becomes 2 De /µe = u av /e . (87.27) 3 In this form it is known as the Einstein relation. Furthermore, if νm is independent of velocity, (87.27) applies regardless of the velocity distribution. Values of De /µe can be determined from experiments where a static electric field E z is applied to electrons drifting under steady state conditions of diffusion. If De is taken to be independent of position, then

∇ 2ne = −

µe ∂n e . Ez De ∂z

(87.28)

Solutions to this equation for n e were for many years used to deduce De /µe from the measurements [87.29]. However, Wagner et al. [87.30] showed that there are two effective diffusion coefficients D L and DT in the longitudinal and transverse directions with D L  DT . For H2 , N2 and He, D L ≈ DT /2, while for Ar, D L ≈ DT /8. See [87.31, 32] for a quantitative discussion. The different coefficients arise because n e and wd are functions of ∂n e /∂z. For H2 , f v0 is fairly well approximated by a Maxwellian distribution. A comparison of the average and characteristic energies u av and De /µe calculated from the Boltzmann transport equation for H2 is shown in Fig. 87.5. Ambipolar Diffusion In the case of electron free diffusion discussed above, diffusion of the electrons (and ions) is unaffected by space-charge fields caused by an imbalance of positive and negative charges. When the charge density is high, this no longer holds. The resulting electric field caused by space-charge separation retards the electron diffusion and enhances the positive ion motion. In the steady state the flows of positive ions and of electrons are equal, i. e., Γ e = Γ + . For electrons in the absence of temperature gradients,

Γ e = −De ∇n e − n e µe E ,

(87.29)

Conduction of Electricity in Gases

where De and µe are given by (87.14) and (87.15) respectively. The analogous equation for the positive ions is Γ + = −D+ ∇n + − n + µ+ E .

The quantity in parentheses is a diffusion coefficient that is applicable both to the electrons and to the ions. Since they are interacting, they diffuse together. This quantity is termed the ambipolar diffusion coefficient: Da =

D+ µe + De µ+ . µe + µ+

(87.32)

Franklin, however, has proposed that the term ambipolar flow is a more correct physical description [87.33]. In the case where the electrons have a temperature Te and the ions a temperature T+ , D+ kB T+ , = µ+ e

De kB Te , = µe e

and since µe  µ+ ,   Te . Da ≈ D+ 1 + T+

(87.33)

(87.34)

DaB =

Da . 1 + µ+ µe B 2

(87.35)

References [87.35, 36] review theoretical and experimental work on diffusion in a magnetic field. Debye Shielding Consider an ensemble of positive ions and electrons in thermal equilibrium. Each ion repels other ions in its

∇ 2φ = −

e (n + − n e ) , 0

(87.36)

where n + is the average ion density. For time independent fields in the absence of collisions, (87.8) becomes v · ∇r f + eE · ∇v f = v · ∇r f − e∇r φ · ∇v f = 0 . (87.37)

Assuming a solution of the form   eφ , f = f 0 exp − kB Te where f 0 is Maxwellian, and using

(87.38)



f 0 d3 v = n e gives

  e v f ≡0. − v · ∇r φ + e∇r φ · kB Te kB T Hence,


  eφ . f d3 v = n e exp − kB Te

(87.39)

(87.40)

Since the assembly is macroscopically neutral, n e = n + , and    eφ e . ∇ 2 φ = − n + 1 − exp − (87.41) 0 kB Te If eφ  kB Te , then ∇ 2φ ≈

e2 φ n+ . 0 kB Te

If spherical symmetry is assumed,   e r , φ= exp − 4π0 r λD

(87.42)

(87.43)

where  λD =

0 kB Te n + e2

1/2 (87.44)

is the Debye shielding length for the Coulomb potential. λD arises from the screening effect of the electron cloud

Part G 87.1

If Te  T+ , then Da ≈ µ − kB Te /e, i. e., the ambipolar diffusion is determined by the ion mobility and the electron energy. Allis and Rose [87.34] have studied the transition from free to ambipolar diffusion as the electron density is increased. For the case in which the electrons and ions are diffusing together in the radial direction across a static Bz , and the electron–atom and ion collision rates may be assumed to be independent of Te , the magneto-ambipolar diffusion coefficient is given by

1325

neighborhood, but attracts electrons. The space charge around each ion (assumed stationary) is e(n + − n e ), which by Poisson’s equation creates a potential φ satisfying

(87.30)

Eliminating E between the two equations and setting n e ≈ n + = n, and Γ e ≈ Γ + = Γ gives   D+ µe + De µ+ ∇n . (87.31) Γ =− µe + µ+

87.1 Electron Scattering and Transport Phenomena

1326

Part G

Applications

about each ion. The dimensions of a plasma must be much larger than λD for the electrons to act collectively, i. e., to undergo plasma oscillations.

106

wd (cm/s)

87.1.6 Cross Sections Derived from Swarm Data

Part G 87.1

The use of transport theory to unfold low energy electron collision cross sections from experimentally measured electron transport data was originally introduced by Townsend [87.29]. Beginning with trial input values for the cross sections as a function of energy, the electron transport coefficients are calculated and compared with experiments (such as wd , De , k , and ionization/attachment coefficients, measured as a function of E/N). The input cross sections are then adjusted in the appropriate energy range, and the comparison procedure is iterated until theory and experiment come into agreement. The above trial and error procedure was greatly advanced by the use of computers to solve the Boltzmann equation [87.37,38]. A large body of fairly complete sets of low energy cross sections has been obtained in this way [87.23]. Assuming that the electron kinetics are accurately described by the solution of the Boltzmann equation, the above unfolding procedure is severely limited by the lack of uniqueness in the derived cross sections (especially in the molecular gases where many inelastic processes dominate in determining the electron energy distribution). For the noble gases, the method works well for He and Ne at low energies, where only elastic scattering occurs [87.24]. For Ar, Kr, and Xe, the method becomes questionable again for electron energies near the RT minimum. The slow redistribution of electron energies leads to lack of sensitivity of the calculated transport coefficients to the σm . It has been shown that the swarm analyses of HeRamsauer noble gas mixtures lead to unique σm of the Ramsauer noble gases (Ar, Kr, Xe) [87.39]. The measured electron drift velocity data in He-Xe mixtures shown in Fig. 87.6 satisfy the two important requirements (1) high sensitivity of the drift velocity data in gas mixtures over that in either pure He and Xe, and (2) the σm of He is very accurately known, to warrant the uniqueness of the derived σm of Xe. Once the cross sections of the Ramsauer noble gases have been more accurately defined by the above approach, the RT minimum can be used to advantage in addressing the uniqueness problem for molecular gases. The electron transport data in molecular-gas– rare-mixtures exhibit large sensitivity due to the low

105 Pure Xe 5 % He-Xe 10 % He-Xe 20 % He-Xe Pure He 104 –19 10

10–18

10–17

10–16 E/N (Vcm2)

Fig. 87.6 Measured electron drift velocities in He–Xe mix-

tures

energy inelastic collisions in the molecular gas (which increase the f1 component of the electron energy distribution) [87.21,40]. Since the cross sections of the buffer noble gas are now accurately defined, the demanding fits to the mixture transport data can be conveniently exploited to enhance the accuracy of the cross sections of the molecular gases. The application of the two-term approximation in many papers treating molecular gases has been criticized [87.41]. Fortunately, methods are now available for the use of multi-term and Monte Carlo approaches to solve the collisional Boltzmann equation. These approaches do require differential cross sections. Schmidt et al. [87.42] report that their experiments using both electric fields and crossed electric and magnetic fields allow the extraction of the drift velocity, the Lorentz deflection angle, the longitudinal and transverse diffusion coefficients, and the ionization coefficient. This approach has lead to a fully automated procedure for extraction of cross sections from accurate experimental transport data. When approximate answers (20% or so) are needed for comparisons with measurements on complex discharges, rather than measurements on transport properties, the program Bolsig and its database provided by Morgan [87.43] and the Paul Sabatier University plasma group [87.44] are justifiably popular. Under very high E/n conditions, approximate answers can be obtained quickly using the completely anisotropic beam assumption.

Conduction of Electricity in Gases

87.2 Glow Discharge Phenomena

1327

87.2 Glow Discharge Phenomena 87.2.1 Cold Cathode Discharges

Ion spectrum

1

2

3

4

1. Cathode sheath f (e) 2.

3. Meniscus

E1

Eb

E2

ε

f (e) 4.

E1

Eb

E2

ε

Fig. 87.7 Illustration of the appearance of a hot cathode

discharge

with the electric field. The ionization waves can be absolutely or convectively stable, and have dispersion properties of forward or backward waves (the latter corresponding to opposite directions of phase and group velocities). At any plasma boundary a positive space charge sheath is usually formed in order to equalize the fluxes of ions and electrons to the boundary. Bohm [87.45], and much earlier Langmuir [87.46], showed that in a collisionless sheath, to avoid oscillatory solutions, the ions must enter the sheath √ with velocities at least equal to the ion sound speed ( kT e /m i ). The Bohm criterion continues to attract research activity. The review paper by Riemann [87.47] treats the issues addressed up to that time. There are important recent studies on satisfying the neutrality boundary conditions in electro-negative gases and in gas discharges with multiple positivelyand negatively-charged species [87.48].

Part G 87.2

The cold cathode discharge operates because of feedback by the electrons that are generated by ion bombardment of the cathode (secondary electron emission) in turn generating sufficient ions that flow to the cathode to keep the process going. There can be contributions to secondary electron emission from photons and metastable atoms which are created by electron collisions with the gas. Since the secondary emission coefficients are functions of the cathode material, ion species, and ion energy, the magnitude of the needed accelerating potential between the plasma and the cathode varies. This potential creates a space charge sheath, and various excitation features occur near the cold cathode. The potential required is typically 200–600 V. The electrons from the cathode (called primaries) are accelerated through the sheath and acquire sufficient energy to excite the gas efficiently, resulting in a region termed the negative glow. As one moves further from the cathode, the light from the negative glow tapers off due to the degradation of the primaries in creating slow electrons (termed secondaries and ultimates). A sufficiently high density of electrons is usually created to satisfy the external circuit continuity at very low fields (or even slight field reversal). Since the low energy electrons do not create visible excitation, the region to the anode side of the negative glow forms the Faraday dark space. At even greater distance from the cathode, where the slow electron density has decayed by drift, diffusion, recombination, and in some gases by attachment, the electric field increases and the discharge develops long diffuse bright regions which occupy the remainder of the inter-electrode space. This is termed the positive column. If one increases the interelectrode spacing, the cathode regions remain essentially constant and the positive column extends with average uniform properties. Temporal variations called moving striations are common in all gases, and spatial variations called standing striations, or ionization waves are frequently observed in molecular gases. The different cathodic regions occur because the abrupt boundary conditions create local discharge conditions of excitation and ionization which are not in equilibrium with the local electric field. Even in the so-called uniform positive column, the ionization waves correspond to nonlocal equilibrium solutions of the discharge equations that permit more efficient discharge ionization and conductivity than would be given by local equilibrium

Atomic spectrum

1328

Part G

Applications

87.2.2 Hot Cathode Discharges Low pressure, high current discharges are important in thyratrons and other high current switches. The hot cathode permits much higher current densities than are usually obtained from a cold cathode. The voltage drop of the cathode sheath is much less, and a discharge will operate if the applied potential exceeds IP for the gas. Hot cathode discharges with low P and Γe show three groups of electrons: the primaries, which are beam electrons that essentially have retained the energy acquired in the cathode sheath; secondaries which are randomized electrons with Te approximately proportional to the energy of the primaries (≈ 5 eV for 30 eV primaries) and approximately independent of the gas; and the ultimates which are the bulk of the electrons with T − E