Advances in Atomic, Molecular, and Optical Physics

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Advances in

ATOMIC, MOLECULAR, AND OPTICAL PHYSICS V O L U M E 51

Editors PAUL R. B ERMAN

University of Michigan Ann Arbor, Michigan C HUN C. L IN

University of Wisconsin Madison, Wisconsin H ERBERT WALTHER

University of Munich and Max-Planck-Institut für Quantenoptik Garching bei München Germany

Editorial Board C. J OACHAIN

Université Libre de Bruxelles Brussels, Belgium M. G AVRILA

F.O.M. Insituut voor Atoom- en Molecuulfysica Amsterdam, The Netherlands M. I NOKUTI

Argonne National Laboratory Argonne, Illinois

Founding Editor S IR DAVID BATES

Supplements 1. Atoms in Intense Laser Fields, Mihai Gavrila, Ed. 2. Cavity Quantum Electrodynamics, Paul R. Berman, Ed. 3. Cross Section Data, Mitio Inokuti, Ed.

ADVANCES IN

ATOMIC, MOLECULAR, AND OPTICAL PHYSICS Edited by

H.H. Stroke DEPARTMENT OF PHYSICS NEW YORK UNIVERSITY NEW YORK

Volume 51

Amsterdam · Boston · Heidelberg · London · New York · Oxford Paris · San Diego · San Francisco · Singapore · Sydney · Tokyo

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Dedicated to B ENJAMIN B EDERSON

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CONTENTS

C ONTRIBUTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

Introduction, by H. Henry Stroke . . . . . . . . . . . . . . . . . . . . . .

3

Appreciation of Ben Bederson as Editor of Advances in Atomic, Molecular, and Optical Physics, by Herbert Walther . . . . . . . . . . .

9

Benjamin Bederson Curriculum Vitae . . . . . . . . . . . . . . . . . . .

11

Research Publications of Benjamin Bederson . . . . . . . . . . . . . . .

17

A Proper Homage to Our Ben, by Harry Lustig . . . . . . . . . . . . . .

23

Benjamin Bederson in the Army, World War II, by Val L. Fitch . . . .

29

Physics Needs Heroes Too, by C. Duncan Rice . . . . . . . . . . . . . . .

35

Two Civic Scientists—Benjamin Bederson and the other Benjamin, by Neal Lane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

An Editor Par Excellence, by Eugen Merzbacher . . . . . . . . . . . . .

49

Ben as APS Editor, by Bernd Crasemann . . . . . . . . . . . . . . . . . .

57

Ben Bederson: Physicist–Historian . . . . . . . . . . . . . . . . . . . . .

65

Roger H. Stuewer 1. 2. 3. 4. 5. 6.

Introduction . . . . . . . . . . . . . . Wartime Reminiscences . . . . . . . . Physics and New York City . . . . . . APS Forum on the History of Physics Conclusion . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . vii

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65 66 69 73 74 74

viii

Contents

Pedagogical Notes on Classical Casimir Effects . . . . . . . . . . . . . .

75

Larry Spruch 1. 2. 3. 4. 5.

Introduction . . . . . . . . . . . . . . . . . . . Dimensional Analysis and Physical Arguments The Vanishing of E¯ Cl . . . . . . . . . . . . . . An Unauthorized Thank You . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . .

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76 77 78 81 81

Polarizabilities of 3 P Atoms and van der Waals Coefficients for Their Interaction with Helium Atoms . . . . . . . . . . . . . . . . . . . . . . .

83

X. Chu and A. Dalgarno 1. 2. 3. 4. 5. 6. 7.

Introduction . . . . . . . . . . . . . . Theory: Dynamic Polarizabilities . . Numerical Method . . . . . . . . . . Results: Static Dipole Polarizabilities Van der Waals Coefficients . . . . . . Acknowledgement . . . . . . . . . . . References . . . . . . . . . . . . . . .

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84 84 86 88 89 90 90

The Two Electron Molecular Bond Revisited: From Bohr Orbits to Two-Center Orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

Goong Chen, Siu A. Chin, Yusheng Dou, Kishore T. Kapale, Moochan Kim, Anatoly A. Svidzinsky, Kerim Urtekin, Han Xiong and Marlan O. Scully 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. A. B. C. D.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recent Progress Based on Bohr’s Model . . . . . . . . . . . . . . . . General Results and Fundamental Properties of Wave Functions . . . Analytical Wave Mechanical Solutions for One Electron Molecules . Two Electron Molecules: Cusp Conditions and Correlation Functions Modelling of Diatomic Molecules . . . . . . . . . . . . . . . . . . . . Alternative Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Separation of Variables for the H+ 2 -like Schrödinger Equation . . . . . The Asymptotic Expansion of (λ) for Large λ . . . . . . . . . . . . The Asymptotic Expansion of (λ) as λ → 1 . . . . . . . . . . . . . Expansions of Solution Near λ ≈ 1 and λ 1: Trial Wave Functions of James and Coolidge . . . . . . . . . . . . . . . . . . . . . . . . . . E. The Many-Centered, One Electron Problem in Momentum Space . . .

95 107 111 135 145 155 184 191 191 192 192 193 194 196 198

Contents

ix

F. Derivation of the Cusp Conditions . . . . . . . . . . . . . . . . . . . . 1 1 ∇ 2 − 2m ∇2 G. Center of Mass Coordinates for the Kinetic Energy − 2m 1 1 2 2 H. Verifications of the Cusp Conditions for Two-Centered Orbitals in Prolate Spheroidal Coordinates . . . . . . . . . . . . . . . . . . . . . . . . I. Integrals with the Heitler–London Wave Functions . . . . . . . . . . . J. Derivations Related to the Laplacian for Section 6.4 . . . . . . . . . . K. Recursion Relations and Their Derivations for Section 6.4 . . . . . . . L. Derivations for the 5-Term Recurrence Relations (6.81) . . . . . . . . M. Dimensional Scaling in Spherical Coordinates . . . . . . . . . . . . . 11. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

203 208 210 216 217 222 231 232 236

Resonance Fluorescence of Two-Level Atoms . . . . . . . . . . . . . . .

239

H. Walther 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Theory of the Spectrum of Resonance Fluorescence . . . . . . . . . . 3. Total Scattered Intensity, Intensity Correlations, and Photon Antibunching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. More Theoretical Results—Variants of the AC Stark Effect . . . . . . 5. Experimental Studies of the Spectrum . . . . . . . . . . . . . . . . . . 6. Spectrum at Low Scattering Intensities and Extremely High Resolution 7. Experiments on the Intensity Correlation—Photon Antibunching . . . 8. Photon Correlation Measured with a Single Trapped Particle . . . . . 9. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

239 240 244 245 247 251 260 266 269 270

Atomic Physics with Radioactive Atoms . . . . . . . . . . . . . . . . . .

273

Jacques Pinard and H. Henry Stroke 1. 2. 3. 4. 5.

Introduction . . . . . . “Off-line” Experiments “On-line” Experiments Bohr–Weisskopf Effect References . . . . . . .

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274 275 280 294 296

Thermal Electron Attachment and Detachment in Gases . . . . . . . .

299

Thomas M. Miller 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. FALP Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Electron Attachment . . . . . . . . . . . . . . . . . . . . . . . . . . .

300 303 309

x

Contents 4. 5. 6. 7. 8. 9.

Electron Detachment Electron Affinity . . . New Plasma Effects . Concluding Remarks Acknowledgements . References . . . . . .

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323 329 330 332 334 336

Recent Developments in the Measurement of Static Electric Dipole Polarizabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

343

Harvey Gould and Thomas M. Miller 1. 2. 3. 4. 5. 6. 7. 8. 9.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deflection in Electric Field Gradients . . . . . . . . . . . . . . . . . . Light Force Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interferometry Experiments . . . . . . . . . . . . . . . . . . . . . . . . Laser-Cooled Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . Alkali Polarizability, Lifetime and the Dispersion Coefficient . . . . . Ionic Polarizabilities from Lifetimes . . . . . . . . . . . . . . . . . . . Core Polarizability from Microwave Spectroscopy of Rydberg Atoms and Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

344 346 347 349 350 352 354 354

Trapping and Moving Atoms on Surfaces . . . . . . . . . . . . . . . . .

363

355 356 357 357

Robert J. Celotta and Joseph A. Stroscio 1. 2. 3. 4. 5. 6. 7.

Introduction . . . . . . . Moving Atoms . . . . . . Atom Dynamics . . . . . Summary and Comments Future Expectations . . . Acknowledgements . . . References . . . . . . . .

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364 365 369 380 380 382 382

Electron-Impact Excitation Cross Sections of Sodium . . . . . . . . . .

385

Chun C. Lin and John B. Boffard 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Excitation out of the Ground State . . . . . . . . . . . . . . . . . . . .

385 387

Contents 3. 4. 5. 6.

Excitation out of Laser Excited States Concluding Remarks . . . . . . . . . Acknowledgements . . . . . . . . . . References . . . . . . . . . . . . . . .

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401 407 409 409

Atomic and Ionic Collisions . . . . . . . . . . . . . . . . . . . . . . . . .

413

Edward Pollack Ben Bederson . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . 2. Collisions Involving Heavy Solar Wind Ions 3. Collisions Involving H0 Projectiles . . . . . . 4. Proton Collisions in the Io Plasma Torus . . 5. Surface Collisions with Highly-Charged Ions 6. Acknowledgement . . . . . . . . . . . . . . . 7. References . . . . . . . . . . . . . . . . . . .

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414 414 416 422 441 443 446 447

Atomic Interactions in Weakly Ionized Gas: Ionizing Shock Waves in Neon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

451

Leposava Vušković and Svetozar Popović 1. 2. 3. 4. 5. 6. 7.

Introduction . . . . . . . . . . . . . . . . . . . Electron Impact Ionization from Excited Neon Energy Pooling Processes in Neon . . . . . . . Ionizing Shock Waves in Neon . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . .

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452 455 457 462 466 466 466

Approaches to Perfect/Complete Scattering Experiments in Atomic and Molecular Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . .

471

H. Kleinpoppen, B. Lohmann, A. Grum-Grzhimailo and U. Becker 1. 2. 3. 4. 5. 6. 7.

Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis of Atomic Collisions . . . . . . . . . . . . . . . . . . . . . . Angle and Spin Resolved Analysis of Resonantly Excited Auger Decay Complete Experiments for Half-Collision; Auger Decay . . . . . . . . Analysis of Molecular Collisions . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

472 474 496 512 519 523 527

xii

Contents

Reflections on Teaching . . . . . . . . . . . . . . . . . . . . . . . . . . .

535

Richard E. Collins Dedication . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . 2. Recollections . . . . . . . . . . . 3. Characteristics of Great Teachers . 4. Characteristics of Great Teaching 5. Rewards of Teaching . . . . . . . 6. Who Should Teach? . . . . . . . . 7. Recognition of Excellent Teaching 8. Evaluation of Teaching . . . . . . 9. Assessment of Students . . . . . . 10. Conclusion . . . . . . . . . . . . . 11. Acknowledgement . . . . . . . . . 12. References . . . . . . . . . . . . .

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535 536 537 543 545 549 550 552 553 555 556 557 557

I NDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C ONTENTS OF VOLUMES IN T HIS S ERIAL . . . . . . . . . . . . . . .

559 575

CONTRIBUTORS

Numbers in parentheses indicate the pages on which the author’s contributions begin.

H. H ENRY S TROKE (3, 273), Department of Physics, New York University, 4 Washington Place, New York, NY 10003, USA H. WALTHER (9, 239), Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany H ARRY L USTIG (23), The City College of the City University of New York, 138th Street & Convent Avenue, New York, NY 10031, USA, and The American Physical Society, One Physics Ellipse, College Park, MD 20740-3844, USA VAL L. F ITCH (29), Department of Physics, Princeton University, Princeton, NJ 08540, USA C. D UNCAN R ICE (35), University of Aberdeen, Kings College, Aberdeen, AB24 3FX, UK N EAL L ANE (41), Department of Physics and Astronomy, Rice University, Houston, TX 77005, USA E UGEN M ERZBACHER (49), Department of Physics and Astronomy, University of North Carolina at Chapel Hill, CB#3255, Chapel Hill, NC 27599-3255, USA B ERND C RASEMANN (57), Physics Department, University of Oregon, Eugene, OR 97403, USA ROGER H. S TUEWER (65), Program in History of Science and Technology, Tate Laboratory of Physics, University of Minnesota, 116 Church Street SE, Minneapolis, MN 55455, USA L ARRY S PRUCH (75), Department of Physics, New York University, 4 Washington Place, New York, NY 10003, USA X. C HU (83), ITAMP, Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA A. DALGARNO (83), ITAMP, Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA G OONG C HEN (93), Institute for Quantum Studies and Dept. of Mathematics, Texas A&M University, College Station, TX 77843, USA xiii

xiv

Contributors

S IU A. C HIN (93), Institute for Quantum Studies and Dept. of Physics, Texas A&M University, College Station, TX 77843, USA Y USHENG D OU (93), Institute for Quantum Studies and Dept. of Physics, Texas A&M University, College Station, TX 77843, USA K ISHORE T. K APALE (93), Institute for Quantum Studies and Dept. of Physics, Texas A&M University, College Station, TX 77843, USA M OOCHAN K IM (93), Institute for Quantum Studies and Dept. of Physics, Texas A&M University, College Station, TX 77843, USA A NATOLY A. S VIDZINSKY (93), Institute for Quantum Studies and Dept. of Physics, Texas A&M University, College Station, TX 77843, USA K ERIM U RTEKIN (93), Institute for Quantum Studies and Dept. of Physics, Texas A&M University, College Station, TX 77843, USA H AN X IONG (93), Institute for Quantum Studies and Dept. of Physics, Texas A&M University, College Station, TX 77843, USA M ARLAN O. S CULLY (93), Institute for Quantum Studies and Dept. of Physics, Texas A&M University, College Station, TX 77843, USA, Depts. of Chemistry and Mechanical and Aerospace Engineering, Princeton University, NJ 08544, USA, and MaxPlanch-Institut für Quantenoptik, D-85748 Garching, Germany JACQUES P INARD (273), Laboratoire Aimé Cotton, Bât. 505, Faculté des Sciences, F91405 Orsay, France T HOMAS M. M ILLER (299, 343), Air Force Research Laboratory, Space Vehicles Directorate, Hanscom Air Force Base, MA 01731-3010, USA H ARVEY G OULD (343), MS: 71-259, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA ROBERT J. C ELOTTA (363), Electron Physics Group, National Institute of Standards and Technology, Gaithersburg, MD 20899-8412, USA J OSEPH A. S TROSCIO (363), Electron Physics Group, National Institute of Standards and Technology, Gaithersburg, MD 20899-8412, USA C HUN C. L IN (385), Department of Physics, University of Wisconsin, Madison, WI 53706, USA J OHN B. B OFFARD (385), Department of Physics, University of Wisconsin, Madison, WI 53706, USA E DWARD P OLLACK† (413), Department of Physics, University of Connecticut, Storrs, CT 06269, USA

† Deceased.

Contributors

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L EPOSAVA V UŠKOVI C´ (451), Department of Physics, Old Dominion University, Norfolk, VA 23529, USA S VETOZAR P OPOVI C´ (451), Department of Physics, Old Dominion University, Norfolk, VA 23529, USA H.H. K LEINPOPPEN (471), Atomic & Molecular Physics, University of Stirling, Stirling FK9 4LA, Scotland B. L OHMANN (471), Westfälische Wilhelms-Universität Münster, Institut für Theoretische Physik, Wilhelm-Klemm-Str. 9, D-48149 Münster, Germany A. G RUM -G RZHIMAILO (471), Institute of Nuclear Physics, Moscow State University, Moscow 119992, Russia U.U. B ECKER (471), Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin/Dahlem, Germany R ICHARD E. C OLLINS (535), The University of Sydney, NSW 2006, Australia


BENJAMIN BEDERSON


ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 51

INTRODUCTION H. HENRY STROKE

I am always suspicious when asked by Ben Bederson to get involved in some editorial work: once it cost me more than two years of intensive work when he persuaded me to edit “The Physical Review: The First 100 Years” [1] for the American Physical Society (APS) and the American Institute of Physics. On a second occasion, he legated to me the editorship of what is now CAMOP [2], and which has kept me occupied for over thirty years. Eugen Merzbacher in his contribution to this volume attests to Ben’s persuasiveness. In spite of these experiences, when asked to edit this book dedicated to him, I accepted with pleasure. For over fifty years Ben has played an important part in my life as a physicist, and also personally. The contributions in this volume are largely devoted to the legacy of Ben’s work, his influence in physics and on students and colleagues with whom he collaborated. Tom Miller, who has worked with Ben, looked into his antecedents [3]. I am grateful to him for providing the Bederson academic lineage: • Ogden Nicolas Rood, BS (Princeton, 1852) Professor and Head of Department, Columbia University • Robert Andrews Millikan, PhD (Columbia University, 1895) “On the Polarization of Light Emitted from the Surfaces of Incandescent Solids and Liquids” • Leonard Benedict Loeb, PhD (University of Chicago, 1916) “On the Mobilities of Gas Ions in High Electric Fields” • Leon Harold Fisher, PhD (University of California, Berkeley, 1944) “Three Problems in Electrical Discharges through Gases” • Benjamin Bederson, PhD (New York University, 1950) “Formative Time Lags of Spark Breakdown in Air” Titles of the theses are listed. Rood apparently did not have a PhD. Among his research fields were acoustics and optics. He authored a book on colorimetry, “Modern Chromatics”, which was valued as a reference by Impressionist painters, and he was a noted painter himself. Ben’s first post-World War II research was in the field of gas discharges. He described this early work in a talk at the 1992 Gaseous Electronics Confer3

© 2005 Elsevier Inc. All rights reserved ISSN 1049-250X DOI 10.1016/S1049-250X(05)51002-4

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H.H. Stroke

ence [4]. The talk also included the ensuing developments that led ultimately to his electron-scattering experiments. In 1950, Ben was on the staff in the Research Laboratory of Electronics (RLE) at the Massachusetts Institute of Technology (MIT) when I first met him in the Atomic Beams Laboratory. I had taken a course in atomic physics with Jerrold Zacharias which inspired me to look for research in the same laboratory. I detail this part of Ben’s career because (1) it entailed physics that he did not pursue afterwards, and (2) it laid the experimental foundations for his physics thereafter. There were many exciting experiments and Ben was involved in a number of them. One of the important ones concerned the anomalous gyromagnetic ratio, gJ , of the electron. Quantum electrodynamics was in its relatively early stages and an atomic beam magnetic resonance (ABMR) experiment on hydrogen would be most important. But, compared to a number of earlier experiments with alkali atoms, where production of the atomic beam and its detection were easy, for hydrogen there were problems on both end. Alkali atomic beams are easily ionized by the process of surface ionization on a hot, high-work function, wire. The resulting ion current is readily measured. Hydrogen, however, is normally a molecule. So, Ben and his coworkers built an “atomizer” based on Wood’s tube [5], where his expertise in gas discharges was put to use in producing the hydrogen atomic beam. The detection part proved to be a great challenge as well: hydrogen has an ionization potential of 13.6 eV and its electron cannot be removed as with alkalis. John King [6] suggested instead the possibility of attaching an electron to the hydrogen atom on a very low work function surface: the electron affinity of hydrogen is only ≈0.7 eV. This was my first real experiment in physics, and it did not succeed. Ben invented a modification of the Pirani gauge, a device in which the incident gas atoms cool a heated wire, causing a measured change in resistance: In Ben’s scheme the incoming beam gas was compressed, increasing the sensitivity by several orders of magnitude. This work is described in the theses of two undergraduate students, Germaine Bousquet and Alan Odian [7] and in the review by King and Zacharias [8]. The other effort in the laboratory was in nuclear structure. This was only about one year after the introduction of the nuclear shell model by Maria Goeppert Mayer and J. Hans D. Jensen, and still before the collective model of the nucleus of Aage Bohr and Ben R. Mottelson. Only about a half dozen spins of odd–odd (odd-proton number, odd-neutron number) nuclei were known at the time and their coupling schemes were a matter of conjecture. After my rookie stage—cleaning oil diffusion pumps and spending months tracking down vacuum leaks in atomic beam apparatus, which by today’s standards would be considered quite primitive—I finally inherited my own ABMR system. With Ben and Vincent Jaccarino we succeeded in measuring the nuclear spin and magnetic moment of 134 Cs, a radioactive isotope with a half life of about 750 days. The experiment was

INTRODUCTION

5

done with a very strong source, the activity of which was close to 1010 becquerels. It would be difficult to obtain approval for such a quantity today! Ben also collaborated on an atomic beam experiment with potassium isotopes to measure the effect of the distributed nuclear magnetization on the atomic electron–nuclear interaction [9]. This effect had been observed with rubidium isotopes for the first time by Francis Bitter only a short while earlier. Ten years before then, Hans Kopfermann, in the first edition of his book on nuclear moments [10], remarked that this effect is much too small ever to be observed! Ben left for New York University (NYU) two years before work started in Zacharias’ laboratory on the first atomic clock, based on an ABMR in cesium. It was the prototype of today’s time and frequency standard. Three years after Ben left MIT, I came to Princeton as a postdoc. Shortly after I arrived, in April 1955, I was asked if I would be willing to meet Albert Einstein and tell him about the efforts that he heard were going on at MIT on a precision clock (the so-called fountain clock) that may be able to detect the gravitational red shift predicted by his theory [11]. I recall this as we are celebrating in 2005 the 100th anniversary of Einstein’s great papers—also commemorating the 50th anniversary of his death. Though I have recounted this event countless times, and will certainly incur Ben’s wrath in telling it again, the day of this memorable meeting was also another occasion for Ben’s (mild) persuasion: he introduced me to my future wife, Norma, who was an apartment-mate of his future wife Betty. Eight years later he exercised his power of persuasion once more that resulted in my giving up another academic offer in favor of NYU. This spans the early part of Ben’s career. The contributions in this volume continue the story. An aspect which is not covered is his importance in organizing scientific meetings. First there was the International Conference on the Physics of Electronic and Atomic Collisions (ICPEAC) which he launched at NYU in the late 1950’s. Then there was the International Conference on Atomic Physics (ICAP), which also had its beginnings at NYU [12], unfortunately marred in the middle of the conference by the news of the assassination of Robert F. Kennedy on 6 June 1968. ICAP was an outgrowth of the “Brookhaven Conferences”, initiated and organized for many years by Victor Cohen. Initially they were devoted exclusively to atomic beam experiments, but they were soon extended to the rapidly growing new fields of atomic physics. ICPEAC and ICAP continue to this day as thriving meetings of the international community of atomic physicists. Ben played a seminal rôle in both of them. The people who have contributed to this volume have had interactions with Ben through his diverse interests and activities. A number were PhD students and postdocs in his laboratory, and colleagues with whom his interests intersected. These contributions constitute the latter part of the volume—first, theoretical then, experimental work. The first part of the volume is more personal, introduced by

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H.H. Stroke

an epitomic poem on Ben’s career by Harry Lustig, emeritus Treasurer of the APS, who held the financial reins on Ben when he was the Editor-in-Chief of the Society. Val Fitch goes back to the early Bederson, at the time when both served at the Los Alamos Laboratory in World War II. Duncan Rice, Principal of the University of Aberdeen, writes from the point of view of a historian who had to be educated in the sciences while he was Dean of the Faculty of Arts and Science at NYU by Ben, then Dean of the Graduate School. Neil Lane writes of Ben as a citizen in the science community. Eugen Merzbacher and Bernd Crasemann recall Ben’s major activity as Editor-in-Chief of the APS. Finally, in this first part, Roger Stuewer comes to Ben’s latest activity, after having attained Emeritus status both at NYU and at the APS: historian of science and Editor of the History of Physics Newsletter (APS). Larry Spruch, long-time Ben’s colleague at NYU, provides a pedagogical paper on Casimir effects. Alex Dalgarno writes on dipole polarizabilities. Ben’s laboratory made significant contributions for their measurements with the introduction of the E–H gradient balance method [13], balancing electric and magnetic moment deflections in congruent magnetic and electric fields. Marlan Scully and collaborators contribute to the Molecular part of the name of this publication with their paper on molecular bonds. The last part groups the more experimental papers. Herbert Walther, who has been sharing the editorship of Advances in Atomic, Molecular and Optical Physics (AAMOP) with Ben, discusses the current status of resonance fluorescence of two-level atoms, a subject that goes back seventy-five years to Victor F. Weisskopf’s doctoral thesis [14]. Parenthetically, in this paper Weisskopf also gives the theory underlying level-crossing spectroscopy. With Jacques Pinard, we give a brief review of atomic physics with radioactive atoms, to which Ben contributed early on. Two topics close to Ben’s interests, gaseous electronics and dipole polarizabilities, are treated extensively by Tom Miller and with Harvey Gould. Robert Celotta, extending Ben’s lineage, and Joseph Stroscio bring us upto-date on single-atom manipulation. Chun Lin, who with Paul Berman and Ennio Arimondo, is taking over as editor of AAMOP from Ben Bederson and Herbert Walther, writes on electron-impact excitations in collaboration with John Borrard. Edward Pollack, who was also a graduate of Ben’s laboratory, contributed his work on collision physics.1 A long-time collaborator of Ben’s, Lepša Vušković, joined by Svetozar Popović, is back to Ben’s first field of research, gaseous electronics. Hans Kleinpoppen and his collaborators return to the subject which some years back Ben described as The “Perfect” Scattering Experiment [13].

1 When I first wrote this sentence, the verb was in the present. It is with great regret that we learned of Ed’s sudden passing. His friends and colleagues, a number of whom are contributors to this volume, express all their sympathy to his wife, Rita, and to the other members of their family.

INTRODUCTION

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At the end, Richard Collins contributes an essay in which Ben’s influence as a teacher is recalled. I thank all the authors for having participated in creating this volume to honor a great colleague and a good friend, one who has contributed much to physics and to many of us as physicists by his insight, enthusiasm, and imagination. Anita Koch, Book Editor at Elsevier, has been most helpful and patient in its preparation. I particularly appreciate the historical input by Tom Miller. Finally, but not least, I am indebted to Eugen Merzbacher for critical reading and many constructive comments which have aided me in the preparation of this work.

References [1] Stroke H.H. (Ed.), “The Physical Review: The First Hundred Years, A Selection of Seminal Papers and Commentaries”, AIP Press, The American Physical Society, Springer-Verlag, New York, 1995. Includes CD-ROM. [2] Comments on Atomic, Molecular and Optical Physics, published by Physica Scripta, Royal Swedish Academy of Science, Stockholm. [3] From library reference series on American Dissertations and R.A. Millikan’s autobiography. [4] Bederson B., Electron–atom scattering using atomic recoil, in: “Gaseous Electronics Conference”, Boston, MA, October 1992. [5] Wood R.W., “Physical Optics”, Dover, New York, 1967, p. 125. [6] King J.G., RLE (MIT) Quarterly Progress Report, 15 October 1950, p. 29. [7] Odian A.C., “An Improved Stern–Pirani Molecular Beam Detector”, S.B. Thesis, Department of Physics, MIT, June 1951; Bousquet A.G., “An Improved Stern–Pirani Detector for a Hydrogen Beam”, Id., June 1952. [8] King J.G., Zacharias J.R., Some new applications and techniques of molecular beams, in: Advances in Electronics and Electron Physics, vol. 8, Academic Press, New York, 1956. [9] Pinard J., Stroke H.H., Atomic physics with radioactive atoms, this volume. [10] Kopfermann H., “Kernmomente”, Akademische Verlagsgesellschaft, Leipzig, 1940. In the translation, “Nuclear Moments” (Academic Press, New York, 1958), this statement has of course vanished. [11] Naumann R., Stroke H., Einstein and the atomic clock, Physics World 9 (1996) 76; Naumann R.A., Stroke H.H., Apparatus upended: A short history of the fountain A-clock, Physics Today 49 (May 1996) 89; Naumann R.A., Stroke H.H., Nobelists played roles in implementation of ‘Fountain’ experiment, Physics Today 51 (February 1998) 15. [12] Bederson B., Cohen V.W., Pichanick F.M.J. (Eds.), “Proceedings of the First International Conference on Atomic Physics”, Plenum Press, New York, 1969. [13] See Research publications of Benjamin Bederson, this volume. [14] See reference in H. Walther, Resonance fluorescence of two-level atoms, this volume.



APPRECIATION OF BEN BEDERSON AS EDITOR OF ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS

This volume of Advances in Atomic, Molecular, and Optical Physics is dedicated to our highly esteemed colleague and friend, Ben Bederson, who was Editor of this series as of Volume 10 (published in 1974), when he succeeded Immanuel Estermann, the second founding Editor of the series together with Sir David Bates. Ben remained Editor up to Volume 50, making 40 volumes and three supplementary volumes shaped by him. During that period the name of the series was changed when Optical Physics was added to the title with Volume 26, published in 1989. This was done to emphasize the various new optics-based techniques playing a constantly increasing role in atomic and molecular physics. Furthermore, an editorial board supported the editors with proposals for new articles and with much other advice. Some board members contributed by generating supplementary volumes. The articles in Advances are a very good indicator of the enormous development of atomic and molecular physics in that period. Starting with Volume 33 (1995), after Sir David retired, I had the pleasure to work with Ben Bederson in editing Advances. It was great co-operating with Ben. He has an extensive view over the entire field resulting from his own research work and then as a member of many organizing bodies for meetings. He was, for example, one of the driving forces when the very successful conference series on Atomic Physics (ICAP) and the International Conference of the Physics of Electronic and Atomic Collisions (ICPEAC) were established. Later, he became Editor-in-Chief of the American Physical Society, where he was confronted with physics publishing in general and problems of the new possibilities afforded by electronic publishing and problems presented by the enormous increase of physics papers during the eighties and at the beginning of the nineties. Ben’s outstanding insight into the entire field was always an enormous advantage when the authors for forthcoming volumes were being selected. His intuition for finding interesting and promising topics was unique and a great help in keeping the series in high demand. The Advances series met a challenge in the late nineties, when book-publishing in general got into financial difficulties. At this 9


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H. Walther

time it was planned to merge Advances, still very successful, with a not so successful journal. Fortunately, this problem could be solved, leaving the series in good shape again. Now, new editors are coming on board so that the future also looks promising and bright. The atomic and molecular physics community thanks Ben for the outstanding work he performed as Editor of this highly successful series.

Herbert Walther

Garching, December 2004


BENJAMIN BEDERSON CURRICULUM VITAE

DATE OF BIRTH 1921

November 15

EDUCATION 1946 1948 1950

B.S. CCNY M.A. Columbia University Ph.D. New York University

FAMILY 1956

PRINCIPAL POSITIONS Scientific Aide 1944–46 Research Assistant and Instructor 1948–50 Staff Member (postdoc) 1950–52 Professonial rank 1952–92 Chair 1973–76 Dean 1986–89 Professor Emeritus 1992– Editor-in-Chief 1993–1996 Editor-in-Chief Emeritus 1997–

Married—Betty Weintraub Bederson Children—Joshua, Geoffrey, Aron, Benjamin

Manhattan Project, Los Alamos, NM (US Army, Special Engineering Detachment) New York University

Research Laboratory of Electronics, MIT (in J.R. Zachariasí atomic beams lab.) New York University, Physics Dept. New York University, Physics Dept. Graduate School of Arts and Science, New York University New York University American Physical Society American Physical Society

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OTHER POSITIONS Visiting Scientist 1962 Visiting Fellow 1968–69 Lecturer 1969 Visiting Fellow 1973 Lecturer 1984 Co-Director, Lecturer 1989 Visiting Fellow 1990 Co-Director, Lecturer 1992 Vice-Chair, Chair, Past-Chair 2000–2002 EDITORIAL Co-Editor (with Wade L. Fite) 1968 Editor 1978–1992

Co-Editor (with Sir David R. Bates) 1974–1992 (with Herbert Walther) 1993–2004 Co-Coordinator 1971–1974 Correspondent 1969–1971

Laboratoria Gas Ionizzati (CNEN) Frascati, Italy Joint Institute for Laboratory Astrophysics, University of Colorado, Boulder, CO Summer School in Atomic Physics, Bari, Italy Center for Theoretical Studies, University of Miami, Coral Gables, FL NATO Advanced Study Institute, Palermo (Santa Flavia), Italy Winter College on Atomic and Molecular Physics: Photon-assisted collisions. International Center for Theoretical Physics, Trieste, Italy Institute for Theoretical Atomic and Molecular Physics, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA Winter College on Coherence in Atom–Radiation Interactions, International Center for Theoretical Physics Forum on History of Physics, American Physical Society

Methods of Experimental Physics, Atomic and Electron Physics, Vol 7A: Atomic Sources and Detectors, Vol 7B: Atomic Interactions, Academic Press Physical Review A, A1: Atomic, Molecular, and Optical Physics; A15: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, American Physical Society Advances in Atomic, Molecular, and Optical Physics, Annual Series, Academic Press

Comments on Atomic and Molecular Physics, Gordon and Breach

BENJAMIN BEDERSON CURRICULUM VITAE Associate Editor 1969–1999 Co-Editor (with F. Pichanik and V.W. Cohen) 1969 Co-Editor, with Sir H.S.W. Massey and E.W. McDaniel 1988 Editor 1999

Editor 2004–2007

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Atomic Data and Nuclear Data Tables, Academic Press Atomic Physics: Proceedings of the First International Conference on Atomic Physics, Plenum Press, NY Applied Atomic Collision Physics, Five Volumes, Academic Press

Reviews of Modern Physics Special Issue in honor of the Centenary of the American Physical Society 71, 1999. Also published in hard cover, under the title “More Things in Heaven and Earth”, Springer-Verlag Forum on History of Physics Newsletter

PhD STUDENTS J.M. Hammer, H. Malamud, K. Rubin, J. Perel, A. Salop, E. Pollack, P. EnglanderGolden, K. Lulla, G. Sunshine, R.E. Collins, M. Goldstein, J. Levine, E.J. Robinson, P. Dittner, I.L. Klavan, R.J. Celotta, A. Kasdan, R. Molof, L. Schwartz, L. Schumann, F. Murray, V. Tarnovsky, S. Ron, R. Dang, P. Weiss, A. Tino, R. Kremens, T. Guella, G.-F. Shen, M. Zuo, T.Y. Jiang, C.H. Yang POSTDOCS AND VISITING COLLEAGUES B.B. Aubrey, D.M. Cox, H.H. Brown Jr., N.D. Bhaskar, J.A.D. Stockdale, T.M. Miller, B. Jaduszliwer, B. Stumpf, L. Vuskovic, J.T. Park RESEARCH INTERESTS Atomic collision phenomena and atomic structure; electronic and atomic interactions with polarized atoms and electrons; interactions of radiation fields, including laser fields, with atoms and molecules. Fundamental concepts in quantum mechanics. Measurements of electric polarizabilities of atoms, molecules and molecular clusters. Interactions of particles and fields with metal and alkali halide clusters. PROFESSIONAL MEMBERSHIPS, AWARDS Fellow, American Physical Society Fellow, American Association of Advancement of Science

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Honorary Member, Sigma Pi Sigma (Physics Honorary Society) Townsend Harris Medal, CCNY, 1993 CONSULTING AND ADVISING POSITIONS National Bureau of Standards Visiting Panel, Atomic Physics Division Advisory to the N.A.S. 1962–64 National Science Foundation Advisory Panel for Graduate Traineeships in Physical Science 1964–65 Member of Various IDA–ARPA panels and committees on atomic processes in upper atmosphere 1960–67 National Academy of Sciences—National Research Council Advisory Committee to Army Research Office ñ Durham 1968–74 Trustee and Member of Board of Directors, Institute for Medical Research and Studies 1960–73 Chairman, Committee on Atomic and Molecular Physics, National Academy of Sciences National Research Council 1960–73 Physical Advisory Committee, Physics Division, National Science Foundation 1973–76 Vice-Chairman, Chairman, Division of Electron and Atomic Physics, American Physical Society 1974–75 Chairman, Subcommittee on Energy, NAS-NRC Committee Atomic and Molecular Science 1975–76 Committee on Atomic and Molecular Science, NAS 1976–79 Committees on Science and Society and Science and Public Policy, NYAS 1977 Chairman, Nominating Committee DEAP-APS; Davisson–Germer Award Nominating Committee 1977 Evaluation Panel for the Joint Institute for Laboratory Astrophysics, Boulder, Quantum Physics Division, NBS (Chairman, 1979) 1978–80 Visiting Committee (Chairman, 1980), Argonne National Laboratory, Physics Division


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1978–80 Board of Governors, N.Y. Academy of Sciences 1977–79 Chairman, Panel for Absolute Physical Quantities, National Measurement Laboratory, National Bureau of Standards (NRC-NAS) 1980–83 Chairman, Survey Subcommittee, Committee on Atomic and Molecular Science (NRC-NAS) 1979–82 Visiting Panel (Physics Representative) Office of Naval Research 1985 Presidential Young Investigators Award Committee, NSF 1983–84 Vice-Chairman, Chairman, Chinese Scholars Program, Chinese Academy of Sciences–APS 1986–91 CONFERENCES Organizer and Secretary First and Second International Conference on the Physics of Electronic and Atomic Collisions (ICPEAC) (New York University and University of Colorado) 1958–61 Member of Program and Steering Committees, ICPEACs 3, 4, 5 1963–67 Organizing Committee First International Conference on Atomic Physics (ICAP), New York University 1968 Vice Chairman, Gordon Conference on New Directions in Atomic Physics 1973–79 Vice Chairman, ICPEAC 13 (Berlin) 1983 Chairman, ICPEAC 14 (Stanford University) 1985 Executive Committee, ICPEAC 15 (Brighton, England) 1987 International Organizing Committee, ICAP 1968–88 International Program Committee, ICAP 1980–92 TEACHING Professor Bederson has taught a significant fraction of the regular undergraduate and graduate courses of the Physics Department over the years. He regularly

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taught the required area course in Advanced Atomic Physics, a third year graduate course, every other year. He has also regularly taught two “popular” physics courses. One is “Sound and Music”, a physics-oriented course on acoustics and music. The other is “Physics and Society”. This course was initiated by Professor Bederson in 1972, the first “popular” course to be offered in the Physics Department. He has taught it on the CBS “Sunrise Semester”. He has also presented a Humanities Seminar entitled “Human Values and Technological Choice”. PATENTS Nonelectric rectangular wave generator—2,706,256 Particle separation apparatus utilizing congruent inhomogeneous magnetostatic and electrostatic fields—3,113,207


RESEARCH PUBLICATIONS OF BENJAMIN BEDERSON

Formative time lags in spark breakdown, with L.H. Fisher, Phys. Rev. 75, 1324 (1949). Also, PhD Thesis, New York U., Dissertation Abstracts International, Source Code SO-14. Search for the transition of streamer to Townsend form of spark in air, with L.H. Fisher, Phys. Rev. 75, 1615 (1949). Further measurements of formative time lags of spark breakdown in air at low overvoltages, with L.H. Fisher, Phys. Rev. 78, 331 (1950). Interpretation of formative time lags of spark breakdown in air at low overvoltages, with L.H. Fisher, Phys. Rev. 78, 331 (1950). Formative time lags of spark breakdown in air in uniform fields at low overvoltages, with L.H. Fisher, Phys. Rev. 81, 109 (1951). Determination of the ratio of the gJ (2 S1/2 ) values for potassium and hydrogen. The gyromagnetic ratio of the electron, with W.B. Pohlman and J.T. Eisinger, Phys. Rev. 83, 475 (1951). Magnetic moment of K40 and the hyperfine structure anomaly of the potassium isotopes, with J.T. Eisinger and B.T. Feld, Phys. Rev. 86, 73 (1952). Nonelectric rectangular wave generator, with M. Silver, Rev. Sci. Inst. 23, 133 (1952). Nuclear spin and magnetic moment of 55 Cs134 , with V. Jaccarino and H.H. Stroke, Phys. Rev. 87, 676 (1952). Low energy scattering of electrons by atoms, with J. Hammer and H. Malamud, Phys. Rev. 100, 1229 (1955). Measurement of the total, differential and exchange cross sections for the scattering of low energy electrons by potassium, with K. Rubin and J. Perel, Phys. Rev. 117, 151 (1960). An atomic beam E–H gradient spectrometer, with J. Eisinger, K. Rubin and A. Salop, Rev. Sci. Inst. 31 (1960); also “Proceedings of the International Symposium on Polarization Phenomena of Nucleons”, Basel, 1961; Helv. Phys. Act., Supp. IV. Measurement of the polarizability of the alkalis using the E–H gradient balance method, with A. Salop and E. Pollack, Phys. Rev. 124, 1431 (1961). 17

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Measurement of total cross sections for the scattering of low-energy electrons by lithium, sodium and potassium, with J. Perel and P. Englander, Phys. Rev. 128, 1148 (1962). Absolute measurements of total cross sections for electron scattering by sodium atoms (0.5 eV–50 eV), with A. Kasdan and T.M. Miller, Phys. Rev. A 8, 1562 (1963). Determination of the polarizability tensors of the magnetic substates of 3 P2 metastable argon, with E. Pollack and E.J. Robinson, Phys. Rev. 134, 1210 (1964). Metastable 3 P2 rare-gas polarizabilities, with E.J. Robinson and J. Levine, Phys. Rev. 146, 95 (1966). Absolute measurements of total cross sections for the scattering of low energy electrons by atomic and molecular oxygen, with G. Sunshine and B.B. Aubrey, Phys. Rev. 154, 1 (1967). Elastic differential spin exchange cross sections for scattering of slow electrons by potassium, with R.E. Collins, M. Goldstein and K. Rubin, Phys. Rev. Lett. 19, 1366 (1967). Crossed-beam electron–neutral experiments, in: B. Bederson and W.L. Fite (Eds.), Atomic Interactions Part A, Atomic and Electron Physics, Vol. 7, Academic Press, 1968, p. 67. Measurement of the electric dipole polarizabilities of metastable mercury, with J. Levine and R. Celotta, Phys. Rev. 171, 31 (1968). Potassium-electron elastic spin-exchange: comparison of experiment with a closecoupling calculation, with R.E. Collins, M. Goldstein and K. Rubin, Phys. Lett. A 27, 440 (1968). Electron-alkali metal inelastic recoil experiments with spin-analysis: experimental method and the small-angle behavior of the 4 2 S1/2 –4 2 P1/2,3/2 excitation of potassium, with K. Rubin, M. Goldstein and R. Collins, Phys. Rev. 182, 201 (1969). The perfect scattering experiment I, II, Comments on Atomic and Molecular Physics 1, 41; 65 (1969). Summary of recent spin-analyzed electron-potassium differential cross section measurements, with R.E. Collins, M. Goldstein and K. Rubin, in: F. Bopp and H. Kleinpoppen (Eds.), “Physics of the One- and Two-Electron Atoms”, NorthHolland, Amsterdam, 1969, p. 642 ff. Report on elastic collisions of low-energy electrons with alkali metals I, II, Comments on Atomic and Molecular Physics 1 (1970). Total electron–atom collision cross sections at low energies—A critical review, with L.J. Kieffer, Rev. Mod. Phys. 43, 601 (1971). Differential spin-exchange and elastic scattering of low-energy electrons by potassium, with R.E. Collins and M. Goldstein, Phys. Rev. A 3, 1976 (1971).


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Measurements of the total cross sections for scattering of low-energy electrons by metastable argon, with R. Celotta, H.H. Brown Jr. and R. Molof, Phys. Rev. A 3, 1622 (1971). Electron–atom excitation with spin-analysis, Comments on Atomic and Molecular Physics 2, 160 (1971). Differential cross sections with spin-analysis for 4 2 S1/2 –4 2 P1/2,3/2 excitation of potassium by electrons, with M. Goldstein and A. Kasdan, Phys. Rev. A 5, 660 (1972). Energy loss of a low energy ion beam in passage through an equilibrium cesium plasma, with L.L. Klavan, D.M. Cox and H.H. Brown Jr., Phys. Rev. Lett. 28, 1254 (1972). Electron polarization behavior in collisions, in: “Atomic Physics III, Proceedings of Third International Conference on Atomic Physics”, Plenum Press, New York, 1972, pp. 401–427. Absolute measurements of total cross sections for electron scattering by sodium atoms (0.5–50 eV), with A. Kasdan and T.M. Miller, Phys. Rev. A 8, 1562 (1973). Production of polarized electrons with the help of a tunable dye laser, Comments on Atomic and Molecular Physics 4, 103 (1973). Measurements of the average dipole polarizabilities of the alkali dimers, with R.W. Molof, T.M. Miller, H.L. Schwartz and J.T. Park, J. Chem. Phys. 61, 1816 (1974). Measurements of electric dipole polarizabilities of the alkali metal atoms and the metastable noble gas atoms, with R.W. Molof, H.L. Schwartz and T.M. Miller, Phys. Rev. A 10, 1131 (1974). Measurement of the static electric dipole polarizabilities of barium and strontium, with H.L. Schwartz and T.M. Miller, Phys. Rev. A 10, 1924 (1974). Energy loss of a low-energy ion beam in passage through an equilibrium cesium plasma, with D.M. Cox, H.H. Brown Jr. and I. Klavan, Phys. Rev. A 10, 1409 (1974). Transport properties of a weakly-ionized cesium plasma, with D.M. Cox, H.H. Brown Jr., L. Schumann and F. Murray, Phys. Rev. A 10, 1711 (1974). Spin-polarization in electron–atom scattering, with T.M. Miller, in: “Electron and Photon Interactions with Atoms, Proceedings of International Symposium”, Stirling, Scotland, Plenum Press, NY, 1976, pp. 191–202. Measurement of the polarizability of calcium, with T.M. Miller, Phys. Rev. A 14, 1572 (1976). Differential spin exchange in the elastic scattering of low energy electrons by rubidium, with B. Jaduszliwer and N.D. Bhaskar, Phys. Rev. A 14, 162 (1976). Atomic and molecular polarizabilities—a review, with T.M. Miller, in: Advances in Atomic and Molecular Physics, Vol. 13, Academic Press, NY, 1977, pp. 1–55.

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Scattering of low energy electrons by excited state sodium atoms using a photon and electron atomic beam recoil technique, with N. Bhaskar and B. Jaduszliwer, Phys. Rev. Lett. 38, 14 (1977). A high temperature atomic beam source, with J.A. Stockdale, L. Schumann and H.H. Brown Jr., Rev. Sci. Inst. 48, 938 (1977). Total cross sections for the scattering of low-energy electrons by excited sodium atoms, with B. Jaduszliwer, R. Dang and P. Weiss, in: “Proceedings of Coherence and Correlation Workshop”, London, Plenum Press, NY, 1979. Total cross sections for the scattering of low-energy electrons by excited sodium atoms in the 3 2 P3/2 , mJ = ±3/2 state, with B. Jaduszliwer, R. Dang and P. Weiss, Phys. Rev. A 21, 808 (1980). Report on ICPEAC 9, Experimental, Comments on Atomic and Molecular Physics 9, 173 (1980). Developments in techniques for the production of quantum-state selected atomic and molecular beams, in: N. Oda and K. Takayanagi (Eds.), “Book of Invited Papers, Proc. of ICPEAC 9”, North-Holland, 1980, pp. 821–830. Absolute total cross sections for the scattering of low-energy electrons by lithium atoms, with B. Jaduszliwer, A. Tino and T.M. Miller, Phys. Rev. A 24, 1249 (1981). Measurements of total cross section for electron scattering by Li2 (0.5–10 eV), with T.M. Miller and A. Kasdan, Phys. Rev. A 25, 1777 (1982). Electric dipole polarizabilities of alkali halide dimers, with B. Jaduszliwer and R. Kremens, in: American Chemical Society Symposium Series, Vol. 179, Ch. 20 (1982). Electron scattering by highly polar molecules: A benchmark experiment, with B. Jaduszliwer, A. Tino and P. Weiss, Phys. Rev. Lett. 51, 1644 (1983). Polarizability of 5s2 5p(2 P1/2 ) atomic indium, with J.A.D. Stockdale, T.M. Miller and T. Guella, Phys. Rev. A 29, 2977 (1984). Small angle (e− , Na) scattering in the 6–25 eV range, with B. Jaduszliwer, P. Weiss and A. Tino, Phys. Rev. A 30, 1255 (1984). Measurements of the electric dipole polarizabilities of the alkali-halide dimers, with R. Kremens, B. Jaduszliwer, J. Stockdale and A. Tino, J. Chem. Phys. 81, 1676 (1984). Electron scattering by highly polar molecules, with B. Jaduszliwer and A. Tino, Phys. Rev. A 30, 1269 (1984). Electron–atom photon interactions in a laser field, with B. Jaduszliwer, G.-F. Shen and J.-L. Cai, in: “Colloque International Atomic and Molecular Collisions in a Laser Field”, Abbaye de Royaumont, France, 1984; Journal de Physique 46, Supp. No. 1, C1-241 (1985). Polarization effects in electron–atom collisions, in: “Proceedings of the NATO Advanced Study Institute, Palermo”, Plenum Press, New York, 1985.


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Total cross sections for electrons scattered by 3 2 P3/2 sodium atoms, with B. Jaduszliwer, G.F. Shen and J.-L. Cai, Phys. Rev. A. 31, 1157 (1985). Quantitative analysis of the deflection of a sodium beam by laser radiation, with B. Jaduszliwer and G.F. Shen, Phys. Rev. A 33, 3792 (1986). Scattering of electrons by alkali-halide molecules: LiBr and CsCl, with L. Vuskovic, M. Zuo, G.F. Shen and B. Stumpf, Phys. Rev. A 40, 133 (1989). Absolute elastic differential cross sections of electrons scattered by 3 2 P3/2 sodium, with M. Zuo, T.Y. Jiang and L. Vuskovic, Phys. Rev. A 41, 2489 (1990). Absolute small angle differential electron excitation cross sections for the resonant transition in sodium, with T.Y. Jiang, C.H. Ying and L. Vuskovic, Phys. Rev. A 42, 3852 (1990). Polarizabilities of the alkali halide dimers II, with T. Guella, T.M. Miller, J.A.D. Stockdale and L. Vuskovic, J. Chem. Phys. 94, 6857 (1991). Absolute cross sections for low energy electron excited-sodium scattering, with T.Y. Jiang, L. Zuo and L. Vuskovic, Phys. Rev. Lett. 68, 915 (1992). Measurements of the dc electric dipole polarizabilities of the alkali dimer molecules, homonuclear and heteronuclear, with V. Tarnovsky, L. Vuskovic and B. Stumpf, J. Chem. Phys. 98, 3894 (1993). Threshold-energy region in the electron excitation cross sections of the sodium resonant transition, with C.H. Ying, F. Perales and L. Vuskovic, Phys. Rev. A 48, 1189 (1993). Superelastic electron scattering by polarized excited sodium, with T.Y. Jiang, Z. Shi, C.H. Yang and L. Vuskovic, Phys. Rev. A 51, 3773 (1995). Communications in Physics, Physics Today 50, 63 (November 1997). OTHER PUBLICATIONS An atomic beam recoil method for studying the scattering of electrons by atoms, with K. Rubin, Technical Report No. 1, N-ONR-285 (15), N.Y.U., 1958. Elastic scattering of electrons by atomic hydrogen, with J. Hammer and H. Malamud, Technical Report No. 2, N-ONR-285 (15), N.Y.U., 1958. Velocity analysis of high speed atomic beams by a Stern–Gerlach magnet and by phased electron beam chopping, with K. Rubin, NYO-10, 117, U.S.A.E.C. Technical Report, N.Y.U., 1962. Atomic and molecular physics, A report by the National Academy of Sciences National Research Council Committee on Atomic and Molecular Physics. (National Academy of Science, 1971, Co-editor, with S.J. Smith and L. T. Crane.) Lasers—from abstraction to reality, Journal of Science College Teachers 3, 322 (1974); also Speaking of science, Conversations with outstanding scientist, Vol. II, 1973, No. 8 (with J.K. Hulm and E. Edelson); A.A.A.S. Cassette.

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Atomic and molecular science and energy-related research, A report of the NASNRC Committee on Atomic and Molecular Science, NAS-NRC, Washington, DC, 1976 (Editor). Atomic physics—a renewed vitality, Physics Today 34, 188 (November 1981) (50th Anniversary of American Institute of Physics issue). Atoms, in: “Encyclopedia of Applied Physics”, VCH Publishers, with AIP, German Physical Society, and Japan Society of Applied Physics, 1990. Atomic Physics, in: “Academic Press Dictionary of Science”, 1991. Atoms, in: “Encyclopedia of Applied Physics”, VCH Publishers, with AIP, German Physical Society, and Japan Society of Applied Physics, 1990. Atomic Physics, in: “Academic Press Dictionary of Science”, 1991. Informal discussion on the history of the International Conference on Atomic Physics, with V. Hughes, H. Shugart et al., AIP Conf. Proc. 233, 585 (1991). Guest Comment: Some Personal Reflections on Physical Review, Am. J. Phys. 60, 873 (1992). SEDís at Los Alamos: a personal memoir, Physics in Perspective 3, 52–75 (2001). “A Century of Physics”, by D.A. Bromley, Book review, Physics Today 55, 53 (October 2002). Sheldon Datz, with Joseph Martinez and Herbert Krause, Physics Today 55, 88 (November 2002). “Selectivity and Discord: Two Problems of Experiment”, by Allan Franklin, Book review, Physics Today 56, 53 (August 2003). The physical tourist: Physics and New York City, Physics in Perspective 5, 87– 121 (2003). Albert Einstein, J. Robert Oppenheimer, Edward Teller, three articles for the “Encyclopedia on Science and Technology Ethics”, Gale-Macmillan Reference USA, Woodbridge, CT, 2005, ISBN: 0-02-865831-0. Fritz Reiche and the International Committee in Aid of Displaced Foreign Scholars, Physics in Perspective 7 (4) (2005).


A PROPER HOMAGE TO OUR BEN1 H. LUSTIG Professor Emeritus of Physics, The City College of the City University of New York Treasurer Emeritus, The American Physical Society

A proper homage to our Ben, To tell about the now and then, To honor one so free of meanness, Would take the tongue of Demosthénes, The dubious sweetness of R. Jastrow,2 Prolixity of Fidel Castro. So as to make it short and terse, I must perforce resort to verse, (I might have said that to save time One is obliged to turn to rhyme.) As Frost showed, if you want to snow ‘em, Don’t be profound, just make a poem. Ben Bederson got grace and knowledge, Where else, at good old City College, Where students were great and staff tried to teach, Where he took years of Public Speech, But never abandoned his Bronxian jargon: De gustibus, there ain’t no arguin’. When war broke out he took his stand; He joined the Army and was sent To work out at Los Alamos. He must have had a ‘mallow boss, 1 This tribute was “inspired” by a doggerel which the author read at the party in 1996, toasting and roasting Ben Bederson, on the occasion of his retirement as Editor-in-Chief of The American Physical Society (APS). It has never been published. This revised and updated version probably shouldn’t be either. 2 Robert Jastrow, space physicist, former director of the Goddard Institute for Space Studies.

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H. Lustig For Bederson in uniform Is stranger by far than cuneiform.3 With peace restored it was his object To launch his own Manhattan project. He ended up at NYU, Got Ph.D. and Betty too. The rest is not a mystery, But (slightly revisionist) history. After a stint at MIT New York was where he chose to be. It wasn’t the money, it wasn’t his group; It was the sweet and sour soup Soon he appeared on all the lists Of great atomic physicists. Then everyone was heard to say “Bederson—Editor—Phys. Rev. A”. For fourteen years he labored on; (See chapter by B. Crasemann4 ). But finally he mused: “Oh bah, Heute gehört mir Phys. Rev. A”. (He actually said: “Oy veh, All I have is Phys. Rev. A”.) “If APS knows what to do, Morgen die ganze Phys. Review”. And so from editorial star, He now became the mighty czar Of all of our publications— A job that calls for drive and patience; For telling quality from kitsch; For spending countless days at Ridge;5 For dealing with Council,6 with E-Pub,7 with REC;8 For listening to wisdom as well as to Dreck.

3 Val Fitch, “Benjamin Bederson in the Army, World War II”, this volume. 4 Bernd Crasemann, “Ben as APS Editor”, this volume. 5 Ridge, Long Island, the location housing The American Physical Society’s publishing operations. 6 The governing body of The American Physical Society. 7 A working group to advance electronic publishing. 8 The Ridge Editorial Council, a group made up of resident editorial staff.

A PROPER HOMAGE TO OUR BEN

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He guided the journals with aplomb, In print, on line, on CD-ROM. His grasp was firm, his reach was clear. Not even Goudsmit9 was his peer. He dealt with staff, with referees, With authors and their fervent pleas. By turns he was profound and funny And even made us lots of money. For years he led our journals’ growth, True to the ephebic oath, Until one day the blessed Merzbacher Task Force caused some Schmerz.10 But this as well he took in stride. So that we could proclaim with pride, When papers didn’t pass the door: “That’s quite all right, for less is more”. It was that insight, let’s confess, That left us then Ben-Bederson-less. Less aggravation, fewer horrors For Ben, and fewer tzores. No more papers from Krajina11 No more threat of a subpoena12

9 Samuel Goudsmit (1902–1978), co-discoverer, with George Uhlenbeck, of electron spin. Scientific

head of the Alsos mission (which was formed during World War II to try to detect the status of the German nuclear weapons effort), and the autocratic, but highly effective, long-term boss (Managing Editor, 1951–1966, Editor-in-Chief, 1967–1975) of The Physical Review. 10 Confronted with the dilemma that the journals were growing and library subscriptions declining, causing prices to be raised, a task force headed by Eugen Merzbacher recommended controlling growth by “ratcheting up” the standards for the acceptance of manuscripts. In spite of this decision, submissions and the journals have continued to grow. 11 The Krajina Region (making up about one third of Croatia) is on the frontier between Croatia and Bosnia-Herzegovina with a university in the city of Knin. Heavily populated by Serbs, Krajina proclaimed itself an autonomous Serbian province after Croatia declared its independence from Yugoslavia in 1991. The authors of a paper submitted from this university demanded that their by-line give Krajina as the country of origin. But no government, nor the United Nations, had recognized Krajina as a country. Resisting significant pressure from both Serbs and Croats, Bederson wisely decided that the location of the authors was simply Knin, period. 12 Beginning during the reign of Ben’s predecessor, David Lazarus, the APS was subjected to frivolous lawsuits and threats of lawsuits, mostly for what was published and not published in its journals.

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H. Lustig No more threats and no more crap From Santilli13 or from Stapp.14 No more cutting to the bone; No more deals with Andy Cohen.15 No more scraping off the barnacles; No more trek with Adams16 chronicles. No POC17 to give him shpilkes No more accolades from Wilkins.18 No more trips to regions rustic;19 No more Franz and no more Lustig.20 But though now living on his Rente For Ben no dolce far niente Imperatively categorical His oeuvre became historical: He writes in PiP21 and is a mover In FHP22 (please read R. Stuewer23 ).

(See Harry Lustig, “To advance and diffuse the knowledge of physics: An account of the one-hundredyear history of the American Physical Society”, American Journal of Physics 68 (7) (July 2000), 595–636.) 13 Ruggero Maria Santilli of The Institute for Basic Research, who complained bitterly about the rejection of his papers “disproving” Einstein’s relativity, which he attributed to Jewish domination of APS’ journals. 14 Henry P. Stapp, who experienced some difficulty in having his manuscript “Theoretical model of a purported empirical violation of the predictions of quantum theory” published in The Physical Review A, 50 (1) (July 1994). 15 Ben’s interlocutor in unsuccessful negotiations to entangle The Physical Review D with Paul Ginsparg’s (Los Alamos) preprint archive. 16 Peter Adams, in a singular arrangement, was the permanent, full-time, in-house editor of The Physical Review B. 17 APS’ Publications Oversight Committee (POC). 18 John Wilkins, the formidable chair of the POC, who seldom dispensed compliments. The rhyming with “shpilkes” is weak, but the interaction was strong. 19 Such as Ashford, Ohio, to negotiate with a small company for the production of Physical Review Letters. 20 Judy Franz was (and still is) the Executive Officer of the APS, and Harry Lustig (the author of this doggerel) was its Treasurer. With Ben Bederson, the Editor-in-Chief, they formed the triumvirate that ran the APS. 21 The journal Physics in Perspective, edited by John S. Rigden and Roger H. Stuewer. 22 The APS Forum on History of Physics. 23 Roger H. Stuewer, “Ben Bederson: Physicist–Historian”, this volume.

A PROPER HOMAGE TO OUR BEN But still, it hardly need be said, The famous Bederson sextet: Ben senior, Betty, and four sons (Of whom not one is yclept Hans) Makes time for family palaver; Bravo to Ben, to Betty brava; Bravi to Josh, to Geoff, to Aron, And to Ben junior, for their carin’. Will our hero still climb higher, From Bederson to Biedermeier? Will there be cause for future cheers, Another Festschrift in ten years? We wish it and we make this toast: To Ben, santé, l’chaim, Prost. This is the best that I can do: A most imperfect clerihew.

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BENJAMIN BEDERSON IN THE ARMY, WORLD WAR II VAL L. FITCH Department of Physics, Princeton University, Princeton, NJ 08540, USA

Many years ago I wrote a brief account of some of my experiences as a GI during WWII working at the Los Alamos Laboratory. I recounted the following story. “Saturday morning inspection also remained on the schedule but became devoid of spit and polish. The new company commander would stride down the length of the barracks at something less than the speed of light and that was it, for another week. On one morning, however, something struck him on the head as he sped down the aisle. He pulled himself to a stop a few strides later, turned around to see some curious object wildly gyrating on a string from the ceiling of the barracks. “What’s this?” asked the captain. “That’s a bagel, sir”, said my fellow platoon member standing at attention beside his bunk. “Well, it’s pretty dangerous up there, you’d better eat it”, said the captain and continued on his way. My friend had received a box of bagels from his home in New York City. In his rather desperate homesickness he had suspended one of the bagels on a string from the ceiling so he could lie back on his bunk and admire it. It just happened to catch the top of the head of the unusually tall captain”. Later in the same essay I revealed that, well after the war, “the bagel lover is now chairman of the physics department of a well-known university, again comfortable in the environment of a big city”. As you have guessed, my fellow platoon member was Ben Bederson. I have been asked to describe what life was like for Ben at Los Alamos during WWII. Ben and I were members of an army group “Special Engineering Detachment”. We were SEDs. The contribution of this group to the Manhattan project appears to be one of the better kept secrets of the war, certainly the least appreciated, despite the fact that, at Los Alamos, by the end of the war, nearly 50% of the technical personnel were SEDs. As one example, in Richard Rhodes’ otherwise excellent account of The Making of the Atomic Bomb, there is but one reference to the SEDs, viz., “The general’s decision started Norris Bradbury and his crews of Special Engineering Detachment GI’s—SEDs, the science-trained recruits were called . . .”. 29


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How did this army group come about? At the end of 1943 recruiting new civilian staff for Los Alamos was becoming impossible. The war had been going full tilt for over two years, all technically trained people in the country were already in positions where their skills were being utilized. In addition, draft boards were under increasing pressure to meet their quotas, and occupational deferments were a thing of the past. Furthermore, there was no possibility that enough adequate housing could be constructed to attract new civilian employees. It was at this time that someone hit on the idea to bring to Los Alamos army people who had had technical training; in chemistry, engineering, physics, mathematics, etc. Not only could the manpower problem be addressed but, since the army personnel could be housed in barracks, the housing could be minimal, recreational facilities, food— everything could be minimal. Living in barracks, the soldiers would need only about 40 sq. feet per person, quite different from the needs of civilians living in apartments. As fast as the barracks were constructed, the ranks of a Special Engineering detachment began to be filled out. From late in 1943 onward their ranks swelled at about forty per month. According to the book, Critical Assembly, Colonel M.H. Trytten would find the SEDs and General Grove’s office, using its power of highest priority, would arrange to have them transferred to Los Alamos. The needs of the laboratory spanned the gamut, from skilled machinists to explosive experts, from chemists to physicists and electrical engineers and the SEDs reflected the needs. Some had advanced degrees, a few had PhDs (draft boards were often hard-pressed to fill their quotas and no one was totally exempt). Many were like Ben and myself, drafted into the army from college where we had had 3 or more years majoring in physics or mathematics. Mainly we functioned as research assistants to the senior scientists who were the core of the Oppenheimer team. Ben was among the first of the SEDs and he discovered that, in the barracks to which he was assigned, he was bunked next to a machinist by the name of David Greenglass who was eventually revealed as a spy for the Soviets and the brother of Ethel Rosenberg. According to Ben, hot political arguments were generated by Greenglass’ extreme pro-Soviet positions. Conveniently for Ben, it was also at this time that the machinists were moved to barracks of their own since they did shift work on the limited number of machines and their hours disrupted the sleep of others. Ben moved to the barracks where I ended up on arrival. It was a curious situation for the SEDs. We were in the army, slept in barracks, and ate in an army mess hall. At the same time, working in the fenced-in technical area, we were beholden to civilians. It was frustrating for the cadre of army officers under whose command we were. We were soldiers, weren’t we? We must be kept in condition to fight a war. Discipline must be maintained. So we were awakened early in the morning for calisthenics—we had spit-and-polish inspections. While we did not have KP, we were assigned duties to keep the latrines clean.

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Perhaps the most apt description of most of the SEDs was that we were intellectual foot soldiers. We were skeptical of anything the army told us if not downright cynical. On one occasion there was a formal military review prompted by a visit by General Groves. The town turned out to watch. Bernice Brode, one of the scientists wives, described the soldierly skills of the SEDs as follows. “The SED boys were terrible, they couldn’t keep in step. Their lines were crooked. They didn’t stand properly. They wore glasses. They waved at friends and grinned. Once, when a sergeant became irritated by his yawning, halfhearted crew doing their morning calisthenics he shouted, ‘if you guys think I like this job, you got another thing coming’, one of the SED boys offered to lead the calisthenics drill in his place. He shouted orders in imitation of the sergeant’s voice: ‘thumbs up, thumbs down, thumbs wiggle-waggle’. Even the sergeant broke down and dismissed them”. The barracks, each housing sixty men, were single floor with double-decker bunks extending out from the walls with an aisle perhaps 10 feet wide running down the center. In this center aisle there were four coal-burning potbellied stoves spaced at equal intervals down the length of the building. A coal bin was outside at the end. When I arrived in the summer of ’44 I looked at those stoves and knew full well, when it got cold, who was going to keep those stoves fired up. We would be assigned, on a rotating basis, stove duty. Later in the summer of ’44 it was George Kistiakowsky who changed all that. Kisty was a famous Harvard chemist, an émigré from Russia, who spoke English with a rich and fascinating accent. He was gregarious, loved people, and clearly had a high regard for those people who worked for him. He was a natural leader and the SEDs under his direction would follow him to the end of the earth A fair fraction of the SEDs worked for Kisty in the development of the high explosive lenses used in the implosion gadget. In fact, some SEDs were group leaders in his division. Kisty felt that the military was going too far, that the attempts to maintain strict military discipline were just petty harassments, and, of course, we all agreed He carried the issue to General Groves who apparently refused to do anything even as Kistiakowsky threatened to resign. Then all at once, we had a new company commander who had some sense of what was important. He was very relaxed about discipline. Now, indigenous labor did not only the KP but also cleaned the latrines, calisthenics were gone. The Saturday morning inspection of quarters was continued but in a much more relaxed manner and I think we all agreed that it should. Some order must be preserved with 60 men living in a single room. After these changes, morale improved significantly. We were still responsible for keeping the barracks heated but in our case, one of our comrades, Bill Davis, volunteered to take the job for a small weekly reimbursement from each of us. He was older than most, married and could use the extra money. He was a graduate of Dartmouth and while in college had been on the ski team. Indeed, he had trained with the ski troops of the 10th Mountain Infantry division

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and while there had been picked out for work at Los Alamos. He was the one who gave me my first skiing lessons. This new, relaxed, atmosphere suited Ben. He had not taken well to army discipline and was always challenging the rules. Consequently, he was frequently short-listed for reprimand and was often saved from demotion by the intercession of his civilian supervisors in the tech area. Now, the relation between the army and the SEDs, especially Ben, relaxed significantly. Ben apparently got to know Kistiakowsky at work. I came to know him on the local ski slope. I have already mentioned that Bill Davis got me interested in skiing and, through the fall, spent Sundays (remember, in WWII everyone worked six days a week) working on the local slope cutting down trees to widen the trails. Kisty came by, saw how laborious the whole process was and deployed some of his crew the next day to wrap explosives around the trees and simply blow them up. It was a marvelous solution. Kisty was an avid skier and it was on the ski slope that I got to know him, occasionally bumming a ride back to town with him instead of bumping along in the back of an army 6 X 6 truck. So Ben and I were housed in the same barracks and had remarkably parallel activities at work. However, we knew each other only casually. We had a different circle of friends. Ben’s interests were much more intellectual than mine. He and his friends, as members of his Mushroom Society, would sit about in the evenings in the tech area listening to Mahler recordings on their home-built hi-fi system. It would be some years before the level of my own music sophistication was up to Mahler. I was in the hiking, skating, skiing, square dancing crowd (as were many of the luminaries; Fermi, Peierls, Weisskopf). Ben worked with Don Hornig (Hornig was later to be President Johnson’s science adviser) on the development of the X-unit which generated the high voltages that went to the detonators on the 32 lenses of the implosion device. I worked for Ernest Titterton, a member of the British mission, and was concerned with measuring the degree of simultaneity of all detonators firing. Part of the program involved flight testing using dummy bombs dropped from B-29s which took off from Wendover, Utah, and released their loads on targets in the Salton Sea in southern California. Enlisted men like Ben and me could not go to Wendover and tell the army air force officers what to do. So we were privileged to travel there in civilian clothes posturing as scientists from Washington. The army gave us each $200 to spend in Santa Fe on civilian clothing. In retrospect, it is hard to imagine what this meant to us. After all, in WWII, if any military person was found in civilian clothes he was automatically a deserter. Of course, we carried special papers giving us the approval to appear as civilians. I made the trip to Wendover in April of ’45 and Ben in May so we didn’t overlap there. Subsequently, Ben became part of the team that went to Tinian to assemble and ready Fat Man, the plutonium implosion gadget, for delivery. I went to Alamogordo to participate in

BENJAMIN BEDERSON IN THE ARMY F IG . 1. The crew from Los Alamos that went to Tinian Island to prepare for delivery thin man (the gun weapon) and fat man (the implosion gadget). Ben is in the third row, fifth from the left. Other familiar figures include, in the first row, Norman Ramsey, fifth from left and George Reynolds, eighth. In the second row, Philip Morrison, second, Robert Serber, eleventh, and Luis Alvarez, fourteenth, all from the left. General T.F. Farrell, 2nd in command to General Groves, is also in the second row, eighth from the left. [The author is grateful to Michael Vickio, president of the Manhattan Project Heritage Preservation Assoc., Inc., for supplying the photograph.]

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the first test at Trinity, where, among other activities, we set about to measure the degree of simultaneity of the firing of the 32 detonators. The SEDs were an eclectic bunch. After the war many went on to get PhDs. Some of them, like Ben, became important figures in physics. There were, among the SED alumni, some who became distinguished mathematicians, e.g., John Kemeny, who invented the Basic programming language, and who later was a highly successful president of Dartmouth. Also, Peter Lax, who became the director of the Courant Institute at New York University, and a colleague of Ben’s. The question inevitably arises, how do you feel about having worked on the development of the atom bomb and its subsequent use? I agree with Ben who addresses this question at the end of his memoir of his days at Los Alamos. He said, “It is virtually obligatory for any atom bomb memoir to include a discussion of the merits of President Harry S. Truman’s decision to drop the bomb. I can take care of this obligation here by simply describing what I have told students at New York University whenever they would confront me with this question. I would point out that had the bombs not been used, many of the men and women in the class would not be there, since one of their parents or grandparents would not have survived the inevitable invasion of Japan. This was, and is, a very compelling argument”.

References [1] L.R. Groves, “Now It Can be Told” (1962). [2] D. Hawkins, “The Los Alamos Story, Toward Trinity” (1961). [3] L. Hoddeson et al., “Critical Assembly. A Technical History of Los Alamos during the Oppenheimer Years” (1993). [4] J. Wilson, “All in Our Time. The Reminiscences of Twelve Nuclear Pioneers” (1975). P. Abelson, L. Alvarez, H. Anderson, K. Bainbridge, F. de Hoffman, V. Fitch, O. Frisch, M. Kamen, J.H. Manley, B. McDaniel, A. Wattenberg, R.R. Wilson. [5] L. Badash et al., “Reminiscences of Los Alamos, 1943–1945” (1980). John Dudley, Edwin McMillan, John Manley, Elsie McMillan, George Kistiakowsky, Joe Hischfelder, Laura Fermi, Richard Feynman, Bernice Brode, Norris Bradbury. [6] B. Bederson, SEDs at Los Alamos: A personal memoir, Physics in Perspective 3 (2001) 52–75. This is Ben’s own fascinating account of his days at Los Alamos, even how he learned to drive a car. And his experiences on Tinian were unique to him. [7] R.C. Sparks, “Twilight Time, A Soldiers Role in the Manhattan Project at Los Alamos”, Los Alamos Historical Society, 2000. The experiences of an SED machinist who was head of a machine shop. [8] F.M. Szasz, “British Scientists and the Manhattan Project” (1992).


PHYSICS NEEDS HEROES TOO C. DUNCAN RICE∗ University of Aberdeen, United Kingdom

It’s a very long time since Ben Bederson and I served at New York University, and I haven’t become a scientist in the intervening years. But I do have a comment on some directions in modern policy debate about science, at least in my own country, and some thoughts on the reason why it’s critical to have people like Ben at the top. The year 2005 has been designated by the United Nations as “The International Year of Physics”. The purpose, as I understand it, is twofold. On the one hand it is to commemorate the hundredth anniversary of Einstein’s publication of three groundbreaking papers on Brownian motion, special relativity and the photoelectric effect. On the other, it is to use the occasion to promote world-wide the importance of the study of physics. In view of my present role, I should perhaps champion James Clerk Maxwell instead of Einstein. He was for a time a Professor of Natural Philosophy at Marischal College, Aberdeen. The embarrassment is that we fired one of Britain’s greatest scientists. Until 1860 Aberdeen consisted of two Colleges, King’s and Marischal. When they merged to form the new University I now lead, it had two professors of physics. Maxwell was at the newer College, Marischal. Even though he was married to the daughter of its Principal, he was the one they let go at the merger. This, it is said, was in part because he was the more junior scientist, and in part because the other was considered a better teacher. I have no son-in-law or daughter-in-law equivalents to trouble my conscience but it is every President’s or Principal’s nightmare to think you have let some future genius such as Maxwell leave on your watch. Actually Maxwell had not been entirely comfortable in the austere culture of Northern Scotland. He once wrote to a friend that “No jokes of any kind are understood here. I haven’t made one for two months, and if I feel ∗ C. Duncan Rice was Dean of the Faculty of Arts and Sciences at NYU between 1985–1994, and also Vice-Chancellor between 1991–1996. He is now Principal and Vice-Chancellor of the University of Aberdeen, in the United Kingdom.

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one coming on I bite my tongue”. Ben Bederson would not have been happy in mid-Victorian Aberdeen. Great academic gaffes aside, Einstein is perhaps the brand name of physics in the popular domain—although I note that Physics World recently voted Maxwell’s Electromagnetism theory as being more significant than the work of Einstein or Newton. I am as quick as anyone inside the academy to write off such events or polls as gimmicks—unless, of course, they give me or my institution a high ranking. But in either case, physics clearly faces a contemporary challenge in demonstrating its importance to the academy and to society—and to that end perhaps we need all the help we can get from gimmicks or any other means. It is now some years since I was a colleague of Ben’s at NYU. I left America in 1996, and am no longer close enough to be able to comment knowledgeably on the position of physics there. Certainly in Britain the declining number of students taking physics courses is an ongoing concern. In Scotland, for example, the numbers of those enrolling in courses where physics is an element have declined by about 10% between 1998 and 20021 . This has at least something to do with widening curricular choices, but together with other comparable trend lines it suggests ignorance about the beauty, the importance, and the accessibility of physics. From a British perspective, however, physics is at risk from a wider challenge to universities engaged in teaching and research on pure science. There is a constant danger that the public, or their political representatives, fall into believing that basic science is a luxury—that knowledge without immediate application is remote from the solution of ‘real-life’, practical problems. On this side of the English speaking Atlantic, higher education debate is biased towards questions about the marketable commodities or direct contributions to economic development produced by scientific teaching. In political debate at least, applications and training have become popular surrogates for research and teaching. Perhaps that has always been so, but it seems ever to get worse. But we have to recognise the dilemma for modern universities. A full-service physics department is extraordinarily expensive, because it would encompass the full spectrum of activity—solid state, atomic, high energy and astronomy, each with a theoretical as well as an experimental wing. It’s not surprising then that as institutions try to manage in conditions of financial restraint, physics has become distributed around the academy. Under severe financial pressures in the eighties, my University, once the employer of James Clark Maxwell, was forced to abolish research physics. But that’s not to say that physics has disappeared or become invisible. There are ways, cheaper but admittedly less attractive, of presenting physics other than through a traditional departmental structure. Physicists can be 1 ‘Science Policy in Scotland’, Institute of Physics Report, Policy Paper 2003-1, p. 15.

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located in applied mathematics, engineering, in chemistry or even medical departments. At Aberdeen, indeed, we have a Department of Biomedical Physics. But it is essential to retain some major physics departments. In Scotland we are having to face this very issue—with the added challenges of having relatively small institutions, in a small and limitedly prosperous democracy. Our Funding Council, the Government’s ‘arms length’ organisation that oversees the distribution of public money to universities, has in the past year begun a targeted effort to bring together respective elements of physics activity across universities in Scotland into a ‘pooling’ arrangement or, as it has been termed, a “Super Department”. This may or may not work, but it’s worth trying because Scotland’s individual universities in general lack the critical mass needed to compete at the very highest level. While there are some significant and significantly sized physics units in Scotland, notably at Edinburgh, the solution may be our best bet in maintaining a high quality physics presence in a small country. Of all disciplines, the tradition of sharing large scale and expensive equipment is perhaps more familiar to physicists than to other scientists. Incidentally, Scottish science faces similar issues in chemistry, and we are taking comparable approaches to cooperation there. But in terms of the public culture of science in a country like this, the physics issue is part of doing everything possible to keep alive in our universities the concept of pure science. Pure science may be expensive, may even seem a luxury to some, but we cannot do without it. That’s especially true of physics, which has dominated the twentieth century. It is as hard to think of a facet of human life that is not touched in some way by physics research as it is easy to reel off a list of technologies it has spawned. The end of the Cold War may have lessened the place of physics in the minds of policy makers. That is the nature of political prioritisation at work, but it ignores the fact that there is so much physicists still don’t know. And what they still have to find out will, among many other things, make huge contributions to preventing disease, and to jump-starting new technologies on which economic growth depends. The problem is that we don’t know where the connections between pure physical research and new applications will unfold—but we do know that many will be based on improved understanding of physics. Our most important work at Aberdeen since Maxwell was Mallard’s success in turning an understanding of nuclear magnetic resonance into the critical diagnostic technology of M.R.I.— but no scientist knew in advance what the application of the pure research would be. Whatever the local position, there is a public interest in governments and funding bodies not losing sight of the importance of pure or curiosity driven research. Since returning to Scotland from New York to become Principal at Aberdeen in 1996 I have frequently found myself making arguments in defense of pure research—most often in the context of the humanities. Much of what I have argued there applies to physics too. The reason is that in public debate, or policy, in

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Britain we increasingly seem to be defining the worth of universities in terms of how useful they can be. Unless we can commercialise research or explain a discipline by its immediacy to graduate employment these areas are often left out of contemporary debate about why our universities matter to society. That can only lead to scholarly impoverishment and ultimately economic weakness if allowed to slip from the work of universities. At NYU I was Dean of a Faculty of Arts and Sciences. These two broad areas are at the intellectual heart of any major university. It’s simply difficult to be a truly great university if you don’t pursue them. That is one argument for why they matter. A more compelling one is that it is the very purpose of a university to be the place where there is room to pursue academic inquiry without knowing what practical application will follow from it. Knowledge, the expansion of knowledge, the understanding and appreciation of the world around us, these are at the core of what a university exists for—and if such things have no place in a university then where do they? And democratic society faced with the myriad of moral debates attached to science, needs more than ever to have academics taking part in that debate instead of leaving it to politicians with short-term and often self-serving agendas. In the context that I describe, physics or pure science, or universities for that matter, have to come out fighting for what they are and why they matter. And we need inspirational academic leaders and figures to do that—inside and outside the academy. In writing that I am deeply conscious that leadership is such a frequently and sloppily used word that it ceases to mean anything. There are foundations dedicated to extending leadership or even teaching it! One of these in the UK states its ‘values’ as being “professionalism, transparency, accountability, respecting equality and diversity, commitment to services of the highest quality, appreciation of the diversity of individual higher education institutions, and responsiveness to our stakeholders”. Where is the reference to qualities such as vision, risk taking, inspiration, and core values? I don’t doubt all this is well intentioned but I simply find these types of ‘leadership’ organisations to be a post-modern fallacy. What I am sure of is that people know when they are led by people that inspire them. It is from that emotional experience that leaderships skills if they can be taught at all are learned. Ben Bederson was first and foremost a scientist, with an old style Bronx personality. But his importance to NYU was in the clarity with which he saw the importance of science as an end in itself, and the centrality of values of excellence and curiosity to the academy. That’s what I mean by the kind of leadership we may now undervalue, and what Ben taught me most about. So that brings me to a question about the nature of personalities needed to lead in science. For our purposes, of course, the question about scientific leadership is not just how it affects other scientists, but how it manages to communicate with people like myself who are from entirely different disciplines. I don’t think that

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kind of skill can be taught, and my recollection of working with Ben was that he had it in spades. I won’t ever forget his patience. When I came to NYU, though I had had extensive experience in a large university at Yale, it was so junior and so focused on the humanities as to be irrelevant. My most recent job had been as academic dean at a small liberal arts college. One of Ben’s first jobs, I am ashamed to say, was to describe to me the established divisions of physics, and to explain gently that having only one solid state scientist probably wouldn’t work very well—or even, I remember to my surprise, that it wasn’t practical (for instance) to have Larry Spruch or Henry Stroke teach astronomy. Once I understood the picture better, of course, his central message was that if something couldn’t be done well it wasn’t worth doing at all. Maybe it was all much easier for him because he is so good at creating commonalties with people from entirely different backgrounds. That in turn was partly because of his transparent humanity. I suppose he and I share an elitism about the importance of very high intellectual quality, but for a young leader in a hurry like myself that sometimes led to being oblivious about real human problems. Again I don’t forget one meeting where he took up the cause of a young—possibly not very able—technician who was in some kind of brawl with our personnel people. Ben brought me over to his side just by pointing to the bureaucracy of a very large institution chiseling away at some minor increase to what he described as this person’s ‘pitiful little salary’. The other part of the communication was the cross-cultural one. In retrospect, that may have been even more important than my initiation into the fundamental structure of the physics profession. I had arrived at NYU with no Yiddish vocabulary at all, and was unable even to conduct myself appropriately in a delicatessen. Ben gave me my first window into the values and the humour of Jewish New York—but that was also the reverse of his own insatiable curiosity about other cultures, his insistence that people of all sorts and backgrounds simply had to be taken for what they are, and his endless wide-eyed amazement at the pomposity and self-importance of Wasps. I have drifted off into the personal. But there are some fundamental lessons to be drawn from Ben’s work as a dean that are much more instructive than the stuff from leadership academies and management journals. First, it isn’t worth being a leader unless every decision is based on believing in bone, blood, and brain—forgive my loose grasp of anatomy—that the only science worth doing is superb science. Secondly, leading the scientists is fine, but it won’t amount to a hill of beans unless the leader can also explain to non-scientists what is going on. And finally, the leader becomes unattractive and therefore ineffective if he or she doesn’t combine respect for the most elite scientists with compassion for the people at the bottom of the system.

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There is only one final point. All my contacts with Ben demonstrated how much he loathed bureaucracy even when he and I had to embody it. I don’t mean by this that he wasn’t acutely aware of the importance of being financially responsible— but he believed that administration was there to serve science and the world of ideas, and not to hamper it. I can’t speak for him, but I think that it was the frustration with the way modern universities aren’t always able to prevent bureaucracy becoming over-rigid that ultimately made him fed up with administration. But of course it is precisely the kind of people who object to the weight of administration who are the ones we need as university leaders. I suspect that most of the readers of this essay will be Americans. The only consolation is that the stranglehold of bureaucracy on British universities is even worse than anything either Ben or I encountered in a private American university.


TWO CIVIC SCIENTISTS—BENJAMIN BEDERSON AND THE OTHER BENJAMIN NEAL LANE Rice University

I have known and greatly admired two Benjamins in my lifetime—Benjamin Franklin, whom I could only read about in the history books, and Benjamin Bederson, whom I had the great good fortune to have as a colleague in the field of atomic physics. The April 2003 meeting of the American Physical Society, held in Philadelphia, included a session on Benjamin Franklin, with contributions by scientists and historians. The following comments about Franklin are based on my contribution to that session and an article in Physics Today that followed.1 These two Benjamins, born 215 years apart, had different backgrounds and experienced different circumstances, but at an important fundamental level, they had much in common. Both have made important contributions to science and to this country. They were both city gentlemen, Franklin in Boston, and later, Philadelphia, and Bederson in New York City. Given their high intellect, easy manner and good humor, I believe they would have greatly enjoyed one another’s company. Perhaps they would have co-authored a paper. Their physical appearance is quite different, though to this author’s eye, neither appears to have placed an unreasonably high priority on having a high density of hair on top. Ben Bederson, in his fascinating article “The Physical Tourist—Physics and New York City”,2 notes that Ben Franklin’s likeness stands among the many American scientists and other dignitaries honored with busts in the Colonnade on the campus of the Bronx Community College, formerly the University Heights campus of New York University (NYU). The reader might conclude that my comparison is a bit of a stretch, given over two hundred years of accumulated lore surrounding Franklin and the large number of books written about his life, including the recent excellent biography by Walter 1 Neal Lane, Benjamin Franklin, civic scientist, Physics Today 56 (10) (2004) 24. 2 Benjamin Bederson, The physical tourist—physics and New York City, Physics in Perspective 5

(2003) 87–121. 41


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Isaacson.3 Well, I would only say, give Bederson some time. This volume is a very good start. The point of the two-Bens comparison in my contribution to this volume is a serious one. It is my way of expressing my great admiration and affection for Ben Bederson and my appreciation for all he has done for science throughout the world, for physics in particular, and through his work as a scientist, teacher, scholar and historian, for our American society. But, my message is also about the uncertain future of science and the responsibility of scientists that goes beyond their work in the laboratory and classroom. This is the role of “civic scientist”, a phrase that I believe was coined in the mid-1990’s by one of the speechwriters at NSF, Ms. Patricia Garfinkel. I’ll use Ben Franklin and Ben Bederson to explain what, I think, the concept of the “civic scientist” is all about. First on my list, a “civic scientist” must truly be a scientist of the first rank. Although most of the history books have understated Ben Franklin’s scientific achievements, he was a scientist of the first rank, known and respected on both sides of the Atlantic. He began his scientific career after he already had established a successful career in business. Even so, his scientific achievements, particularly his contributions to understanding electricity, were recognized in his time with perhaps what would be comparable to the distinction of a Nobel Prize today. His careful laboratory experiments led to his “elastic” fluid model of electricity, which suggested a particulate aspect of electricity made up of two kinds of matter, having opposite charges. Franklin suggested a wave-like aspect of light as well.4 He was a practical scientist, in the mode of Bacon and Descartes, applying his new understanding of electricity to invent the lightning rod, among other new devices. Ben Bederson, without question, is an outstanding, internationally acclaimed scientist, as other contributions to this volume confirm. When he was awarded the Townsend Harris Medal (CCNY) in 1993, a portion of the citation, which was printed in the announcement summarizing his accomplishments, read: “During a research career spanning some five decades, Benjamin Bederson has been and is among the dozen or so leading experimental atomic physicists in the world. . . At NYU he developed an atomic beams laboratory, one of the first in the world to study atomic collisions and structure using beam techniques.” And Bederson’s patents testify to his commitment to insure that the science provides benefits to the public that pays for it. Franklin might have been particularly interested in Bederson’s patent on a “particle separation apparatus utilizing congruent inhomogeneous magnetostatic and electrostatic fields”.5 3 Walter Isaacson, “Benjamin Franklin—An American Life”, Simon and Schuser, New York, Lon-

don, Toronto, Sydney, Singapore, 2003. 4 Bernard Cohen, “Science and the Founding Fathers”, W.W. Norton and Company, New York and London, 1995, pp. 141–147. 5 Benjamin Bederson Curriculum Vitae, this volume.

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Second on my list of attributes of a “civic scientist” is wisdom. Ben Franklin was much older than his fellow patriots but he did seem wiser beyond his years on matters of state and everyday life. About the latter, he once wrote, only partly in jest: “If man could have half his wishes he would double his troubles!” 6 Instead of waxing philosophical about future forms of government, Franklin tended to apply what he knew from experience to solve specific problems. He freely shared opinions on a variety of matters, and made his points with humor and clarity. Franklin carried his scientific inclinations to government when, over a year before the Constitutional Convention, he wrote to a colleague “We are, I think, on the right Road of Improvement, for we are making Experiments!”7 Ben Bederson has never been said to lack opinions or good ideas. And, he articulated them so effectively, in his New York friendly way, that he usually won the argument. Throughout his career, organizations, communities and individuals have come to him seeking wisdom, for example, through his many editorial appointments, his selection as chair of the National Academies of Sciences— National Research Council Committee on Atomic and Molecular Sciences, during its formative years, as well as his tenure as Dean of the Graduate School of Arts and Sciences, at NYU. Bederson’s legendary good humor conveys a warmth that is obviously genuine and has allowed him to deal with unhappy authors, reviewers and, I’m sure, students, faculty and administrators. Age aside, Ben Bederson has always been regarded as one of the most original thinkers and wisest colleagues in his field. Third on the list, a “civic scientist” should be able to effectively communicate with the public. Franklin’s well honed skills as a writer on various topics for the public, for example, his “Poor Richard’s Almanac”, also gave him the ability to explain science. The distinguished science historian, Bernard Cohen has noted that Franklin’s book “Observations and Experiments on Electricity” “ranks among the most notable books on science of that age and of any other age”.8 It was published in ten editions and four different languages. And, it was read not only by scientists but by the general public.9 Ben Bederson’s writing is at the same time informative, well documented, and simply a pleasure to read. This is true of his scientific publications as well as his works on history. His article2 on physics and NYC is a literary gem and a treat for

6 James C. Humes, “The Wit and Wisdom of Benjamin Franklin”, Harper Collins, New York, 1995,

p. 76. 7 Cohen (ref. 4), pp. 189–190. 8 Cohen (ref. 4), p. 139. 9 Cohen (ref. 4), p. 144.

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anyone who is interested in the history of science or this extraordinary American city. He is a stickler for clarity and the highest standards in writing and publishing. Fourth on the list, a “civic scientist” should understand the value of compromise and be able to build consensus. Consistent with his pragmatic nature, Franklin possessed the ability to form consensus and negotiate a positive outcome. In Cohen’s words: “Above all, he [Franklin] was an important force in effecting compromise on certain crucial issues”.7 It is Franklin who is credited with the compromise of equal-state representation in the Senate, reversing his previous position, and population-based representation in the House of Representatives. Ben Bederson certainly shared with Franklin an ability to bring people together to reach a positive outcome. It served him well in his many editorial responsibilities over the years, as he made important, but sometimes, controversial changes in policy. His personal warmth and collegial style engendered trust, respect, and affection among all those he worked with, at home and abroad. He was a key figure in the founding of the International Conference on the Physics of Electronic and Atomic Collisions (ICPEAC), in the late 1950’s. In his article in Comments on Atomic and Molecular Physics,10 Bederson described the early days of this important conference, which still thrives today. While not saying so himself, I would speculate that Ben’s skills in building consensus and establishing trust were important to successfully reaching some of those early challenging agreements, including the decision to invite physicists from the U.S.S.R. to the conference in the midst of the cold war. Indeed, physicists from the former Soviet Union became regular participants, which allowed a peaceful dialogue to occur—at least in physics—between the U.S. and U.S.S.R. throughout the cold war. ICPEAC was held in Leningrad in 1967. Finally—and perhaps this sums it all up—a “civic scientist” is a scientist who uses his or her special scientific knowledge and skills to influence policy and inform the public. Ben Franklin advanced the standards of journalism and enhanced the power of the press as a means of getting the truth out to the people, along with some witty common sense. He had a special relationship with the print medium, as a printer, author, bookseller, and founder of a public library.11 Ben Bederson, through his writings and editorial responsibilities, including editor of Physical Review A: Atomic, Molecular and Optical Physics, and later, editor-in-chief of The American Physical Society (APS), where he was responsible for all publications of the APS, focused on raising the standards of scientific

10 Benjamin Bederson, ICPEAC—the early days, Comments on Atomic and Molecular Physics 9

(1980) 173. 11 Cohen (ref. 4), p. 140.


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publication, leading the way in the use of modern information technology and establishing a higher level of fairness in the review process. And as editor of the Forum on the History of Physics Newsletter and the Newsletter of the APS, he is putting a historically accurate human face on physics. Ben Franklin served in government policymaking, not as an elected official, but in other official capacities, including that of advisor to President George Washington. (One could say that Franklin was the first “Presidential Science Advisor”.) Ben Bederson has also served government by giving freely of his good advice, chairing or serving as a member of many advisory and review committees of federal agencies and national laboratories throughout his career. He was a key advisor to the National Science Foundation’s physics programs for several years. Ben Franklin established the Junto, a mutual-aid association of young tradesmen and artisans, who, along with their other activities, performed scientific experiments.12 Ben Bederson was instrumental in organizing a number of international scientific conferences, which were somewhat akin to mutual-aid associations for physicists from all over the world, including the U.S.S.R., who wished to understand atomic physics. Even in the years Franklin was spending most of his time on government matters, he considered himself a scientist—of the practical kind. He took both comfort and delight in his scientific experiments, expressing the desire to continue unfettered on this path, until the duties of patriot and diplomat took him in another direction. Ben Franklin was not simply a good scientist who engaged in public service. He was a scientist who used his scientific nature, intuition and analysis in ways only a scientist could do, thus making him especially effective in the policy and public domain. Ben Bederson was born two centuries too late to discover, first hand, the physical nature of electricity, that having been done by Franklin. But, he was able to use that knowledge of electromagnetic forces, in the most creative ways, to generate and control atomic beams and probe complex atomic systems, particularly under the disturbed conditions that occur during collisions. Ben Bederson did not have the opportunity to help establish a new Republic or fight a king and country across the Atlantic. But early in his life, he was called upon to defend the U.S. During WWII, Ben was a member of the U.S. Army Special Engineering Detachment, serving as a Scientific Aide to the Manhattan Project in Los Alamos. He did not have to give up his physics research career for public service, as Franklin had done, because Bederson’s call to service came before he had established a physics career.

12 Cohen (ref. 4), p. 148.

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Ben Franklin, in playful modesty, referred to himself as “a printer”, when asked his profession. I don’t know how Ben Bederson would answer such a question. But, he would certainly say “physicist”, and then could follow with “teacher, scholar, administrator, historian, editor and writer, and humble public servant”. Ben Franklin lived in a world of dramatic change. He was at the forefront of overturning the old ideas—old paradigms—of government, a time of enlightenment in politics complementing that in science. We, as scientists, face a similar situation in our own time. We are not, of course, fighting to remove a colonial master. But over 200 years after the founding of this great country and the gaining of independence and freedom for generations of our ancestors, we are again threatened by forces and ideologies that would take away the very freedoms that Franklin and his contemporaries fought so hard to obtain. At the same time, the discoveries, advances and future promise of science— including atomic, molecular and optical physics—have never been more exciting. Nothing should be allowed to hold back future discoveries and deny the nation and the world the benefits that will flow from them.The nation will need its scientists to continue to make the great scientific breakthroughs. But they will be called upon to do other things as well. A renewed focus on the unique contributions and responsibilities of scientists seems particularly timely, given the present state of the world, disturbing trends in how science is seen by society and government, and potential harm that can be done to people living today and those of future generations by the misuse of science and technology. The world’s problems will not be solved by science and technology alone; but without question, science and technology must be a significant part of the solution. And scientists will increasingly be called upon not only to continue to make the great scientific discoveries but to work with policy makers and the American public to insure that science and technology do, indeed, provide the benefits they promise. Following the horrible attacks of 9/11 and the collapse of the twin towers of the World Trade Center, observed by Ben Bederson from the window of his Manhattan apartment, we are a nation living in fear and confusion about who our friends and enemies are and how to protect our families and friends from future horrors. The 9/11 attacks did not use the modern developments of science and technology, although the situation may be different in the future and prudent steps should be taken to counter terrorist activity. But an unreasonable fear of terrorism can easily result in government actions that remove long cherished rights and freedoms (including scientific freedoms) from U.S. citizens and others who wish to visit, study and work in our country. Some of this is happening at the present time. Freedoms, once removed, are not so easy to win back. Quite aside from reactions to genuine or perceived concerns about national and domestic security, science appears to be losing out to attacks by special interest groups—business, religious, ideological—that have strong political muscle at the


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present time and agendas that are in conflict with the pursuit of scientific truth. Many scientists feel that the very integrity of science is under attack. There are other areas where science is threatened; perhaps the most striking is in K-12 education, if only because we seem to have made so little progress in recent decades. The most serious problem is that the science and math teachers have insufficient knowledge in the subjects they are asked to teach, are paid too little to make teaching an attractive career, and often have to work in poor conditions with inadequate resources and too little time to do their job. To make matters worse, teachers must cope with political pressures, even laws in some states that attack science, e.g., the teaching of evolution in biology. Even at the university level, the challenge of reaching more than a tiny fraction of the university undergraduates with some understanding of physics has been a formidable one that remains largely unmet on most campuses. However, Ben Bederson, through his “popular” physics courses “Sound and Music” and “Physics and Society”, which he established in 1972 and which he has taught on CBS “Sunrise Semester”, as well as the Humanities Seminar entitled “Human Values and Technological Choice”, has provided a model we should all attempt to emulate. The attitude toward science held by the American public remains positive, according to the National Science Board.13 But, far too few citizens feel they have any real understanding of scientific and technical issues. While we live in a time when the power of science and technology to change people’s lives is so immense, ironically, science is under attack from some influential quarters. There is the risk that a society that continues to be uninformed about science will begin to mistrust science and fail to support it, financially and philosophically. It is a crucial time for scientists, in the role of “civic scientists”, to become more visible in society. Ben Franklin and Ben Bederson are, in my opinion, model “civic scientists”. Both Bens had—and in Bederson’s case, still does have—an enormously productive and influential scientific life, which broadened into a life of public service, as they put all of their knowledge and skills to work for the improvement of society. I did not have the chance to meet Benjamin Franklin. But, Benjamin Bederson was a leader in his field when I finished my PhD at Oklahoma University and set out to begin my career. He was immediately helpful, friendly and welcoming to a “kid from Oklahoma” who had so much to learn. I feel enormously grateful to Ben Bederson for helping me in those early years, as he helped so many others, and, of course, for his enormous contributions to physics, to the history and literature of physics, to education, and to society.

13 National Science Board (of the National Science Foundation), “Science and Engineering Indicators”, 2004, Chapter 7.



AN EDITOR PAR EXCELLENCE EUGEN MERZBACHER Department of Physics and Astronomy, University of North Carolina at Chapel Hill, CB#3255, Chapel Hill, NC 27599-3255, USA, e-mail: [email protected]

Over the years Ben Bederson has called me up from time to time, out of the blue, announcing his arrival for a visit in my office or at my home the next day, or even the same day. He would fly in and out within a few hours, sometimes en route from or to Alaska (where one of his sons lives)—allowing no more time than needed for lunch in one of our college eateries. More to the point, his schedule would provide for just enough leisure to talk me into some new assignment in the physics community or into letting him place my name in nomination on some election ballot. His persuasive powers were, and are, astonishing. Superficially, on these visits Ben would appear relaxed as if he had nothing else to do in the foreseeable future, but by the time we reached dessert he would have delivered his message and his “invitation”. I never thought of him as very athletic, but he certainly knew how to twist my arm. He was always forthright about the time and effort required for the duties that he had in mind for me, but he stressed the support that he could make available to ease the burden. Needless to say, I (almost) always capitulated to his dulcet entreaties, and shortly thereafter we were in the car again on the way back to the airport, usually talking about the “weirdness” of quantum mechanics that bothered Ben long before it became a favorite topic for science writers. Ben was true to his promises: Anyone who succeeded him, whether in the position of chair of DAMOP (Division of Atomic, Molecular, and Optical Physics of the APS) or ICPEAC (International Conference on the Physics of Electronic and Atomic Collisions) or any other of his many roles in our community, would find the house in good order. He usually had resolved messy old problems and identified important new issues on the horizon. I was always in awe of Ben’s people skills and tried to learn from him, but I could never come close to achieving his deftness in charming his quarry. Finally, the day of revenge came in 1991, when I was fortunate to participate in recruiting him to a five-year term (1992–1996) as Editor-in-Chief of The American Physical 49


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Society (APS).1 The physics community owes him a profound debt of gratitude for his far-sighted stewardship of the premier research journals that serve all of us. This essay, in his honor, recounts my interactions with Ben during this period. I should have known Ben! As soon as he accepted the position of Editor-inChief, he established an APS Task Force on Publication Policy2 and twisted my arm to serve as its chair. Our job was to analyze some urgent problems that all sections of the Physical Review (PRA through PRD, alphabetically, shortly thereafter augmented by PRE)3 and the other APS research journals, Physical Review Letters (PRL) and Reviews of Modern Physics, were facing, and to formulate recommendations for ensuring their future into the early 21st century.4 We were asked to consider: (1) a policy on growth of the journals; (2) a policy on launching new journals in emerging areas of physics; (3) the implications of all this for space owned by the APS at Ridge, NY, and leased at Woodbury, NY, housing editorial and production facilities; (4) the future of editorial operations, evaluating the pros and cons of in-house and remote editing and their relative costs, as well as the optimal level of copy and technical editing; (5) the implications for acquiring new hardware (computers, etc.); and last but certainly not least, (6) the inexorable drive toward electronic publishing. These questions and issues reflected the wisdom of David Lazarus, Ben’s predecessor as Editor-in-Chief (1981–1991), and Ben’s farsighted vision of the scientific publishing business, for which his long-term editorship of Advances in Atomic, Molecular, and Optical Physics and of PRA (1978–1992) had prepared him.5 Ben himself was an indispensable ex officio member of the Task Force, as were the other two Operating Officers of the Society, the Executive Secretary and the Treasurer. Ben asked questions and listened carefully to the discussion by the members, most of whom had been selected because of their extensive experience with the operations of the APS publishing enterprise. He was savvy enough not to push his own agenda for the future of the journals—but of course in the end 1 Letter of 27 June 1991, from E. Merzbacher to B. Bederson, with details of Bederson’s appointment as Editor-in-Chief of the research publications of The American Physical Society. 2 This group was also known as the Task Force on Publication Operations Planning. Its members were H. H. Barschall, Benjamin Bederson, Alan A. Chodos, James A. Krumhansl, David Lazarus, Harry Lustig, Barrett H. Ripin, Jack Sandweiss, Myriam Sarachik, N. Richard Werthamer, and Eugen Merzbacher (chair). Its final report and recommendation to the APS Council is dated 31 March 1992. 3 The five sections of the Physical Review represent the following subjects: PRA, Atomic, Molecular, and Optical Physics; PRB, Condensed Matter and Materials Physics; PRC, Nuclear Physics; PRD, Particles, Fields, Gravitation, and Cosmology; PRE, Statistical, Nonlinear, and Soft Matter Physics. 4 For a concise account of the history of the journals and their status near the end of the twentieth century, see the Preface by Benjamin Bederson in “The Physical Review, The First Hundred Years, A Selection of Seminal Papers and Commentaries”, H. Henry Stroke, ed., American Institute of Physics, New York, 1995, pp. xi–xiii. 5 See Bernd Crasemann’s article on Ben as APS Editor in this volume.

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most of our recommendations conformed to his well-considered plans and provided him with the organizational, i.e., political, support that he needed for their successful implementation. One might wish that our country’s leadership would include more people as astute as Ben! Much of the Task Force’s advice was of a technical nature and directed to the Publications Oversight Committee and the management of the APS and its publications. One central issue, however, called for a more formal policy resolution by the governing bodies of the Society, because of its profound implications for the financial health of the APS: the near-explosive growth of the Physical Review, both with regard to submission and acceptance of manuscripts. Although the late Heinz Barschall expressed some technical reservations, based on his long experience as Editor of PRC, the Task Force submitted a unanimous report to the APS Council for endorsement in late March 1992 in the form of a resolution: It is the policy of the American Physical Society that the Physical Review shall continue to accept for publication those manuscripts that advance physics and have been found to be scientifically sound, of high quality, and in satisfactory form. No arbitrary quantitative limits [i.e., quotas] will be set on the acceptance of papers published in the Physical Review. Only editorial judgments of scientific merit, arrived at through the editorial process utilizing the peer review (refereeing) system, may be used to control the content of the Physical Review. As in the past, the [American Physical] Society will implement this policy as fairly and efficiently as possible and without regard to national boundaries.

As Editor-in-Chief, Ben spared no effort to ensure, preferably by gentle persuasion but by decree if necessary, that the editorial process of APS would adhere to this policy statement. The Task Force conducted its review with an awareness of the economic implications and costs of the proposed publication policy, but was not asked to spell out how the expansion in staff, space, and facilities, required by the projected continuing growth of the journals, would be funded. When it comes to implementing lofty principles that have budgetary consequences, tensions inevitably arise between the guardians of editorial purity and the Society’s Treasurer, who is also the Publisher of the Physical Review and responsible for keeping the organization solvent. The priorities of the Editor-in-Chief had to be reconciled with the fiduciary responsibilities of the Treasurer. In the event, their shared keen understanding of the physics community, their mutual commitment to high standards of quality, and their enlightened views of the intricacies of the global system of scientific publishing guided the officers in making decisions that have proved to be sound and prescient. Fortunately, those were relatively rosy economic times, allowing the realization, on his watch, of many of Ben’s initiatives, including the addition of new space for new editors working with new technologies, without impinging adversely on the Society’s activities that benefit the physics community but bring in little or no revenue (e.g., educational programs, international outreach, public affairs initiatives).

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Journals, however, are dynamic systems, and there is never a time when the leadership of APS can afford the luxury of ignoring the pressures on the income side of the ledger sheet. While adhering to the principle that an APS member subscriber should not have to pay more than necessary to defray the incremental cost of subscription fulfillment and mailing of one copy of the journal, how much of a surcharge over the raw production costs might APS reasonably impose on its primary customers, the world’s science libraries? On the other hand, what fraction of the cost of producing research articles should be borne by authors as a legitimate levy against their research budgets, and what form should such charges take (e.g., page charges, article charges, abstract charges, etc.)? Editor-in-Chief Bederson was constantly involved in discussions of these questions.6 His outlook was invariably optimistic, and for the good of physics he was willing to take risks, without being reckless. His heart was always on the side of the individual author, the member subscriber, and the consumer, meaning the reader, but he never lost sight of the overriding objective—to go for the best physics and to foster the highest standards of quality in our research journals. Meanwhile, the flood of submissions of manuscripts from all over the world to Physical Review and PRL continued to rise unabated, causing severe strains in the editorial system. It seemed as if authors regarded every new hurdle a challenge, intended to enhance the appeal of publishing in our journals. At the same time, library budgets and library shelf space could no longer keep up with the rapid inflation in numbers of journals, volume of printed pages, and the attendant skyrocketing costs. The journals were growing and becoming ever more expensive, while the number of subscribers was steadily dropping—two trends that, if not soon reversed, were bound eventually to lead to a catastrophe. The Physical Review was in danger of becoming the victim of its very success as the premier physics research publication. Ben found himself in the middle of the gathering storm. Early in 1994 Ben Bederson paid me again one of his dreaded visits,7 and before I knew it I had agreed to chair an APS Task Force on Journal Growth. In the charge to the new Task Force, Ben, together with Michael Turner (then Chair of the APS Publications Oversight Committee), laid out the problem: For many years now submissions to our research journals have grown inexorably, more or less linearly with time. While we are appreciative of the fact that growth generally implies a healthy situation, growth also carries with it a number of problems. It happens that essentially all of our growth is from non-US sources at the present time. But perhaps even more importantly, we simply do not understand the reasons for our growth,

6 Today these issues are again hotly debated, especially in the life sciences community. See the model proposed by the Public Library of Science at its website, http://www. publiclibraryofscience.org/. 7 My memory is fuzzy here. This visit may have been virtual, by telephone.


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and, not understanding them, we might equally well not anticipate significant decreases in submissions, should they occur. You should also consider the question of limitation on growth . . . Bear in mind that the coming decade must surely witness the arrival of electronic journals . . . You should consider the likely impact these developments will have on our journals.8

Ben provided the new Task Force with constant encouragement and support, but scrupulously avoided any personal interference in its independent deliberations. Unlike the earlier Task Force on Publication Policy, which relied entirely on existing data, the new Task Force undertook several studies and surveys of its own.9 1. Members of the Task Force made a quality study of the Physical Review to assess the possibility of controlling the growth of the APS journals without compromising their excellence. They concluded that the application of more stringent acceptance criteria might stem the tide, at least temporarily. 2. A survey of foreign-address authors found that these authors have every intention of submitting even more papers to the APS journals, which enjoy the greatest international prestige, are regarded in many subfields as the best vehicle to publicize and validate an author’s work, and are universally accessible. 3. A study on The Growth of Physics Journals was conduced by Heinz Barschall as a follow-up on his earlier study of The Cost of Physics Journals.10 The data again confirmed what a relative bargain the APS journals (and the journals published by the American Institute of Physics) are for the libraries, compared to most other journals in our field, although few of them experienced the same explosive growth that confronted Ben Bederson. Ben understood clearly, and the Task Force concurred, that in the midst of a revolution in Information Technology, adjustments in policy and procedure cannot be expected to be current for more than a few years and must be flexible enough to accommodate to the impending advances in electronic publishing. Almost everyone agreed that the system of anonymous peer review should be preserved, because it confers significant value upon a published paper. In the short term, the Task Force saw no need for radical measures (such as fixed annual page budgets), but it urged that

8 Memo of 10 March 1994, from Benjamin Bederson to APS Task Force on Journal Growth. The

members of the Task Force on Journal Growth were Heinz H. Barschall, Wick Haxton, John R. Kirtley, T. Maurice Rice, Barrett H. Ripin, Jin-Joo Song, Erick J. Weinberg, and Eugen Merzbacher (chair). 9 Report of APS Task Force on Journal Growth, 8 February 1995. 10 Bull. Am. Phys. Soc. 33 (1988): 1437–1447. For a fascinating account of the protracted litigation caused by this report, see the carefully documented article by Harry Lustig, Am. J. Phys. 68 (2002): 595–636.

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• the standards of scientific quality be raised by applying stricter acceptance criteria, • the appeals procedures be streamlined and guidelines be adopted that, while still safeguarding the rights of authors, would cut back on time-consuming exchanges between editors and authors, and • incentives should be offered to authors who submit articles electronically and disincentives be imposed on those whose papers require manual processing and heavy copyediting. Building on his experience as editor of Physical Review A from 1978 until 1992,5 Ben introduced during his five-year term as Editor-in-Chief many of these and other reforms. At every step, he worked tirelessly to justify and maintain the reputation of the APS journals for scientific excellence. He took initiatives that kept the journals in tune with the changing landscape of physics,11 and he insisted on scrupulous editorial fairness. The American Physical Society has made it easy to evaluate the impact of Ben’s efforts on the journals. A well-organized website12 provides a cornucopia of statistical information on the Physical Review and Physical Review Letters, over a forty-year period, including some instructive graphs. After a steep rise in submissions in the early part of the Bederson tenure, there was a notable slowing down in the second half of Ben’s term, but recently the upward trend has resumed, reaching a growth rate of about six percent per year. Since 1988, more manuscripts have originated outside than within the U.S., and there are now over three times as many foreign submissions as domestic ones. Amazingly, and to the delight of the hard-strapped libraries, in 2005 the subscription prices for APS journals will decrease for the first time in memory, despite continuing growth.13 The shift to electronic information technology, implemented by a cost-conscious editorial and operational staff, is largely responsible for the savings that are making this extraordinary turn-around possible. While he could not have done it alone, Ben deserves much credit for preparing the APS publications system for the challenges of the twenty-first century. To appreciate this accomplishment one only has to look at the Physical Review OnLine Archive (PROLA),14 which goes back to the inaugural issue of The Physical Review in 1893—an event whose hundredth anniversary fell into Ben’s tenure as Editor-in-Chief.4 Even as he has managed to hold numerous responsible administrative positions —often several of them simultaneously—Ben has never become the proverbial 11 For example, in 1993 Physical Review A15 became Physical Review E, a separate section, now covering statistical physics, nonlinear, and soft matter physics. 12 http://forms.aps.org/general.html#stats. 13 http://librarians.aps.org/2005pricing.html. 14 http://prola.aps.org/.


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bureaucrat. His deep love for physics, history, music, good food, New York City and the Berkshires, and his devotion to his family and many friends have always been on the front burner. What a wonderful man we have in our midst! I am grateful to Bernd Crasemann and Roger Stuewer for sharing drafts of their contributions to this volume with me, to Harry Lustig for helpful advice, and to George Basbas for useful information. This is also a good place to acknowledge the valuable and cheerful support that Nicole Wachsman provided to the APS task forces whose work is summarized in this article. The title of this essay was subliminally suggested by Roger Stuewer’s concluding paragraph.



BEN AS APS EDITOR BERND CRASEMANN Physics Department, University of Oregon, Eugene, OR 97403, USA

Ben Bederson served as Editor of Physical Review A (PRA) from 1978 until 1992, when he was appointed Editor-in-Chief of the American Physical Society (1992– 1996), to deal with problems of a generally different nature. I succeeded Ben as Editor of PRA and I still have that job. This sketch is intended to relate some of the highlights of Ben’s contributions as Editor and some personal impressions of his style and the impact he made on APS’ editorial procedures and guiding philosophy. Many colleagues kindly contributed vignettes from memory. Ben relates that his introduction to the Editorial Office started “with a bang”: The (then) Editor-in-Chief overruled the PRA Editor’s decision to reject a manuscript and had it reinstated. This incident led Ben immediately to start steps that would bring about greater accountability in the editorial process and lead to authors’ right to appeal editors’ decisions formally to an Editorial Board (which he created) for adjudication; a further appeal, regarding fairness of procedure only, could then be lodged with the Editor-in-Chief. Ben does not recall overruling editors as Editor-in-Chief at any time, “an affirmation that editors [are] generally fair and conscientious in exercising their authority”. The appeals recourse, probably unique among scientific journals, is praiseworthy for being very democratic. Yet, with the number of submissions to PRA having risen to some 3000 per year at present, appeals (often used by authors with little understanding of science) have become increasingly like the ring about the cormorant’s neck, taxing eminent Editorial Board members’ and overworked editors’ time to the limit. During a current reassessment of future plans for the Journal, modification of this provision may need to be addressed. The foregoing episode is an example of Ben’s attitude toward his editorial staff, authors and readers. The overriding aspect was his collegial approach, perhaps grounded in his long-time association with academia, as a member of the Physics faculty and, in 1986–1989, Dean of the Graduate School of Arts and Sciences of New York University. He trusted editors and staff and carefully listened to their advice on proposed changes of substance in Journal policy and operations. Twice 57


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a year Editors Meetings were held, in which significant items were debated and often subjected to a vote by participants. This approach enhanced loyalty and dedication of the “A-Team”, whose members came to consider themselves participants in the noble aims of the Society. These aims were distilled in Article 2 of a Constitution drafted by a representative Council, soon after the founding of the APS in 1899: “The objective of the Society shall be the advancement and diffusion of the knowledge of physics.” Implicit in this proclamation and an amendment added in 1971 is the volunteerism of the many persons engaged in carrying out and overseeing the Society’s tasks—a spirit that persists to this day, despite growth of the tasks. In the editorial milieu, this volunteerism has been a special boon as exercised beyond Editors and Staff by our referees, of whom now some 6000 lend advice every year to PRA editors. Ben’s years in the Editorial Office spanned a period of drastic changes, in spirit as well as in technology. Carol Kraner, who led the PRA in-house staff until her retirement in 1996, recalls: “When Ben was first appointed Editor of PRA, the editorial offices were located on the third floor of the Physics building at Brookhaven National Laboratory. The journals were able to use the mainframe computer of the Physics Department; in fact my first job was proofreading punch cards and then, if I was really lucky, I could take the trays of cards down to the computer room and try to run the job. This was done at the end of the work day and on several occasions the job wouldn’t run because of errors, which meant that the offending cards had to be routed out and corrected and then the job be taken down to be run again”. Continues Carol: “I mention this because one of Ben’s ideas when he first came was to give a more personal touch to communications with both authors and referees. Adding editorial notes to the computer-generated forms and letters wasn’t possible except for a few already programmed sentences. A battle ensued between secretarial services and Ben’s wishes. It was even suggested that he be “trained”. Of course that was impossible, so some letters and notes were hand-typed. The software to handle these requests was eventually developed, but not for a long time; the pressure was constant . . .” Ben’s personal touch extended not only to authors and referees, but also very much to Associate and Assistant Editors and Staff. Margaret Malloy, who succeeded Carol upon her retirement as in-house manager and has continued to be the guardian of the “A-Team” spirit, recalls: “Memorable, to me, was when I was teasing Ben (he had just taken over as Editor-in-Chief) about his ‘bonding’ with the Editorial Assistants and he asked what I thought he could do to ‘bond’ better. I teasingly told him that he could take us all out to eat at the Russian Tea Room. Well, he smiled and said: ‘I can do that’. True to his word, he invited the 20 of us to his apartment (Georgetown Plaza, overlooking the East River and the skyline) for drinks followed by a dinner at an elegant restaurant (the One Fifth Avenue, the

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Russian Tea Room being closed). This was on December 6, 1995. It was a memorable affair; I still have a picture, hanging on my office door, of Ben surrounded by the Assistants. It was taken that night in his apartment. We used to joke about playing duets together (we both play violin) . . .” Adds Carol to the theme: “We did manage to have fun together outside the job too. Henry Stroke may remember when we borrowed his bicycle one noon at NYU. With Ben, on his decrepit rusty bike which had only one pedal (but he could make it work), we rode to Chinatown for a dim sum lunch. And we managed to play tennis a few times, once on the roof of the athletics building at NYU and once or twice at meetings of the APS Division of Atomic and Molecular Physics (DAMOP). He was a pretty cagey player with a bagful of spins, slices, and drop shots that made me run a lot. Even as our friendship grew, however, our professional relationship stayed formal. It was really the challenge of editing that we loved and that remained the most important part of our relationship”. A reminiscence from Tom Miller, who worked at NYU at the time, further illustrates the toil exerted by editorial staff under circumstances that now appear quite primitive. “For the selection of referees, a computer database was not constructed until the early 80’s. Before, Ben used a private set of dog-eared 3 × 5-in. cards he leafed through . . .” [In fact, when I took over Ben’s job in 1992, I inherited a copy of this set which I still consult on occasion, largely for historical interest.] “The computer terminal that communicated with the Brookhaven office was in a supply closet in the [NYU] Graduate Dean’s (Ben’s) office, having something to do with phone lines and power. Once a day, you’d open the closet door, pull up a chair, assign referees, and deal with staff questions”. Interacting with referees is by no means dull. Tom recalls a referee report from J.S. Bell at Queen’s University in Belfast, whom Ben had asked to evaluate a manuscript that dealt with Bell’s Inequality. The review started off: “J.S. Bell would turn over in his grave if he were to see this paper”. When Physical Review celebrated its Centennial in 1993, Ben with characteristically stern conviction of the importance of referees’ anonymity, turned down a suggestion that he tell about this incident, saying “No, no, we can’t use a referee’s report”. (The statute of limitations may have expired by now.) Carol’s successor Margaret Malloy at times cheers up the Editorial Board at one of its staid meetings with passages from recent referee reports. Here are a few examples: From a PRB referee: “I cannot review this paper as it is wrong and I did it first”. Brute honesty: “This paper should be rejected for the following reasons: (1) No one cares about this anymore; (2) anyone who could referee it is probably dead; (3) all who read it will wish they were”. Team spirit: “I have downloaded the manuscript to review. However, it looks terribly familiar. In fact, I saw it the other night in bed. My husband (the delin-

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quent reviewer) was reading it. Can I suggest that you let me apply some pressure to him to return a review, rather than have me begin reading this lengthy manuscript?” Many changes that came about during Ben’s dozen years at the helm, innovative at the time, have grown symbiotically into the modus operandi of the Editorial Office. From the time of his appointment in 1978, Ben was a “remote” editor, retaining his base at NYU while working closely with the journal editorial staff in Ridge, Long Island. The staff was headed by Carol Kraner, assisted by Margaret Malloy who rose to the position when Carol retired. Explains Ben, being a “remote” editor did not imply a lack of involvement, but indicated that I remained a faculty member at New York University and conducted editing business by mail, telephone and email, with virtually daily interaction with the staff, interspersed with occasional visits to Ridge. On the whole this system worked very smoothly, attributable in no small part to the skill and dedication of the Ridge editorial staff, although of course we had our occasional foul-ups, discontented authors and dilatory referees. These are problems which will never go away.” The “remote” approach in fact worked so well that I was appointed on a similar basis to succeed Ben, the distance from Oregon to Long Island being even greater than from Manhattan, the ease of communication leaping ahead with more sophisticated technology, and especially with the World-Wide Web coming into use. These advances and their application through Journal Information Systems Director Bob Kelly and his ingenious staff have led to a new age of editing. Long-time APS colleague Amy Halsted has illustrated this, citing a 1993 piece by Gary Taubes in Science (259, 1246) entitled “Publication by electronic mail takes physics by storm”. Completion of the “PRISM” code now allows APS editors and staff (and, to a limited extent, referees) to view manuscripts and related correspondence electronically—thus propelling us toward the long-time aim of a “paperless office”. The electronic approach has made it possible to engage six of the present seven PRA Associate Editors on a “remote” basis, as consultants, at considerable savings—except for Margaret Malloy, who had to remain in Ridge to coordinate the whole effort. Other APS journals benefit similarly. It also seems likely that this dispersal of editors keeps them closer to researchers and places where scientific frontiers are actually being expanded—an antidote against Ridge becoming an ivory tower! The move toward electronic communication, and especially to help get the journals and archives to appear online, is considered by Ben to have been “one of [his] proudest achievements, though hardly performed alone”. The earliest effort in this direction was unsuccessful: A major library journal distributor, OCLC, was hired to produce Physical Review Letters online. They soon reneged on the deal, however, after discovering that they “were not quite ready for prime time”. Fortunately, the American Institute of Physics rescued the process.

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Another advance, way overdue, was introduced in Ben’s second year on the job: a more logical division of subjects, followed by concomitant new titles and splitting of the Journal. Writes Ben: “When I first became Editor, Physical Review A had the subtitle ‘General Physics’. I never cared for this designation. It gave the impression of being a catch-all for anything that could not fit properly into a truly specialized journal, implying, it seemed, second-class status. In fact, the topics covered under the rubric ‘General Physics’ have turned out to be as lively, cutting-edge and stimulating as any other subfields of physics”. More realistic labeling and some redistribution of subjects led, in July 1990, to a split between Physical Review A, dealing with Atomic, Molecular and Optical Physics, and a new Physical Review E, now subtitled Statistical, Nonlinear, and Soft Matter Physics. In a subsequent “very revolutionary action”, instituted in 1993, Ben’s first year as Editor-in-Chief, the covers of APS journals, previously referred to as ‘the Jolly Green Giant’ or more derogatorily as ‘the Green Monster’, were all changed, with each of the Physical Review journals given its own color. PRA became, and remains, silver. “An expected outcry against the violation of our glorious green tradition never materialized” states Ben. There were some dark periods. While the APS has never sued anybody, the Society since 1987 has had to defend itself in the courts in four cases, all connected with publishing. (A detailed account of these fascinating episodes is given by Harry Lustig in his splendid article on the history of the American Physical Society (see [4]). Two plaintiffs sued the Society for not publishing their work, one sought to punish APS for publishing an author’s work, and in one case a third party attempted to compel the Society to reveal the identity of a referee of a manuscript that was not accepted for publication. Ben and I became personally involved when the APS was accused of improperly refusing to publish a manuscript. Physical Review A had accepted an article by a University of Maryland research associate. When the author’s colleagues found out about it, through prepublication of the abstract, they complained that the work had in fact been done jointly with them and that publishing the article (a Brief Report) without recognition of them would be improper. The Journal decided to suspend publication unless and until the matter could be resolved. The author thereupon sued his collaborators and the APS, as well as Ben and me personally, for $1,000,000 each, claiming that the refusal by the Physical Review to proceed with the publication of his manuscript after he had received notification of its acceptance constituted a breach of contract. The APS responded that there was no contract, that the Physical Review’s letter of acceptance contained conditions that were never satisfied, and that even if the parties had created a contract, there was no breach because the journal had not refused to publish the manuscript. The court agreed and dismissed the case. The lead lawyer for the APS had been Richard Meserve, a physicist and Fellow of the APS who has successfully represented the Society in all of the cases

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until 1999, when he was confirmed as head of the Nuclear Regulatory Commission. Ben was amused by the large sum for which we were being sued, purportedly because we had prevented the author’s receiving a Nobel Prize, and commented: “If a Brief Report is worth a million dollars, what would be the price of a Rapid Communication?” Another trying time took place when Ben became a party to a huge struggle over the issue of page charges, particularly in Physical Review D, which ultimately led the APS to eliminate all required page charges for all its journals except for Physical Review Letters. The highlights upon which we have touched made Ben’s career as Editor singularly fruitful in solid accomplishments and also, perhaps more importantly, in the subtle ways in which he set the tone, built up the morale and fostered the team spirit among editorial workers. States Carol Kraner: “I believe that Ben’s greatest contribution to the PR journals was that he never settled for the status quo but was always thinking of new ways to improve and enrich the editorial process. What comes through in my recollections is my great admiration for Ben’s breadth of thought and creativity as an editor as well as for the man himself”. Adds Margaret Malloy: “Ben always had a down-to-earth charm about him that was very endearing. He is a brilliant man who also has a great sense of humor”. Those of us who had the privilege of working with Ben cherish the memory of an inspiring and productive relationship, seasoned with mutual respect and devotion to the aims of the APS publishing enterprise. We are happy that Ben appears to have similar recollections when he writes, “My over 12 years involvement with the APS journals has become one of the most interesting and rewarding periods of my life”. The writer is much indebted to Ben Bederson for sharing his recollections from the period covered in this chapter [1]. Friends and colleagues have been generous in communicating relevant experiences and observations. Notably, recollections by Carol Kraner [2] and Margaret Malloy [3] have been refreshing. Harry Lustig’s carefully researched article on APS’ history has been an important resource on context and details of many events related here [4]. Tom Miller contributed valuable recollections from early days of Physical Review A editorial operations [5]. Amy Halsted kindly made her Master’s Thesis available in which she describes the decision-making process that ended up with the APS moving from New York to College Park, Maryland [6]; this contains much valuable background material. Numerous pleasant conversations with other friends at meetings, in the APS Editorial Office and, particularly, on the “A-Team” have helped to round out my own impressions of this fragment of APS publishing. Needless to say, the writer is responsible for any errors or omissions.

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References [1] [2] [3] [4]

B. Bederson, private communication. C. Kraner, private communication. M. Malloy, private communication. H. Lustig, To advance and diffuse the knowledge of physics: an account of the one-hundred-year history of the American Physical Society, Am. J. Phys. 68 (2000) 595. [5] T.M. Miller, private communication. [6] A. Halsted, “From Manhattan to Maryland—The American Physical Society and its Relocation”, Master Thesis, Baruch College of the City University of New York, 1993 (unpublished).



BEN BEDERSON: PHYSICIST–HISTORIAN ROGER H. STUEWER1 Program in History of Science and Technology, Tate Laboratory of Physics, University of Minnesota, 116 Church Street SE, Minneapolis, MN 55455, USA, e-mail: [email protected] 1. 2. 3. 4. 5. 6.

Introduction . . . . . . . . . . . . . . . Wartime Reminiscences . . . . . . . . Physics and New York City . . . . . . APS Forum on the History of Physics Conclusion . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . .

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Abstract I discuss Ben Bederson’s contributions to the history of physics, focusing particularly on two articles he published in Physics in Perspective, one on his wartime experiences as a soldier in the U.S. Army’s Special Engineering Detachment, the other on physics in New York City. I also discuss his work for the Forum on the History of Physics of the American Physical Society.

1. Introduction Four years after P.W. Bridgman won the Nobel Prize in Physics for 1946 for his work in high-pressure physics, he offered what he called “impertinent reflections” on the history of science. It “seems to me”, Bridgman wrote, “that the one most important thing to realize about science is that it is a human activity”, a point of view that will be stressed “automatically” if “science is taught with a large admixture of history”, which will serve a purpose “that is increasingly important in our present day, namely to impart an adequate appreciation of the fundamental conditions under which science flourishes”. One of the lessons to be learned from the 1 Roger H. Stuewer is Professor Emeritus of the History of Science and Technology at the University of Minnesota. He came to know Ben as a member of the Executive Committee of the APS Forum on the History of Physics and as Co-Editor-in-Chief of the journal Physics in Perspective, as he describes in this article.

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history of science, Bridgman continued, is that instead of scientific ideas following a “logical order” of development, “the impartial historian will be compelled to invert the process by showing how seldom the course of scientific development has been the logical course”. The insight that the progress of science is often illogical is perhaps more important to the scientist himself than to the layman, and constitutes one of the reasons why the scientist is concerned with his history. This suggests that perhaps one of the most important fields of service of history of science is to the scientist . . . .

Bridgman concluded by challenging both scientists and historians: The upshot of all this skepticism and criticism is my conviction that the most profitable sort of history of science, at least for the scientist himself and perhaps for others, is contemporary history . . . . This will involve not only scanning contemporary literature . . . but intimate personal contact with the scientists themselves, attendance at scientific meetings and prodding of the individual scientists by correspondence or personal interview to disclose the more personal aspects which a false scientific modesty often inclines him to believe are irrelevant. This is going to demand a high degree of technical competence on the part of the historian himself, the acquiring of which will put serious demands on his abilities and demand long preparation. For this reason it seems to me that perhaps it will prove that the historians of science best able to serve this purpose will be scientists themselves who in their later years and out of the fullness of their experience are moved, because of an intrinsic interest, to set down their reminiscences. I believe, further, that an informed and intelligent recording of contemporary scientific history will be of value not only for the scientist himself, but will also have cultural and educational value for the layman (Bridgman, 1955 [1950]).

Bridgman’s words are as true today as they were a half century ago, and they relate directly to Ben Bederson, who is exactly the kind of distinguished physicisthistorian to whom Bridgman was referring.

2. Wartime Reminiscences Four years ago, because of his intrinsic interest in the history of physics, Ben published his reminiscences about his wartime service in the U.S. Army’s Special Engineering Detachment (SED) at Los Alamos in Physics in Perspective (Bederson, 2001), a journal that John S. Rigden and I founded three years earlier. Ben tells how in 1942, after two and a half years as a physics major at the City College of New York (CCNY), he was drafted into the army, was sent to take an electrical-engineering course in an Army Specialized Training Program at Ohio State University in Columbus, and then in January 1944 was assigned to the U.S. Army’s SED, serving first for a brief period at Oak Ridge, Tennessee, and then at Los Alamos. At Los Alamos Ben worked essentially as a research assistant to senior scientists on various aspects of explosives, first under the British physicist Philip B.

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Moon, testing the “strain gauges” attached to cylindrical hollow tubes containing explosives. Then, after the Russian–American chemist George B. Kistiakowsky, head of the Explosives Division, gave a small group of SEDs, including Ben, a complete overview of the bomb work (far beyond their “need to know”, thus belying the later myth of strict compartmentalization for security purposes), Ben worked under the American chemist Donald Hornig, developing and testing the switches that had to be triggered simultaneously to ignite the 32 detonation fuses at the apexes of the implosion lenses in the plutonium bomb (Fat Man). Because of the expertise he acquired in this work, in the spring of 1945 Ben had to fly frequently to the Air Force base at Wendover Field on the Utah–Nevada border to help train B-29 bomber crews in the assembly and use of these switches and other gear.2 That also was the reason he was among the SEDs who received orders to go to the South Pacific in May, after the defeat of Germany, to assemble, test, and select the best spark-igniter triggers for Fat Man. He flew out of Albuquerque on July 9 on a B-29 (the famous Green Hornet), which after several stopovers for refueling landed on the island of Tinian in the Marianas, the launching pad for the American bombing raids on the Japanese mainland. Fat Man was dropped on Nagasaki on August 9, 1945, and at the end of that month Ben received orders to return to Los Alamos, where he worked as a nuclear-reactor inspector until he was discharged from the army at Fort Bliss, Texas, in January 1946. Ben makes clear that his wartime experiences, as was the case for so many others, were crucial for his future career as a physicist. The element of luck can never be discounted. His modest educational background in physics by 1942 might well have been overlooked by the U.S. Army (which is not famous for its perceptiveness), and he could easily have been sent into combat in the European or Far Eastern theaters of war, perhaps getting wounded or even killed, as a close friend of his was at the end of 1944 in the Battle of the Bulge. Instead, his commanding officer in Columbus, Ohio, recognized Ben’s talent in physics and engineering and set him on a course to serve, eventually along with about 1800 other soldiers at Oak Ridge and Los Alamos, in the U.S. Army’s Special Engineering Detachment. Given that break, however, Ben then began to shine: At Los Alamos his “blue badge”, a kind of “second-class admission ticket”, was replaced with a “white badge”, giving him almost unlimited access to the Tech Area, where the main bomb work was being carried out. It also gained him admission to the weekly Tuesday-evening colloquia where progress reports were given by Enrico Fermi (the first one Ben heard) and other eminent scientists, enabling him to acquire “a pretty complete picture of most of the important aspects of the bomb work” and

2 Ben reveals that he used these opportunities to occasionally walk across the border in the evenings to the Nevada side to legally play blackjack, being “convinced at the time [that this game] could be conquered by counting cards”.

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to learn a great deal of science and engineering along the way. Further, despite constant homesickness, so familiar to every GI, Ben met and established lifelong friendships with many of his fellow SEDs as well as with senior scientists at Los Alamos. At the same time, he was given pause when he later learned that two of his fellow SEDs, David Greenglass, his communist-propagandizing bunk neighbor for a time, and Ted Hall, an occasional third participant in the Mushroom Society that he and Norman Greenspan formed to play classical music loudly at night,3 had been exposed as Soviet spies. Ben, like other SEDs, smarted under the knowledge that his army paycheck was under $100 per month and thus much less than that received by civilians at Los Alamos with lesser responsibilities and duties who, moreover, did not have to perform early-morning calisthenics and stand for Saturday-morning inspections. Still, Ben fondly remembered that he was excused from latrine duty and KP (which as a former GI myself, though a decade later and not in wartime, filled my heart with envy). That lesson in the venerable army tradition of “rank hath its privileges” was followed by others, as when Ben was given special authorization to wear civilian clothes, as well as a shoe-ration certificate for a pair of civilian shoes to wear on his flights to Wendover Field so as not to reveal to Air Force officers that a lowly soldier, far below their military ranks, was empowered to give them instructions. This also exposed a clue to another aspect of Ben’s future professional life, his commitment to history, since he carefully preserved these authorization documents for posterity. Similarly, he began keeping a diary when he left Albuquerque for Tinian on July 9, 1945, making entries into it almost every day until he returned to the United States at the end of August. He opened a further window into his personality when he recalled that among the items he packed into his footlocker for later shipping to Tinian was a novel to reread, Thomas Mann’s The Magic Mountain, “partly because of my identification with the protagonist Hans Castrop”. He carried with himself, however, the Modern Library edition of The Pocket Book of Verse, which he would pull out and read when standing in line— an experience familiar to every GI. “To this day”, Ben said, “I can recite by heart many of the poems in this splendid volume”. Ben’s homeward flight was marked by another significant event, a conversation he had with a fellow passenger, Edward C. Stevenson, who soon would become professor of physics at the University of Virginia, who advised him to go to graduate school after completing college: “I had not thought of this before”, Ben recalled. Ben took Stevenson’s advice: He declined to take part in the nuclear

3 After returning to Los Alamos from Tinian, Ben built his own 50-watt hi-fi amplifier, which he called the Bederson Belchmaster, and which he was able to get through army inspection and ship home to New York when he was discharged in January 1946.

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tests scheduled to take place in the South Pacific in 1946, even at the amazing salary of $1000 per month for six months—more than ten times his army pay. Instead, he returned to New York, completed his B.S. degree at City College that year, his M.A. degree at Columbia University in 1948, and his Ph.D. degree in physics at New York University in 1950, writing his thesis under Leon Fisher, a civilian friend of his at Los Alamos.4

3. Physics and New York City Ben witnessed another kind of war on September 11, 2001, when he saw the destruction of the South Tower of the World Trade Center from the window of his apartment in New York City. At that time he was about midway through a draft of another article that John S. Rigden and I had asked him to write for Physics in Perspective, an article on “Physics and New York City”, for its Physical Tourist section (Bederson, 2003). People throughout the world were horrified by the images they saw either directly, as Ben did, or on their television screens, and along with Ben people throughout the world knew that New York City, America, and the world had been forever transformed. Ben’s task immediately assumed great poignancy, and instead of writing “a relatively straightforward tour of the many sites in New York related to physics”, he wrote what he “unabashedly could call a story in praise of both my city and its physics, and how they have worked together to produce the remarkable pattern that stands unique in the world”. A New Yorker his entire life, Ben began by discussing the achievements of “three extraordinary people”, Thomas Alva Edison (1847–1931), an American born in the Midwest, and two Serbs, Nikola Tesla (1857–1943) and Michael Idvorsky Pupin (1858–1935), pointing out the locations of Edison’s powergenerating and distribution plants, Tesla’s several laboratories and living quarters, and Pupin’s educational roots (Cooper Union and Columbia University), bust, and apartment where he lived in his later years. Ben then took his readers to the Colonnade of the Hall of Fame for Great Americans overlooking the Harlem River in the Bronx, where 98 busts of noteworthy Americans have been erected (with 4 more to come after funds for them have been secured). These include busts of Benjamin Franklin (1706–1790), Peter Cooper (1791–1883), Samuel F.B. Morse (1791–1872), George Westinghouse (1846–1914), Alexander Graham Bell (1847–1922), Edison, the physicists Joseph Henry (1797–1878), Josiah Willard

4 In 1949 Ben needed one additional course in statistical mechanics as a requirement for his Ph.D. degree, which turned out to be given by the German-born theoretical physicist Fritz Reiche (1883– 1969) and was the most memorable course he ever took at NYU. Ben has presented a warm portrait of Reiche’s life and work in yet another article in Physics in Perspective (Bederson, 2005).

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Gibbs (1839–1903), and Albert A. Michelson (1852–1931), and the astronomers Maria Mitchell (1818–1889) and Simon Newcomb (1835–1909). Ben particularly admires the words of Gibbs that accompany his bust: “One of the principal objects of theoretical research is to find the point of view from which the subject appears in its greatest simplicity”. Ben noted next that Albert Einstein (1879–1955), after arriving at Ellis Island with his wife Elsa on April 2, 1921, gave the first lecture he ever delivered in America five days later at City College. He then told a heartwarming story about how Max Born (1882–1970) secured financial support for the famous atomicbeam experiments of Otto Stern (1888–1969) and Walther Gerlach (1889–1979) in Frankfurt in the early 1920s from Henry Goldman (1857–1937), head of the Goldman Sachs & Co. investment house; how Born later secured further financial support from Goldman for his own institute in Göttingen; and how after 1933 Born intervened with Goldman to secure financial support for Einstein’s work in assisting Jewish refugees from Germany. Continuing his focus on prominent physicists, Ben next sketched the deep personal and educational roots of Isidor I. Rabi (1898–1988), J. Robert Oppenheimer (1904–1967), Richard P. Feynman (1918–1988), and Julian Schwinger (1918–1994) in New York City. All except Rabi were born in New York City, although he spent his entire career there, and all except Oppenheimer later won Nobel Prizes in Physics. New York City high schools, Ben averred, “have been a breeding ground for physicists for at least three quarters of a century” and have produced a large number of future Nobel Laureates, as well as a vastly greater number of physicists who have had distinguished careers, with assorted accompanying honors . . . almost all of them within the free public school system. Many of these physicists were children of immigrants, or immigrants themselves, some coming from families living in poverty, at the lowest rungs of the economic ladder.

Three high schools “are particularly oriented to science, and to which admission is determined by competitive examinations throughout the city”, namely, the Bronx High School of Science, the Stuyvesant High School in Manhattan, and the Brooklyn Technical High School. The first is the leader in producing future Nobel Laureates (Leon N. Cooper, Melvin Schwartz, Sheldon L. Glashow, Steven Weinberg, Russell A. Hulse), but the second and third can claim one each (Roald Hoffman, Arno Penzias). Others too make the list: James Monroe High School and Walton High School, both in the Bronx (Leon Lederman, Rosalyn Yalow), the Manual Training High School (now John Jay High School) in Brooklyn (I.I. Rabi), Columbia Grammar (Murray Gell-Mann), and Far Rockaway High School (Richard Feynman and Burton Richter). Townsend Harris High School, which was established as part of City College, also can claim among its graduates one future Nobel Laureate (chemist Herbert Hauptman) and was long regarded as the elite of the elite—until the beloved Mayor Fiorello La Guardia abolished it in 1942. (“I don’t often make mistakes”, La Guardia said later, “but when I do it’s

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a beaut”.) Now resuscitated in Queens in 1984 by a group of alumni, Townsend Harris is today “as dynamic as it was in its first incarnation”. The New York Academy of Sciences in Manhattan, which was founded under another name in 1817 and assumed its present one in 1876, has sponsored numerous meetings, conferences, and symposia over the years, but perhaps its “most important role” has been its leadership, particularly by physicists, in advocating human rights. Its most noteworthy case was that of Andrei Sakharov (1921–1989), “one that played a significant role in loosening the tight grip of communism on the U.S.S.R.”. Sakharov first visited the Academy in 1988 and credited it with his release from exile in Gorki two years earlier. Turning to colleges and universities and concentrating only on those with major physics departments, Ben first calls our attention to Fordham University in the Bronx, where Victor F. Hess (1883–1964), Nobel Laureate of 1936 for his discovery of cosmic rays, taught after his immigration to the United States following the Anschluss of Austria in 1938 until his retirement in 1956. Unique among New York physicists, in 1947 Hess convinced the New York Board of Transportation to allow him to carry out experiments on the radiation emitted from rocks at the bottom of the 191st Street subway station, 180 feet below street level, a site well-protected from cosmic rays. Unique in another respect, this time as an educational institution, is the Cooper Union for the Advancement of Science and Art (Cooper Union for short), which was founded in 1859 and subsequently nurtured by the self-educated inventor and industrialist Peter Cooper (1791–1883). It remains unique to this day as a completely tuition-free college whose students are selected competitively; in fact, it was recently ranked as the “hardest school to get into” in the United States. Next on Ben’s list is the City University of New York (CUNY), the administrative umbrella organization for all of the public city colleges in New York, the major ones being City College (CCNY), Brooklyn College, Hunter College, Queens College, and Lehman College, each of which has its own undergraduate physics program but whose graduate programs are unified by the CUNY Graduate School at 365 Fifth Avenue, where some graduate classes also are taught. While all of CUNY’s colleges can claim well-known students and distinguished faculty members, both past and present, the “jewel in the crown” is City College in upper Manhattan, which was founded as the Free Academy in 1847. The American diplomat Townsend Harris (1804–1878) was instrumental in its founding, with its mission “to educate the whole people”. Almost all of its students came “from the city’s poorer population, were very talented and came from striving families having high ambitions for their children”, who “themselves were aware that the best way to escape working-class constraints was through the professions”. Prior to World War II, they were taught by a “small but select faculty” who worked with them to turn them into “generations of physicists who have become virtually legendary in their number and accomplishments”, as is evident from the partial list

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of them that Ben provided. In the postwar period, the transformation to a department with a flourishing Ph.D. program in physics began under the chairmanship of Harry Lustig in 1965–1970 and “reached its peak” under the presidency of Robert Marshak in 1970–1979. The department remains today “extraordinarily vital and productive”. Ben’s home institution, New York University at Washington Square in Manhattan, was founded as the University of the City of New York in 1831 “by a number of wealthy and influential New Yorkers who believed that their city should have an institution of higher education that would be accessible to children of middleclass and even working-class parents” who could not afford to send their children to prestigious Ivy League colleges. From its beginning, NYU was defined as a university, thus granting advanced degrees much earlier than Columbia University, for example. Around the beginning of the last century, NYU acquired a second campus at University Heights in the Bronx, and after World War II the Courant Institute of Mathematical Science, named after the distinguished German–American mathematician Richard Courant (1888–1972), also was established at Washington Square, all together thus effectively forming for a time three NYU physics departments, each with a distinguished history. Two future Nobel Laureates, Clifford G. Shull (1915–2001) and Frederick Reines (1883–1998), did their graduate work at the University Heights campus. Later, when that campus was sold to New York State, physics was consolidated at Washington Square, where it flourishes today in many of its subfields. Ben finally discusses Polytechnic University (founded as Brooklyn Polytechnic in 1854), the dominant engineering school in New York City and the second oldest one in the nation, and Columbia University, which is in “a class by itself on the New York physics scene” and “has had a glorious history that stretches back over a century”. To substantiate his claim, Ben lists 28 Nobel Laureates in Physics who were visiting professors at Columbia, who received their Ph.D. degrees there, or who were or are on the faculty there. Noteworthy also is that just prior to World War II, John R. Dunning (1907–1975) and his colleagues carried out pioneering research on nuclear fission in the basement of Pupin Hall, and during the war they, together with prominent physicists from other universities, did the same on the gaseous-diffusion process for separating the uranium isotopes, which later found massive implementation in the enormous Manhattan Project plants at Oak Ridge, Tennessee. Ben closes by giving a guided tour of Manhattan,5 pointing out the sites he discussed earlier as well as additional ones, such as that extraordinary national 5 Ben notes that in Manhattan, except for lower Manhattan and Greenwich Village, 20 street blocks equal 1 mile and distances between avenues equal 2.5 city blocks or 1/8 mile, the former being walkable in about 30 minutes “if you obey the traffic lights—which, of course would set you off immediately as a tourist”.

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treasure, the New York Public Library.6 It is “inspiring to note”, Ben reflects, how a “few visionaries” made such a great difference in the life of New York City, especially singling out two, Peter Cooper and Townsend Harris, both of whom “had an intense belief in free education as the best way to help children from lower and middle-class families to live creative and fruitful lives”. While they “represented the best of New York”, they also “serve as symbols for the multitude of other idealistic and far-sighted business leaders, teachers, and parents who contributed so much to the environment” of New York City. “And, of paramount importance, from the very beginning the arms of America embraced and welcomed the magnificent influx of talented people, physicists and others alike, into New York and beyond”. As his final thought, Ben leaves his readers with the words of the Ephebic Oath, once taken by young males in ancient Athens and now taken by all entering freshmen at Townsend Harris High School, urging that they are particularly appropriate after the attack of September 11, 2001, on New York City: I shall never bring disgrace to my city, nor shall I desert my comrades in the ranks; but I, both alone and with my many comrades, shall fight for the ideals and sacred things of the city. I shall willingly pay heed to whoever renders judgment with wisdom and shall obey both the laws already established and whatever laws the people in their wisdom shall establish. I, alone and with my comrades, shall resist anyone who destroys the laws or disobeys them. I shall not leave my city any less but rather greater than I found it.

4. APS Forum on the History of Physics Ben’s intrinsic interest in and commitment to history found further expression when he was elected as Chair of the Forum on the History of Physics of the American Physical Society, serving in 2001–2002. Then, when the Forum decided to establish a committee charged to define the nature of a new prize to recognize outstanding scholarship in the history of physics, and to raise the substantial funds required to support it, he also agreed to serve as chair of this committee. The American Institute of Physics soon became a cosponsor of the prize, which with the approval of both organizations has been named after another distinguished physicist-historian as the Abraham Pais Prize for the History of Physics. I was privileged to serve as a member of the APS-AIP Prize Committee during 2002– 2005,7 and I thus can attest personally to the outstanding leadership that Ben 6 Among the countless treasures deposited in the Manuscripts and Archives Division of the New York Public Library are the records of the Emergency Committee in Aid of Displaced Foreign Scholars (1933–1945), which Ben used in his research on Fritz Reiche (Bederson, 2005). 7 The other members are Stephen G. Brush (who resigned in 2004), Gloria B. Lubkin, Harry Lustig (who has now replaced Ben as Chair), Michael Riordan, and Spencer R. Weart.

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provided to it. We met at the APS meetings in Albuquerque in April 2002 and in Philadelphia in April 2003, and both before and after these meetings we had more or less monthly conference calls, defining and refining the nature of the prize, exploring means of raising funds for it, and determining the criteria to be used in the selection of its recipients. Under Ben’s leadership, all of these aspects of the prize were brought to successful conclusions. The first recipient of the Abraham Pais Prize for the History of Physics is Martin J. Klein (Yale University), who delivered his Pais Prize Lecture, “Physics, History, and the History of Physics”, at the APS meeting in Tampa, Florida, in April 2005, appropriately during the World Year of Physics 2005. Ben’s stellar leadership here, however, did not truncate his service to the APS Forum on the History of Physics: In early 2004, he agreed to become Editor of the Forum’s Newsletter—a position that no one is more qualified to occupy than Ben.

5. Conclusion P.W. Bridgman was right: Physicist–historians play a vital role in writing “informed and intelligent . . . contemporary scientific history [that] will be of value not only for the scientist himself, but will also have cultural and educational value for the layman”. Ben Bederson is a physicist–historian who satisfies Bridgman’s imperative par excellence. I, like many others, have had the great good fortune that my worldline intersected with his.

6. References Bederson, B. (2001). SEDs at Los Alamos: A Personal Memoir. Physics in Perspective 3, 52–75. All quotations appear in this article. Bederson, B. (2003). Physics and New York City. Physics in Perspective 5, 87–121. All quotations appear in this article. Bederson, B. (2005). Fritz Reiche and the Emergency Committee in Aid of Displaced Foreign Scholars. Physics in Perspective 7 (4). Bridgman, P.W. (1955). Impertinent reflections on history of science [1950]. In: Reflections of a Physicist. Philosophical Library, New York, pp. 338–360. Quotations on pp. 346–347, 353–355, 360.


PEDAGOGICAL NOTES ON CLASSICAL CASIMIR EFFECTS LARRY SPRUCH Department of Physics, New York University, 4 Washington Place, New York, NY 10003, USA

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Introduction . . . . . . . . . . . . . . . . . . . . Dimensional Analysis and Physical Arguments The Vanishing of E¯ Cl . . . . . . . . . . . . . . An Unauthorized Thank You . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . .

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Abstract Quantum electrodynamics (QED) gives finite values for properties for which preQED calculations give infinity. One can take the difference between two infinities. For the (Casimir) interactions between two ideal uncapped parallel plates at a tem¯ perature T = 0, for example, the (renormalized) energy per unit area, E(T = 0), the difference between the unrenormalized energies of the virtual photons in the region between the plates in the presence and absence of the plates, is finite. The same is true for T > 0, where real photons are also present: one then considers both ¯ ). For T ∼ ∞, the real ¯ ) and the (renormalized) free energy per unit area, E(T E(T ¯ photons dominate and E(T ∼ ∞) becomes classical (h-independent). The (renor¯ malized) classical result E¯Cl can also be obtained within a classical framework, the ¯ )) is zero. We ¯ ) (not E(T ultraviolet catastrophe notwithstanding. One finds that E(T give a more “physical” proof of that fact than the original proof. Thus, since the ¯ ∼ ∞) = kB T N¯ , where N¯ is the difclassical energy per mode is kB T , E(T ference between the number of modes per unit area in the region between the plates in the presence and absence of the plates. That N¯ = 0 is not obvious; the plates are uncapped and the adiabatic theorem is not immediately applicable. The evaluation of N¯ reduces to a one-dimensional problem for which the adiabatic theorem is immediately applicable, so that N¯ = 0 and therefore E¯ Cl = 0. We comment briefly on a few other Casimir effects. 75


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1. Introduction Classical physics is bedevilled by infinities. Thus, the Rayleigh–Jeans law—more appropriately the Rayleigh law—for the electromagnetic energy density in free space at temperature T in the frequency interval ω to ω + dω is dω (1) , 2π 2 c3 where kB T is the energy for a mode. The energy density, the integral over ω, is therefore infinite, “the ultraviolet catastrophe”. Quantum electrodynamics, QED, made it possible to obtain finite values for the Lamb shift, Casimir effect, . . . . An ¯ ), with its associated interesting Casimir effect is the free energy per unit area, E(T force per unit area, F¯ (T ), between two open-ended ideal parallel plates at temperature T , one at z = 0 and one at z = l3 . (Throughout, an overline on a quantity ¯ represents the value per unit area of the quantity.) E(0) has long been known; it ¯ was obtained by Casimir (1948). One approach in calculating E(0) is to take the difference between the energy per unit area in the region between the plates in the presence and absence of the plates. Each energy is infinite, but if one recognizes that the plates are transparent to high energy photons and introduces a cut-off, that is, a convergence factor, the difference is finite.( Infinities can be avoided, as shown for example, by Brown and Maclay (1969), Schaden and Spruch (1998) and Gutzwiller (1990); the results in the last reference are based on Gutzwiller’s semi-classical approximation to the energy Green’s function.) At T = 0 only virtual photons, with zero-point energies h¯ ω/2, are present. For T > 0, real photons are also present. If the separation of the walls is changed ¯ 3 . With the high frequency photons slowly, with T constant, F¯ (T ) = −∂ E/∂l having been cut-off, one can interpret T ∼ ∞ as the inequality kB T h¯ ω for all ω. It has been found for a number of systems that, as is to be expected, ¯ E(T ∼ ∞) and F¯ (T ∼ ∞) are independent of h¯ . See for example, Milonni (1994). The results will be written as E¯Cl (T ) and F¯Cl (T ). It is natural to ask if the classical results can be derived within the framework of classical physics, that is, without the detour into QED. (It might seem obvious that it would be simpler, if possible, to avoid the detour, but obvious or not it isn’t necessarily true. In the Planck distribution frequencies with h¯ ω kB T fall of exponentially; there is no natural cut-off for the Rayleigh–Jeans distribution, u(ω) dω. We note, parenthetically, that u(ω) dω follows immediately from the distribution given by the uncertainty principle, 2d 3 pd 3 x/(2π h) ¯ 3 , on using 3 3 3 3 3 2 3 d p/h¯ = d k = d ω/c = 4πω dω/c .) In any event, it has been shown by Schaden and Spruch (2002) that the detour can be avoided, at least for parallel plates. A theoretical physicist in the 1890’s could have determined F¯ (T ) for the plates, the ultraviolet catastrophe notwithstanding. Of course, without a knowledge of QED or even of quantum mechanics, u(ω)dω = 2kB T ω2

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it would not have been possible to estimate the range of validity of the result, namely, that kB T hc/ ¯ l3 . (c/ l3 is the only frequency that can be formed from c and l3 .) Furthermore, experimental confirmation at that time would have been inconceivable. ¯ ) can be obtained in a clasWe will not consider the proof that for T ∼ ∞, E(T sical context—we have just above given the reference to the proof—but we will ¯ ) = 0, give a new argument for an essential element of the proof, namely, that E(T ¯ )). ¯ where E(T ) is the energy per unit area (not the free energy per unit area E(T The important contribution is the original rigorous proof by Feinberg et al. (2001), but our argument may be helpful. An essential point is the following. For E¯ Cl = 0, the standard relation E¯Cl = ¯ ECl −T S¯Cl reduces to E¯Cl = −T S¯Cl , where S¯Cl , the classical entropy per unit area, is not uniquely defined. However, F¯Cl = −T ∂ S¯Cl /∂l3 , and ∂ S¯Cl /∂l3 is uniquely defined. ¯ ) = 0 for plates, we comment Before turning to a more detailed proof that E(T briefly on the energies of some systems.

2. Dimensional Analysis and Physical Arguments Some insights into Casimir energies can be gained by studying a few simple systems. Consider a massless scalar meson field at T = 0 in the one-dimensional region 0 ≤ x3 ≤ l3 , satisfying Dirichlet or Neumann boundary conditions at each end. On dimensional grounds one finds the quantum result EQu (T = 0) = EQu (h¯ , c, l3 ) = C1 h¯ c/ l3 ;

(2)

the dimensionless constant C1 has not here been determined. For arbitrary T one cannot determine even the form of EQu (h¯ , c, kB T , l3 ) by dimensional analysis alone. However, as T ∼ ∞, the real photons dominate completely, EQu becomes independent of h¯ (and c), and one finds the classical result ECl (c, l3 ) ∼ C2 kB T , independent of l3 . (We need not include T ∼ ∞ in the argument; the subscript Cl implies that T ∼ ∞.) From the experimental viewpoint this is not an interesting system, because one normally measures forces, not energies, and the force is here zero, but there is the interesting feature that one can obtain at least the forms of ECl without having obtained EQu (T ). The situation is much the same for an electromagnetic (EM) field within a thin conducting spherical shell. The shell manifests itself via the boundary conditions it imposes; neither the charge nor the mass of the electron enters.The forms are then EQu (T = 0) = C3 h¯ c/R and ECl = C4 kB T . Indeed, the argument is the same for any obstacle or cavity completely defined by one length. For an EM field in a conducting box, l1 × l2 × l3 , with l1 l3 and l2 l3 , it follows on physical grounds that EQu and ECl are each proportional to l1 l2 . One

78

L. Spruch

[3

then deduces, for T = 0, that EQu (h¯ , c, l1 , l2 , l3 ) = l1 l2 EQu (h¯ , c, l3 ) = C5 hc ¯

l1 l2 l33

,

(3)

while ECl = C6 kB T l1 l2 / l32 . We note that for each of the above cases ECl has the same form as ECl ; the coefficients will differ. The argument can be extended somewhat. Consider a cavity completely characterized by lengths l1 , l2 , . . . , lM . The surface can be arbitrarily curved, for M = ∞ is allowed. (The results may not be valid for a non-ideal surface, nor in the presence of atoms.) We can then write EQu (T = 0) = hcf ¯ (l1 , . . . , lM ),

(4)

where f has the dimensions of 1/length but is otherwise arbitrary. If we scale each length by the factor K, EQu will be changed by a factor 1/K. The classical form is ECl = kB T g(l1 , . . . , lM ),

(5)

where g is dimensionless but otherwise arbitrary. ECl does not change if each length is scaled. We have not faced the serious problem of determining even the signs (but see later) of the E’s, let alone their magnitudes. (As noted above, Casimir (1948) determined C5 .)

¯ Cl 3. The Vanishing of E Since classically each mode has an energy kB T , it follows that E¯ Cl = kB T N¯ , where N¯ is the difference between the number of modes per unit area in the region between the plates in the presence and absence of the plates. Furthermore, we need not determine the frequency distribution of N¯ ; we need only know N¯ itself, the difference in the total number of modes per unit area. The modes, with wave functions described in terms of x1 , x2 , x3 , are uncoupled. The (unnormalized) wave function is ψ(x) = ψ(x1 , x2 )

∞

eiπn3 x3 / l3 − e−iπn3 x3 / l3 ,

(6)

n3 =1

where +∞ ψ(x1 , x2 ) =

+∞

−∞

dk2 eik2 x2 =

dk1 eik1 x1 −∞

dk eik ·x

(7)

3]


79

with k = (k1 , k2 ) and x = (x1 , x2 ). We therefore have N¯ = N¯ 12 N3 , where N¯ 12 = N1 N2 is the number of x -dependent modes per unit area and N3 is the number (not the number per unit area) of x3 -dependent modes. Since N¯ 12 is unchanged as l3 is changed, the evaluation of N¯ reduces to the evaluation of N3 , that is, to a one-dimensional problem, with 0 ≤ x3 ≤ l3 , where l3 is slowly changed. Since the energy levels for the one-dimensional problem are discrete, N3 = 0, and therefore N¯ = 0, and finally E¯ Cl = 0, which we set out to show. There is however, a proviso, that N¯ 12 be finite; if N¯ 12 = ∞ the statement that N¯ 12 = 0 would not be meaningful. Now we assumed there is a frequency ω0 such that the “ideal” walls are perfectly reflecting for ω < ω0 and perfectly transparent for ω > ω0 . (The value of ω0 need not concern us. It is of order 1016 Hz for a real metal.) The index of refraction in the vacuum is, of course, unity. Let n(ω) be the index of refraction of the wall. For a wave normal to the surface the transmission coefficient is T = 2/(1 + n). We must therefore assign the values n = ∞ for ω < ω0 and n = 1 for ω > ω0 . But then, by Eqs. (7.39a) and (7.41a) of Jackson (1975), T = 1 for any incident angle θ other than π/2. Since the wave vector k has a component k3 = ±n3 π/ l3 , θ will not be π/2. Only waves with ω < ω0 propagate forever between the walls, so that the number of such waves per unit area is finite. N¯ 12 is indeed finite. There is still a difficulty. We invoked the fact that k3 = 0 to show that waves could not travel forever parallel to the modes, and that represents a coupling between the walls, a coupling which we stated did not exist. We can bypass the difficulty by assuming that the region between the plates is not an ideal vacuum but contains an infinitesimal number density of atoms. There would then be some absorption and propagation over infinite lengths would not be possible. We hope that most physicists will feel comfortable with the argument, but note that Feinberg et al. (2001) did not need any such assumption. If one is to obtain the frequency distribution, one cannot factor N¯ . ω for the mode for ψ given by Eq. (6) is 1/2 π 2 2 + k c. ω = n3 (8) l3 ω is not a monotonic function of l3 . Some frequencies increase and some decrease as l3 is decreased, with thresholds at ωn = n3 πc/ l3 . The frequency distribution is known and was used by Feinberg et al. (2001) in their proof that ECl = 0. We note with some disbelief that an argument that persisted in the literature for years was ¯ that E(T = 0) < 0 because there were, presumably, fewer modes in the region between the plates in the presence than in the absence of walls. The argument was shown to be wrong by Boyer (1974). (That the argument cannot be true for T ∼ ∞ follows from the validity of the adiabatic theorem even for uncapped plates.) It should be mentioned that since the energy per mode for T ∼ ∞ is kB T for almost any system, the determination of ECl reduces to the determination

80

L. Spruch

[3

of N ; one need not know the frequency distribution. This opens up the very interesting possibility of a quite different approach to the determination of ECl . The following discussion is very crude and preliminary, ignoring the distinction between the energy internal and external to the sphere. We are interested in the possibilities of the method just noted rather then in the results, and we study massless scalar mesons satisfying Dirichlet boundary conditions within a spherical shell of radius R; the shell need not be thin and is not ideal. The number density n(k) of modes with wave number k in a volume V of area S and curvature K, given by McKean and Singer (1967), is 1 S V 2 k − K + O k −2 . Sk + (9) 8π 2π 2 6π 2 (The k dependence of the various terms follows on dimensional grounds. The volume term is that due to Weyl.) The number density nfree (k) in free space of volume V is the volume term in Eq. (9). When applied to the sphere, under the assumption that, as for the electromagnetic radiation, the shell is transparent to mesons with k > k0 —the k0 is not that for EM radiation—one finds for the difference N in the total number of modes that propagate within the region occupied by the sphere, in the presence and absence of the sphere, n(k) =

k0 N =

4πR 2 1 1 2 + · · · dk − 4πR k + 8π 6π 2 R

0

2 (10) (k0 R) + · · · . 3π [The sphere is now characterized not by R alone but by R and a length l0 = 1/k0 . (Our previous argument, which gave ECl = const · kB T , is no longer valid.) It is important to recognize that in our earlier arguments one assumed that a cutoff existed but that the details of the cutoff were irrelevant. In the present context the nature of the cutoff plays a more significant role.] For k0 R 1, the first term in Eq. (10) dominates, and N < 0. The estimate of N is crude—higher order terms were neglected, and the cutoff was rough—but we might well expect the result N < 0 to be valid, and therefore ECl = kB T N for the sphere to be negative. For applications to an electromagnetic field within a non-ideal shell, k0 is defined by k0 h¯ c ≈ 10 eV, or k0 ≈ (1/20) nm−1 . The form of n(k) differs somewhat from that of Eq. (9); in particular, there is no surface term. In line with a remark in Section 2, we mention that Schaden and Spruch, unpublished, have shown that the sign of a number of Casimir energies follows from a knowledge of the form of the semi-classical approximation to the energy Green’s function G(E), as given by Gutzwiller (1990), for the problem at hand. (The phase of G(E) plays a central role.) The energy itself needs not be calculated. = − 14 (k0 R)2 +

4]


81

(A side remark. It would be nice if, as a variation of an ancient Roman tradition, physicists granted posthumous awards—in state lie—to those who were not, but might well have been, Nobelized. My suggestions would include Casimir, on whose work this and some thousand other papers have been based, and R.E. Peierls and V.F. Weisskopf.) I thank M. Schaden for a very helpful discussion.

4. An Unauthorized Thank You Professor Bederson asked that one refrains from personal comments, but it would be remiss on my part were I not to mention the enormous influence Ben has had on my career. I had been trained as a nuclear theorist but on coming to New York University, my interests gradually switched to atomic theory. My work (with L. Rosenberg, T.F. O’Malley, and others) on atomic scattering theory (and, in particular on bounds on scattering lengths) and on the effect of induced electric dipole moments on the energy- and angular-dependence of low-energy electron– atom scattering was stimulated by Ben’s work on electron–hydrogen scattering and on the polarizabilities of atoms. His constant encouragement has been much appreciated.

5. References Boyer, T.H. (1974). Van der Waals forces and zero-point energy for dielectric and permeable materials. Phys. Rev. A 9, 2078–2084. Brown, L.S., Maclay, G.J. (1969). Vacuum stress between conducting plates: An image solution. Phys. Rev. 184, 1272–1279. Casimir, H.B.G. (1948). On the attraction between two perfectly conducting plates. Jon. Ned. Akad. Wetensch. Proc. 51, 793–795. Feinberg, J., Mann, A., Revzen, M. (2001). Casimir effect: The classical limit. Ann. Phys. 288, 103– 136. Gutzwiller, M.C. (1990). “Chaos in Classical and Quantum Mechanics”. Springer, Berlin. The results in Ref. 3 are based on Gutzwiller’s semiclassical approximation to the energy Green’s function. Jackson, J.D. (1975). “Classical Electrodynamics”, 2nd Edn. Wiley, New York. McKean, H.P., Singer, I.M. (1967). Curvature and the eigenvalues of the Laplacian. J. Differential Geom. 1, 43–69. Milonni, P.W. (1994). “The Quantum Vacuum”. Academic, New York. Schaden, M., Spruch, L. (1998). Infinity-free semi-classical evaluation of Casimir effects. Phys. Rev. A 58, 935–953. Schaden, M., Spruch, L. (2002). A classical Casimir effect: The interaction of ideal parallel walls at a finite temperature. Phys. Rev. A 65, 034101(4).



POLARIZABILITIES OF 3P ATOMS AND VAN DER WAALS COEFFICIENTS FOR THEIR INTERACTION WITH HELIUM ATOMS X. CHU and A. DALGARNO∗ ITAMP, Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA

1. 2. 3. 4. 5. 6. 7.

Introduction . . . . . . . . . . . . . . Theory: Dynamic Polarizabilities . . Numerical Method . . . . . . . . . . Results: Static Dipole Polarizabilities Van der Waals Coefficients . . . . . Acknowledgement . . . . . . . . . . References . . . . . . . . . . . . . .

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84 84 86 88 89 90 90

Abstract Time-dependent density functional theory with self-interaction correction is applied to obtain the scalar and tensor dynamic polarizabilities of atoms in 3 P states. The static polarizabilities are in good agreement with previous values. The dynamic polarizabilities are then used to evaluate the leading term in the long range interaction between the 3 P atoms and helium. They are significant in determining the likelihood that the 3 P atoms can be trapped in a helium buffer gas.

∗ I (AD) met Ben in New York in 1959 at the first International Conference on the Physics of Electronic and Atomic Collisions. Ben was the inspiration for the conference and the series that followed. The 24th will be held this year in Argentina. Amongst his many national and international activities, Ben was introducing clever ways of measuring atomic polarizabilities. His experiments were a great stimulus to my efforts to devise methods for their calculation. I see from my paper with Xi Chu that I am still working on the same problem. It is all Ben’s fault.

83


84

X. Chu and A. Dalgarno

[2

1. Introduction The effectiveness of the buffer gas loading technique introduced by Doyle and his collaborators [1–4] in creating an ultra-cold gas of atoms depends on the relative efficiency of elastic and inelastic collisions of the atoms with 3 He atoms at temperatures below 0.4 K. Inelastic collisions lead to trap loss. For atoms in states of non-zero orbital angular momentum inelastic collisions may be probable, as is the case for collisions of O(3 P) oxygen atoms [5], and the resulting gain in energy would then limit the range of atoms that can be cooled and trapped. The inelastic cross sections are controlled by the energy separations of the interaction potentials of the different symmetries of the molecular system formed by the approach of the helium atom and the gas atom [6]. It has been shown by explicit calculations of the interaction potentials that the electronic anisotropy between Ti(3 F) and He atoms and Sc(2 D) atoms and He atoms is suppressed by the shielding of the valence electrons and the resulting inelastic to elastic cross section ratio is small [7], a prediction confirmed by measurements [4]. Thus a wide range of complex atoms in non-zero angular momentum states may be trappable. Scattering at low temperatures is sensitive to the interactions at large internuclear distances R and the range of candidate atoms can be explored through calculations of the long range anisotropy. At large R the interaction potentials vary as C6 R −6 . We have developed a version of time-dependent density functional theory (TDDFT) for the calculation of the long range interactions and we have applied it to the calculation of the scalar value of the coefficient C6 for many complex atoms [8]. For atoms in P states the molecular states have or symmetry and the scalar values of C6 are averages over the two symmetries. We extend the theory to obtain the values of C6 of the individual symmetries. The method makes use of the expression of C6 as an integral over the products of dynamic polarizabilities over imaginary frequencies. The accuracy of the results can be assessed by comparing the static polarizabilities with measurements, many of which have been carried out by Bederson and his associates [9–14]. We present results for C, O, Si, S, Ge, and Se interacting with helium.

2. Theory: Dynamic Polarizabilities If the electric field is taken parallel to the z-axis, the dynamic electric dipole poz (iω) at imaginary frequencies iω of a P state atom with angular larizabilities αM momentum quantum number L = 1, magnetic quantum number M, eigenstate

2]

3P

ATOMS INTERACTION WITH HELIUM ATOMS

85

|P M, and eigenvalue Ep may be written z (iω) = 2 αM

(Et − Ep )|tM|Dz |PM|2 , (Et − Ep )2 + ω2 t

(1)

where Dz is the component along the z-axis of the electric dipole moment, Et is the eigenvalue of state with eigenfunction |tM. In Eq. (1) all quantities are expressed in atomic units. If the field is perpendicular to the z-axis, the polarizx (iω) or α y (iω) [α x (iω) = α y (iω)] is given by Eq. (1) with D or D ability αM x y M M M z in place of Dz and with tM ± 1| in place of tM|. For P state atoms, α±1 (iω) = α0x (iω). The scalar polarizability is given by α0 (ω) = {α0z (ω) + 2α1z (ω)}/3 and the tensor polarizability by α2 (ω) = {α0x (ω) − α0z (ω)}/3. We evaluate the dynamic polarizabilities by applying time-dependent density functional theory (TDDFT) [8] to an atom perturbed by a frequency-dependent external field vext (r, ω) and extracting the linear response function χσ,M (r, r , ω), where r and r are electron position vectors, σ is the spin label, and ω is the frequency. The response function may be expressed in the form χσ,M (r, r , ω) = χkσ,M (r, r , ω) k

=

∗ (r)φ (r)φ (r )φ ∗ (r ) φkσ jσ kσ jσ

ω + (j σ − kσ )

k,j

−

∗ (r ) φj∗σ (r)φkσ (r)φj σ (r )φkσ

ω − (j σ − kσ )

,

(2)

where φkσ and φj σ are orbital wave functions of the occupied and unoccupied orbitals respectively, and kσ and j σ are the corresponding orbital energies. The spin-orbitals are solutions of the equation 2 2 h¯ ∇ − (3) + Vσ (r) φiσ (r) = iσ φiσ (r). 2me The potential Vσ (r) is the optimized effective potential with an explicit selfinteraction correction. It has the form [8] ρ(r ) + Vxc ρα (r), ρβ (r) + VσS (r), Vσ (r) = VN (r) + dr (4) |r − r | where VN (r) is the electron–nuclear interaction, Vxc is the exchange-correlation potential, VσS (r) is the self-interaction correction, and ρσ (r) is the electronic spindensity ρσ (r) =

Nσ i=1

ρiσ (r) =

Nσ

φiσ (r) 2 , i=1

(5)

86


[3

where Nσ is the number of orbitals whose spin quantum number is σ and ρ(r) = ρα (r) + ρβ (r). The dynamic polarizabilities are given by z z χσ,M r, r , iω δvσ,M αM (iω) = (6) r , iω z dr dr , σ z δvσ,M (r , iω)

is the frequency-dependent scalar potential that reflects the where shielding of the applied field by the electron–electron interaction [8]. The polarz is obtained by setting vext (r, iω) ≡ z and solving self-consistent izability αM equations z z δvσ,M r , ω dr (r, ω) = z + Kαα,M r, r , ω δvα,M z r , ω dr , + Kαβ,M r, r , ω δvβ,M (7) in which Kαα,M

∂Vxc

χα,M (r , r , ω) r, r , ω = χα,M r, r , ω dr +

|r − r | ∂ρα (r) ρα (r) n σ ρiσ (r) χiσ,M (r , r , ω) + − dr ρσ (r) |r − r | i=1

∂Vxc,iσ

− χiσ,M r, r , ω , (8) ∂ρiσ (r) ρσ (r)

χβ,M (r , r , ω) dr . |r − r |

Kαβ,M r, r , ω =

(9)

y

x or α may be obtained by replacing z by x or y and δv z The polarizability αM M σ,M y x by δvσ,M or δvσ,M .

3. Numerical Method To evaluate αM (iω), we use cylindrical coordinates (r, ϕ, z) defined in terms of Cartesian coordinates (x, y, z) by x = r cos ϕ,

y = r sin θ,

z = z.

(10)

The atomic orbitals take the form eimj σ ϕ φj σ (r) = √ φj σ (r, z), 2π

mj σ = 0, ±1, ±2, . . . ,

(11)

3P

3]


σ where σ N j =1 mj σ = M. Equation (3) becomes 1 ∂ m2 ∂2 1 ∂2 + − (r, z) φj σ (r, z) + + V − σ 2 ∂r 2 r ∂r ∂z2 r2 = j σ φj σ (r, z), where

87

(12)

ρ(r, z) + VSIC,σ (r, z), (13) |r − r | 1 Nσ 2 in which Z is the nuclear charge and ρ(r, z) = 2π σ i=1 |φiσ (r, z)| , and VSIC,σ (r, z) is the self-interaction correction term. Because the atomic system is rotationally invariant, the linear response function can be re-written in the following form, m χσ,M r, r , iω = (14) r, z, r , z , iω , eimϕ e−imϕ χσ,M Z + Vσ (r, z) = − √ 2 r + z2

d 3r

m=0,±1

where φkσ (r, z)φj σ (r, z)φkσ (r , z )φj σ (r , z ) m r, z, r , z , iω = χσ,M iω + (j σ − kσ ) k,j

−

φj σ (r, z)φkσ (r, z)φj σ (r , z )φkσ (r , z ) iω − (j σ − kσ )

with the restriction that mj σ ≡ mkσ + m. Accordingly, Kσ1 σ2 ,M r, r , ω = eimϕ e−imϕ Kσm1 σ2 ,M r, z, r , z , ω .

(15)

(16)

m=0,±1

We numerically integrate z (iω) = αM

∞ ∞ ∞ ∞

0 r, z, r , z , iω χσ,M

σ

0 −∞ 0 −∞ z × δvσ,M r , z , iω z dr

x αM (iω)

1 = 2 σ

∞ ∞ ∞ ∞ 0 −∞ 0 −∞

dz dr dz ,

1 r, z, r , z , iω χσ,M

x r , z , iω r dr dz dr dz × δvσ,M

(17)

88

X. Chu and A. Dalgarno 1 + 2 σ x × δvσ,M

∞ ∞ ∞ ∞

[4

−1 r, z, r , z , iω χσ,M

0 −∞ 0 −∞

r , z , iω r dr dz dr dz ,

(18)

where ∞ ∞ z (r, z, ω) δvα,M

=z+

z 0 r, z, r , z , ω δvα,M r z , ω dr dz Kαα,M

0 −∞

∞ ∞

z 0 r, z, r , z , ω δvβ,M r , z , ω dr dz , Kαβ,M

+ 0 −∞

(19)

and ∞ ∞ x (r, z, ω) δvσ,M

=r+

−1 1 Kαα,M r, z, r , z , ω + Kαα,M r, z, r , z , ω

0 −∞

x r , z , ω dr dz × δvα,M ∞ ∞ +

−1 1 Kαβ,M r, z, r , z , ω + Kαβ,M r, z, r , z , ω

0 −∞ x r , z , ω dr dz . × δvβ,M

(20)

A two-dimensional generalized pseudo-spectral method [15,16] is used for the numerical calculations. A non-uniform spatial grid distribution is generated with a denser mesh near the origin. Many fewer grid points are needed compared to equal spatial distribution methods. The kinetic energy matrix elements take simple analytical forms and the potential energy matrix is diagonal in the coordinate representation. The convergence with respect to the number of grid points is exponential and high accuracy and efficiency are achieved. The calculated polarizabilities at imaginary frequency iω behave correctly at large frequencies, varying asymptotically as N/ω2 where N is the total number of electrons of the system.

4. Results: Static Dipole Polarizabilities In Table I we present the calculated static isotropic polarizabilities α0 (ω = 0) of 3 P atoms from carbon to selenium in units of a03 . The table includes the range of values that have been derived earlier by a diversity of theoretical methods.

3P

5]


89

Table I Isotropic static dipole polarizabilities α0 (ω = 0) in units of a03 . Atom Range This paper

C

O

Si

S

Ge

Se

(11.0, 11.8) 11.3

(4.73, 5.61) 5.29

(36.2, 36.8) 37.9

(19.1, 19.6) 19.3

(41.6, 41.7) 41.1

(25.3, 26.1) 25.5

Table II Tensor static dipole polarizabilities α2 (ω = 0) in units of a03 . Atom

C

O

Si

S

Ge

Se

Range (0.66, 0.97) (−0.15, −0.37) (2.97, 3.00) (−1.07, −1.50) (3.66, 4.65) (−1.47, −2.01) This paper 0.72 −0.36 2.93 −1.21 4.65 −1.92

They are taken from lists assembled by Andersson and Sadlej [17], Stiehler and Hinze [18], Rerat et al. [19], Medved et al. [20], Lupinetti and Thakkar [21], and Chu and Dalgarno [8]. The values obtained here lie within the range of previous calculations and are probably as reliable as any. The calculated values of the tensor polarizability are compared in Table II with the results of [17–21]. The agreement is satisfactory. The tensor polarizability is negative for atoms with more than half-filled shells and positive for atoms with less than half-filled shells.

5. Van der Waals Coefficients At large values of R, the interaction potential of the 3 and 3 molecular states formed by the approach of the 3 P atom and the helium atom vary as C6M /R 6 , where M = 0 corresponds to the state and M = ±1 to the state. The van der Waals coefficients may be expressed as integrals over the product of the frequency-dependent polarizabilities of the 3 P atom and the frequency-dependent polarizabilities αHe in the form 2 1 z x (iω)αHe (iω) dω + (iω)αHe (iω) dω. αM αM C6M = (21) π π Precise values of αHe have been calculated by Jamieson et al. [22]. The values of C6M for the and states are presented in Table III. The fractional anisotropy γ =3

C60 − C6±1

C60 + 2C6±1

(22)

is also listed in Table III. It ranges between −0.049 for carbon to 0.061 for germanium.

90


[7

Table III C6M for the X 3 and A 3 states of 3 P atoms interacting with helium in atomic units. Atom

γ

C

O

Si

S

Ge

Se

7.15 7.51 −0.049

4.85 4.67 0.038

16.5 17.31 −0.048

13.1 12.7 0.031

17.4 18.5 −0.061

16.7 15.8 0.056

The inelastic rate coefficients depend on the anisotropy also at intermediate and small separations and cannot be predicted without explicit solution of the scattering equations. Calculations have been carried out for oxygen which may provide an indication of the order of magnitude to be expected. For O(3 P) colliding with 3 He at 1 K in a magnetic field of 1 T the inelastic rate coefficient is 5 × 10−11 cm3 s−1 [5].

6. Acknowledgement This work is supported by the Chemical Sciences, Geosciences and Biosciences Division of the Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy.

7. References [1] Doyle J.M., Friedrich B., Kim J., Patterson D., Nature 431 (2004) 281. [2] Weinstein J.D., deCarvalho R., Guillet T., Friedrich B., Doyle J.M., Nature 395 (1998) 148. [3] Weinstein J.D., deCarvalho R., Amar K., Boca A., Odom B.C., Friedrich B., Doyle J.M., J. Chem. Phys. 109 (2004) 2656. [4] Hancox C.I., Doret S.C., Hummon M.T., Luo L., Doyle J.M., Nature 431 (2004) 281. [5] Krems R.V., Dalgarno A., Phys. Rev. A 68 (2003) 013406. [6] Krems R., Groenenboom G.C., Dalgarno A., J. Phys. Chem. A 108 (2004) 8941. [7] Krems R., Klos J., Rode M.F., Szczeńiak M.M., Chalasiński G., Dalgarno A., Phys. Rev. Lett. 94 (2005) 013202. [8] Chu X., Dalgarno A., J. Chem. Phys. 121 (2004) 4083. [9] Bederson B., Robinson E.J., Adv. Chem. Phys. 10 (1966) 1. [10] Schwartz H.L., Miller T.M., Bederson B., Phys. Rev. A 10 (1974) 1924. [11] Molof R.W., Schwartz H.L., Miller T.M., Bederson B., Phys. Rev. A 10 (1974) 1131. [12] Miller T.M., Bederson B., Stockdale J.A.D., Jaduzliwer B., Phys. Rev. A 14 (1976) 1572. [13] Miller T.M., Bederson B., Adv. At. Mol. Phys. 13 (1977) 1. [14] Guella T.P., Miller T.M., Bederson B., Stockdale J.A.D., Jaduzliwer B., Phys. Rev. A 29 (1984) 2977. [15] Chu X., Chu S.I., Phys. Rev. A 63 (2001) 013414. [16] Chu X., Chu S.I., Phys. Rev. A 63 (2001) 023411. [17] Andersson Q., Sadlej Q., Phys. Rev. A 46 (1992) 2356.

7] [18] [19] [20] [21] [22]

3P


Stiehler J., Hinze J., J. Phys. B 28 (1995) 4055. Rerat M., Bussery B., Freçón M., J. Mol. Spectrosc. 182 (1997) 260. Medved M., Fowler P.W., Hutson J.M., Mol. Phys. 98 (2000) 453. Lupinetti C., Thakkar A.J., J. Chem. Phys. 122 (2005). Jamieson M.J., Drake G.W.F., Dalgarno A., Phys. Rev. A 51 (1995) 3358.

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THE TWO ELECTRON MOLECULAR BOND REVISITED: FROM BOHR ORBITS TO TWO-CENTER ORBITALS∗ GOONG CHEN1 , SIU A. CHIN2 , YUSHENG DOU2,† , KISHORE T. KAPALE2,3,4 , MOOCHAN KIM2,4 , ANATOLY A. SVIDZINSKY2,4 , KERIM URTEKIN2 , HAN XIONG2 and MARLAN O. SCULLY2,4,5,6 1 Institute for Quantum Studies and Dept. of Mathematics, Texas A&M University, College Station,

TX 77843, USA 2 Institute for Quantum Studies and Dept. of Physics, Texas A&M University, College Station, TX 77843, USA 3 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA 4 Depts. of Chemistry, and Mechanical and Aerospace Engineering, Princeton University, Princeton,

NJ 08544, USA 5 Institute for Quantum Studies and Dept. of Chemical and Electrical Engineering, Texas A&M

University, College Station, TX 77843, USA 6 Max-Planck-Institut für Quantenoptik, D-85748 Garching, Germany

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. The Bohr Molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Simple Correlation Energy from the Bohr Model . . . . . . . . . . . . . . . . . . 1.4. Correlated Two-Center Orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5. Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6. Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Recent Progress Based on Bohr’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Interpolated Bohr Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. General Results and Fundamental Properties of Wave Functions . . . . . . . . . . . . 3.1. The Born–Oppenheimer Separation . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Variational Properties: The Virial Theorem and the Feynman–Hellman Theorem . 3.3. Fundamental Properties of One and Two-Electron Wave Functions . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

95 95 96 100 101 103 106 107 108 111 111 114 117

∗ MOS wishes to express his appreciation and admiration for Prof. Benjamin Bederson for his insights into and nurturing of AMO physics. Most recently, Ben’s work focused on the experimental determination of atomic polarizabilities. The classic determination was in bulk material. However, Ben developed a beautiful atomic beam method which he brought to marvelous perfection. It is a pleasure to dedicate this article to him. † Now at Dept. of Physical Sciences, Nicholls State University, Thibodaux, LA 70310, USA.

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4. Analytical Wave Mechanical Solutions for One Electron Molecules . . . . . . . . . . . . . 4.1. The Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. H+ 2 -like Molecular Ion in Prolate Spheroidal Coordinates . . . . . . . . . . . . . . . 4.3. The Many-Centered, One-Electron Problem . . . . . . . . . . . . . . . . . . . . . . . 5. Two Electron Molecules: Cusp Conditions and Correlation Functions . . . . . . . . . . . 5.1. The Cusp Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Various Forms of the Correlation Function f (r12 ) . . . . . . . . . . . . . . . . . . . 6. Modelling of Diatomic Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. The Heitler–London Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. The Hund–Mulliken Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. The Hartree–Fock Self-Consistent Method . . . . . . . . . . . . . . . . . . . . . . . . 6.4. The James–Coolidge Wave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5. Two-Centered Orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Alternative Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Improvement of Hartree–Fock Results Using the Bohr Model . . . . . . . . . . . . . 7.2. Dimensional Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Separation of Variables for the H+ 2 -like Schrödinger Equation . . . . . . . . . . . . . . . . B. The Asymptotic Expansion of (λ) for Large λ . . . . . . . . . . . . . . . . . . . . . . . C. The Asymptotic Expansion of (λ) as λ → 1 . . . . . . . . . . . . . . . . . . . . . . . . D. Expansions of Solution Near λ ≈ 1 and λ 1: Trial Wave Functions of James and Coolidge E. The Many-Centered, One Electron Problem in Momentum Space . . . . . . . . . . . . . . F. Derivation of the Cusp Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 ∇2 − 1 ∇2 . . . . . . . . . . G. Center of Mass Coordinates for the Kinetic Energy − 2m 2m2 2 1 1 H. Verifications of the Cusp Conditions for Two-Centered Orbitals in Prolate Spheroidal Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Integrals with the Heitler–London Wave Functions . . . . . . . . . . . . . . . . . . . . . . J. Derivations Related to the Laplacian for Section 6.4 . . . . . . . . . . . . . . . . . . . . . K. Recursion Relations and Their Derivations for Section 6.4 . . . . . . . . . . . . . . . . . . K.1. A(m; α) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K.2. F (m; α) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K.3. S(m, n; α) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K.4. T (m, n; α) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K.5. H0 (m, n; α) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K.6. H1 (m, n; α) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K.7. Hτ (m, n; α) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1) K.8. Hτ (m, n; α) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2) K.9. Hτ (m, n; α) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. Derivations for the 5-Term Recurrence Relations (6.81) . . . . . . . . . . . . . . . . . . . M. Dimensional Scaling in Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 11. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

135 137 138 143 145 145 147 155 155 158 162 166 174 184 184 185 191 191 192 192 193 194 196 198 203 208 210 216 217 222 222 222 223 225 225 226 227 229 230 231 232 236

Abstract Niels Bohr originally applied his approach to quantum mechanics to the H atom with great success. He then went on to show in 1913 how the same “planetaryorbit” model can predict binding for the H2 molecule. However, he misidentified the correct dissociation energy of his model at large internuclear separation, forcing him

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to give up on a “Bohr’s model for molecules”. Recently, we have found the correct dissociation limit of Bohr’s model for H2 and obtained good potential energy curves at all internuclear separations. This work is a natural extension of Bohr’s original paper and corresponds to the D = ∞ limit of a dimensional scaling (D-scaling) analysis, as developed by Herschbach and coworkers. In a separate but synergetic approach to the two-electron problem, we summarize recent advances in constructing analytical models for describing the two-electron bond. The emphasis here is not maximally attainable numerical accuracy, but beyond textbook accuracy as informed by physical insights. We demonstrate how the interplay of the cusp condition, the asymptotic condition, the electron-correlation, configuration interaction, and the exact one electron two-center orbitals, can produce energy results approaching chemical accuracy. To this end, we provide a tutorial on using the Riccati form of the ground state wave function as a unified way of understanding the two-electron wave function and collect a detailed account of mathematical derivations on the exact one-electron two-center wave functions. Reviews of more traditional calculational approaches, such as Hartree–Fock, are also given. The inclusion of electron correlation via Hylleraas type functions is well known to be important, but difficult to implement for more than two electrons. The use of the D-scaled Bohr model offers the tantalizing possibility of obtaining electron correlation energy in a non-traditional way.

1. Introduction 1.1. OVERVIEW We are in the midst of a revolution at the interface between chemistry and physics, largely due to the interplay between quantum optics and quantum chemistry. For example, the explicit control of molecules afforded by modern femtosecond lasers and adaptive computer feedback [1] has opened new frontiers in molecular science. In such studies, molecules are controlled by sculpting the amplitude and phase of femtosecond pulses, forcing the molecule into predetermined electronic and rotational–vibrational states. This holds great promise for vital applications, from the trace detection of molecular impurities, such as dipicolinic acid as it appears in anthrax [2], to the utilization of molecular excited states for quantum information storage and retrieval [3]. We are thus motivated to rethink certain aspects of molecular physics and quantum chemistry especially with regard to the excited state dynamics and coherent processes of molecules. The usual discussions of molecular structure are based on solving the many-particle Schrödinger equation with varying degree of sophistication, from exacting Diffusion Monte Carlo methods, coupled cluster expansion,

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G. Chen et al. Table I The binding energy of H2 molecule based on “exact” two-center H+ 2 orbitals.

Orbital Jaffé (6.55) Hylleraas (6.58)

Binding energy (eV) No free parameter

1 free (screening) parameter

4.50 4.51

4.60 4.62

When we allow the α and B parameters of Eqs. (6.55) and (6.58) to vary, we obtain a binding energy of 4.7 eV. The binding energy is comparable to the experimental value of 4.7 eV.

configuration interactions, to density functional theory. All are intensely numerical. Despite these successful tools of modern computational chemistry, there remains the need for understanding electron correlations in some relatively simple way so that we may describe excited states dynamics with reasonable accuracy. In this work, we propose to reexamine these questions in two complementary ways. One approach is based on the recently resurrected Bohr’s model for molecules [4]. In particular, we show that by modifying the original Bohr’s model [5] in a simple way, specially when augmented by dimensional scaling (D-scaling), we can describe both the singlet and triplet potential of H2 with remarkable accuracy (see Figs. 2, 5). In another approach, following the lead of the French school of Le Sech [6,7], we use correlated two-center orbitals of the H+ 2 molecule to model H2 ’s ground and excited state. This approach worked well, even when only a simple electron correlation function is used, see Table I. The Bohr model and D-scaling technique taken together with good (uncorrelated) molecular orbitals is especially interesting and promising. As discussed in Section 1.3, the Bohr model yields a good approximation to the electron–electron Coulomb energy, which can be used to choose a renormalized nuclear charge and a much improved (correlated) two electron wave function.

1.2. T HE B OHR M OLECULE Figure 1 displays the Bohr model for a hydrogen molecule [5,4], in which two nuclei with charges Z|e| are separated by a fixed distance R (adiabatic approximation) and the two electrons move in the space between them. The model assumes that the electrons move with constant speed on circular trajectories of radii ρ1 = ρ2 = ρ. The circle centers lie on the molecule axis z at the coordinates z1 = ±z2 = z. The separation between the electrons is constant. The net force on each electron consists of three contributions: attractive interaction between an electron and the two nuclei, the Coulomb repulsion between electrons, and the centrifugal force on the electron. We proceed by writing the energy function E = T + V , where the kinetic energy T = p12 /2m + p22 /2m for electrons 1 and 2


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F IG . 1. Cylindrical coordinates (top) and electronic distances (bottom) in H2 molecule. The nuclei Z are fixed at a distance R apart. In the Bohr model, the two electrons rotate about the internuclear axis z with coordinates ρ1 , z1 and ρ2 , z2 , respectively; the dihedral angle φ between the (ρ1 , z1 ) and (ρ2 , z2 ) planes remains constant at either φ = π or φ = 0. The sketch corresponds to configuration 2 of Fig. 2, with φ = π .

can be obtained from the quantization condition that the circumference is equal to the integer number n of the electron de Broglie wavelengths 2πρ = nh/p, so that we have T = p 2 /2m = n2 h¯ 2 /2mρ 2 ; the unit of distance is taken to be the Bohr radius a0 = h¯ 2 /me2 , and the unit of energy the atomic energy, e2 /a0 , where m and e are, respectively, the mass and charge of the electron. The Coulomb potential Energy V is given by V =−

Z Z Z 1 Z2 Z − − − + + , ra1 rb1 ra2 rb2 r12 R

(1.1)

where rai (i = 1, 2) and rbi are the distances of the ith electron from nuclei A and B, as shown in Fig. 1 (bottom), r12 is the separation between electrons. In cylindrical coordinates the distances are 2 R R 2 , rbi = ρi2 + zi + , rai = ρi2 + zi − 2 2 r12 = (z1 − z2 )2 + ρ12 + ρ22 − 2ρ1 ρ2 cos φ, here R is the internuclear spacing and φ is the dihedral angle between the planes containing the electrons and the internuclear axis. The Bohr model energy for a

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homonuclear molecule having charge Z is then given by n22 1 n21 E= + + V (ρ1 , ρ2 , z1 , z2 , φ). 2 ρ12 ρ22

(1.2)

Possible electron configurations correspond to extrema of the energy function (1.2). For n1 = n2 = 1 the energy has extrema at ρ1 = ρ2 = ρ, z1 = ±z2 = z and φ = π, 0. These four configurations are pictured in Fig. 2 (upper panel). For example, for configuration 2, with z1 = −z2 = z, φ = π, the extremum equations ∂E/∂z = 0 and ∂E/∂ρ = 0 read Z(R/2 − z) z Z(R/2 + z) + − 2 = 0, [ρ 2 + (R/2 − z)2 ]3/2 4[ρ 2 + z2 ]3/2 [ρ + (R/2 + z)2 ]3/2 (1.3) Zρ Zρ ρ 1 + 2 − = 3, [ρ 2 + (R/2 − z)2 ]3/2 [ρ + (R/2 + z)2 ]3/2 4[ρ 2 + z2 ]3/2 ρ (1.4) which are seen to be equivalent to Newton’s second law applied to the motion of each electron. Eq. (1.3) specifies that the total Coulomb force on the electron along the z-axis is equal to zero; Eq. (1.4) specifies that the projection of the Coulomb force toward the molecular axis equals the centrifugal force. At any fixed internuclear distance R, these equations determine the constant values of ρ and z that describe the electron trajectories. Similar force equations pertain for the other extremum configurations. In Fig. 2 (lower panel) we plot E(R) for the four Bohr model configurations (solid curves), together with “exact” results (dots) obtained from extensive variational calculations for the singlet ground state 1 g+ , and the lowest triplet state, 3 + [8]. In the model, the three configurations 1, 2, 3 with the electrons on opu posite sides of the internuclear axis (φ = π) are seen to correspond to the 1 g+ singlet ground states, whereas the other solution 4 with the electrons on the same side (φ = 0) corresponds to the first excited, 3 u+ triplet state. At small internuclear distances, the symmetric configuration 1 originally considered by Bohr agrees well with the “exact” ground state quantum energy; at larger R, however, this configuration’s energy rises far above that of the ground state and ultimately dissociates to the doubly ionized limit, 2H+ + 2e. In contrast, the solution for the asymmetric configuration 2 appears only for R > 1.20 and in the large R limit dissociates to two H atoms. The solution for asymmetric configuration 3 exists only for R > 1.68 and climbs steeply to dissociate to an ion pair, H+ + H− . The asymmetric solution 4 exists for all R and corresponds throughout to repulsive interaction of two H atoms. We then extend these “Bohr molecule” studies in several ways. In particular, we use a variant of the dimensional scaling (D-scaling) theory as it was originally developed in quantum chromodynamics and applied with great success to molecular and statistical physics [9,10]. This is based on an analysis in which the usual


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F IG . 2. Energy E(R) of H2 molecule for four electron configurations (top) as a function of internuclear distance R calculated within the Bohr model (solid lines) and the “exact” ground 1 g+ and first excited 3 u+ state energy of [8] (dots). Unit of energy is 1 a.u. = 27.21 eV, and unit of distance is the Bohr radius.

kinetic energy terms in the Schrödinger equation are written in D dimensions, i.e., −

h¯ 2 ∂ 2 h¯ 2 ∂ 2 →− . 2 2m 2m ∂xi ∂xi2 3

D

i=1

i=1

(1.5)

This provides another avenue into the interface between the old (Bohr–Sommerfeld) and the new (Heisenberg–Schrödinger) quantum mechanics. In particular, when D → ∞ the two electron Schrödinger equation can be scaled and sculpted into a form which is when D → ∞ identical to the Bohr theory of the H2 molecule, and much better than when 1/D or other corrections are included.

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1.3. S IMPLE C ORRELATION E NERGY FROM THE B OHR M ODEL The Bohr model offers an effective way to treat most of the correlation energy absent in the conventional Hartree–Fock (HF) approximation. Here we show how a charge renormalization method can be applied to improve the ground state energy obtained in the HF approximation. We start from He-like ions and consider a nucleus with charge Z and two electrons moving around it. According to the Bohr model the ground state energy is given by the minimum of the following expression 1 Z 1 Z 1 1 , + − + − E= (1.6) 2 2 2 ρ1 ρ1 ρ2 ρ2 ρ12 + ρ22 − 2ρ1 ρ2 cos φ where φ is the dihedral angle between electrons. At the minimum φ = π and Eq. (1.6) reduces to 1 1 1 Z 1 Z E= (1.7) + − + . − 2 2 2 ρ1 ρ1 ρ2 ρ1 + ρ2 ρ2 Optimization with respect to ρ1 and ρ2 yields 1 1 2 ρ1 = ρ 2 = . , EB = − Z − Z − 1/4 4

(1.8)

The HF approximation in the framework of the Bohr model means that optimum parameters ρ1 and ρ2 are determined by minimization of Eq. (1.7) with no electron repulsion term, i.e., omitting the electron–electron correlation. In the HF approximation the Bohr model gives 1 Z , EB-HF = −Z 2 + . (1.9) Z 2 For the He atom (Z = 2) we obtain EB-HF = −3 a.u., while EB = −3.0625 a.u. Thus, the inclusion of correlation shifts the ground state energy down by ρ 1 = ρ2 =

1 = −0.0625 a.u. (1.10) 16 The Bohr model itself is quasiclassical and, as a consequence, it predicts the He ground state energy with only 5.4% accuracy (Eexact = −2.9037 a.u.). However, the Bohr model provides a quantitative way to include the correlation energy. Let us consider the He ground state energy calculated using the HF (effective charge) variational wave function 1 + r2 ) , (r1 , r2 ) = C exp −Z(r (1.11) B =− Ecorr

is a variational parameter (effective charge), which is determined by where Z + 5Z/8, = Z − 5/16. The wave 2 − 2Z Z Z minimizing the energy E = Z

ELECTRON MOLECULAR BONDS mechanical HF energy is 5 2 W EHF = − Z − . 16

101

(1.12)

W = −2.8476 a.u. The difference between E W and the For Z = 2 we obtain EHF HF exact value is due to the correlation energy missing in the HF treatment. One can W we obtain notice that if we add the correlation energy (1.10) to EHF W B EHF + Ecorr = −2.9101 a.u.,

which substantially improves the answer and deviates by only 0.2% from the exact value. Such an idea can be incorporated by renormalization of the nuclear charge [11]. Let us define an effective charge Zeff by the condition W EB-HF (Zeff ) = EHF (Z),

which yields Zeff

1 = + 4

1 5 2 . + Z− 16 16

(1.13)

(1.14)

The effective charge improves the Bohr model energy by taking into account the difference between the quasiclassical and fully quantum mechanical description. The effective charge is calculated from the correspondence between the Bohr model in the HF approximation and the quantum mechanical HF answer. Now, if we take the Bohr model energy EB (Z) (1.8) (that includes correlation) but with Zeff instead of Z it improves the quantum mechanical HF answer: 5 2 1 1 W EB (Zeff ) = − Z − (1.15) − (Z) − . = EHF 16 16 16 The correction energy −1/16 is independent of Z and coincides with Eq. (1.10). Table II compares the quantum mechanical HF answer for the ground state energy of He-like ions and the improved value (1.15). Depending on Z Eq. (1.15) improves the accuracy 10–20 times.

1.4. C ORRELATED T WO -C ENTER O RBITALS From the preceding discussion it is clear that we need good (hopefully simple) HF wave functions. There is, of course, a great deal of work on this problem but we find the two-center orbital approach of Le Sech and coworkers [6,7,12–14] and of Patil [15] to be especially useful. In a previous publication [16], we attempted a

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Table II W and the value improved by the Ground state energy of the He-like ions in the HF approximation EHF Bohr model EB . The last two columns compare the accuracy of the HF and the improved result. Z

W (Z) EHF

2 3 4 5 6 7 8 9 10

−2.8476 −7.2226 −13.597 −21.9726 −32.3476 −44.7226 −59.0976 −75.4726 −93.8476

Z − Zeff

EB (Zeff )

Eexact

EHF (%)

EB (Zeff ) (%)

0.0441 0.0508 0.0540 0.0558 0.0570 0.0578 0.0584 0.0589 0.0592

−2.9101 −7.2851 −13.6602 −22.0351 −32.4102 −44.7852 −59.1602 −75.5352 −93.9102

−2.9037 −7.2799 −13.6555 −22.0309 −32.4062 −44.7814 −59.1566 −75.5317 −93.9068

1.93 0.79 0.42 0.26 0.18 0.13 0.10 0.08 0.06

0.22 0.072 0.033 0.019 0.012 0.008 0.006 0.004 0.003

first principle (semi-tutorial) presentation employing that the exact two-center orbitals obtained from solving the Schrödinger equation for the H+ 2 ion. As shown by Le Sech these are the most useful building blocks for constructing the electronic wave functions of the homonuclear H2 molecule. One simple form of the electronic ground state constructed with two-center orbitals is 1 H2 (1, 2) = H+ ,1σ (1)H+ ,1σ (2)χ00 1 + r12 , (1.16) 2 2 2 where H+ ,1σ is the solution of the Schrödinger equation for the H+ in prolate√2 2 spheroidal (ellipsoidal) coordinates, χ00 = [|↑1 ↓2 − |↓1 ↑2 ]/ 2 is singlet spin function, and (1 + 12 r12 ) is the Hylleraas correlation factor. See more detailed discussions of (1.16) in Sections 4–6 below. This wave function yields a binding energy of 4.5 eV for H2 molecule without any variational parameters. Variation with respect to a couple of parameters in the function (1.16) shifts the binding energy to 4.7 eV, giving remarkable agreement with the experimental value. To achieve the same result, sums over many one-centered atomic orbitals or Hylleraas type of wave function (cf. (4.30) below) that explicitly include the interelectronic distance are usually used. This has been demonstrated by the earlier work of Kolos and Wolniewicz [17]. Kolos and Szalewicz [18], and that of James and Coolidge [19], respectively. In these studies the introduction of a correlation factor, taking into account the Coulomb interaction between the two electrons, is naturally motivated by considering the trial wave function as broken into three parts, we write (r1 , r2 ) = (r1 )(r2 )f (r1 , r2 ),

(1.17)

where (r1 ) and (r2 ) are exact one electron solutions in the absence of interaction between electrons. For (r1 , r2 ) the Schrödinger equation (in atomic units)


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gives ∇2 Zb Za (r2 )f (r1 , r2 ) − 1 − − (r1 ) 2 ra1 rb1 ∇22 Zb Za − − (r2 ) + (r1 )f (r1 , r2 ) − 2 ra2 rb2 ∇2 ∇2 1 f (r1 , r2 ) + cross terms + (r1 )(r2 ) − 1 − 2 + 2 2 r12 Za Zb (r1 )(r2 )f (r1 , r2 ), = E− R

(1.18)

where cross terms mean terms that go as ∇1 (r1 ) · ∇1 f (r1 , r2 ), etc. The solution with only the first two terms is just that for H+ 2 . The functions (r1 ) and (r2 ) exponentially decay at large distances from the nuclei. The third term corresponds to the Schrödinger equation for two free electrons, ∇2 ∇2 1 f (r1 , r2 ) = εf (r1 , r2 ) − 1 − 2 + (1.19) 2 2 r12 that is the solution to Eq. (1.19) is well known [21] and is given in terms of Coulomb wave functions, i.e., confluent hypergeometric functions (see more detailed discussions in Section 5.2), which yields the (1 + r12 /2) Hylleraas factor as an asymptotic at small r12 . In order to place this part of the present review in perspective and be ready for the paradigm shift from the old quasi quantummechanical Bohr model to the new fully wave-mechanical Schrödinger–Born– Oppenheimer model, we next give a brief history of the molecular orbital concepts and computations.

1.5. C ONTEXT Molecular quantum chemistry is a fascinating success story in the annals of 20th Century science. In 1926, Schrödinger introduced the all-important wave equation which soon bore his name. In the following year Schrödinger’s new theory was applied to the simplest molecular systems of the hydrogen molecular ion H+ 2 by Burrau [22] and to the hydrogen molecule H2 by Heitler and London [23] and Condon [24]. In the same year, Born and Oppenheimer [25] published their important paper dealing with molecular nuclear motion. Further, in 1928, Hund and Mulliken [26] presented their venerable molecular orbital (MO) theory, which provided a powerful computational tool for chemistry and a foundation for the subsequent development of modern molecular science. Diatomic molecules such as H2 and HeH+ are the simplest of all molecules. Their analysis, modelling and computation constitute the bedrock of the study of

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chemical bonds in molecular structures. To quote a recent insightful article by Cotton and Nocera [27]: “It can be said without fear of contradiction that the two-electron bond is the single most important stereoelectronic feature of chemistry.”

Indeed, the description of the covalent bond in diatomics, based on the methods of Heitler–London and Hund–Mulliken, is one of the crowning achievements of quantum mechanics and fundamental physics. Computational quantum chemistry dawned in 1927 with the advent of the Heitler–London method. However, the accuracy of these early numerical results were not satisfactory, as can be seen from the following quotation (Hinchliffe [28, p. 254]): “The calculated bond length and dissociation energy [of MO theory] are in poor agreement with experiment than those obtained from simple VB [valence bond] treatment (Table 15.3), and this puzzled many people at the time. It has also led them to believe that the VB method was the correct way forward for the description of molecular structure . . . .”

New ideas were then proposed to improve the numerical accuracy of the Heitler–London and Hund–Mulliken method. The first idea of configuration interaction (CI) is to incorporate excited states into the wave function. The second idea of correlation introduces explicit dependence of the interelectronic distance in the wave function. The idea of correlation was first demonstrated by Hylleraas [29] in 1929 for the helium atom and by James and Coolidge [19] in 1933 for H2 . The use of configuration interaction and correlation are key evolutionary steps in improving the original ideas of Heitler–London and Hund– Mulliken. We can quote the following from Rychlewski [30]: “. . . Very soon it has been realized that inclusion of interelectronic distance into the wave function is a powerful way of improving the accuracy of calculated results . . . . Today, methods based on explicitly correlated wave functions are capable of yielding the “spectroscopic” accuracy in molecular energy calculations (errors less than the orders of one µ Hartree) . . . .”

For more than two electrons, it is difficult for most numerical methods to include electron correlations directly except in Monte Carlo simulations. When it is possible, as in the two electron case, excellent results can be achieved with very compact wave functions. Molecular calculations are inherently more difficult than atomic calculations. The fundamental difficulty is well stated by Teller and Sahlin [31]: “The molecular problem has a greater inherent complexity than the corresponding atomic problem . . . . In atoms, degeneracy due to spherical symmetry causes many levels to have nearly the same energy. This grouping of levels is responsible for the presence of a shell structure in atoms, and this shell structure is in turn the primary reason for the striking and simple behavior of atoms and the consequent successes of


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the independent-electron approximation for atomic systems. In passing from atoms to molecules the symmetry is reduced and the amount of degeneracy for the electronic levels becomes smaller, and, as a consequence, the power of the independent-particle model is decreased relative to the atomic case.”

Nothing illustrates this loss of symmetry, and its consequent loss of validity of the independent-particle picture better than the complete failure of the molecular orbital picture to account for the correct dissociation energy of H2 . At large internuclear separation, the symmetry is greatly reduced, and the independent occupation of single-particle molecular orbitals fails catastrophically. This failure can of course be averted by configuration-interaction, but this extra work makes it plain that molecular problems are inherently more complicated than atomic problems. Fortunately, for the investigation of ground and excited molecular states near equilibrium, one is far from the dissociation limit; the loss of symmetry complicates the calculation of the molecular orbital, but the independent particle model remains a good approximation. In the case of H2 , a natural candidate is the orbital of the two-center oneelectron molecular ion. Such orbitals will be referred to as diatomic orbitals (DO) or, in more complicated cases, shielded diatomic orbitals (SDO) when shielding is a factor to be considered. The early (1930s) ansatz wave functions of James and Coolidge [19] are expressed in terms of prolate spheroidal coordinates of the two electrons with respect to the two centers of the diatomic nuclei. However, their wave functions are not DOs in that they are not expansions of the exact one electron H+ 2 states. Rather, their approach is CI with a basis conveniently chosen for numerical evaluation. Their work is the forerunner of the Polish quantum chemistry group [8,17,18] of Kolos, Wolniewicz, etc., which have achieved the highest accuracy in numerical computation of two-electron molecules. The high accuracy obtained by Kolos and Wolniewicz in [17] is admirable, but as noted by Patil et al. [15], “. . . It is, however, perhaps somewhat unfortunate that these very impressive accomplishments have largely discouraged further fundamental studies on novel approaches to obtain accurate wave functions more directly . . . .”

A similar comment was made much earlier by Mulliken [32]: “[T]he more accurate the calculations become the more concepts tend to vanish into thin air.”

Thus the human quest for comprehension remains, and the recent research on novel approaches to obtain accurate wave functions have indeed yielded accurate, physically motivated, and compact two-electron wave functions. Patil et al. [33,34,36–39] have advocated the construction of coalescence wave functions by incorporating both cusp and asymptotic conditions. We have provided a detailed review with simplified derivations of this development in Section 3. The other approach is the use of diatomic orbitals. Historically, the original idea of using

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DOs as basis for diatomic molecules seems to begin from the work of Wallis and Hulburt [40]. More extensive references and history can be found in the works of Mclean et al. [41], Teller and Sahlin [31] and Shull [42]. Wallis and Hulburt’s result was not particularly successful, because there was no explicit electron correlation and solving the two-center wave function was difficult. According to Aubert et al. [12, Part I, p. 51]: “. . . the use of these functions, i.e., diatomic orbitals (DOs), within the oneconfiguration molecular-orbital scheme has not been very successful, owing to the difficulty of taking into account the interelectronic interactions and, moreover, owing to the complexity of calculations.”

The calculation of H+ 2 wave function improved over the years, culminating in the extensive tabulations by Teller and Sahlin [31]. In 1974–75, Aubert, Bessis and Bessis published a three-part series [12] detailing how to determine SDOs for diatomic molecules. These three papers emphasize the determination of shielded DOs. Surprisingly, the use of DO with correlation to study H2 was not undertaken until 1981 by Aubert-Frécon and Le Sech [13]. Le Sech et al. have since then made further refinements to the method (Siebbeles and Le Sech [14], Le Sech [6]). Our study of the DO’s approach was motivated by our strong interest in the modelling and computation of molecules. We were especially attracted by DOs as a natural and accurate description of chemical bonds. In Scully et al. [16], largely unaware of the prior work done by Aubert-Frécon and Le Sech of the French school, we obtained simple correlated DOs for diatomic molecules with good accuracy. The present paper represents part of our continued efforts in this direction. In this work, we will first study the mathematical properties of wave functions such as their cusp conditions, asymptotic behaviors, and forms of correlation functions. We summarize methods, techniques and formulas in a tutorial style, interspersed with some unpublished results of our own. It it not our intention to complete an exhaustive review on this vast subject, rather, only to record developments relevant to our interest in sufficient details. We apologize in advance for any inadvertent omissions.

1.6. O UTLINE The present paper is organized as follows: (i) In Section 2, we present some recent progress of an interpolated Bohr model. (ii) In Section 3, we discuss some general and fundamental properties of atoms and molecules, including the Born–Oppenheimer separation, the Feynman– Hellman Theorem, Riccati form of the ground state wave function, proximal and asymptotic conditions, the coalescent construction and the dissociation limit.


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(iii) In Section 4, we introduce the basics of the 1-electron two-centered orbitals from the classic explicit solutions of Hylleraas and Jaffé. The one-centered and multi-centered orbitals will also be reviewed. (iv) In Section 5, we present the details of the derivations of the all-important cusp conditions of Kato, and show examples as to how to verify them in prolate spheroidal coordinates. We also provide a glossary of various correlation functions that satisfy the interelectronic cusp condition. (v) In Section 6, we discuss numerical modelling of diatomic molecules and compare results with the classic methods such as the Heitler–London, Hund– Mulliken, Hartree–Fock and James–Coolidge, and the new approach of twocentered orbitals by Le Sech et al. (vi) In Section 7 we discuss alternative methods for molecular calculations based on the generalized Bohr model and the dimensional scaling. (vii) Finally, in Section 8 we give our conclusions and present an outlook.

2. Recent Progress Based on Bohr’s Model The diatomic molecules in a fully quantum mechanical treatment addressed in Section 1.5 requires solution of the many-particle Schrödinger equation. However, such an approach also requires complicated numerical algorithms. As a consequence, for a few electron problem the results become less accurate and sometimes unreliable. This is pronounced for excited electron states when the application of the variational principle is much less involved. Therefore, invention of simple and, at the same time, relatively accurate methods of molecule description is quite desirable. In this section we discuss a method which is based on the Bohr model and its modification [4]. In particular, we show that for H2 a simple extension of the original Bohr model [5] describes the potential energy curves for the lowest singlet and triplet states just about as nicely as those from the wave mechanical treatment. The simplistic Bohr model provides surprisingly accurate energies for the ground singlet state at large and small internuclear distances and for the triplet state over the full range of R. Also, the model predicts the ground state is bound with an equilibrium separation Re ≈ 1.10 and gives the binding energy as EB ≈ 0.100 a.u. = 2.73 eV. The Heitler–London calculation, obtained from a two-term variational function, yields Re = 1.51 and EB = 3.14 eV [23], whereas the “exact” results are Re = 1.401 and EB = 4.74 eV [8]. For the triplet state, as seen in Fig. 2, the Bohr model gives a remarkably close agreement with the “exact” potential curve and is in fact much better than the Heitler–London result (which, e.g., is 30% higher for R = 2). One should mention that in 1913, Bohr found only the symmetric configuration solution, which fails drastically to describe the ground state dissociation limit.

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The simple Bohr model offers valuable insights for the description of other diatomic molecules. For N electrons the model reduces to finding configurations that deliver extrema of the energy function 1 n2i E= + V (r1 , r2 , . . . , rN , R), 2 ρi2 N

(2.1)

i=1

where the first term is the electron kinetic energy, while V is the Coulomb potential energy. In such formulation of the model there is no need to specify electron trajectories nor to incorporate nonstationary electron motion. Eq. (2.1) assumes that only at some moment in time the electron angular momentum equals an integer number of h¯ and the energy is minimized under this constraint. In the general case, the angular momentum of each electron changes with time; nevertheless, the total energy remains a conserved quantity. Let us now discuss the ground state potential curve of HeH. To incorporate the Pauli exclusion principle one can use a prescription based on the sequential filling of the electron levels. In the case of HeH the three electrons can not occupy the same lowest level of HeH++ . As a result, we must disregard the lowest potential energy curve E(R) obtained by the minimization of Eq. (2.1) and take the next possible electron configuration, which is shown in Fig. 3 (upper panel). For this configuration, n1 = n2 = n3 = 1, however, the right energy corresponds to a saddle point rather than to a global minimum. In order to obtain the correct eff = 1.954. The dissociation limit we assign the helium an effective charge ZHe charge matches the difference between the exact ground state energy of the He atom and the Bohr model result. Fig. 3 shows the ground state potential curve of HeH in the Bohr model (solid curve) and the “exact” result (dots) obtained from extensive variational calculations [43]. The Bohr model gives a remarkably close agreement with the “exact” potential curve. 2.1. I NTERPOLATED B OHR M ODEL The original Bohr model assumes quantization of the electron angular momentum relative to the molecular axis. This yields a quite accurate description of the H2 ground state E(R) at small R. However, E(R) becomes less accurate at larger internuclear separation as seen in Fig. 2. To obtain a better result one can use the following observation. At large R each electron in H2 feels only the nearest nuclear charge because the remaining charges form a neutral H atom. Therefore, at large R the momentum quantization relative to the nearest nuclei, rather than to the molecular axis, must yield a better answer. This leads to the following expression for the energy of the H2 molecule n22 1 n21 Z Z Z 1 Z2 Z E= (2.2) + − − − + + . − 2 2 2 ra1 ra1 rb1 ra2 rb2 r12 R rb2


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F IG . 3. Energy E(R) of HeH molecule for the electron configuration shown on the upper panel as a function of internuclear spacing R calculated within the Bohr model for n1 = n2 = n3 = 1, eff = 1.954 (solid line) and the “exact” ground state energy of [43] (dots). ZHe

For n1 = n2 = 1 and R > 2.8 the expression (2.2) has a local minimum for the asymmetric configuration 2 of Fig. 2. We plot the corresponding E(R) without the 1/R term in Fig. 4 (curve 2). At R < 2.8 the electrons collapse into nuclei. At small R we apply the quantization condition relative to the molecular axis which yields curve 1 in Fig. 4. To find E(R) at intermediate separation we connect smoothly the two regions by a third order polynomial (thin line). Addition of the 1/R term yields the final potential curve, plotted in Fig. 5. The simple interpolated Bohr model provides a remarkably close agreement with the “exact” potential curve over the full range of R. As an example of application of the interpolated Bohr model to other diatomic molecules, we discuss the ground state potential curve of LiH. The Li atom contains three electrons, two of which fill the inner shell. Only the outer electron with the principal quantum number n = 2 is important in forming the molecular bond. This makes the description of LiH similar to the excited state of H2 in which two electrons possess n1 = 1 and n2 = 2. So, we start from the H2 excited state and apply the interpolated Bohr model as described above. Then, to obtain E(R) for

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F IG . 4. The Bohr model E(R) for H2 molecule without 1/R term. Curves 1 and 2 are obtained based on the quantization relative to the molecular axis (small R) and the nearest nuclei (large R), respectively. Thin line is the interpolation between two regions.

F IG . 5. Ground state E(R) of H2 molecule as a function of internuclear distance R calculated within the interpolated Bohr model (solid line) and the “exact” energy of [8] (dots).

LiH, we take the H2 potential curve and add the difference between the ground state energy of Li (−7.4780 a.u.) and H in the n = 2 state, i.e., add −7.3530 a.u.


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F IG . 6. Ground (1 + ) state energy E(R) of LiH molecule as a function of internuclear distance R calculated within the interpolated Bohr model (lower solid line) and the “exact” energy of [44] (dots). Upper solid curve is the first excited (3 + ) state energy of LiH obtained from the Bohr model 3 g+ E(R) of H2 molecule by adding the difference between the ground state energy of Li and H.

The final result is shown in Fig. 6 (lower solid line), while dots are the “exact” numerical answer from [44]. One can see that the simple interpolated Bohr model provides quite good quantitative description of the potential curve of LiH, which is already a relatively complex four electron system.

3. General Results and Fundamental Properties of Wave Functions Having examined the quasi-quantum mechanical Bohr model in the preceding two sections, we now attend to the fully quantum mechanical model and its approximation and analysis.

3.1. T HE B ORN –O PPENHEIMER S EPARATION The underlying theoretical basis for problems involving a few particles in atomic and molecular physics is the Schrödinger equation for the electrons and nuclei,

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which provides satisfactory explanations of the chemical, electromagnetic and spectroscopic properties of the atoms and molecules. Assume that the system under consideration has N1 nuclei with masses MK and charges eZK , e being the 1 electron charge, for K = 1, 2, . . . , N1 , and N2 = N K=1 ZK is the number of electrons. The position vector of the Kth nucleus will be denoted as R K , while that of the kth electron will be r k , for k = 1, 2, . . . , N2 . Let m be the mass of the electron. The Schrödinger equation for the overall system is given by

N2 N1 N2 N1 2 2 2 ZK e2 2 ∇K − ∇k − 2MK 2m |R K − r k | K=1 k=1 K=1 k=1 N2 N1 1 e2 ZK ZK e2 1 + + (R, r) 2 |r k − r k | 2 |R K − R K |

H (R, r) = −

k,k =1 k=k

K,K =1 K=K

= E(R, r),

(3.1)

where H is the Hamiltonian and R = (R 1 , R 2 , . . . , R N1 ),

r = (r 1 , r 2 , . . . , r N2 ).

The above equation is often too complex for practical purposes of studying molecular problems. Born and Oppenheimer [25] provide a reduced order model by approximation, permitting a particularly accurate decoupling of the motions of the electrons and the nuclei. The main idea is to assume that in (3.1) takes the form of a product (R, r) = G(R)F (R, r).

(3.2)

Substituting (3.2) into (3.1), we obtain

N2 N2 N1 2 2 ZK e2 ∇k − 2m |R K − r k | k=1 K=1 k=1 N2 1 e2 + F (R, r) 2 |r k − r k |

−

G(R)

k,k =1 k=k

+

N1 2 1 2 ∇K + − 2MK 2 K=1

N1 K,K =1 K=K

+ T1 + T2 = EG(R)F (R, r),

ZK ZK e2 G(R) F (R, r) |R K − R K | (3.3)


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where T1 ≡ −G(R)

N1 2 ∇K G(R) · ∇K F (R, r), MK

(3.4)

N1 2 ∇ 2 F (R, r). 2MK K

(3.5)

K=1

T2 ≡ −G(R)

K=1

The essential step in the Born–Oppenheimer separation consists in dropping T1 and T2 in (3.3). This leads to the separation of the electronic wave function

N2 N2 N2 N1 2 2 2 e Z e 1 K − ∇k2 + − F (R, r) 2m 2 |r k − r k | |R K − r k | k=1

k,k =1 k=k

K=1 k=1

= Ee (R)F (R, r),

(3.6)

and a second equation for the nuclear wave function

N1 2 1 − ∇2 + 2MK K 2 K=1

N1 K,K =1 K=K

ZK ZK e2 + Ee (R) G(R) |R K − R K |

= EG(R), where Ee (R) is the constant of separation. In “typical” molecules, the time scale for the valence electrons to orbit about the nuclei is about once every 10−15 s (and that of the inner-shell electrons is even smaller), that of the molecular vibration is about once every 10−14 s, and that of the molecule rotation is every 10−12 s. This difference of time scale is what make T1 and T2 in (3.4) and (3.5) negligible, as the electrons move so fast that they can instantaneously adjust their motions with respect to the vibration and rotation movements of the slower and much heavier nuclei. The Born–Oppenheimer separation breaks down in several cases, chief among them when the nuclear motion is strongly coupled to electronic motions, e.g., when the Jahn–Teller effects [45,46] are present. It also requires corrections for loosely held electrons such as those in Rydberg atoms. Research work on non-Born–Oppenheimer effects, the inclusion of the spinorbit coupling, and on models without using the Born–Oppenheimer separation by treating the coupled dynamical motions of the electrons and nuclei simultaneously, may be found in Yarkony [47] and Öhrn [48], for example.

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3.2. VARIATIONAL P ROPERTIES : T HE V IRIAL T HEOREM AND THE F EYNMAN –H ELLMAN T HEOREM The variational form of (3.1) is min |H | = E,

(3.7)

|=1

where the trial wave functions = (R, r) belong to a proper function space (which is actually the Sobolev space H 1 (R3(N1 +N2 ) ) in the mathematical theory of partial differential equations [20, Chapter 2]). The (unique) solution 0 attaining the minimum of |H | is called the ground state and the associated value E0 ≡ 0 |H |0 is the corresponding ground state energy. Excited states k with successively higher energy levels may be obtained recursively through min|H | ≡ Ek = k |H |k , where is subject to the constraints

(3.8)

| = 1, |0 = |1 = · · · = |k−1 = 0,

for k = 1, 2, 3, . . . .

There are two useful theorems related to the above variational formulation: the virial theorem and the Feynman–Hellman theorem. We discuss them below. Let ψ be any trial wave function. The expectation value of the kinetic energy is

N N2 1

2 2 2

2 Ekin ≡ − (3.9) ∇ − ∇ .

2MK K 2m k K=1

k=1

Now, consider the scaling of all spatial variables by 1 + λ: R ≡ (R, r) −→ (1 + λ)(R, r) ≡ (1 + λ)R.

(3.10)

Subject to the above transformation (3.10), we have 1 Ekin . (1 + λ)2 kin with respect to λ, we see that it is the same as By taking the variation of E taking the variation of Ekin with respect to . Thus

d

= δEkin , Ekin dλ λ=0

1 d

E = −2(1 + λ)−3 λ=0 · Ekin = −2Ekin = δEkin . kin 2 dλ (1 + λ) λ=0 (3.11) kin = E

For the potential energy, the expectation is given by Epot ≡ |V | ≡ V ,

(3.12)


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where V consists of the 3 summations of Coulomb potentials inside the bracket of (3.1) (but V can be allowed to be any potential whose negative gradient is force). The spatial scaling (3.10) implies the change of displacement δ = λ, which is now regarded as infinitesimal. Therefore,

d δEpot = ψ ∗ (R)V (1 + λ)R ψ(R) dR)

dλ λ=0

R3N

ψ ∗ (R) R · ∇V (R) ψ(R) dR

= R3N

= R · ∇V ≡ Virial,

(3.13)

where Virial is the quantum form of the classical virial. For = 0 , the variational form (3.7) demands that δH = δEkin + δEpot = 0.

(3.14)

Substituting (3.11) and (3.13) into (3.14), we obtain −2Ekin + Virial = 0.

(3.15)

This is the general form of the virial theorem for an isolated system of an atom or a molecule. In the particular case that N2 N1

V (R) = −

K=1 k=1

+

1 2

N2 Zk e2 e2 1 + |R K − r k | 2 |r k − r k |

N1 K,K =1 K=K

k,k =1 k=k

ZK ZK e2 , |R K − R K |

which is the power law Rn (where R is the distance) with n = −1, the classical virial property holds: Virial = nEpot = −Epot

(3.16)

as R · ∇Rn = nRn−1 R · ∇R = nRn . From (3.15) and (3.16), we obtain the virial theorem for an (exact, not trial) wave function: 2Ekin + Epot = 0,

(3.17)

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or 1 Ekin = − Epot . (3.18) 2 This property is used as a check for accuracy of calculations and the properness of the choices of trial wave functions. The next, Feynman–Hellman theorem, shows how the energy of a system varies when the Hamiltonian changes. Assume that the Hamiltonian of an atom or molecule system depends on a parameter α, H = H (α). For example, α may represent the internuclear distance of the molecular ion H+ 2 . The exact wave function = (α) also depends on α, so does the energy of the system E = E(α). Let’s see how E(α) changes with respect to α, i.e., dE(α) d ∗ (α)H (α)(α) dr = dα dα ∂ ∗ (α) ∂H (α) = H (α)(α) dr + ∗ (α) (α) dr ∂α ∂α ∂(α) dr + ∗ (α)H (α) ∂α ∂ ∗ (α) ∂H (α) = E(α) (α) dr + ∗ (α) (α) dr ∂α ∂α ∂(α) dr + E(α) ∗ (α) ∂α

∂H (α) d (α) (α) + ∗ (α) (α) dr = E(α) dα ∂α

∂H (α)

(α) = (α)

∂α as (α)|(α) = 1 and is independent of α. The above is the Feynman–Hellman theorem. Its advantage is that oftentimes ∂H (α)/∂α is of a very simple form. For example, for the H+ 2 -like equation (4.13), upon taking α = R we have Zb rb Za Zb ∂H =− 3 R+ 3 ∂R R rb and the average force on the nucleus B is Zb rb Za Zb ∂E R− = . F=− ∂R R3 rb3 That is the force and, hence, the potential energy curve requires calculation of Zb rb /rb3 only. This substantially simplifies the problem since the matrix element from the ∇ 2 is no longer necessary.


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3.3. F UNDAMENTAL P ROPERTIES OF O NE AND T WO -E LECTRON WAVE F UNCTIONS 3.3.1. Riccati Form, Proximal and Asymptotic Conditions In this subsection, we introduce the Riccati form of the ground state wave functions as a unified way of understanding and deriving various cusp, asymptotic and correlation functions. This forms the basis by which compact wave functions for diatomic molecules can be derived. Consider the Schrödinger equation with a spherically symmetric potential in reduced units, −

1 2 ∇ ψ(r) + V (r)ψ(r) = Eψ(r), 2µ

where µ = 1 is the central-force case, and µ = coordinate case. We will be primary interested in

(3.19) 1 2

is the equal-mass, relative

Z (cf. (4.12) below), (3.20) r however, the long range Coulomb potential is special in many ways and we can best understand the Coulomb-potential wave function by comparing and contrasting it to the short-range, Lennard-Jones potential 1 1 VLJ (r) = 0 12 − 6 . (3.21) r r Vc (r) = −

Since the ground state of (3.19) is strictly positive and spherically symmetric, it can always be written as (unnormalized) ψ(r) = e−S(r) .

(3.22)

Substituting this into (3.19) gives the Riccati equation for S(r) 1 2 1 ∇ S(r) − ∇S(r) · ∇S(r) + V (r) = E0 . 2µ 2µ

(3.23)

Since ∇S(r) = S (r)ˆr and ∇ · rˆ = 2/r, we have simply, 1 1 1 2 S + S − S + V (r) = E0 . 2µ µr 2µ

(3.24)

The advantage of this equation is that since the RHS is a constant E0 , all the singularities of V (r) must be cancelled by the derivatives of S(r). In particular, if we are only seeking an approximate ground state, then a reasonable criterion would be to require S(r) to cancel the most singular term in V (r). For both the Coulomb and the Lennard-Jones case, the potential is most singular as r → 0. We will refer to this as the proximal limit. For both cases, the singularity of V (r)

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is only polynomial in r and therefore we can assume S(r) = ar n ,

(3.25)

giving explicitly 1 1 2 2 2n−2 + V (r) = E0 . an(n + 1)r n−2 − a n r 2µ 2µ

(3.26)

Note the structure of this equation, if the V (r) has a repulsive polynomial singularity, then it can always be cancelled by the r 2n−2 term, leaving a less singular term r n−2 behind. For example, in the Lennard-Jones case, the most singular term is cancelled if we set −

0 1 2 2 2n−2 + 12 = 0, a n r 2µ r

(3.27)

yielding, n = −5, and for µ = 12 , the famous McMillan correlation function for quantum liquid helium, 1√ 0 r −5 . 5 However, if V (r) has an attractive singularity, then it can only be cancelled by the r n−2 term, and if n ≤ 0, would leave behind a more singular term instead. This means that an attractive potential V (r) cannot be as singular, or more singular, than −1/r 2 , otherwise, the Schrödinger equation has no solutions. Fortunately, for the Coulomb attraction (3.20), the −Z/r singularity can be cancelled by setting S(r) =

Z 1 an(n + 1)r n−2 − = 0, 2µ r

(3.28)

yielding, for µ = 1, n = 1, S(r) = Zr. For electron–electron repulsion with µ =

1 2

and Z = −1, we would have instead

1 S(r) = − r. 2 In the Coulomb case, these proximal conditions are known as cusp conditions. A more thorough treatment of the cusp condition for the general case will be given in Section 5.1 and Appendices F and H. The proximal criterion of cancelling the leading singularity of V (r) can be applied generally to any V (r). The cusp conditions are just special cases for the Coulomb potential. Note that in (3.28), the Coulomb singularity is actually cancelled only by the S /(µr) term of the


119

Riccati equation, 1 Z S − = 0, µr r since requiring S to be a constant at the singular point forces S = 0. Thus the cusp condition can be stated most succinctly in term of the Riccati function S(r): wherever the nuclear charge is located, the radial derivative of S(r) at that point must be equal to the nuclear charge. Next, we consider the asymptotic limit of r → ∞. In the case of short range potential, such that V (r) → 0 faster than 1/r, we can just completely ignore V (r) in (3.24). Substituting in S(r) = αr + β ln(r) gives 1 β 1 1 β 2 β − − + α+ α+ = E0 . 2µ r 2 2µ r µr r

(3.29)

The constant terms determine α = −2µE0 = 2µ|E0 |. Setting the sum of 1/r terms zero gives, 1 α(1 − β) = 0, µr

(3.30)

which fixes β = 1. The remaining terms will decay faster than 1/r and can be neglected in the large r limit. Thus the asymptotic wave function for any shortranged potential (decays faster than 1/r) must be of the form ψ(r) →

1 −αr e . r

(3.31)

(we wish to reserve the possibility that However, for the Coulomb potential −Z/r Z can be distinct from Z and unrelated to E0 ), we must retain it among the 1/r terms in (3.30), Z 1 α(1 − β) − = 0, µr r

(3.32)

resulting in β =1−

µZ . α

(3.33)

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Thus the general asymptotic wave function for a Coulomb potential has a slower decay, ψ(r) →

1 −αr e . rβ

(3.34)

= Z, α = Z, β = 0, For a single electron in a central Coulomb field, µ = 1, Z and ψ(r) → e−Zr ,

(3.35)

the decay is the slowest, very different from (3.31). For more than one electron, β does not vanish and the correct asor more than one nuclear charge, α = Z, ymptotic wave function is (3.34). We tend to forget this fact because we are too familiar with the single-electron wave function, which is the exception, rather than the norm. The proximal and asymptotic conditions are very stringent constraints: wave functions that can satisfy both are inevitably close to the exact wave functions. Satisfying the proximal condition alone is sufficient to guarantee an excellent approximate ground state for all radial symmetric potentials such as the LennardJones, the Yukawa (V = 1r e−αr , α > 0), and the Morse potential (V = e−2αr − 2e−αr , α > 0). Needless to say, the proximal condition alone determines the exact ground state for the Coulomb and the harmonic oscillator potential. The significance of the proximal condition has always been recognized. The current interest [34] in deriving compact wave functions for small atoms and molecules is based on a renewed appreciation of the importance of the correct asymptotics wave functions. 3.3.2. The Coalescence Wave Function Consider the case of two electrons orbiting a central Coulomb field, 1 1 Z Z 1 − ∇12 − ∇22 − − + ψ(r1 , r2 ) = Eψ(r1 , r2 ). 2 2 r1 r2 r12

(3.36)

Imagine that we assemble this atom one electron at a time. When we bring in the first electron, its energy is E1 = −Z 2 /2, with wave function ψ(r1 ) = exp(−Zr1 ) localizing it near the origin. When electron 2 is still very far away, we can write the two-electron wave function as ψ(r1 , r2 ) = e−Zr1 −S(r2 ) .

(3.37)

Substituting this into (3.36) yields the Riccati equation for S(r2 ), 1 Z 1 1 1 1 2 + = E0 + Z 2 . S − S + S − 2 2 r2 r2 r12 2

(3.38)


121

The RHS defines the second electron’s energy, E2 ≡ E0 + 12 Z 2 , whose magnitude is just the first removal or ionization energy. Since electron 2 is far away, and electron 1 is close to the origin, r12 ≈ r2 . Thus electron 2 “sees” an effective Coulomb √ field −Z/r2 with Z = Z − 1. This is the case envisioned in (3.32) with α = −2E2 . The asymptotic wave function for the second electron is, therefore, −β

ψ(r2 ) → r2 e−αr2 ,

(3.39) −β

with β = 1 − (Z − 1)/α. Since β is always less than one, the r2 term is only a minor correction. Its effect can be accounted for by slightly altering α. The important point here is that the second electron need not have the same Coulomb wave function as the first electron. This coalescence scenario would suggest, after symmetrizing (3.37), the following two-electron wave function: ψ(r1 , r2 ) = e−Zr1 −αr2 + e−αr1 −Zr2 .

(3.40)

For the case where there are more than two electrons, one can imagine building up the atom or molecule sequentially one electron at a time. Each electron would then acquire a different Coulomb-potential wave function. This sequential, or coalescence scenario of approximating the ground state, in many cases, resulted in better wave functions than considering all the electrons simultaneously, which is the traditional Hartree–Fock point of view; see Section 6.3. In the case of He, the simple effective charge approximation ψ(r1 , r2 ) = e−Zeff r1 e−Zeff r2 with Zeff = Z −

5 = 1.6875, 16

(3.41)

2 = −2.8476, while the “exact” value is −2.9037. The standard gives Evar = −Zeff HF wave function of the form

ψ(r1 , r2 ) = φ(r1 )φ(r2 ) improves [35] the energy to Evar = −2.8617. For comparison, the coalescent wave function (3.40) can achieve Evar = −2.8674 at α = 1.286. Patil [34], by √ restricting α to be consistent with the output variational energy via α = −2Evar − 4, obtained Evar = −2.8671 at α =√1.317. All√these values of α are very close to the exact asymptotic value of α = −2E2 = 2(0.9037) = 1.344, lending credence to the coalescence construction. For arbitrary Z, by approximat2 , we can estimate α by ing E0 by −Zeff 2 − Z 2 (≈ Z − 5/8). α = 2Zeff (3.42)

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G. Chen et al. Table III Ground state energy of the He-like ions as calculated from various wave functions.

Z

E(Coales) E(Approx.) E(Abs.Min.) E(exact) fLS

fPJ

1 + 12 r12

2 3 4 5 6 7 8 9 10

−2.8673 −7.2355 −13.6072 −21.9802 −32.3539 −44.7280 −59.1023 −75.4768 −93.8514

−2.9017(4) −7.271(1) −13.642(1) −22.014(2) −32.386(3) −44.755(3) −59.129(4) −75.502(4) −93.875(5)

−2.8898(7) −7.264(1) −13.634(2) −22.011(2) −32.382(3) −44.753(3) −59.127(4) −75.494(4) −93.885(5)

−2.8757 −7.2490 −13.6232 −21.9978 −32.3725 −44.7473 −59.1221 −75.4970 −93.8719

−2.8757 −7.2488 −13.6230 −21.9975 −32.3723 −44.7471 −59.1220 −75.4969 −93.8718

−2.9037 −7.2799 −13.6555 −22.0309 −32.4062 −44.7814 −59.1566 −75.5317 −93.9068

−2.9016(3) −7.268(1) −13.637(2) −22.004(2) −32.376(3) −44.740(4) −59.115(4) −75.490(6) −93.859(6)

For Z = 2, this gives α = 1.30 (for small Z, we need to use the full expression rather than the approximation), an excellent estimate. This obviates the need for Patil’s self-consistent procedure to determine α, and produces even slightly better results. The coalescent wave function (3.40) with this choice for α, defines a set of parameter-free two-electron wave functions for all Z. The resulting energy for Z = 2 − 10 is given in Table III. In 1930, Eckart [49] has used wave functions of the form ψ(r1 , r2 ) = e−ar1 −br2 + e−br1 −ar2

(3.43)

to compute the energy of a two-electron Z-atom. His resulting energy functional is 1 EEck (Z, a, b) = −Z(a + b) + 1 + C(a, b) 1 2 2 × K(a, b) + a + b + abC(a, b) , (3.44) 2 where 64a 3 b3 . (a + b)6 (3.45) He obtained an energy minimum −2.8756613 for He at a = 2.1832 and b = 1.1885. (We have used his energy expression to re-determine the energy minimum more accurately.) While the improvement in energy is a welcoming contribution, it seems difficult to interpret the resulting wave function physically. How can an electron “sees” a nucleus with charge greater than 2? To gain further insight into Eckart’s result, and coalescence wave function in general, we note that (3.43) can be rewritten as χ(r1 , r2 ) = e−A(r1 +r2 ) cosh B(r1 − r2 ) , (3.46) K(a, b) =

20a 3 b3 a 2 b2 ab + + a + b (a + b)3 (a + b)5

and

C(a, b) =


123

with A = (a+b)/2 and B = (a−b)/2. This form has the HF part e−A(r1 +r2 ) . If we substitute Eckart’s values, we see that A = (2.1832+1.1885)/2 = 1.6859, which is nearly identical to the effective charge value (3.41). This is not accidental. If we use the approximation (3.42) for α, then the coalescence wave function (3.40) would automatically predict 1 5 (3.47) (Z + α) = Z − ! 2 16 Thus within the class of Eckart wave function (3.43), the coalescence scenario correctly predicts the path of optimal energy as being along A = Zeff . Moreover, this improvement in energy, which has laid dormant in Eckart’s result for three quarters of a century, can now be understood as due to the radial correlation cosh term in (3.46), built-in automatically by the coalescence construction. This term is the smallest (=1) when r1 = r2 , but is large when the separation r1 − r2 is large, i.e., it encourages the two electrons to be separated in the radial direction. This suggests that we should reexamine Eckart’s energy functional in terms of parameters A and B. Expanding (3.44) to fourth order in B yields, A=

3 1 2 3 2 EEck (Z, A, B) = −Zeff + (A − Zeff )2 − y + y 2 − y , 8 2 2A

(3.48)

where y ≡ B 2 /A. If B = 0, A = Zeff this yields the effective charge energy 2 . Regarding the effect of B as perturbing on this fixed choice of A, the −Zeff 1/A ≈ 1/Z term can first be ignored. Minimizing y simply yields EEck (Z, A, B) = −Zeff −

3 1 − 128 128Zeff

(3.49)

at 1 B2 = , (3.50) A 8 where we have restored the 1/A term. This remarkably simple result is the content of Eckart’s energy functional. The energy is lower from the effective charge value by a nearly constant amount 3/128. In column three of Table III, we compare this approximate energy (3.49), with the absolute minimum of the Eckart’s energy functional on the fourth column. The agreement is uniformly excellent. By comparison, we see that the coalescence construction, without invoking any minimization process, also gives a very good account of the energy minimum. y=

3.3.3. Electron Correlation Functions Coalescence wave functions are better than HF wave function because they have built-in radial correlations. To further improve our description of He, as first realized by Hylleraas [29], one can introduce electron–electron correlation directly

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by forcing the two-electron wave function to depend on r12 explicitly. (A more detailed discussion of correlation functions will be given in Section 5.2 below.) Again, our analysis is simplified by use of the Riccati function. Let ψ(r1 , r2 ) = e−S(r1 ,r2 ,r12 ) ,

(3.51)

but now consider S(r1 , r2 , r12 ) = Zr1 + Zr2 + g(r12 ).

(3.52)

We have ∇1 S = Zˆr1 + g rˆ 12 , 2Z 2g ∇12 S = + + g , r1 r12 and (3.36) in terms of S reads

∇2 S = Zˆr2 + g rˆ 21 , 2Z 2g ∇22 S = + + g , r2 r12

2g + 1 − (g )2 − Zg (ˆr1 − rˆ 2 ) · rˆ 12 − Z 2 = E0 . r12 In order to eliminate the 1/r12 singularity, we must have g +

(3.53)

1 lim g (r12 ) = − . r12 →0 2 Thus, one can consider a series expansion for g(r12 ) starting out as 1 1 2 + ···. g(r12 ) = − r12 + Cr12 (3.54) 2 2 Keeping only up to the quadratic term, in the limit of r12 → 0, (3.53) reads 1 − Z 2 + O(r12 ) = E0 . (3.55) 4 If g(r12 ) were exact, the LHS above would be the constant ground state energy for all values of r12 . Inverting the argument, we can exploit this fact to determine C at r12 = 0, provided that we can estimate the ground state energy E0 . The simplest estimate for a two electron atom would be E0 = −Z 2 , implying that 3C −

1 . (3.56) 12 However, since the effective charge approximation for the energy is much better, 2 , thus fixing we should take instead, E0 = −Zeff 1 1 2 5 5 1 2 + Z − Zeff = + Z− . C= (3.57) 12 3 12 24 32 C=

The determination of the quadratic term of g(r12 ) was advanced only recently by Kleinekathofer et al. [50]. (If we also improve the one-electron wave function


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from exp(−Zr) → exp(−Zeff r), then C must go back to the value (3.56). The coefficient C therefore depends on the quality of the single electron wave function.) The wave function (3.51) can now be written as ψ(r1 , r2 ) = e−Zr1 e−Zr2 f (r12 ),

(3.58)

where

1 f (r12 ) = exp r12 (1 − Cr12 ) . 2

(3.59)

Since the above argument is only valid for small r12 , the large r12 behavior of f (r12 ) is not determined. It seems reasonable, however, barring any long range Coulomb effect, to assume lim f (r12 ) −→ constant.

(3.60)

r12 →0

The form (3.59) can have behavior (3.60) if we just rewrite it as r12 . fPJ (r12 ) = exp 2(1 + Cr12 )

(3.61)

This Padé–Jastrow form has been used extensively in Monte Carlo calculations of atomic systems [51]. Alternatively, to achieve (3.60), Patil’s group [34,50] have suggested the form 1 1 + 2λ − e−λr12 ; 2λ which has the small r12 expansion fP (r12 ) =

cf. (5.2.3) and Fig. 12,

1 λ 2 fP (r12 ) = 1 + r12 − r12 + ···. 2 4 By comparing this with similar expansion of (3.59), we can identify 5 5 1 1 Z− − . λ = 2C − = 2 12 32 3

(3.62)

(3.63)

(3.64)

Le Sech’s group [14] have employed the form 1 fLS (r12 ) = 1 + r12 e−ar12 , 2

(3.65)

with 1 λ. 2 This function is not monotonic; it reaches a maximum at r12 = 1/a before level off back to unity; cf. Fig. 11 in Section 5.2. However, this point may not be practically relevant, since most electron separations do not reach beyond the maximum. a=

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F IG . 7. Comparison of three electron–electron correlation functions.

For the sake of comparison, we may use the maximum of Le Sech’s correlation function as its asymptotic limit. This is the natural thing to do because all three functions can now be characterized by their asymptotic value as r12 → ∞: 1 1 fPJ (r12 ) → exp ≈ 1+ , 2C λ + 1/2 1 fP (r12 ) → 1 + , 2λ 1 fLS (r12 ) → 1 + . eλ Their approaches toward unity are approximately λ−1 , 12 λ−1 , and 13 λ−1 , respectively. Note also that as λ increases with Z, the asymptotic value of f (r12 ) decreases, this is the correct trend long observed in Monte Carlo calculations on atomic systems [51]. For Z = 2, we take C = 1/2, λ = 1/2, a = 1/4 and compare all three correlation functions in Fig. 7. Also plotted is the simple linear and quadratic forms. The simplest linear correlation function, (1 + 12 r12 ), is very distinct from the other three. We now have all the ingredients needed to construct an optimal wave function for He. First, with the introduction of explicit electron correlation function, there is no need for the radial correlation introduced by the coalescence wave function,


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i.e., we are free to abandon the radial correlation function cosh(B(r1 − r2 )). Second, the asymptotic form of the wave function exp(−αr) should be maintained. However, this asymptotic wave function, when extended back to small r, violates the cusp condition. These concerns can be simultaneously alleviated by replacing the second electron’s wave function via e−αr2 → e−Zr2 cosh(βr2 ). At small r2 , the cosh function is second order in r2 and therefore will not affect the cusp condition. At large r2 , cosh(βr2 ) → exp(βr2 ), and the choice β =Z−α

(3.66)

will give back the correct asymptotic wave function. Upon symmetrization, we finally arrived at the following compact wave function for He: ψ(r1 , r2 , r12 ) = e−Zr1 −Zr2 cosh(βr1 ) + cosh(βr2 ) f (r12 ). (3.67) This wave function, first derived by Le Sech [52], satisfies all the cusp and asymptotic conditions. We fixed α, β and C by (3.42), (3.66) and (3.57), respectively, and there are no free parameters. The only arbitrariness is the form the correlation function f (r12 ). Since fP (r12 ) is bracketed by fPJ (r12 ) and fLS (r12 ), we only need to consider the latter two cases. For Z = 2–10, the resulting ground state energy for the two electron atoms are given in Table III. 3.3.4. The One-Electron Homonuclear Wave Function The one-electron, homonuclear two-center Schrödinger equation 1 Z Z − ∇2 − − ψ(r) = Eψ(r), 2 ra rb

(3.68)

where ra = |r + R/2|, rb = |r − R/2|, can be solved exactly, as will be shown in Section 4.2 below. However, its ground state wave function can also be accurately prescribed by proximal and asymptotic conditions. For Z = 1, this is hydrogen molecular ion problem. As first pointed out by Guillemin and Zener (GZ) [53], when R = 0, the exact wave function is ψ(r) = e−2Zr

(3.69)

and when R → ∞, the exact wave function is ψ(r) = e−Zra + e−Zrb .

(3.70)

They, therefore, propose a wave function that can interpolate between the two, ψGZ (ra , ra ) = e−Z1 ra −Z2 rb + e−Z2 ra −Z1 rb .

(3.71)

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At R = 0, ra = rb = r, one can take Z1 = Z2 = Z. At R → ∞, one can choose Z1 = Z and Z2 = 0. At intermediate values of R, Z1 and Z2 can be determined variationally. This GZ wave function gives an excellent description [53] of the ground state of H+ 2 . As explained by Patil et al. [15], another reason why this is a good wave function is that (3.71) can satisfy both the cusp and the asymptotic condition. We can simplify Patil et al.’s discussion by rewriting the GZ wave function, again, in the form B(ra − rb ) , ψGZ (ra , rb ) = e−A(ra +rb )/2 cosh (3.72) 2 when R = 0, A = 2Z, and when R → ∞, A = B = Z. The imposition of the cusp condition can be done most easily in terms of the Riccati function. We, therefore, write ψGZ (ra , rb ) = e−S(ra ,rb ) ,

(3.73)

with

A(ra + rb ) B(ra − rb ) S(ra , rb ) = − ln cosh . 2 2 The cusp condition at ra = 0 is then easily computed,

BR ∂S

= Z, A + B tanh = 2Z. ∂r 2

(3.74)

a ra =0,rb =R

One can verify that this is also the cusp condition at rb = 0. From (3.74), one sees easily that at R = 0, A = 2Z, and when R → ∞, A + B = 2Z. At finite R, the asymptotic limit r → ∞ means that ra = rb = r and the GZ wave function approaches ψGZ (ra , rb ) → e−Ar . On the other hand, the exact wave function must be of the form (3.34) √

ψ(ra , rb ) → e− 2|E0 |r−β ln(r) , (3.75) √ √ with β = 1 − 2Z/ 2|E0√ |. Since 2Z ≥ 2|E0 | ≥ Z, we can estimate that at intermediate values of R, 2|E0 | ≈ 32 Z, suggesting a negligible β ≈ − 13 . Thus it is suffice to take A ≈ 2|E0 |. (3.76) Guillemin and Zener have allowed both A and B to be variational parameters. Patil et al.’s estimate [15] of A is essentially that of (3.76) but with slight improvement to incorporate the variation due to β = 0. We adhere to the cusp condition (3.74) but allow A to vary. In practice, it is easier to just let B vary and fix A via the cusp condition (3.74). In all cases, this wave function can provide an excellent description of the hydrogen molecular ion, with energy derivation only on the order of 10−3 Hartree over the range of R = 0 − 5.


129

3.3.5. The Two-Electron Homonuclear Wave Function The two-electron homonuclear Schrödinger equation is, 1 2 1 2 Z Z Z Z 1 − ∇1 − ∇2 − − − − + ψ(r1 , r2 ) 2 2 r1a r1b r2a r2b r12 = Eψ(r1 , r2 ).

(3.77)

Let’s denote the one-electron two-center GZ wave function as φ(r) = e−Aσ cosh(Bδ), where we have defined ra + rb ra − rb σ = , δ= . 2 2 (The variables σ and δ here will correspond, respectively, to λ and µ of the prolate spheroidal coordinates in Section 4.2.) To describe the two-electron wave function, if one were to follow the usual approach, one would begin by defining the Hartree–Fock-like wave function ψ(r1 , r2 ) = φ(r1 )φ(r2 ) = e−A(σ1 +σ2 ) cosh(Bδ1 ) cosh(Bδ2 ).

(3.78)

However, this molecular orbital approach is well known not to give the correct dissociation limit of H2 . In the limit of R → ∞, we know that φ(r) → e−Zra + e−Zrb

(3.79)

and therefore ψ(r1 , r2 ) → e−Zr1a + e−Zr1b e−Zr2a + e−Zr2b → e−Zr1a e−Zr2b + e−Zr1b e−Zr2a + e−Zr1a e−Zr2a + e−Zr1b e−Zr2b .

(3.80)

Only the first parenthesis, the Heilter–London wave function, gives the correct energy of two well separated atoms with energy E = 2(− 12 Z 2 ). The remaining parenthesis describes, in the case of H2 , the ionic configuration of H− H+ , which has higher energy than two separated neutral hydrogen atoms. Thus the molecular orbital approach (3.78) will always overshoot the correct dissociation limit. This is a fundamental shortcoming of the molecular orbital approach and cannot be cured by merely improving the one-electron wave function, i.e., by use of the exact one-electron, two-center wave function. Even the coalescent construction cannot overcome this fundamental problem. In the large R limit, the inner electron’s wave function must be (3.79), and hence no matter how one constructs the outer electron’s asymptotic wave function, one can never reproduces the Heilter– London wave function. In both the molecular orbital and the coalescent approach,

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one must resort to configuration interaction to achieve the correct dissociation limit. Even if one were to use the exact one-electron wave function in doing configuration interaction, as it was done by Siebbeles and Le Sech [14], the energy still overshoots the correct dissociation limit if the correlation (1 + 12 r12 ) is used. This is because we must have f (r12 ) → const in order to reproduce the Heilter– London limit. However, one can learn from Guillemin and Zener’s approach, and insist on a wave function that is correct in both the R = 0 and R → ∞ limit. This seemed a very stringent requirement, but surprisingly, it is possible. The wave function is ψ(r1 , r2 ) = e−A(σ1 +σ2 ) cosh B(δ1 − δ2 ) f (r12 ). (3.81) For R = 0, σ1 = r1 , σ2 = r2 , and the above function reduces to ψ(r1 , r2 ) = e−A(r1 +r2 ) f (r12 ). which is not a bad description of He. In the limit of R → ∞, if we take A = B = Z, we have ψ(r1 , r2 ) = e−Zr1a e−Zr2b + e−Zr1b e−Zr2a f (r12 ), = e−Zr1a e−Zr2b + e−Zr1b e−Zr2a , (3.82) since f (∞) → 1. Thus wave function (3.81) is the simplest homonuclear two electron wave function that can describe both limits adequately. The wave function (3.81) for H2 without f (r12 ) has been derived some time ago by Inui [54] and Nordsieck [55]. However, they were only interested in improving the wave function and energy at the equilibrium separation and were not concerned with whether the wave function can yield the correct dissociation limit. To estimate the form of the correlation function f (r12 ), we repeat our analysis as in the Helium case. The two-electron wave function can again be written in the Riccati form (3.51), but now with S(r1 , r2 , r12 ) = Zr1a + Zr2b + g(r12 ),

(3.83)

where we have assumed the unsymmetrized form of the Heilter–London wave function. The resulting equation for g is also similar to (3.53), 2g + 1 2 1 g + − g − Zg (ˆr1a − rˆ 2b ) · rˆ 12 − Z 2 + O = E0 . r12 R (3.84) In the Helium case, the dot product term vanishes in the limit of r12 → 0, here it does not. In the case where the two electrons meet along the molecular axis, rˆ 1a = zˆ , rˆ 2b = −ˆz, rˆ 12 = −ˆz, the resulting equation 2g + 1 2 1 2 g + (3.85) − g + 2Zg − Z + O = E0 r12 R


131

can be solved by setting E0 = −Z 2 , and expanding 1 1 2 + ···. g(r12 ) = − r12 + Cr12 2 2 In the limit of r12 → 0, (3.85) reads 3C −

1 − Z + O(r12 ) = 0, 4

(3.86)

(3.87)

giving Z 1 + . 3 12 This agrees with Patil et al.’s result [15] of C=

(3.88)

1 1 = (2Z − 1), 2 3 but without the need of consulting hypergeometric functions. For the H2 case, C = 5/12 = 0.42. In our calculation with wave function (3.81), with fPJ (r12 ) given by (3.61), the energy minimum at intermediate values of R is at C = 0.40, in excellent agreement with the predicted value. Since C ≈ 0.50 for Helium, C’s variation with R is very mild. The resulting energy for the wave function (3.81), is given in Fig. 8 (solid line). We vary the parameter B, while the other parameters A and C are fixed by Eqs. (3.74) and (3.88). The parameter B is 0.8 for R < 2, and moves graduately toward one at larger values of R. The energy at equilibrium is as good as Siebbeles and Le Sech’s calculation [14] with unscaled H+ 2 wave functions and correlation 1 function (1 + 2 r12 ) (triangles). Without configuration interaction, Siebbeles and Le Sech’s energy overshot the dissociation limit as shown. The wave function (3.81) can be further improved by adding a coalescence component à la Patil et al. [15]. This will be detailed in the next subsection. λ = 2C −

3.3.6. Construction of Trial Wave Functions by Patil and Coworkers The previous discussion demonstrates that relatively simple wave functions which incorporate the cusp conditions and the large distance asymptotics, having no or only a few variational parameters, can be constructed to yield fairly accurate results. Here we mention other similar trial wave functions studied in the literature. The importance of the local properties in the calculation of the chemical bond has been emphasized by Patil and coworkers [36,15,56,37–39]. Their analysis is in the spirit of our previous discussion, however, yields more complicated trial wave functions. Here we briefly mention the main aspects of the construction scheme and, to be specific, consider the ground state of H2 molecule described by the Schrödinger equation (3.77).

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F IG . 8. Ground state potential curve E(R) of H2 for different trial wave functions. Triangles correspond to “Le Sech” molecular orbital calculation with exact H+ 2 one-electron orbitals. Our wave function (3.81) (solid line) gives the correct dissociation limit. Patil et al.’s results are shown as small dot (functions (3.93), (3.96), (3.97)) and dash (function (3.107)) lines. Large dots are “exact” values of [8].

Let us assume that r2 r1 , R, 1. Then r12 ≈ r2a ≈ r2b ≈ r2 and the Hamiltonian reads 2 = − 1 ∇12 − Z − Z + Z − 1 ∇22 − (2Z − 1) . H 2 r1a r1b R 2 r2

(3.89)

The first four terms in (3.89) yield the H+ 2 problem, while the last two terms correspond to motion of a particle in a Coulomb potential with an effective charge 2Z − 1. This Hamiltonian allows us to separate variables and write (r1 , r2 ) = H+ (r1 )ϕ(r2 ), where the function ϕ(r2 ) satisfies the equation 2 1 (2Z − 1) − ∇22 − (3.90) ϕ(r2 ) = −εϕ(r2 ), 2 r2 ε = EH+ − E > 0 is the ionization energy of the H2 molecule, and E is the 2 ground state energy of H2 . As a result, the asymptotic behavior of (r1 , r2 ) at r2 r1 , R, 1, is √ √ (2Z−1)/ 2ε−1 (r1 , r2 ) ≈ r2 (3.91) exp − 2εr2 H+ (r1 ), 2


133

which is similar to the coalescence wave function (3.39) for He. Now assume that r2 r1 R, 1, then H+ (r1 ) is given by Eq. (3.75) and, therefore, 2

√ 2Z/ 2ε1 −1 (2Z−1)/ 2ε−1 r2 exp − 2ε1 r1

(r1 , r2 ) ≈ r1

√

−

√ 2εr2 ,

(3.92)

where ε1 = Z 2 /R − EH+ > 0 is the separation energy of electron in H+ 2 . The 2 power-law factor slowly varies as compared to the exponential decaying contribution. Hence, one can assume the power-law factor to be a constant or approximate the combination r a exp(−br) as (r − r0 )a a r exp(−br) = exp(−br + a ln r) ≈ exp −br + a ln r0 + , r0 where r0 can be determined as a variational parameter or chosen to be r0 = R + 1/b [15]. To incorporate the cusp conditions and the large distance asymptotic the trial wave function is separated into two parts (r1 , r2 ) = (r1 , r2 )fP (r12 ),

(3.93)

where fP (r12 ) is the Patil et al. electron–electron correlation function given by Eq. (3.62). Roothaan and Weiss [57] have made a very accurate numerical investigation of the desired correlation function for the ground state of the He atom. In the vicinity of r12 = 0, the correlation function is linear and satisfies the cusp condition. It monotonically increases and approaches a constant as r12 becomes very large. Clearly the function fP (r12 ) satisfies these conditions (see Fig. 12). In the united atom limit (R = 0) it was found that the energies computed with the variationally determined λ are essentially the same as given by the analytical expression, 5 1 (3.94) Z− 12 3 derived from a theory in which 1/r12 is treated as a perturbation [36]. In a molecular system, as R increases, one should expect λ to decrease monotonically and become vanishingly small for R → ∞. The small and large R behavior is satisfied provided [33] λ=

λ=

5Z/6 − 1/3 . 1 + 10Z 3 R 2 /(15Z − 6)

(3.95)

The electron–nucleus cusp conditions do not uniquely define the space wave function (r1 , r2 ). If one wishes to maintain the electronic configuration idea with an independent particle picture, one can adopt the following form of (r1 , r2 ): (r1 , r2 ) = φ(r1 )φ(r2 ),

(3.96)

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with φ(rj ), j = 1, 2, being the Guillemin–Zener [53] trial wave function for H+ 2 φ(rj ) = exp(−z1 rj a − z2 rj b ) + exp(−z2 rj a − z1 rj b ),

(3.97)

where z1 > 0, z2 > 0 are variational parameters. Alternatively, z1 and z2 can be determined from the cusp conditions at rj a = 0 and rj b = 0 for j = 1, 2. The wave function (3.96), (3.97) is identical to (3.78) which are known not to give the correct dissociation limit of H2 . However, for the pedagogical reason we briefly discuss it here. As r1a approaches zero (r1 , r2 ) → φ(r2 ) (1 − z1 r1a ) exp(−z2 R) + (1 − z2 r1a ) exp(−z1 R) . (3.98) Imposing the cusp condition (r1 , r2 ) → G(r2 )(1−Zr1a ) we obtain an equation for z1 and z2 : z1 = Z + (Z − z2 ) exp −(z1 − z2 )R . (3.99) Thus, if z1 and z2 are related as in Eq. (3.99), then the electron–nucleus cusp conditions are automatically satisfied. The second equation for z1 and z2 can be determined by the asymptotic condition. For r2 r1 R, 1 Eqs. (3.96), (3.97) yield (r1 , r2 ) ≈ exp −(z1 + z2 )r1 − (z1 + z2 )r2 . (3.100) From the other hand, according to Eq. (3.92), the wave √ must have the √ function following exponential behavior (r1 , r2 ) ∼ exp(− 2ε1 r1 − 2εr2 ). The two parameters z1 and z2 do not allow to match the asymptotic exactly. However, one can approximately choose [33] Z2 (3.101) − E. R Equations (3.99), (3.101) determine z1 and z2 self-consistently together with the ground state energy E. Kleinekathöfer et al. [33] used the trial function (3.93), (3.96), (3.97) with λ, z1 and z2 determined from Eqs. (3.95), (3.99), (3.101). The wave function has no free parameters and yields 4.661 eV for the binding energy of the H2 molecule which is very close to the exact value of 4.745 eV. However, E(R) becomes less accurate at large R and fails to describe the dissociation limit. The corresponding E(R) is shown as a small dot line (Patil et al. 1) in Fig. 8. Both the cusp conditions and the large distance asymptotic can be satisfied exactly provided more sophisticated trial functions are introduced. For example, for small and intermediate R, Patil et al. [15] suggested to use a combination of “inner” and “outer” molecular orbitals which are build from the Guillemin–Zener z1 + z2 = ε1 + ε =

ELECTRON MOLECULAR BONDS one-electron wave functions: m (r1 , r2 ) = φin (r1 )φout (r2 ) + φin (r2 )φout (r1 ) fP (r12 ),

135

(3.102)

where the “inner” orbital is φin (rj ) = exp(−z1 rj a − z2 rj b ) + exp(−z2 rj a − z1 rj b ).

(3.103)

Analogously, an “outer” orbital is defined as φout (rj ) = exp(−z3 rj a − z4 rj b ) + exp(−z4 rj a − z3 rj b ).

(3.104)

All the parameters z1 , z2 , z3 and z4 are determined by the cusp and asymptotic conditions. At large R, the atomic orbital wave function provides a better description of the two electron system. The appropriate wave function is [15] a (r1 , r2 ) = (r1 , r2 ) + (r2 , r1 ) fP (r12 ), (3.105) where

(r1 , r2 ) = exp −Z(r1a + r1b + r2a + r2b )

! × cosh(z5 r1b ) cosh(z6 r2a ) + cosh(z6 r1b ) cosh(z5 r2a ) . (3.106) Eq. (3.105) satisfies all the electron–nucleus cusp conditions. At the same time, it has two free parameters z5 and z6 which can be used to satisfy the two asymptotic conditions. For a description in the entire range of internuclear distances, one can use a linear combination of the two wave functions just discussed = m + Da ,

(3.107)

where D is a variational parameter. For H2 the molecular orbital m dominates in the region R < 1.7, while the atomic orbital a dominates at R > 1.7. With this complicated one parameter wave function Patil et al. [15] obtained 4.716 eV for the binding energy of H2 molecule and a very accurate potential curve in the entire range of R. The corresponding E(R) is shown as a dash line (Patil et al. 2) in Fig. 8. Similar wave functions which take full advantage of the asymptotic and proximal boundary conditions are useful in variational calculations of larger systems [38].

4. Analytical Wave Mechanical Solutions for One Electron Molecules From now on throughout the Sections 4, 5 and 6, unless otherwise noted, we assume the Born–Oppenheimer separation, where there are N nuclei, containing

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Zk protons located at R k , respectively, for k = 1, 2, . . . , N , and Ne electrons. Each electron’s coordinates are denoted as r j , j = 1, 2, . . . , Ne , where r j = (xj , yj , zj ). The steady-state equation, in atomic units, can be written as H ψ = Eψ, H =− +

e 1 ∇j2 + 2

N

j =1

1≤j 1 but λ ≈ 1. In such a limit we have (see Appendix C) (λ) ≈ (λ − 1)|m|/2

∞

ck (λ − 1)k .

(4.23)

k=0

Our results in (4.21) and (4.23) suggest that the form (λ) = e−pλ (λ − 1)|m|/2 λβ f (λ),

for some function f (λ),

(4.24)

would contain the right asymptotics for both λ 1 and λ ≈ 1. Here, obviously, f (λ) must satisfy

f (1) = 0, lim f (λ) ≤ C, for some constant C > 0. (4.25) λ→∞

Actually, in the literature [59–61], two improved or variant forms of the substitution of (4.24) are found to be most useful: (i) (Jaffé’s solution [60]) ∞ |m|/2 λ−1 n (λ) = e−pλ λ2 − 1 (λ + 1)σ gn , λ+1 n=0

σ ≡

R1 − |m| − 1. p

(4.26)

This leads to a 3-term recurrence relation αn gn−1 − βn gn + γn gn+1 = 0, where

n = 0, 1, 2, . . . ;

g−1 ≡ 0,

(4.27)


141

αn = (n − 1 − σ )(n − 1 − σ − m), βn = 2n2 + (4p − 2σ )n − A + p 2 − 2pσ − (m + 1)(m + σ ),

(4.28)

γn = (n + 1)(n + m + 1), and, consequently, the continued fraction β0 α1 = γ1 α2 γ0 β1 − α3 β2 − βγ32−···

(4.29)

for A and p. (ii) (Hylleraas’ solution [29]) (λ) = e−p(λ−1) (λ2 − 1)|m|/2

∞ n=0

cn Lm (x), (m + n)! m+n

x ≡ 2p(λ − 1),

(4.30) where Lm is the associated Laguerre polynomial and c satisfy the 3-term ren m+n currence relation αn cn−1 − βn cn + γn cn+1 = 0,

n = 0, 1, 2, . . . ;

c−1 ≡ 0,

(4.31)

where αn = (n − m)(n − m − 1 − σ ), βn = 2(n − m)2 + 2(n − m)(2p − σ ) − A − p 2 + 2pσ + (m + 1)(m + σ ) ,

(4.32)

γn = (n + 1)(n − 2m − σ ), and the same form of continued fractions (4.29). 4.2.2. Solution of the M-Equation (4.17) Equation (4.17) has close resemblance in form with (4.16) and, thus, it can almost be expected that the way to solve (4.16) will be similar to that of (4.16). First, we make the following substitution M(µ) = e±pµ M(µ),

−1 ≤ µ ≤ 1,

in order to eliminate the p2 µ2 term in (4.17). We obtain ± 2p 1 − µ2 M 1 − µ2 M 2 m2 = 0. + (−2R2 ∓ 2p)µ + p − A − M 1 − µ2

(4.33)

(4.34)

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To simplify notation, let us just consider the case M(µ) = e−pµ M(µ), but note pµ that for M = e M(µ), we need only make the changes of p → −p in (4.37) below. Write M(µ) = e−pµ

∞

m fk Pm+k (µ),

(4.35)

k=0

Pnm (µ)

where are the associated Legendre polynomials, and substitute (4.35) into (4.17). We obtain a 3-term recurrence relation αn fn−1 − βn fn + γn fn+1 = 0,

n = 0, 1, 2, . . . ;

f−1 ≡ 0,

(4.36)

where αn =

1 −2nR2 + 2pn(m + n) , 2(m + n) − 1

βn = A − p 2 + (m + n)(m + n + 1), ! 2m + n + 1 γn = −2R2 − 2p(m + n + 1) , 2(m + n) + 3

(4.37)

and, consequently, again the continued fractions of the same form as (4.29). The continued fractions obtained here should be coupled with the continued fraction (4.29) for the variable µ to solve A and p. In the homonuclear case, R2 = R(Za − Zb )/2 = 0, Eq. (4.17) reduces to m2 2 2 2 1 − µ M + −A + p µ − M = 0. 1 − µ2 In this case, several different optional representations of M can be used: (a)

∞ |m|/2 M(µ) = 1 − µ2 ck µ2k , k=0 ∞ 2 |m|/2 M(µ) = 1 − µ ck µ2k+1 ;

(4.38)

k=0

(b)

∞ |m|/2 m M(µ) = 1 − µ2 ck Pm+2k (µ),

|m|/2 M(µ) = 1 − µ2

k=0 ∞

(4.39) m ck Pm+2k+1 (µ);

k=0

(c)

∞ |m|/2 ck (1 ∓ µ)k . M(µ) = e±pµ 1 − µ2 k=0


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In Appendix D we discuss expansions of solution near λ ≈ 1 and λ 1 and their connection with the James–Coolidge trial wave functions. As a conclusion of this section, we note that the eigenstates of the hydrogen atom given in the preceding subsection can also be easily represented in terms of the prolate spheroidal coordinates. We let the nucleus of H (i.e., a proton) sit at location a where (0, 0, −R/2) with Za = 1 while at location b where (0, 0, R/2) we let Zb = 0. Thus, the hydrogen atom satisfies Eq. (4.13) in the form 1 2 1 Hψ = − ∇ − (4.40) ψ = Eψ. 2 ra Now, in terms of the prolate spheroidal coordinates (A.3) in Appendix A, and ψ(λ, µ, φ) = (λ)M(µ)(φ),

where (φ) = eimφ

(4.41)

in the form of separated variables, we have ∂ 1 ∂ 4 ∂ 2 2 ∂ − λ −1 M+ 1−µ M 2 R 2 (λ2 − µ2 ) ∂λ ∂λ ∂µ ∂µ

2 1 (λ2 − µ2 )m2 M − M = EM, − 2 2 Rλ−µ (λ − 1)(1 − µ ) which has two fewer terms than (A.6) does as now Zb = 0. Set p 2 = −R 2 E/2, we again have (4.16) and (4.17) except that now R1 = R2 = R/2 therein. The rest of the procedures follows in the same way with some minor adjustments as noted above. The above discussion also leads to a sequence of identities between (4.6) and (4.41), as R ψ (1) (x, y, z) = ψ (2) x, y, z + , 2 where ψ (1) (x, y, z) is an eigenstate of the hydrogen atom obtained from (4.2) but expressed in terms of the Cartesian coordinates (x, y, z) while ψ (2) (x, y, z) is that for the solution of (4.40).

4.3. T HE M ANY-C ENTERED , O NE -E LECTRON P ROBLEM When Ne = 1 and N ≥ 3 in (4.1), we have a molecular ion with three or more nuclei sharing one electron. A simple example is a CO2 -like structure, with N = 3. For such a problem, separable closed-form solutions are extremely difficult to come by from the traditional line of attack. However, we want to describe an elegant analysis by Shibuya and Wulfman [58] (see also the book by Judd [62]) which works in momentum space and expand electron’s eigenfunction as a linear combination of 4-dimensional spherical harmonics. This analysis may offer

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useful help to the modelling and computation of complex molecules after proper numerical realization. The model equation reads N 1 2 Zj − ∇ − ψ(r) = Eψ(r), r = (x, y, z) ∈ R3 , (4.42) 2 |r − R j | j =1

where Rj are positions of the nuclei. Appendix E shows how to reduce the problem to a matrix form. Here we provide the answer for the energy; it is determined from the solution of the eigenvalue equation √ P c = −2Ec, (4.43) where c is an infinite-dimensional vector, P is an infinite matrix with entries ∗ 1 nm Pnnm Snnmm (Rj ) Zj S (Rj ), m = n nm j

n m

n, n , n = 1, 2, 3, . . . ; , , = 0, 1, 2, . . . ; m, m , m = −, − + 1, . . . , −1, 0, 1, . . . , − 1, ;

(4.44)

the matrix S is given by an integral over 4-dimensional unit hypersphere S3 with the surface element d = sin2 χ sin θdχdθdφ, nm Sn m (Rj ) = exp(iRj · p)Ynm ()Yn m () d, (4.45) S3

Ynm () is a product of the spherical function Ym (θ, φ) and the associated Gegenbauer function Cn (χ) Ynm () = (−i) Cn (χ)Ym (θ, φ). The 3-dimensional vector p in Eq. (4.45) has components px = p sin θ cos φ, py = p sin θ sin φ, √ where p = −2E tan(χ/2).

pz = p cos θ,

In practice, the infinite matrix P in (4.44) is truncated to a finite size square matrix according to the quantum numbers (nm) for which the restriction n ≤ n0 is specified for some positive integer n0 . In the derivation, if we restrict N = 1, Z1 = 1 and set R 1 = 0, then the matrix P is diagonal and we recover the hydrogen atom as derived in Section 4.1. Obviously, if N = 2, by setting R 1 = (0, 0, −R/2) and R 2 = (0, 0, R/2), we should also be able to recover those H+ 2 -like solutions given in Section 4.2.


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The 1-electron one-centered or two-centered orbitals derived in Section 4.1 and 4.2 will be utilized frequently in the rest of the paper. At the present time, there is very limited knowledge about the 1-electron many-centered orbitals as discussed in Section 4.3 There seems to be abundant space for their exploitation in molecular modelling and computation in the future.

5. Two Electron Molecules: Cusp Conditions and Correlation Functions 5.1. T HE C USP C ONDITIONS In the study of any linear partial differential equations with singular coefficients, it is well known to the theorists that solutions will have important peculiar behavior at and near the locations of the singularities. We have first encountered such singularities in Section 3.3.1. Here we give singularities of the Coulomb type a more systematic treatment. The critical mathematical analysis was first made by Kato [63] in the form of cusp conditions for the Born–Oppenheimer separation. Consider the following slightly more general form of the Schrödinger equation for a 2-particle system − E ψ = 0, H (5.1) where = − 1 ∇12 − 1 ∇22 − Za − Zb − Za − Zb + q1 q2 + Za Zb . H 2m1 2m2 r1a r1b r2a r2b r12 R (5.2) The operator H has five sets of singularities, at r1a = 0,

r1b = 0,

r2a = 0,

r2b = 0

and r12 = 0.

(5.3)

It has been proved by Kato [63] that the wave function ψ is Hölder continuous, with bounded first order partial derivatives. However, these first order partial derivatives ∂ψ/∂xi , etc., i = 1, 2, . . . , 6, are discontinuous at (5.3). In the terminology of the mathematical theory of partial differential equations, (5.2) is said to have a nontrivial solution in the Sobolev space H 1 (R6 ). We now discuss the cusp conditions at these singularities. What is a cusp condition? It can be simply explained in the following paragraph. Let us elucidate it for the two particle Hamiltonian (5.2); for a multi-particle Hamiltonian the idea is the same. In order for the wave function ψ to satisfy the eigenvalue problem (5.1) at the singularities (5.3), the kinetic energy operators −∇12 /2m1 and −∇22 /2m2 , after acting on ψ, must produce terms that exactly cancel those singularity terms

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in the potential in order to give us back just a constant E times ψ, because the wave function ψ is bounded everywhere in space, including the points where the nuclei are located, without exception. One can see that, if the cusp conditions are not satisfied, then there is some unboundedness at the singularities (5.3) which can affect the accuracy in numerical computations. Conversely, if the cusp conditions are satisfied, this normally improves the numerical accuracy. In case we don’t know the exact eigenstate, but only a certain trial wave func − E)φ = 0 will not be satisfied, in general. Rather, we have tion, say φ, then (H − E φ(r 1 , r 2 ) = f (r 1 , r 2 ) H for some function f depending on the spatial variables r 1 and r 2 . However, we can insist on choosing parameters in φ such that the residual f (r 1 , r 2 ) is a bounded function everywhere; in particular, f (r 1 , r 2 ) cannot contain any singularity at (5.3). We say that the trial wave function φ satisfies (i) the electron–nucleus cusp condition at a (resp. b) if f is not singular at a (resp. b); (ii) the interelectronic (or electron–electron) cusp condition if f is not singular when r12 = 0. For example, in the simple case of a hydrogen atom, = − 1 ∇2 − 1 , H 2 r let φ(r) = Ce−αr be a trial wave function. Then for any E,

(α − 1)e−αr α 2 −αr H −E φ =C − E+ e . r 2 The singularity 1/r can be eliminated only by choosing α = 1. This is the cusp condition, which actually forces φ to be the ground state (with E = −1/2). The profile of φ, as shown in Fig. 10 illustrates the appearance of a cusp at the origin. In Appendix F we derive the cusp conditions for the two particle electron wave function ψ of (5.1):

∂ψ

∂ψ

= −m1 Za ψ(r1a = 0), = −m1 Zb ψ(r1b = 0), ∂r1a r1a =0 ∂r1b r1b =0 (5.4)

∂ψ

∂ψ

= −m2 Za ψ(r2a = 0), = −m2 Zb ψ(r2b = 0), ∂r ∂r 2a r2a =0

∂ψ

m1 m2 = q1 q2 ψ(r12 = 0).

∂r12 r12 =0 m1 + m2

2b r2b =0

(5.5) (5.6)


147

F IG . 10. A 1-dimensional cross section of φ(r) = e−r showing a cusp.

Equations (5.4), (5.5) are the electron–nucleus cusp condition, while Eq. (5.6) is the interelectronic condition. In forming trial wave functions from one-centered or two-centered orbitals for a homonuclear diatomic molecule a commonly used wave function is 1 f (r12 ) = 1 + r12 (cf. (1.17)) (5.7) 2 where φ(r i ), i = 1, 2, is an orbital for the molecular ion. In Appendix F we show that (5.7) satisfies the interelectronic cusp condition. If, however, φ1 = φ2 , then the trial wave function 1 ψ(r 1 , r 2 ) = φ1 (r 1 )φ2 (r 2 ) 1 + r12 2 ψ(r 1 , r 2 ) = φ(r 1 )φ(r 2 )f (r12 ),

satisfies the interelectronic cusp condition if and only if φ2 (r)∇φ1 (r) − φ1 (r)∇φ2 (r) = 0. The actual verifications of cusp conditions for specifically given examples of trial wave functions in the cases of one-centered orbitals or their products are not difficult. But such work is nontrivial when the trial wave functions are expressed in terms of prolate spheroidal coordinates. In Appendix H we illustrate through concrete examples how to carry out this task.

5.2. VARIOUS F ORMS OF THE C ORRELATION F UNCTION f (r12 ) We have learned the importance of the interelectronic cusp condition in Section 5.1. But there are, in addition, three important constructs that are crucial for diatomic calculations: orbitals, configurations and electronic correlation. In

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this section, we compile a list of often cited correlation functions f which help the satisfaction of the interelectronic cusp conditions in the context of Eq. (5.6). The study of the other two, i.e., orbitals and configurations, will be addressed in the next section. 5.2.1. f (r12 ) = 1 + 12 r12 This is the simplest possible interelectronic configuration function. The specific form is due to the correlation cusp condition only. We derive it as follows (see Patil et al. [15]). Consider two charged particles which are described by the Schrödinger equation 1 2 1 2 q1 q2 − (5.8) ∇ − ∇ + ψ = ψ. 2m1 1 2m2 2 r12 First, transform the above equation to the center-of-mass coordinates (cf. Appendix G) 1 2 q1 q2 1 2 − (5.9) ∇ − ∇ + ψ = ψ, 2M S 2µ r12 r12 where M = m1 + m2 , m1 r 1 + m2 r 2 , S= m1 + m2

µ=

m1 m2 , m1 + m2

(5.10)

r 12 = r 1 − r 2 .

In the ground state ψ is independent of S. Near the singularity point r12 = 0, the wave function ψ has a local representation as a power series 2 ψ = C0 + C1 r12 + O r12 (5.11) . ∂ ∂ Substituting (5.11) into (5.9) and using ∇r2 = r12 ∂r (r 2 ∂r ) we obtain C1 3C2 1 + q1 q 2 C 0 + − + C1 q 1 q 2 + · · · − r12 µ µ 2 = C0 + C1 r12 + O r12 .

The above mandates that the coefficient of 1/r12 must vanish, that is C1 = µq1 q2 C0 , which for µ = 1/2 (i.e., m1 = m2 ), and q1 = q2 = 1, yields C1 =

1 C0 . 2

(5.12)

ELECTRON MOLECULAR BONDS This gives us the small r12 behavior 1 ψ = C0 1 + r12 , 2

149

(5.13)

2 ) terms. The asymptotic expression (5.13) where we have dropped all the O(r12 motivates the choice of f (r12 ) in such particular form. This simple f (r12 ) captures short distance interelectronic interaction very well. It offers elegant representations of molecular orbitals and great facility to computation. Nevertheless, its asymptotic behavior of linear growth for large r12 is not physically correct. When we write molecular orbitals as

ψ(r 1 , r 2 , r12 ) = φ(r 1 , r 2 )f (r12 ), if the function φ(r 1 , r 2 ) is already quite small in the region where r12 becomes large compared to 1, then this simple f (r12 ) = 1 + 12 r12 can work quite well (Kleinekathöfer et al. [36, pp. 2841–2842]). 5.2.2. f (r12 ) = 1 +

r12 −r12 /d 2 e

(d > 0)

This function was proposed by Hirschfelder [64] where d is a variational parameter. Its profiles is shown in Fig. 11. It satisfies the cusp condition near r12 = 0. This function was used by Siebbles et al. [65]. Nevertheless, Le Sech et al. [65,7] reported that in performing variational calculations by writing f (r12 ) = 1 + r212 e−αr12 , they found that α is computed to be very close to 0 for small and intermediate R. That is, it virtually degenerates into f (r12 ) = 1 + 12 r12 .

F IG . 11. Graph of f (r12 ) = 1 + (r12 /2)e−r12 /d , where the maximum happens at r12 = d.

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However, at large R one should use α = 0 in order to obtain the correct dissociation limit. From physical considerations, there is no reason to believe why f (r12 ) should have a local maximum as shown in Fig. 11. Even though the choice of this f (r12 ) seems satisfactory asymptotically for both r12 small and large, it may not be satisfactory for medium values of r12 . 5.2.3. f (r12 ) = 1 −

1 −λr12 1+2λ e

(λ > 0)

This correlation function is partly motivated by Hirschfelder’s work [64], and partly by Hylleraas study of the helium atom [29]. It was introduced by Kleinekathöfer et al. [36]. At small r12 we have the expansion f (r12 ) =

2λ λ 2 1 ± ··· . 1 + r12 − r12 1 + 2λ 2 4

Therefore the cusp condition is satisfied for any λ > 0. This λ can be either used as a variational parameter, or be determined from a given Hamiltonian. For example, for a helium-like 2-electron atom with nuclear charge Z, using a perturbation argument, Kleinekathöfer et al. [36] analyzed that the best value for λ is λ=

5 1 Z− . 12 3

This f (r12 ) has a monotone profile and correct asymptotics for both r12 small and large. See Fig. 12.

F IG . 12. Graph of f (r12 ) = 1 − 1/(1 + 2λ)e−λr12 .


151

5.2.4. f (r12 ) = e(1/2)r12 This f (r12 ) satisfies the correlation-wave equation 1 2 1 2 1 1 − ∇1 − ∇2 + e(1/2)r12 = − e(1/2)r12 2 2 r12 4 with a negative energy −1/4. For small r12 , its expansion is 2 1 . e(1/2)r12 = 1 + r12 + O r12 2 Therefore, it satisfies the correlation-cusp condition and its asymptotics for small r12 is good. However, this f (r12 ) has exponential growth for large r12 and is thus physically incorrect. 5.2.5. f (r, r12 ) =

sinh(tr) tr

·

F0 (1/(2k),kr12 ) r12

Where Fj (η, ρ) is the Coulomb wave function regular at the origin, j is an integer, t, k are separation of variables constants and r = |r1 + r2 |. This is perhaps the most complex form of the correlation function in the literature, given by Aubert-Frécon and Le Sech [13]. It comes from solving

1 2 1 2 1 − ∇1 − ∇2 + (5.14) f = f. 2 2 r12 Rewrite the above in center-of-mass coordinates: 1 1 f = f. − ∇S2 − ∇r212 + 4 r12

(5.15)

Assume that f depends only on r, r12 and γ , where γ is the angle between S and r 12 (see Fig. 13).

F IG . 13. The variables r, r12 and angle γ in the center-of-mass coordinates.

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Then Eq. (5.15) can be written as ∂ 1 ∂ ∂f 1 ∂ 1 ∂ 2 ∂f − 2 r2 − 2 1 − q2 f r12 − 2 ∂r12 ∂r ∂q r ∂r r12 ∂r12 r12 ∂q ∂f 1 1 ∂ f = f, q ≡ cos γ . 1 − q2 + − 2 (5.16) ∂q r12 r ∂q Equation (5.16) can be separated by writing f = Pj (q)g j (r)uj (r12 ), where ∂Pj ∂ 1 − q2 = −j (j + 1)Pj , ∂q ∂q

(5.17)

Pj is the Legendre polynomial of degree j and g j (r) and uj (r12 ) satisfy, respectively, 2

d 2 d j (j + 1) (5.18) + + g g j (r) = 0, − r dr dr 2 r2

2 2 d j (j + 1) 1 d + − − + uj (r12 ) = 0, (5.19) u 2 2 r12 dr12 r12 dr12 r12 with = g + u . Take g = −t 2 ,

u = k 2 ,

j = 0.

(5.20)

Then P0 (q) = 1,

gt0 (r) =

sinh(tr) , tr

u0k =

F0 (1/(2k), kr12 ) , r12

where the function F0 (η, ρ) belongs to the class of regular Coulomb wave function FL (η, ρ), satisfying d2 2η L(L + 1) FL (η, ρ) + 1 − − FL (η, ρ) = 0 ρ dρ 2 ρ2 for the two parameters η and L. Note that t and k in (5.20) can be used as variational parameters. For small r12 , 1 1 2 , kr12 = C · kr12 1 + r12 + O (kr12 ) F0 2k 2

ELECTRON MOLECULAR BONDS and, thus, u0k (r12 )

153

F0 (1/(2k), kr12 ) 1 2 = = C · k 1 + r12 + O (kr12 ) r12 2

satisfies the correlation-cusp condition for any given k. For large r12 , the Coulomb wave function FL (η, ρ) has an asymptotic expansion FL = g cos θL + f sin θL , π θL ≡ ρ − η ln zρ − L + σL , 2 σL ≡ arg (L + 1 + iη), ∞ ∞ f ∼ fk , g∼ gk , k=0

f0 = 1,

k=0

g0 = 1,

fk+1 = ak fk − bk gk , (2k + 1)η , ak = (2k + 2)ρ

gk+1 = ak gk + bk fk , L(L + 1) − k(k + 1) + η2 bk = . (2k + 2)ρ

Thus lim u0 (r12 ) r12 →∞ k

= lim

r12 →∞

F0 (1/(2k), kr12 ) = 0. r12

1 , 2, −2Zr12 5.2.6. f (r12 ) = 1F1 − 2Z Here 1F1 (a; b; x) is the confluent hypergeometric function satisfying the differential equation dw d 2w + (b − x) − aw = 0. (5.21) dx dx 2 This f (r12 ) was given by Patil et al. [15] for the case when the internuclear separation R is large: x

3 . Z Its derivation can be motivated as follows. Consider Z Z Z Z Z2 1 1 2 1 2 + + + + + H = − ∇1 − ∇2 − 2 2 r1a r1b r2a r2b r12 R R

(5.22)

(5.23)

for an H2 -like molecule. Guillemin–Zener-type one-electron wave functions [53] suggest the molecular orbital for large R: = e−Zr1a −Zr2b + e−Zr1b −Zr2a f (r12 ). (5.24)

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F IG . 14. For large R, electron e1 is localized near A, and electron e2 is localized near B.

When R is large, electron 1 is localized around A and electron 2 is localized around B, as shown in Fig. 14. We have r1b r1a ,

r2a r2b .

(5.25)

Because of (5.25), we have, from (5.24), = e−Zr1a −Zr2b + e−Zr1b −Zr2a f (r12 ) ≈ e−Zr1a −Zr2b f (r12 ).

(5.26)

ψ = Eψ, we obtain Substituting (5.26) into H

1 2 1 1 2 − ∇1 + ∇2 f + Z(∇1 f ) · (ˆr 1a + rˆ 2b ) + f =O . 2 r12 R

(5.27)

Note that Eq. (5.27) contains the effect of cross terms. For large R, the electron– electron correlation is most significant when the two electrons are collinear and in between the two nuclei. In this situation, either r12 is antiparallel to r 1 − r a , or r12 is parallel to r 2 − r b . Now, using the center-of-mass coordinates and dropping all the O(1/R) terms, we obtain 2 ∂ 2 ∂ ∂f 1 − (5.28) + f − 2Z = 0. f+ 2 r ∂r r ∂r ∂r12 12 12 12 12 The solution to (5.28), after setting it into the form of (5.21) is 1 f (r12 ) = 1F1 − , 2, −2Zr12 . 2Z For small r12 , the expansion is 1 (2Z − 1) 2 r12 + · · · . f (r12 ) = 1 + r12 − 2 12


155

Therefore, the correlation-cusp condition is satisfied. For large r12 , the asymptotics is f (r12 ) ∼ C(−2Zr12 )1/(2Z) (cf. [66, p. 508, 13.5.1]).

6. Modelling of Diatomic Molecules In this section, we give a survey of major existing methods for the numerical modelling of diatomic molecules. These methods provide approximations of wave functions either in explicit form through properly selected ansatzs, or in implicit form through iterations as numerical solutions of integro-partial differential equations. The methods and ansatzs to be described below are (1) (2) (3) (4) (5)

The Heitler–London method; The Hund–Mulliken method; The Hartree–Fock (self-consistent) method; The James–Coolidge wave function; Two-centered orbitals.

Items (1)–(3) above historically are associated with one-centered orbitals, while (4), (5) are based on two-centered orbitals. But this dichotomy is not inflexible. An example is a hybrid type containing both one-centered and two-centered orbitals considered in Section 3.3.6. We now discuss them in sequential order below. Each approach has a set of modelling parameters which can be optimized through calculus of variations. In particular, we will point out what these parameters are.

6.1. T HE H EITLER –L ONDON M ETHOD This method has the longest history. It was developed by Heitler and London during the 1920s soon after Heisenberg laid the quantum mechanical foundation of ferromagnetism. The method is usually called the valence-bond (or atomic orbital) method. In this method, each molecule is thought of as composed of atoms, and the electronic structure is described using atomic orbitals of these atoms. Here we present a version of refined Heitler–London approach due to Slater [67]. In the method, electron spin-orbitals are taken from a determinant

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G. Chen et al. Table IV Spin-orbital wave functions of singlet and triplet states. Ms (total spin)

Spin-orbital wave functions Singlet state

[a(1)b(2) + b(1)a(2)][α(1)β(2) − β(1)α(2)]

0

Triplet states

[a(1)b(2) − b(1)a(2)][α(1)α(2)] [a(1)b(2) − b(1)a(2)][α(1)β(2) + β(1)α(2)] [a(1)b(2) − b(1)a(2)][β(1)β(2)]

1 0 −1

u1 (1)

u2 (1)

.

.

.

un (1)

u1 (2) u2 (2) .. .

... ... .. .

un (2) . . .

u1 (n)

u2 (n) ..

, .

un (n)

(6.1)

which satisfy the Fermionic property of the Pauli exclusion principle. The orbitals in (6.1) are called the Slater orbitals. Let us consider the 2-electron case, i.e., n = 2 in (6.1). The spin-orbital ui (j ), i, j = 1, 2, consists of (i) the electron-atomic orbital part α 3 −αr1a α 3 −αr2a , a(2) = , e e a(1) = π π α 3 −αr1b α 3 −αr2b b(1) = , b(2) = , e e π π

(6.2)

(6.3)

where in (6.2) and (6.3), the atomic electron wave functions are centered at, respectively, a and b; (ii) the spin part spin α(1), α(2), α(j ) = |s, ms ↑,

for j = 1, 2,

spin β(1), β(2), β(j ) = |s, ms ↓,

for j = 1, 2.

(6.4)

The linear combinations of the total spin-orbital wave function that are antisymmetric are tabulated in Table IV. The singlet state has lower energy than the triplets. For example, if we aim to calculate the ground state of H2 , we use Table IV as the trial wave function to minimize the total energy , subject to | = 1, H (6.5) where = − 1 ∇12 − 1 ∇22 − 1 − 1 − 1 − 1 + 1 + 1 , H 2 2 r1a r r2a r2b r12 R 1b = N a(1)b(2) + b(1)a(2) , N = a normalization factor.

(6.6) (6.7)


157

in (6.6) can Heisenberg, and Heitler–London’s basic point of view is that H be written as = − 1 ∇12 − 1 + − 1 ∇22 − 1 H 2 r1a 2 r2b 1 1 1 1 + − − + + , r1b r2a r12 R where the terms inside the third pair of parentheses above can be viewed as a “perturbation”. Since the Heitler–London method is quite fundamental in molecular chemistry, let us give some details about the calculation of (6.5) given (6.6) and (6.7). Define

S ≡ the overlap ≡ a(1) b(1) α3 = e−αr1a e−αr1b dx · (dx = dx1 dx2 dx3 ) π R3

(αR)2 = e−αR 1 + αR + . 3 Then a(1)b(2)|a(2)b(1) = a(1)|b(1)b(2)|a(2) = S 2 , and the normalized state for the singlet or the triplets, without spin, is =

a(1)b(2) ± a(2)b(1) , 2(1 ± S 2 )

such that | = 1.

The total energy of the singlet and triplet states can now be written as E± =

1 1 1 a(1)b(2) ± a(2)b(1) − ∇12 − ∇22 2 2 2 2(1 ± S ) 1 1 1 1 1 1

− − − − + + a(1)b(2) ± a(2)b(1) . (6.8) r1a r1b r2a r2b r12 R

The integrals involved are given in Appendix I. Using these integrals, we are able to write down the total energy E = KE + P E as follows: KE± = kinetic energy

1 1 = a(1)b(2) ± a(2)b(1) − ∇12 2 2 2(1 ± S ) 1 2

− ∇2 a(1)b(2) ± a(2)b(1) 2 α2 = 1 ∓ 2KS ∓ S 2 ; 1 ± S2

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P E± = potential energy

1 1 1 = − a(1)b(2) ± a(2)b(1) − 2 r1a r1b 2(1 ± S ) 1 1 1 1

− − + + a(1)b(2) ± a(2)b(1) r2a r2b r12 R α α = −2 + 2J + J ± 4KS ± K + , 2 w (1 ± S ) where w = αR, the other symbols are defined in Appendix I. The parameter α can be used as the variational parameter to minimize (6.5) [68]. Consider the special case when α = 1. Then E± = −1 + where

H0 = R6

H0 ± H1 , (1 ± S 2 )

1 1 1 1 a 2 (1)b2 (1) − − + + dx dy r1b r2a r12 R

1 R is the Coulomb integral, and 1 1 1 1 H1 = a(1)b(1)a(2)b(2) − − + + dx dy r1b r1a r12 R = 2J + J +

R6

S2 R is the exchange integral. According to numerical values computed in Slater [67, Table 3.2], e.g., it is known that H1 is usually many times larger than H0 , and is largely responsible for the attraction between atoms in forming a molecule. In Fig. 15 we plot the ground 1 g+ and first excited 3 u+ state potential energy curves E(R) of the H2 molecule. When α = 1 the ground state curve yields the binding energy of 0.116 a.u. = 3.16 eV; the value must be compared with 4.748 eV obtained by Kolos and Roothaan [8]. When α (effective charge) is treated as a variational parameter [69] the calculation yields the binding energy of 0.139 a.u. = 3.78 eV and the bond length of 1.41 Bohr radii. For the 3 u+ state the effective charge and the α = 1 curves are practically indistinguishable. = 2KS + K +

6.2. T HE H UND –M ULLIKEN M ETHOD In 1927, Robert Mulliken worked with Friedrich Hund and developed the Hund– Mulliken molecular orbital theory in which electrons are assigned to states over


159

F IG . 15. Ground and first excited state energy E(R) of H2 molecule for the Heitler–London wave function (solid lines) and the “exact” energy of [8] (dots).

an entire molecule. Hund–Mulliken’s molecular orbital method was more flexible and applicable than the traditional Valence-Bond theory that had previously prevailed. Because of this, Mulliken received the Nobel Prize in Chemistry in 1966. The approach has some similarity to the Heitler–London’s, so we can inherit the notation from there. Its special feature is that a linear combination of the molecular gerade (g) and ungerade (u) states are used: 1 + g : 1 − g :

g+g− u+ u−

u : g + u− − g − u+

1

3 + u : g + u+

u : g + u− + g − u+

3

3 − u :

g − u−

a(1) + b(1) a(2) + b(2) ↑1 ↓2 − ↓1 ↑2 , √ √ √ 2(1 + S) 2(1 + S) 2 a(1) − b(1) a(2) − b(2) ↑1 ↓2 − ↓1 ↑2 , √ √ √ 2(1 − S) 2(1 − S) 2 a(1)a(2) − b(1)b(2) ↑1 ↓2 − ↓1 ↑2 , √ 2 2(1 − S 2 )

(6.9) a(1)b(2) − b(1)a(2) ↑1 ↑2 (Ms = 1), 2(1 − S 2 ) a(1)b(2) − b(1)a(2) ↑1 ↓2 + ↓1 ↑2 (Ms = 0), √ 2 2(1 − S 2 ) a(1)b(2) − b(1)a(2) ↓1 ↓2 2(1 − S 2 )

(Ms = −1),

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with

α 3 −αria a(i) = , e π a(i) + b(i) g: √ 2(1 + S)

α 3 −αrib , e π a(i) − b(i) , √ 2(1 − S)

b(i) =

i = 1, 2,

u:

i = 1, 2.

Especially, note that the first three states in (6.9), i.e., 1 g+ , 1 g− and 1 g− signify the possibility of double occupancy of the two electrons at a single nucleus as products of a(1)a(2) and b(1)b(2) appear in the wave functions. Such states, chemically, represent ionic bonds. On the other hand, the last three states in (6.9) i.e., 3 u+ , 3 u0 , 3 u− , agree with the triplet states in Table IV. We denote the six molecular orbitals in (6.9) in sequential order as 1 , 2 , . . . , 6 . Then the energy of any linear combination 6j =1 cj j corresponds to a quadratic form:

6 6 6

cj j H ck k = Hj k c¯j ck ,

j =1

k=1

j,k=1

where Hj k are the (j, k)-entry of the following symmetric matrix  H11 H12   H12 H22      H1 u H= ,   H3 u   H3 u (6.10) H3

1

1 u 1 1

H22 = g − H g− , H11 = g + H g+ , 1 1 3 j 3 j H1 u = u H u , H3 u = u H u , j = +, 0, −. Specifically,

1 ∓ S ∓ 2K H11 , H22 = α 1±S −2 + 2J ± 4K 1 5/8 + J + 2K ± 4L +α + + , 1±S w 2(1 ± S)2

1 5/8 − J H12 = g + H g− = α , 2(1 − S 2 ) 2 1 5/4 − 2K −2 + 2J − 4KS 2 1 + 2KS + S + + , +α H1 u = α w 1 − S2 1 − S2 2(1 − S 2 ) 1 1 + 2KS + S 2 −2 + 2J + J − 4KS − K + , + α H3 u = α 2 w 1 − S2 (1 − S 2 )

2


161

F IG . 16. Ground and first excited state E(R) of H2 molecule for the Hund–Mulliken wave function (solid lines) and the “exact” energy of [8] (dots).

where

1 1 a 2 (1)a(2)b(2) dx dy L= α r12 5 5 1 −w −2w 1 −e + . w+ + =e 8 16w 8 16w

The other symbols are defined in Appendix I. In the calculation of the ground state energy, the sub-block of 2 × 2 matrix in the upper left corner of (6.10) plays the exclusive role as the remaining diagonal 4 × 4 block in (6.10) contributes no effect. We thus determine the ground state energy E by the determinant

H11 − E H12

= 0,

H12 H22 − E ) 1( 2 . H11 + H22 ± (H11 − H22 )2 + 4H12 E± = 2 The value E− will correspond to the ground state energy. For the Hund–Mulliken method discussed above, again α is the variational parameter. In Fig. 16 we plot the ground 1 g+ and first excited 3 u+ state potential energy curves E(R) of the H2 molecule. When α = 1 the ground state curve yields the binding energy of 0.119 a.u. = 3.23 eV. The effective charge calculation yields the binding energy of 0.148 a.u. = 4.03 eV and the bond length of 1.43 Bohr radii. The 3 u+ curve is identical to the Heitler–London E(R) (see Fig. 15).

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6.3. T HE H ARTREE –F OCK S ELF -C ONSISTENT M ETHOD This is perhaps the best known method in molecular quantum chemistry and it works for multi-electron and multi-center cases. In computational physics, the Hartree–Fock calculation scheme is a self-consistent iterative procedure to calculate the optimal single-particle determinant solution to the time-independent Schrödinger equation. As a consequence to this, while it calculates the exchange energy exactly, it does not calculate the effect of electron correlation at all. The name is for Douglas Hartree, who devised the self consistent field method, and Vladimir Fock who reformulated it into the matrix form used today and introduced the exchange energy. The starting point for the Hartree–Fock method is a set of approximate orbitals. For an atomic calculation, these are typically hydrogenic orbitals. For a molecular calculation, the initial approximate wave functions are typically a linear combination of atomic orbitals. This gives a collections of one electron orbitals, which due to the Fermionic nature of electrons must be antisymmetric; the antisymmetry is achieved through the use of a Slater determinant. Once an initial wave function is constructed, an electron is selected. The effect of all the other electrons is summed up, and used to generate a potential. This is why the procedure is sometimes called a mean-field procedure. This gives a single electron in a defined potential, for which the Schrödinger equation can be solved, giving a slightly different wave function for that electron. This process is then repeated for all the other electrons, which complete one iteration of the procedure. The whole procedure is then repeated until the self-consistent solution is obtained. Here we discuss the method in detail. Consider the Hamiltonian of a multielectron and multi-center molecular system (with n electrons and N nuclei, respectively) under the Born–Oppenheimer separation in the following form = −1 H 2

n j =1

∇j2

+

1≤j 0, the coefficients α, and Cmnj kp for finitely many indices m, n, j, k and p, can be used as variational parameters to minimize the energy ψ|H |ψ, subject to the normalization condition ψ|ψ = 1.

(6.19)

For example, for fixed α and R in (6.17) and (6.18), introduce a Lagrange multiplier λ for the constraint (6.19) and choose only a total of s terms in (6.17) by truncation, then the variational problem ψ + λ ψ|ψ − 1 min ψ H (6.20) ψ

leads to a set of s linear equations [19, p. 826] for the coefficients Cj , j = 1, 2, . . . , s,  (H11 − λS11 )C1 + (H12 − λS12 )C2 + · · · + (H1s − λS1s )Cs = 0,    (H12 − λS12 )C1 + (H22 − λS22 )C2 + · · · + (H2s − λS2s )C2 = 0, ..    . (H1s − λS1s )C1 + (H2s − λS2s )C2 + · · · + (Hss − λSss )Cs = 0 (6.21) where φj , Hij = φi H Sij = φi |φj , φi is a typical summand term in (6.17) without the coefficient Cj , for i, j = 1, 2, . . . , s. Solving (6.21) then leads to the ground state ψ0 and its energy E0 . The trial wave function (6.17) may be further adapted to ψ=

1 −α1 λ1 −α2 λ2 −β1 µ1 −β2 µ2 e e 2π ∞ p n j k n m k j × Cmnj kp λm 1 λ 2 µ 1 µ 2 ± λ1 λ 2 µ 1 µ 2 ρ m,n,j,k,p=1

for the calculation of heteronuclear cases and excited states.

(6.22)

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Next, we address the analytic treatment and provide a complete compendium for the integrals given in (6.17)–(6.22), which constitutes the keystone in the twocentered variational treatment. The model Hamiltonian that we will be using here is the (homonuclear case) H2 as given in (6.6). Let us rewrite the James–Coolidge wave function as (1, 2) = (6.23) Cr r (1, 2), r

where 1 −α(λ1 +λ2 ) mr nr jr kr r (6.24) λ1 λ2 µ1 µ2 r12 . e 2π We remark that for the heteronuclear case and for excited states, the trial wave function (6.23), (6.24) can be easily re-adjusted to the form (6.22) with relative ease. In the evaluation of the energy for given two-electron trial wave functions, a typical term is of the form 1 n j k · · · e−α(λ1 +λ2 ) λm Z ν (m, n, j, k; ) = 1 λ2 µ1 µ2 r12 4π 2 × M ν cosν (φ1 − φ2 ) dλ1 dλ2 dµ1 dµ2 dφ1 dφ2 , (6.25) where 1/2 , M = λ21 − 1 λ22 − 1 1 − µ21 1 − µ22 (6.26) r (1, 2) =

and ν, m, n, j , k and are integers ( ≥ −1 and others are nonnegative integers). Using the above function, we can write every integration in terms of Z ν (m, n, j, k; ). For example, the Coulomb interaction energy between the nuclei and electrons may be easily written as follows: 1 ra1 1 rb1 1 ra2 1 rb2

2 , R(λ1 + µ1 ) 2 = , R(λ1 − µ1 ) 2 = , R(λ2 + µ2 ) 2 = . R(λ2 − µ2 ) =

(6.27) (6.28) (6.29) (6.30)

The evaluation of the Coulomb interactions can be done in terms of Z. The evaluation of the Laplacian matrix element, however, somewhat more involved. But it also can be expressed via Z ν . We provide details of the derivation in Appendix J.


169

Since we can write every term in the energy in terms of Z ν , let us express Z ν via simpler functions defined by recurrence relations. Such relations are given in Appendix K. For ≥ 1 and ν = 0, one can reduce to 0 or −1 using the following identity: R2 2 λ1 + λ22 + µ21 + µ22 − 2 − 2λ1 λ2 µ1 µ2 − 2M cos(φ1 − φ2 ) . 4 (6.31) For ν = 0 and = 0, the integration can be evaluated as follows. Given 2 r12 =

Z 0 (m, n, j, k; 0) ≡ Z(m, n, j, k; 0) 1 n . . . e−2α(λ1 +λ2 ) λm = 1 λ2 4π 2 j

× µ1 µk2 dλ1 dλ2 dµ1 dµ2 dφ1 dφ2 ,

(6.32)

note that the φ integrals are trivial to evaluate and the µ integrals survive only for even integers j and k, i.e., +1 µj dµ = −1

2 , j +1

we arrive at Z(m, n, j, k; 0) =

for even j ;

  A(m; α)A(n; α) 

0,

(6.33)

4 , (j + 1)(k + 1)

for even j, k, for odd j or k, (6.34)

where we have introduced the function ∞ A(m; α) =

e−αλ λm dλ,

(6.35)

1

for the λ integration. Using integration by parts one can show that the A(m; α) satisfies the recursion relation 1 −α e + mA(m − 1; α) , A(m; α) = (6.36) α with e−α . (6.37) α Thus we have given a recipe for the evaluation of Z(m, n, j, k; 0) for arbitrary values of the power parameters m, n, j , and k. A(0; α) =

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Next, consider the case where = −1, i.e., Z(m, n, j, k; −1) 1 n j k 1 dλ1 dλ2 dµ1 dµ2 dφ1 dφ2 . . . . e−2α(λ1 +λ2 ) λm = 1 λ2 µ1 µ2 2 r12 4π (6.38) Recalling the Neumann expansion for 1/r12 : ∞ τ 1 2 (τ − |ν|)! 2 ν = (−1) (2τ + 1) r12 R (τ + |ν|)! ν=−τ τ =0

× Pτν (λ< )Qντ (λ> )Pτν (µ1 )Pτν (µ2 )eiν(φ1 −φ2 ) ,

(6.39)

where λ< = min(λ1 , λ2 ), λ> = max(λ1 , λ2 ) and Pτν , Qντ are the associated Legendre functions of the 1st and 2nd kind, respectively. After the angular integration, only terms corresponding to ν = 0 survive, as long as the wave function has no angular or r12 dependence. Separating the λ and µ integrals, we arrive at Z(m, n, j, k; −1) =

∞ 2 (2τ + 1)Rτ (j )Rτ (k)Hτ (m, n; α), R

(6.40)

τ =0

where Rτ and Hτ are defined by 1 Rτ (j ) ≡

µj Pτ (µ) dµ, −1 ∞ ∞

Hτ (m, n; α) ≡ 1

n e−α(λ1 +λ2 ) λm 1 λ2 Pτ (λ< )Qτ (λ> ) dλ1 dλ2 .

(6.41)

(6.42)

1

In the discussion to follow we give recursion relations for the evaluation of the various auxiliary functions. For τ = 0, H0 (m, n; α) = A(m; α)F (n; α) + A(n; α)F (m; α) − T (m, n; α) − T (n, m; α).

(6.43)

Here, F (m; α) can be evaluated for arbitrary m by noting its recursion relation ∞ F (m; α) =

e−αλ λm Q0 (λ) dλ

1

1 mF (m − 1; α) α − (m − 2)F (m − 3; α) − A(m − 2; α)

= F (m − 2; α) +

ELECTRON MOLECULAR BONDS and the initial values 1 e−α eα F (0; α) = (ln 2α + γ ) − Ei [−2α] , 2 α α

171

(6.44)

and

1 1 1 1 −α 1 α F (1; α) = (ln 2α + γ )e + 2 − Ei [−2α]e − + 2 , 2 α α α α (6.45) where ∞ Ei(−x) = −

e−t dt t

(6.46)

x

and γ = 0.577216 . . . is the Euler constant; see Appendix I. Similarly the quantity T (m, n; α) can be determined for arbitrary values of m, n through T (m, n; α) ≡

m m! α ν F (n + ν; 2α) ν! α m+1 ν=0

1 mT (m − 1, n; α) + F (m + n; 2α) = α with the initial value 1 F (n; 2α). (6.47) α Note that we have so far considered only a special case where τ = 0. We turn to the case where τ = 1. Once again we note the recursion relation for H1 (m, n; α): T (0, n; α) =

H1 (m, n; α) = H0 (m + 1, n + 1; α) − S(m, n + 1; α) − S(n, m + 1; α) (6.48) where S(m, n; α) can be determined according to S(m, n; α) ≡

m m! α ν A(n + ν; 2α) ν! α m+1 ν=0

1 mS(m − 1, n; α) + A(m + n; 2α) = α with initial value S(0, n; α) =

1 A(n; 2α). α

(6.49)

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We now have shown relations needed to evaluate Hτ (m, n; α) for particular values of τ = 0, 1. Here we summarize the evaluation for τ > 1 through the following recursion relations: Hτ (m, n; α) =

1. (2τ − 1)2 Hτ −1 (m + 1, n + 1; α) τ2 + (τ − 1)2 Hτ −2 (m, n; α) − (2τ − 1)(2τ − 3) ! × Hτ −2 (m + 2, n; α) + Hτ −2 (m, n + 2; α) + 2(2τ − 1)(2τ − 5)Hτ −3 (m + 1, n + 1; α) − (2τ − 1)(2τ − 7)

! × Hτ −4 (m + 2, n; α) + Hτ −4 (m, n + 2; α) + 2(2τ − 1)(2τ − 9)Hτ −5 (m + 1, n + 1; α) + one of the following:

 (even τ )     

+ 2; α) − (2τ − 1) H0 (m + 2, n; α) + H0 (m, n !/ − S(m + 1, n; α) − S(n + 1, m; α) ,

(odd τ )     

1, n + 1; α) − S(m, n + 1; α) + (2τ − 1) 2H0 (m + !/ − S(n, m + 1; α) ,

where the starting values H0 (m, n; α) and H1 (m, n; α) are already given in (6.43) and (6.48). For nonzero ν, the integral Z ν can be written in terms of various simple function as discussed below. It is fairly straightforward to obtain the recursion relation for the Z ν function, just by inspection of the equation. For ≥ 1, Z ν (m, n, j, k; ) = Z ν (m + 2, n, j, k; − 2) + Z ν (m, n + 2, j, k; − 2) + Z ν (m, n, j + 2, k; − 2) + Z ν (m, n, j, k + 2; − 2) − 2Z ν (m, n, j, k; − 2) − 2Z ν (m + 1, n + 1, j + 1, k + 1; − 2) − 2Z ν+1 (m, n, j, k; − 2).

(6.50)

The above recursion relation can be proved in a straightforward manner by using 2 in the elliptical coordinates introduced in (6.31). the expansion of r12 Higher order terms of Z ν are given by Z 1 (m, n, j, k, −1) = −

∞ 2 (2τ + 1) 1 R (j )Rτ1 (k)Hτ1 (m, n; α) (6.51) R τ 2 (τ + 1)2 τ τ =1


173

and Z 2 (m, n, j, k, −1) ∞ 2 (2τ + 1) = R 2 (j )Rτ2 (k)Hτ2 (m, n) R (τ − 1)2 τ 2 (τ + 1)2 (τ + 2)2 τ τ =2

∞ 1 (2τ + 1) Rτ (j ) − Rτ (j + 2) Rτ (k) − Rτ (k + 2) R τ =0 × Hτ (m + 2, n + 2; α) − Hτ (m + 2, n; α) − Hτ (m, n + 2; α) + Hτ (m, n; α) .

+

For the = 0 case, Z 1 (m, n, j, k; 0) = 0, Z 2 (m, n, j, k; 0) = A(m + 2; α) − A(m; α) A(n + 2; α) − A(n; α) 8 , × (j + 1)(j + 3)(k + 1)(k + 3) when j and k are even; otherwise Z 2 (m, n, j, k; 0) vanishes. The definitions and recurrence relations for Rτν and Hτν are given below: 1 Rτν (j )

≡

ν/2 j ν 1 − µ2 µ Pτ (µ)dµ,

−1

Hτν (m, n; α)

∞∞ ≡ 1 1

ν/2 n 2 e−α(λ1 +λ2 ) λm 1 λ2 λ1 − 1 ν/2 ν × λ22 − 1 Pτ (λ< )Qντ (λ> ) dλ1 dλ2 .

The higher order recursions for Hτν (m, n; α) for ν = 1 and ν = 2 are listed below: Hτ1 (m, n; α) =

τ (τ + 1)2 Hτ +1 (m, n; α) − τ (τ + 1)Hτ (m + 1, n + 1; α) (2τ + 1) τ 2 (τ + 1) Hτ −1 (m, n; α), + 2τ + 1

and Hτ2 (m, n; α) =

τ 2 (τ − 1)2 1 H (m, n; α) − (τ + 2)(τ − 1) (2τ + 1) τ +1 (τ + 2)(τ + 1)2 1 Hτ −1 (m, n; α). × Hτ1 (m + 1, n + 1; α) + 2τ + 1

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F IG . 18. Schematic correlation diagram for He, the united atom limit, and H2 , the separated-atom case.

The James–Coolidge wave functions may be said to have the most “brawny” power as thousands of coefficients Cmnj kp have been calculated with automated computer programs, yielding very accurate values for the binding energy of diatomic molecules. Nevertheless, there are certain associated shortcomings: (i) The physical insights about molecular bonding seem to be lost; (ii) The wave functions in general do not satisfy the correlation cusp condition; (iii) For large λ1 and λ2 , the asymptotic conditions are violated (see Appendix D).

6.5. T WO -C ENTERED O RBITALS Historically, the first use of two-centered orbitals for molecular calculations is attributable to Wallis and Hulbert’s paper [40] published in 1954. For a historical summary, see [31,42]. New push and progress have been made by a school of French researchers [6,12–14,72,73] since 1976. Their work has made the twocenter orbital approach to diatomic modelling and computation an admirable success. The contributions by the French school are manifold, generalizing most of the aspects of one-centered orbitals to two-centered ones. To introduce its basic elements, let us utilize some of the semi-tutorial material from Scully et al. [16]. Recall the correlation diagram for H2 molecule shown in Fig. 18 [16]. A typical form of the trial wave function for H2 , e.g., may be represented as H2 ,1sσ + c2 ψ H2 ,2pσ f (r12 ), ψ = c1 ψ (6.52)


175

H2 ,1sσ and ψ H2 ,2pσ are chosen to be, respectively, where ψ H2 ,1sσ = ψ + (1)ψ + (2), ψ H ,1σ H ,1σ 2

2

H2 ,2pσ = ψ + (1)ψ + (2). ψ H ,2p H ,2p 2

(6.53)

2

From (6.52) and (6.53), we see that the trial wave function (6.52) (i) is uncorrelated if f (r12 ) ≡ 1. If f (r12 ) has explicit dependence on r12 , then (6.52) is correlated. A simple choice of f (r12 ) was 1 f (r12 ) = 1 + r12 , (6.54) 2 in Scully et al. [16], but many such correlation functions f satisfying the interelectronic cusp condition as given in Section 5.2 may be used also; (ii) has configuration interaction if c1 c2 = 0 in (6.52). It is found by [16] that: (a) For uncorrelated orbitals and without configuration interaction, i.e., f (r12 ) ≡ 1 and c2 = 0 in (6.52), the choice of ψH + ,1σ (j ) = N e−α1 λj 1 + B2 P2 (µj ) , j = 1, 2, (6.55) 2

in (6.53)1 , where P2 is a Legendre polynomial, gives the binding energy EB = 0.132 a.u. = 3.59 eV for H2 . Here, N is a normalization constant, and α1 and B2 are two variational parameters. (b) For correlated orbitals but without configuration interaction, i.e., f (r12 ) = 1 + 12 r12 and c2 = 0 in (6.52), the choice of (6.55) yields the binding energy EB = 0.1710 a.u. = 4.653 eV for H2 . The choice f (r12 ) = 1 + κr12 , where κ is a variational parameter, slightly improves the answer and gives EB = 0.1713 a.u. = 4.661 eV. (c) For correlated orbitals with configuration interaction, i.e., f (r12 ) = 1 + 12 r12 and c1 c2 = 0 in (6.52), the choice of (6.55) along with ψH + ,2p (j ) = N e−α2 λj P1 (µj ) + B3 P3 (µj ) (6.56) 2

in (6.53) (α1 , B2 , α2 and B3 are variational parameters) yields the binding energy EB = 0.1712 a.u. = 4.658 eV. For the correlation factor f (r12 ) with the variational parameter κ we obtain EB = 0.1721 a.u. = 4.682 eV. What is quite striking here is that if, instead of (6.55), we choose a finer twocentered orbital ψH + ,1σ (j ) = N e−α1 λj 1 + B2 P2 (µj ) + B4 P4 (µj ) , j = 1, 2, (6.57) 2

and perform calculations using three variational parameters α1 , B2 and B4 , then numerical results manifest that B2 totally dominates B4 , with a ratio |B4 /B2 | ≈

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2 × 10−3 . This shows that the simple orbital (6.52) with (6.55) is able to capture the essence of chemical bonding of H2 with very good accuracy. For a heteronuclear molecule such as HeH+ , a simple adaptation of the above scheme works equally well. Nevertheless, the authors have found that for large R, more terms involving both λ and µ variables should be included in (6.55), such as ψH + ,1σ (j ) = N e−α1 λj 1 + B2 P2 (µj ) + · · · + B2m Pm (µj ) 2 × 1 + A1 L1 (xj ) + · · · + An Ln (xj ) , xj = 2α1 (λj − 1), (6.58) where Ln (x) are Laguerre polynomials, so that good accuracy can be maintained in the calculation of E. Another choice of simple two-centered wave functions, in the same spirit of this subsection, was given by Patil [56], where he has generalized the one-centered Guillemin–Zener-type one-centered molecular orbitals (see Item (3)) by considering the gerade state ψg = N (1 + bλ)β e−aλ cosh(aµ)

(6.59)

and ungerade state ψu = N (1 + bλ)β e−aλ sinh(aµ)

(6.60)

+ for H+ 2 (or any homonuclear) ionic orbitals. Asymptotic behavior of H2 -like orbitals can be built in through β, which plays the same role as β in the Jaffé solution (4.22).

6.5.1. Le Sech’s Simplification of Integrals Involving Cross-Terms of Correlated Wave Functions Siebbeles and Le Sech [14] developed an ingenious approach to the evaluation of energy by applying integration by parts (or, Green’s Theorem) to avoid the quadratures of cross terms of the type ∇i (i j ) · ∇i f,

i, j = 1, 2,

(6.61)

where f is a correlation function, and i j is a product of two-centered orbitals. This is a fine feature of their approach. They choose wave functions of the form ψ(r 1 , r 2 ) = φ(1, 2)(1, 2),

(6.62)

where (1, 2) plays the role of the correlation function (but may be more general than the) f discussed in (6.52), while φ(1, 2) = (1)(2),

(6.63)


177

where (i), i = 1, 2, are the Hylleraas-type two-centered orbitals; cf. (6.77) below. Note here that each of 1 and 2 can have higher order molecular configurations such as 2sσg , 3pσu , 3dσg and 4f σu (in the computation of He, e.g., in [6]). Let the diatomic molecule’s Hamiltonian be H =−

1 2 ∇ + ∇22 + V (1, 2), 2 1

(6.64)

where

with

(1, 2) + 1 V (1, 2) = V r12

(6.65)

(1, 2) = − Za + Zb + Za + Zb . V r1a r1b r2a r2b

(6.66)

Let n (i), i = 1, 2, be eigenstates of the two center problem Zb 1 2 Za − n (i) = n n (i), i = 1, 2. − ∇i − 2 ria rib Denote φj (1, 2) to be solutions of 1 2 2 − ∇1 + ∇2 + V (1, 2) φj (1, 2) = Ej φj (1, 2). 2

(6.67)

(6.68)

Then 1 φj (1, 2) ≡ n1 (1)n2 (2) · √ α(1)β(2) − α(2)β(1) 2

(6.69)

satisfies (6.68) with Ej = n1 + n2 .

(6.70)

Now consider a trial wave function for (6.64) in the form φ(1, 2)(1, 2), where φ(1, 2) is of the form (6.69) while (1, 2) is intended to model the Coulombic repulsive effect from the 1/r12 term and, therefore, plays a similar role as (but may be more general than) the correlation function f (r12 ). Without loss of generality, φ(1, 2) and (1, 2) are assumed to be real. Denote the 6-dimensional Laplacian ∇62 ≡ ∇12 + ∇22 .

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Consider the matrix element Hij ≡ φi | − (1/2)∇62 + V (1, 2)|φj 1 = dr 1 dr 2 − φi ∇62 (φj ) + φi φj 2 V (1, 2) 2 R3 R3

=

1 dr 1 dr 2 − 2 φi ∇62 φj + φi φj ∇62 + 2φi ∇6 φj · ∇6 12 3 0 2

2 + φi φj V (1, 2) .

T1

(6.71)

To treat T1 , write

T1 = 2φi ∇6 φj · ∇6 = φi ∇6 φj · ∇6 2 ,

and note through an easy verification the following: 2φi ∇φj · ∇ 2 = − φi φj ∇ 2 2 + 2 ∇ · (φi ∇φj − φj ∇φi ) + ∇ · φi φj ∇ 2 + (φi ∇φj − φj ∇φi )2 .

(6.72)

But the LHS of (6.72) is equal to twice of T1 . So we now substitute one-half of the RHS of (6.72) for T1 in (6.71), obtaining 1 Hij = dr 1 dr 2 − 2 φi ∇62 φj +φi φj ∇62 2 0 12 3 T2

1 + φi φj ∇62 2 + 2 ∇6 · (φi ∇6 φj −φj ∇6 φi ) + φi φj 2 V (1, 2) , 04 12 3 T3

(6.73) where the divergence term ∇ · [. . .] in (6.72) disappears after integration in the 6-dimensional space provided that φi , φj , ∇(2 ) and 2 decay fast enough as |r 1 |, |r 2 | → ∞. The RHS of (6.73) is further simplified by utilizing 1 2 2 1 1 ∇ = |∇|2 + ∇ 2 , 4 2 2 and by combining T2 with T3 terms therein, leading to 1 1 dr 1 dr 2 φi φj |∇6 |2 − 2 φi ∇62 φj + φj ∇62 φi Hij = 2 4

+ φi φj 2 V (1, 2) .

(6.74)


179

Now, with φ satisfying (6.68) we obtain

1 1 1 Hij = dr 1 dr 2 φi φj 2 (Ei + Ej ) + + φi φj |∇6 |2 . 2 r12 2 (6.75) Comparing (6.75) with (6.71), we see that all the cross-derivative terms ∇6 φj × ∇6 have been eliminated, and the only “burden of differentiation” has been placed only on |∇6 |2 in (6.75). If we choose = 1 + 12 r12 , then |∇6 |2 = |∇1 |2 + |∇2 |2 = 1/2 and (6.75) becomes

1 1 1 Hij = (6.76) dr 1 dr 2 φi φj 2 (Ei + Ej ) + + φi φj . 2 r12 4 But doesn’t have to be chosen as = 1 + 12 r12 . Le Sech has preferred a special form of as given in Section 5.2.2. Note that g and u can be used as variational parameters, so can be d in Section 5.2.2. In a personal communication from Le Sech to Scully (9/6/2004), Le Sech pointed out that the trial wave functions in the form of (6.62) with (1, 2) chosen as Section 5.2.2 is particularly suitable for the diffusion Monte Carlo method [73]. Regarding the two-centered orbitals n1 (1) and n2 (2) in (6.67) and (6.69), Aubert-Frécon and Le Sech [13, (4)] used a truncated summation from the Hylleraas series solution, cf. (4.30): = (λ, µ, φ) K

N

m/2 m cn e−p(λ−1) 2p(λ − 1) Ln 2p(λ − 1) , k=m n=0 (6.77) m where p is used as a variational parameter, and fk and cn are coefficients which can be determined from the Killingbeck procedures [74]. =

fkm Ykm (µ, φ)

6.5.2. Generalized Correlated or Uncorrelated Two-Centered Wave Functions The two-centered orbitals we have been using in this section to build up the molecular orbitals, whether they be correlated, uncorrelated, with or without configuration interaction, have been heavily influenced by the classic solutions due to Hylleraas and Jaffé, cf. (4.30) and (4.26). These two famous solutions were derived during the 1920s and 1930s with great ingenuity, their greatest advantages being that a 3-term recurrent relation of the series coefficients. The 3-term recurrences further lead to continued fractions which have enabled mathematicians to perform asymptotic analysis. That was during a time when no electronic computers were available and it was perhaps the only way to perform any theoretical analysis at all. Nowadays, fast computers are readily available so we don’t have

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to rely overly on 3-term recurrence relations. Five-term recurrence relations are just as good and can be treated with relative ease also by the Killingbeck algorithm [74], for example. Recall that for the single-electron, two-centered heteronuclear problem, separation of variables leads to 2 m2 = 0, λ − 1 + A + 2R1 λ − p 2 λ2 − 2 λ −1 R(Za + Zb ) R1 ≡ (6.78) 2 m2 M = 0, 1 − µ2 M + −A − 2R2 µ + p 2 µ2 − 1 − µ2 R(Za − Zb ) . R2 ≡ (6.79) 2 The appearances of the above two equations suggest the ansatz (λ) =

∞

fk,1 Pkm (λ),

k=0

M(µ) =

∞

fk,2 Pkm (µ),

(6.80)

k=0

where Pkm (λ) are the associated Legendre polynomials. Substituting (6.80) into (6.78) and (6.79) and equate coefficients of Pkm (λ) and Pkm (µ) to zero, see Appendix L, we obtain 5-term recurrence relations Akj fk−2,j + Bkj fk−1,j + Ckj fk,j + Dkj fk+1,j + Ekj fk+2,j = 0, k = 1, 2; j = 0, 1, 2, . . . ,

(6.81)

where p 2 (k − m − 1)(k − m) , (2k − 3)(2k − 1) 2Rj (k − m) =− , j = 1, 2, 2k − 1 p 2 (k − m + 1)(k + m + 1) = Ck2 = − k(k + 1) − A + (2k + 1)(2k + 3) p 2 (k − m)(k + m) , + (2k + 1)(2k − 1) 2Rj (k + m + 1) =− , j = 1, 2, 2k + 3 p 2 (k + m + 1)(k + m + 2) = Ek2 = . (2k + 3)(2k + 5)

Ak1 = Ak2 = Bkj Ck1

Dkj Ek1

(6.82)


181

The two-centered orbitals derived above differ from those of Hylleraas and Jaffé. The ansatz (6.80) is not the only way to obtain two-centered orbitals. It is possible to derive several other solutions in different forms using different orthogonal polynomials. Numerical results indicate that these two-centered orbitals here give accuracy compatible to that corresponding to Hylleraas and Jaffé-type solutions. 6.5.3. Numerical Algorithm In contrast to the explicit analytic formulas for the quadratures of James–Coolidge wave functions given in Section 6.4 and in the affiliated Appendices J and K, here the integrals will be computed using the Gaussian quadrature routines. (The analytic formulas given earlier in Section 6.4 can then be compared with those obtained here as a good check.) For the two-electron problem, the number of coordinates are six, so that the energy calculation requires the 6-dimensional integration. Using the cylindrical symmetry, we can reduce two angular variables to one. The next simplification is the choice of the wave functions. If we only restrict the wave functions to be the James–Coolidge type, the 5-dimensional integration can be divided as one two-dimensional integration and two three-dimensional integration. However, if we include the exponential function of inter-electron distance, that is e−κr12 , this 5-dimensional integration should be evaluated as multidimensional integration, and the numerical accuracy should be carefully investigated. Here, we only restrict the wave functions to be the James–Coolidge type. The usual integration has the following form . . . 1 (λ1 )2 (λ2 )M1 (µ1 )M2 (µ2 ) n dλ1 dλ2 dµ1 dµ2 dφ1 dφ2 . × (φ1 − φ2 )r12

(6.83)

2 -identity, cf. (6.31), the exponent of r can be reduced into 0 or −1. Using the r12 12 For n = 0, the whole integration is divided into only 5 one-dimensional integrations. However, for 1/r12 , by the Neumann expansion the form of wave function includes sum of Legendre Polynomials with proper coefficients; cf. (6.39). Especially, the form of the λ-part is P (λ1 )Q(λ2 ), λ1 < λ2 , P (λ< )Q(λ> ) = (6.84) P (λ2 )Q(λ1 ), λ2 > λ1 .

This makes the form of integration over the λ variable as ∞∞

∞λ1 ··· =

0 0

∞∞ ··· +

0 0

··· 0 λ1

(6.85)

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F IG . 19. Comparison of potential curves of the H2 molecule computed with no correlation factor using the exact solution of the one-electron wave function (6.88) (small dots) and the Patil’s wave function (6.89) (solid line). The inner (outer) curves are the result without (with) configuration interaction. Squares are the “exact” ground state E(R) from [8]. Upper curves correspond to “excited states”.

The implementation of these numerical integration is automated by standard computer software. Let us describe the numerics for a homonuclear dimer H2 molecule and a heteronuclear dimer HeH+ molecular ion. The approximation of wave function of two-electron system into a multiplication of one-electron system can be easily made in the natural orbit expansion (r1 , r2 ) = (6.86) ck χk (r1 )χk (r2 ) where χk (r1 ) is the wave function of one-electron two-center problem. For the ground state, it’s well known that c1 ∼ 1, or (r1 , r2 ) χ1 (r1 )χ1 (r2 ).

(6.87)

There are a few alternatives to approximate the χ1 (r1 ). One is using the exact solution of one-electron two-center problem, 1sσg state. The Jaffé’s form is λ − 1 n −αλ σ (1 + λ) gn f2m P2m (µ), χ1 (r1 ) = e (6.88) λ+1 n m cf. (4.26).


183

F IG . 20. Same computations as with Fig. 19, but with the correlation factor.

And the other is Patil’s wave function χ1 (r1 ) = (1 + bλ)β e−αP λ cosh(aµ);

cf. (6.59).

(6.89)

We may also include configuration interaction, such as (r1 , r2 ) = c1 χ1 (r1 )χ1 (r2 ) + c2 χ2 (r1 )χ2 (r2 ),

(6.90)

where χ1 is 1sσg state and χ2 is 2pσu state. The result is shown in Fig. 19. Before the diagonalization (that is, CI), the asymptotic behavior of the ground state E(R) is monotonically increasing. By diagonalizing, the behavior at large R becomes almost flat, however, E(R) is slightly above the exact asymptotic value, −1 (htr). Next, we perform computations with correlation by using f (r12 ) = 1 + 12 r12 . Improvement can be readily seen in Fig. 20 which is much closer to the exact calculation done by Kolos [8]. In the calculations below (Figs. 19 and 20), “exact” solutions of H+ 2 -solutions were used (whose coefficients are obtained through truncated Killingbeck [74] procedures). Instead, one can use (6.55), (6.56), (6.57) by optimizing the coefficients α and B therein, or, for the Patil’s wave function, by optimizing the coefficients a and β in (6.59) and (6.60). By doing so, we obtain the energy curve

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F IG . 21. Ground state potential energy curve of the H2 obtained using the truncated exact wave functions of one-electron system with the correlation factor and several variational parameters. The curve yields the binding energy of EB = −0.1721 a.u. = 4.684 eV. Squares are the “exact” ground state E(R).

of the ground state as shown in Figs. 21 and 22, respectively. The energy curves are improved over a wide range of R values, and we obtain the binding energy of 0.171 a.u. = 4.65 eV, close to the experimental value. However, E(R) in Fig. 21 overshoots the dissociation limit.

7. Alternative Approaches 7.1. I MPROVEMENT OF H ARTREE –F OCK R ESULTS U SING THE B OHR M ODEL The Bohr model can also be applied to calculate the correlation energy for molecules and then improve the HF treatment. Figure 23 shows the ground state potential curve for H2 molecule calculated in the Bohr-HF approximation. Such an approximation omits the electron repulsion term 1/r12 in finding the electron configuration from Eq. (1.2). The difference between the Bohr and Bohr-HF potential curves yields the correlation energy plotted in the insert of Fig. 23.


185

F IG . 22. Ground state potential curve of H2 computed from the Patil’s wave function with the correlation factor and variational parameters. The curve yields the binding energy of EB = −0.1713 a.u. = 4.662 eV. Squares are the “exact” ground state E(R).

In Fig. 24 we draw the ground state E(R) for the H2 molecule obtained with the Heitler–London trial function that has the form of the combination of the atomic orbitals [67]: ! = C exp −α(ra1 + rb2 ) + exp −α(rb1 + ra2 ) , where α is a variational parameter. Addition of the correlation energy from Fig. 23 improves the Heitler–London result and shifts E(R) close to the “exact” values. The improved potential curve yields the binding energy of 4.63 eV.

7.2. D IMENSIONAL S CALING Dimensional scaling offers promising new computational strategies for the study of few electron systems. This is exemplified by recent applications to atoms, as well as H+ 2 and H2 molecules [10,4]. D-scaling emulates a standard method of quantum chromodynamics [9], by generalizing the Schrödinger equation

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F IG . 23. Ground state E(R) for the H2 molecule in the Bohr and Bohr-HF models. Insert shows the correlation energy as a function of R.

F IG . 24. Ground state energy E(R) of the H2 molecule in the Heitler–London method and the improved E(R) after the addition of the correlation energy. Dots are the “exact” result from [8].


187

to D dimensions and rescaling coordinates [10]. The D → ∞ limit corresponds to infinitely heavy electrons and reduces to a classic electrostatic problem of finding an electron configuration that minimizes a known effective potential. = E for a two particle wave We start from the Schrödinger equation H function (r1 , r2 ). The H2 Hamiltonian in atomic units reads = − 1 ∇12 − 1 ∇22 + V (ρ1 , ρ2 , z1 , z2 , φ, R), H 2 2 where the Coulomb potential energy V is given by Eq. (1.1). In a traditional dimensional scaling approach the Laplacian is treated in D-dimension and the wave function is transformed by incorporating the square root of the Jacobian via → J −1/2 , where J = (ρ1 ρ2 )D−2 (sin φ)D−3 in cylindrical coordinates, and φ is the dihedral angle specifying relative azimuthal orientation of electrons about the molecular axis [75]. D-scaling in spherical coordinates is discussed in Appendix M. On scaling the coordinates by f 2 and the energy by 1/f 2 , with f = (D − 1)/2, the Schrödinger equation in the limit D → ∞ leads to minimization of the effective potential [75] 1 1 1 1 + + V (ρ1 , ρ2 , z1 , z2 , φ, R). E= (7.1) 2 2 2 ρ1 ρ2 sin2 φ The obtained electron configuration is sometimes called the Lewis structure because it provides a rigorous version of the familiar electron-dot formulas introduced by Lewis in 1916 [76]. Figure 25 (upper curve) displays the D → ∞ potential curve E(R) that exhibits no binding and substantially deviates from the D = 3 “exact” energy (dots). In the limit D → 1 the dimensional scaling reduces the Hamiltonian to the delta function model [77] 2 2 R R R = −1 d + d + − + − δ x − δ x − δ x H 1 1 2 2 dx12 2 2 2 dx22 R − δ x2 − (7.2) + δ(x1 − x2 ). 2 A variational solution of the one-dimensional wave equation [78] is pictured in Fig. 25 (lower curve). Also shown is an approximation to E(R) for D = 3 obtained by interpolating linearly in 1/D between the dimensional limits: 2 1 E∞ (R) + E1 (R). (7.3) 3 3 The interpolated D = 3 curve exhibits binding which indicates the feasibility of extending dimensional interpolation to molecules. E3 (R) =

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F IG . 25. Ground state energy E(R) of the H2 molecule in D = ∞, D = 1 and three-dimensional interpolation (solid lines). Dots show the “exact” D = 3 energy from [8].

A proper choice of scaling can improve the method. Here we discuss a dimensional scaling transformation of the Schrödinger equation that yields the Bohr model of H2 in the limit of infinite dimensionality [4]. For such a transformation, the large-D limit provides a link between the old (Bohr–Sommerfeld) and the new (Heisenberg–Schrödinger) quantum mechanics. The first-order correction in 1/D substantially improves the agreement with the exact ground state E(R). In the modified scaling the Laplacian depends on a continuous parameter D as follows ∂ 1 ∂ ∂2 1 ∂2 ρ D−2 + 2 2 + 2. ∇ 2 = D−2 (7.4) ∂ρ ∂ρ ρ ρ ∂φ ∂z For D = 3, Eq. (7.4) reproduces the 3D Laplacian. On scaling the coordinates by f 2 , the energy by 1/f 2 (recalling f = (D − 1)/2) and transforming the electronic wave function by = (ρ1 ρ2 )−(D−2)/2 ,

(7.5)


189

the Schrödinger equation is recast as (K + U + V ) = E,

(7.6)

where 2

∂2 ∂2 ∂2 1 1 ∂2 ∂ 2 + 2+ 2+ 2+ + 2 , K=− (D − 1)2 ∂ρ12 ∂ρ2 ∂z1 ∂z2 ρ12 ρ2 ∂φ 2 (7.7) 1 (D − 2)(D − 4) 1 + 2 . U= 2(D − 1)2 ρ12 ρ2 In the D → ∞ limit the derivative terms in K are quenched. The corresponding energy E∞ for any given internuclear distance R is then obtained simply as the minimum of an effective potential 1 1 1 E= (7.8) + + V (ρ1 , ρ2 , z1 , z2 , φ, R). 2 ρ12 ρ22 This is identical in form to that for the Bohr model, Eq. (1.2), and we thus obtain for E∞ (R) the same solutions depicted in Fig. 2. This result for E∞ (R) differs in an interesting way from that obtained in a traditional study of the D → ∞ limit for the H2 molecule [75]. Here, in order to connect with the Bohr model, it is necessary to incorporate only the radial portion of the Jacobian in transforming the electronic wave function via Eq. (7.5). The customary practice, which employs the full Jacobian, introduces a factor of 1/(sin2 φ) into the centrifugal potential, as seen from Eq. (7.1). The modified procedure yields directly a good zeroth-order approximation. The ground state E(R) can be substantially improved by use of a perturbation expansion in powers of 1/D, developed by expanding the effective potential of Eq. (7.8) in powers of the displacement from the minimum [10]; for He and H+ 2 this has yielded highly accurate results [79]. Terms quadratic in the displacement describe harmonic oscillations about the minimum and give a 1/D correction to the energy. Anharmonic cubic and quartic terms give a 1/D 2 contribution. For the He ground state energy (the R = 0 limit for H2 ) a first-order approximation yields [4]1 4E∞ 0.1532 E(0) = (7.9) , 1 − D (D − 1)2

1 To improve convergence of the 1/D expansion a variant scaling involving a factor (2D + 9)/5D is used in front of the ∂ 2 /∂φ 2 term in Eqs. (7.4), (7.7).

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F IG . 26. Ground state energy E of H2 molecule as a function of internuclear distance R calculated within the dimensional scaling approach (solid lines) and the “exact” energy (dots).

where E∞ = −3.062 a.u. is the Bohr model He result [5,80]. For D = 3 Eq. (7.9) improves the ground state energy of He to E(0) = −2.906 a.u., which differs by 0.07% only from the “exact” value of −2.9037 a.u. [81]. To evaluate the 1/D correction for arbitrary R, it is convenient to introduce new coordinates 1 z˜ 1 = √ (z1 − z2 ), 2

1 z˜ 2 = √ (z1 + z2 ). 2

The effective potential of Eq. (7.8) then has a minimum at ρ1 = ρ2 = ρ0 , φ = π, z˜ 2 = 0. Along the coordinates ρ1 , ρ2 , z˜ 2 and φ the potential has a single well structure no matter what the internuclear spacing R is. However, along the z˜ 1 direction at R = 1.2 the potential changes shape from a single to a double well; such symmetry breaking is a typical feature exhibited at large D [82]. To avoid the inaccuracy of approximation by a single quadratic form one can solve the Schrödinger equation numerically along the z˜ 1 direction for the exact potential and add contributions from the harmonic motion associated with the other coordinates ρ1 , ρ2 , z˜ 2 and φ. The result is shown in Fig. 26 [4]. The 1/D correction improves E(R) and predicts the equilibrium separation to be Re = 1.62 with binding energy EB = 4.38 eV.


191

8. Conclusions and Outlook Many major topics on diatomic molecules, and some other atoms and molecules in general, have been addressed in this article, giving it a very “locally diverse” or perhaps a somewhat disjoint look. But a simple, “global” picture may be viewed and understood in/from the following diagram:

This “unification” is by no means an easy task. Nevertheless, it was a goal we somehow envisioned to achieve when this research was started and we hope we have made at least some success. At present, in the study of ultrafast laser applications to chemical physics and molecular chemistry problems, there is a pressing need to understand the quantum-dynamical behavior of molecules and the associated properties of excited states and their computations. Many challenges are lying ahead and awaiting elegant solutions.

9. Acknowledgements We wish to thank Professor D. Herschbach for lectures given at Princeton in Fall 2003 which stimulate our interest in D-scaling, and Professor C. Le Sech for the personal communication cited in Section 6.5.1. We also thank Professor G. Hunter, who read the manuscript and offered constructive criticisms. This research is partially supported by grants from ONR (N00014-03-1-0693 and N00014-041-0336), NSF DMS 0310580, TITF of Texas A&M University, and the Robert A. Welch Foundation (#A1261).

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10. Appendices Appendix A. Separation of Variables for the H+ 2 -like Schrödinger Equation Let us consider (4.13). Here we show how to separate the variables through the use of the ellipsoidal (or, prolate spheroidal) coordinates (see Fig. 9) R 2 R 2 x= λ − 1 1 − µ2 cos φ, y= λ − 1 1 − µ2 sin φ, 2 2 (A.1) R z = λµ. 2 Note that the coordinates λ, µ and φ are orthogonal, and we have the first fundamental form ds 2 = dx 2 + dy 2 + dz2 = h2λ dλ2 + h2µ dµ2 + h2φ dφ 2 , where

2 ∂y 2 ∂z R 2 1 − µ2 , = + + = ∂λ ∂λ 4 λ2 − 1 2 2 2 ∂x ∂y ∂z R 2 λ2 − 1 h2µ = + + = , ∂µ ∂µ ∂µ 4 1 − µ2 2 2 2 ∂x ∂y ∂z R2 2 2 + + = λ − 1 1 − µ2 . hφ = ∂φ ∂φ ∂φ 4 h2λ

Thus

∂x ∂λ

2

1 ∂ hλ hφ ∂ ∂ hµ hφ ∂ + hλ hµ hφ ∂λ hλ ∂λ ∂µ hµ ∂µ ∂ hλ hµ ∂ + ∂φ hφ ∂φ ∂ 4 ∂ ∂ 2 2 ∂ = 2 2 λ −1 + 1−µ ∂λ ∂µ ∂µ R (λ − µ2 ) ∂λ

2 2 2 λ −µ ∂ + 2 . 2 (λ − 1)(1 − µ ) ∂φ 2

∇ 2 =

Note that through the coordinate transformation (A.1), we have   R r + rb    ra = (λ + µ), λ = a , 2 R equivalently, ra − rb R    r = (λ − µ). µ = , b R 2

(A.2)

(A.3)


193

Also, we have λ ≥ 1, −1 ≤ µ ≤ 1. Write = (λ)M(µ)(φ).

(A.4)

(φ) must be periodic with period 2π. Therefore (φ) = eimφ ,

m = 0, ±1, ±2, . . . .

(A.5) eimφ :

Substitute (A.2), (A.4) and (A.5) into (4.13), and then divide by ∂ 4 ∂ 1 ∂ 2 2 ∂ − 1 λ M + 1 − µ M − 2 R 2 (λ2 − µ2 ) ∂λ ∂λ ∂µ ∂µ

2 Zb (λ2 − µ2 )m2 M − M − 2 Rλ+µ (λ − 1)(1 − µ2 ) 2 Za Za Zb − (A.6) M + M = EM. Rλ−µ R 2

Further multiplying every term by − R2 (λ2 − µ2 ), we obtain ∂ ∂ 2 ∂ λ2 − µ2 2 ∂ m2 M λ −1 M+ 1−µ M − 2 ∂λ ∂λ ∂µ ∂µ (λ − 1)(1 − µ2 ) RZa Zb R2E − (λ2 − µ2 ) M = 0. + RZb (λ − µ) + RZa (λ + µ) − 2 2 0 12 3

R 2 E RZa Zb 2 − 2

(λ2 −µ2 )+R[(Za +Zb )λ+(Za −Zb )µ]

(A.7)

Set p2 =

1 −R 2 E + RZa Zb > 0. 2

(A.8)

We have p 2 > 0 here due to the fact that we are mainly interested in the electronic states that are bound states, i.e., not ionized. Let the constant of separation of variables be A. Then from (A.7) and (A.8) we obtain (4.16) and (4.17) in Section 4.2.

Appendix B. The Asymptotic Expansion of (λ) for Large λ Here, we provide a quick proof of Eq. (4.22). From (4.19), we have 2λ A + 2R1 λ 2 (λ) + (λ) − p (λ) = − 2 λ −1 λ2 − 1

2 m2 λ 2 −1 − 2 −p (λ) . λ2 − 1 (λ − 1)2

(B.1)

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We now find the Laurent expansions of the coefficient functions on the right-hand side of (B.1) as follows: 2λ 2 2 0 2 0 = + 2 + 3 + 4 + 5 + ···, λ λ λ2 − 1 λ λ λ 1 A + 2R1 λ 1 1 = (A + 2R λ) 1 + + + · · · 1 λ2 − 1 λ2 λ2 λ4 2R1 2R1 A A = + 2 + 3 + 4 + ···, λ λ λ λ 2 2 0 0 p2 p λ −p 2 2 − 1 = − 2 + 3 − 4 ± ··· λ λ −1 λ λ λ −

(λ2

m2 0 0 m2 0 2m2 0 = + 2 + 3 − 4 + 5 − 6 ± ···. 2 λ λ − 1) λ λ λ λ

From these expansions above, we now use the ansatz c2 c1 + 2 + ··· (λ) = a0 e−pλ λβ 1 + λ λ by substituting it into (B.1) and equating all the coefficients of λ−n to zero for n = 0, 1, 2, . . . . We easily obtain β=

R1 − 1, p

c1 =

p 2 − β 2 − 2β . 2R1 − 2pβ − p

All the other coefficients cn can be determined in a straightforward way. Note that the asymptotic expansion (4.22) also gives (λ) − a1 e−pλ λβ

n cj = O e−pλ λβ(n+1) , j λ

for λ 1.

(B.2)

j =0

Appendix C. The Asymptotic Expansion of (λ) as λ → 1 Here we provide a proof of Eq. (4.23). Multiply (4.18) by (λ − 1)/(λ + 1) and rewrite it as 2λ 0 = (λ − 1)2 (λ) + (λ − 1) (λ) λ+1 (A + 2R1 λ − p 2 λ2 ) m2 + (λ − 1) − (λ) λ+1 (λ + 1)2 ≈ (λ − 1)2 (λ) + (λ − 1) (λ) −

m2 (λ), 4

for λ ≈ 1.

(C.1)


195

A differential equation (such as (C.1)) set in the form (x − 1)2 y (x) + (x − 1)q(x)y (x) + r(x)y(x) = 0,

(C.2)

near x = 1, where q(x) and r(x) are analytic functions at x = 1, is said to have a regular singular point at x = 1. The solution’s behavior near x = 1 hinges largely on the roots ν of the indicial equation ν(ν − 1) + q(1)ν + r(1) = 0

(C.3)

because the solution y(x) of (C.2) is expressible as y(x) = b1 (x − 1)ν1

∞

ck (x − 1)k + b2 (x − 1)ν2

k=0

∞

dk (x − 1)k

k=0

(c0 = d0 = 1), where ν1 and ν2 are the two roots of the indicial equation (C.3), under the assumptions that ν1 > ν2 ,

ν1 − ν2 is not a positive integer.

(C.4)

However, if (C.4) is violated, then there are two possibilities and two different forms of solutions arise: (a) ν1 = ν2 . Then y(x) = b1 y1 (x) + b2 y2 (x),

(C.5)

where y1 (x) = (x − 1)ν1

∞

ck (x − 1)k

(c0 = 1)

(C.6)

k=0

and y2 (x) = (x − 1)ν1

∞

dk (x − 1)k + ln(x − 1) y1 (x)

(d1 = 1).

k=1

(C.7) Solution y2 in (C.7) should be discarded because it becomes unbounded at x = 1. (b) ν1 − ν2 = a positive integer. Then case (a) holds except with the modification that y2 (x) = (x − 1)

ν2

∞

dk (x − 1)k + c ln(x − 1) y1 (x)

k=0

(d0 = 1, c is a fixed constant but may be 0).

(C.8)

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Applying the above and (C.3) to Eq. (C.1): (λ − 1)2 (λ) + (λ − 1) (λ) −

m2 (λ) ≈ 0, 4

(C.9)

we obtain the indicial equation ν(ν − 1) + ν −

m2 = 0, 4

(C.10)

with roots ν1 =

|m| , 2

ν2 = −

|m| 2

(m can be either a positive or a negative integer). (C.11)

Thus either ν1 = ν2

(when m = 0)

or ν1 − ν2 = |m| = a positive integer,

where m = 0.

Again, we see that solution y2 in (C.8) must be discarded because it becomes unbounded at x = 1. Thus, from (C.6) and (C.9), we have (λ) ≈ (λ − 1)|m|/2

∞

ck (λ − 1)k .

(C.12)

k=0

Appendix D. Expansions of Solution Near λ ≈ 1 and λ 1: Trial Wave Functions of James and Coolidge In the pioneering work of James and Coolidge [19], the two-centered trial wave functions for H2 are chosen to be (ground state) ψ(λ1 , λ2 , µ1 , µ2 , ρ) 1 −α(λ1 +λ2 ) = e 2π

p n j k n m k j Cmnj kp λm 1 λ 2 µ 1 µ 2 + λ1 λ 2 µ 1 µ 2 ρ ,

(D.1)

m,n,j,k,p

and (excited state) ψ(λ1 , λ2 , µ1 , µ2 , ρ) = e−α(λ1 +λ2 ) m,n,j,k,p

p n j k n m k j Cmnj kp λm 1 λ 2 µ 1 µ 2 − λ1 λ 2 µ 1 µ 2 ρ ,

(D.2)


197

where ρ = r12 is the distance between the two electrons of H2 , the coefficients Cmnj kp are computed from the minimization of the energy, plus possible orthogonality conditions. We examine the asymptotics of the trial solutions (D.1) or (D.2) based on the discussions in Section 4.2. We see that the exponent e−αλ1 in e−α(λ1 +λ2 ) would correspond to the factor e−pλ in (4.21) or (4.26). This is excellent as it reflects the exponential decay in the radial variable of the (first) electron. However, the other non-constant polynomial terms of the form j

k 1 n1 λm 1 λ2 µ1 µ2 ,

j

2 k λn1 2 λm 2 µ1 µ2 ,

with |m | + |n | ≥ 1, = 1, 2, (D.3)

possess polynomial growth rates in either λ1 or λ2 , which are at odds with the asymptotics in (4.22) since for large λ (which may be either λ1 or λ2 ), there should n2 m2 1 n1 not be any polynomial growth λm 1 λ2 or λ1 λ2 in (D.3) besides the exponential 1 n1 −pλ −pλ 1 2 decay factor e ·e . (One might argue that the polynomial growth λm 1 λ2 n2 m2 −pλ −pλ 1 2 or λ1 λ2 would be killed by the exponential decay e ·e . This can be true, however, only by increasing p and thus it causes the loss of accuracy.) One might still argue that the typical term in the series (4.26) for m = 0 (ground state) satisfies k σ λ−1 (λ + 1) λ+1 (σ − 1)(σ − k − 1) 1 2 1 = λσ 1 + (σ − k) + ··· + λ 1·2 λ σ = O λ , for λ 1. This means that either m or n in (D.3) should never exceed σ . Can’t we, at least, n2 m2 1 n1 use terms λm 1 λ2 or λ1 λ2 with some restrictions such as 0 ≤ m , n ≤ σ ;

σ ≡

R1 − 1, for m = 0, = 1, 2. p

The most important behavior of happens near λ = 1. For λ ≈ 1, the typical term in (4.26) satisfies λ−1 k (λ2 − 1)|m|/2 (λ + 1)σ λ+1 λ−1 |m| = 2|m|/2+σ −k (λ − 1)|m|/2+k 1 + +σ −k 2 2 2 (|m|/2 + σ − k)(|m|/2 + σ − k − 1) λ − 1 + + ··· 1·2 2 = (λ − 1)|m|/2 2|m|/2+σ −k (λ − 1)k + O (λ − 1)k+1 , for λ ≈ 1, λ > 1.

(D.4)

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When m = 0, the case of the ground state, the above expansion in terms of powers of λ is consistent with the terms involving powers of λ1 or λ2 in (D.3) since those powers in (D.3) can always be re-expanded in terms of powers of λ1 − 1 or λ2 − 1. However, when m = 1 (or m = ± odd integer, for that matter) for the excited state, the function in (D.3) is no longer consistent with those in (D.4) as far as the λ-variable is concerned because the factor (λ − 1)|m|/2 in (D.4) is unaccountable for those in (D.3). Indeed, we have indicated in (4.23) through asymptotic analysis that the factor (λ − 1)|m|/2 is inherent in the solution and, thus, must be properly taken into account. The discussion in this section points out some weaknesses in the choice of basis functions (D.1) or (D.2) based on the asymptotic arguments for λ 1 and λ ≈ 1. Such weaknesses may have contributed to the fact that many terms are required in (D.1) in order to calculate or to do the variational analysis of the energy E accurately for H2 by James and Coolidge [19]. Our conclusion for this subsection is: because of the vastly different asymptotic behaviors of (λ) for λ ≈ 1 and λ 1, the best strategy for numerical computation is to use two different representations for (λ), one for λ ≈ 1 and another for λ 1 and match them, say, at a medium-size value such as λ = 5 or 10. This, however, will invoke more computational work and is beyond the interest of the authors for the time being.

Appendix E. The Many-Centered, One Electron Problem in Momentum Space Here we derive the eigenvalue equation (4.43). We start from the model equation N 1 2 Zj − ∇ − (E.1) ψ(r) = Eψ(r), r = (x, y, z) ∈ R3 . 2 |r − R j | j =1

Recall the Fourier transform 1 (p) = e−ir·p ψ(r) dr. (2π)3/2

(E.2)

R3

Note that the Fourier transform of the potential term 1 1 e−ir·p dr |r − R j | (2π)3/2

(E.3)

R3

is a divergent integral in the classical sense. However, the modern mathematical theory of the “regularization of divergent integrals” [20, Chapter 3] makes (E.3)

ELECTRON MOLECULAR BONDS well defined: (E.3) = (2π)−3/2

e−i(r +R j )·p

R3 −3/2 −iR j ·p

≡ (2π)

e

1 dr |r |

lim

e

−ir ·p e

α→0 R3 ∞ π 2π

= (2π)−3/2 e−iRj ·p lim

α→0 0

0

199

(r ≡ r − R j ) −αr

r

dr

e−ir p cos θ−αr r 2 dr sin θ dθ dφ

0

−iR j ·p 4π −1/2 e = (2π)−3/2 e−iRj ·p lim 2 = 2(2π) . α→0 p + α 2 p2

(E.4)

Applying the Fourier transform to (E.1) by utilizing (E.4) and other well known properties such as convolution, we obtain N 1 2 Zj p − E (p) − (2π)−3/2 · 2(2π)−1/2 2 j =1 −iR j ·(p−p ) e × p dp = 0. (E.5) |p − p |2 R3

Define p02 = −2E,

(E.6)

then we obtain the integral equation

N Zj p 2 + p02 (p) = π2 j =1

R3

e−iR j ·(p−p ) p dp . |p − p |2

(E.7)

Now, we project the 3-dimensional momentum vector p onto the surface of the unit sphere, S3 , of the 4-dimensional space, the 1-1 correspondence ξ ↔ p through −1 ξ1 = 2p0 px p 2 + p02 = sin χ sin θ cos φ, 2 −1 = sin χ sin θ sin φ, ξ2 = 2p0 py p + p02 2 −1 ξ ∈ R4 , ξ = |ξ | = 1, (E.8) = sin χ cos θ, ξ3 = 2p0 pz p + p02 −1 = cos χ, ξ4 = p02 − p 2 p 2 + p02 0 ≤ χ ≤ π, 0 ≤ θ ≤ 2π, 0 ≤ φ ≤ π,

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while keeping in mind that px = p sin θ cos φ,

Then for ξ ↔ p, ξ ↔

p ,

py = p sin θ sin φ,

pz = p cos θ.

(E.9)

and κ be the angle between ξ and ξ , we have

cos κ = ξ · ξ ,

ξ − ξ 2 = ξ 2 + ξ 2 − 2ξ · ξ = 2 − 2 cos κ = 4 sin2 (κ/2).

(E.10)

Also, it is straightforward to verify that

ξ − ξ 2 =

4p02 |p − p |2 (p 2 + p02 )(p 2 + p02 )

.

Hence p02 1 = . 2 2 |p − p | (p 2 + p0 )(p 2 + p02 ) sin2 (κ/2)

(E.11)

Let d be the infinitesimal surface area of the 4-dimensional hypersphere S3 . Then from (E.8), d = sin2 χ sin θ dχ dθ dφ = − sin2 χ dχ d(cos θ ) dφ

(E.12)

while (E.9) gives the standard dp = p 2 sin θ dp dθ dφ = −p 2 dp d(cos θ ) dφ.

(E.13)

From (E.8)4 , we further obtain p 2 + p02 dp . = dχ 2p0

(E.14)

Equations (E.8) and (E.12)–(E.14) now give 2 p + p02 3 p 2 dp d. d = dp = 2p0 sin2 χ dχ

(E.15)

We can now use (E.11) to eliminate the denominator inside the Fourier integral of (E.7). Moreover, define ϕ() =

(p 2 + p02 )2 5/2

(p),

(E.16)

4p0

where ∈ S3 is the point ξ ↔ p. Then the integral equation (E.7) simplifies to n Zj e−iRj (p−p ) 1 p() = (E.17) ϕ d . p0 8π 2 sin2 (κ/2) j =1 S3


201

At this point, we need to introduce the hyperspherical harmonics on O3 , which constitute an orthonormal basis for square summable functions on S3 and are given by Ynm () = (−i) Cn (χ)Ym (θ, φ), n = 1, 2, . . . , = 0, 1, 2, . . . , m = −, − + 1, . . . , 0, 1, . . . , , (E.18) cf. (4.6)–(4.9) for Ym (θ, φ), where

Cn (χ) ≡

2n(n − − 1)! π(n + )!

1/2

sin χ

d d(cos χ)

Cn−1 (cos χ) ,

(E.19) are the associated Gegenbauer functions, and the Ck ’s are the Gegenbauer functions, with the generating function ∞

1 ≡ hj Cj (µ), 2 1 − 2µh + h

|h| < 1.

(E.20)

j =0

For ξ , ξ ∈ R4 ,

 1 1    ξ 2 1 − 2 cos κ(ξ /ξ ) + (ξ /ξ )2

1 =  1 1 |ξ − ξ |2   ξ 2 1 − 2 cos κ(ξ/ξ ) + (ξ/ξ )2

ξ < 1, ξ ξ if < 1. ξ if

(E.21)

Denote ξ> = max(ξ, ξ ), ξ< = min(ξ, ξ ). Then (E.20) and (E.21) give the Neumann expansion ∞ ∞ 1 ξ ξ> |ξ − ξ | ξ>n+1 n=0

which, in the limit as

ξ, ξ

n=1

→ 1, yields

1 1 (by (E.10)) 2 = 2 |ξ − ξ | 4 sin (κ/2) ∞ Cn−1 (cos κ). =

(E.22)

n=1

But by the addition theorem of angles, we have Cn−1 (cos κ) =

2π 2 ∗ Ynm Ynm (), n m

(E.23)

202

G. Chen et al. n ≥ 1, = 0, 1, 2, . . . , m = −, − + 1, . . . , 0, 1, . . . , ,

(E.24)

so from (E.22) we obtain 1 2

4 sin (κ/2)

= 2π 2

∞ n=1 ,m

1 ∗ , Ynm () Ynm n

and, thus, the kernel of the integral equation (E.17) can be written as

Zj e−iRj ·(p−p ) 8π 2 sin2 (κ/2)

= Zj

t

1 −iR ·p ∗ e−iR j ·p Yt () e j Yt , n

(E.25)

where t = (nm) runs triple summation indices according to (E.24). We now use the orthonormal basis functions (E.18) to make a re-expansion eiR j ·p Yt () = Sτ+ (R j )Yτ (), j = 1, 2, . . . , N, τ

where τ =

(n m )

Sτt (R j ) =

runs triple summation indices similarly to t. Then eiRj ·p Yt ()Yτ () d

(E.26)

S3

and (E.25) becomes

Zj e−iRj ·(p−p ) 8π 2 sin2 (κ/2)

= Zj

tτ τ

∗ 1 τ Stτ (R j ) St (R j )Yτ ()Yτ∗ . n

The orthonormal expansion of ϕ() in (E.17) is denoted as ct Yt (). ϕ() =

(E.27)

(E.28)

t

Substituting (E.27) and (E.28) into (E.17) and equating coefficients, we obtain ∗ 1 τ 1 t ct = St (R j ) Zj S (R j )cτ . p0 n t tτ j

This is an eigenvalue problem P c = p0 c,

(E.29)

where P is an infinite matrix with entries ∗ 1 t Ptt = Sτt (R j ) Zj S (R j ), n τ τ j

where c is an infinite-dimensional vector with entries ct . The value of p0 will yield the energy E from (E.6). One can obtain the wave function ψ(r) from applying the inverse Fourier transform to (p) through (E.16) and (E.8).


203

Appendix F. Derivation of the Cusp Conditions Here we derive the cusp conditions at the singularities of Eq. (5.3). Since the idea is the same for each of the five sets of singularities, we will only treat the most complicated case, r12 = 0. Use the center-of-mass coordinate system, see Appendix G, we can transform (5.2) into the form 2Za 2Zb = − 1 ∇S2 − 1 ∇r2 − − H 2M 2µ 12 |2R a + r 12 | |2R b + r 12 | Za Zb 2Za 2Zb q1 q 2 + − − + , |2R a − r 12 | |2R b − r 12 | r12 R

(F.1)

where S is the center of mass coordinate of two electrons, see Appendix G for the details of the notation. We now define the spherical means of a function. Given a point r 0 ∈ R3 , the spherical means of a function u(r) at r 0 on the sphere with radius ρ is defined to be 1 u(r 0 + ρν) dω, uav,ρ (r 0 ) ≡ (F.2) 4π S1

where Sρ is the sphere with radius ρ centered at r 0 , here ρ = 1; dω = sin θ dθ dφ, where ω represents all the angular variables; ν = (ν1 , ν2 , ν3 ), ν12 + ν22 + ν32 = 1; ν is the unit outward pointing normal vector on S1 . It is easy to see that if u(r) is continuous in a neighborhood of r 0 , then the spherical means just converge to the pointwise value: lim uav,ρ (r 0 ) = u(r 0 ).

ρ→0

over a small 3-dimensional ball Bρ0 with radius ρ0 centered We now integrate H at r12 = 0, for any S ∈ R3 , S = 0: 1 2 2Za 1 2 ∇ ψ(r 12 , S) − ∇ ψ(r 12 , S) − ψ(r 12 , S) − 2M S 2µ 12 |2R a + r 12 | Bρ0

2Zb 2Za ψ(r 12 , S) − ψ(r 12 , S) |2R b + r 12 | |2R a − r 12 | 2Zb − ψ(r 12 , S) |2R b − r 12 |

Za Zb q1 q2 ψ(r 12 , S) + ψ(r 12 , S) dr 12 = 0. + r12 R −

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Note that as ρ0 → 0, the integrals of all the terms above vanish (by their continuity at r12 = 0), except possibly those of −

q1 q2 ψ(r 12 , S). r12

1 2 ∇ ψ(r 12 , S), 2µ 12

Thus, we need only consider −

lim

ρ0 →0 Bρ0

1 2 q1 q2 ψ(r 12 , S) dr 12 = 0. ∇12 ψ(r 12 , S) + 2µ r12

(F.3)

Apply the Gauss Divergence Theorem to the first term of the integral to get 1 ∂ψ 1 2 − ∇ ψ(r 12 , S) dr 12 = − (r 12 , S) dSρ0 2µ 12 2µ ∂r12 Sρ0

Bρ0

=− Bρ0

q1 q2 ψ(r 12 , S) dr 12 . r12

(F.4)

But dSρ0 = ρ02 dω; thus Sρ0

∂ψ dSρ0 = ∂r12

S1

∂ψ dω1 · ρ02 ∂r12

=

ρ02 S1

∂ψ(r 12 , S) 2 ∂ψ(0, S) dω = 4πρ0 , (F.5) ∂r12 ∂r12 av,ρ0

2 dω dr , we where S1 = Sρ0 |ρ0 =1 . By using spherical coordinates, dr 12 = r12 12 have

Bρ0

q1 q2 ψ(r 12 , S) dr 12 = r12

ρ0 0

ρ0 = q1 q2 0

Sρ

q1 q2 2 ψ(r 12 , S) dω r12 dr12 r12

ψ(0, S) + ε(r 12 , S) dω r12 dr12 ,

Sρ

where ε(r 12 , S) → 0 as r 12 → 0. This follows from the fact that ψ(r 12 , S) is continuous at r 12 = 0 for any S.

ELECTRON MOLECULAR BONDS (Continuing from the above) ρ0

ρ2 r12 dr12 + ε˜ (ρ0 , S)4π 0 2

≡ q1 q2 ψ(0, S)

205

0

ρ02 + 2πρ02 ε˜ (ρ0 , S). 2 From (F.3)–(F.6), we now get ρ2 1 1 2 ∂ψ(0, S) = − ρ02 q1 q2 ψ(0, S) + 0 ε˜ (ρ0 , S). − ρ0 2µ ∂r12 2 2 av,ρ0 = q1 q2 ψ(0, S)4π

(F.6)

Dividing all the terms above by ρ02 and let ρ0 → 0, we have ε˜ (ρ0 , S) → 0 and, therefore, ∂ψ(0, S) = µq1 q2 ψ(0, S), for all S ∈ R3 . lim (F.7) ρ0 →0 ∂r12 av,ρ0 (This is Eq. (2.11) in Patil et al. [15].) Similarly, one can derive the electron– nucleus cusp condition at, e.g., r1a = 0, for (5.1) to be

∂ψ(r 1 , r 2 ) = −m1 Za ψ(r 1 , r 2 ) r =0 , for all r 2 . lim (F.8) 1a ρ0 →0 ∂r1a av,ρ0 T HEOREM F.1. Assume that m1 = m2 and q1 = q2 = −1 in (5.1). Let (S, r 12 ) denote the CM and relative coordinates and ω12 denote the angular variables of the vector r 12 . Let ψ be a nontrivial solution of (5.1) such that its local Taylor expansion near r12 = 0 is of the form 2 , for r12 small, ψ(r 1 , r 2 ) = C0 (S) + C1 (S)r12 + O r12 (F.9) where C0 (S) and C1 (S) are independent of ω12 while the remainder satisfies 2 ) = C (r , S)r 2 for some bounded function C which depends on r O(r12 3 12 3 12 12 and S. Then C1 (S) = 12 C0 (S) for all S and, consequently, 2 1 ψ(r 1 , r 2 ) = C0 (S) 1 + r12 + O r12 (F.10) . 2 P ROOF. We substitute the RHS of (F.9) into (F.7). There is no need to take the spherical mean on the left of (F.7) anymore as the dominant term on the RHS of (F.10) does not depend on the angular variables of r12 . We therefore obtain C1 (S) = µq1 q2 C0 (S) = q1 = q2 = −1.

1 C0 (S), 2

as µ =

m1 m2 1 = , m1 + m2 2

206

G. Chen et al.

Hence 2 1 ψ(r 1 , r 2 ) = C0 (S) + C0 (S)r12 + O r12 2 2 1 , = C0 (S) 1 + r12 + O r12 2

which is (F.9).

It looks as though the condition (F.9) is somewhat contrived. However, useful application can be seen shortly, in Theorem F.4. Similarly, we can obtain the cusp conditions at r1a = 0, r1b = 0, r2a = 0 and r2b = 0, as given in the following. T HEOREM F.2. Let ψ be either a nontrivial solution of (5.1) or a trial wave function for (5.1). Let r1a and ω1a denote, respectively, the radial and angular variables of the vector r 1a . Assume that for r1a sufficiently small, ψ satisfies the Taylor expansion 2 ψ(r 1 , r 2 ) = C0 (r 2 ) + C1 (r 2 )r1a + O r1a (F.11) , r1a small, for some functions C0 and C1 depending on r 2 only, where the remainder satisfies 2 ) = C (r , r )r 2 for some bounded function C depending on r and r . O(r1a 3 1 2 1a 3 1 2 Then 2 ψ(r 1 , r 2 ) = C0 (r 2 )(1 − m1 Za r1a ) + O r1a (F.12) , r1a small, 2 ) = C (r , r )r 2 . where O(r1a 3 1 2 1a

P ROOF. The kinetic energy term −1/(2m1 )∇12 , after the translation z1 → z1 +

R , 2

cf. (G.1) and Fig. 27 for notation,

becomes −1/(2m1 )∇r21a which is centered at (x1 , y1 , z1 ) = (0, 0, −R/2). Now apply the Hamiltonian = − 1 ∇r2 − Za H 2m1 1a r1a 1 2 Zb Za Zb q1 q 2 Za Zb + − ∇2 − − − + + 2m2 r1b r2a r2b r12 R

(F.13)

to (F.11). We need only focus our attention locally near r1a = 0 and ignore the terms in the bracket of (F.13) as they have no effect on the singularity at r1a = 0.


207

We obtain 2 Za − E ψ = − 1 1 ∂ r1a H C0 (r 2 ) + C1 (r 2 )r1a · C1 (r 2 ) − 2 2m1 r1a ∂r1a r1a

+ higher order terms in r1a 1 2C1 (r 2 ) Za · C0 (r 2 ) =− − + higher order terms in r1a . 2m1 r1a r1a

To eliminate the singularity at r1a = 0 above, it is necessary that C1 (r) + Za C0 (r 2 ) = 0. m1

The above gives (F.12), as desired.

E XAMPLE F.3. The trial wave function ψ(r 1 , r 2 ) = φ(r 1a )φ(r 2b ), where φ(r) = e−αr is a one-centered orbital, satisfies condition (F.11) of Theorem F.2. One can easily apply Theorem F.2 to other singularities at r1b = 0, r2a = 0 and r2b = 0. T HEOREM F.4. Let φ(r) be a sufficiently smooth function. Let the Hamiltonian represents a homonuclear case: 2 = − 1 ∇12 − 1 ∇22 − Z − Z − Z − Z + 1 + Z . H 2 2 r1a r1b r2a r2b r12 R

Then the product function

1 ψ(r 1 , r 2 ) = φ(r 1 )φ(r 2 ) 1 + r12 2

satisfies the interelectronic cusp condition as given in (F.9). P ROOF. We represent the variables r 1 and r 2 in terms of the CM coordinates 1 r 1 = S + (r 1 − r 2 ) = S + 2 1 r 2 = S − (r 1 − r 2 ) = S − 2

1 r 12 , 2 1 r 12 , 2

where S = 12 (r 1 + r 2 ). Then for r12 small, by Taylor’s expansion, 1 φ(r 1 ) = φ S + r 12 2 11 T 1 r · D 2 φ(S) · r 12 + · · · , = φ(S) + ∇φ(S) · r 12 + 2 2! 4 12

208

G. Chen et al.

1 φ(r 2 ) = φ S − r 12 2 1 11 T = φ(S) − ∇φ(S) · r 12 + r · D 2 φ(S) · r 12 − · · · . 2 2! 4 12 Therefore 1 ψ = φ(r 1 )φ(r 2 ) 1 + r12 2 1 11 T = φ(S) + ∇φ(S) · r 12 + r 12 · D 2 φ(S) · r 12 + · · · , 2 2! 4 11 T 1 1 r 12 · D 2 φ(S) · r 12 ± · · · 1 + r12 φ(S) − ∇φ(S) · r 12 + 2 2! 4 2 1 = φ 2 (S) 1 + r12 + quadratic or higher order terms involving r12 . 2

Hence condition (F.9) is satisfied.

Appendix G. Center of Mass Coordinates for the Kinetic 1 1 ∇ 21 − 2m ∇ 22 Energy − 2m 1 2 The coordinates for electron 1 and 2 are, respectively, r 1 = (x1 , y1 , z1 ),

r 2 = (x2 , y2 , z2 ).

(G.1)

The kinetic energy operator is = − 1 ∇12 − 1 ∇22 , H 2m1 2m2 where ∇j2 =

∂2 ∂2 ∂2 + 2 + 2, 2 ∂xj ∂yj ∂zj

j = 1, 2.

Define the CM (center-of-mass) coordinate S: m1 r 1 + m2 r 2 S= m1 + m2 and (relative coordinate) r 12 = r 1 − r 2 . Here we derive the kinetic energy term in coordinates S, r12 . For any differentiable scalar function f (r 1 , r 2 ), we have df = ∇1 f · dr 1 + ∇2 f · dr 2 = ∇S f · dS + ∇12 f · dr 12 ,


209

F IG . 27. Various vectors are defined in this diagram.

where ∇S and ∇12 are the gradient operators with respect to the variables of S and r 12 . Then dr 1 dS [∇1 f, ∇2 f ] (G.2) = [∇S f, ∇12 f ] . dr 12 dr 2 But

dS dr 12

=

m1 m1 +m2

m2 m1 +m2

1

−1

Hence from (G.2) and (G.3), [∇1 f, ∇2 f ] = [∇S f, ∇12 f ]

dr 1 . dr 2

(G.3)

m1 m1 +m2

m2 m1 +m2

1

−1

;

1 − 2m 0 1 2 1 2 ∇1 1 H =− ∇ − ∇ = [∇1 , ∇2 ] ∇2 2m1 1 2m2 2 0 − 2m1 2 m1 m2 − 1 0 2m1 m1 +m2 m1 +m2 = [∇S , ∇12 ] 1 1 −1 0 − 2m 2 m1 1 ∇S m1 +m2 × m2 ∇12 −1 m1 +m2 − 2(m11+m2 ) 0 ∇S = [∇S , ∇12 ] ; ∇12 0 −1 1 + 1 2 m1

m2

210 or

G. Chen et al. 1 1 1 1 2 + ∇S2 − ∇12 2(m1 + m2 ) 2 m1 m2 1 2 1 2 =− ∇S − ∇ , 2M 2µ 12

=− H

where M ≡ m1 + m2 is the total mass of electrons and µ ≡ mass.

m1 m2 m1 +m2

is the reduced

Appendix H. Verifications of the Cusp Conditions for Two-Centered Orbitals in Prolate Spheroidal Coordinates In the work of Patil (see Eq. (2.15) in [56]), he indicated that for the ground state ψ (i.e., with azimuth quantum number m = 0) for the molecular ion with Hamiltonian = − 1 ∇12 − Za − Zb , H (H.1) 2 ra rb the “coelescense” condition at rb = 0 can be expressed as

1 ∂ψ

1 1 ∂ψ

+ = −Zb , (H.2) 2 ψ ∂rb θ=0 ψ ∂rb θ=π rb =0 the angle θ is introduced in Fig. 28. He then indicates that the above is “essentially the same” as Kato’s cusp condition. Actually, at least two ways are viable, which are going to be described below through some concrete examples. The first way takes a limiting approach in a similar spirit as Patil’s (H.2). The second way utilizes an explicit representation.

F IG . 28. A local spherical coordinate system (ra , θ, φ).


211

E XAMPLE H.1. For the Hamiltonian (H.1), motivated by the Jaffé solution (4.26), let us consider a trial wave function for the ground state: ψ(λ, µ) = e−αλ 1 + B2 P2 (µ) . (H.3) We want to consider the cusp condition in terms of the undetermined coefficients α and B2 in (H.3). Recall from (A.3) that (i)

ra → 0

is equivalent to

λ → 1, µ → −1;

(ii) rb → 0 is equivalent to λ → 1, µ → 1.

(H.4) (H.5)

The singularities in the Laplacian ∇ 2 in prolate spheroidal coordinates, according to (A.3), are discerned to be contained in λ2

1 1 1 = · . 2 λ+µ λ−µ −µ

, the following: Thus, we deduce that for (H.1), after substituting (H.3) into H (i) ψ ∼ H

1 F1 (λ, µ), λ+µ

for ra ≈ 0, where λ 1, µ ≈ −1,

(H.6)

where we have dropped terms not containing the singularity (λ + µ)−1 and collected the dominant terms corresponding to singularity (λ + µ)−1 in 2 F1 (λ, µ) = − 2 −2αλe−αλ 1 + B2 P2 (µ) + λ2 − 1 α 2 R (λ − µ) × e−αλ 1 + B2 P2 (µ) + e−αλ 3B2 1 − 3µ2 2Za −αλ − 1 + B2 P2 (µ) e R (ii) 1 G1 (λ, µ), λ−µ where, similarly, Hψ ∼

for rb ≈ 0, where λ 0, µ ≈ 1,

(H.7)

−2αλe−αλ 1 + B2 P2 (µ) + λ2 − 1 α 2 e−αλ × 1 + B2 P2 (µ) + e−αλ 3B2 1 − 3µ2 2Zb −αλ − 1 + B2 P2 (µ) . e R For (H.6) and (H.7) to stay bounded, it is necessary that F1 (1, −1) = 0 and G1 (1, 1) = 0. 2Za −α 1 F1 (1, −1) = − 2 −2αe−α (1 + B2 ) − 6B2 e−α − e (1 + B2 ) = 0, R R G1 (λ, µ) = −

2

R 2 (λ + µ)

212

G. Chen et al.

i.e., 3B2 (H.8) . 1 + B2 2Zb −α 1 G1 (1, 1) = − 2 −2αe−α (1 + B2 ) − 6B2 e−α − e (1 + B2 ) R R = 0,

α = RZa −

i.e., α = RZb −

3B2 . 1 + B2

(H.9)

These are the cusp conditions at ra = 0 and rb = 0. However, we need to remark that F1 (1, −1) = 0 and G1 (1, 1) = 0 are only necessary conditions for the desired boundedness because the limits in (H.6) and (H.7) do not exist in general as λ and µ may be related in infinitely many different ways to yield totally different limits as λ → 1 and µ → ±1. Thus, the above estimation approach has an ad hoc nature. Nevertheless, (H.8) and (H.9) do provide correct answers as to be cross-validified with (H.16) and (H.17) in Example H.2. In inspecting (H.8) and (H.9), we see that they are consistent when and only when Za = Zb , i.e., the homonuclear case. Therefore, (H.3) would not be a good choice of a trial wave function in the heteronuclear case. For the heteronuclear case, taking hints from the exact solutions (4.30) and (4.35), we choose the trial wave function (λ, µ) = e−αλ eβµ 1 + B1 P1 (µ) + B2 P2 (µ) . Here, α may be different to β even though the exact solution says α = β. We have ∼ H

1 F2 (λ, µ), λ+µ

where the singular terms are collected in −2αλe−αλ eβµ 1 + B1 P1 (µ) + B2 P2 (µ) + λ2 − 1 α 2 e−αλ eβµ 1 + B1 P1 (µ) + B2 P2 (µ) + e−αλ eβµ β 2 − β 2 µ2 − 2βµ 1 + B1 P1 (µ) + B2 P2 (µ) + 2β − 2βµ2 − 2µ (B1 + 3B2 µ) ! 2Za −αλ βµ e 1 + B1 P1 (µ) + B2 P2 (µ) , e + 1 − µ2 3B2 − R

F2 (λ, µ) = −

2

R 2 (λ − µ)


213

with the terms corresponding to the dominant singularity in 1 F2 (1, −1) = − 2 e−α−β −2α(1 − B1 + B2 ) + 2β(1 − B1 + B2 ) R 2Za −α−β e + 2(B1 − 3B2 ) − (1 − B1 + B2 ), R i.e., (B1 − 3B2 ) . 1 − B1 + B2 Similarly, the behavior near rb = 0 gives α − β = RZa +

∼ H

1 G2 (λ, µ), λ−µ

where 2 −2αλe−αλ eβµ 1 + B1 P1 (µ) + B2 P2 (µ) R 2 (λ + µ) + λ2 − 1 α 2 e−αλ eβµ 1 + B1 P1 (µ) + B2 P2 (µ) + e−αλ eβµ β 2 − β 2 µ2 − 2βµ 1 + B1 P1 (µ) + B2 P2 (µ) + 2β − 2βµ2 − 2µ (B1 + 3B2 µ) ! 2Zb −αλ βµ e e 1 + B1 P1 (µ) + B2 P2 (µ) ; + 1 − µ2 3B2 − R 1 −α+β −2α(1 + B1 + B2 ) − 2β(1 + B1 + B2 ) G2 (1, 1) = − 2 e R 2Zb −α+β e − 2(B1 + 3B2 ) − (1 + B1 + B2 ), R

G2 (λ, µ) = −

i.e., (B1 + 3B2 ) . 1 + B1 + B2 So, the cusp conditions give α + β = RZb −

B1 = −

3(RZa + 2β − RZb ) , 2(3 + 4α − 2RZa − 2RZb − RZa α − RZa β + R 2 Za Zb − αRZb + βRZb + α 2 − β 2 )

B2 = −

(2α 2 + 2α − 2RZa α − 2αRZb − RZa + 2βRZb − 2β 2 + 2R 2 Za Zb − RZb − 2RZa β) . 2(3 + 4α − 2RZa − 2RZb − RZa α − RZa β + R 2 Za Zb − αRZb + βRZb + α 2 − β 2 )

If, in addition, we set α = β, then 3(RZa + 2α − RZb ) , 2(2RZa α − R 2 Za Zb + 2RZa − 4α + 2RZb − 3) (−2R 2 Za Zb + RZa + 4RZa α + RZb − 2α) B2 = − . 2(2RZa α − R 2 Za Zb + 2RZa − 4α + 2RZb − 3)

B1 =

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G. Chen et al.

E XAMPLE H.2 (Verification of the cusp conditions via explicit calculations). Consider the same Hamiltonian as given in (H.1). We translate the origin from (0, 0, 0) to (0, 0, −R/2) and set up spherical coordinates (ra , θ, φ) as shown in Fig. 28. From the cosine law, cf. Fig. 28, 1/2 rb = ra2 + R 2 − 2Rra cos θ 1/2 2 cos θ 1 cos θ =R 1− =R 1− ra + 2 ra2 ra + O ra2 . (H.10) R R R ψ near ra = 0, we need only As we are exploring the singularity behavior of H concentrate on the dominant terms of ra and, thus, we drop O(ra2 ) in (H.10) and approximate rb by cos θ rb = R 1 − ra . R Using (A.3), we therefore obtain  ra   λ = (1 − cos θ ) + 1, R r   µ = a (1 + sin θ ) − 1. R The transformation (H.11) will greatly facilitate our calculations. Consider the trial wave function (H.3) again here. We have

∂ψ ∂µ 3 ∂λ B2 −αλ 2 =e + B2 µ + 3B2 µ −α 1− . ∂ra ∂ra 2 2 ∂ra From (H.11), we have  1 λ−1 ∂λ   = (1 − cos θ ) = ,  ∂ra R ra ∂µ 1 µ+1    = (1 + cos θ ) = , ∂ra R ra

(H.11)

(H.12)

(H.13)

so we can use (H.11) and (H.13) to rewrite (H.12) as 2 ∂ψ 3 λ−1 B2 −α[(λ−1)+1] 2 =e + B2 (µ + 1) − 1 −α 1− ∂ra ra 2 2

µ+1 + 3B2 (µ + 1) − 1 · ra −α e = · e−α(λ−1) − α(1 + B2 )(λ − 1) + 3B2 (µ + 1) ra 12 3 0 O(ra )

ELECTRON MOLECULAR BONDS 3 2 + αB2 2(λ − 1)(µ + 1) + 3B2 (µ + 1) 2 0 12 3

215

O(ra2 )

3 2 − B2 α(λ − 1)(µ + 1) . 2 0 12 3

(H.14)

O(ra3 )

If we set ra = ρ and then substitute (H.14) into the left-hand side (LHS) of (F.8), the two rightmost brackets on the RHS of (H.14) will become O(ra ) = O(ρ) and O(ra2 ) = O(ρ 2 ), and then vanish after ρ → 0. So we only need to consider the spherical mean of ∂ψ/∂ra over a sphere with radius ρ which is given by −α e ρ 1 −αρ/R(1−cos θ) e = −α(1 + B2 ) (1 − cos θ ) ρ R 4πρ 2 Sρ

ρ − 3B2 (1 + cos θ ) ρ 2 dω R 2ππ 1 1 1 −α −αρ/R αρ/R cos θ e α(1 + B2 ) · + 3B2 · =− e e 4π R R 0 0

1 1 cos θ sin θ dθ dφ + −α(1 + B2 ) + 3B2 R R 1 1 k 1 e − e−k α(1 + B2 ) + 3B2 = − e−α−αρ/R 2 k R

1 1 k 1 k −k −k + − 2 e −e + e +e −α(1 + B2 ) + 3B2 · , (H.15) k R k with k ≡ αρ/R. Letting ρ → 0, we have k → 0, and 1 (RHS) of (H.15) = −e−α α(1 + B2 ) + 3B2 · R ≡ (RHS) of (F.8) (modified for the case of molecular ion) = −Za e−α (1 + B2 ), i.e., α(1 + B2 ) + 3B2 = RZa (1 + B2 ),

(H.16)

which is exactly (H.8). Similarly, at rb = 0, we can obtain α(1 + B2 ) + 3B2 = RZb (1 + B2 ), which is exactly (H.9).

(H.17)

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G. Chen et al.

Appendix I. Integrals with the Heitler–London Wave Functions The integrals involved in Eq. (6.8) can be separated into eight types of elementary integrals: 1 1 (i) a(1) −∇12 a(1) dx = b(2) −∇22 b(2) dy 2 2 R3

1 = 2 (ii)

1 2

1 a(1) −∇12 b(1) dx = 2 α2

αJ = R3

(vi)

αK = R3

(vii)

b(1) −∇12 b(1) dx

R3

a(2) −∇12 b(2) dx

=

α2 ; 2

R3

αJ =

1 1 1 a 2 (1) − ; dx = α − + e−2w 1 + r1b w w 1 a(1)b(1) − dx = −αe−w (1 + w); r1b

R6

(viii)

w2 a(1)b(1) dx = e−w 1 + w + ; 3

S= R3

(v)

1 2 =− 1+w− w ; 2 2 1 a 2 (1) − dx = −α; r1a

R3

(iv)

1 a(2) −∇22 a(2) dy = 2

R3

R3

(iii)

R3

1 a 2 (1)b2 (2) dx dy r12

1 11 3 1 1 =α − e−2w + + w + w2 ; w w 8 4 6 1 αK = a(1)b(1)a(2)b(2) dx dy r12 R6

=

1 1 25 23 α −e−2w − + w + 3w 2 + w 3 5 8 4 3

6 2 + S (γ + ln w) + S 2 Ei (−4w) − 2SS Ei (−2w) , w


217

where w ≡ αR; 1 S ≡ ew 1 − w + w 2 ; 3 1 γ = Euler’s constant =

1 − e−t dt − t

∞

0

e−t dt = 0.57722 . . . ; t

1

∞ Ei (x) = integral logarithm = −(P.V.) −x

e−t dt t

(for x > 0),

here P.V. means “principal value” of a singular integral.

Appendix J. Derivations Related to the Laplacian for Section 6.4 Evaluation of the Laplacian in the prolate spheroidal coordinates for the general form of the wave function which includes electron–electron correlations explicitly is a demanding task. Here we provide the details of the calculations. The general form of each term in the wave function is s (1, 2) =

1 s (1)s (2)Ps (r12 ) 2π

(J.1)

with s (1) = Fs (λ1 )Gs (µ1 )

(J.2)

s (λ2 )M s (µ2 ). s (2) = F

(J.3)

and

It can readily be seen that the Laplacian ∇12 operates only on s (1). According to (A.6): ∂ (λ21 − µ21 ) ∂2 4 ∂ 2 λ1 − 1 ∇12 s (1) = + 2 2 2 2 2 2 ∂λ1 ∂λ1 R (λ1 − µ1 ) (λ1 − 1)(1 − µ1 ) ∂φ1 +

∂ ∂ 1 − µ21 s (1) ∂µ1 ∂µ1

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G. Chen et al.

2 (λ1 − µ21 )Fs Gs ∂ 2 Ps 4 2 2 R 2 (λ1 − µ1 ) (λ21 − 1)(1 − µ21 ) ∂φ12 ∂(Fs Ps ) ∂ 2 ∂ 2 ∂(Gs Ps ) λ −1 1 − µ1 + Fs . + Gs ∂λ1 1 ∂λ1 ∂µ1 ∂µ1 (J.4) We first single out part of the second term in the square bracket above for further evaluation to obtain

∂(Fs Ps ) ∂ 2 ∂Fs ∂Ps ∂ 2 λ1 − 1 λ1 − 1 Ps = + Fs ∂λ1 ∂λ1 ∂λ1 ∂λ1 ∂λ1 ∂Fs 2 ∂Fs ∂Ps ∂ 2 = Ps + 2 λ1 − 1 λ1 − 1 ∂λ1 ∂λ1 ∂λ1 ∂λ1 ∂Ps ∂ 2 λ −1 + Fs . (J.5) ∂λ1 1 ∂λ1 =

Similarly, part of the third term inside the square bracket of (J.4) becomes

∂(Gs Ps ) ∂Gs ∂Ps ∂ ∂ = + Gs 1 − µ21 1 − µ21 Ps ∂µ1 ∂µ1 ∂µ1 ∂µ1 ∂µ1 ∂ ∂G ∂Gs ∂Ps s 1 − µ21 + 2(1 − µ21 ) = Ps ∂µ1 ∂µ1 ∂µ1 ∂µ1 ∂ ∂P s + Gs . (J.6) 1 − µ21 ∂µ1 ∂µ1 We can rewrite the Laplacian to take into account the separable functional dependence of the three parts of the wave function, that is, ∇ 2 Fs ∇ 2 Gs ∇ 2 Ps 1 2∇1 Fs · ∇1 Ps + 1 + ∇12 s (1) = 1 + 1 s (1) Fs Gs Ps Fs Ps 2∇1 Gs · ∇1 Ps + , Gs Ps

(J.7)

since ∇1 s (1) =

2(λ21 − 1)1/2 ∂s (1) 2(1 − µ21 )1/2 ∂s (1) eλ1 + e µ1 2 2 R(λ1 − µ1 )1/2 ∂λ1 R(λ21 − µ21 )1/2 ∂µ1 ∂s (1) 2 + e φ1 , (J.8) 2 2 1/2 1/2 ∂φ1 R(λ1 − 1) (1 − µ1 )

where, eλ1 , eµ1 , and eφ1 are the unit vectors pointing to the respective directions. Now, we use the actual functional forms of various parts of the wave function to complete the evaluation of the expectation value of the Laplacian. Setting


219

s Fs (λ1 ) = e−αλ1 λm 1 , we obtain

∇12 Fs = = =

=

=

∂ ∂ 2 4 Fs λ1 − 1 2 ∂λ1 − µ1 ) ∂λ1 ! ∂ 2 4 s λ1 − 1 −αe−αλ1 λm + ms e−αλ1 λ1ms −1 1 2 2 R 2 (λ1 − µ1 ) ∂λ1 ∂ 4 s −αe−αλ1 λ1ms +2 − λm 1 2 2 2 ∂λ R (λ1 − µ1 ) 1 ! + ms e−αλ1 λ1ms +1 − λ1ms −1 4 s α 2 e−αλ1 λ1ms +2 − λm 1 2 2 R 2 (λ1 − µ1 ) − αe−αλ1 (ms + 2)λ1ms +1 − ms λ1ms −1 ms −2 s + ms e−αλ1 (ms + 1)λm 1 − (ms − 1)λ1 ! − ms αe−αλ1 λ1ms +1 − λ1ms −1 1 4Fs 2 2 α − 1 − 2α (m + 1)λ − m λ s 1 s 1 λ1 R 2 (λ21 − µ21 )

1 + ms (ms + 1) − (ms − 1) 2 . λ1 R 2 (λ21

j

Similarly, setting Gs (µ1 ) = µ1s , we obtain ∂ ∂ 4 1 − µ21 Gs 2 ∂µ1 − µ1 ) ∂µ1 ∂ 4 j −1 1 − µ21 js µ1s = 2 2 2 R (λ1 − µ1 ) ∂µ1 4Gs 1 js (js − 1) 2 − js (js + 1) . = R 2 (λ21 − µ21 ) µ1

∇12 Gs =

R 2 (λ21

(J.9)

Since the effect of the Laplacian on a given function is coordinate-free, we can consider the evaluation of ∇12 Ps (r12 ), through the spherical coordinates for a general function f (r12 ) of r12 to obtain ∇12 f (r12 ) = ∇r212 f (r12 ) =

1 d 2 df (r12 ) r12 . 2 dr dr12 r12 12

(J.10)

, Thus, with Ps (r12 ) = r12

∇12 Ps =

( + 1) 1 2 −2 −1 r12 l(l − 1)r12 = + 2r12 r12 Ps . r12 r12

(J.11)

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G. Chen et al.

The other terms involved are ∂Ps −1 ∂r12 = lr12 ∂λ1 ∂λ1 2 R lPs M 2λ1 − 2λ2 µ1 µ2 − 2 cos(φ1 − φ2 )2λ1 = 2 8r12 λ1 − 1 Mλ1 R 2 lPs λ cos(φ − λ µ µ − − φ ) = 1 2 1 2 1 2 2 4r12 (λ21 − 1) 2 (λ21 − 1) λ1 R 2 lPs λ = − 1 − λ µ µ − M cos(φ − φ ) 2 1 2 1 2 1 2 (λ2 − 1) λ1 4r12 1 4 2 λ1 2λ2 µ1 µ2 R 2 lPs 2 2 2 2 , r + λ − λ − µ − µ + = 1 2 1 2 2 (λ2 − 1) R 2 12 λ1 8r12 1 (J.12) and

∂Ps R 2 lPs M 2µ1 − 2λ1 λ2 µ2 − cos(φ1 − φ2 )(−2µ1 ) = 2 ∂µ1 8r12 (1 − µ21 ) Mµ1 R 2 lPs µ cos(φ − λ λ µ + − φ ) = 1 1 2 2 1 2 2 4r12 (1 − µ21 ) (1 − µ21 ) µ1 R 2 lPs − 1 − µ21 + λ1 λ2 µ2 =− 2 2 µ1 4r12 (1 − µ1 ) − M cos(φ1 − φ2 )

4 2 µ1 2λ1 λ2 µ2 R 2 lPs 2 2 2 2 . r − λ − λ + µ − µ + 1 2 1 2 2 (1 − µ2 ) R 2 12 µ1 8r12 1 (J.13) Upon putting the above together, the Laplacian operation part of the Hamiltonian takes the form =−

1 ∇ 2 s (1) s (1) 1

2 ∇ 2 Gs ∇ 2 Ps ∇1 Fs 2∇1 Fs · ∇1 Ps 2∇1 Gs · ∇1 Ps = + 1 + 1 + + Fs Gs Ps Fs Ps Gs Ps 4 1 2 2 = α λ − 1 − 2α (m + 1)λ − m s 1 s 1 λ1 R 2 (λ21 − µ21 )

1 + ms (ms + 1) − (ms − 1) 2 λ1

ELECTRON MOLECULAR BONDS 221 4 1 ( + 1) + js (js − 1) 2 − js (js + 1) + 2 2 2 2 R (λ1 − µ1 ) µ1 r12 1 (−αλ1 + ms ) + 2 (λ1 − µ21 ) 2λ2 µ1 µ2 4 2 2 2 2 2 r12 + λ1 − λ2 − µ1 − µ2 + × 2 λ1 r12 R 2 1 4 2 2λ1 λ2 µ2 2 2 2 2 . j − 2 r − λ − λ + µ − µ + s 1 2 1 2 2 R 2 12 µ1 (λ1 − µ21 ) r12 (J.14) As before we can construct the inter-term expectation value integral for the Laplacian using the above relation. Introducing the function X ν (m, n, j, k; ) = Z ν (m, n + 2, j, k; ) − Z ν (m, n, j, k + 2; ),

(J.15)

and defining (m, n, j, k; ) = (mr , nr , jr , kr ; r ) + (ms , ns , js , ks ; s ),

(J.16)

we obtain r nr jr kr r 2 ms ns js ks s 2

∇1 r,s = λm 1 λ2 µ1 µ2 r12 ∇1 λ1 λ2 µ1 µ2 r12 4 = 2 α 2 X ν (m + 2, n, j, k; ) R ! + −α 2 + (ms − js )(ms + js + 1 + s ) X ν (m, n, j, k; ) − 2α(ns + 1)X ν (m + 1, n, j, k; ) + 2αns X ν (m − 1, n, j, k; ) + ms (ms − 1)X ν (m − 2, n, j, k; ) + js (js − 1) × X ν (m, n, j − 2, k; ) + s (s + 1) X ν (m, n + 2, j, k; − 2) ! − X ν (m, n, j, k + 2; − 2) − s αX ν (m + 1, n, j, k; ) − s αX ν (m + 3, n, j, k; − 2) + s αX ν (m + 1, n + 2, j, k; − 2) + s αX ν (m + 1, n, j + 2, k; − 2) + s αX ν (m + 1, n, j, k + 2; − 2) − 2s αX ν (m, n + 1, j + 1, k + 1; − 2) + s (ms + js )X ν (m + 2, n, j, k; − 2) − s (ms − js )X ν (m, n + 2, j, k, − 2) − s (ms + js )X ν (m, n, j + 2, k; − 2) − s (ms − js )X ν (m, n, j, k + 2; − 2)

222

G. Chen et al. − 2s ms X ν (m − 1, n + 1, j + 1, k + 1; − 2) − 2s js X ν (m + 1, n + 1, j − 1, k + 1; − 2).

Thus we have furnished complete details of the electronic kinetic energy calculations.

Appendix K. Recursion Relations and Their Derivations for Section 6.4 In this appendix we provide simple proofs of the recursion relations which are needed in the analytical calculations.

A PPENDIX K.1. A(m; α) ∞ A(m; α) ≡

m −αλ

λ e

dλ = λ

∞

+ m λm−1 e−αλ dλ −α 1 α

me

1

−αλ ∞

1

1 −α e + mA(m − 1; α) . = α

(K.1)

When m = 0, ∞ A(0; α) =

e−αλ dλ =

e−α . α

(K.2)

1

The recurrence relation can be used in succession to give m e−α m! 1 A(m; α) = . α (m − ν)! α ν

(K.3)

ν=0

A PPENDIX K.2. F (m; α) The definition is ∞ F (m; α) ≡

λm e−αλ Q0 (λ) dλ.

1

To prove the recurrence relation for F (m; α),

(K.4)


223

mF (m − 1; α) − (m − 2)F (m − 3; α) ∞ m−1 = mλ − (m − 2)λm−3 e−αλ Q0 (λ) dλ 1

= λ −λ m

m−2

e

−αλ

∞ Q0 (λ) 0 + α

∞

dλ λm − λm−2 e−αλ Q0 (λ)

1

∞ − 1

1 1 1 dλ λm − λm−2 e−αλ − 2 λ+1 λ−1

= α F (m; α) − F (m − 2; α) +

∞

dλλm−2 e−αλ .

1

So, the recurrence relation is 1 mF (m − 1; α) α − (m − 2)F (m − 3; α) − A(m − 2; α) .

F (m; α) = F (m − 2; α) +

(K.5)

The initial conditions for F (m; α) are already given in (6.44) and (6.45).

A PPENDIX K.3. S(m, n; α) S(m, n; α) is defined by ∞ S(m, n; α) ≡

λ1 dλ1

1

n −α(λ1 +λ2 ) dλ2 λm . 1 λ2 e

(K.6)

1

The recurrence relation is S(m, n; α) =

1 mS(m − 1, n; α) + A(m + n; 2α) , α

(K.7)

S(0, n; α) =

1 A(n; 2α). α

(K.8)

with

By using S(m, n; α), the following integrals can be represented in terms of A(m; α) and S(m, n; α):

224

G. Chen et al. ∞

λ1 dλ1

1

n −α(λ1 +λ2 ) dλ2 λm 1 λ2 e

1

∞ =

−αλ1 dλ1 λm 1 e

1

λ1 1

∞ =

−αλ1 dλ1 λm 1 e

1

dλ2 λn2 e−αλ2

λ1 λn2 −αλ2

λ1 n n−1 −αλ2 e

+ α dλ2 λ2 e −α 1 1

e−α

A(m + n; 2α) + A(m; α) α α ∞ λ1 n n−1 −α(λ1 +λ2 ) + dλ1 dλ2 λm e 1 λ2 α

=−

1

1

n 1 1 n! =− A(m + n − ν; 2α) α αν (n − ν)! ν=0

e−α

+ =−

α

n

n!

ν=0

α ν (n − ν)!

A(m; α)

n n n! 1 αs n! e−α + A(m; α) A(m + s; 2α) n ν α α s! α α (n − ν)! s=0

ν=0

= −S(n, m; α) + A(n; α)A(m; α).

(K.9)

Furthermore, ∞

∞ dλ1

1

n −α(λ1 +λ2 ) dλ2 λm 1 λ2 e

1

= A(m; α)A(n; α) ∞ λ1 ∞ ∞ m n −α(λ1 +λ2 ) n −α(λ1 +λ2 ) + dλ1 dλ2 λm = dλ1 dλ2 λ1 λ2 e 1 λ2 e 1

1

1

λ1

∞

λ1

∞

λ1

=

dλ1 1

1

n −α(λ1 +λ2 ) dλ2 λm 1 λ2 e

+

dλ1 1

−α(λ1 +λ2 ) dλ2 λn1 λm 2 e

1

= −S(n, m; α) + A(n; α)A(m; α) − S(m, n; α) + A(m; α)A(n; α). (K.10)


225

So S(m, n; α) + S(n, m; α) = A(m; α)A(n; α).

(K.11)

A PPENDIX K.4. T (m, n; α) The definition is T (m, n; α) ≡

m m! α ν F (n + ν; 2α). ν! α m+1

(K.12)

ν=0

The recurrence relation and T (0, n; α) are T (m, n; α) =

1 mT (m − 1, n; α) + F (m + n; 2α) α

(K.13)

T (0, n; α) =

1 F (n; 2α). α

(K.14)

and

A PPENDIX K.5. H0 (m, n; α) By definition, ∞ H0 (m, n, α) ≡

∞ dλ1

1

1

∞

λ1

=

dλ1 1

n −α(λ1 +λ2 ) dλ2 λm Q0 (λ> ) 1 λ2 e

n −α(λ1 +λ2 ) dλ2 λm Q0 (λ1 ) 1 λ2 e

1

∞ +

λ1 dλ1

1

−α(λ1 +λ2 ) dλ2 λn1 λm Q0 (λ1 ). 2 e

1

The first term on the right-hand side above yields ∞

λ1 dλ1

1

n −α(λ1 +λ2 ) dλ2 λm Q0 (λ1 ) 1 λ2 e

1

∞ = 1

−αλ1 dλ1 λm Q0 (λ1 ) 1 e

λ1 1

dλ2 λn2 e−αλ2

(K.15)

226

G. Chen et al. ∞ =

−αλ1 dλ1 λm Q0 (λ1 ) 1 e

1

λ1 λn2 −αλ2

λ1 n n−1 −αλ2 e

+ α dλ2 λ2 e −α 1 1

F (m + n; 2α) e−α + F (m; α) + =− α α F (m + n; 2α) e−α =− + F (m; α) + α α

n H0 (m, n − 1; α) α n F (m + n − 1; 2α) − α α −α e n−1 + F (m; α) + H0 (m, n − 2; α) α α

=−

n n! 1 1 F (m + n − ν; 2α) ν α α (n − ν)! ν=0

+ =−

e−α α

n

n!

ν=0

α ν (n − ν)!

F (m; α)

n n n! n! e−α 1 αs F (m + s; 2α) + F (m; α) α αn s! α α ν (n − ν)! s=0

ν=0

= −T (n, m; α) + A(n; α)F (m; α),

(K.16)

while the second term in (K.15) is the same as the first term if we interchange m and n. Therefore, H0 (m, n; α) = −T (n, m; α) + A(n; α)F (m; α) − T (m, n; α) + A(m; α)F (n; α).

(K.17)

A PPENDIX K.6. H1 (m, n; α) By definition, ∞ H1 (m, n; α) ≡

∞ dλ1

1

1

∞

λ1

=

dλ1 1

n −α(λ1 +λ2 ) dλ2 λm P1 (λ< )Q1 (λ> ) 1 λ2 e

n+1 −α(λ1 +λ2 ) dλ2 λm e Q1 (λ1 ) 1 λ2

1

+ (same as left, with m ↔ n).

(K.18)


227

The first term in (K.18) is evaluated as ∞

λ1 dλ1

1

n+1 −α(λ1 +λ2 ) dλ2 λm e Q1 (λ1 ) 1 λ2

1

∞ =

λ1 dλ1

1

1

∞ −

λ1 dλ1

1

∞ =

n+1 −α(λ1 +λ2 ) dλ2 λm e 1 λ2

1

λ1 dλ1

1

dλ2 λm+1 λn+1 e−α(λ1 +λ2 ) Q0 (λ1 ) 1 2

dλ2 λm+1 λn+1 e−α(λ1 +λ2 ) Q0 (λ1 ) − S(m, n + 1; α). (K.19) 1 2

1

By combining the two terms in (K.18), we obtain H1 (m, n; α) = H0 (m + 1, n + 1; α) − S(n, m + 1; α) − S(m, n + 1; α). (K.20) A PPENDIX K.7. Hτ (m, n; α) The recurrence relations for the Legendre polynomials are (τ + 1)Pτ +1 = (2τ + 1)xPτ − τ Pτ −1 ,

(K.21)

(τ + 1)Qτ +1 = (2τ + 1)xQτ − τ Qτ −1 .

(K.22)

We then have

n −α(λ1 +λ2 ) λm dλ1 dλ2 1 λ2 Pτ (λ< )Qτ (λ> )e 1 n λm = 2 1 λ2 (2τ − 1)λ< Pτ −1 − (τ − 1)Pτ −2 τ × (2τ − 1)λ> Qτ −1 − (τ − 1)Qτ −2 e−α(λ1 +λ2 ) dλ1 dλ2 ; (K.23)

Hτ (m, n; α) =

τ 2 Hτ (m, n; α) = (2τ − 1)2 Hτ −1 (m + 1, n + 1; α) + (τ − 1)2 Hτ −2 (m, n; α) n − (2τ − 1)(τ − 1) λm 1 λ2 (λ< Pτ −1 Qτ −2 + λ> Pτ −2 Qτ −1 ) × e−α(λ1 +λ2 ) dλ1 dλ2

(K.24)

228

G. Chen et al. = (2τ − 1)2 Hτ −1 (m + 1, n + 1; α) + (τ − 1)2 Hτ −2 (m, n; α) n − (2τ − 1) λm 1 λ2 λ< (2τ − 3)λ< Pτ −2 − (τ − 2)Pτ −3 Qτ −2 + λ> Pτ −2 (2τ − 3)λ> Qτ −2 − (τ − 2)Qτ −3 e−α(λ1 +λ2 ) , dλ1 dλ2 = (2τ − 1)2 Hτ −1 (m + 1, n + 1; α) + (τ − 1)2 Hτ −2 (m, n; α) n 2 2 − (2τ − 1)(2τ − 3) λm 1 λ 2 λ< + λ> × Pτ −2 Qτ −2 e−α(λ1 +λ2 ) dλ1 dλ2 n + (2τ − 1)(τ − 2) λm 1 λ2 (λ< Pτ −3 Qτ −2 + λ> Pτ −2 Qτ −3 ) × e−α(λ1 +λ2 ) dλ1 dλ2

(K.25)

= (2τ − 1)2 Hτ −1 (m + 1, n + 1; α) + (τ − 1)2 Hτ −2 (m, n; α) − (2τ − 1)(2τ − 3) Hτ −2 (m + 2, n; α) + Hτ −2 (m, n + 2; α) n + (2τ − 1) λm 1 λ2 λ< Pτ −3 (2τ − 5)λ> Qτ −3 − (τ − 3)Qτ −4 + λ> (2τ − 5)λ< Pτ −3 − (τ − 3)Pτ −4 Qτ −3 e−α(λ1 +λ2 ) dλ dλ2 = (2τ − 1)2 Hτ −1 (m + 1, n + 1; α) + (τ − 1)2 Hτ −2 (m, n; α) − (2τ − 1)(2τ − 3) Hτ −2 (m + 2, n; α) + Hτ −2 (m, n + 2; α) n −α(λ1 +λ2 ) dλ1 dλ2 + 2(2τ − 1)(2τ − 5) λm 1 λ2 λ< λ> Pτ −3 Qτ −3 e n − (2τ − 1)(τ − 3) λm 1 λ2 (λ< Pτ −3 Qτ −4 + λ> Pτ −4 Qτ −3 ) × e−α(λ1 +λ2 ) dλ1 dλ2 .

(K.26)

As shown in the equations (K.26)–(K.26) above, the same patterns are repeated until τ is reduced to 0. So, let’s consider the last term. If τ is even, the last term is n −α(λ1 +λ2 ) dλ1 dλ2 −(2τ − 1) λm 1 λ2 (λ< P1 Q0 + λ> P0 Q1 )e −α(λ +λ ) n 2 2 1 2 dλ dλ = −(2τ − 1) λm 1 2 1 λ2 λ< Q0 + λ> Q0 − λ> e = −(2τ − 1) H0 (m + 2, n; α) + H0 (m, n + 2; α) ∞ + (2τ − 1)

λ1 dλ1

1

1

dλ2 λm+1 λn2 e−α(λ1 +λ2 ) 1

ELECTRON MOLECULAR BONDS ∞ + (2τ − 1)

∞ dλ1

229

n+1 −α(λ1 +λ2 ) dλ2 λm e 1 λ2

λ1

1

= −(2τ − 1) H0 (m + 2, n; α) + H0 (m, n + 2; α) − S(m + 1, n; α) − S(n + 1, m; α) .

(K.27)

If τ is odd, the last term then is n −α(λ1 +λ2 ) (2τ − 1) λm dλ1 dλ2 1 λ2 (λ< P0 Q1 + λ> P1 Q0 ) e n −α(λ1 +λ2 ) = (2τ − 1) λm dλ1 dλ2 1 λ2 (λ< λ> Q0 − λ< + λ> λ< Q0 ) e = 2(2τ − 1)H0 (m + 1, n + 1; α) ∞ λ1 n+1 −α(λ1 +λ2 ) e − (2τ − 1) dλ1 dλ2 λm 1 λ2 1

1

∞

∞

− (2τ − 1)

dλ1 1

dλ2 λm+1 λn2 e−α(λ1 +λ2 ) 1

λ1

= (2τ − 1) 2H0 (m + 1, n + 1; α) − S(m, n + 1; α) − S(n, m + 1; α) . (K.28) (1)

A PPENDIX K.8. Hτ (m, n; α) The recurrence relations for the associated Legendre polynomial are 1/2 ν+1 2 Pτ (x) = (τ − ν)xPτν (x) − (τ + ν)Pτν−1 (x), x −1 1/2 ν+1 2 Qτ (x) = (τ − ν)xQντ (x) − (τ + ν)Qντ −1 (x); x −1

(K.29) (K.30)

Hτ(1) (m, n; α) n −α(λ1 +λ2 ) = τ2 dλ1 dλ2 λm 1 λ2 (λ< Pτ − Pτ −1 )(λ> Qτ − Qτ −1 ) e = τ 2 Hτ (m + 1, n + 1; α) + Hτ −1 (m, n; α) n −α(λ1 +λ2 ) − τ2 dλ1 dλ2 λm 1 λ2 (λ< Pτ Qτ −1 + λ> Pτ −1 Qτ ) e = τ 2 Hτ (m + 1, n + 1; α) + Hτ −1 (m, n; α) −

τ2 (2τ + 1)2 Hτ (m + 1, n + 1; α) + τ 2 Hτ −1 (m, n; α) (2τ + 1)τ

230

G. Chen et al. − (τ + 1)2 Hτ +1 (m, n; α) =

τ (τ + 1)2 Hτ +1 (m, n; α) − τ (τ + 1)Hτ (m + 1, n + 1; α) 2τ + 1 τ 2 (τ + 1) + Hτ −1 (m, n; α). 2τ + 1

(K.31)

So 2τ + 1 (1) H (m, n; α) τ (τ + 1) τ = (τ + 1)Hτ +1 (m, n; α) − (2τ + 1)Hτ (m + 1, n + 1; α) + τ Hτ −1 (m, n; α).

(K.32)

(2)

A PPENDIX K.9. Hτ (m, n; α) Similarly to the preceding paragraph, Hτ(2) (m, n; α) 2 1/2 n 2 (τ − 1)λ< Pτ1 − (τ + 1)Pτ1−1 = λm 1 λ2 (λ1 − 1)(λ2 − 1) × (τ − 1)λ> Q1τ − (τ + 1)Q1τ −1 e−α(λ1 +λ2 ) dλ1 dλ2 (1) = (τ − 1)2 Hτ(1) (m + 1, n + 1; α) + (τ + 1)2 Hτ −1 (m, n; α) 2 2 1/2 2 n − (τ − 1) λm 1 λ2 λ1 − 1 λ2 − 1 × λ< Pτ1 Q1τ −1 + λ> Pτ1−1 Q1τ e−α(λ1 +λ2 ) dλ1 dλ2 , (K.33) and (1)

Hτ +1 (m, n; α) 2 2 1/2 1 n (2τ + 1)λ< Pτ1 − (τ + 1)Pτ1−1 λm = 2 1 λ2 λ1 − 1 λ2 − 1 τ × (2τ + 1)λ> Q1τ − (τ + 1)Q1τ −1 e−α(λ1 +λ2 ) dλ1 dλ2 1 (1) = 2 (2τ + 1)2 Hτ(1) (m + 1, n + 1; α) + (τ + 1)2 Hτ −1 (m, n; α) τ 2 1/2 2 (2τ + 1)(τ + 1) n λm − 1 λ2 λ1 − 1 λ2 − 1 2 τ 1 1 × λ< Pτ Qτ −1 + λ> Pτ1−1 Q1τ e−α(λ1 +λ2 ) dλ1 dλ2 . (K.34)


231

Therefore, (2τ + 1)Hτ(2) (m, n; α) (1)

= τ 2 (τ − 1)Hτ +1 (m, n; α) − (τ + 2)(τ − 1)(2τ + 1) (1)

× Hτ(1) (m + 1, n + 1; α) + (τ + 1)2 (τ + 2)Hτ −1 (m, n; α).

(K.35)

Appendix L. Derivations for the 5-Term Recurrence Relations (6.81) First, we cast both equations (6.78) and (6.79) into the form m2 2 2 2 1 − x φ + −A − 2Rj x + p x − φ = 0, 1 − x2 j = 1, 2,

(L.1)

where for j = 1,

x = λ,

φ = (λ),

for j = 2,

x = µ,

φ = M(µ),

R(Za + Zb ) ; 2 R(Za − Zb ) . R2 = 2 R1 =

Set φ(x) =

∞

fk Pkm (x)

(L.2)

k=0

as in (6.80) and substitute (L.2) into (L.1): ∞ dPkm (x) d fk 1 − x2 dx dx k=0 ∞ + fk −A − 2Rj x + p 2 x 2 − k=0 ∞

fk

k=0

+

d m2 d 1 − x2 − dx dx 1 − x2

∞

m2 P m (x) = 0, 1 − x2 k − A Pkm (x)

fk −2Rj x + p 2 x 2 Pkm (x) = 0,

k=0 ∞

∞ ∞ fk −k(k + 1) − A Pkm (x) − fk Rj xPkm (x) + fk p 2 x 2 Pkm (x)

k=0

= 0,

k=0

k=0

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G. Chen et al. ∞

fk [−k(k + 1) − A]Pkm (x)

k=0

1 m m (x) + (k − m + 1)Pk+1 (x)] [(k + m)Pk−1 2k + 1 k=0 ∞ (k − m + 1)(k − m + 2) m + fk p 2 · Pk+2 (x) (2k + 1)(2k + 3) k=0 (k − m + 1)(k + m + 1) (k − m)(k + m) + + P m (x) (2k + 1)(2k + 3) (2k − 1)(2k + 1) k

(k + m − 1)(k + m) m Pk−2 (x) = 0. + (2k − 1)(2k + 1) ∞

−

f k Rj

m (x), P m (x), P m (x) and P m (x) to We an now shift indices to convert Pk+2 k+1 k−1 k−2 m Pk (x). We obtain ∞

fk−2

k=0

2Rj (k − m) p 2 (k − m − 1)(k − m) − fk−1 + f k Ck (2k − 3)(2k − 1) 2k − 1

− fk+1

2Rj (k + m + 1) p 2 (k + m + 1)(k + m + 2) + fk+2 2k + 3 (2k + 3)(2k + 5)

= 0.

The terms inside the parentheses above are exactly the 5-term recurrence relations (6.81).

Appendix M. Dimensional Scaling in Spherical Coordinates For description of diatomic molecules cylindrical coordinates provide a natural way of making a dimensional (D-) scaling transformation. Here we show how to do the D-scaling transformation in spherical coordinates, which is useful for description of atoms. Let us first consider the Laplacian in the D-dimensional hyperspherical coordinates x1 = r cos θ1 sin θ2 sin θ3 · · · sin θD−1 , x2 = r sin θ1 sin θ2 sin θ3 · · · sin θD−1 , x3 = r cos θ2 sin θ3 sin θ4 · · · sin θD−1 , x4 = r cos θ3 sin θ4 sin θ5 · · · sin θD−1 , .. .

(M.1)


233

xj = r cos θj −1 sin θj sin θj +1 · · · sin θD−1 , .. . xD−1 = r cos θD−2 sin θD−1 , xD = r cos θD−1 , 0 ≤ θ1 ≤ 2π,

0 ≤ θj ≤ π

for j − 2, 3, . . . , D − 1,

where D is a positive integer and D ≥ 3. Define h=

D−1 4

hj ,

(M.2)

j =0

where h2k

=

D ∂xj 2 j =1

∂θk

.

(M.3)

Then the scaling factors are h0 = 1, h1 = r sin θ2 sin θ3 · · · sin θD−1 , h2 = r sin θ3 sin θ4 · · · sin θD−1 , .. . hk = r sin θk+1 sin θk+2 · · · sin θD−1 , .. .

(M.4)

hD−2 = r sin θD−1 , hD−1 = r, h = r D−1 sin θ2 sin2 θ3 sin3 θ4 · · · sink−1 θk · · · sinD−2 θD−1 . The D-dimensional Laplacian now becomes 2 = ∇D

1 ∂ D−1 ∂ 1 r + 2 D−1 2 2 ∂r ∂r r r sin θk+1 sin θk+2 · · · sin2 θD−1 k=1

∂ ∂ 1 k−1 × sin θ k ∂θk sink−1 θk ∂θk

1 ∂ ∂ 1 D−2 . + 2 (M.5) sin θ D−1 ∂θD−1 r sinD−2 θD−1 ∂θD−1 1

D−2

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Define the generalized orbital angular momentum operators by L21 = −

∂2 , ∂θ12

L22 = −

L21 ∂ ∂ 1 sin θ2 + , sin θ2 ∂θ2 ∂θ2 sin2 θ2

(M.6)

.. . L2k = −

L2k−1 ∂ ∂ k−1 sin θ + . k ∂θk sink−1 θk ∂θk sin2 θk 1

Then we have 2 ∇D = KD−1 (r) −

where KD−1 (r) ≡

L2D−1 r2

,

∂ D−1 ∂ r . ∂r r D−1 ∂r 1

(M.7)

(M.8)

Let us consider Schrödinger equation for a particle moving in D-dimensions in a central potential V (r): ∇2 − D + V (r) D = ED . 2 To eliminate the angular dependence we separate the variables by writing D (r, D−1 ) = R(r)Y (D−1 ).

(M.9)

Near the origin r = 0,

or

D ∼ r l Y (D−1 ),

(M.10)

2 l ∇D r Y (D−1 ) = l(l + D − 2) − C r l−2 Y (D−1 ) = 0

(M.11)

L2D−1 Y (D−1 ) = CY (D−1 ).

(M.12)

with

The effective Hamiltonian is given by 1 l(l + D − 2) + V (r). HD = − KD−1 (r) + 2 2r 2 With the following transformation D = r −(D−1)/2 D

(M.13)

(M.14)


235

the corresponding equation for D reads 1 ∂2 ( + 1) − + + V (r) D = ED D , 2 ∂r 2 2r 2

(M.15)

where 1 = l + (D − 3). 2

(M.16)

Equation (M.15) is the Schrödinger equation in D-dimensions for the function D . As an example, consider the Schr ödinger equation for the H-atom in D-dimensions 1 2 Z − ∇ − (M.17) = E. 2 r In the scaled variables 3 r , 2 r0 1 E Es = , 2 E0 rs =

(M.18) (M.19)

with r0 = D(D − 1)/4 and E0 = 2/(D − 1)2 , the Schrödinger equation reads 1 3 2 2 3 D−1Z − ∇s − = Es . 2 D D 2 rs

(M.20)

Now, let us write the Laplacian in spherical coordinates and transform the wave function according to (M.14). We obtain 1 3 2 d2 (D − 1)(D − 3) 3(D − 1) Z − − − = Es . 2 D 2D rs drs2 4rs2 (M.21) In the limit D → ∞ Eq. (M.21) reduces to a simple algebraic problem of minimization the expression Es =

9 3Z − , 2 rs 8rs2

(M.22)

which yields rs = 3/2Z and Es = −Z 2 /2. This value coincides with the ground state energy of the hydrogen atom in 3 dimensions.

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RESONANCE FLUORESCENCE OF TWO-LEVEL ATOMS∗ H. WALTHER Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theory of the Spectrum of Resonance Fluorescence . . . . . . . . . . . . . Total Scattered Intensity, Intensity Correlations, and Photon Antibunching More Theoretical Results—Variants of the AC Stark Effect . . . . . . . . . Experimental Studies of the Spectrum . . . . . . . . . . . . . . . . . . . . Spectrum at Low Scattering Intensities and Extremely High Resolution . . Experiments on the Intensity Correlation—Photon Antibunching . . . . . Photon Correlation Measured with a Single Trapped Particle . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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239 240 244 245 247 251 260 266 269 270

1. Introduction Resonance fluorescence of atoms is a basic process in radiation–atom interactions, and has therefore always generated considerable interest. The methods of experimental investigation have changed continuously with the advent of new experimental tools. A considerable step forward was taken when tunable and narrow-band dye laser radiation became available. These laser sources are sufficiently intense to easily saturate an atomic transition. In addition, such lasers provide highly monochromatic light with coherence times much longer than typical natural lifetimes of excited atomic states. Excitation spectra with laser light using well-collimated atomic beams lead to a width that is practically the natural width of the resonance transition; it became possible therefore to investigate the frequency spectrum of the fluorescence radiation with high resolution. However, ∗ This paper is dedicated to my highly esteemed colleague and friend Ben Bederson. I had the pleasure to edit the Advances jointly with him for almost ten years. It was a great pleasure to cooperate with Ben. His extensive view on the entire field of Atomic, Molecular and Optical Physics was of great help when the new volumes were planned.

239


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[2

the spectrograph used to analyze the re-emitted radiation was a Fabry–Perot interferometer, the resolution of which did reach the natural width of the atoms, but was insufficient to reach the laser linewidth, see, for example (Hartig et al., 1976; Cresser et al., 1982). Considerable progress in this direction was achieved by investigating the fluorescence spectrum of ultra-cold atoms in an optical lattice in a heterodyne experiment of Jessen et al. (1992). In these measurements a linewidth of 1 kHz was achieved, but the quantum aspects of the resonance fluorescence such as antibunched photon statistics cannot be investigated under these conditions, since they wash out when more than one atom is involved. The ideal experiment thus requires a single atom to be investigated. For some time it has been known that ion traps allow one to study the fluorescence from a single laser-cooled particle practically at rest, thus providing the ideal case for spectroscopic investigation of resonance fluorescence. The other essential ingredient for achieving high resolution is measurement of the frequency spectrum by heterodyning the scattered radiation with laser light as demonstrated with many cold atoms. Such an optimal experiment with a single trapped ion was conducted and is reviewed in this paper together with other laser experiments on resonance fluorescence. In addition, photon correlation in resonance fluorescence is discussed, leading historically to the first demonstration of nonclassical radiation. We start the review with a survey of the theory of the spectrum of resonance fluorescence (see also (Cresser et al., 1982) for comparison and more details).

2. Theory of the Spectrum of Resonance Fluorescence The theoretical treatment of resonance fluorescence of a two-level atom irradiated by a completely monochromatic light field in the low-intensity limit was first reviewed by Heitler (1954). A scattered field spectrum was predicted which was very sharply peaked around the incident field frequency. The high-intensity limit was first considered by Apanasevich (1964), who, by numerical calculations based on earlier theoretical work (Apanasevich, 1963), predicted a three-peak spectrum. Subsequently, Newstein (1968) also examined the problem with collisional rather than radiation damping providing the relaxation mechanism. He also predicted a three-peak spectrum in the high-intensity limit, though owing to a different damping mechanism the widths and heights of the three peaks differed from those later found in the pure radiation damping case. However, the first complete theoretical treatment in which exact expressions were obtained for the scattered field spectrum when radiation damping is present is the work of Mollow (1969). In his work the scattering atom was driven near resonance by a monochromatic classical electric field. The atom came into equilibrium with this field through the effects of radiation damping, this being included in the theory by explicitly coupling the atom to the quantized electromagnetic field. The solution was based

2]

RESONANCE FLUORESCENCE OF TWO-LEVEL ATOMS

241

on deriving the optical Bloch equations for the elements of the (2 × 2) reduced density matrix of the atomic system. These equations were obtained from a master equation approach in the derivation of which the Markov approximation was made. The diagonal elements of this reduced density matrix are just the probabilities of the atom being found in its ground or excited states, while the off-diagonal elements essentially give the mean dipole moment of the radiating atom. However, it is not the mean dipole moment of the atom that acts as the source of the radiated field; rather it is the instantaneous value of the dipole moment, i.e., its mean value plus quantum fluctuations. This is recognized in Mollow’s work, in which rather than calculating the correlation function of the mean dipole moment, and hence, by a Fourier transform, the spectrum of the radiated field, it is the correlation function of the dipole moment operator that is found, so that quantum fluctuations are not averaged out. This latter correlation function is obtained from the optical Bloch equations by means of the quantum regression theorem (Lax, 1963). Since Mollow used a classical description of the incident field, his method was originally not believed to be a fully quantum-electrodynamic treatment (Stroud, 1977), although it was later shown by Mollow (1975) that this work was in fact equivalent to such a description. Following the work of Mollow, Stroud (1977) made the first attempt at deriving a solution for the case in which the incident field was described quantum electrodynamically (QED). Stroud’s work was prompted as much by the need to avoid the semiclassical approach of Mollow as by a desire to compare the results of a QED calculation with those obtained from the so-called neoclassical theory (Stroud and Jaynes, 1970), in which the concept of a quantized electromagnetic field was avoided altogether. It was later found that the predictions of the QED approach were fully vindicated by experiment. Stroud’s work introduced the “dressed atom” method later popularized by Cohen-Tannoudji (1975, 1977). This method amounts to making a judicious choice of basis states, these states being eigenstates of the coupled atom-driving field system. The energy eigenvalue spectrum of the dressed-atom system assumes the form of a series of doublets, the frequency separation between the members of a doublet being just the (off-resonance) Rabi frequency = (2 + 2 )1/2 , where is the on-resonance Rabi frequency and is the detuning of the driving field from resonance. The frequency separation between corresponding levels in successive doublets is just the frequency of the driving field. The scattering of photons can then be visualized as a sequence (or cascade) of spontaneous decays down through the states of the dressed-atom system. Stroud, however, truncated this problem by considering only a single spontaneous transition and obtained a result similar to Mollow’s, but differing from Mollow’s results as regards the widths and relative heights of the three peaks. In the real physical situation there is a cascade through the successive energy levels of the dressed-atom system that is accompanied by spontaneous emission of many pho-

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tons, and in a correct calculation of the spectrum, proper account must be taken of these photon cascades. Unfortunately, this direct approach is very difficult since, among other problems, quantum interference effects associated with the different possible orders of emission of the radiated photons must be properly included in the calculations. Such explicit photon descriptions of the AC Stark effect will be discussed later. The first fully QED treatment was not based on explicitly following the photon cascades. A Markovian master equation approach was used in which the reduced density operator for the dressed-atom system was obtained (Oliver et al., 1971). This reduced density matrix is of a far more complex form than the 2 × 2 matrix obtained by Mollow (1969) since the dressed-atom system consists of a large number of states. Oliver et al. (1971) used the quantum regression theorem to obtain the spectrum of the fluorescent field. However, no explicit expressions for this spectrum were reported, only computed plots of the spectrum were given. Carmichael and Walls (1975; 1976) also made use of the dressed-atom picture for a driving field exactly on resonance to obtain, by a fully QED method, expressions for the spectrum in agreement with Mollow’s results. Alternative fully QED treatments have also been given subsequently to Carmichael and Wall’s work (Hassan and Bullough, 1975; Kimble and Mandel, 1976; Wodkiewicz and Eberly, 1976; Renaud et al., 1976, 1977). These methods were based on the use of a Heisenberg equation of motion approach in which the equations of motion of the atomic and field operators are obtained from the Hamiltonian of the total system and, by eliminating unwanted variables, are reduced to equations involving only atomic and free-field operators. The atomic operators in these equations evolve (in the Heisenberg picture) under the action of the total Hamiltonian of an atom plus fields plus interaction, while the free-field operators evolve under the Hamiltonian of the free-field only. Hassan and Bullough (1975) actually derive equations of motion for the atomic operators averaged over the initial state of the field, taken to be given by a coherent state, while in the other treatments this approximation was avoided. Nevertheless, results in complete agreement with Mollow’s were obtained. In the above Heisenberg equation of motion approaches, in almost all cases a Markovian-type approximation was made in the derivation of the equations (an exception is Wodkiewicz and Eberly, 1976), although it was referred to by different names, viz. the adiabatic approximation or the harmonic approximation. A claim by Kimble and Mandel (1976) that such an approximation was not made in their work was shown by Ackerhalt (1978) to be in error. Cohen-Tannoudji (1977) made use of a Langevin equation of motion approach. In this method, the optical Bloch equations, which are linear differential equations, relating the mean values of atomic-system operators for the two-level atom, were formally replaced by operator equations, with delta-function-correlated random force operators added to each equation to take account of quantum fluctuations. This is in accordance with an approach to quantum noise problems

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developed by Lax (1967). Cohen-Tannoudji obtained the usual (i.e., Mollow) result for the spectrum. All of the foregoing methods relied on making a Markov approximation or atom-field statistical factorization assumption. The validity of these approximations was open to question (Stroud, 1977) so that an approach to the problem was required which either avoided the approximations or else rigorously established their validity. To do this it was necessary to turn to the more difficult problem of working directly with the photon cascades. The first such method was developed by Mollow (1975). In this paper he assumed a coherent state description for the incident field and was able to show by a canonical transformation that this fully QED description of the model was exactly equivalent to one in which the field was treated classically, with the initial state of the quantized field transformed into the vacuum state. The equivalence of the QED approach and his original semiclassical approach (Mollow, 1969) was thus rigorously established. Moreover, within the context of a well-defined set of approximations he then showed how all the photon reabsorption processes could be allowed for, leading to a new (non-Hermitian) Hamiltonian in which the energy of the upper state was assigned an imaginary part – its natural linewidth. The interaction term in this Hamiltonian only creates photons, so that under the action of this Hamiltonian the transformed initial state, i.e. the vacuum state, evolves into a linear combination of Fock states containing multiphoton contributions of all orders. The photon cascade effect mentioned above was thus fully included in the theory. Of course, the usual expression for the spectrum was obtained. An important aspect of this work is that the validity of all approximations made was very carefully investigated. Mollow was able to establish, therefore, the validity of the Markov approximation made in other approaches to this problem, and was also able to derive the quantum regression theorem, thereby placing its use in calculating the spectrum on firm ground. Mollow (1975) has investigated the consequences of not making the Markov approximation, and was able to show that a very slight asymmetry was to be expected in the scattered light spectrum though the effect was shown to be very small and difficult to observe. Other calculations of the spectrum based on working directly with the photon states were also made by Smithers and Freedhoff (1975), Swain (1975a), Ballagh (1978). Mollow (1975) did question the correctness of the method used in (Smithers and Freedhoff, 1975). Swain based his calculations on his continued fraction method (Swain, 1975b). Ballagh used the dressed atom picture and a Feynman diagram technique, while Cresser made use of a generalization of the formalism of Mower (Cresser and Dalton, 1980). Of interest is the fact that these methods all showed the importance of quantum interference effects in determining the final form of the spectrum, and in fact it was shown that the coherent (delta-function) contribution to this spectrum is entirely due to interference effects. In this regard, Ballagh gave a very complete description of how this co-

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herent contribution builds up through the succession of spontaneous decays in the dressed-atom picture.

3. Total Scattered Intensity, Intensity Correlations, and Photon Antibunching Calculation of the total intensity of the scattered field was made in greatest detail by Kimble and Mandel (1976). They gave computer plots of the intensity of the scattered field as a function of time for arbitrary driving field intensities and detunings. They showed that the scattered field intensity exhibited oscillatory behavior that became more apparent at high driving field intensities and increased detuning. The intensity was also shown to always settle down to a constant steady-state value after a sufficiently long time had elapsed. The intensity correlations are of far more interest, however, than the intensity itself. The intensity correlation function was investigated in (Kimble and Mandel, 1976; Carmichael and Walls, 1975, 1976). Carmichael and Walls (1975, 1976) considered the case of an on-resonance driving field only, while Cohen-Tannoudji (1977), Kimble and Mandel (1976) provided a generalization for arbitrary detuning of this field away from resonance. The importance of this work is that the intensity correlation was found to exhibit a behavior which has no classical counterpart. For usual light fields (e.g., thermal light) it is found that the intensities of the field at two neighboring instants in time are strongly correlated, i.e. if a photomultiplier irradiated by this field emits an electron at some instant in time, the probability is high for a second photoemission to occur a short time later. This phenomenon is known as photon bunching, and can be explained by means of either a classical or quantum-mechanical description of the light field. However, it was found that for the field scattered by a single two-level atom, the intensity correlation function was of a form that showed that if a photon was detected (by a photomultiplier) at some instant in time, then the probability of detecting another photon during a short time interval following the first detection remained close to zero. This phenomenon is the reverse of that described earlier and is known as photon antibunching. This behavior can be explained quantum mechanically by the fact that the process of detecting a scattered photon also prepares the scattering atom in its ground state. No further photons can thus be emitted and hence detected until the atom has had sufficient time to be pumped back up to its excited state (the only state from which emission can occur) by the driving field. It should be pointed out that fields exhibiting antibunching can be generated in other ways, i.e. by multiphoton absorption in which two or more photons are simultaneously absorbed (Simeon and London, 1975; McNeil and Walls, 1974), or else as a result of nonlinear optical effects in the

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degenerate parametric process first discussed by Stoler (1974) and further developed by Paul and Brunner (1980), Bandilla and Ritze (1979), Ritze and Bandilla (1979). However, we will confine our attention here to antibunching in the case of resonance fluorescence only. The significance of antibunching is that a classical field which exhibits this behavior does not exist. The existence of photon antibunching can thus be taken as a test of the validity of QED. However, the theoretical model on which the above result is based is not directly representative of a true experimental situation. All experimental studies of the AC Stark effect and related phenomena involve a beam of atoms passing perpendicularly through a laser field. However, the antibunching effect can be washed out if a number of atoms are simultaneously interacting with the laser field (Carmichael and Walls, 1975, 1976). To observe the phenomenon ideally, the intensity of the atomic beam, must thus be sufficiently low that only a single atom at a time passes through the field. In the real situation, however, even for low beam intensities, there is a statistical fluctuation in the number of atoms in the field at any time. For the purposes of comparing theory and experiment, the above theory must thus be modified to allow for fluctuations in the number of atoms in the field at any time, and also, as it turns out, the effect of the finite transit time of the atoms through the field must also be accounted for (Jakeman et al., 1977; Kimble et al., 1978; Carmichael et al., 1978). In addition, the possible effects of nonzero laser bandwidth need to be examined, the pertinent work in this case being that of Wodkiewicz (1980), who used a phase diffusion model (PDM) of the laser field to calculate improved expressions for the intensity correlation. These corrections were found to be important at low laser field intensities (Section 6). Detailed examinations of laser bandwidth effects have also been made for both the PDM and chaotic field model for the incident light (Schubert et al., 1979, 1980). Although differences in detail were found in these investigations, the antibunching phenomenon was still present and detectable. It is shown that the results of the experiment (Kimble et al., 1977; Walls, 1979) are in agreement with the improved theory. The experiments will be discussed in some detail in Section 5.

4. More Theoretical Results—Variants of the AC Stark Effect Theoretical investigation has also been conducted into variants of the simple form of the AC Stark effect problem. Table 3.3 lists a representative sample of the papers in which such variations of the basic problem have been studied. Renaud et al. (1977), using a fully quantum-electrodynamic treatment developed in an earlier paper (Renaud et al., 1976), investigated the spectrum of the scattered field detected during an observation period of finite length. One of the

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important results of their investigations was that for a weak driving field tuned offresonance the spectrum of the transient field, i.e. the field radiated at the start of the atom-field interaction, was asymmetric with the sideband closest to the atomic transition frequency enhanced. This transient behavior was later found to play an important role in determining the spectrum for a nonzero bandwidth driving field. A number of publications have been devoted to examining the effects of a driving field having a nonzero bandwidth (Agarwal, 1977; Eberly, 1976; Avan and Cohen-Tannoudji, 1977; Kimble and Mandel, 1977; Zoller, 1977; Knight et al., 1978; Agarwal, 1976; Zoller and Ehlotzky, 1977; Raymer and Cooper, 1979; Georges et al., 1979; Georges, 1980; Le Berre-Rousseau et al., 1980). In all these treatments, the nonzero bandwidth of the driving field was introduced by supposing that either the phase or both the phase and amplitude of the field were subject to random fluctuations. Avan and Cohen-Tannoudji (1977) and Agarwal (1977) both examined the on-resonance case and showed that the spectrum still had a symmetric three-peak structure with, however, each of the peaks broadened. Kimble and Mandel (1977) used a generalized version of the Heisenberg operator technique developed in an earlier publication (Kimble and Mandel, 1976) which enabled them to treat both the on- and off-resonance situations. They showed that for an off-resonance driving field the scattered spectrum became markedly asymmetric, an effect noted in some experimental work. Knight et al. (1978) were able to explain physically the origin of this antisymmetric structure. They used a simple Lorentz model appropriate to a weak driving field and used a method from (Eberly, 1976) to take account of the random nature of this field. They were able to show that the scattering atom cannot settle down to a steady-state phase relation with the fluctuating applied field. The atom continually goes out of phase with the field and is repeatedly returned to its transient interaction regime. As shown by Renaud et al. (1977), it was just in this transient regime that an asymmetric spectrum was to be expected; however, in contrast to the transient effect discussed there, which was a once only affair associated with the turning-on of the atom-field interaction, the transient effect described by Knight et al. (1978) was an intrinsic property of the driving field and existed independently of the turn-on of the field. They also showed that in the transient scattered field it was the nonelastic contribution at the transition frequency of the atom due to the overlap of the excitation spectrum with the atomic absorption line that produced the enhanced sideband. Other variations of the basic problem involve considerably more complex models for the scattering atom, e.g. Cohen-Tannoudji and Reynaud (1977a) investigated the case of a multilevel atom acting as the scattering center, and used this theory to examine resonance Raman scattering in very intense fields (Cohen-Tannoudji and Reynaud, 1977b). The particular case of a three-level atom has been examined in (Sobolewska, 1976; Sobolewska and Sobolewski, 1978;

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Kornblith and Eberly, 1978). The expressions for the spectra are very complex. For instance, Kornblith and Eberly (1978) obtained a seven-peak spectrum. Study of the total fluorescent intensity radiated by a three-level atom is perhaps of more interest, in certain cases, than that of the spectrum of the light. An example of this kind is the study of the total fluorescent light when a static field is scanned around the value corresponding to a crossing between two excited sub-levels (level crossing or Hanle effect). The theory of the Hanle effect for monochromatic excitation was developed by Avan and Cohen-Tannoudji (1975). The Hanle effect is well known in classical experiments involving broadband excitation of the atomic states, but a fundamental difference occurs when the atom is irradiated by monochromatic light. This is best seen by examining the elements of the reduced density matrix of the atomic system, which now contains nonzero off-diagonal elements coupling the excited and ground states (optical coherences), thus representing a coherent superposition of the ground and excited states. For broadband excitation, these coherences are completely washed out. However, as the optical coherences are not negligible for monochromatic excitation, more complex behavior is expected in this case than in the case of broadband excitation. A detailed discussion of the experimental investigations in comparison with the theory (Avan and Cohen-Tannoudji, 1975) is given in the paper by Cresser et al. (1982).

5. Experimental Studies of the Spectrum The spectrum of the scattered fluorescent light is, as discussed above, related to the Fourier transform of the first-order correlation function of the atomic operators. We shall summarize the theoretical results as follows: For low laser intensities the atom remains very close to its ground state and behaves like a classical oscillator. The light is therefore scattered elastically, and for a monochromatic driving field one observes a sharp spectrum at the same frequency as the driving field. As the intensity of the exciting light increases the atom spends more time in the upper state and the effect of the vacuum fluctuations due to spontaneous emission comes into play. An inelastic component enters the spectrum, and the magnitude of the elastic scattering component is correspondingly reduced. The spectrum gradually broadens as the Rabi frequency increases, until exceeds /4; then sidebands begin to appear. For the saturated atom the form of the spectrum shows three well-separated Lorentzian peaks. The central peak has a width /2 and the sidebands, which are displaced from the central peak by the Rabi frequency, are broadened to 3/4. The ratio of the height of the central peak to the sidebands is 3 : 1. Experimental study of the problem requires that Doppler broadening be almost completely excluded. The laser light has therefore to be scattered by the

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free atoms of a strongly collimated atomic beam. In order to measure the frequency distribution, the fluorescent light has to be investigated by means of a highly resolving spectrometer. The first experiments of this type were conducted by Stroud (1977) and later by Walther (1975), Hartig et al. (1976), Wu et al. (1975), and Grove et al. (1977). In all experiments the excitation was performed by single-mode dye lasers and the scattered radiation was analyzed with Fabry– Perot interferometers. In the following the experiments of Hartig et al. (1976) are discussed in more detail. The results for the fluorescence spectrum at low laser intensities using a single trapped ion and heterodyning of the fluorescence radiation are described in the subsequent chapter. For the experiment of Hartig et al. (1976) the atomic beam was collimated by circular apertures with a collimation ratio of about 1 : 500. This corresponds to a residual Doppler width of about 2 MHz. The direction of the atomic beam, the axis of excitation, and the line of observation were mutually perpendicular. The interaction region between the atomic beam and laser light was inside a confocal Fabry–Perot. With this arrangement the fluorescence signal is enhanced by a factor which is almost equal to the finesse of the interferometer. The total linewidth observable in the experiment is determined by the collimation ratio of the atomic beam, the opening angle for observation, the laser linewidth, and the finesse of the Fabry– Perot. With the numbers given above a total linewidth of about 10 MHz had to be expected. The experiments with high laser intensity were conducted at the F = 3 − F = 2 hyperfine transition of the sodium D2 line. This transition is suitable since the upper F = 3 level can only decay into the F = 2 level of the ground state from where the excitation is performed; multiple excitations are therefore possible and, in addition, no hyperfine pumping can occur. The transition has, of course, the disadvantage that it deviates from the two-level system usually considered in the theoretical treatments since the two hyperfine levels are degenerate. Furthermore, the other hyperfine levels of 2 P3/2 are so close that their influence cannot be neglected at high laser intensities, because they overlap with the F = 3 level due to power broadening. As a consequence, spontaneous decay to the 2 S1/2 , F = 1 can occur besides the decay to the F = 2 level. Since the hyperfine splitting between F = 1 and F = 2 is 1772 MHz, re-excitation of those atoms is impossible. The degeneracy of the hyperfine levels has an influence on the frequency distribution of the fluorescent light since the Rabi nutation frequency, which determines the distance of the side components, depends on the transition probability; the different Zeeman substates therefore contribute in different ways to the spectrum. In order to reduce this effect the number of Zeeman states involved has to be reduced. This is possible by using circularly polarized light for excitation. Due to optical pumping during the first excitation processes when the atoms enter the laser beam, almost all are pumped into the mF = 2 or mF = −2 Zeeman sublevel

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of the F = 2 hyperfine level for right-handed or left-handed circular polarization, respectively. Finally, only the transition 2 S1/2 , F = 2, mF = ±2 → 2 P3/2 , F = 3, mF = ±3 is excited and no other substate has to be taken into account. In addition, no coupling with other 2 P3/2 hyperfine levels is possible since there is no other allowed transition starting from 2 S1/2 , F = 2, mF = ±2 which may be populated by the laser light; as a consequence, the two-level system is closely approached (Schieder and Walther, 1974). For the experiments two single-mode CW dye lasers were used (Kogelnik et al., 1972; Walther, 1974). The linewidth of both lasers was less than 1 MHz when measured with a Fabry–Perot interferometer in a time of about 30 s. One laser was stabilized to the 2 P3/2 , F = 3 → 2 S1/2 , F = 2 transition of sodium using a separate atomic beam. The second laser was free-running with an output power of up to 60 mW in a single mode. The second laser beam was heterodyned with the first one in order to determine the frequency of the second laser with respect to the atomic transition frequency. The beat signal was analyzed with a radio frequency spectrum analyzer. The difference frequency between the two lasers could thus be accurately determined. Figure 1 shows the spectra of the fluorescent light for different laser powers. The laser frequency was tuned to the center of the resonance line. It is evident that the positions of the side maxima vary with the laser power. The signal intensity of the side components is about one-third of that of the central peak. This is in agreement with the predictions of the theory of (Mollow, 1969) and others (see Section 2). The separation of the two side maxima from the central component, for excitation on resonance according to Mollow, is given by the Rabi nutation frequency 2 2 = ω − , 4 where is the natural decay constant of the atomic transition and ω2 = 4|µ e|2 |E0 |n2 , where µ is the value of the electric dipole moment of the transition, e the polarization vector of the electric field, and E0 the amplitude of the electric field. The separation of the side maxima deduced from the experiment is in very good agreement with theory (Hartig et al., 1976). The dependence of the position of the side maxima on the detuning of the laser with respect to the accurate transition frequency is shown in Fig. 2. The laser power was 30 mW. For larger detuning the side maxima move further away from the central component and become smaller. In (Hartig et al., 1976) the distance between the side and central components was studied in detail as a function of the detuning. This comparison gives excellent agreement, too. In (Hartig et al., 1976; Grove et al., 1977) the convolution of the theoretical spectra with the instrumental line shape (linewidth about 10 MHz in both

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F IG . 1. Spectra of the fluorescent light from the F = 3 − F = 2 hyperfine transition for different laser powers.

experiments) was also compared with the measured spectra. There is good agreement. However, one small discrepancy between the measurements of Grove et al. (1977) and Hartig et al. (1976) should be briefly discussed. The experimental results show under some conditions asymmetries, i.e. one side peak is smaller than the other. In (Hartig et al., 1976) results are presented which indicate that the use of linearly polarized light leads to asymmetry due to the possibility of exciting transitions to more than one Zeeman substate. However, Grove et al. (1977) demonstrated that asymmetric spectra may be produced by observing atoms in a nonuniform field, regardless of polarization. Hence, misalignment may be responsible for the observed deviation. It is quite probable that there are contributions of

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F IG . 2. Variation of the spectrum of the fluorescence for different detunings of the laser frequency with respect to the transition frequency. The laser power for all measurements was 30 mW.

both effects; however, these small discrepancies are of no basic importance. The agreement between theory and experiment can be considered to be established in the essential features.

6. Spectrum at Low Scattering Intensities and Extremely High Resolution Present theory on the spectra of fluorescent radiation following monochromatic laser excitation can be summarized as follows: fluorescence radiation obtained

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with low incident intensity is also monochromatic owing to energy conservation. In this case, elastic scattering dominates the spectrum and thus one should measure a monochromatic line at the same frequency as the driving laser field. The atom stays in the ground state most of the time and absorption and emission must be considered as one process with the atom in principle behaving as a classical oscillator. This case was treated on the basis of a quantized field many years ago by Heitler (1954). With increasing intensity upper and lower states become more strongly coupled, leading to an inelastic component which increases with the square of the intensity. At low intensities, the elastic part dominates since it depends linearly on the intensity. As the intensity of the exciting light increases, the atom spends more time in the upper state and the effect of the vacuum fluctuations comes into play through spontaneous emission. The inelastic component is added to the spectrum, and the elastic component goes through a maximum where √ the Rabi flopping frequency is = / 2 ( is the natural linewidth), and then disappears with growing . The inelastic part of the spectrum gradually broadens as increases and for > /2 sidebands begin to appear (see Section 2 of this paper). The experimental study of the problem requires, as mentioned earlier, a Doppler-free observation. In order to measure the frequency distribution, the fluorescent light has to be investigated by means of a high-resolution spectrometer (see Section 5 of this paper). Experiments to investigate the elastic part of the resonance fluorescence giving a resolution better than the natural linewidth have been performed by Gibbs and Vencatesan (1976) and Cresser et al. (1982). In the following an experiment on the basis of a trapped single ion is described giving the highest possible resolution for the elastic component of the fluorescent light. The experiments were conducted with a single 24 Mg+ ion confined in a radiofrequency trap. To detect the fluorescence intensity of a single ion with high efficiency, a large solid angle for the collection of the emitted light is required. To this end, we developed a quadrupole trap (Schrama et al., 1993) which consists of two cylindrical endcap electrodes. Separate tubes around each endcap replace the single hyperbolic ring electrode in the original Paul trap design. A schematic drawing of the trap is shown in Fig. 3. The open structure allows us to capture 15% of the fluorescent light and inject laser beams at different angles for excitation, cooling, and diagnostics of the ion’s motion. By positioning the endcap electrodes (diameter r0 = 0.5 mm) at a distance of 0.56 mm, anharmonicities of the trapping field are minimized. The trap is driven at a RF frequency of /2π = 35 MHz and a typical ac-amplitude V0 = 500 V, while the DC voltage is U0 = −50 V. The resulting secular frequency in the plane perpendicular to the electrodes is ν/π = 5 MHz. The 3 2S1/2 − 32 P3/2 transition in 24 Mg (with a wavelength of 280 nm) is excited by coherent radiation from a frequency-doubled dye laser. The natural

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F IG . 3. Electrode configuration of the endcap trap employed to store a single 24 Mg+ ion. The RF field is coupled to the inner vertical cylinders. DC voltages may be applied to the outer cylinders at RF ground (UU , UL ) and two additional electrodes (U1 , U2 ) to compensate electric stray fields which push the ion from the node of the RF field.

linewidth of this transition (/2π = 43 MHz) is larger than the secular frequency ν of the trapped ion, thus the experiment operates in the non-resolved sideband limit. The laser frequency ωL is slightly red-detuned from the atomic resonance ω0 ( = ωL − ω0 < 0) in order to laser-cool the motion of the ion in the pseudo-potential. Part of the fluorescent light is imaged on a photo-multiplier for diagnostic purposes. Single ion count rates of up to 105 photons s−1 were obtained. High-resolution spectroscopy of the fluorescent light of an ion requires two particular preconditions. First, the residual thermal motion of the ion in the trapping potential must be minimized, since it leads to Doppler-generated sidebands in the fluorescence spectrum separated by the secular frequency of the trap (Cirac et al., 1993). The intensity of these sidebands depends on the ratio η = 2πa/λ of the amplitude a of the ion’s motion and the wavelength λ of the

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optical transition. In the limit of small η (Lamb–Dicke limit (Dicke, 1953)), optical transitions changing the vibrational state of the ion are effectively suppressed and only a small fraction of the fluorescent light is emitted in the sidebands. In our experiment this condition was achieved by laser cooling close to the Doppler limit, which corresponds to a temperature of 1 mK for magnesium. The resulting motional amplitude was below λ/5, which reduces the losses in sidebands to less than 50%. In addition to the oscillation in the confining potential, the ion is subject to a coherent motion driven by the trapping radio-frequency field (micromotion). Since it leads to additional sidebands and may cause heating of the ion, the second essential requirement for high-resolution spectroscopy is to minimize micromotion. For a single ion in a quadrupole trap, this is achieved by controlling the ion’s position, making use of the fact that the amplitude of the micromotion vanishes if the ion is located exactly at the node of the RF field. In the actual experiment the ion can be displaced from this ideal position by, for example, contact potentials due to coating of the electrodes with magnesium or surface charges accumulated during loading of the trap. The resulting stray electric fields in the trap center were compensated for by fields generated by additional electrodes (see Fig. 3). The following procedure was employed to minimize the micromotion along all three axes of the trap. The amplitude of micromotion in the direction of a probe laser was determined from the intensity modulation of the emitted fluorescent light, resulting from the periodic Doppler shift of the incident light in the rest frame of an ion undergoing micromotion. For highest sensitivity the laser was tuned to the steepest gradient of the atomic line profile. By recording fluorescence counts in correlation with the RF phase, a modulated signal was obtained with the depth of modulation proportional to the micromotion amplitude. By minimizing the modulation a set of compensating voltages was determined, which cancelled the micromotion along the direction of the probe laser employed. In order to minimize micromotion in three dimensions, the procedure was applied with three non-coplanar probe laser beams to be sensitive to all spatial components of the micromotion. In this manner the amplitude of the micromotion was reduced to a value smaller than λ/8. Once the Doppler effects of the secular and micromotion were effectively reduced, the fluorescence spectrum of a single ion was observed by the heterodyne detection technique (Fig. 4). A strong local oscillator beam was derived from the laser exciting the ion, and frequency-shifted by 137 MHz with an acoustooptic modulator. This beam was superimposed and carefully mode-matched with the imaged fluorescent light from the ion using a beam splitter. The combined light field was focused on a photodiode, where mixing of the two optical frequencies occurred. The resulting beat signal was amplified by a low-noise, narrowband amplifier and frequency down-converted to the kHz range. The signal was recorded for a few seconds and a fast Fourier transform (FFT) was used to gen-

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F IG . 4. Setup for single-ion heterodyne spectroscopy. Fluorescent light and the local oscillator (LO), frequency-shifted with an acousto-optic modulator (AOM), are superimposed on a beam splitter (BS) and detected with a photodiode (PD). With a RF mixer the spectrum is finally mapped to the kHz range, where it is analyzed by fast Fourier transform (FFT).

erate the final spectrum. Shot-noise-limited detection was achieved with a local oscillator power exceeding 100 µW. In this case, the signal-to-noise ratio (SNR) is given by the ratio of the fluorescence count rate to the resolution bandwidth of the Fourier transform (Cummins and Swinney, 1970). The best possible SNR is reached if the resolution bandwidth is smaller than or equal to the spectral width of the signal. However, in practice the SNR is degraded by limited mixing efficiency, for example due to imperfect mode matching of the Gaussian local oscillator beam with the imaged fluorescence radiation. In the low intensity limit, the fluorescence spectrum is dominated by elastic scattering of the incident radiation. More precisely, the coherent component of the emitted light is strongest at a saturation parameter (Mollow, 1969). Figure 5 shows the dependence of elastic and inelastic component as it follows from the theory of Mollow as a function of the saturation parameter. The measurements for the coherent part of the spectrum were obtained for s = 1. s=

2 /2 2 /4 + 2

= 1,

(1)

where is the Rabi frequency of the exciting laser light. In order to provide stable conditions for spectroscopy of the coherent peak and at the same time achieve sufficient cooling of the ion’s thermal motion, we used a detuning in the range of −0.5 to −3. The Rabi frequency was chosen between and 4.5 in order to optimize the heterodyne signal. The maximum intensity was observed around s ≈ 1 in agreement with Eq. (1). A heterodyne measurement recorded with a resolution bandwidth of 0.9 Hz is displayed in Fig. 6(a). A narrow central line together with a spectrum of equally

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F IG . 5. Mollow triplet and the contribution of the coherent part of the spectrum (Mollow, 1969). The lower part shows the dependence of the coherent peak on the saturation parameter. At low intensities the coherent scattered part is dominant and diminishes at higher intensities.

narrow sidebands at a separation of 28 Hz is visible. These sidebands are due to the operation of rotary vacuum pumps which generate a mechanical oscillation in the mounting assembly of the trap. Figure 6(b) shows a heterodyne spectrum after the offending pumps were isolated and shut down. The sidebands are completely suppressed and only a single peak remains at zero detuning from the exciting laser light. The smallest linewidths observed were of the order of 0.7 Hz using a resolution bandwidth of 0.5 Hz (Fig. 7), which is the narrowest optical heterodyne spectrum of resonance fluorescence reported to date. Our experiment thus provides the most compelling confirmation of Weisskopf’s prediction of a coherent component in resonance fluorescence (Weisskopf, 1931). The linewidth observed

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F IG . 6. (a) Heterodyne spectrum of the coherent peak with sidebands generated by mechanical vibration of the mount holding the trap (resolution bandwidth 0.9 Hz). (b) Heterodyne spectrum of the coherent peak with all mechanical pumps shut off (resolution bandwidth 4 Hz). For details see Höffges et al. (1997b).

implies that exciting laser and fluorescent light are coherent over a length of 400,000 km. The results also provide a test of the highest resolution achievable with our heterodyne setup. A relevant question in this respect is what determines the limiting residual linewidth of 0.7 Hz. It must be stressed that there is no observed frequency shift or broadening due to the photon recoil momentum, which is absorbed by the combined system of ion and trap, by analogy with the Mössbauer effect. Phase fluctuations of the exciting laser (linewidth 2 MHz) do not limit the spectral resolution either, since they are common to the coherently scattered light and the local oscillator beam and cancel in the heterodyne mixing process. We found that the main contribution to the linewidth are random phase fluctuations

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F IG . 7. Narrowest heterodyne spectrum observed with a 0.7 Hz linewidth of the coherent peak (resolution bandwidth 0.47 Hz). The solid line is a Lorentzian fit to the experimental data. The peak appears on top of a small pedestal 4 Hz wide. For details see Höffges et al. (1997b).

in the spatially separated parts of the local oscillator and fluorescent light paths. They are largely generated by variable air currents in the laboratory. The elastic fluorescence peak is a feature which is independent of the motional state of the ion. However, due to the oscillatory motion of the ion in the trap, sharp sidebands should appear in the spectrum, shifted by the secular frequency of the trap. In the Lamb–Dicke limit, only a single pair of sidebands is expected, the contribution of higher-order sidebands being negligible. Motional sidebands are of particular experimental interest, because they provide information on the ion’s dynamics in the trap. In the limit of very low intensity excitation, the lineshape of the sidebands should approach a δ-function like the central coherent peak. In this case the probability of transitions between vibrational levels of different quantum number is low and thus the oscillatory states of the ion are nearly stable (Lindberg, 1986). In the presence of laser cooling this is no longer true, since each oscillator state is damped at a rate corresponding to the cooling rate of the ion. The predicted linewidth of the sidebands is therefore given by the Doppler cooling rate γD , which in the limit of weak excitation may be written as (Lindberg, 1986; Cirac et al., 1993) 2 /2 /2 2 γD = η (2) − 2 . 4 2 /4 + ( + ν)2 /4( − ν)2 (see also the discussion in (Phillips and Westbrook, 1997)). In order to reduce the width of the motional sidebands in the fluorescence spectrum, the laser cooling

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rate of the ion must be decreased. Experimentally, a trade-off between a narrow linewidth of the sidebands and the need for cooling has to be found. If cooling is not efficient enough, e.g., if too small a detuning is chosen, then the ion’s kinetic energy becomes too large and the Lamb–Dicke condition η < 1 will no longer be fulfilled. This leads to higher-order sidebands and a reduced amplitude for each individual sideband. For our system, it may be realistic to reduce the cooling rate to the kHz range without excessively increasing the Lamb–Dicke parameter. Lower Doppler cooling is best achieved by tuning the exciting laser close to resonance (0 < − /2), since in this way the fluorescence intensity is maximized. Hitherto, experimental observation of the motional sidebands has only been achieved by means of a probe-field absorption technique (Diedrich et al., 1989; Monroe et al., 1995). In contrast, our heterodyne setup allows us to detect directly the narrow sidebands in the weak-field fluorescence spectrum of a single-ion. Heterodyne efficiency should be optimized to compensate for the decrease in SNR resulting from the increased signal linewidth. In addition, the position of the sidebands, i.e. the secular frequency ν of the trap, must be known and stabilized to within an accuracy given by the ratio ν/γD . In particular, the RF frequency as well as the AC voltage must be stabilized to this accuracy. If these conditions are met, detection of the fluorescence sidebands in a heterodyne measurement is feasible. The linewidth of the motional sidebands contains information on the dynamics of the ion’s motion in the trapping potential, in particular cooling and heating processes. In thermal equilibrium, the height of the upper sideband is proportional to the mean excitation n of the harmonic oscillator levels, the lower sideband proportional to (n + 1). From their ratio one can deduce the temperature of the ion. However, one must take into account the fact that the intensity of the sidebands nontrivially depends on the angle ψ between the directions of propagation of the incident and the scattered photon, resulting in an additional asymmetry between the height of the upper and lower motional sidebands (Cirac et al., 1993; Javanainen, 1980). This is due to the fact that two quantum paths contribute to each sideband, namely |n, g → |n, e → |n ± 1, g

and

|n, g → |n, ±1, e → |n ± 1, g, with |n, g denoting the ground state in |n, e the excited state of the ion in vibrational level n. Interference of the two corresponding transition amplitudes creates a spatially asymmetric radiation pattern for each sideband.

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7. Experiments on the Intensity Correlation—Photon Antibunching Further information on the nature of the fluorescent light can be obtained via the second-order correlation function of the light field defined by g 2 (τ ) =

E (−) (t)E (−) (t + τ )E (+) (t + τ )E (+) (t) , [E (−) (t)E (+) (t)]2

where E (+) (t) and E (−) (t) are the positive and negative frequency components of the electromagnetic field, respectively. The second-order correlation function was introduced by Glauber (1963) in his formulation of optical coherence theory. In essence, g 2 (τ ) is a measure of the probability that a second photon will be measured at time t + τ in a light beam after detection of one at time t. The result of a photon correlation experiment (Hanbury-Brown and Twis, 1956, 1957) for a chaotic light source with Gaussian frequency distribution (e.g., discharge lamp) is a Gaussian with a peak at τ = 0 and a width given by the inverse bandwidth of the light source. This means that there is a tendency for the photons to arrive in pairs, or in bunches. If instead of a discharge lamp, a highly stabilized laser is used for the correlation experiment, one obtains a constant g (2) = (τ ). This result holds even when the laser and discharge lamp have the same bandwidth. Hence there is some fundamental difference between a laser and a chaotic light source which may not be apparent in the spectrum or the first-order correlation function, but which may be seen in the second-order correlation function. The second-order correlation function of the light in resonance fluorescence has been calculated by Carmichael and Walls and others (Cohen-Tannoudji, 1977; Kimble and Mandel, 1976; Carmichael and Walls, 1975, 1976). The result in the steady state for the saturated atom ( > ) is g (2) (τ ) = 1 − e−3τ/4 cos τ . We see that this function exhibits damped oscillations at the Rabi frequency. Moreover, the extraordinary feature of this correlation function is that it begins at zero and increases. This is quite unknown in electromagnetic fields produced by classical sources. For example, the g (2) (τ ) for chaotic light fields as measured in the Hanbury-Brown and Twiss experiment begins at two and decreases to one. For coherent light such as produced by ideal lasers g (2) (τ ) has a constant value of one. The function g (2) (0) can be interpreted as a measure of the probability of photons arriving in pairs. Hence for chaotic light where g (2) (0) is twice the random background, this effect has been termed “photon bunching”. The behavior evident in g (2) (τ ) for resonance fluorescence is a manifestation of quite the opposite

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effect, i.e. “photon antibunching”. It is known that the quantum theory of the electromagnetic field allows such behavior. lf we evaluate the correlation function via a probability distribution P (E) for the complex field amplitude E, we find (see, for example, (Walls, 1979)). 5 P (E)(|E|2 − |E|2 )2 d 2 E (2) g (0) − 1 = . |E|2 2 Classically P (E) is a probability distribution and g (2) (0) − 1 is always positive, ensuring g (2) (0) ≥ 1. For a chaotic field with a Gaussian distribution for P (E) we find g (2) (0) = 2, whereas for a coherent field with a stabilized amplitude P (E) is a delta function and hence g (2) (0) = 1. The photon correlation for chaotic fields and for coherent laser fields is therefore adequately described by classical theory. Since g (2) (0) ≥ 1 must be fulfilled in the classical case, antibunching is not allowed. However, in the quantum formulation of the electromagnetic field P (E) is a quasi-probability distribution and may take on negative values allowing a g (2) (0) < 1. The interpretation of this effect in the phenomenon of resonance fluorescence goes as follows: The first photon detected implies the atom has undergone an emission process and so is now in its ground state. In order to register a correlation, a second photon must be detected. It is clear that there must be a time lapse for the atom to regain its excited state. In fact, g (2) (τ ) is just proportional to the probability of observing the atom in the upper state when it was initially prepared in the ground state (Mollow, 1975). We note that as the probability of the atom being in the upper state increases, g (2) (τ ) may exceed 1, and we see photon bunching and photon antibunching exhibited in the same phenomenon. In order to observe the photon antibunching, one requires a very small number of atoms in the observation region (preferably ≤ 1). In the presence of a large number of atoms the heterodyne signal from the beating of light from different atoms will completely obscure the photon antibunching in the light from a single atom. In a dilute atomic beam with a mean number of atoms in the observation region ≤ 1, one still has the problem of atomic number fluctuations. As pointed out by Jakeman et al. (1977), the correlation measurement will include the statistics of the atomic number fluctuations. For Poissonian number fluctuations this implies that g (2) (0) = 1. The photon antibunching then cannot be directly observed but is superposed on the atomic number fluctuations. It has been suggested (Jackson and Tuan, 1966) that an alternative normalization scheme of the correlation function which reduces the effect of the atomic number fluctuations may enable photon antibunching with a g (2) (0) < 1 to be observed directly. At the end of the seventies two photon correlation experiments were conducted on Na atoms in order to observe the photon antibunching. The first one was published by Mandel and co-workers (Kimble et al., 1978; Dagenais and Mandel, 1978; Kimble et al., 1977), and the second one was conducted in our laboratory

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in the frame of the thesis of Rateike, which was later published in (Cresser et al., 1982). The initial result obtained by Kimble et al. (1977) showed that the second-order correlation function g (2) (t) had a positive slope, which is characteristic of photon antibunching. However, g (2) (0) was larger than g (2) (t) for t → ∞ due to number fluctuations in the atomic beam and to the finite interaction time of the atoms (Jakeman et al., 1977; Kimble et al., 1978). Further refinement of the analysis of the experiment was provided by Dagenais and Mandel (1978). In the thesis of Rateike longer interaction times of the atoms were used so that antibunching could be seen directly; furthermore, the photon correlation was investigated at very low light intensities (see Cresser et al., 1982 for a review). Later, photon antibunching was measured using a single trapped ion, thus avoiding the disadvantages of atom number statistics and finite interaction time between atom and laser field (Diedrich and Walther, 1987). This experiment is discussed in a new version later in this review. As pointed out in many papers, photon antibunching is a purely quantum phenomenon (see, for example, Cresser et al., 1982; Walls, 1979). The fluorescence of a single ion displays the additional nonclassical property that the variance of the photon number is smaller than its mean value (i.e. it is sub-Poissonian) (see Short and Mandel, 1983; Diedrich and Walther, 1987). Antibunching and subPoissonian statistics are often associated. They are nevertheless distinct properties and need not necessarily be simultaneously observed, as is the case in the experiment with a single trapped ion described later. Although there is evidence of antibunching in the atomic-beam experiments, the photon counts were not subPoissonian as a result of the Poissonian fluctuations of the number of atoms. In the experiment by Short and Mandel this effect was excluded by means of a special trigger scheme for the single-atom events (Short and Mandel, 1983). In the ion experiment (Diedrich and Walther, 1987) described later these precautions are not necessary since there are no fluctuations in the atomic number. The dye laser used in the correlation experiment was the same as that employed for the measurements of the fluorescence spectrum (Section 5). The cavity length had to be stabilized to compensate for frequency drifts, since signal averaging up to two hours was necessary in the experiment. The standard way of stabilizing a laser is to modulate the cavity length. This results in a modulation of the output frequency, which is a disadvantage for the experiment since the effective linewidth of the laser is broadened in this way. The resonance frequency of the reference atomic beam was therefore modulated instead. The reference atomic beam with a collimation ratio of 1 : 500 was exposed to an oscillating magnetic field produced by a pair of Helmholtz coils. Part of the dye laser beam having an intensity low enough not to saturate the atomic transition was circularly polarized and directed at right angles onto the reference atomic beam, inducing the F = 2 → F = 3 hyperfine transition of the D2 line. Because of the different g factors of the two

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levels the atomic transition frequency was modulated by the oscillating magnetic field. As a result the atomic fluorescence was also modulated. With the magnetic field oscillating around B = 0, the laser was stabilized exactly on resonance. By introducing an offset B = 0 the laser could also be stabilized to frequencies slightly off resonance. To study the photon correlation of the fluorescence, the frequency-stabilized dye laser beam is directed at right angles onto the highly collimated atomic sodium beam (collimation ratio 1 : 1000). Using circularly polarized light the Zeeman sublevels of the atoms are optically pumped and after about 100 spontaneous emissions only the 32 S1/2 , F = 2, mF = 2 level is populated, which can then be coupled to the 32 P3/2 , F = 3, mF = 3 level. The sodium atoms thus represent a two-level system to a good approximation, as discussed above. In order to allow Rabi oscillations in the photon correlation to be observed, it is essential that the atoms in the observation region be subject to a constant laser intensity. The spatial distribution of the laser beam, however, is described by a Gaussian profile. The observation has therefore to be limited to the maximum of the Gaussian profile, only allowing laser intensity changes of /4) the Rabi frequency varies with the square root of the laser power. The measurement for 10 mw (Fig. 8) shows the Rabi oscillation to level out for longer times to a value larger than the minimum at τ = 0, which clearly exhibits the antibunching at τ = 0. This is in contrast to the experiments conducted by Kimble et al. (1978), where the number of coincidences gets smaller rather fast for large delay values due to the short transit time (≈100 ns) of the atoms through the observation volume. In the present experiment such a decrease has not been observed, since the transit time was 250 ns. There is still interest in the photon correlation at low laser intensities (Wodkiewicz, 1980) where / < 1. In this limit the laser bandwidth changes the photon correlation in a way different to that in the case / > 1. The result for low laser intensity gives a generalization of the Heitler–Weisskopf effect (Section 5) applied to photon correlations. The signal is described by g (2) (τ ) = 1 + e−τ

1 + 2δ/ e−τ /2−δτ −2 , 1 − 2δ/ 1 − 2δ/

where δ is the diffusion coefficient of the phase of the laser (Wodkiewicz, 1980), i.e. the laser linewidth. For the limit δ = 0 (monochromatic source) it follows that 2 g (2) (τ ) = 1 − e−τ /2 ; this is in reasonable agreement with the measurement shown in Fig. 9 when in addition the finite transit time of the atoms through the observation region is considered. A new type of correlation experiment was conducted by Aspect et al. (1980). In contrast to the experiments on resonance fluorescence described in Section 5, emphasizing either the frequency or the time features, the new experiment involves a mixed analysis. It investigates the time correlation between fluorescence photons selected by frequency filters. If the three components of the fluorescence triplet are well separated, one can use filters centered at any one of these components. It is then possible to study the time correlation of the filtered fluorescence. In the experiment a Sr atomic beam is excited by a laser line which is 28 Å off resonance. It was seen that the photons of the two sidebands of the fluorescence triplet are emitted in a well-defined time order which can be explained in terms of

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F IG . 9. Photon correlation for low laser intensities / < 1. The theoretical curve (solid line) is corrected for finite transition time effects.

the sequence of fluorescence decays down the energy diagram of a dressed atom. This experiment gives, despite the fact that the basic features of the resonance fluorescence in a strong monochromatic laser field are understood, an interesting new view of the processes involved.

8. Photon Correlation Measured with a Single Trapped Particle The theory of the spectrum of resonance fluorescence as discussed in the previous sections does not in principle require quantization of the electromagnetic field to end up with the right result. However, quantum properties of the fluorescent light do appear if higher-order correlations of the radiation field are investigated. One of the most striking quantum effects in single-atom resonance fluorescence is the phenomenon of antibunching. It refers to fields with a reduced photon coincidence rate on short time scales, characterized by a second-order correlation function g (2) (τ ) with a positive slope for small τ > 0. Such an intensity anticorrelation cannot occur with classical fields, as was pointed out above. In single-atom fluorescence it arises due to the delay associated with the need

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(a)

(b) F IG . 10. (a) Hanbury-Brown and Twiss setup for detecting intensity correlations of the fluorescent light. (b) Setup for the homodyne cross-correlation scheme. At the beam splitter BS the fluorescent light and a local oscillator beam of the same intensity are superimposed.

for re-excitation of the atom between two consecutive photon detection events. This phenomenon was observed using a dilute atomic beam (Kimble et al., 1977; Cresser et al., 1982). Later it was measured using a single laser-cooled ion (Diedrich and Walther, 1987). Photon correlations are detected, as discussed above, with a Hanbury-Brown and Twiss setup (Diedrich and Walther, 1987) in which a beam splitter divides the fluorescent light between two photon counters (Fig. 10(a)). From the delay between photon events registered in the two detectors, the intensity correlation function g (2) (τ ) is obtained. We measured the intensity correlation of the fluorescent light under the same experimental conditions as for heterodyne spectroscopy, in particular with minimized micromotion. The results presented in Fig. 11 clearly show the antibunching effect with a vanishing probability of consecutive detection of two photons at τ = 0. For increasing intensity of the exciting laser, the function displays oscillations at the Rabi frequency . Another fundamental quantum phenomenon predicted for the fluorescent radiation of a single ion is quadrature squeezing (Walls and Zoller, 1981). For

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F IG . 11. Intensity correlation measurements for a single Mg+ ion for three different detunings and intensities of the incident field. (a) = −2.3, = 2.8 ; (b) = −1.1 , = 1.0 ; (c) = −0.5 , = 0.6 . The parameters were obtained from a theoretical fit (solid line) to the data, using the relevant formula from (Dagenais and Mandel, 1978). The values for s are (a): s = 0.7; (b): s = 0.3 and (c): s = 0.4. For details see Höffges et al. (1997a).

suitable intensity and detuning of the driving field, fluctuations in one quadrature component of the scattered field are reduced below the classically allowed limit, while the other quadrature component shows increased fluctuations. Squeezing in resonance fluorescence has not yet been observed in the laboratory, because the experimental requirements are much more difficult to meet than, for example, those required to observe antibunching.

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A precondition for measuring squeezing is the elimination of phase shifts associated with atomic motion, since they would destroy any squeezing effect. In our system, these phase shifts are avoided by cancelling the micromotion in three dimensions, which localizes the ion to the Lamb–Dicke regime. The observation of coherently scattered radiation with a linewidth below 1 Hz is an indication of the excellent phase stability that was achieved. Further improvement can be expected from active phase stabilization. The observation of squeezing requires phase-sensitive detection of the electromagnetic field, which is usually accomplished with a homodyne scheme (Slusher et al., 1985; Wu et al., 1986). However, for single-ion fluorescence, the squeezing signal obtained in a homodyne measurement is sharply reduced by the limited collection and detection efficiencies, practically eliminating any observable effect (Mandel, 1982). To solve this problem, an alternative measurement scheme was proposed (Ou et al., 1987; Vogel 1991, 1995), based on observation of intensity cross-correlations. It can be implemented in our experiment with only slight modifications of the existing setup. As in the antibunching experiment, a pair of photon counters are used as detectors. To obtain phase sensitivity, the fluorescent light from the ion is superimposed upon by a local oscillator at the input beam splitter (Fig. 10(b)) and the intensity cross-correlation function of the combined fields is recorded. In this scheme the maximum signal due to squeezing is obtained if the fluorescent light and the local oscillator have the same intensity (weak local oscillator). The main advantage of the method, compared with the homodyne detection scheme, is that the detection efficiency appears only as a constant multiplicative factor and hence cancels when the cross-correlation function is normalized. The magnitude of squeezing is extracted from the modulation of the cross-correlation function at zero delay when the phase of the local oscillator is varied. As was the case for the resonance fluorescence spectrum, squeezing is also influenced by motion of the ion in the trapping potential (De Matos Filho and Vogel, 1994). For example, the quantum fluctuations of the fluorescent light depend on the direction of observation if realistic values of the Lamb–Dicke parameter are used.

9. Conclusion This review discusses laser experiments on the resonance fluorescence of atoms. Of special interest are the experiments using trapped and laser-cooled ions where a heterodyne experiment on the elastic peak of the resonance fluorescence is possible. At similar experimental parameters we also measured antibunching in the photon correlation of the scattered field. Together, the two measurements show that, in the limit of weak excitation, the fluorescence light differs from the excitation radiation in the second-order correlation but not in the first-order correlation. However, the elastic component of resonance fluorescence combines an extremely

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narrow frequency spectrum with antibunched photon statistics, which means that the fluorescence radiation is not second-order coherent as expected from a classical point of view. This apparent contradiction can be easily explained by taking into account the quantum nature of light, since first-order coherence does not imply second-order coherence for quantized fields (Loudon, 1980). The heterodyne and the photon correlation measurements are complementary, since they emphasize either the classical wave properties or the quantum properties of resonance fluorescence, respectively. The heterodyne measurement corresponds to wave detection of the radiation, whereas the measurement of the photon correlation is a particle detection scheme. Wave detection provides the properties of a classical atom, i.e. a driven oscillator, whereas particle or photon detection displays the quantum properties of the atom. Whether the atom shows classical or quantum properties thus depends on the method of observation. With the combination of heterodyne and photon correlation measurement, it will be possible to detect squeezing in resonance fluorescence. For this purpose, the antibunched fluorescent radiation must be superimposed upon by a modematched fraction of the exciting laser beam which has been attenuated to the same intensity as the fluorescent light. The photon correlation of the combined beams will still show antibunching. The signature of squeezing is an enhancement or reduction in the level of antibunching, depending on the phase difference between the local oscillator and the fluorescent light (Vogel, 1991, 1995). A single ion is needed to observe squeezing in resonance fluorescence since the antibunching effect in the photon correlation washes out if more than one ion contributes to fluorescent radiation (Jessen et al., 1992; Diedrich and Walther, 1987). For an experiment where squeezing in resonance fluorescence has been observed see (Lu et al., 1998). In a recent treatment of a quantized trapped particle (Glauber, 1992) it was shown that a trapped ion in the vibrational ground state of the trap will also show the influence of the micromotion since the wavefunction distribution of the ion is pulsating at the trap frequency. This means that a trapped particle completely at rest will also scatter light into the micromotion sidebands. Investigation of the heterodyne spectrum at the sidebands may afford a chance of confirming these findings.

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ATOMIC PHYSICS WITH RADIOACTIVE ATOMS∗ JACQUES PINARD1 and H. HENRY STROKE2 1 Laboratoire Aimé Cotton, Bât. 505, Faculté des Sciences, F91405 Orsay, France 2 Department of Physics, New York University, 4 Washington Place, New York, NY 10003, USA

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. “Off-line” Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Atomic Spectroscopy: the Actinides . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Hyperfine Structure and Isotope Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . 3. “On-line” Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. A Challenge for Atomic Spectroscopy: the Search for Optical Transitions in Francium 3.2. Isotope Shifts; Pre-laser Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. High-Resolution Laser Spectroscopy Work On-line with an ISOL . . . . . . . . . . . 4. Bohr–Weisskopf Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract A review is given of atomic spectroscopy experiments with radioisotopes and their impact on atomic and nuclear structure studies. These range from dispersive and interference techniques to radiofrequency and laser spectroscopy. An extensive study has been made of spectra of actinides. Availability of beams of isotopes produced at accelerators and used “on line” permit studies of nuclides with sub-second halflives. The atomic properties of francium could thus be obtained. Systematic measurements by these atomic techniques over a range of isotopes can yield data on nuclear charge and magnetization properties not readily available by nuclear spectroscopic methods.

∗ The authors have been collaborating for many years on atomic beam magnetic resonance experiments, particularly with radioisotopes, at the Laboratoire Aimé Cotton and at CERN. Jacques Pinard, who early on worked on high-resolution Fourier spectroscopy with Pierre Connes, has been engaged, more recently, in experiments that combine laser and nuclear techniques for measuring atomic-electron-nuclear interactions, as well as in atomic structure experiments on Rydberg atoms perturbed by strong electric or magnetic fields. Henry Stroke has studied spectra of radioactive atoms with different techniques, from ordinary spectroscopy to resonance methods, starting with his very first experiment in collaboration with Ben Bederson, to whom the authors dedicate this article.

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1. Introduction The celebration of the achievements of Ben Bederson offers us an opportunity to give an aperçu of atomic spectroscopy with radioactive atoms. Since Ben’s scientific career took place for many years at New York University (NYU), we will touch on a little pertinent history. We will not be complete by any means in the selection of experiments that we take as examples, nor will we be able to cite the many colleagues who contributed to this field over the years. They were in many laboratories, in Europe, North America, and Japan. Since the 1896 discovery by Henri Becquerel of radioactivity in uranium, the chart of the elements grew rapidly with the addition of new radioactive atoms to the table of known stable ones. These fall into two categories. The first includes the discovery of new unstable elements in the heavy mass region of the chart, the so-called actinides. The second group consists of the natural or artificiallyproduced radioactive isotopes of known stable elements. It may be hard to imagine the extent of the work generated by the study of all these new atoms from the point of view of both nuclear and atomic physics. Optical spectroscopy of radioisotopes started in fact around 1926 with 209 Bi which, however, was not recognized to be radioactive until recently [1], though with a half-life of 1.9 × 1019 years! Radon and radium were the next ones to be measured in the early 1930’s [2]. Theorists were already making many calculations. The study of resonance lines was pursued actively at NYU at the time [3]. Herman Yagoda, in particular, worked on the next member of the alkali series [4], ekacesium, with atomic number Z = 87, an element as yet undiscovered. He made predictions for the wavelengths of its resonance lines (apparently named at the time raies ultimes or “ultimate lines”) [5]. The element, to be named francium, was not discovered until 1939 by Marguerite Perey, who had worked with Marie Curie at the Institut du Radium. It took another forty years until the first spectral lines of this element, the longest lived isotope of which has a lifetime of 22 minutes, were observed at CERN with use of atomic beam and laser spectroscopic techniques by research groups from the Laboratoire Aimé Cotton and the Laboratoire René Bernas from Orsay [6]. Despite a number of subsequent more sophisticated calculations, the values of Yagoda proved to be the closest to the experimental wavelengths (see Section 3.1). This brings us to the post-World-War II era. We shall discuss the work in two groups: “off-line” and “on-line”. Globally, the physics underlying the experimental work that is presented here spans atomic structure, hyperfine structure, and isotope shifts. Before leaving the period around 1930, we note the seminal theoretical isotope-shift work by Jenny Rosenthal and Gregory Breit [7], that also took place at NYU, in which appeared the first calculation of the effect of the extended nuclear charge volume on the hfs and the

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isotope shift.1 [In addition to this volume, or “field shift”, which is of nuclear structure interest, we recall that there are mass-dependent terms that make up the isotope shift. These arise from the elimination of the center-of-mass motion in the Hamiltonian: (a) the “normal” or Bohr (reduced-mass) shift, and (b) the “specific mass” shift (a many-electron correlation term). For heavy nuclei, the mass shifts are almost negligible, but for light nuclei they become relatively important and make the extraction of nuclear-shape data less reliable.]

2. “Off-line” Experiments To begin, we define “off-line” experiments as those where the radioactive isotopes are available in nature, or are produced at some accelerator or at a nuclear reactor, and are then used to fabricate sources, e.g. spectral lamps or cells, or atomic beam sources. These frequently entail radiochemistry and possibly mass separations of the isotopes, all of which has limited such work to nuclides with half-lives not much shorter than one hour.

2.1. ATOMIC S PECTROSCOPY : THE ACTINIDES This group of elements, from Z = 89 to 103, by analogy to the name for francium before its discovery, could have been named ekalanthanides. These fifteen elements, by now all identified and named, are the actinides. Some of these elements have half-lives that are less than one hour. Spectral studies have been performed using mainly the classical methods of atomic spectroscopy with instruments such as the large grating spectrograph at Argonne National Laboratory, and the Fourier spectrometer at the Laboratoire Aimé Cotton. The light sources were either electrodeless quartz lamps or hollow cathodes. The interpretation of the enormous number of measured spectral lines represented (and still represents) a life-long challenge for many atomic physicists: the lines regrettably do not come with initial and final state labels! Identifications rely on a number of tools. First, there is the “combination principle” proposed by Johannes Robert Rydberg (1896) at Lund and Walter Ritz (1908). The combination principle, postulated a number of years before the Bohr atom, states that 1 The effect on the hfs still bears the name “Breit–Rosenthal correction” (to the interaction with a point-like nucleus). Following this work, Jenny Rosenthal (later—Bramley) changed fields. Half a century later one of us (H.H.S.) had occasion to meet her and told her that her work was still current: she never knew before then that in the literature her name was attached to this correction. En passant, after undergraduate studies in Paris, she did her graduate work at NYU, and became the first woman in the United States to receive a PhD in physics; at that time she was only 19.

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the frequencies of sets of observed spectral lines could be explained effectively in terms of differences of sets of spectral terms. (The work at Lund, applied to spectra of highly-ionized atoms of astrophysical interest, continued for many years under the direction of Bengt Edlén.) Ritz, a Swiss from the Valais, did much of his work in Göttingen as a mathematician and mathematical physicist. He is well known for his variational method of solving differential equations. He used complex mathematical models in an attempt to account for the relations between the observed spectral frequencies. But, apparently, he arrived at the formulation of the combination principle that bears his name only after doing some spectroscopic work himself in the laboratory of Aimé Cotton, just about one hundred years ago. Second, Zeeman effect studies permit the determination of Landé g values and relevant angular momenta. Hyperfine structure and isotope shifts can provide additional clues: transitions in which the isotope shifts appear to involve penetrating (s or relativistic p1/2 ) electrons, which have a non-zero probability at the nucleus. In heavy nuclei the isotope shifts in the atomic spectra are caused mainly by variations in the nuclear charge radius from isotope to isotope. The analysis of the actinide spectra continues very actively. An excellent database site has been established at the Laboratoire Aimé Cotton [8] by Jean Blaise and Jean-François Wyart, both of whom contributed substantially to the level classifications of these elements. The information, frequently updated, gives energy levels and line wavelengths, as well as brief historical accounts and references.

2.2. H YPERFINE S TRUCTURE AND I SOTOPE S HIFTS We will first look at the early experiments which were possible with a specific isotope, or a small set of isotopes, to extract individual data on nuclear spins and hfs interactions. We will then focus our attention on the extensive applications to the spectroscopy of radioactive atoms provided by the discovery and development of single-mode tunable lasers in the 1970’s and 80’s. Basically two problems have to be addressed: adequate accuracy to extract the required high-resolution data, and the concomitant production of the radioisotopes. In all the experiments the line width obtainable by the various experimental techniques is a crucial consideration: the “natural” line width that corresponds to the lifetime of the atomic states involved in a transition could be obscured by the Doppler broadening of the line 6 T −7 νD = 7.16 × 10 ν , M where ν is the frequency of the transition, T , the absolute temperature, and M, the molecular weight of the radiating atom. By going from optical to rf transitions, the linewidth narrowing is seen to be an immediate advantage for hfs

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measurements. Resonance methods, however, are not applicable to isotope-shift (IS) measurements, as we are dealing with different atomic systems. It is thus not surprising that rf resonance experiments with radioisotopes preceded dispersive optical spectroscopy. We will see in Section 3.3.1 how laser-spectroscopic methods can provide a new way to limit the Doppler effect even for isotope shift measurements. 2.2.1. Early Atomic Beam Magnetic Resonance Experiments We can probably date the first radioactive atomic beam magnetic resonance (ABMR) to the measurement of 40 K by Jerrold Zacharias [10]. Though this isotope occurs naturally with an abundance of 0.0117%, it is radioactive with a half life of 1.277 × 109 years. Experimentally it represented a crucial advance. In the original ABMR method of I.I. Rabi [11] the atomic beam traversed two inhomogeneous (“Stern–Gerlach”, or two-wire field) magnets oriented in opposite directions: in the absence of a magnetic resonance in the intervening region, the full atomic beam was detected. The resonance was observed as a small decrease in the large-intensity background. This was not suitable for detecting the small amount of 40 K in the presence of the stable isotope components. The clever modification made by Zacharias consisted in having the two deflecting magnets oriented in the same direction, so that in the absence of resonance, the detected atomic beam intensity was zero. This is known by the name of “flop-in” technique. The gain in signal-to-noise ratio now allowed working with very small quantities of atoms, including radioisotopes. After the end of World War II, Zacharias established an atomic beam laboratory at MIT where bona fide radioactive experiments were begun, with 2.6-year 22 Na [12]. Experiments were done on long-lived 135 Cs and 137 Cs, but in those early days of nuclear structure studies interest was particularly great in the spins of odd-Z, odd-N isotopes. Ben Bederson was thus involved in the measurement of 134 Cs [13]. (We note that the value of the nuclear spin was confirmed at the same time by the earlier, non-resonance, “zero-moment” method [14], which locates the magnetic fields at which the effective magnetic moments, µeff = ∂E/∂B, in the Zeeman pattern are zero, allowing the value of the nuclear spin to be extracted.) Following this experiment, it was proposed by one of the authors that it would be interesting to measure similarly the 2.903-hour nuclear isomer 134m Cs: this suggestion was derisively shot down by an esteemed professor at MIT: “Impossible by many orders of magnitude”! Not long afterwards, this experiment became, in fact, the first one to be done simultaneously by two new ABMR groups, at Argonne National Laboratory and at Brookhaven National Laboratory. The gathering of new information on nuclear magnetism also became possible with the use of precision ABMR: the measurement of the distribution of nuclear magnetism in the nucleus, now known as the “Bohr–Weisskopf effect” (BW) [15].

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We return to this in Section 4 as current efforts continue in this study, but with online experiments. A measurement of this effect, in 40 K and the stable potassium isotopes, was done by Bederson and his coworkers [16]. It continued with measurements of BW in four cesium isotopes, of which three are radioactive. Work in this direction in the MIT Atomic Beams Laboratory stopped in 1954, after an unfortunate major spill of long-lived radioisotopes! However, this was only the beginning of ABMR work with radioisotopes in a number of laboratories in the world, with many important modifications. Some of these, particularly in conjunction with the on-line experiments, are described separately. There was the introduction by Donald Hamilton and his group at Princeton University of focussing hexapole deflecting magnets to gain beam intensity. In one of his lesser-known works, Steven Weinberg, as a graduate student, made calculations of the atom orbits in these deflecting fields [17]. Work with relatively short-lived isotopes was pursued at Princeton, aided by the proximity of the cyclotron that was used to produce them. A similar situation existed at the University of California at Berkeley, where the group of William Nierenberg [18] was an important source of nuclear spins and associated data. One of the graduates of this group, Vernon Ehlers, took his training in radioactive beam spectroscopy to become eventually a Member of Congress of the US. (The other physicist in the Congress, Rush Holt, a student of one of us (H.H.S.), also worked in spectroscopy.) 2.2.2. Optical Methods in Radioisotope Spectroscopy; Isotope and Isomer Shifts Almost contemporaneously with the MIT ABMR experiments, work was begun on hfs studies in excited states of atoms by combining resonance radiation optical excitation with rf transitions in mercury. After reading on, one may ask the question “why are so many of the different experiments done on mercury”? When hfs- and isotope-shift studies began, mercury stood out as having a combination of outstanding advantages: (1) it has a relatively simple spectrum, (2) it has seven stable isotopes, two of which have odd-mass number with differing nuclear spins, (3) there is a further dozen isotopes and nuclear isomers with sufficiently long half-lives for “off-line” experiments, (4) it is a heavy element where the isotope shifts reflect nearly entirely variations in the nuclear charge distribution, (5) its resonance radiation at 253.7 nm is easily transmitted through the walls of quartz lamps and cells, and (6) it is liquid at room temperature and is easy to vaporize. In these first experiments, the appropriately tunable light sources were singleisotope electrodeless lamps placed in magnetic fields, suitably tuned by the Zee-

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man effect to excite transitions in the radioisotopes. The rf resonances were detected by changes in intensity of the fluorescent polarized light. Determination of hyperfine structure was also made with the use of the “level-crossing” technique [19], in which observations were made of coherences in the fluorescence when levels became degenerate at specific magnetic fields, determined by the hfs of the atom. We note that the level-crossing determinations of hfs are Dopplerfree. These methods allowed the hfs measurement of a series of radioisotopes of mercury from 193 Hg to 203 Hg, and included the structures of the first nuclear isomers. These isomers are characterized by large differences in angular momentum from those in the ground state, leading to small transition probabilities. In Section 3.3.3, we discuss another type of isomer. This was the first systematic study by such optical means of nuclear properties over an extended range of isotopes. They are considered further for mercury in Section 3.2.1. Two other avenues of radioisotope studies by optical techniques were also pursued. The first was “optical pumping”, a method due to Alfred Kastler: The fluorescence that follows excitation of the atom by polarized resonance radiation causes a selective redistribution of atomic ground-state magnetic sublevels. In the case of the diamagnetic ground state of mercury, these sublevels correspond to the nuclear magnetic substates. The equilibrium population can be reestablished (and detected) by applying the nuclear magnetic resonance frequency. Nuclear magnetic moments of the odd mass-number stable and radioisotopes and isomers of mercury in the above range could thus be measured. The second approach was high-resolution spectroscopy with use of the MIT 10 m focal length two mirror monochromator in which a 25 cm wide diffraction grating was used in autocollimation. With this, the optical spectra and isotope shifts of the mercury radioisotopes could be measured (as well as those of other elements). The importance of these experiments is that they allowed the determination of isotope and isomer shifts which, as we noted in Section 2.2, are not accessible to resonance experiments. Their significance is discussed in Sections 3.2.1 and 3.2.2 in the light of nuclear structure studies. A review of these early optical measurements was compiled by Francis Bitter [20]. These pioneering optical experiments were followed many years later by highprecision microwave and rf spectroscopy of ions in traps [21]. Paul traps (with static electric and rf fields applied to the electrode caps) or Penning traps (with superposed electric and magnetic fields) were used, depending on whether hfs, near zero magnetic field, or g-values, in high fields, were to be studied. In contradistinction to the double-resonance (optical-rf) method of Jean Brossel and Francis Bitter [22], in which the rf transitions are induced in the excited states, in the trap experiments the resonant transitions are in the ground state. The resonances are detected by monitoring the fluorescence that follows ground-state hfs optical pumping by laser excitation [23]. The very long trapping times allow the resonant frequencies to be determined with very high precision: the smaller contributions to

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the hfs interactions, i.e. of the nuclear magnetic octupole and the electric hexadecapole, could thus be measured. The sensitivity was adequate to measure the hfs of several long-lived europium and barium isotopes [24].

3. “On-line” Experiments As we indicated earlier, “on-line” experiments were marked by the introduction of the tunable laser into the spectroscopy of radioactive atoms to measure isotope shifts and hyperfine structures with sufficient accuracy to extract nuclear data. Several review papers have already been published on this subject [25,26] and on some experiments which we describe below. These are presented here to recall the different steps from the early “on-line” experiments to the present day. Most of these experiments were performed at ISOLDE (Isotope Separator with On-Line DEtection) at CERN. ISOLDE was conceived in 1964 by a collaboration of several European mass-spectrometer laboratories. It received the important support of Victor F. Weisskopf, Director General of CERN at that time, in its implementation. Experiments started three years later. René Bernas, who headed the Orsay mass-spectroscopy laboratory (in whose memory the laboratory was later named), was one of the ISOLDE Collaboration initiators. This had an important bearing later on the Orsay laboratories becoming engaged in “on-line” atomic spectroscopy. The radioisotopes at ISOLDE are mostly produced artificially by spallation, fission reactions, or fragmentation, when an appropriate target is bombarded with protons or light nuclei. The ionized products are mass separated and the proper masses of the desired element selected. Care must be exercised to avoid the inclusion of undesirable isobars in the resulting ion beam which is directed to the experimental apparatus. For some isotope productions, a laser resonance ionization source has been used. It permits, in certain difficult cases, to selectively increase the ionization efficiency, and greatly reduce the quantity of parasitic isobars produced. A facility such as ISOLDE represents a good prototype for on-line studies. Though it was the first, a number of such mass-separator facilities at accelerator sites now exist and contribute to this research effort. As an example, Fig. 1 shows the yields for the production of cesium isotopes. It can be seen from the yield curves that in favorable cases one can obtain some 1010 ions per second. A second consideration is again the required precision. The signals from which one wants to extract data for spectral analysis are rather small and are generally embedded in much larger effects. The experimental methods must therefore give very high resolution spectra, not limited by the Doppler effect. The requirement of high spectral sensitivity is generally not compatible with the condition of low production rate. We described in Section 2 how good resolution could be achieved

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F IG . 1. ISOLDE production yields of cesium isotopes: full circles, spallation of La, open circles, U fission.

with ABMR. By combining ABMR with optical pumping, new non-optical detection methods for the resonance could be obtained. The advent of the tunable laser was essential in the development of a number of the new techniques which will be discussed in conjunction with the experiments in which they were used to obtain hfs and isotope shift data. The availability of long chains of radioactive isotopes (at present, no less than 40 isotopes of cesium are known) provides the very attractive possibility of studying in a systematic way many isotopes of a single element, and of looking for changes in properties from one isotope to the next. It was shown early on that atomic spectroscopy may play an important role in these studies, as the isotope

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F IG . 2. Hyperfine structure and isotope shifts in the D1 line of 20–31 Na. (From G. Huber, F. Touchard, S. Büttgenbach, C. Thibault, R. Klapisch, H.T. Duong, S. Liberman, J. Pinard, J.L. Vialle, P. Juncar, P. Jacquinot, Phys. Rev. C 18 (1978) 2342.)

shift of atomic energy levels was connected with the mass and shape of the nucleus. Furthermore, hyperfine structure, first suggested by Wolfgang Pauli [27] and measured and interpreted in the spectrum of bismuth by Ernst E.A. Back (Tübingen) and Samuel Abraham Goudsmit (Leiden) [28], can provide important information on the angular momentum, magnetic dipole, and, eventually, electric quadrupole and higher moments of the nucleus. As an example, Fig. 2 shows a series of recordings of the structure of the well-known D1 line of sodium for ten consecutive isotopes: one can see in the plot the hyperfine structures of the lines as well as the shifts of the centers of gravity from one isotope to the other. Before describing the different experimental methods which allowed the acquisition of considerable nuclear data through hfs and isotope-shift measurements, we first give an account of a purely atomic physics experiment performed on line at the limit of feasibility, the spectroscopy of francium, a subject which we already introduced at the outset.

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3.1. A C HALLENGE FOR ATOMIC S PECTROSCOPY: THE S EARCH FOR O PTICAL T RANSITIONS IN F RANCIUM In Section 1, we mentioned that the element 87, named francium, was discovered by Perey as a decay product of actinium; this is the last element of the alkali series, and its optical spectrum was expected to be very simple. Because of its rarity in nature (an estimate gives only 30 g in the entire earth’s crust), it was quite impossible to make a light source containing sufficient francium atoms to be able to detect its optical lines in the background of the very complex and intense spectrum of actinium and other spurious elements. It was not until 1975 when all the conditions were met to consider a spectroscopy experiment on this element. Several years earlier, Bernas reported back on the exciting on-line atomic spectroscopy that was already under way at ISOLDE. This led, in 1973, to the start of the collaboration of Pierre Jacquinot (Aimé Cotton) and Robert Klapisch (Centre de Spectroscopie Nucléaire et de Spectroscopie de Masses (CSNSM)) and their groups in future on-line hfs and isotope-shift experiments. The early experiments at ISOLDE were made by Ernst Otten (Mainz) [29] on mercury isotopes (Section 3.2.1), and on a chain of cesium isotopes by Curt Ekström (Göteborg) [30], and by their respective collaborators. With the success obtained on other radioactive alkali isotopes, Pierre Jacquinot, in 1977, launched the experiments to measure the wavelengths of the D1 and D2 lines of francium. In his words, ‘This is a challenge, but it would be a great première!’. In 1977, semi-empirical as well as theoretical predictions for the frequencies of the D1 and D2 lines of francium already existed. We noted in Section 1 that as early as 1931 Yagoda gave an estimate with very small error bars (±3 nm), but more recent values were less optimistic and ranged between 710 and 780 nm, with generally much larger error bars. Pure theoretical calculations (Hartree–Fock) gave results that differed by more than 100 nm from the previous values. All this describes the situation in 1977, and searching for an optical line over a range of at least 100 nm with techniques of high-resolution spectroscopy, such as those already developed for experiments on alkali hyperfine interactions described below, was like ‘looking for a needle in a haystack’. It required a modification of the technique to record, in a first step, a rough spectrum without losing detection sensitivity. But the optical pumping used in the experiments requires that the excitation linewidth be less than the ground-state hfs. A laser source was therefore conceived to provide five or six equidistant packets of laser modes, each packet containing two or three modes. Thus, by scanning the laser 9 GHz (the spacing between packets), one covered a spectral range of, at least, 36 GHz with a resolution of 0.5 GHz: this clearly resulted in a great reduction of the time required to record the anticipated spectral range. The francium was produced at the ISOLDE facility. A beam of 108 ions per second was obtained with use of the spallation of uranium by 600 MeV protons furnished by the synchrocyclotron, the first CERN accelerator.

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F IG . 3. Recording of the ion signal revealing the optical resonance of the D2 francium line. The multiple traces ((a): with laser light, (b): without, (c) = (a) − (b)) are needed to eliminate the spurious signal caused by proton flux fluctuations [9].

After many days of search, a signal corresponding to the excitation of the francium D2 line was recorded (Fig. 3), and the frequency of the laser measured using a spectrograph, with a neon source as the reference. It was remarkable that the measured wavelength, 717.97 ± 0.01 nm was in good agreement with the value predicted by Yagoda [9]! The hfs of this line was subsequently measured with high resolution for all the francium isotopes available at ISOLDE (Fig. 4). The experimental method is described in Section 3.3.

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F IG . 4. A high-resolution recording of the ground-state hfs of the Fr D2 line: the two groups of lines are ∼ 50 GHz apart [9].

Later, the D1 line was also found [31], but with new calculations [32] of the fine structure, and taking into account the D2 measurement, the uncertainty in the position was much lower and the difficulty of locating it was considerably reduced. The spectroscopic studies were then extended to higher energy levels, first by detecting the blue resonance line, and then transitions to many levels of the S and D Rydberg series. This was done in collaboration with the Mainz group with use of collinear laser spectroscopy (Section 3.3.1) on a fast atom beam, a technique which they had developed previously [33]. Because of its heavy mass and simple spectrum, francium was soon found to be the best element for studying fundamental physics phenomena, such as parity non-conservation [34] and, even, time-reversal invariance violation [35]. For this work, many groups were led to try to catch francium atoms in magneto-optical traps, as is currently done with stable alkalis. At the time of this writing, several of these experiment have succeeded [36,37]. We note that, since 1996, a group at the State University of New York (Stony Brook) has pursued the atomic spectroscopy of francium [38], with impressive high-precision measurements on the cloud of cold francium atoms in the trap (lifetimes, hfs measurements), giving them the possibility of studying the Bohr–Weisskopf effect (Section 4) in this heavy ele-

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ment and analyzing the result in the light of the neutron radial distribution in the nucleus [39].

3.2. I SOTOPE S HIFTS ; P RE - LASER E XPERIMENTS 3.2.1. The RADOP Experiment: Discovery of a Nuclear Shape Transition Isotope shift measurements in a given element along a series of isotopes can reveal systematics in the changes in mean-square charge radii, r 2 , of nuclei, and allow furthermore to characterize the shape of the nucleus. We introduced the beginnings of this effort, off-line, in Section 2.2.1. A most spectacular demonstration of the importance of such studies for nuclear physics was given by the experiments of the Mainz group [40] on the very long chain of mercury isotopes produced at ISOLDE. As early as 1964, thirteen isotopes of the chain, mainly stable and long-lived radioactive ones, were already analyzed [41], showing clearly the existence of “odd–even staggering”, the observation that the addition of an odd neutron to an even-neutron isotope generally leads to less than one half of the shift produced by the addition of a neutron pair. In studying systematically the isotope shifts further into the highly neutron-deficient region, Otten et al. discovered a sudden discontinuity that they attributed to a shape transition when the neutron number N changed from 107 to 105 (see Figs. 6 and 10). At this time, only odd isotopes could be analyzed; later on, it was shown that the nucleus alternated in shape from nearly spherical, for even N , to strongly deformed for odd N [29]. This was a remarkable observation, and became rapidly a fascinating problem for theoreticians, which led to the idea of shape coexistence in these nuclei. We first describe some of the features of the experiment on these short-lived mercury isotopes done in 1971 on-line at ISOLDE, just before the advent of the tunable dye laser. The method was called “RADOP” (Radioactive Detection of Optical Pumping), Fig. 5. The principle is the following: when the frequency of a light source is in resonance with atoms accumulated in a vapor cell in the presence of a buffer gas, these atoms may undergo hyperfine or Zeeman optical pumping, i.e. selective repopulation of the ground state magnetic sublevels. In the case of mercury, the ground-state electron angular momentum is J = 0, so only Zeeman optical pumping can take place. This results in an assembly of oriented atoms with respect to the quantization axis defined by an external magnetic field. Excitation by circularly-polarized light, σ + , or σ − , will lead, respectively, to atoms with mF = +I or mF = −I (F is the total angular momentum, and I the nuclear spin). The atomic electron-nuclear coupling transfers this orientation to the nucleus. The ensuing asymmetry in the beta emission of the decaying mercury (a similar type of asymmetry exists in gamma emission, but under different conditions of excitation) is detected with high efficiency with the use of the difference signal of two detectors, 180◦ apart, placed along the direction of

3] ATOMIC PHYSICS WITH RADIOACTIVE ATOMS 287

F IG . 5. Schematic of the apparatus used for RADOP experiments on line at ISOLDE. The diagram on the right shows the Zeeman scan of the hfs of 189 Hg using a 198 Hg light source and the corresponding recorded signal [40,26].

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F IG . 6. Early isotonic shifts observed for mercury, thallium, lead, and bismuth. (From M. Barboza-Flores, O. Redi, H.H. Stroke, Hyperfine spectrum of 207 Bi by absorption spectroscopy, Z. Phys. A 321 (1985) 85, Fig. 2. © Springer-Verlag, 1985.) Some francium lines are shown for comparison.

the magnetic field. One can also observe the optical resonance by switching alternately between the σ + and σ − excitation. In this early experimental work, the “tunable” light source was the Zeeman-tuned (Section 2.2.2) 253.7 nm line of a mono-isotopic 198 Hg discharge lamp. Unfortunately, this sensitive method was not applicable to even isotopes for which I = 0. Consequently, the experiment was repeated some years later by using laser-induced fluorescence in a gas cell. The resolution was poor, but sufficient to analyze the isotope shifts and show that the nuclei with N < 107 pass alternately from a nearly spherical shape for even N to strongly deformed for odd N . The variations in r 2 are presented in Fig. 10, Section 3.3.2. This experiment contributed greatly to opening this field of research at the frontier between atomic and nuclear physics. From 1972 on, the tunable dye laser stimulated the development of new methods of resonance detection, always more precise in frequency determination and more sensitive, permitting the extension of radioactive isotope studies further and further from the region of nuclear stability.

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3.2.2. Isotonic shifts In the preceding section we noted the remarkable jumps in the nuclear charge radii that occur in highly neutron-deficient mercury isotopes. A particularly useful way of studying isotope shifts, when a large body of systematic data provided by the experiments with radioisotopes is available, is to look at isotone shifts. By normalizing the isotope shifts in different elements to a particular pair of nuclei with the same neutron numbers, we can compare the relative effects that changes of their neutron numbers and orbits have on variations of the nuclear charge distribution. We recall that in heavy nuclei the isotope shifts reflect essentially the changes in r 2 of the charge distribution. In Fig. 6 this is shown for the first set of isotonic shifts obtained with grating spectroscopy [41] for mercury (Z = 80) and several neighboring elements. There are seen to be remarkable similarities in the isotone shifts. Since these initial measurements, a vast amount of data has been obtained for seven elements, extending the range to radium (Z = 88). The similarities appear to persist to a large extent over this region [42]. Such systematic measurements have also allowed new possible explanations of the “odd–even staggering” of the isotope shifts. A picture of the polarization of the nucleons in the nucleus by the addition of neutrons appears to be able to account qualitatively for this staggering [43]. A calculation of an isomer shift, i.e. the change in the charge radius upon excitation of a neutron, was able to give reasonable agreement with the use of an appropriate internucleon potential [44].

3.3. H IGH -R ESOLUTION L ASER S PECTROSCOPY W ORK O N - LINE WITH AN ISOL The unexpected results on the neutron-deficient isotope shifts in mercury, which we described above, led to other similar searches, in particular in sodium. It was known that such a deformation may appear in the vicinity of 31 Na. This isotope is at the extreme limit of the isotopic series, so its lifetime is quite short (17 ms), and the production was very low: this represented a challenge. At that time (∼1973), sodium was used extensively at the Laboratoire Aimé Cotton to test different new methods of high-resolution spectroscopy with home made single mode tunable dye lasers. The choice of sodium was governed principally by the fact that its resonance line at 590 nm falls in the range of the most efficient existing laser dye, Rhodamine 6G. With this “ideal” in mind, a collaboration was formed between Aimé Cotton and the CSNSM, and a first successful experiment performed at the synchrocyclotron of the Institut de Physique Nucléaire in Orsay in 1974. We describe briefly the experiment with reference to Fig. 7, a prototype for alkali studies done subsequently at ISOLDE.

290 J. Pinard and H.H. Stroke [3

F IG . 7. (a) Schematic of the apparatus used for alkali studies (e.g. francium); (b) signal recorded for each of the five resonances of the hfs of the D2 line, exhibiting an optical pumping effect. (From C. Thibault, F. Touchard, S. Büttgenbach, R. Klapisch, M. de Saint Simon, H.T. Duong, P. Jacquinot, P. Juncar, S. Liberman, P. Pillet. J. Pinard, J.L. Vialle, A. Pesnelle and G. Huber, Phys. Rev. C 23 (1981) 2720.)

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For alkalis, the ground J = 1/2 level is split by the hyperfine interaction into two levels, F+ = I + 1/2 and F− = I − 1/2 (I ≥ 1/2). At normal temperature, their populations are in the ratio of their level degeneracies, 2(I + 1) and 2I . If laser light is in resonance with one of the transitions of the group n2 S1/2 → 2 P1/2 , optical pumping between the two ground-state levels occurs: the initial level from which the excitation begins will be completely depopulated to the benefit of the other level. For atoms in a beam, this change can be analyzed with use of an inhomogeneous magnetic field, actually, a six-pole magnet: it acts as a focusing element for the “low-field seeker state”, i.e. states with mJ = +1/2, and a defocussing element for the others. The experiment is performed as follows [45]: The 60 keV ion beam of a selected isotope from ISOLDE is first transformed, with use of a neutralizer, into a thermal atomic beam. The atoms of this collimated beam are excited by the light of a tunable dye laser at a right angle, in order to reduce Doppler line broadening. The atoms then pass through a hexapole magnet which focuses those with mJ = +1/2 (which means half of the atoms when the light is out of resonance) on the entrance aperture of a mass spectrometer. Here, as in a number of previous ABMR experiments with alkalis, the atoms are reionized, and then sorted by mass before detection, thereby avoiding signals from spurious ions. A result is shown in Fig. 7(b). The signal can be either positive or negative, depending on where the optical transition starts, at F+ or F− . All the results on the alkalis, including francium, were obtained with use of this method. For the particular case of the sodium experiments, the atoms were produced directly from a uranium target bombarded by the 20 GeV protons from the CERN proton synchrotron. The overall efficiency was of the order of 10−5 , which resulted in a counting rate as low as 10 ions per proton pulse for 30 Na, and only 3 for 31 Na (Fig. 2).

3.3.1. Collinear Laser Spectroscopy: a More General Spectroscopic Method The preceding method had some major disadvantages: (1) it is neither universal nor applicable to all atoms, and (2) the reduction of the Doppler broadening is obtained by collimation of the beam, which leads to a major loss in efficiency. In 1976, Stanley L. Kaufman [46] pointed out a very interesting property of an accelerated ion beam, which can be summarized by the fact that the energy spread in the beam, which comes from the ion source itself, Es , is a constant of the beam, whatever its acceleration. In the source, in the direction of the beam excitation, 12 mvs2 = kT . For the accelerated beam in the potential U, 12 mvb2 = eU , and δE = mvb δvb . We thus obtain for the ratio of the velocity spread in the beam to

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F IG . 8. Schematic of the collinear-laser-spectroscopy system with fluorescence detection. (From A.C. Mueller, F. Buchinger, W. Klempt, E.W. Otten, R. Neugart, C. Ekström, and J. Heinemeier, Nucl. Phys. A 403 (1983) 234.)

the velocity in the source (identical to the ratio of the Doppler widths): δvb 1 kT 1/2 = . vs 2 eU With a source at 2000 K and a 60 keV beam, the reduction factor is of the order of 10−3 . The accelerated ion beam is neutralized by passing through a charge exchange cell, Fig. 8. The atoms are then excited by a laser beam, collinear with the atomic beam. The laser frequency is kept constant, and the atomic levels are brought into resonance by corresponding Doppler shifts brought about by precise voltage variations impressed on the ion beam. The resulting fluorescence is detected with use of a large collector and a cooled photomultiplier. Elsewhere, depending on the element studied, a number of other detection methods were used, e.g. selective re-ionization of the atoms in a gas cell or RADOP (Section 3.2.1). These allowed the extension of the method to include noble gases and atoms produced with low yields. There are clear advantages with this method: All the atoms produced in the source participate in the experiment, so that, in principle, maximum efficiency in their use is achieved, and the use of the accelerated beam allows, as noted above, electrostatic scanning of the resonance, avoiding the delicate frequency scanning of the laser. Another non-negligible interest in this method is that metastable levels can be populated in the charge exchange with the fast ion beam. making it possible to study, for example, spectra of the noble gases. There are also some difficulties. Much care must be taken because optical pumping is a limiting factor in this experiment: When an atom is on resonance, it re-emits only a few fluorescence photons, reducing substantially the detection efficiency. Atoms, which are optically pumped before entering the detection region are lost for detection. This can be avoided by applying a magnetic field between the charge exchange cell and the detection region, which produces a Zeeman shift on the atoms and switches them off resonance. Stray light from the laser, or from

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the fluorescence of the beam itself, which cannot be completely suppressed by filters, contributes background noise which reduces the detection limit. The number of results obtained with this method is impressive (an apparatus is still on line at ISOLDE); many of these are summarized in [26]. We recall that collinear laser spectroscopy was also the means for the measurement of highlying states in francium: in these, two-step laser excitation was used [47,48]. It has also allowed us to measure the isomeric shift between the well-known high spin nuclear isomer of 178 Hf and its ground state [49]. 3.3.2. Resonance Ionization Spectroscopy: the Study of Refractory Elements Refractory elements belong to a special class of atoms which cannot be extracted from the proton-bombarded target. They can, however, be obtained as daughter products in the decay of a parent element that can be produced by an ISOL. This was the case of the series gold, platinum, iridium, osmium . . . , all obtained starting from mercury. They are of obvious interest for systematic studies in this region. The technique consists of implanting the appropriate mercury isotope in a substrate; after a time period that corresponds to its half-life, the implanted material is desorbed by heating with use of an energetic laser pulse (e.g. Nd:Yag). This creates an atomic cloud, which is irradiated at a right angle by two (or more) crossed pulsed laser beams: the first one is weak, to avoid saturation, but is single mode to obtain a high-precision spectrum. The second one must have sufficient energy to photoionize the excited atoms with maximum efficiency. When the first laser is in resonance, the resulting photoions are detected with use of a time-of-flight mass spectrometer, which allows the simultaneous analysis of many isotopes, and provides a perfect calibration of the isotope shift. A first experiment of this type was performed by Wallmeroth et al. in 1986 [50] on a chain of gold and platinum isotopes. In Fig. 9 we show another version of this experiment, called COMPLIS (COllaboration for spectroscopy Measurements using a Pulsed Laser Ion Source), conceived by a new collaboration, which includes McGill University (Montreal), IPN, Laboratoire Aimé Cotton, and Johannes Gutenberg Universität (Mainz) [51]. The main advantage of this new arrangement is that it allows an implantation of the ISOLDE ions in a graphite substrate at a much lower energy (∼1 keV), leading to a substantially higher desorption efficiency. This is achieved with use of a decelerating lens which focuses the ion beam on the target. The single mode tunable pulsed dye laser system, which was developed at the Laboratoire Aimé Cotton, relies on frequency doubling or tripling, and allows excitations at wavelengths as low as 200 nm. (The most recent studies with tellurium required a wavelength of about 214 nm.) Following re-ionization, the ions are re-accelerated by a lens and directed towards a multichannel-plate detector. A magnet provides the means to separate the incident ISOLDE beam from the emerging photoion beam. As described above, the different isotopes are separated by a time-of-flight detector.

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F IG . 9. Schematic of the COMPLIS system used for the studies of refractory elements. On the upper right is the level excitation scheme used for the platinum isotope studies [51].

In Fig. 10 we summarize some of the results obtained with this method used to study the mercury decay products. 3.3.3. Fission Isomers In Section 2.2.2 we described observations of shifts of high-spin isomers. It was an even greater challenge to measure the shape isomer shift, i.e. of the metastable state before the nucleus fissions. The first observation was made on 240m Am at Oak Ridge National Laboratory [52]. Subsequently, the hfs and fission-isomer shift of the 14 ms 242f Am were measured by a Mainz, Heidelberg, Kassel collaboration [53]. The arrival rate of the fission isomers at the optical cell in this on-line experiment was only ≈6 per second.

4. Bohr–Weisskopf Effect We return now to the Bohr–Weisskopf effect. In fact, Ben Bederson worked on one of the first experiments in this field more than fifty years ago. The electron– nuclear hfs interaction is sensitive to the extended distributions of both the nuclear charge and the nuclear magnetization. Electrons for which the wave function does not vanish inside the nucleus are sensitive to these structure effects. For valence electrons of neutral atoms the effect is relatively small, but the high precision of atomic hfs experiments allows BW to be determined with reasonable accuracy.

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F IG . 10. A typical summary of isotope-shift data resulting from laser spectroscopy experiments with RIS on the daughter products of mercury. The peculiar case of the mercury isotopes is also shown [51].

So why the continued interest in BW? First, for example, experiments done at the storage ring at GSI, Darmstadt, have measured the hfs of hydrogenic bismuth, 209 Bi82+ [54]. Here, accounting for BW is essential to extracting QED effects and nuclear structure information. We note that for muonic bismuth BW represents a 50% effect in the hfs interaction. When combined with independent nuclear magnetic moment data, BW allows an improved determination of neutron wave functions [55]. We have seen that isotope shifts also reveal data on neutron orbits. Such knowledge of what the neutrons are doing in the nucleus is of potential use in the study of parity nonconservation (PNC) effects in atomic interactions. It has been suggested that the determination of PNC caused by uncertainties in the electron wave func-

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tion can be removed by measuring it in isotopic ratios [56]. The cost, however is the knowledge of the variation in neutron distributions. Atomic spectroscopy can thus shed light on this requirement. Another instance of the potential use of neutron distribution data is their interest for neutron stars [57]: Neutron radii, and in particular neutron skin thickness, have been suggested to be relevant to the star structure [58]. This renewed interest in BW has stimulated resumption of experiments: our first measurement toward the goal of a systematic study of BW in a series of radioactive cesium isotopes has recently been completed [59]. ABMR was used for this experiment at ISOLDE. Measurements at SUNY (Stony Brook) (Section 3.1) have been successful in measuring BW in francium [34]. The use of rf spectroscopy of trapped radioactive ions has also been initiated for determining the BW effect [24]; no results with radioactive atoms have been obtained to date. In closing, we note that not only BW, but also isotope shifts are finding their way recently into some unexpected areas. They include studies such as stellar chemistry [60] and variation of fundamental constants [61]. In both of these cases, a better understanding of isotope shifts, the separation of mass- and fielddependent effects, and in particular of the many-electron “specific mass shift”, is important. Systematic measurements of the isotope shifts, particularly in the lighter elements, can be expected to bring about needed elucidation.

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THERMAL ELECTRON ATTACHMENT AND DETACHMENT IN GASES THOMAS M. MILLER Air Force Research Laboratory, Space Vehicles Directorate, Hanscom Air Force Base, MA 01731-3010, USA, e-mail: [email protected]

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2. FALP Apparatus . . . . . . . . . . . . . . . . . . . . . . . 2.1. Basic Description . . . . . . . . . . . . . . . . . . . . 2.2. Langmuir Probe Operation . . . . . . . . . . . . . . . 3. Electron Attachment . . . . . . . . . . . . . . . . . . . . . 3.1. Transition-Metal Trifluorophosphines and Carbonyls 3.2. Sulfur-Fluoride Compounds . . . . . . . . . . . . . . 3.3. Single-Center Hexafluorides . . . . . . . . . . . . . . 3.4. NF3 and Phosphorous Compounds . . . . . . . . . . 3.5. Other Organic Compounds . . . . . . . . . . . . . . . 3.6. Ozone, Sulfur Trioxide, and Chlorine Nitrate . . . . . 4. Electron Detachment . . . . . . . . . . . . . . . . . . . . . 5. Electron Affinity (EA) . . . . . . . . . . . . . . . . . . . . 6. New Plasma Effects . . . . . . . . . . . . . . . . . . . . . 7. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . 8. Acknowledgements . . . . . . . . . . . . . . . . . . . . . 9. References . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract This article describes measurements of electron attachment and detachment rate constants made in the thermal environment of a flowing-afterglow Langmuir-probe apparatus (FALP) at the Air Force Research Laboratory (AFRL). If the molecular electron affinity is low enough that electron detachment from the parent anion occurs in the temperature range of the apparatus (298–550 K), the attachment/detachment equilibrium constant allows accurate determination of the electron affinity. Electron attachment reactions to a variety of molecules is described, from simple hexafluorides to transition-metal trifluorophosphines and carbonyls. 299


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1. Introduction The first measurement of an electron attachment rate was reported in the Physical Review in 1921 by Benjamin Bederson’s academic grandfather, Leonard Benedict Loeb (1891–1978, Fig. 1), who was also the godfather of gaseous electronics in the United States.1 Loeb carried out this work (Loeb, 1921) at the Ryerson Laboratory at the University of Chicago, where he had received his PhD in 1916 under

F IG . 1. Leonard B. Loeb (at age 86, with Bob Crompton’s daughter Cathy, and Tom Miller), near his home at Pacific Grove, California, 1978.

1 Benjamin Bederson, as a New York University graduate student, gave his first physics talk at the first Gaseous Electronics Conference (Gas Discharges, then), held at Brookhaven, New York, in 1948, where he went with his PhD advisor and former Loeb student at Berkeley, Leon H. Fisher. Bederson’s PhD research, timing the initiation of electric discharges, was carried out on an apparatus donated by Leonard Loeb, for an experiment suggested by Loeb. See Physical Review for 1949, vol. 75, page 1615, and for 1951, Vol. 81, pages 109–114.

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THERMAL ELECTRON ATTACHMENT AND DETACHMENT 301

Robert A. Millikan, and the paper acknowledges Millikan’s advice and provision of laboratory facilities.2 Loeb’s work played on earlier mobility measurements of negative charge carriers produced in gases which showed large changes in mobilities, depending on gas pressure and electric field strength. Thomson (1916) had proposed that some molecular types could attach electrons to form negative ions, while others could not, and that the chemical nature of the molecule determined the probability of attachment. Loeb’s 1921 paper reported that (a) it is the oxygen molecule that is responsible for electron attachment in air, (b) the nitrogen molecule does not attach electrons, and (c) that O2 attaches electrons about once in every 50,000 collisions. Loeb’s conclusions regarding O2 and N2 were correct, but his rate measurement is very approximate because he had deduced that there was no electron energy dependence even up to the ionization limit. A few years later he used an improved apparatus to observe the electron energy dependence of attachment (see (Loeb, 1955)). Since Loeb’s time, many experimental techniques have been applied to the electron attachment problem. Notable among electron swarm experiments are the drift tube measurements made by Phelps and Pack (1961), Chanin et al. (1962), and Pack and Phelps (1966) on two-body (low pressures) and three-body (high pressures) electron attachment to O2 and thermal detachment from O− 2 . From the attachment/detachment equilibrium constant, these authors were able to obtain the electron affinity of O2 , namely, EA(O2 ) = 0.46 ± 0.05 eV. This result is quite good (the best value today is 0.450 ± 0.002 eV, from (Schiedt and Weinkauf, 1995), and was obtained at a time when other experiments yielded quite disparate values (Miller, 1980). Similar drift tube experiments by Parkes and Sugden (1972) on attachment and detachment with NO molecules led to EA(NO) = 0.026 ± 0.018 eV, in excellent agreement with the photoelectron spectroscopy result obtained by Travers et al. (1989). The 1970s and onward saw more thermal attachment methods coming on line: the Oak Ridge/University of Tennessee drift tube work (Davis et al., 1973; Christophorou 1984, 1980); highly accurate measurements at the Australian National University using the Cavalleri diffusion cell (Crompton and Haddad, 1983); the pulse-radiolysis microwavecavity method (Shimamori et al., 1992); extremely low-temperature measurements at Rennes (Speck et al., 1997); flowing-afterglow Langmuir-probe (FALP) research (Smith and Španˇel, 1994a); and flowing afterglow measurements of electron attachment and detachment by the NOAA group (McFarland et al., 1972; Fehsenfeld, 1970), high-pressure mass spectrometry work of Grimsrud and

2 Leonard Loeb’s father Jacques (1859–1924) was a noted German biologist who immigrated to the USA in 1891 and served on the faculty at Chicago for 10 years. When Ludwig Boltzmann came to the USA in 1905, he made a point of visiting Jacques Loeb and his marine biology laboratory, then at Monterey, Calif.

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F IG . 2. Members of the Atomic Beams and Plasma Physics Laboratory at NYU in 1971. From left to right: Abe Kasdan, Lee Schulmann, Shlomo Ron, Bob Molof, Don Cox, Zohreh Parsa, Tom Miller, Bob Pai, Ben Bederson, and Irv Klavan. Not present is Prof. Howard Brown.

Knighton (Williamson et al., 2000), electron paramagnetic resonance spectrometry by McFadden and students (Morris et al., 1983), and drift tube work in Mayhew’s laboratory (Kennedy and Mayhew, 2001; Liu et al., 1996) and by Barszczewska et al. (2004). The length restrictions on this chapter prevent a review of thermal electron attachment research beyond that in which I have participated, with AFRL scientists and a host of visitors, but I will occasionally refer to other experiments carried out with thermal and near-thermal electrons in gases. More comprehensive reviews of have appeared in the past decade, including techniques such as electron beam, Rydberg electron, and photoionization electron experiments: the excellent one by Hotop et al. (2003) in this same Advances series, emphasizing attachment to clusters; a review emphasizing experimental methods and applications, by Chutjian et al. (1996); and a review of flowing afterglow work by Smith and Španˇel (1994a, 1994b). A more specialized review, regarding gas laser media, was written by Chantry (1992). An earlier, and still good, review is (Christophorou, 1984), which is especially useful for its data tabulations. It was in Benjamin Bederson’s laboratory that I learned about working with electrons, beginning in 1968, carrying out scattering experiments in which we measured total, differential, and spin-exchange cross sections (Bederson and Miller, 1976). Figure 2 shows people in the Atomic Beams and Plasma Physics Laboratory at New York University (NYU) in 1971. Figure 3 is a photograph one of Ben Bederson’s laboratories at NYU at that time. The present article concerns electron interactions with molecules, but in a weak thermal plasma rather than in beams.

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F IG . 3. One laboratory in Ben Bederson’s Atomic Beams and Plasma Physics Laboratory on the 10th floor of Meyer Hall of Physics at NYU in 1971. On the left is a crossed electron and atomic beam apparatus in which total and spin-exchange measurements were made. On the right is an apparatus for the scattering of polarized electrons off mercury atoms.

2. FALP Apparatus 2.1. BASIC D ESCRIPTION I am fortunate to have spent 5 months at the University of Birmingham in England in 1977 working with David Smith (now at Keele University) and Nigel G. Adams (now at the University of Georgia) on the original FALP and original selected ion flow tube (SIFT) apparatuses that they had developed (Smith et al., 1978a, 1978b, 1975; Adams and Smith, 1976).3 A copy of the Birmingham FALP was 3 The circumstances of this research visit are worth noting. Ion–ion recombination is the most important process in the neutralization of a plasma created in the atmosphere, e.g., by a nuclear blast. Early beam measurements of ion–ion recombination rate constants gave numbers from 1–8×10−7 cm3 s−1 , and these figures were being used to model atmospheric disturbances. The Birmingham flowingafterglow data showed rate constants around 5 × 10−8 cm3 s−1 . The U.S. Air Force sent me to work with David Smith and Nigel Adams to evaluate their results.

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F IG . 4. A sketch of the FALP apparatus. The gas flow is from left to right.

constructed at the University of Oklahoma, and moved to the Air Force Research Laboratory in 1990.4 A sketch of the AFRL FALP is shown in Fig. 4. Its utility was greatly improved with software developed by Španˇel (1995). Flow tube experiments are an improvement over stationary afterglow experiments because of their continuous operation mode—no pulsing is required—and the reactant gases are not present in the discharge region. The distance scale along the flow tube provides the time scale for rate constants, provided the plasma velocity is measured. Impurities are much less of a problem in flowing systems as well. On the other hand, there are gas pressure limitations at present with the flowing system. The AFRL FALP consists of a stainless steel flow tube (3.65-cm I.D.) in which a fast flow of He gas normally at 133-Pa pressure is established with a large mechanical pump. The He gas is passed through liquid nitrogen traps to remove possible impurities. A 10-W microwave discharge produces an electron-He+ plasma. The electron temperature is thousands of degrees in the discharge, but thermalizes in the afterglow within 1 ms, which also gives enough time for higher-order diffusion modes to die out. A small amount of Ar gas is added downstream in order to quench metastable He via Penning ionization and to remove He+ 2 by charge transfer, so that a thermalized, electron-He+ , Ar+ plasma obtains. A movable Langmuir probe allows one to measure the electron density along the axis of the flow tube. The ambipolar diffusion decay rate is thus readily determined. The plasma velocity is measured by timing the propagation of a pulsed disturbance of the microwave power along the flow tube axis using the Langmuir probe. The plasma velocity is nominally such that every 10 cm along the flow tube represents a reaction time of 1 ms. Note that the plasma velocity is greater than the bulk He velocity because fundamental-mode diffusion yields an ion distribution across 4 The AFRL FALP utilizes CEC diffusion pumps and gate valves, close to 50 years old, taken from Benjamin Bederson’s laboratory at New York University. More extensive equipment (35 years old) from Ben’s laboratory is in use by Lepsha Vušković at Old Dominion University.

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the flow tube diameter which overlaps more with the higher-velocity part of the parabolic, laminar flow velocity distribution of the He gas (Ferguson et al., 1969). An electron-attaching gas enters at a point halfway along the length of the flow tube, through three hollow glass needles. Electron attachment causes the electron density to decay faster than through ambipolar diffusion alone. If there were no diffusion, the electron density decay would simply be exponential. But the rate equation includes loss of electrons from both processes, and the solution shows the coupled effects of diffusion and attachment (Smith and Španˇel, 1994a): ne (t) =

ne (0) νa exp(−νa t) − νD exp(−νD t) , (νa − νD )

(1)

where ne (t) is the electron density as a function of time t, with ne (0) being the initial electron density, and νa and νD are the electron decay frequencies for attachment and diffusion, respectively. The electron attachment rate constant, ka , is νa times the concentration of reactant molecules, nr . The ambipolar diffusion coefficient, D, is νD divided by the square of the characteristic diffusion length

F IG . 5. An example of electron attachment data with the FALP apparatus. These data are for an SF5 CF3 at a concentration of 8.13 × 109 cm−3 (250 parts-per-billion in He gas) at 296 K. The fit to the data yielded an attachment frequency of 718 s−1 (ka = 8.8 × 10−8 cm3 s−1 ). The diffusion decay frequency νD = 290 s−1 was measured in absence of SF5 CF3 gas.

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of the flow tube, (1.51 cm, in this case, for fundamental-mode diffusion). An example of the electron attachment data is given in Fig. 5. 5 The electron attachment rate constant determined from data as shown in Fig. 5 is independent of the absolute value of ne , i.e., use of a different ne (0) will simply move the curve up or down on the graph. However, knowledge of the absolute value of ne is needed (a) to ensure that ne is low enough that electron recombination with molecular positive ions (which depends on n2e and on the production rate of molecular positive ions from the reactant gas) is negligible, and (b) to ensure that ne nr so that first order kinetics applies. The ion products of attachment are determined using a mass spectrometer at the downstream end of the flow tube, as shown in Fig. 4. These determinations are made with very low flows of reactant gas in order to eliminate or minimize secondary ion chemistry. Once in a while, when the attachment rate constant is small and the concentration of reactant gas is consequently high, secondary chemistry is impossible to avoid. In this case, modelling has been used to elucidate the chemistry issue. Neutral products of attachment are not detected in the experiment.

2.2. L ANGMUIR P ROBE O PERATION The cylindrical Langmuir probe used in the FALP is conceptually simple. A thin wire in the plasma collects positive or negative charges from the plasma constituents, depending on the bias potential, and probe theory due to Langmuir and Mott-Smith (1924) is used to deduce the plasma density from the probe current. The FALP Langmuir probe consists of a tungsten wire that is 25 µm in diameter and is 4 mm in length. It is mounted from a thin glass tube or ceramic insulation so that it is centered about the flow tube axis, perpendicular to that axis. For our typical plasma densities of 109 cm−3 , the probe collects several µA of electron current, or several nA of positive ion current. The plasma forms a sheath around the probe, a region in which charge neutrality does not hold. Charged particles can reach the probe only by traversing the cylindrical sheath. Probe theory applicable to the FALP system assumes that the radius of the probe is much less than that of the plasma sheath, and that the charged particles make no collisions as they spiral toward the probe surface. The relation between the probe current I and the plasma density n is (Langmuir and Mott-Smith, 1924): !1/2 I = Aenπ −1 2m−1 kTe + e(V − Vp ) (2) , where A is the probe surface area, e is the electron charge, m is the chargedparticle mass, k is Boltzmann’s constant, Te is the electron temperature (equal to 5 Figure 5 is reprinted from Fig. 2 of (Miller et al., 2002b) with permission. Copyright (2002) by the American Institute of Physics.

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F IG . 6. A typical Langmuir probe current–voltage characteristic in the electron-He+ , Ar+ plasma. The squared-current plot has a slope in the electron region that is proportional to the electron density squared.

the gas temperature in most of the results presented here), V is the bias potential on the probe, and Vp is the plasma potential (or space potential). In operation, the probe bias potential is swept from a few volts negative to a few volts positive, and back again, to map out the current–voltage characteristic. Because the plasma density is proportional to I 2 in Eq. (2), it is customary to plot I 2 vs V , as shown in Fig. 6. The slope of the linear portion of the I 2 –V characteristic, and from that the electron density ne , may be obtained without knowledge of Vp or Te . The electron attachment data shown in Fig. 5 were obtained by moving the probe along the axis of the flow tube and recording a current–voltage characteristic every 1.5 cm following the port where SF5 CF3 gas was admitted. Were the vertical scale to be expanded, the positive-ion portion on the left side of the figure would show a sloping behavior similar to that of the electron region on the right side, but with much smaller slope because of the higher mass of the positive ions. In further discussion we will focus on the electron parts of the probe characteristic. However, both regions, and the case of collecting negative ions in a positive-ion/negative-ion plasma, are important for ion–ion recombination measurements with the FALP (Smith et al., 1975). A few other notes about the probe operation are important. (a) The plots shown here are against relative potential applied to the probe; the actual potential on the

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F IG . 7. An analysis of the electron-retarding region, from which the electron temperature may be determined.

surface of the probe depends on surface conditions. (b) The I –V characteristic of the probe will show hysteresis if the probe surface is not clean. The probe is cleaned before a data run by applying a large positive bias sufficient to cause the probe to glow white hot from electron bombardment, for a second. (c) The I 2 –V characteristic will show a slight upcurving behavior at low ne because the plasma sheath around the probe may become large enough relative to the length of the probe that the probe begins to take on the look of a spherical probe to the plasma, rather than a cylindrical one. In FALP work, a few percent correction is needed for electron densities around 108 cm−3 . The correction for this behavior has been described by Španˇel (1995). (d) The greatest uncertainty in ne is related to A in Eq. (2) because the effective length of the collecting surface of the probe is difficult to estimate at the point where the probe is supported by glass or ceramic, as the plasma sheath is larger around the larger-diameter supporting structure. The uncertainty in A is estimated to be 10%. However, the relative uncertainty in the resulting ne values is smaller (3%), and electron attachment and detachment rate constants are determined from relative values of ne . Referring to the Langmuir probe characteristic in Fig. 6: generally speaking, for negative potential, positive ions are attracted to the probe and electrons are repelled; for positive potential, the opposite happens. But the probe

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acts as a retarding potential energy analyzer in the region around (actual) zero potential. If Te is small, the transition from the positive-ion to electron portions is sharp. If Te is large, that transition is more curved. The second derivative of the probe current shows an exponential segment in the retarding region, which yields Te (Španˇel, 1995). An example of this analysis is illustrated in Fig. 7. Even deviations from a Boltzmann distribution have been seen using this method, for a case where electron attachment removes electrons of a particular energy (Španˇel and Smith, 1993). Te may be made as large as 4000 K in the FALP by flowing pure Ar instead of He. The Ramsauer–Townsend minimum in the electron-Ar scattering cross section inhibits the cooling of Te in this case. The AFRL FALP has thus far only utilized this capability to examine the change in product-ion branching fractions with Te (Van Doren et al., 2005a; Miller et al., 2004a). Though outside the scope of the present article, the FALP method has been applied to a variety of problems such as electron–ion recombination and ion–ion recombination, or mutual neutralization (Smith and Španˇel, 1994b), observation of optical emissions from recombining ions (Španˇel and Smith, 1996), diffusion cooling (Trunec et al., 1994), electron attachment studies in which the gas temperature Tg and electron temperature Te were varied independently (Španˇel et al., 1995), and studies in which electron attachment was observed to distort the Boltzmann distribution of electron energies (Španˇel and Smith, 1993).

3. Electron Attachment Electron attachment is one of several possible outcomes of the electron-molecule interaction, and must be viewed in that general context (Hotop et al., 2003). Attachment processes are divided into two categories: nondissociative attachment, in which the parent anion is formed, and dissociative attachment, in which various fragment anions are formed. Both processes may be observed for the same reaction. The usual picture of electron attachment is that the incoming electron is trapped into a temporary resonant state, which then either autodetaches or is stabilized radiatively or collisionally. The resonance time explains the large cross sections often observed for electron attachment reactions. Dissociation into products may occur either by attachment into a dissociating state or by vibrational excitation induced by the incoming electron. A molecule will tend to undergo dissociative electron attachment if energetically possible, though an activation energy may be required to access the dissociating state. A molecule may form the parent anion in a binary collision with an electron if the molecule is floppy enough to take up some of the electron energy into internal energy, long enough for collisional or radiative stabilization to occur. Usually, this means that small molecules will not undergo nondissociative electron attachment. For example, O3 does not

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form the parent anion in binary thermal electron attachment, but SO3 does, though I must say that I was on the wrong side in the office pool on whether SO3 would attach electrons; the bonds are too strong to permit dissociative attachment, and the molecule seemed to me too solid to take up much of the electron energy on impact. In simplest terms, the maximum cross section (for a point molecule) is given by the “size” of the electron, π times the reduced de Broglie wavelength of the electron. That maximum cross section is easily integrated over a Maxwell– Boltzmann electron energy distribution to give a 300 K attachment rate constant −1/2 temperature dependence. Klots (1976) proof 5 × 10−7 cm3 s−1 , with a Te posed a simple formula which gives a more realistic maximum electron capture cross section: σc (E) =

! πa02 1 − exp −(32αE)1/2 , 2E

(3)

where E is the electron energy and α is the polarizability of the molecule. The evolution of this expression from the Vogt–Wannier theory for s-wave capture has been well explained by Hotop et al. (2003), along with other theoretical aspects of the attachment problem. The issue of the collisional stabilization of the resonant state means that different rate constants may be measured in low pressure and high pressure experiments, depending on the lifetime of the resonant state. Two AFRL studies were carried out on product branching fractions in collaboration with Montana State University, in order to cover a wide range of buffer-gas pressures (Knighton et al., 2004; Miller et al., 1998; Williamson et al., 2000). A nice comparison of rate constants measured with the truly-thermal FALP and in nonthermal apparatuses has been given by Smith et al. (1989). Table I Transition-metal trifluorophosphines and carbonyls for which electron attachment rate constants have been measured with the AFRL FALP apparatus in 133 Pa of He gas. Molecule

Ref.

ka at 298 K (cm3 s−1 )

T range studied (K)

Ionic products

HCo(PF3 )4 HRh(PF3 )4 HMn(CO)5 HRe(CO)5 Ni(PF3 )4 Pt(PF3 )4

a

2.0 × 10−8 2.4 × 10−8 1.9 × 10−7 1.4 × 10−8 1.9 × 10−7 5.3 × 10−8

297–554 300 299–521 300–523 298–492 298–448

− HCo(PF3 )− 3 , Co(PF3 )4 − Rh(PF3 )− , HRh(PF ) 3 3 4 − HMn(CO)4 HRe(CO)− 4 Ni(PF3 )− 3 Pt(PF3 )− 3

a a a b b

a (A.E.S. Miller et al., 2005a). b (A.E.S. Miller et al., 2005b).

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Table II Sulfur fluoride compounds for which electron attachment rate constants have been measured with the AFRL FALP apparatus in 133 Pa of He gas. Molecule

Ref.

ka at 298 K (cm3 s−1 )

T range studied (K)

Ionic products

Other ka at 300 K

SF4 SF6 SF5 Cl SF5 CF3 SF5 C6 H5 SF5 C2 H3

a

2.5 ± 0.6 × 10−8 2.3 ± 0.6 × 10−7 4.8 ± 1.2 × 10−8 8.6 ± 2.2 × 10−8 9.9 ± 3.0 × 10−8 7.2 ± 1.8 × 10−9

300–557 300–545 300–550 296–563 296–550 299–550

SF− 4 − SF− 6 , SF5 SF− 5 SF− 5 SF− 5 SF− 5

Note e Note f Note g Note h – –

a b c d d

a (Miller et al., 1994b). b (Van Doren et al., 2005a). c (Miller et al., 2002a). d (T.M. Miller et al., 2005b). e A lower value was given by Babcock and Streit (1982). f Many other measurements have been made; see (Miller et al., 1994b). The most accurate is that of (Crompton and Haddad, 1983): 2.27 ± 0.07 × 10−7 cm3 s−1 at 295 K. g (Mayhew et al., 2004) obtained a lower value using a drift tube. h A value of 7.7±0.6×10−8 cm3 s−1 was extrapolated from non-thermal data (Kennedy and Mayhew,

2001). Table III Single-center hexafluoride compounds for which electron attachment rate constants have been measured with the AFRL FALP apparatus in 133 Pa of He gas. Molecule SF6 MoF6 ReF6 WF6

Ref.

ka at 298 K (cm3 s−1 )

T range studied (K)

Ionic products

Other ka at 300 K

a

2.3 ± 0.6 × 10−7 2.3 ± 0.6 × 10−9 2.4 ± 0.6 × 10−9 ∼10−12

300–545 297–385 297 297–522

− SF− 6 , SF5 − MoF6 ReF− 6 –

Note c – – –

b b b

a (Miller et al., 1994b). b (T.M. Miller et al., 2005a). c See footnote f in Table II.

The molecular systems that have been studied with the AFRL FALP apparatus are listed in Tables I–VI. The most interesting features of these reactions will be discussed here.

3.1. T RANSITION -M ETAL T RIFLUOROPHOSPHINES AND C ARBONYLS Table I lists results for transition-metal acids synthesized by Amy Stevens Miller (Miller et al., 1990; A.E.S. Miller and T.M. Miller, 1992). HCo(PF3 )4 and HRh(PF3 )4 are gas-phase superacids, meaning that the acids, denoted by AH, have deprotonation enthalpies, Hacid , less than 13.6 eV, in which case the reac-

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Table IV NF3 and phosphorus compounds for which electron attachment rate constants have been measured with the AFRL FALP apparatus in 133 Pa of He gas. Molecule NF3 PF3 PF5 PCl3 POCl3 PSCl3

Ref.

ka at 298 K (cm3 s−1 )

T range studied (K)

Ionic products

Other ka at 300 K

a

7 ± 4 × 10−12 3) cross sections.

2.4. S UMMATION R ELATIONS FOR D IRECT E XCITATION C ROSS S ECTIONS Another quantity that can be determined readily from the optical excitation cross section data is the total inelastic excitation cross section defined as the sum of the direct excitation cross sections for all excited states. Note that this quantity includes only excitation into atomic levels, and excludes any ionization components. At first it appears that the high angular momentum levels would be a serious obstacle since they emit only in the far infrared making their cross sections inaccessible by the optical method. However, it has pointed out that each event of direct excitation is followed by a sequence of emission transitions ultimately reaching the ground state through an nP → 3S transition. Thus the sum of the direct excitation cross sections for all excited levels is simply equal to the sum of

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the optical emission cross sections for the nP → 3S series, i.e. Qdir Qopt (nP → 3S). j = j

395

(8)

n≥3

Values of the total inelastic excitation cross section are included in Table I. The first member Qopt (3P → 3S) accounts for about 99% of the full summation. Within an uncertainty of about a percent, the total inelastic excitation cross section is virtually no different from the optical emission cross section of the first resonance line. A more interesting relation is found if we remove from the above summation the direct excitation cross section of the 3P level which is related to the optical cross sections as opt Qdir Qopt (nS → 3P) 3P = Q (3P → 3S) − −

n>3 opt

Q

(nD → 3P).

(9)

n≥3

Subtracting Eq. (9) from Eq. (8) gives Qdir Qopt (nP → 3S) + Qopt (nS → 3P) j = j =3P

n≥4

+

n>3

Qopt (nD → 3P),

(10)

n≥3

where the prime on the summation is to emphasize the exclusion of the 3P member. All members of the right hand side of this equation are measured with the highest precision in the experimental method adopted here since they are the raw products of direct measurements from Eq. (1) and are free of the uncertainties introduced in elucidating direct excitation cross sections from the optical emission cross section data. It can be used to test, for example, the overall accuracy of a set of theoretical cross sections for excited levels.

2.5. A LTERNATIVE M ETHODS FOR A BSOLUTE C ALIBRATION Enemark and Gallagher (1972) initiated the approach of by-passing measurements of absolute atom number density by normalizing the experimentally measured relative cross sections to the theoretical Born-approximation cross section at 1000 eV. They used a crossed-beam apparatus in which an electron beam intersects a sodium-atom beam at right angles and the 3P → 3S resonance fluorescence is observed along the third orthogonal axis. Upon applying the polarization correction, the 3P → 3S emission signal observed at different incident electron energy yields a relative apparent excitation function for the 3P level. To

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make contact with theoretical direct excitation cross sections, Enemark and Gallagher (1972) used the absolute emission cross sections for the nS → 3P and nD → 3P lines reported by Zapesochnyi and Shimon (1965) to estimate the cascade into the 3P level. The relative apparent cross section was then normalized to the cascade estimate plus the Born-approximation cross section for direct excitation into the 3P level at 1000 eV calculated by Karule and Peterkop (1965). Once the absolute calibration of the apparent excitation cross section is established by normalization to the Born approximation, the direct excitation cross sections are obtained by subtracting the cascade estimates from the apparent excitation cross sections. Some comments are of interest in connection with the use of the Born approximation for absolute calibration. The first step toward validating this approach is to confirm the straight line relationship at high energies in the Bethe plot (the product of the relative apparent excitation cross section times energy versus energy in logarithmic scale) as demonstrated by Enemark and Gallagher. One may also question the accuracy of the value of the Born cross section used for normalization which was obtained by extending the calculation of Karule and Peterkop with an oscillator strength of 0.98. The sodium 3S–3P line has one of the most accurately known oscillator strengths. The uncertainty of the Born-approximation 3P cross sections of Karule and Peterkop (1965), though not given, presumably can be assessed by repeating the calculations using more refined wave functions. Another interesting point is that in the normalization procedure a cascade estimate is added to the Born cross section to match the experimental relative apparent excitation cross section at 1000 eV. Fortunately the resulting direct excitation cross sections are not strongly influenced by the uncertainty of the cascade estimate. This is because the cascade was first added in to normalize the apparent excitation cross section at 1000 eV and then subtracted out. While there is no dependence on the cascade estimate at 1000 eV, the correction to the direct excitation cross section at other energies varies in accordance with the energy dependence of the cascade estimate. Thus, the more crucial quantity is the energy dependence of the cascade estimate rather than its magnitude. Figure 4 compares the direct excitation cross section determined by this method with those of Phelps and Lin (1981) where the absolute calibration was done entirely by experiment. Numerical values of these two sets of cross sections are also included in Table I. Above 50 eV the two sets of data agree to within typically 7%. The deviation is larger at lower energies averaging to about 12%. Also included in Fig. 4 are the angle-integrated differential cross sections of Marinković et al. (1992) (see Section 2.7) and the results of theoretical calculation which are discussed in the following section. Using a similar absolute calibration based on the Born approximation, Stumpf and Gallagher (1985) measured the excitation cross section for the 3D level. After correcting for polarization and cascades, their 3D cross sections agree with those

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F IG . 4. Comparison of excitation cross section values for the 3P level. Experimental values include optical measurements of Phelps and Lin (1981) and Enemark and Gallagher (1972), and integrated differential cross section measurements of Marinković et al. (1992); theoretical values include the CCC results of Bray (1994), the 10CC R-matrix results of Trail et al. (1994), and values based upon the Born-approximation.

of Phelps and Lin (1981) to within typically 8% as can be seen in Table I. It is gratifying to see such a close agreement between the two sets of experiments on both the 3P and 3D cross sections considering the difference in calibration schemes, one based on Born theory and the other by a rather elaborate experimental method to measure the atom number density. 2.6. C OMPARISON WITH T HEORETICAL C ALCULATIONS Theoretical calculations of electron excitation in the early days were dominated by versions of the Born approximation. In 1972 Moores and Norcross (1972) published a four-state close coupling (CC) calculation. The continual advent of computer capability made it possible to include more target states in the expansion manifold to examine convergence. A series of CC (R-matrix) calculations for excitation cross sections using an expansion containing four, seven, nine, ten, and eleven states (4CC, 7CC, . . . ) has been reported for energies from threshold to 5.2 eV (Trail et al., 1994). Another recent development is the method of convergent close coupling (CCC) which addresses the role of the continuum states in the CC expansion (Bray et al., 2002).

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At 5 eV the 10CC calculation of Trail et al. (1994) is 20% below the experiment of Phelps and Lin (1981) for the 3P cross section, 30% below the experimental value for both the 3D and 4S cross sections, and 10% below for the 4P cross section. In the case of the CCC calculation the theoretical excitation cross section at 10 eV is 13% below the measured value for the 3P level, 15% below for 3D, 24% below for 4S, but 5% above for 4P (Bray, 1994). The measured 3P cross sections of Enemark and Gallagher (1972) agree very well with the 10CC calculations (10% at 5 eV) and with the CCC calculations (3% at 10 eV). On the other hand, a similar comparison of the 3D cross sections measured by Stumpf and Gallagher (1985), shows a 23% difference from the 10CC calculation at 5 eV and a 13% difference from the CCC calculation at 10 eV. 2.7. D IFFERENTIAL C ROSS S ECTIONS : ATOMIC B EAM R ECOIL M ETHOD Measurements of differential cross sections for electron-impact excitation of sodium have been conducted in different laboratories over two decades (see, for example, Buckman and Teubner, 1979; Srivastava and Vušković, 1980). In a conventional experiment one measures the number of scattered electrons that have suffered an energy loss corresponding to excitation into a particular excited state as a function of scattering angle of the electron. For example, Marinković et al. (1992) have reported differential cross sections for excitation of sodium into the 3P, 3D, 4S and 4P levels from 2◦ to 150◦ at 10, 20 and 54.4 eV. In the case of the 3P excitation cross section, which has been studied extensively over the years, significant discrepancies were found among the results of different experimental groups (see, for example, Teubner et al., 1986). To some degree this is an indication of the difficulty of these experiments. The differential cross sections often vary sharply with scattering angle aggravating such problems as low detection signal and angular resolution. Furthermore absolute calibration is a serious problem. It is a common practice to extrapolate the measured differential cross sections to cover the full range of scattering angles (θ = 0◦ to 180◦ ) and normalize the integral over θ to an experimentally measured integrated cross section (if available) or to a theoretical value. At small scattering angles the differential cross section decreases steeply with increasing θ , thus extrapolation of the measured differential cross section to θ = 0◦ may present problems. Bederson and his co-workers pioneered the method of atomic beam recoil in which one measured the spatial distribution of the recoiled atoms and use the kinematic relations to unfold the distribution of the scattered electrons (Bhaskar et al., 1977; Jaduszliwer et al., 1980). One main advantage of this method is that the cross sections are given by the fraction of the atoms recoiled in various directions and no absolute atom number density is needed. However, there are some complications in the implementation of this method because the measured recoil distribution is a superposition of the results of elastic and inelastic collisions.

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Techniques for isolating the cross sections for individual inelastic channels have been discussed (Jaduszliwer et al., 1984). Also the velocity spread in the atomic and electron beams and the finite size of the atom detector causes complications so that special deconvolutions must be done to extract the differential cross sections from atom recoil measurements. The atomic beam recoil method has been applied to determine the differential scattering cross sections for electron excitation into the 3P level of Na from 1◦ to 20◦ for 10 eV electrons (Jiang et al., 1990) and from 1◦ to 60◦ at energies of 2.3, 2.4, 2.5, 2.6, 3.0, 3.3, and 3.7 eV (Ying et al., 1993). At 10 eV where differential excitation cross sections by both methods of electron energy loss (Marinković et al., 1992) and atomic beam recoil (Jaduszliwer et al., 1984) are available, the agreement is generally about 50%. Comparison of the experimental cross sections at 2.6, 3.0 and 3.7 eV with CC calculations of Trail et al. (1994) have been discussed by Ying et al. (1993). Agreement is seen only in the dominance of very small angle scattering but not in the qualitative shape at intermediate and larger angles.

2.8. PARTIAL C ROSS S ECTIONS For an S → P excitation the integral cross sections obtained from measurements of the resulting fluorescence includes excitation into all magnetic sublevels with ml = 0, ±1. Cross sections for excitation into each individual magnetic sublevel can be segregated by measuring the polarization of the fluorescence with the assumption of equal cross sections for ml = ±1. For the case of sodium electron excitation can also cause a change of the spin quantum number ms = 0, ±1. Han et al. (1988) have developed an elegant method with which they measure the “partial” cross sections for 3 2 S → 3 2 P excitation corresponding to a given ml andms . They started with Na(3S) atoms in a pure ms state produced by optical pumping. An electron beam excited these atoms into the Zeeman-split 3 2 P3/2 group in a 200 G magnetic field. The atoms in each Zeeman component, 3 2 P3/2 (mJ ), are pumped to the 5 2 S(ms ) state by a laser tuned to the appropriate frequency (ν2 ) for that mJ , as illustrated in Fig. 5. The 4P → 3S fluorescence resulting from radiative decay of the 5 2 S(ms ) state is related to the 3 2 P3/2 (mJ ) population, and therefore can be used to determine the relative 3S1/2 (ms ) → 3P3/2 (mJ ) cross section for the four different values of mJ . They are placed on an absolute basis by normalization to the known total 3S → 3P cross section. Transformation to the ml , ms representation gives the (integrated) s cross section Qm mL associated with the given changes in the ms value (ms ) and mL value (mL ) resulting from the 3S → 3P excitation. Barring explicit spin-dependent forces, symmetry considerations require invariance of these partial cross sections with respect to change of the signs of ms or mL . From this

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F IG . 5. Energy levels involved in partial cross section measurements. Atoms are first optically pumped into the 3 2 S1/2 (MS = 1/2) level. Atoms excited to the different mJ sublevels of the 3 2 P3/2 level are monitored by laser excitation to the 5 2 S1/2 level, and then detecting the 4P → 3S emissions in the second step of the decay chain back to the ground state. A 220 Gauss magnetic field produces a ∼ 450 MHz splitting between the individual 3 2 P3/2 Zeeman sublevels.

experiment Han et al. (1988, 1990) obtained the partial cross sections Q00 , Q10 , Q01 , and Q11 for the 3S → 3P excitation for several electron energies between 2 and 4 eV. Collisions with ms = ±1 are attributed to electron exchange and such cross sections approach zero when exchange is neglected. Comparing the ratio of cross sections for |ms | = ±1 versus those for |ms | = 0 for the same |mL | (0 or 1), one finds Q10 /Q00 ∼ 0.35 and Q11 /Q01 ∼ 0.1. These ratios reflect the relative importance of electron exchange in the excitation process. The experimental results are generally in reasonable agreement with CC calculated values except the Q11 cross section which shows a discrepancy as large as 50% at electron energies below 3 eV. Han et al. (1988, 1990) have extended their technique to extract differential cross sections for a given set of ms and mL from the same optical experiment. Like the method of Bederson described in the preceding section, they used atomic recoil to infer the electron scattering angle. Instead of monitoring the spatial distribution of the recoil atoms, however, they effectively measured the spread of recoil velocities picked up by the atoms. The recoil velocities introduced a Doppler shift in the frequency (ν2 ) needed for laser pumping the atoms from 3P to 5S. The dependence of the observed 4P → 3S fluorescence signal on the probe laser frequency ν2 is a function of the natural width of the transition, the Doppler width due to the velocity distribution of the atoms in the undisturbed beam, and the Doppler-shift introduced by the electron–atom collision recoil velocity. The last one can be mapped into the angular distribution of the scattered electrons, thus information about differential cross sections is embedded in the ν2 -spectrum.

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Han et al. (1988, 1990) have presented a complicated de-convolution procedure to extract the differential partial cross sections, i.e., three sets of differential cross sections corresponding to (a) mL = ms = 0, (b) mL = 1 and ms = 0, and (c) mL = 0 and ms = 1. Comparing these results to CC calculations, one find qualitatively similar shape in the angular dependence of the differential cross sections but significant discrepancy on a quantitative level (Zhou et al., 1990).

3. Excitation out of Laser Excited States 3.1. P RODUCTION OF E XCITED S TATES The necessary first step in measuring cross sections out of excited states is the preparation of the target atoms in the excited states. Since the lifetime of the 3P level is only 16 ns, to create a 3P target requires a laser to continuously re-excite atoms that have decayed to the ground state. In a typical experiment, an oven is used to create a thermal beam of ground state Na atoms with a number density in the range of 109 to 1010 cm−3 . A fraction of the atoms in the beam are then excited to the 3P level by a laser tuned to the 589.2 nm 3 2 S1/2 –3 2 P3/2 transition. Usually a CW dye-laser (using the ubiquitous rhodamine 6G dye) pumped by an argon–ion laser is used to produce a beam with a power on the order of 100 mW. If the 3S–3P cycling transition were a closed two-level system, about half the target atoms could be kept in the 3P excited level by a high intensity laser beam. Unfortunately, the I = 3/2 nuclear spin of 23 Na splits the 3 2 S1/2 ground state into two hyperfine levels. The 1720 MHz splitting of the F = 1 and F = 2 hyperfine levels is generally much greater than the linewidth (typically < 50 MHz) of the pump laser. As a result, in most experiments only the 3 2 S1/2 (F = 2)– 3 2 P3/2 (F = 3) transition is pumped. This precludes any excitation of the ground state atoms that enter the laser interaction region in the F = 1 ground state, limiting the possible excited state fraction that can be obtained. Furthermore, absorption in the tail of the 3 2 S1/2 (F = 2)–3 2 P3/2 (F = 2) line profile, 120 MHz below the 3 2 S1/2 (F = 2)–3 2 P3/2 (F = 3) transition leads to some atoms that are initially in the F = 2 ground state the chance to decay to 3 2 S1/2 (F = 1) ground level thus removing them from the cycling laser transition. While this has a low probability of happening on any single 3S–3P excitation cycle, an atom may undergo 102 –103 such cycles while in the laser beam. Depending upon the polarization of the laser light (circular or linearly polarized), the laser intensity and frequency stability of the laser, typical experimental conditions produces a 3 2 P3/2 (F = 3) excited state fraction in the target beam of 4 to 30%. To give one particular example, in a laser excited state target used by Stumpf and Gallagher (1985) 20% of the atoms were in the desired 3 2 P3/2 (F = 3), with 25% in the 3 2 S1/2 (F = 2) ground state and 55% in the non-cycling 3 2 S1/2 (F = 1) ground state.

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A dramatic increase in the 3 2 P3/2 (F = 3) excited state fraction can be achieved by also pumping atoms out of 3 2 S1/2 (F = 1) ground state. This can be achieved by adding a second laser frequency that pumps atoms out of the 3 2 S1/2 (F = 1) dark state to the 3 2 P3/2 (F = 2) excited state. Atoms excited to this level then decay (with some probability) to the F = 2 ground state and thus re-enter the primary 3 2 S1/2 (F = 2)−3 2 P3/2 (F = 3) recycling transition. This can be accomplished by adding a second dedicated repump laser, adding frequency sidebands to the existing laser, or running the laser multi-mode. For example, Dorn et al. (1997) increased the 3 2 P3/2 excited state fraction from 9% to 40% by modifying their single-mode dye-laser to run double-mode with a frequency spacing of 1792 MHz. Numerical calculations by Hall et al. (1996) indicate that a F = 1 repump intensity of only one percent of the primary transition intensity is enough to produce significant gains in the excited state fraction. Such a low intensity sideband can be created by electro-optical modulation of a single-mode laser. In addition to studying excitation processes out of the 3 2 P3/2 level, a second laser can be used to create atom targets with even higher energy levels. A group at Rice University has used two N2 pumped dye lasers to study electron impact excitation of high lying nD Rydberg levels (Foltz et al., 1982). The first dye-laser was used to excite atoms to the 3 2 P3/2 level. The second dye-laser was tuned to excite atoms to a particular n 2 D level with 36 ≤ n ≤ 50. In the previously mentioned experiment of Dorn et al. (1997), the high density 3 2 P3/2 target was further excited with a second single-mode dye laser to either the 5 2 S1/2 level with a 616.2 nm pump beam, or to the 4 2 D5/2 level with a 569.0 nm pump wavelength. The subsequent decay of these laser-pumped levels also lead to non-trivial population levels (2 to 7%) in the 4 2 S and 4 2 P levels. Standage and co-workers have used a single mode Ti:sapphire operating at 819.7 nm to create a 3 2 D5/2 target for use in electron–atom collision experiments (Hall et al., 1996). Returning to the simpler case of an experiment with only a single 589 nm laser, the atomic target will consist of atoms in both the 3S and 3P levels. Hence the measured cross section for whatever process (x) is investigated is equal to Q(x) = fe Q(3P → x) + (1 − fe )Q(3S → x),

(11)

where fe is the fraction of atoms in the 3P laser-excited level. The extraction of the 3P laser excited cross section thus requires two corrections to the measured signal rate, one for the less than complete excited state fraction of the target, and one for any background signal introduced by the ground state component of the target. The second correction is often unnecessary for electron-scattering measurements since the energy resolution of such experiments is almost always sufficient to differentiate between scattering from the two initial states which differ by 2.1 eV. The first of these corrections, however, is almost always required and necessitates a good knowledge of the excited-state fraction of the target.

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At least three techniques have been used to determine the excited state fraction in electron-scattering experiments using excited Na atoms. The first is to calculate the number densities of atoms in the various states using rate equations or a density matrix formulation to describe the transitions possible among the hyperfine levels for the known laser parameters (polarization, intensity, frequency, etc.). A second technique used to obtain the excited state fraction, fe , is to measure (1 − fe ) by observing the decrease in the signal rate for a reaction from the ground state. As an example, let us consider the pioneering electron-energy loss measurements Hertel and Stoll (1974) used to study excitation out of the 3P level into the 3S, 3D and 4S levels. Hemispherical electrostatic analyzers were used to record the energy of scattered electrons with 60 meV energy resolution. Electron energy loss spectra were recorded with the 3S–3P pump laser on (Ron ) and with the laser off (Roff ). For the 3S → 3P inelastic peak at an energy loss of 2.1 eV, it can be derived from Eq. (11) that Roff − Ron (12) . Roff Another innovative technique was used by Bederson and co-workers to measure the excited state fraction in their atomic beam recoil measurements (Bhaskar et al., 1977). The atomic beam recoil technique as applied to measuring ground state cross sections was described in Section 2.7. By adding a laser beam at right angles to the atomic and electron beams, an excite state target can be created (see Fig. 6). Just as scattering of electrons leads to a deflection of the atomic beam, so too does the scattering of resonant photons lead to a deflection of the beam. By translating their detector along the direction of the laser beam instead of the direction of the electron beam, the excited state fraction could be derived from the spatial profile of the atomic beam. fe =

3.2. 3P TO 3S S UPERELASTIC C ROSS S ECTIONS The most widely studied electron–atom collision process with laser excited sodium atoms is the 3P → 3S de-excitation process. This is often referred to as super-elastic scattering since the incident electron gains 2.1 eV of energy in such a collision. It is also the time-reversed equivalent of the 3S → 3P excitation process (see Section 2). In this context, most work with laserexcited atoms has been in pursuit of studying scattering amplitudes in the quest for the complete scattering experiment (Andersen and Bartschat, 1996; Hertel and Stoll, 1977). For studying 3S → 3P excitation, this requires measuring the scattered electron in coincidence with the photon from the decay of the excited atom. The state of the excited atom is further probed by measuring the polarization of the fluorescence. Such experiments typically have very low counting rates. In contrast, in the time-reversed 3P → 3S de-excitation process,

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F IG . 6. Schematic diagram of atomic beam recoil method for measurement of cross sections out of Na(3P) level (Zuo et al., 1990). Without the mirror below the interaction region, the atomic beam is deflected by the laser beam allowing measurements of the 3P excited state fraction of the target. Since atoms in the 3S(F = 1) ground state are non-resonant with the laser, these atoms are undeflected. With the mirror in place, there in no net deflection of the atomic beam, except for the electron–atom collision induced recoil.

the 3P state is preselected via the polarization of the laser beam, requiring only the detection of the super-elastically scattered electron. This results in two orders of magnitude gain in signal rates versus the corresponding photon-coincidence experiments (McClelland et al., 1986; Scholten et al., 1988). As a result, almost all of the 3P → 3S measurements done with laser excited atoms determine scattering amplitudes and atomic coherence parameters (Teubner and Scholten, 1992; Sang et al., 1994), which are beyond the scope of this paper, and are more properly addressed in the context of the “perfect scattering experiment” (Bederson, 1969). The only absolute 3P → 3S cross section measurements made to date are a set of differential cross section measurements made by Bederson and co-workers (Jiang et al., 1992, 1995). They used the atomic beam recoil technique they developed for ground state excitation (Section 2.7) to measure differential cross sections at 3, 5, 10 and 12 eV incident energies for scattering angles from 1 to 30◦ . As described in Section 3.1, they could measure the 3P excited state fraction by monitoring the photon-induced deflection of the atomic beam. While desirable for measuring the excited state fraction, this deflection complicates the electron-

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induced deflection measurements. As a result, for cross sections measurements the laser beam was retro-reflected through the collision region, leading to zero net deflection of the atomic beam (in the direction of the laser beam). Since the laser-induced fluorescence of the atomic beam is proportional to the excited state fraction, fluorescence measurements were used to cross calibrate the unknown excited state fraction for the standing wave configuration to the known value for the traveling wave configuration. Their measurements were found to be in reasonable agreement with theoretical calculations using the distorted-wave (Madison et al., 1992), CC (Trail et al., 1994) and CCC (Bray et al., 1994) methods.

3.3. D IFFERENTIAL C ROSS S ECTIONS As was the case for 3P → 3S super-elastic scattering, most of the work on excitation of laser-excited Na atoms into higher energy levels have mainly been directed at the study of atomic coherence parameters in various forms, with only a couple of experiments measuring differential cross sections directly. Hertel and Stoll (1974) measured the ratios of the differential cross sections for excitation into the 4 2 S and 3 2 D levels to the that of the 3 2 S super-elastic value at a scattering angle of 10◦ and an incident energy of 6 eV for various pump laser polarizations. They also obtain ratio measurements for the 3 2 D at 35 eV. Considering only the case of a circularly-polarized pump beam (which only excites the 3 2 P3/2 ML = ±1 levels), they found that the 4 2 S/3 2 S differential cross section ratio was 0.55 ± 0.10, while the 3 2 D/3 2 S ratio was 1.8 ± 0.1 at 6 eV and 1.4 ± 0.07 at 35 eV. These values are broadly consistent with the values of an early 4-state CC calculation of Moores et al. (1974). The only absolute differential cross section results are from the Bederson and co-workers experiment described in Section 3.2. In addition to studying the superelastic process, they also obtained values for the differential cross section into the 4 2 S level at 2 eV (Jiang et al., 1992). Within the approximately ±50% experimental uncertainties, their results are consistent with a 10-state CC calculation of (Zhou et al., 1992). While they did not obtain data at the same energies as Hertel and Stoll, (1974, 1977) their 4 2 S/3 2 S differential cross section ratio at 10◦ and ∼2 eV incident energy is about the same.

3.4. I NTEGRAL C ROSS S ECTIONS Integral cross sections for electron–atom processes involving Na atoms in the 3P laser-excited level include measurements of the total scattering cross section (Jaduszliwer et al., 1980), ionization cross section (Tan et al., 1996), and excitation into the 3D level (Stumpf and Gallagher, 1985). Using a 2-laser excitation scheme, Foltz et al. (1982) have studied state changing collisions for

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high-lying Rydberg states for an incident electron energy of 25 eV. In keeping within the limits of this paper, only the later two experiments will be discussed at any length. In the experiment of Stumpf and Gallagher (1985) the 3P → 3D excitation cross section was measured by optical means for electron energies from threshold to 1000 eV. Similar to the ground state experiment of Enemark and Gallagher (1972) (see Section 2.5), an atomic beam (∼109 cm−3 ) was crossed at right angles with an electron beam and the 819 nm fluorescence from the decay of Na(3D) atoms was detected with an optical system composed of an interference filter and photomultiplier tube orientated at right angles to both beams. Along the same optical axis, in the opposite direction, another interference filter and PMT was used to monitor the Na(3P) density using the 589 nm resonance emissions. To create the excited Na(3P) target, a circularly polarized laser beam was introduced collinearly with the electron beam through a hole in the back of the Faraday cup. The single-mode CW dye laser produced ∼100 mW of power with a ∼1 MHz linewidth. Due to leakage to the 3 2 S1/2 (F = 1) ground state, the excited state fraction was 0.20 in the 3 2 P3/2 (F = 3, MF = 3) state. The difference in the 819 nm fluorescence signal with the laser on and off was measured as a function of the incident electron energy. As in the ground state work of Enemark and Gallagher (1972) the relative optical emission cross sections are placed on an absolute scale by normalizing the results at high energies (200–1000 eV) to theoretical values obtained with the Born approximation. In addition to including corrections for cascades from higher levels, however, the authors also had to correct for the 3S → 3D contribution due to the ground state atoms in the target, and an additional correction for the fact that only the ML = 1 atoms in the 3P excited state target are populated. They found a peak cross section of 38 × 10−16 cm2 at 7 eV. This is a factor of eight larger than the peak 3S → 3D excitation cross section of 4.8 × 10−16 cm2 . Their results at low energies are consistent with the 4-state CC results of (Moores et al., 1974) at 2.9 eV (see Fig. 7). In the only other experiment to measure excitation cross sections using laserexcited Na atoms Foltz et al. (1982) used selective field ionization (SFI) to study electron-induced state changing collisions with nd (36 ≤ n ≤ 50) Rydberg atoms. As mentioned in Section 3.2, two lasers were used to sequential excite ground state atoms first to the 3P level, and then to a particular nd level. Pulsed N2 pumped dye lasers were used for both lasers. As in the experiment of Stumpf and Gallagher (1985) the electron beam and laser beams were collinear and counter propagating. After the Rydberg atoms are created, an electric field is ramped up and the ionization signal is recorded. SFI spectra are obtained with and without an electron beam pulse. Assuming that the electric-field required to ionize an atom that has undergone a state-changing collision is different than the field required for the target atoms, the difference in the two spectra will be proportional to the

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F IG . 7. Plot of cross section values for excitation of the 3D level from the laser excited 3P level.

l-changing cross section. The Na(nd) target population is obtained from the integrated SFI spectrum with the electron beam off, allowing absolute cross section results to be obtained.

4. Concluding Remarks It is often stated that the sodium atom, because of its simplicity in electronic structure, furnishes an excellent testing ground for comparing theory versus experiment in the field of electron collisions. For integral cross sections this is illustrated in Fig. 4 for 3S → 3P excitation. Here we have three mutually consistent sets of experimental data (two by the optical method and one by the electron energy-loss technique) and they are in agreement with two sets of theoretical calculations that cover two energy ranges (2–5 eV and 4–54 eV). Because of its exceptionally large cross section, the 3S → 3P excitation is not a typical case. Thus we make a similar comparison for the 3S → 3D excitation in Fig. 8. Here the two sets of experimental cross sections are again in excellent agreement, but significant discrepancies with theory become apparent. For instance the 10-state R-matrix calculation shows a decrease in cross section from 4.1 to 5 eV whereas an opposite trend is found experimentally. For the higher levels one sees in Table I reasonable overall agreement between experimental 4S, 4P, 4D cross sections

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F IG . 8. Comparison of excitation cross section values for the 3D level. Experimental values include optical measurements of Phelps and Lin (1981) and Stumpf and Gallagher (1985); theoretical values include the CCC results of Bray (1994), the 10CC R-matrix results of Trail et al. (1994), and values based upon the Born-approximation.

with the CCC theory, however the experimental uncertainty are larger for these levels. Even at the level of integral excitation cross sections, the goal of a definitive comparison between theory and experiment is only partially fulfilled at this time. The situation becomes even less optimistic as we move on to the differential cross sections. Here one compares with theory the curves of measured differential cross sections for excitation into a particular level as a function of the scattering angle at a given energy. For the 3S → 3P excitation, measurements have been made by three different methods; electron energy loss, atom recoil (Section 2.7), and partial cross sections (Section 2.8). The three sets of data show varying degrees of agreement. Differential cross sections for 3S → 4S and 3S → 4P excitation have also been measured using the electron-energy loss method. Comparison between theory and experiment is complicated by the fact that within each cross section curve (versus scattering angle) the difference varies significantly from one part of the curve to another. A succinct and unequivocal answer to the issue of testing theory versus experiment is still elusive. The use of laser-excited sodium atoms as targets for electron collisions has provided new directions for excitation studies. In this paper we have surveyed experimental results for excitation out of the laser-excited 3P level. However, as

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mentioned in Section 3.1, sodium atoms in the 4S, 4P, 3D, and some Rydberg levels have been prepared by laser pumping. Such highly excited sodium atoms could be possible candidates as targets for future excitation experiments. The production of laser-excited alkali atoms has been vastly simplified by the widespread introduction of tunable solid-state Ti:sapphire lasers and low-cost diode lasers. Unfortunately, the 590 nm wavelength of the 3S → 3P transition (the first step in exciting these higher levels), still generally requires a dye-laser. With diode lasers now available at the wavelengths of the resonance transitions for all of the alkali atoms except sodium, we have a complete reversal of the situation that existed in the 1970’s and 1980’s when laser-pumping of sodium was the easiest among the alkalis. For example, atom trapping of rubidium with low-power, low-cost, diode lasers provides an effective way to produce targets with high excited state fractions (Schappe et al., 2002). Thus the use of atom-trapping techniques to measure electron excitation cross sections out of laser excited levels was relegated to the Rb atom (Keeler et al., 2000), a much less popular target atom in the circle of electron collision practitioners. If, however, commercial diode lasers at 560 nm become available in the future, one may look forward to a dramatic increase in studies involving laser-excited sodium atoms. Low-pressure sodium lamps exemplify technological applications of electron excitation processes. Analysis of a glow discharge plasma requires detailed knowledge of multi-step processes involving electron impact excitation and ionization from excited levels. An extensive set of excitation cross sections out of both the ground and 3P levels into higher levels is most important for a fundamental understanding of the atomic processes in these discharge systems. The authors appreciate the opportunity to contribute to this special volume. One of us (CCL) has greatly treasured the long-standing friendship with Professor Bederson through years of research in electron collisions and service on committees. Equally memorable were the excursions to restaurants in New York’s Chinatown and the Metropolitan Opera House. It is a great honor for CCL to succeed Professor Bederson as a co-editor of the Advances in Atomic, Molecular and Optical Physics.

5. Acknowledgements This work was supported by the National Science Foundation and the U.S. Air Force Office of Scientific Research.

6. References Andersen, N., Bartschat, K. (1996). Adv. Atom. Molec. Opt. Phys. 36, 1. Bederson, B. (1969). Comm. At. Mol. Phys. 1, 41–65.

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ATOMIC AND IONIC COLLISIONS EDWARD POLLACK† Department of Physics, University of Connecticut, Storrs, CT 06269, USA

Ben Bederson . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . 2. Collisions Involving Heavy Solar Wind Ions . 3. Collisions Involving H0 Projectiles . . . . . . 4. Proton Collisions in the Io Plasma Torus . . . 5. Surface Collisions with Highly-Charged Ions 6. Acknowledgement . . . . . . . . . . . . . . . 7. References . . . . . . . . . . . . . . . . . . .

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Abstract Experimental and theoretical studies of collision processes provide insights into the basic interactions between particles. This review addresses collisions between both ionic and atomic projectiles with selected targets. A brief introduction outlines some of the experimental techniques used in the laboratory and suggests how the results can be presented. A substantial amount of data is being acquired by unmanned space missions but much of the data cannot be interpreted because relevant experimental results are not yet available. Many of the important collision processes involve multi-charged ions in the solar wind as well as the keV energy hydrogen atoms generated by the proton component. In this context, the role of the solar wind in the X-ray emissions found from the “atmospheres” of comets as they approach the sun is briefly discussed. Particular attention is given to the X-ray spectrum of the important O6+ + CO → O5+ collision. Processes involving keV energy H0 collisions with H2 , O2 , and N2 molecules are important in the aurora and discussed with particular emphasis on how the results can best be presented. Collisions involving H+ with SO2 are important in the atmosphere of the Jovian Moon Io and optical results are presented. The concluding section is on the interactions of highly charged ions with surfaces and collisions of highly-charged S ions with a highly oriented pyrolytic graphite surface are reviewed.

† As we go to press, we were saddened by the news of the untimely death of Ed Pollack: he was a good friend of many of us, and we value the more having his last work as a contribution to this volume. I thank Professor W.W. Smith for kindly looking after the proofs of Ed Pollack’s paper.

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Ben Bederson I have known Ben Bederson since my graduate student days at New York University. I was particularly fortunate to have Ben as my adviser, to learn from him, and to also have an opportunity to closely follow the innovative work done by the several groups in his laboratory. I greatly benefited by working directly with three of his students, Arthur Salop, Ed Robinson, and Judah Levine on the atomic polarizability experiments. In thinking back I fondly recall the excitement associated with our groups attendance at the Brookhaven Conference and my introduction to ICPEAC.

1. Introduction Collisions play important roles in many areas of interest to physics. They underlie the establishment and maintain the equilibrium in a gas, initiate chemical reactions, generate the aurora, and result in the X-ray emission found in the coma of comets as examples. Experimental work on collisions is generally done to guide our understanding of interactions, test the underlying theory and approximations, and provide cross sections for modeling. There are a number of experimental approaches, each having a particular advantage, which are used to study collisions on systems above the nuclear level. The experimental techniques include measurements of the energy dependence of total cross sections, determinations of the angular differential cross sections for elastic scattering and selected inelastic processes, studies of optical and X-ray emission, as examples. A brief discussion of some of these is presented in this introduction. There is a substantial heritage for the studies of collisions and many reviews of the field are available. A major goal of many atomic/molecular collision experiments is to understand the excitation processes. How does the excitation of a particular state depend on the initial charge, state, kinetic energy of the projectile and the distance of closest approach during the collision? The studies undertaken can involve energy and angular distribution measurements on the scattered projectiles (or recoil targets) as well as on the emitted radiation. Angular distribution measurements provide a stringent test of the underlying theory which generally determines σ (θ), the differential cross section for a particular process as a function of scattering angle. Laboratory data at low keV energies presented in terms of the reduced differential cross section, ρ = θ sin θ σ (θ ) ≈ θ 2 σ (θ) (at small angles), as a function of τ = Eθ , the reduced scattering angle at an energy E, have a particular advantage in probing the excitation process (Smith et al., 1966; Kessel et al., 1978; Pollack and Hahn, 1986). The reduced scattering angle is a function of the distance of closest approach during a collision and is the same in the laboratory and center of mass frames (Pollack and Hahn, 1986). When data at different energies are plotted in a

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ρ vs τ format they are found to lie on a common curve for the excitation resulting from a curve crossing. The technique was used to study low keV energy He+ +D2 collisions (see Bray et al., 1977 as an example). One experimental approach to the collisional excitation problem involves measurements of E, the projectile energy loss as a function of θ, the laboratory scattering angle. The quantity of interest however is Q, the excitation energy (of either the projectile or the target or of both) which is determined from E and θ as well as from the masses of the projectile and target. A projectile of initial energy E0 , mass mp which is scattered through an angle θ by a target of mass mt loses an energy E = (mp /mt )E0 θ 2 + Q (at small angles). The target mass is known only in the case of an atomic target. In the case of a molecular target it may not be the molecular mass but by a suitable analysis of the data (Bray, 1975; Bray et al., 1977) it can be determined. As an example in He+ + D2 collisions at small angles the measured E values, for electronically elastic collisions, were consistent with collisions involving a target having the mass of a D2 molecule (Bray, 1975; Bray et al., 1977). Beyond a “critical” point however the measured E values as a function of laboratory scattering angle were found to lie above the expected value for a D2 mass. The difference between the measured and “expected” E values indicated vibro-rotational excitation of the target molecule (Bray et al., 1977; Anderson et al., 1980). The results of this type of measurement can provide tests of predicted potential energy surfaces for a number of triatomic molecules (Jakacky et al., 1985; Snyder and Russek, 1982; Anderson et al., 1980). An additional important point is that for a collision between ground state partners only the kinetic energy in the center of mass system is available for excitation. As an example, electrons with laboratory energies of 10.2 eV can excite a hydrogen atom to the n = 2 level in e− + H collisions. In H+ + H collisions the minimum laboratory energy required, by the H+ beam, would be 20.4 eV. In atomic/molecular collisions although quantum mechanical considerations are generally required to understand the underlying interactions, conservation of momentum (in billiard ball type classical collisions) is always valid. An alternate approach to the excitation problem involves measurements on the emitted radiation. The high resolution optical data generally acquired allows for accurate determinations of the emitting states but there is no information provided on important collision parameters such as the distance of closest approach. Photons which originate from a particular state generally result from excitations corresponding to a range of distances of closest approach, by the projectile, to the target and therefore to projectiles scattered over a range of angles. This problem was addressed (Kessel et al., 1978; Smick, 1973) and a number of coincidence experiments between emitted photons and the scattered projectiles have been done but with very low count rates. An important advantage of the optical experiments however is that the projectile beam can have a relatively broad energy spread. Energy loss measurements on the scattered projectiles must use beams

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with small energy spreads but they provide additional information. Unlike the energy loss measurements, which directly probe the excitation process, the emitted spectra may include contributions from cascade processes and therefore do not always provide direct information on the initial excitation. Measurements on the radiation resulting from a collision require careful interpretation. The lifetime of the emitting state must also be known to determine the fraction of excited atoms which decay in the region probed by the spectrometer. Since collisional excitation by a beam generally results in anisotropic states, the emitted radiation has an angular distribution reflecting this anisotropy (Kessel et al., 1978; Macek and Jaecks, 1971; Fano and Macek, 1973; Clark, 1978). The emitted light is polarized and the polarization must be taken into account in determining total cross sections from experimental data. Only measurements made by detectors positioned at an angle θm the “magic angle” with respect to the beam direction can be properly interpreted without polarization analysis. At this angle (where cos2 θm = 1/3, θm = 54.7 deg.) the measured intensity, without the generally required polarization analysis, yields the total emission intensity. At all other angles the intensity must be obtained separately for each polarization state (Clark, 1978). For experiments involving optical measurements, the efficiency of the detector must also be known as a function of wavelength for absolute cross section determinations.

2. Collisions Involving Heavy Solar Wind Ions Collisions involving ions in the solar wind contribute to processes that must be understood for applications to astrophysics and atmospheric physics problems. Although large quantities of data on these processes are being acquired by space missions the underlying interactions involving the ions are not fully understood. The major ionic components of the solar wind (there are also electrons) are energetic protons (92%) and helium ions (8%) (Cravens, 2002). Heavy multi-charged ions account for only about 0.1% of the ion flux and the most important of these include O6+ , O7+ , C5+ , N5+ , and Ne8+ (Schwarenden and Cravens, 2000). Although the multi-charged components of the solar-wind constitute only a small fraction of the flux they are the primary source of the discrete lines found in the X-ray emission spectra from comets. These heavy ions fall into two velocity ranges of about 400 and 750 km/sec (Geiss et al., 1995). The higher velocity ions are attributed to polar coronal holes and the lower velocity ions to emission from coronal holes close to the equator (Neugebauer et al., 1998). The ion compositions in the two velocity ranges are different and in addition both are time dependent. In 1996 the ROSAT satellite (Röntgen satellite) observed X-ray emission from comet Hyakutake (Lisse et al., 1996). The 109 Watt X-ray power associated with these emissions (Cravens, 2002) was far greater than expected since comets were

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known to be cold bodies of ice and dust. They predate the solar system and their composition reflects the primordial matter from which it was created (LevasseurRegourd, 2004). X-rays were soon found for other comets (Dennerl et al., 1997) and by now it is well established that these emissions are a general feature of comets as they approach the sun. A number of models were proposed to account for the data. Currently the most promising one attributes the observed X-rays to electron capture (McDaniel et al., 1993) by the highly-charged ions in the solar wind from molecules in the cometary “atmospheres” (Wegmann et al., 1998). The important solar wind ions are cited above. H2 O is the dominant volatile species accounting for about 80% of the cometary atmosphere. Other molecules found include CO, CO2 , CH4 , NH3 , and N2 (Krankowsky, 1991). On the basis of the model, X-ray emission is attributed to electron-capture collisions such as O6+ + CO → O5+∗ . The excited O5+∗ product then decays to lower lying levels. To model the observed emissions it is essential to know both the lines and their relative intensities that result from the underlying collisions as well as the cross sections for electron capture by the multi-charged ions from the atoms and molecules in the cometary atmospheres. With the availability of both high resolution spectral data and theoretical predictions, the solar wind composition can be inferred from the observed X-ray spectra (Kharchenko and Dalgarno, 2001). The X-ray spectra resulting from electron capture by solar wind ions from “cometary” molecules were calculated in recent papers (Kharchenko and Dalgarno, 2000, 2001). Typical solar wind compositions were assumed and the results were shown to be in good agreement with the data. A problem for modeling with “laboratory astrophysics” data is that forbidden transitions from long-lived metastable states can give rise to intense emission lines. Because of the long lifetimes these lines cannot be found in the laboratory since the states do not decay while the projectiles are in the spectrometer acceptance region (Greenwood et al., 2000). Although the multi-charged projectiles can capture more than one electron, the X-ray emission results primarily from single electron capture. Projectiles capturing several electrons typically emit Auger electrons (Posthumus and Morgenstern, 1992; Beirsdorfer et al., 2003). Single electron capture occurs preferentially into highly excited states which decay by cascading transitions and result in the emission of the characteristic X-ray (and optical) lines. Once the initial state into which the electron capture occurs is identified the resulting emission spectra can be predicted using existing tables. The “Ion Abundance and Species Table” in (Schwarenden and Cravens, 2000) lists 42 species (there are more) in the solar wind that should be considered in constructing “approximate line spectra”. Even though a particular multi-charged ion has a small abundance in the solar wind, it may make an important contribution to the X-ray spectrum by having a large cross section for single electron capture. In addition to the large number of incident ions the cometary atmospheres include a number of molecular species. Modeling of the resulting emissions would

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therefore require using several hundred single electron capture cross sections and knowing the states into which the electrons are captured. There are only a limited number of calculations of single electron capture cross sections (see Kimura and Lane, 1989 as an example) and simple models must be used. It is therefore important to determine the reliability of these. The simplest starting model to identify the initial state into which the electron is captured assumes energy resonance at infinite separation, where the binding energy of the electron in the target is the same as its binding energy after capture. The model neglects changes in the binding energies which result from the presence of two interactions (due to the projectile and target) with the active electrons during a collision. The classical over-thebarrier model is more realistic and in addition allows both the cross section and state to be approximated (Niehaus, 1986; Hoekstra and Morgenstern, 2001; Bransden and McDowell, 1992). On the basis of this model the active electron interacts with the electric fields of both the projectile having a charge +q and the target having a charge +1 and is associated with a net potential barrier (in atomic units) V (r, R) = −1/r − q/(R − r) where r is its coordinate on the target and R the changing internuclear distance during a collision. At a projectile–target separation R, the binding energy of a target electron is ItR = It + q/R (It is the target binding energy at infinite separation). When this energy is equal to the electron binding energy at the maximum barrier height, the target electron can transfer to the projectile. This occurs when R = Rc = [2q 1/2 +1]/It (Hoekstra and Morgenstern, 2001). Assuming the absorbing sphere model of Olsen and Salop (1976), electron transfer occurs for all R < Rc and the total cross section for electron capture σ = πRc2 . The electron is then captured to a state with a binding energy Ip = It + {(q − 1)/Rc } (Hoekstra and Morgenstern, 2001). The solar wind ions capture electrons into highly-excited states which then undergo transitions resulting in both X-ray and optical emissions. Along these lines a number of collision systems important to the cometary problem have recently been investigated in the laboratory (see Beirsdorfer et al., 2003; Ehrenreich et al., 2004a, 2004b as examples). One of these (Ehrenreich et al., 2004a, 2004b) involved the X-ray spectra resulting from collisions of multi-charged ions with molecular targets. The experimental arrangement used the 14.0-GHz “Caprice” electron-cyclotron resonance ion source at the JPL/ Caltech HCI facility (Chutjian et al., 1999). The ion beams were extracted, charge to mass ratio analyzed, collimated, and directed into a cell containing the scattering gas. Spectra of the emitted X-rays were acquired using a spectrometer-CCD camera arrangement. O6+ is the dominant multi-charged ion in the solar wind. Figure 1 shows an X-ray spectrum from O6+ + CO → O5+ collisions at 36 keV. The spectral wavelength resolution in the figure is about 0.05 nm. Figure 2 shows the transitions and corresponding energy levels resulting in the strong lines observed. It can be seen that the O5+ (4s, 4p, 4d) states are at the highest lying levels into which the electron is captured in the initial process. Although X-ray transitions from higher

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F IG . 1. An X-ray spectrum from O6+ + CO collisions at 36 keV in a wavelength range from 7 to 18 nm. The dominant peaks are labeled and the corresponding transitions due to electron transfer from the target (CO) to the projectile (O6+ ) and resulting in O5+ are identified in Fig. 2. The clearly present but unlabeled peaks in the spectrum are due to other processes such as O6+ + CO → O4+ .

levels are possible and would result in emission lines that lie in the wavelength acceptance range of the spectrometer, they are not seen. Thus it is concluded that n = 4 is the highest level into which electron capture to O5+ occurs in the collision. The binding energy of an outer electron in CO is 14.1 eV. On the basis of the simplest resonant-electron-capture model cited above, capture is expected into the n = 6 level of O5+ , where the binding energy is approximately 14 eV. The collision would result in X-ray emission and branching to lower-lying states. The resulting CO+ target would then be in its ground electronic and vibrational states. For electron-capture to levels with binding energies a few eV more than the 14 eV, the CO+ target molecule would still be in the ground electronic state but would have rovibrationally excited states. The Grotrian diagram in Fig. 2, however, shows that n = 4 (binding energy approximately 32 eV) is the highest lying level into which the electron is captured. On the basis of the more complex “over-the-

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F IG . 2. An energy level (Grotrian) diagram showing the labeled transitions in Fig. 1. Although X-ray transitions from levels above the principal quantum number n = 4 would fall in the wavelength range of the spectrometer, they are not seen. This clearly indicates that the highest level into which an electron is captured in this collision is n = 4. On the basis of energy considerations, the collision can be identified as O6+ + CO → O5+ + C+ + O∗ (or C∗ + O+ ). There are also lines originating from the n = 3 level. These may result from the direct capture to n = 3 or from an n = 4 to n = 3 cascade. The transitions labeled “F” are not in the wavelength range covered in Fig. 1 but were seen in the spectrum (not shown here) covering an adjacent wavelength range.

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barrier” model cited above (Hoekstra and Morgenstern, 2001) the electron capture in O6+ + CO collisions occurs when Rc , the critical projectile–target separation is about 6 × 10−8 cm with a corresponding cross section of 1.1 × 10−14 cm2 . The model predicts electron capture to a level with a binding energy of 26 eV. This binding energy lies approximately half way between those for the n = 4 and n = 5 levels in O5+ . Independent of the model the experimental data are then consistent with O6+ + CO → O5+ + C+ + O (or O+ + C), indicating dissociation of the target molecule. One or both of the dissociation products would be electronically excited. Optical results showing emission from excited C+ and O as an example would be consistent with the interpretation. The collision would involve a multi-electron process which results in the simultaneous electron capture and dissociation of the target molecule. Alternately, a more tightly bound CO electron may be captured into the n = 4 level with the target undergoing simultaneous dissociation. The electron capture into n = 4 is also consistent with collisions resulting in O5+ + CO+ with an O5+ product having a larger kinetic energy than that of the incident O6+ . The process involved cannot be uniquely established only on the basis of the current experimental results. However for the purpose of modeling the cometary X-ray emissions, they provide all the necessary information. Once n = 4 is established as the level into which the electron is captured, the resulting spectrum can be determined from existing tables. Results on O6+ + CO collisions at energies of 18, 24, 30, 36, and 42 keV show the same emission spectra as Fig. 1 but small differences in the relative intensities of the lines. This is due in part to the different lifetimes and decay lengths of the emitting states. Since the transit time of the projectile through the region probed by the spectrometer depends on its velocity, states with short lifetimes (that decay while the projectile is still in the target region accessible to the spectrometer) would result in a stronger recorded line than longer lived states. An additional important result of this study is that although X-ray emission from possible final target product states is energetically allowed, only projectile X-ray lines are seen. A more detailed analysis of collisions involving multi-charged ions with molecules can be made by studying the optical spectra. Results of such studies will be useful in interpreting X-ray results and in probing the underlying collision mechanisms. As an example, Fig. 3 shows the optical emission spectrum in the visible range from 625 to 715 nm for O5+ + CO collisions (Ehrenreich et al., 2004a, 2004b). The O5+ beams were obtained from the University of Connecticut Van de Graaff and the spectrum acquired with an optical-CCD camera system. Matches within 0.1–0.2 nm to the NIST wavelength tables were found for lines labeled A–K. The ion beam energy affects the relative line intensities but not their wavelengths. Lines B and C match emission from 2s2 2p2 nl multiplets in excited O+ ; D, G, and K match emission from 2s2p3p and 2s2p3d multiplets in excited C+ . Line A may be attributed to emission from a blend of excited O4+ and C2+ ions, while weak lines E and F match O+ , O2+ , C+ , and perhaps O3+ excitation. Line H

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F IG . 3. The optical emission spectrum in the visible range from 625 to 715 nm for O5+ + CO collisions. The peaks labeled A through K are identified in the text.

is probably due to excited O3+ 2s2p4d emission, from which state cascade emission in the soft X-ray region can also occur. The lines can be attributed to: (a) the collisional dissociation of the molecular target following electron capture as well as to (b) the direct excitation (no change of charge state by the projectile) of dissociating states of the molecular target O5+ + CO → O5+ + C + O+ as an example. It is likely that the collisional dissociation, ionization, and excitation of the target CO molecule would also occur in O6+ + CO collisions. Using coincidence techniques between a detected optical emission line and the charge state of the related scattered projectile, an unambiguous assignment to the above cited processes (a) or (b) (or both) can be made in future measurements.

3. Collisions Involving H0 Projectiles The earlier discussion cited the role of heavy multi-charged solar wind ions in X-ray emission from cometary atmospheres. For the light components in the wind the role played by protons is now discussed, with examples of their contributions to the auroral optical emissions in the earth’s atmosphere. The emissions are due

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to collisions of low keV energy H0 with atmospheric molecules. The H0 has its origin in the much higher energy H+ ions which enter the upper atmosphere. Inelastic collisions then cause excitation and ionization of the target atoms and molecules. The H+ ions are primarily scattered through small angles in these collisions, degrade to energies where electron capture processes become important and result in H0 and H− (McNeal and Birely, 1973). At auroral altitudes, low keV energy H0 is the dominant hydrogen projectile species and at energies less than 10 keV causes most of the observed optical emissions. Modeling of upper atmospheric processes clearly requires a basic understanding of collisions involving H0 projectiles. As examples: (1) it is known that the Lyman α radiation in the hydrogen aurora primarily results from the collisional excitation of H0 (1s) to H0 (2p) rather than electron capture by H+ (McNeal and Birely, 1973). (2) H0 + O2 collisions cross sections must be known for modeling and interpreting the behavior of auroral and night air-glow phenomena (Noxon, 1970). Collisions involving low keV energy H0 with molecules were generally assumed to be electronically elastic. It was shown however that in collisions of H0 atoms with N2 , H2 , and O2 molecules, the inelastic channels were dominant beyond the smallest scattering angles (Quintana, 1995; Quintana and Pollack, 1996a, 1996b). As an example even at a relatively low energy of 1.0 keV, measurements of the H0 projectile energy loss over a range of scattering angles showed that the inelastic channels dominate for scattering angles beyond 0.2◦ in H0 + N2 collisions. The time-of-flight (TOF) experimental techniques used for these measurements on H0 were more complex than the electrostatic analysis for energy loss generally used for ions. To obtain the necessary resolutions, the TOF measurements require relatively long flight paths and primary ion beams which are cleanly chopped (generally by an electric field). The detectors must be carefully designed so that the flight path distance from the collision region is basically the same to all parts of the detector. As an example channeltrons having a conical detection surface for the incident H0 cannot be used since the flight time differences to positions along the surface would generally result in an unacceptable broadening of the energy loss spectra. The experimental setups and techniques used for the H0 on H2 , N2 , and O2 measurements have been previously described (Kessel et al., 1978; Pollack and Hahn, 1986) and are only outlined here. Briefly H+ is extracted from an ion source which is biased at the required voltage. The H+ ions are focused by an Einzel lens and pass between two small gold-plated electrodes where a voltage pulsed at 0.3 MHz “chops” the beam to provide a time reference for the TOF energy-loss measurements. The beam is analyzed by a Wien filter and then passes through a charge-exchange cell filled with H2 where electron capture in H+ + H2 collisions generates a composite H0 and H+ beam. The residual H+ component of the beam is electrically deflected leaving an H0 beam which is basically at the same energy as that of the initially incident H+ . It enters a scattering cell containing the target gas maintained at a pressure in the “single-collision”

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range. The scattered projectiles enter an electrostatic energy analyzer for energyloss measurements on the H+ and H− which result from the collision. The H0 component of the scattered beam passes through the analyzer to a TOF detector positioned 4.2 m from the scattering cell. There have been a number of studies of keV energy ion–molecule and atom– molecule collisions but our understanding of them remains very limited. Except for a few collision systems, the interpretation of experimental data on electronically inelastic processes requires qualitative approaches which are basically similar to those used successfully for atomic targets (Hodge et al., 1977; Dowek et al., 1983). As examples: (1) Electronic excitation is treated on the basis of a “Demkov process” (Demkov, 1964) between an initial and final state for which the corresponding potential-energy curves are parallel. (2) The excitation may be associated with molecular orbital or state crossings (Dowek et al., 1983). In He+ collisions with CO and NO targets it was shown that both excitation and electron capture involve a correlated two-electron transfer process (Dowek et al., 1983) similar to that successfully used for atomic targets. Although low keV energy H0 + N2 collisions are important in the aurora, only limited work has been done on this collision system to date. Experimental charge production cross sections and differential scattering calculations were reported by Van Zyl et al. (1987). Electron capture and loss studies were made by Smith et al. (1991), differential and total cross sections for stripping were reported by Cisneros et al. (1976), and the absolute differential cross sections at very small angles were reported by Johnson et al. (1988). A study of the elastic and electronically inelastic channels at small angles was made of the direct scattering (H0 (1s) + N2 → H0 ), stripping (H0 + N2 → H+ ), and electron capture (H0 + N2 → H− ) processes in H0 + N2 at energies of 1.0, 2.0, and 3.0 keV (Quintana and Pollack, 1996a). The importance of electronically inelastic collisions in H0 + N2 → H0 was established and is seen in Fig. 4 which shows an energy loss spectrum at an energy E = 1.0 keV and scattering angle θ = 0.18◦ . Peak A corresponds to elastic collisions. Using Gilmore’s (1965) potential-energy curves for N2 and assuming that the electronic excitations occur via Franck–Condon transitions from the N2 (X 1 g+ , ν = 0) ground state, peak B is consistent with the electronic excitation of any in a group of N2 states (a 1 g , a 1 u− , B 3 u− , and w 1 u ). Contributions to the spectrum at an energy loss near 8 eV may result from excitation of the N2 (B 3 g ) state. It is also interesting to note that the N2 (A 3 u+ ) state is excited (but only weakly). The excitation of triplet states from an initial singlet state in N2 is not allowed with H+ projectiles but can occur with H0 . At energy losses between 11 and 12 eV the shape of the peak suggests possible contributions from both the N2 C 3 u and E 3 g+ states although the excitation of H0 (n = 2) at an energy loss of 10.2 eV cannot be discounted. The small structure in peak C which is seen primarily at larger angles is attributed to the simultaneous ionization of the molecule and a one elec-

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F IG . 4. An energy loss spectrum for the direct scattering in H0 + N2 . The peak labeled A corresponds to electronically elastic scattering. Peaks B and C result from inelastic processes. Reprinted figure with permission from (Quintana and Pollack, 1996a). Copyright 1996 by the American Physical Society.

tron excitation of higher lying Rydberg series of the N2 target. The peak is also consistent with the simultaneous excitation of single electron states in both the H0 and N2 . Using the areas under the peaks in Fig. 4 PA , the probability of an elastic collision, is found to be 0.4 even at a small angle θ = 0.18◦ . PA continues to decrease with increasing scattering angle and is 0.2 at θ = 0.8◦ . The collision cannot be assumed to be elastic even at very small angles. In addition to the direct scattering results just outlined, there are collisions where an electron is stripped from the initial H0 projectile or captured by it. The final products are charged and for these cases the energy loss spectra are obtained by electrostatic energy analysis rather than by time-of-flight. The basic advantage in using electrostatic analysis is the marked improvement in energy resolution and substantial increase in beam intensity since the beams are not chopped. A problem does arise

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however from the change in charge state. In the direct collision measurements the time-of-flight energy loss spectrum at small angles used the incident beam as an energy reference. For stripping and electron capture collisions there are no energy references available and these must be found from “reference” collisions where the processes are known. Figure 5(a) shows composite spectra (with E = E − 13.6 eV) taken with the same incident beam for H0 + N2 → H+ and the reference H0 + Ar → H+ + e− collisions. The main peaks for both the N2 and reference spectra have the same basic shapes and maxima for E ≈ 0 showing that the dominant stripping process in N2 results in H+ + N2 (X) + e− . The FWHM is greater than the 0.5 eV of the incident beam at 1.0 keV and is attributed in part to the kinetic energy carried off by the ejected electron. Figure 5(b) shows a magnified plot of peak b with energy losses in the range of 6.5–12 eV relative to peak a which result from stripping of the electron from the H0 with simultaneous excitation of the N2 target. H+ due to electron capture by the target in H0 (1s)+N2 → H+ +N− 2 (E = 1.6 eV relative to stripping at threshold to the continuum) is seen to be weak. Although the N− 2 state is unstable (Gilmore, 1965) its excitation would be imprinted on the H+ spectra (as stated in the introduction, measurements on the beam yield information on the excitation processes). The probabilities for stripping, at energies of 1.0, 2.0, and 3.0 keV, as a function of reduced scattering angle are shown in Fig. 6(a) for H0 + N2 → H+ + N2 (X) + e− (with the N2 in its ground X state) and Fig. 6(b) for H0 + N2 → H0 + N∗2 + e− (N∗2 is electronically excited) at energies of 1.0, 2.0, and 3.0 keV. The probabilities are seen to be strongly energy dependent. Figure 7 shows reduced cross sections as a function of reduced scattering angle for the simple stripping (ρa ) and stripping with target excitation (ρb ) at the same energies. The results are plotted in different arbitrary units and are seen to be well approximated by common curves at the three energies. The common curves are consistent with excitations that occur at a particular inter-particle separation (Pollack and Hahn, 1986). In H0 +N2 → H− +N+ 2 collisions the electron transfer involves a minimum energy loss E of at least 14.7 eV (for capture to the ground state). Figure 8 shows a typical energy spectrum from this collision. The energy E (= E − 15 eV) is based on an H0 + Ar → H− + Ar+ energy reference. The dominant process (α) corresponds to H0 + N2 → H− (1s2 ) + N2 (X). The peak labeled β corre2 + 2 sponds to H− + N+ 2 (C u and D g ) with a tail labeled γ . Figure 9 shows the reduced differential cross sections plotted as a function of reduced scattering angle at energies of 1.0, 2.0, and 3.0 keV. The experimental results for each of the three processes again basically fall on common curves when plotted in this format. As in the case for N2 molecules H0 + H2 , collisions are also of interest in connection with the aurora. But another important problem is associated with H3 . The ground state of this simplest triatomic molecule is repulsive and collision experiments can provide the most direct test of approximation techniques used for

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F IG . 5. Stripping in H0 + N2 → H+ collisions. (a) Spectra from H0 + N2 → H+ and from a reference H0 + Ar → H+ collision at θ = 0.18 deg. E (= E − 13.6 eV) is set to zero at the maximum of the peak labeled a. On the basis of the reference the dominant process in N2 is identified as H0 (1s) + N2 (X) → H+ + N2 (X) + e− . (b) An energy loss spectrum for stripping in H0 + N2 → H+ at θ = 0.27 deg. Two peaks labeled a and b are seen and correspond to H+ + N2 (X) and H+ + N∗2 respectively. Reprinted figure with permission from (Quintana and Pollack, 1996a). Copyright 1996 by the American Physical Society.

calculating the potential energy surface. Along these lines the absolute “summed” differential cross section (without energy analysis) for H0 + H2 → H0 were reported by Johnson et al. (1988) at 0.5, 1.5, and 5.0 keV for scattering angles in the range of 0.05◦ to 0.5◦ . To date this collision system has been studied primar-

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F IG . 6. The probabilities for stripping as a function of reduced scattering angle in H0 + N2 collisions. Pa corresponds to H+ + N2 (X) where the target remains in its ground state and Pb to H+ + N∗2 where the target is electronically excited. Reprinted figure with permission from (Quintana and Pollack, 1996a). Copyright 1996 by the American Physical Society.

ily at thermal energies for applications to chemical processes. Very little work has been done at keV energies to probe the short range part of the potential energy surface. Differential cross section results on the electronically elastic channel can best reveal the most important features of this surface.

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F IG . 7. The reduced differential cross sections as a function of reduced scattering angle for simple stripping (ρa ) and stripping with target excitation (ρb ) in H0 + N2 collisions. The results at the three energies are plotted to different arbitrary units and show that they are well approximated by common curves when presented in this format. Reprinted figure with permission from (Quintana and Pollack, 1996a). Copyright 1996 by the American Physical Society.

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F IG . 8. Electron capture in H0 + N2 → H− collisions. (a) The H0 + Ar → H− + Ar+ serving as an energy reference for electron capture shows that the dominant process in N2 is H0 + N2 → H− (1s2 ) + N+ 2 . E (= E − 15 eV) is set to zero at the maximum of peak α. (b) A typical energy spectrum for electron capture in H0 + N2 . Two main peaks α corresponding to +∗ − 2 H− (1s2 ) + N+ 2 (X) and β from H (1s ) + N2 as well as a tail γ are seen. Possible contributions to cannot be ruled out. The energy loss scale is reversed relative to the stripping peak β from H−* + N+ 2 scale since the detected beam is negatively charged. Reprinted figure with permission from (Quintana and Pollack, 1996a). Copyright 1996 by the American Physical Society.

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(c) F IG . 9. The reduced differential cross sections for electron capture to (a) H− (1s2 ) + N+ 2 (X), − (1s2 ) + N+∗ as a function of the reduced scattering angle in H0 + N (b) H− (1s2 ) + N+∗ , and (c) H 2 2 2 collisions. The results at the three energies studied are plotted to different arbitrary units and are seen to be reasonably well fitted by common curves. Reprinted figure with permission from (Quintana and Pollack, 1996a). Copyright 1996 by the American Physical Society.

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F IG . 10. An energy-loss spectrum for the direct scattering in H0 + H2 → H0 collisions at E = 1.0 keV and θ = 0.30◦ . Peak A results from electronically elastic collisions and B from electronically inelastic processes. Reprinted figure with permission from (Quintana and Pollack, 1996b). Copyright 1996 by the American Physical Society.

The H0 +H2 collision was studied at energies of 1.0, 2.0, and 3.0 keV (Quintana and Pollack, 1996b). Figure 10 shows an energy loss spectrum at E = 1.0 keV and scattering angle θ = 0.3◦ . Peak A corresponds to electronically elastic scattering and peak B results from a one-electron excitation of the projectile to H0 (n = 2) or the target to H2 (B 1 u+ ) at a threshold energy loss close to 11.5 eV. Contributions from both of these channels most probably result in peak B whose maximum is at 10.6 eV. Structure in the energy loss range 18 < E < 30 eV which is observed primarily at larger τ is attributed to two electron excitation processes. The probability of electronically elastic scattering in the direct channel is shown as a function of τ in Fig. 11(a) and is seen to fall below PA = 0.5 for τ > 0.2 keV deg. The elastic scattering is in fact weaker than this since there are inelastic charge changing processes in addition to the direct elastic and inelastic collisions. The reduced cross section for elastic scattering is shown in Fig. 11(b).

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(b) F IG . 11. Electronically elastic scattering in H0 + H2 . (a) The probability of electronically elastic scattering as a function of the reduced scattering angle for projectile energies E = 1.0, 2.0 and 3.0 keV. The plot shows that for τ > 0.2 keV deg the electronically inelastic processes dominate the scattering. (b) The reduced differential cross section for elastic scattering, ρA [= PA × ρsummed ], as a function of the reduced scattering angle. The results, at the three energies, are plotted to different arbitrary units and show that they are well approximated by a common curve in the angular range studied. Reprinted figure with permission from (Quintana and Pollack, 1996b). Copyright 1996 by the American Physical Society.

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For the direct scattering (no change in the charge state), the energy loss measurements were made with time-of-flight techniques and the incident beam served as an energy reference. For stripping collisions H0 + H2 → H+ , an electrostatic energy analyzer was used and the incident beam could not provide the necessary reference. In this case H0 + Ar → H+ + Ar + e− was again used as a reference. Figure 12(a) shows composite results taken by rapidly switching the target gas to acquire energy loss spectra for H0 + Ar → H+ + Ar + e− and H0 + H2 → H+ + H2 + e− collisions. The main peaks in these spectra at E = E − 13.6 = 0 eV have the same basic shape and position. In H0 + H2 collisions, this peak is attributed to the stripping of a projectile electron (requiring a minimum energy loss E = 13.6 eV) by a target which remains in the ground state. Figure 12(b) shows a spectrum for H0 + H2 → H+ + H2 + e− at E = 1.0 keV and θ = 0.53◦ . Peak a results from stripping with the target remaining in its ground state and an electron ejected at threshold energy. Peak b, which is also shown multiplied by six times, corresponds to stripping with simultaneous excitation of the H2 target. Pa , the probability of the target remaining in its ground state after stripping is plotted as a function of τ in Fig. 13(a). Figure 13(b) shows that the normalized reduced cross sections for this channel at the three energies lie on a common curve when plotted as a function of τ . Electron capture by the H0 projectile to H− is generally much weaker than the direct and stripping processes. Energy references are again needed and Fig. 14(a) shows spectra from H0 + H2 → H− (1s2 ) + Ar+ and H0 + H2 → H− . The polarity of the voltage applied to the electrostatic analyzer is different for the H+ and H− energy analysis and the energy loss scale E is reversed in the plot. The ionization potentials of Ar and H2 are less than one third of an eV apart and therefore the peak labeled α can be attributed to H0 + H2 → H− (1s2 ) + H+ 2 (X). The actual energy loss E = E + 14.8 eV for the electron capture. Figure 14(b) shows the spectrum at an energy of 2.0 keV and the presence of a secondary peak β separated by 9.6 eV from α. Although the tail in β is at energies consistent with capture to excited states of H− , these states are thought to be short lived and not reach the detector. The singly excited levels, H− (1s2s) at E ≈ 2.49 eV and H− (1s2p) at E ≈ 3.11 have not been observed in the laboratory but are important in astrophysical studies (Ingemann-Hilberg and Rudkjobing, 1970). The auto-detaching H− (n = 2) at E ≈ 10.5 eV and H− (n = 3) at E ≈ 12.5 eV states which have been identified in electron spectra for H− scattering experiments (Risley et al., 1974) are too short lived to be seen in this 2 + study. The simple transfer of a single electron resulting in H− (1s2 ) + H+ 2 (X g ) is found to dominate the collision in the angular region studied (Quintana and Pollack, 1996b). The reduced cross section as a function of τ at the three energies studied was again found to scale well. H0 +O2 collisions are of particular interest in modeling atmospheric processes. As examples, the direct scattering was shown to result in dissociating states of the

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F IG . 12. Stripping in H0 + H2 → H+ . (a) The main peaks (labeled a) are attributed to H0 + Ar → H+ + Ar + e− and to H0 + H2 → H+ + H2 (X) + e− collisions. These are seen to basically have the same excitation energy (E = E + 13.6 eV) and shape (FWHM ≈ 2.5 eV). (b) The energy loss spectrum for stripping in H0 + H2 collisions includes two peaks separated by 10.0 eV and attributed to stripping with H2 remaining in its ground state (peak a) and stripping with simultaneous excitation of the H2 target (peak b). Reprinted figure with permission from (Quintana and Pollack, 1996b). Copyright 1996 by the American Physical Society.

O2 , yielding atmospheric atomic oxygen (Quintana and Pollack, 1996b). In addition, the H0 + O2 → H+ collision was dominated by the capture of the stripped electron into an O− 2 bound state rather than by its ejection into the continuum. In this collision, the electronically inelastic processes were found to dominate the direct collision channel for τ > 0.3 keV deg. At τ = 1.0 keV deg the electronically inelastic processes accounted for as much as 80% of the direct scattering

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(b) F IG . 13. (a) The probability for stripping with the H2 target remaining in its ground state as a function of the reduced scattering angle. (b) The reduced cross section for stripping, with the resulting H2 target remaining in the ground state. The normalized ρa vs. τ results are well approximated by a common curve at the three energies. Reprinted figure with permission from (Quintana and Pollack, 1996b). Copyright 1996 by the American Physical Society.

(Quintana and Pollack, 1996b). The acquisition of the energy loss spectra for H0 + O2 → H+ again required using H0 + Ar → H+ + Ar + e− collisions as an energy reference. As cited above, the dominant process was found to re2 sult in H+ + O− 2 (X g ) rather than electron loss to the continuum. Weaker structures found at higher energy losses were attributed to the stripping of an

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F IG . 14. Electron capture in H0 + H2 → H− . (a) With H0 + Ar → H− (1s2 ) + Ar+ as an energy reference peak α is attributed to H0 + H2 → H− (1s2 ) + H+ 2 (X) with an excitation energy E = E + 14.8 eV. (b) The two main peaks (α and β) for electron capture in H0 + H2 collisions are separated by 9.6 eV. Peak β is consistent with a number of possible electronic excitations of the H− or H+ 2 . The energy loss scale is reversed relative to the stripping scale since the detected beam is negatively charged. Reprinted figure with permission from (Quintana and Pollack, 1996b). Copyright 1996 by the American Physical Society.

electron into the continuum with simultaneous excitation of the O2 (a 1 g ) and O2 (b 1 g+ ) states. Figure 15(a) shows the probability of H0 + O2 going primarily 2 to H+ + O− 2 (X g ). The probability is seen to be energy dependent in this plot. When presented in terms of the reduced cross section, the normalized results are seen to be well fitted by the common curve shown in Fig. 15(b).

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(b) F IG . 15. (a) The probability for stripping in H0 + O2 collisions as a function of the reduced 2 scattering angle. The stripping primarily results in H+ + O− 2 (X g ) with the electron captured by the target rather than being ejected into the continuum. (b) The corresponding reduced cross section for peak α as a function of the reduced scattering angle. The normalized results for ρa are well fitted by a common curve at the three collision energies studied. Reprinted figure with permission from (Quintana and Pollack, 1996b). Copyright 1996 by the American Physical Society.

The transfer of an electron from the O2 target to the H0 projectile involves an energy loss of at least 11.8 eV. Figure 16(a) shows spectra from the reference H0 + Ar → H− (1s2 ) + Ar+ and H0 + O2 → H− collisions. The 3.35 eV separation in E (= E − 11.8 eV) between the reference peak and the main peak α 2 identifies the process as H0 + O2 → H− (1s2 ) + O+ 2 (X g ). Figure 16(b) shows ◦ the spectrum at E = 3.0 keV and θ = 0.27 . The peak labeled β with its max-

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F IG . 16. Electron capture in H0 + O2 → H− . (a) The 3.35 eV energy separation between the reference H0 + Ar → H− (1s2 ) + Ar+ and H0 + O2 spectra shows that peak α corresponds to 2 H0 + O2 → H− (1s2 ) + O+ 2 (X g ). (b) The energy loss spectrum for electron capture in small angle 0 H + O2 collisions involves two main peaks separated by 4.8 eV. The peak labeled β is attributed 2 primarily to H0 + O2 → H− (1s2 ) + O+ 2 (A u ). Reprinted figure with permission from (Quintana and Pollack, 1996b). Copyright 1996 by the American Physical Society. 2 imum near 4.8 eV is attributed to H0 + O2 → H− (1s2 ) + O+ 2 (A u ) and the + 4 − 0 − 2 structure near E ≈ 6.0 eV with H + O2 → H (1s ) + O2 (b g ). Other exit channels corresponding to singly excited H− are also seen. The probability, Pα , 2 for exciting the H− (1s2 ) + O+ 2 (X g ) channel in an electron capture collision is shown as a function of reduced scattering angle in Fig. 17(a). The corresponding reduced cross section is shown in Fig. 17(b) and the results again fall on a common curve.

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(b) F IG . 17. (a) The probability for projectile electron capture to the ground state in H0 + O2 collisions, as a function of the reduced scattering angle. (b) The corresponding normalized reduced cross sections as a function of the reduced scattering angle. The data are again seen to be reasonably well fitted by a curve common to the three energies studied. Reprinted figure with permission from (Quintana and Pollack, 1996b). Copyright 1996 by the American Physical Society.

The low keV energy H0 collisions with H2 , O2 , and N2 show very similar results in the direct collision channel. In these systems the inelastic processes dominate beyond the smallest scattering angles. This is in sharp contrast to the atom–atom collision case, where the elastic channels generally dominate the collision at small angles. In reporting on charge production in H0 + N2 Van Zyl

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et al. (1987) discussed the possible role that ionic states may play in the collision. The direct scattering results on H2 , O2 , and N2 support this. As an example, at infinite separation the lowest H− + H+ 2 level is approximately 15 eV above the incident H3 ground state surface. As the collision progresses and the inter-particle separation decreases the repulsive ground state surface crosses the attractive ionic curve. At large inter-particle separation the ionic states excited at these crossings can then couple to the close lying H0 + H∗2 and H∗ + H2 channels observed. Since the intermediate ionic states are attractive they result in decreased scattering angles and the dominance of the electronically inelastic collisions at small angles. The results on the three molecules studied show that the reduced scattering angle, which is related to the distance of closest approach in the collision, is generally a useful variable for plotting experimental results.

4. Proton Collisions in the Io Plasma Torus As was shown above, protons in the solar wind play an important role in generating the aurora. Collisions involving protons may also contribute to the sulfur and oxygen UV and visible radiation first found in the Io plasma torus during the 1979 Voyager mission. More recent results obtained by the Galileo spacecraft (Williams et al., 1996) have identified H+ , O+ , and S+ ions as the primary constituents of the plasma torus. The H+ ions are predicted to have energies of several hundred keV. The presence of O+ and S+ ions is due to the dissociation of SO2 and SO molecules which result from volcanic action on Io. Additional results from the Hubble space telescope have also shown near VUV radiation from both neutral oxygen and sulfur. The radiation was at first attributed to electron–SO2 /S/O collisions but the model was shown to fail by several orders of magnitude which suggested the possible role of H+ in the underlying collisions and resulting emission (Keihling et al., 2001). In this context H+ + SO2 collisions were studied in an energy range from 50 to 250 keV. The H+ was obtained from a Pelletron accelerator, mass and energy analyzed, and passed through the target gas cell. The transmitted beam current was measured by a Faraday cup. Photons emitted perpendicular to the beam path were passed through a lens system, which also included a linear polarizer, to a spectrometer–CCD system. Figure 18 shows the spectrum from H+ + SO2 collisions in the 600 to 700 nm wavelength range. The most important line is due to electron capture by the projectile and the resulting emission of Hα . Spectra in the other wavelength ranges show Hβ and Hγ as well emission from SO2 , S+ , O+ , S, and O (Keihling et al., 2001). The spectral lines and terms are identified in a Table 1 in (Keihling et al., 2001). The most intense line seen in the collision process was an oxygen line at 777 nm. A plot of the emission cross section for this line as a

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F IG . 18. An optical emission spectrum from H+ + SO2 collisions at an energy of 200 keV. Reprinted figure with permission from (Keihling et al., 2001). Copyright 2001 by the American Geophysical Union.

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function of 1/energy showed a linear 1/E dependence. This 1/v 2 energy dependence is consistent with electron capture by the H+ at lower energies (< 400 keV) as shown by calculations (Ray and Saha, 1979, 1981; Sural and Sil, 1965; Schultz and Olsen, 1988) and laboratory measurements (Barnett, 1977). Continued studies along these lines would provide important information on the role of protons in the plasma torus.

5. Surface Collisions with Highly-Charged Ions The review to this point has discussed collisions with gaseous targets. The solar wind however also interacts with solid target surfaces. As an example the “solid” targets include cometary dust particles in a “gas and dust coma around the nucleus” (Levasseur-Regourd, 2004). Laboratory studies involve collisions with both insulating and conducting targets. Conducting surfaces can be grounded but insulating surface targets present a special challenge for ion beam experiments since they can rapidly charge up to the “ion accelerating voltage”. This brief review of surface collisions emphasizes conducting surfaces. Using ion sources based on the electron beam ion trap (EBIT) or on electron cyclotron resonance (ECR), it became possible to study collisions with projectiles having relatively low kinetic but high potential energies. These features are particularly useful in probing ion–surface interactions. In an early experiment using a high resolution crystal spectrometer, X-ray lines from highlyexcited states of argon were found from Ar17+ collisions at normal incidence on a solid-silver target (Briand et al., 1990). The experimental results indicated that a number of electrons from the surface were captured into highlyexcited outer shells of the projectile resulting in a “hollow atom”. Additional experimental and theoretical work (Burgdorfer et al., 1995; Andra et al., 1991; Clark et al., 1993; Meyer et al., 1991; Reaves, 1977; Folkerts et al., 1995; Schneider and Briere, 1996) has resulted in a model of the collision process where an incident highly-charged ion generates an image charge on a conducting target. The ion is then accelerated toward the surface and attracts electrons from it thus neutralizing the projectile. In the classical “over-the barrier” model (Burgdorfer et al., 1995), the image charge lowers the highly-charged-ion surface potential barrier and conduction band electrons from the Fermi level cross the barrier into the highly-excited outer levels of the projectile. Because of energy resonance the electrons are preferentially captured into high-lying projectile states. These can then decay and result in ejected Auger electrons and emissions in the optical and X-ray regions, as well as in additional electron capture from the surface. On entering the surface the loosely bound electrons are stripped. The electrons ejected by projectiles both before and after the collision with the surface have been observed (Meyer et al., 1991).

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Another aspect of the collision involves the negative ion production following interactions with the target. Along these lines, collisions of highly-charged N, O, F, and S ions with a highly oriented pyrolytic graphite (HOPG) surface were studied (Reaves et al., 1997). In collisions of ions with conducting surfaces the acceleration of the incoming projectile by the image charge limits both the neutralization and relaxation times. The effect is particularly important at normal incidence but can be minimized by glancing collisions with the surface. For glancing collisions the surface penetration is less important and a high degree of neutralization and creation of negative ions is expected. On this basis negative ion production should be found, especially for atoms having positive electron affinities where there are bound states for an extra electron. This effect was seen by Folkerts et al. (1995) who investigated the fast neutralization and final charges of Oq+ in grazing collisions with Au crystal surfaces. The results showed the creation of O− ions and in addition the fractions of all the other charge states were in agreement with calculations (Burgdorfer et al., 1995). Of particular significance is that these calculations predicted that almost 100% of the scattered projectiles would be negative ions when in the vicinity of the surface. The highly charged N, O, F, and S ions for the study were extracted from the Lawrence Livermore National Laboratory’s EBIT II (Schneider et al., 1990) and accelerated through a potential difference of 7 kV. They were then magnetically analyzed and directed toward the target. The ions were incident on the C(0001) plane of the HOPG target which was positioned with its normal at 89 degrees to the beam corresponding to grazing incidence at 1 degree. Calculations with the Marlow program (Robinson and Torrens, 1974) showed that at these low angles surface penetration rarely occurs. Collimating slits were set to detect ions scattered at 1.35 degrees with respect to the surface. The scattered projectiles were found to have narrow angular distributions consistent with specular reflection of the beam. The scattered ions and atoms were electro-statically charge-state analyzed and detected by a position-sensitive channel plate detector. The experimental arrangement also used a second channel plate detector mounted parallel to the target surface to detect photons and secondary electrons. Output from the second detector was used to gate the ion and atom detector and thereby minimize the background. The most detailed studies were made on S7+ , S9+ , S13+ , and S15+ beams at 7 kV corresponding to velocities, v, of 0.25, 0.28, 0.34, and 0.36 a.u. respectively. Figure 19 shows the resulting data points for P − , the negative ion yield as a function of the incident projectile velocity parallel to the surface. The solid curve is plotted following a formulation, based on the Saha–Langmuir equation (McDaniel, 1964), by Winter (1991) with P− ≈

1

1 + (g − /g + ) exp

Eg (ys ) +

v2 2vc v

.

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F IG . 19. The probability of negative ion (S− ) production in the small angle scattering of highly charged S ions from highly oriented pyrolytic graphite, as a function of the beam velocity component parallel to the surface. The solid curve is a fit to the data following the procedure of Winter (1991). Reprinted figure with permission from (Reaves et al., 1997). Copyright 1997 AIP press, New York.

Here Eg (ys ), is the energy gap, in atomic units, at a distance ys from the image plane (the “freezing distance” defined in Eq. (9) in (Winter, 1991). For y < ys the transition rates are larger and then become negligible for y > ys . At infinite separation Eg is the energy difference between the Fermi level and atomic level and decreases as the ion approaches the surface. The g + and g − take spin and the degeneracy of the atomic level into account and the ratio g − /g + was taken to be 1/3 for S. The vc is used as a parameter to fit the experimental data. To evaluate Eg (ys ) and vc a quantity f (v) = (1/vc )v 2 /2 + Eg /vc is defined and plotted versus v 2 /2 in Fig. 20. The straight line fit in this figure and the fit to the data in Fig. 19 show good agreement between the experimental results and Winter’s (1991) description of neutralization and negative ion creation in glancing collisions. Using the fit in Fig. 20 the energy gap, Eg (ys ), at the distance of closest approach to the image plane is found to be 0.25 eV. Assuming a work function for the surface to be 4.81 eV and the electron affinity of S to be 2.08 eV the energy gap at infinite separation is 4.81 − 2.08 = 2.73 eV. This implies an increase in the effective electron affinity to 4.81 − 0.25 = 4.56 eV or a change in electron affinity of 2.48 eV. This in turn implies a distance of closest approach to the image plane of about 2.7 a.u. (Eq. (1) in Winter, 1991). As the image plane is likely to lie about 4 a.u. above the top layer of surface atoms, the apparent distance for

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F IG . 20. A least squares fit of f (v) plotted as a function of v 2 /2. The statistical counting errors in the data are smaller than the size of the plotted data points. Reprinted figure with permission from (Reaves et al., 1997). Copyright 1997 AIP Press, New York.

resonant electron capture is approximately 1 to 2 a.u. above the surface (Reaves et al., 1997). In the above approach to the problem, the image interaction between the ion and surface increases the affinity energies of negative ions. Electron capture then can take place via resonant one-electron tunneling and shows a well defined dependence on the velocity component parallel to the surface. The fitted curve in Fig. 20 is sensitive to the work function of the target, and electron affinity of the ion, as well as to the parallel velocity. The curve is particularly sensitive to the energy gap between the Fermi level and the atomic level which changes along the trajectory as the electron affinity increases. If the atomic level comes into energy resonance with the Fermi level, then the transfer of an electron to the neutralized projectile is likely to occur. Additional experimental results show that for S7+ , S9+ , S13+ , and S15+ (with corresponding velocities of 0.25, 0.28, 0.34, and 0.36 a.u.) the probabilities of neutralization are 0.60, 0.58, 0.54, and 0.56, respectively. The probabilities of singly charged ion production are 0.15, 0.20, 0.26, 0.27 and for doubly charged ions they are 0.02, 0.04, 0.05, and 0.40. In these studies of N, O, F, and S collisions with the HOPG surfaces, no doubly charged negative ions or N− (which exists only in an excited state) were found.

6. Acknowledgement The work discussed in “collisions involving heavy solar wind ions” and “proton collisions in the Io plasma torus” sections was supported by NASA Grant NCC5-

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601. The work on “collisions involving heavy solar wind ions” was carried out at JPL/Caltech, through agreement with NASA, and also received support from the University of Connecticut Research Foundation.

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ATOMIC INTERACTIONS IN WEAKLY IONIZED GAS: IONIZING SHOCK WAVES IN NEON∗ LEPOSAVA VUŠKOVIC´ and SVETOZAR POPOVIC´ Department of Physics, Old Dominion University, Norfolk, VA 23529, USA

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Introduction . . . . . . . . . . . . . . . . . . . Electron Impact Ionization from Excited Neon Energy Pooling Processes in Neon . . . . . . Ionizing Shock Waves in Neon . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . .

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Abstract The study of weakly ionized gas at atmospheric and near-atmospheric pressure has been attracting an increased interest due to the enormous potential for applications. Atomic and molecular processes drive its complex phenomenology. Excited atoms, especially ones with a long lifetime, have a very important role acting as both an energy reservoir and as an agent of energy transfer. We are using the exemplary case of ionizing shock waves in neon to illustrate how a complete knowledge of the collisional dynamics involving excited atoms is crucial for fundamental understanding of the properties of high-pressure, weakly ionized gas.

∗ One of authors (LV) would like to express her appreciation for the great privilege of having had the opportunity of working with Ben Bederson from 1985 to 1993 in the Atomic Beams Laboratory at New York University. In the summer of 1993 she transferred the laboratory to the Department of Physics at Old Dominion University, Norfolk, Virginia. Various configurations of the equipment are still in operation and innovative experiments continue to be performed.

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1. Introduction Weakly ionized gas (WIG) is a common name for gaseous media with a finite degree of ionization, but much lower than unity. It can be sustained by external electrical power (various forms of electrical discharges), electromagnetic, optical or other form of ionizing radiation, or by a strong shock wave. The WIG often has a rather complex chemistry and conditions that are far from the local thermodynamic equilibrium. The electron energy distribution reflects the lack of thermal equilibrium, which adds to the difficulties in defining the properties of the particular type of WIG. These properties, such as detailed particle balance, ionization–recombination kinetics and balance, and transport of radicals, determine the complex WIG phenomenology that is defined fundamentally by the involved atomic and molecular collision processes. Important background information for development of the basic understanding of the properties of WIG was developed over several decades by the systematic work of Benjamin Bederson and his coworkers at the New York University on measurements of basic atomic properties and collision processes. This work includes a long list of experiments that employed methods of observation developed in Bederson’s laboratory. Atomic polarizabilities were measured using an E–H balance technique or electric deflection method (Robinson et al., 1966; Levine et al., 1968; Molof et al., 1974; Miller and Bederson, 1988; Guella et al., 1991; Tarnovsky et al., 1993) and absolute electron collision cross sections were measured employing recoil atom technique (Jaduszliver et al., 1980, 1981, 1985; Vušković et al., 1989; Zuo et al., 1990; Jiang et al., 1990, 1992, 1995; Ying et al., 1993). These measurements included the first electron impact ionization studies from an excited state (Tan et al., 1996). The pioneering work in determining absolute cross sections without direct measurements of the number of target particles employing a recoil atom technique was later extended in the work of C.C. Lin and his coworkers on absolute electron collision cross section measurements with the use of a magneto-optical trap (Schappe et al., 1995, 1996, 2002). Research on atomic collision processes of interest in ionized gas physics had a number of highlights in the last half of century. One of them was extensive research conducted on the radiative properties of electrical discharges, which promoted the development of new types of lasers and non-coherent light sources. This development mobilized over a relatively short time the research community to work on specific atomic and molecular processes of interest for given applications. In the same period, the systematic experimental work on collision processes has practically ceased to exist. The consequence is that the list of atomic parameters and collision data needed in WIG studies is far from being complete. Therefore, WIG research is still more the art than the exact science. This situation is amplified by the use of oversimplified models for collisional processes in computer simulations. Initial success of the simple single-line models in symmetric

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discharge conditions at, or close to, equilibrium had the effect of diminishing most of the atomic collision research, as a redundant and repetitive effort. However, recent experiments involving complex electric discharges with simultaneous action of multiple physical agents, such as lasers, microwave beams, and permanent external fields (Exton et al., 2001), require basic knowledge of collision dynamics in order to explain the observed effects. The use of simplified models became inadequate to explain the non-equilibrium, anisotropic, and non-stationary character of these effects. One of the consequences is frequent report on discoveries of an “anomaly”, “extremity”, or a “paradox”, which later become explained by a not-so-obvious combination of known properties of the discharges based on the full knowledge of governed collisional processes (Joviˇcević et al., 2004). Another consequence of this situation is that every attempt to produce an overview of the role of excited atoms in electric discharges is incomplete and too specific to be of general interest. Relevant information on collisions involving excited atoms, important for the understanding of high-pressure discharges, can be found in the following reviews. Trajmar and Nickel (1992) and Christophorou and Olthoff (2000) gave a comprehensive review of the work on electron impact by excited atom species. Lin and Anderson (1991) have presented a review of the work on the electron excitation processes of rare-gas atoms. Siska (1993) gave an extensive review of Penning ionization and energy-pooling processes. Data on collisions involving excited atoms of the same species however are still quite sparse. Kolokolov and Blagoev (1993) reviewed the work on the subject performed before 1990. Weiner et al. (1989) studied associative ionization. Smirnov (1977, 1981) reviewed ion conversion, ion cluster growth, as well as homo-nuclear associative ionization processes. Wuilleumier et al. (1987) reviewed collisional ionization of excited atoms, including the analysis of the homo-nuclear associative ionization process. Our work in the field of high-pressure discharges was inspired by the emerging interest in other physical characteristics of these discharges, which are even less understood than radiative properties. These are the interaction of high-pressure discharges with external electromagnetic fields (Kwan and Ahlborn, 1984), generation of internal electromagnetic effects closely related to the interaction of the discharges with acoustic waves (Bletzinger et al., 2003), general aerodynamic properties, and the effects of chemical reactions and impurities on the parameters and kinetics in WIG (Glass and Liu, 1978). Research on phenomenology of non-radiative, hydrodynamic properties is sometimes called plasma aerodynamics. Typical objects of study in plasma aerodynamics are the phenomena usually described as dispersion of acoustic shock waves in the WIG, and the related group of aerodynamic phenomena associated with strong, ionizing shock waves. Hydrodynamic effects in ionized gas was a rather neglected subject of investigation until a series of laboratory experiments revealed the complex phenomenology of dispersion and propagation of acoustic shock waves (Klimov et al., 1982;

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Mishin et al., 1991; Popović and Vušković, 1999; Bletzinger et al., 2000; Exton et al., 2001). Satisfactory explanation for most of the effects was found in the heat transfer between charged and neutral particles. However, many questions still remain unresolved. One of them is the problem of the accumulated concentration of particles in long-lived electronically excited states in shock layers and its relation to the dispersion of the acoustic shock waves. A necessary tool for understanding these phenomena would be an accurate ionization–recombination model. There are still no systematic spectroscopic measurements of local excitedstate populations in leader, precursor, and relaxation regions of weak shocks in WIG. However, there is some circumstantial evidence for the link between the excessive excited-state population and the observed dispersion (Bletzinger et al., 2003). Experimental results of McIntyre et al. (1991) on strong, ionizing shock waves in neon show the excessive population of excited states in the precursor region that could not be accounted for using the standard ionization– recombination model based on local thermodynamic equilibrium. In both cases, the ionization–recombination process departs from the model based on the ratecontrolling excitation to the first excited state (Raizer, 1991). In WIG, one has to adopt a non-equilibrium presentation of ionization, where several competitive processes are involved. To evaluate the ionization rates, one has to consider at least three terms of ionization: (1) electron impact ionization from the ground state or, (2) from an excited state, and (3) ionization in the collision of two excited states. Ionization from ground state is important at very low pressures, below the limit of interest here. Electron impact ionization from an excited state is the dominating process at high pressures, while ionization by an excited pair collision is important in a wide intermediary range of pressures. Implementing available data into an ionization–recombination model in a consistent manner proves to be a difficult task. The reason is twofold: data are very scarce and consequently their implementation ambiguous, and thermodynamic equilibrium conditions are not fulfilled. This situation has not been a serious problem in modeling low-pressure discharges, where processes involving excited atoms are less important (Ferreira et al., 1985; Bogaerts and Gijbels, 1995; Sommerer, 1996). However, at a pressure of 1 kPa and above, relative contribution of individual elementary processes may differ substantially from the low-pressure discharge, due to pressure and temperature dependence of rate coefficients and increased number densities of involved excited species. In order to illustrate the relevance of atomic interactions in WIG, we chose here to analyze a case of WIG generated by a strong shock in neon. Therefore, we will focus only on electron impact ionization from neon excited states presented in Section 2 and the energy pooling ionization processes between neon atoms in excited states presented in Section 3. In Section 4 we show how these data were

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used to explain experimental results (McIntyre et al., 1991) on excessive excited state populations in ionizing shock waves in neon.

2. Electron Impact Ionization from Excited Neon There are very few experimental data on the electron impact ionization from excited neon. Only one direct measurement refers to an integral ionization cross section involving metastable targets (Dixon et al., 1973). A single experimental value of total electron impact ionization from a single excited state, although it is usually the most populous one at low densities, is not satisfactory—especially at higher densities where radiation trapping makes the radiative states constantly populated and all effectively metastable. In order to produce a correct description with a well-estimated error for the total electron impact ionization of excited atoms, one has to rely on approximate calculations verified by benchmark experiments. There are several sets of approximate calculated data available for average cross sections for higher-lying excited states, but there are few experimental values that would present a realistic test of the theory. However, there are available data for the ground state and 3 P state of sodium (Tan et al., 1996). These states have similar configuration and comparable binding energies to 3s and 3p states of neon. The measurements of absolute total ionization cross-section for ground state and laser-excited 3 P state of sodium in the incident electron energy range from threshold to 30 eV were performed in Bederson’s laboratory using a modified recoil-atom technique (Tan et al., 1996). Our measurements of electron impact ionization of the laser-excited sodium 3p state (Tan et al., 1996), and the experiment of Trajmar et al. (1986) on barium, still remain the only ionization measurements on short-lived excited states. Excited-state sodium results are given in Fig. 1 together with calculations (Bray, 1994; McGuire, 1977; Vriens, 1969; Erwin and Kunc, 2004). The Convergent Close-Coupling (CCC) calculations of total ionization cross section (CS) (Bray, 1994) correctly describe the cross section trend above 10 eV, but underestimate absolute values of the ionization cross section by 50 to 100% at higher energies. Calculated CS values from the energy-corrected binary-encounter approximation (ECBEA) show the same location of the maximum as CCC—that is 20% below the energy found in the experiment. It tends to overestimate CS. The Born approximation (McGuire, 1977) shows the maximum at the same energy as the experiment but overestimated by 20 to 30%. An empirical expression (Vriens, 1969) has the best quantitative agreement with the experiment. All calculated results tend to overestimate experimental data near threshold. One can draw a similar conclusion from comparison between calculations and experiment for the electron impact ionization cross section of the 3s states in neon

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F IG . 1. Total ionization cross section for the 3p state of Na: •, Tan et al. (1996); thick black line, Vriens (1969); medium black line, Bray (1994); thin black line, McGuire (1977); gray line, Erwin and Kunc (2004).

F IG . 2. Total ionization cross section for the neon 3s states: •, Dixon et al. (1973); black line, Vriens (1969); gray line, Erwin and Kunc (2004). In both calculations an average binding energy of all four states was used.

(see Fig. 2). Calculated values are in a satisfactory quantitative agreement with the experimental data at higher energies. However, calculated results do not reproduce threshold behavior or the location of maximum of the data. Since cross section behavior at energies close to threshold are of critical importance for calculation of ionization rates in the discharge plasmas, we inspected how this discrepancy is reflected on the accuracy of ionization rates. For this purpose we used a calcu-

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lated set of electron energy distribution functions for the range of reduced electric field E/N between 1 and 300 Td. Using different sets of cross section data, we found that the calculated ionization rates are within ±10% for 3s and 3p states of sodium.

3. Energy Pooling Processes in Neon Ionization in a collision involving two atoms in excited states has two types of ion products: kEP

A∗ + A∗ −→ A+ + A + e− (E), kAI

− A∗ + A∗ −→ A+ 2 + e (E)

(1a) (1b)

where kEP and kAI are the rate coefficients for energy pooling (atomic ion production) and for associative ionization (molecular ion production), respectively. In noble gas plasmas ionization takes place practically in all collisions between atoms in excited state since the sum of excitation energies exceeds the ionization potential. In both ionization channels, the ejected electrons carry out most of the excess kinetic energy. These features of the process contribute to the restoring of the excited state population and affect the electron energy distribution in the discharge. However, this aspect of the energy pooling process has usually been neglected in most models of gas discharges. Since there are practically no experimental data on the energy pooling processes, models usually use arbitrary values derived from a few reported observations (i.e. Phelps and Molnar, 1953). The energy pooling ionization process is closely related to the mutual Van der Waals interaction of two excited atoms. In order to estimate the cross section for this kind of processes, one has to know the dispersion constant C6 , which depends on the polarizabilities. However, neither of these is available. Thus, we have used the comprehensive review on polarizabilities by Bonin and Kresin (1997) and generated the needed data. We present here three kinds of data: (1) experimental, (2) analytical based on exact theories involving ab initio calculation of the potential and auto-ionization shift curves (Derevianko and Dalgarno, 2000), and (3) approximate analytical data, based on simplified collision models (Smirnov, 1981; Bell et al., 1968). When ion potential curves are known, one can also determine the cross section for the associative ionization. The approximate analytical data based on the simplified models require knowledge of the Van der Waals interaction potential between the excited states and the corresponding values of dispersion constants, C6 . One can generate these data in the following ways:

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(a) calculate dipole polarizability of each particular excited state based on compiled experimental data on oscillator strengths, (b) evaluate an approximate Van der Waals constant C6 , (c) evaluate total ionization cross section, and (d) evaluate total ionization rate by averaging over the excited atom velocity distribution function. In our analyzes we rely exclusively on the experimental values of parameters involved in the calculation. The polarizability and dispersion constants of the metastable state of neon, 3s 3 P2 , are well known (Molof et al., 1974). In order to properly account for the influence of all excited states in the ionization by energy pooling we have to evaluate the polarizabilities of other 3s states, and all 3p, 3d, 4s, and 4p states. We used the existing oscillator strength databases (Wiese et al., 1996; Hartmetz and Schmoranzer, 1983) and completed the list with our own measurements of line intensities in a controlled microwave discharge using an absolutely calibrated detection system. These measurements have produced the quality of data comparable to those listed in the database. Typical error margins were about 25% for all states above the 3s. The measured values were also compared to the published calculated values (Seaton, 1998). The average discrepancy is less than 10%, which is well within the expanded margin of error of the estimate. This data set of oscillator strengths was then used for evaluating polarizabilities. Table I shows the individual polarizabilities and dispersion constants of neon for all 3s and 3p states and average polarizabilities of states in the 3d, 4p, and 4s group. Contributions from higher excited states are negligible due to their low number densities at WIG discharge conditions. Further, we tested our results with the experimental values for the 3 P2 state. Our polarizability values are within 5–10% of the experimental values of Robinson et al. (1966) and Molof et al. (1974), which reflect the state of accuracy of the oscillator strength data. At this point it is important to emphasize that the experimental values of Bederson’s group (Miller and Bederson, 1988) for metastable noble gas polarizability have endured over the time period of 40 years and are still used for comparison with exact calculations of Van der Waals dispersion constants (Zhu et al., 2004). Using the calculated dipole polarizabilities, the Van der Waals constant C6 was evaluated using the Slater–Kirkwood approximation for atoms with one valence electron e2 η2 αAαB 3 C6AB ≈ − (2) √ √ . 2 me α A + α B Accurate theoretical values of the dispersion constant C6 are available for the interaction between two 3 P2 metastable neon states (Derevianko and Dalgarno, 2000). An average value over all states of the excited Ne*2 complex formed

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Table I Polarizability α, constant C6 , and energy pooling ionization constant CEP of selected neon excited states. State

α (10−24 cm−3 )

C6 (a.u.)

CEP (cm3 s−1 K−1/6 )

1s5 , 3 P2 1s4 , 3 P1 1s3 , 3 P0 1s2 , 1 P1 2p10 , 3 S1 2p9 , 3 D3 2p8 , 3 D2 2p7 , 3 D1 2p6 , 1 D2 2p5 , 1 P1 2p4 , 3 P2 2p3 , 3 P0 2p2 , 3 P1 2p1 , 1 S0 4s 4p 3d

27.8 ± 1 28.4±0.7 28.9±0.3 30.8±0.9 41.3±0.6 73.7±1.1 59.2±6.5 62.1±1.1 67.0±3.2 77.4±1.4 69.1±0.7 66.7±4.6 78.2±1.3 96.6±2.5 260 390 900

1930 1990 2045 2250 3490 8330 5990 6440 7220 8960 7560 7170 9100 12500 5.5e4 1.01e5 3.6e5

4.12E−10 4.16E−10 4.2E−10 4.34E−10 5.02E−10 6.71E−10 6.01E−10 6.16E−10 6.4E−10 6.88E−10 6.5E−10 6.38E−10 6.91E−10 7.68E−10 1.26E−9 1.54E−9 2.34E−9

by these two metastable states gives C6 = 1938 a.u. The present result of C6 = 1930 a.u. is again in a very good agreement, considering the uncertainties of the experimental values of oscillator strengths. At present there is no accurate reference data for higher excited states of neon. The total ionization cross section then can be estimated based on the assumption that the auto-ionization of the intermediary molecular excited state can occur if the collision energy E is greater than the relative maximum in the effective radial potential. The total ionization cross section σtot is given by (Garrison et al., 1973) C6 V0 (R) 2 σtot (E) = πR 1 − (3) ; V0 (R) = − 6 E R where R = R(E) is the larger root of the equation E = V0 (R) + 12 RV0 (R)

(4)

where V0 (R) is the derivative of V0 over inter-atomic distance. The above approach leads to the well-known expression for the binary capture cross section (Bell et al., 1968; Smirnov, 1981) 3π C6 1/3 σtot = (5) . 2 E

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In WIG plasmas one can distinguish two types of energy pooling reactions. One is due to relative motion of excited atoms and we can call it “thermal” energy pooling. It takes place even in the absence of the gas flow. In this case thermal equilibrium between heavy particles is satisfied and the ionization rate has the form (Smirnov, 1981; Kolokolov and Blagoev, 1993) 1/3

thermal = cC6 m1/2 T 1/6 = CEP T 1/6 kEP

(6)

where T is the gas temperature in the discharge, m is the reduced mass of excited atoms in atomic units, C6 is the dispersion constant in atomic units, and c is a numerical constant fully explained in Kolokolov and Blagoev (1993). Constants C6 and CEP are given in Table I for the most important groups of excited states of neon. Equation (6) is numerically equivalent to the expression given by Bell et al. (1968) who evaluated m in units of electron mass. At supersonic gas flow conditions, additional term to the energy pooling ionization rate has to be applied reflecting directional motion of excited particles in the flow. We can call it “directional” energy pooling. In this case f (E) = δ(E − E0 ) and kEP = σ (E0 ) · v0 , where v0 is the shock velocity. Since E0 is the kinetic energy of the excited atom, the ionization rate in the supersonic flow is 3 2C6 v0 1/3 directional = π . kEP (7) 2 m Energy pooling ionization rates due to the supersonic flow for three representative groups of neon states (3s, 3p, and 3d) are calculated using Eq. (7) and presented in Fig. 3. Shock velocity is given in Mach number. One can see that for strong shock waves (M > 10) the directional energy pooling ionization rate is almost an order of magnitude larger than thermal energy pooling ionization rate. The thermal energy pooling ionization rate for a collision between two metastable Ne(3 P2 ) states in WIG is given in Fig. 4. The discrepancy between the present calculation and Kolokolov and Blagoev (1993), as well as the result obtained by Salinger and Rowe (1968) at T = 573 K can be attributed to the uncertainty in the oscillator strength data used in each calculation. The experimental results from Kolokolov and Blagoev (1993), obtained with the plasma electron spectroscopy technique in a discharge afterglow at T = 300 K are in fairly good agreement with calculation. The approximate character of the Slater–Kirkwood relation, which tends to underestimate the Van der Waals interaction constant, is much more pronounced in the ground state. In the excited states, it proved to be reasonably accurate. We were able also to reproduce the static polarizability and dispersion constant for alkali–noble gas complexes calculated by Derevianko and Dalgarno (2000). The dynamics of the intermediary excited complex in the ionization process is unknown and the uncertainties about the spin conservation rule in its decay may lead

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F IG . 3. Directional ionization rates in the WIG supersonic flow for energy-pooling collisions between two excited neon atoms: 3s, thick line; 3p, medium line; 3d, thin line.

F IG . 4. Thermal ionization rates for energy-pooling collisions between two excited neon atoms: thick line, 3 P2 –3 P2 ; thin line, 3 P2 –3 P1 ; circle, 3 P2 –3 P2 , Kolokolov and Blagoev (1993); square, 3 P –3 P , Kolokolov and Blagoev (1993); triangle, Salinger and Rowe (1968). 2 1

to additional error in calculation. Finally, the uncertainties in oscillator strength data certainly contribute to the error. It is a challenging task to test the evaluated ionization rates at real plasma conditions. Development of various advanced laser absorption spectroscopy techniques gives an opportunity to probe the population of excited atoms in plasma. Combined with plasma electron spectroscopy (Kolokolov and Blagoev, 1993) that has more capability to discriminate between the excited states, one can obtain the ex-

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perimental values of the energy pooling ionization rates of excited states other than metastables. This would provide the ultimate test for the role of energy pooling processes in ionization–recombination process.

4. Ionizing Shock Waves in Neon In order to demonstrate the effect of electron scattering on weakly ionized flows we simulated the conditions of the experiments performed with ionizing shock waves in neon (McIntyre et al., 1991). There are several reasons for this choice. First, the experiments are well documented and the population of single excited species was measured at well-defined conditions. Excited neon has strong absorption lines in the visible region of the spectrum. These lines offered the possibility for more accurate measurements of excited-state population using absorption spectroscopy techniques. Second, the experiments were performed with planar shock waves and a successful one-dimensional model could be developed, with the ability to describe correctly most relaxation and recombination processes. Main features of the shock structure could be simulated with high accuracy in a one-dimensional description, although there were some issues related to the transverse instabilities of the flow requiring a multi-dimensional treatment. Third, the experimental results were still not explained in full detail. In particular, the standard equilibrium ionization–recombination model used by McIntyre et al. (1991) could not explain the excessive excited-state population in precursor and relaxation regions of the shock wave. The experiment consists of a shock tube filled with neon at 1.33 kPa where strong shock waves with Mach number of M = 15–20 were generated (see Fig. 5). The experiment clearly revealed four different regions in shock structures. The precursor region covers the long part of the tube in front of the shock. It turns out that this region is a WIG with very low degree of ionization, sustained by the resonant radiation from the ionization wave (electron cascade front) that follows the translational shock. It contains a large density of excited states, mostly 3s. A small amount of charged particles is produced in the energy pooling collision processes. The electron number density is about two orders of magnitude lower than the population of atoms in 3s states, which are effectively all metastable due to the radiation trapping mechanism. Since most of the electrons are produced in the energy pooling collisions, their energy distribution is dispersed with a cut-off at around 12 eV. It results in a highly non-equilibrium WIG reminiscent of glow discharges with weak cathode sheath. The shock layer is characterized by a strong jump in temperature and density of heavy particles, intense excitation of ground state atoms, and increase of electron density due to energy pooling collisions. This region is still far from equilibrium

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F IG . 5. Scheme of the shock structure: 1, precursor region; 2, shock layer; 3, relaxation zone; 4, ionizing wave.

since the electrons are not directly affected by the shock, except for the secondary effect of the double electric layer generated by separation of charges around the shock front. Behind the shock layer there is a relatively wide shock relaxation zone, characterized with mutual thermalization process between charged and neutral particles. It eventually develops into the ionization wave (electron cascade front) characterized by a high degree of ionization and local thermodynamic equilibrium with temperature of about 1 eV. Therefore, the whole shock structure consists of four regions where three could be characterized as non-equilibrium WIG and only the fourth as partially ionized gas in local thermodynamic equilibrium. Representative set of data characterizing neon in our ionization–recombination model consisted of four groups of excited states, 2s2 2p5 3s, . . . 3p, . . . 4s, . . . 3d, in the precursor region. 3s states and small quantities of 3p, 4s, and 3d states (about 3% of 3s population) are sustained by absorption of resonant radiation from the strongly ionized plasma in the relaxation zone of the shock wave. Near the shock layer, . . . 4p, 4d, 5s, and 4f states were added. Necessary ionization rates were calculated according to the procedure outlined in Sections 2 and 3. The uncertainties observed in the cross section data did not exceed 10% uncertainty in ionization rates. As pointed out, three regions of the shock structure are not in thermodynamical equilibrium. The often used equilibrium models (Hoffert and Lien, 1967) do not give appropriate description of detailed particle balance and excited state distribution. We are using the ionization–recombination balance dNe = dt

dNe dt

+ αNe2 , ion

(8)

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where Ne is electron density and α is the recombination coefficient. Charge particles loss due to diffusion to the wall is neglected. dNe = kdi · Ne · N + kj · Nj · N e + kij · Ni · Nj , (9) dt ion j

ij

where kdi , kj , and kij are the ionization rates of electron impact ionization from the ground state, the j th excited state, and ionization in collision between excited atoms in ith and j th states, respectively; N , Ni , and Nj are number densities of the ground state, ith and j th excited states, respectively. Energy pooling ionization rate includes the thermal and directional component defined by Eqs. (6) and (7). The recombination coefficient is (Stevefelt et al., 1975; in cm3 s−1 ) αcr = c1 × 10−10 u−0.63 + c2 × 10−9 u−2.18 Ne0.37 + c3 × 10−9 u−4.5 Ne ,

(10)

where electron temperature was substituted with the average electron energy u = f (E/N), and c1 , c2 , and c3 are numerical constants fully explained in (Stevefelt et al., 1975). Electron and heavy particle number density distributions across the shock have shown a fairly good agreement between the model and experimental data of McIntyre et al. (1991). The electron number density profile correctly described the location, the amplitude and the gradient of the electron cascade front. The decay of electron number density in the equilibrium region indicates that the gas cooling due to radiation losses was taken into account correctly. The electron cascade front is better described with the present model than with the previously used one, reflecting the correct description of an ionization process that included the role of excited states in more detail. The effect of absorption of resonant radiation on the population of first resonant level is illustrated in Fig. 6. It is obvious that the results of conventional models based on the thermal equilibrium with ground state (Boltzmann equilibrium) or thermal equilibrium with free electrons (Saha equilibrium) cannot reproduce the excited-state population profile in the first three regions of the shock structure. By inclusion of the absorption of resonant radiation, this profile was reproduced with the present model both qualitatively and quantitatively behind the translational shock. A basic assumption in the diffusion approximation of resonant radiative transfer is that the density of irradiating and absorbing atoms is comparable, so that the effective width of the deexciting states is the same as the effective width of the downstream excited state. However, in reality the downstream excited state has a narrower width since the absorbing atoms are in the lower density region. Therefore, the resonant radiation is absorbed with partial redistribution. The result is a slowing of radiation transfer. Hence, the density of excited atoms in the vicinity of density gradients is higher. Both experiment and calculation with the

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F IG . 6. Excited state population across a strong shock wave in neon, us = 9.14 × 103 m/s, p0 = 10 Torr: thick black line, experiment (McIntyre et al., 1991); gray line, non-equilibrium model with partial redistribution approximation for radiation transfer; thin black line, non-equilibrium model with diffusion approximation for radiation transfer; dashed line, Saha equilibrium model; dot-dashed line, Boltzmann thermal equilibrium model.

non-equilibrium model show clearly the elongation of the shock layer and dispersion of the translational shock. This effect could not be obtained using equilibrium models. In the latter are not included communications between electron cascade front and precursor, which are opened by radiation transfer. The communication between the upstream and downstream portions of the shock wave is one of the characteristic properties of its interaction with the WIG. Application of the new ionization–recombination model to the case of ionizing shock in neon (McIntyre et al., 1991) has demonstrated that the electron and ion production, charge separation, non-steady state recombination as well as transport and radiation processes were included with satisfactory accuracy. The choice of McIntyre et al. (1991) data to exemplify the performance of the present model was made because a well-documented and accurate experiment was available, the flow is relatively simple, and there were some unresolved issues related to the excited states in the relaxation and the precursor regions. The model simulated correctly the main features of the flow. An induced electric field associated with the charge separation may be a possible reason for the observed disturbances of the flow pattern. Substantial modification of the precursor and relaxation region due to diffusion of resonant radiation lead to a conclusion that initial conditions related to the precursor region may differ substantially from the unaffected free stream. Increase of excited state population leads to visible changes of the ionization process. The observed increase of excited states is partly due to the non-steady

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recombination process and partly due to the action of physical agents such as radiation. In this particular case the other mechanisms are of less importance.

5. Concluding Remarks Research in the field of plasma aerodynamics is full of exciting effects, governed by atomic collision processes, exposed in the macroscopic behavior of weakly ionized gas. In this article we presented how the ionization processes involving excited atoms affect their distribution across the shock structure in the case of an ionizing shock wave in neon. We have applied a non-equilibrium ionization– recombination model to reproduce the excited state population in the precursor region of the ionizing shock wave propagating in neon. Models based on equilibrium between charged particles and excited atoms were not able to predict the presence of excited atoms and a weakly ionized state of the precursor region. We show that electron impact ionization and energy pooling processes involving excited atoms are crucially important for the production of charged particles in the precursor region.

6. Acknowledgements This article is dedicated to Benjamin Bederson on his retirement as Editor of the Advances in Atomic and Molecular Physics series. The authors thank Charles Sukenik of Old Dominion University for the helpful suggestions. The authors wish to acknowledge continuing support for their work in the field of plasma aerodynamics from NASA Langley Research Center.

7. References Bell, K.L., Dalgarno, A., Kingston, A.E. (1968). Penning ionization by metastable helium atoms. J. Phys. B 1, 18–22. Bletzinger, P., Ganguly, B.N., Garscadden, A. (2000). Electric field and plasma emission responses in a low pressure positive column discharge exposed to a low Mach number shock wave. Phys. Plasmas 7, 4341–4346. Bletzinger, P., Ganguly, B.N., Garscadden, A. (2003). Strong double-layer by shock waves in nonequilibrium plasma. Phys. Rev. E 67, 047401. Bogaerts, A., Gijbels, R. (1995). Modeling of metastable argon atoms in a direct-current glow discharge. Phys. Rev. A 52, 3743–3751. Bonin, K.D., Kresin, V.V. (1997). “Electric-Dipole Polarizabilities of Atoms, Molecules, and Clusters”. World Scientific, Singapore. Bray, I. (1994). Calculation of electron impact total, ionization, and non-breakup cross sections from the 3 S and 3 P states of sodium. Phys. Rev. Lett. 73, 1088–1091, and private communication.

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Christophorou, L.G., Olthoff, J.K. (2000). Electron interactions with excited atoms and molecules. Adv. Atom. Mol. Opt. Phys. 44, 155–293. Derevianko, A., Dalgarno, A. (2000). Long-range interaction of two metastable rare-gas atoms. Phys. Rev. A 62, 062501-1-5. Dixon, A.J., Harrison, M.F.A., Smith, A.C.H. (1973). Ionization of metastable rare gas atoms by electron impact. In: Cobic, B.C., Kurepa, M.V. (Eds.), “Abstracts of Papers, Eight International Conference on the Physics of Electronic and Atomic Collisions”, vol. I, Institute of Physics, Beograd, Yugoslavia, pp. 405–406. Erwin, D.A., Kunc, J.A. (2004). Ionization of excited xenon atoms by electrons. Phys. Rev. A 70, 022705-1-6. Exton, R.J., Balla, R.J., Shirinzadeh, B., Brauckmann, G.J., Herring, G.C., Kelliher, W.C., Fugitt, J., Lazard, C.J., Khodataev, K.V. (2001). On-board projection of a microwave plasma upstream of a Mach 6 bow shock. Phys. Plasmas 8, 5013–5017. Ferreira, C.M., Louriero, J., Ricard, A. (1985). Populations in the metastable and the resonance levels of argon and stepwise ionization effects in a low-pressure argon positive column. J. Appl. Phys. 57, 82–90. Garrison, B.J., Miller, W.H., Schaefer, H.F. (1973). Penning and associative ionization of triplet metastable helium atoms. J. Chem. Phys. 59, 3193–3198. Glass, I.I., Liu, W.S. (1978). Effects of hydrogen impurities on shock structure and stability in ionizing monatomic gases. J. Fluid Mech. 84, 55–77. Guella, T., Miller, T.M., Stockdale, J.A.D., Bederson, B., Vušković, L. (1991). Polarizabilities of the alkali halide dimers II. J. Chem. Phys. 94, 6857–6861. Hartmetz, P., Schmoranzer, H. (1983). Absolute transition probabilities in the NeI 3p–3s fine structure by beam–gas–day laser spectroscopy. Phys. Lett. A 93, 405–408. Hoffert, M.I., Lien, H. (1967). Quasi-one-dimensional, non-equilibrium gas dynamics of partially ionized two-temperature argon. Phys. Fluids 10, 1769–1777. Jaduszliver, B., Dang, R., Weiss, P., Bederson, B. (1980). Total cross sections for the scattering of low-energy electrons by excited sodium atoms in the 3 2 P3/2 , mJ = ±3/2 state. Phys. Rev. A 21, 808–818. Jaduszliver, B., Tino, A., Bederson, B. (1981). Absolute total cross sections for the scattering of lowenergy electrons by lithium atoms. Phys. Rev. A 24, 1249–1253. Jaduszliver, B., Shen, G.F., Cai, J.-L., Bederson, B. (1985). Total cross sections for electrons scattered by 3 2 P3/2 sodium atoms. Phys. Rev. A 31, 1157–1159. Jiang, T.Y., Ying, C.H., Vušković, L., Bederson, B. (1990). Absolute small angle electron excitation cross sections for the resonant transition in sodium. Phys. Rev. A 42, 3852–3860. Jiang, T.Y., Zuo, M., Vušković, L., Bederson, B. (1992). Absolute cross sections for low-energy scattering of electrons by excited sodium. Phys. Rev. Lett. 68, 915–918. Jiang, T.Y., Shi, Z., Ying, C.H., Vušković, L., Bederson, B. (1995). Super-elastic electron scattering by polarized excited sodium. Phys. Rev. A 51, 3773–3782. Joviˇcević, S., Ivković, M., Konjević, N., Popović, S., Vušković, L. (2004). Excessive Balmer line broadening in microwave-induced discharges. J. Appl. Phys. 95, 24–29. Klimov, A.I., Kobolov, A.N., Mishin, G.I., Serov, Yu.I., Yavor, I.P. (1982). Shock wave propagation in decaying plasma. Sov. Tech. Phys. Lett. 8, 192–194. Kolokolov, N.B., Blagoev, A.B. (1993). Ionization and quenching of excited atoms with the production of fast electrons. Phys. Usp. 36, 152–170. Kwan, J., Ahlborn, B. (1984). Electrical potentials at probes in supersonic plasma flow. Phys. Fluids 27, 499–505. Levine, J., Celotta, R.J., Bederson, B. (1968). Measurement of the electric dipole polarizabilities of metastable mercury. Phys. Rev. 171, 31–35. Lin, C.C., Anderson, L.W. (1991). Studies of electron impact excitation of rare-gas atoms into and out of metastable levels using optical and laser techniques. Adv. Atom. Mol. Opt. Phys. 29, 1–32.

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McGuire, E.J. (1977). Electron ionization cross sections in the Born approximation. Phys. Rev. 16, 62–72, and private communication. McIntyre, T.J., Houwing, A.F.P., Sandeman, R.J., Bachor, H.-A. (1991). Relaxation behind shock waves in ionizing neon. J. Fluid Mech. 227, 617–640. Miller, T.M., Bederson, B. (1988). Electric dipole polarizability measurements. Adv. Atom. Mol. Opt. Phys. 25, 37–60. Mishin, G.I., Serov, Yu.L., Yavor, I.P. (1991). Flow around a sphere moving supersonically in a gas discharge plasma. Sov. Tech. Phys. Lett. 17, 413–416. Molof, R.W., Schwartz, H.L., Miller, T.M., Bederson, B. (1974). Measurements of electric dipole polarizabilities of the alkali-metal atoms and the metastable noble-gas atoms. Phys. Rev. A 10, 1131–1140. Phelps, A.V., Molnar, J.P. (1953). Lifetime of metastable states of noble gases. Phys. Rev. 89, 1202– 1208. Popović, S., Vušković, L. (1999). Anomalous propagation of planar shock wave in weakly ionized gas. Phys. Plasmas 6, 1448–1454. Raizer, Y.P. (1991). “Gas Discharge Physics”. Springer, New York. Robinson, E.J., Levine, J., Bederson, B. (1966). Metastable 3 P2 rare-gas polarizabilities. Phys. Rev. 146, 95–100. Salinger, S.N., Rowe, J.E. (1968). Determination of the predominant ionization and loss mechanisms for the low-voltage arc mode in a neon plasma diode. J. Appl. Phys. 39, 4299–4307. Schappe, R.S., Feng, P., Anderson, L.W., Lin, C.C., Walker, T. (1995). Electron collision crosssections measured with the use of a magneto-optical trap. Europhys. Lett. 29, 439–444. Schappe, R.S., Walker, T., Anderson, L.W., Lin, C.C. (1996). Absolute electron-impact ionization cross section measurements using a magneto-optical trap. Phys. Rev. Lett. 76, 4328–4331. Schappe, R.S., Keeler, M.L, Zimmerman, T.A., Larsen, M., Feng, P., Nesnidal, R.C., Boffard, J.B., Walker, T.G., Anderson, L.W., Lin, C.C. (2002). Methods of measuring electron–atom collision cross sections with an atom trap. Adv. Atom. Mol. Opt. Phys. 48, 357–390. Seaton, M.J. (1998). Oscillator strengths in Ne I. J. Phys. B: At. Mol. Opt. Phys. 31, 5315–5336. Siska, P.E. (1993). Molecular-beam studies of Penning ionization. Rev. Mod. Phys. 65, 337–412. Smirnov, B.M. (1977). Cluster ions in gases. Sov. Phys. Usp. 20, 119–133. Smirnov, B.M. (1981). Ionization in low-energy atomic collisions. Sov. Phys. Usp. 24, 251–275. Sommerer, T.J. (1996). Model of a weakly ionized, low-pressure xenon dc positive column discharge plasma. J. Phys. D: Appl. Phys. 29, 769–778. Stevefelt, J., Boulmer, J., Delpech, J.-F. (1975). Collisional-radiative recombination in cold plasmas. Phys. Rev. A 12, 1246–1251. Tan, W., Shi, Z., Ying, C.H., Vušković, L. (1996). Electron-impact ionization of laser-excited sodium atom. Phys. Rev. A 54, R3710–R3713. Tarnovsky, V., Bunimovicz, M., Vušković, L., Stumpf, B., Bederson, B. (1993). Measurements of the dc electric dipole polarizabilities of the alkali dimer molecules, homo-nuclear and hetero-nuclear. J. Chem. Phys. 98, 3894–3904. Trajmar, S., Nickel, J.C., Antony, T. (1986). Electron-impact ionization of laser-excited 138 Ba( . . . 5p6 6s6p) and 138 Ba( . . . 5p6 6s5d) atoms. Phys. Rev. A 34, 5154–5157. Trajmar, S., Nickel, J.C. (1992). Cross-section measurements for electron impact on excited atomic species. Adv. Atom. Mol. Opt. Phys. 30, 45–103. Vriens, L. (1969). Binary-encounter and classical collision theories. In: McDaniel, E.W., McDowel, M.R. (Eds.), “Case Studies in Atomic Collision Physics I”, North-Holland, Amsterdam, pp. 335– 398. Vušković, L., Zuo, M., Shen, G.F., Stumpf, B., Bederson, B. (1989). Scattering of electrons by alkalihalide molecules: LiBr and CsCl. Phys. Rev. A 40, 133–149. Weiner, J., Masnou-Seeuws, F., Giusti-Suzor, A. (1989). Associative ionization: Experiments, potentials, and dynamics. Adv. Atom. Mol. Opt. Phys. 26, 209–296.

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APPROACHES TO PERFECT/COMPLETE SCATTERING EXPERIMENTS IN ATOMIC AND MOLECULAR PHYSICS H. KLEINPOPPEN1,2,* , B. LOHMANN3,2 , A. GRUM-GRZHIMAILO4 and U. BECKER2 1 Atomic & Molecular Physics, University of Stirling, Stirling FK9 4LA, Scotland 2 Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin/Dahlem,

Germany 3 Westfälische Wilhelms-Universität Münster, Institut für Theoretische Physik, Wilhelm-Klemm-Str. 9,

D-48149 Münster, Germany 4 Institute of Nuclear Physics, Moscow State University, Moscow 119992, Russia

1. Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Analysis of Atomic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Classification of Atomic Collision Processes . . . . . . . . . . . . . . . . . . . 2.2. Examples of Approaches to Complete/Perfect Scattering Experiments . . . . . 2.3. Analysis of Spin and Coincidence Experiments Including Photon Polarization tion in Atomic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Spin Effects in Atomic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . 3. Angle and Spin Resolved Analysis of Resonantly Excited Auger Decay . . . . . . . 3.1. General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Theory of Angle and Spin Resolved Auger Processes . . . . . . . . . . . . . . 3.3. Numerical Calculation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Experimental Details and Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Analysis and Comparison of Theoretical and Experimental Data . . . . . . . . 3.6. Final Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Complete Experiments for Half-Collision; Auger Decay . . . . . . . . . . . . . . . 4.1. General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Approaches to Complete Experiment for Auger Decay . . . . . . . . . . . . . . 4.3. Examples of Complete Experiments . . . . . . . . . . . . . . . . . . . . . . . . 5. Analysis of Molecular Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Photoionization Dynamics: 4σ −1 Photoemission of NO . . . . . . . . . . . . . 6. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . Detec. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

472 474 474 476 477 488 496 496 500 504 505 506 512 512 512 515 516 519 519 520 523 527

* E-mail: [email protected]

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H. Kleinpoppen et al.

[1

Abstract So-called perfect or complete scattering experiments in atomic and molecular physics have extensively been carried out applying polarized interacting particles and/or coincidence techniques in order to detect the resulting particles. The description of scattering amplitudes and their relevant phases can be referred to Ben Bederson’s (New York) initial papers at the end of the sixties and beginning of the seventies. In his analysis of theoretical and experimental data, Fano’s work of 1957 resulted already in a physical picture, which related scattering amplitudes to state multipoles of excited atoms, e.g., orientation and alignment. The article starts with Perfect Scattering Experiments in atomic collisions, including special topics in the areas of electronic, ionic, atomic collisions and photoionization processes. These examples of special topics do not restrict the striking enormous consequences for the advancement of physics as initially introduced by Ben Bederson. Angle and spin resolved experiments and theory, and their analysis will be discussed with a particular emphasis on electron emission processes. This will be illustrated for the resonantly excited Auger decay of argon atoms. For the generation of the intermediate excited Rydberg state two different mechanisms, photoexcitation and electron impact excitation, will be considered. Main ideas of complete experiments for the Auger decay are presented together with selected results obtained in this comparatively new field of research. In a subsequent chapter, a selection of fundamental achievements connected to perfect scattering processes within molecules have been described.

1. Introductory Remarks Structural analysis of atoms and molecules in their ground and excited states belong to topics of atomic and molecular spectroscopy including actions of static or dynamical fields on free atoms and molecules. Atoms, molecules, ions, electrons, photons can also collide with each other which causes a large series of phenomena that we call atomic and molecular collision processes. Physically important properties of such collision processes are their development as a function of time, the various types of interactions between the colliding partners, the intensities of all these processes as a function of initial conditions, such as the relative energies of the colliding atomic particles, their mutual potential energies and their quantum numbers before and after the collision. The total entity of all these physical processes of atomic collisions is determined both by atomic structure as well as by parameters of the collision processes. While atomic and molecular structure is an important parameter in collision processes,

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atomic collision dynamics is the central problem of the physics of atomic collision processes. A huge number of possible atomic and molecular collision processes exists; any arbitrary atom or ion can collide with another atom or ion or with an electron or photon. The energies of the colliding atomic particles can be chosen arbitrarily; after the collision, the particles involved propagate in all possible directions. The manifold character of collisions requires a classification and ordering system based upon physical processes; these will be described up to a certain point. A more complete description of atomic and molecular collision processes can be found in the specialized literature. Since about the sixties, the field of research on atomic and molecular collisions has advanced through a variety of new development of experimental and theoretical methods; these have led to a more detailed understanding of quantummechanical collision dynamics. In analogy with nuclear and elementary-particle physics, especially the technology of coincidence and spin experiments have contributed to the advancement of new knowledge of atomic and molecular collision physics. Applications of atomic and molecular collision processes are found in astrophysics, atmospheric physics, plasma physics, nuclear fusion physics and chemical reactions. We want to emphasize that this article in honor of Ben Bederson does not claim to represent an approach to review on Complete/Perfect Scattering Experiments; it only encloses a selection of topics of our personal choices. More detailed reviews are available and partly listed in the references, e.g., Andersen and Bartschat (2000); Becker and Crowe (2001); Hanne et al. (2003). The existing data provide a deep insight into physical mechanisms of atomic and molecular scattering processes. They reveal what types of processes and interactions occur or compete with each other in the collisional process. Scattering amplitudes and their phase differences and also atomic target parameters extracted from above type experiments have successfully been applied as most sensitive tests of modern collision theories. Coincidence and spin experiments do not, in selected cases, average or sum over atomic cross sections for various subprocesses or interactions of the collisional processes. Such coincidence and spin experiments resulting in collision amplitudes, phases and target parameters have been classified as third generation type of experiments going well beyond the more limited kind of information obtained from differential (second generation type of experiments) or total (first generation type of experiments) cross section measurements. In a detailed analysis of atomic and molecular collisions we are dividing the task into the various subparts, namely on electron–photon coincidence experiments, on atomic and electron spin experiments and comparisons between electron and positron scattering.

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Only recently a start has been made to combine spin and coincidence experiments; we will briefly refer to this newest development. We will reflect on angle and spin resolved experiments which yield more information about the electron emission process. In particular, information about the scattering phase differences can be obtained. This can be seen as a step toward a complete experiment, i.e., to determine all elements of the describing density matrix, as has been first requested by Bederson (1969a, 1969b). This will be demonstrated for the particular process of electron emission via so-called resonant Auger decay. We will discuss the example of resonantly excited Ar∗ (2p−1 3/2 4s1/2 )J =1 states and the subsequent L3 M2,3 M2,3 Auger emission in detail. For the generation of the intermediate excited Rydberg state two different mechanisms will be considered. The process of photoexcitation which reveals several advantages with respect to its theoretical description and, due to the availability of 3rd generation synchrotron beam sources, allows for a large variety of experimental approaches. As an alternative process electron impact excitation will be considered theoretically. Though, only a sparse number of experiments have been carried out applying this method, it yields information about the Auger process which cannot be accessed in a photoionization/excitation experiment. The angle and spin resolved studies of the Auger decay in a combination with detection of the polarization state of the residual ion leads to complete experiments for the Auger decay; i.e. all the complex decay amplitudes can be determined. We will discuss such experiments and present their first results for both resonant and normal Auger processes. Having the validity of a two-step model of the reaction as a necessary prerequisite, the complete experiments for the Auger decay provide a showcase for complete experiments for so-called half-collision processes. The last and most challenging task of performing quantum mechanically complete experiments is molecular photoionization. This is because the anisotropic molecular potential seen by the photoelectron causes an admixture of an unlimited number of outgoing partial waves. This process may be visualized for localized core electrons as partial wave mixing due to the intra-atomic–molecular scattering of the photoelectron on its way out.

2. Analysis of Atomic Collisions 2.1. C LASSIFICATION OF ATOMIC C OLLISION P ROCESSES An atomic collision process can be illustrated and classified by its geometry. An atomic particle A may collide with an atomic particle B. Particle A propagates in a collimated aligned beam of atoms, ions, electrons or photons. It may then hit particle B of an atomic target. Often the target B is also produced by a collimated beam of particles (crossed-beam technique).

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Further classifications can be obtained as follows: In elastic collision processes the energy states of the collision partners are unchanged, e.g. both collision partners remain in their ground states. According to the laws of mechanics, however, kinetic energy can be transferred between the colliding particles. It is common to include electron spin exchange in elastic processes. In inelastic processes the energy state of one or both colliding partners can be changed; for example, the kinetic energy of the projectile can be transferred into excitation energy of the target atom. During such processes, kinetic energy can also be used to excite both atoms (A∗ , B∗ ), i.e., or A + B −→ A∗ + B or −→ A + B∗ −→ A∗ + B∗ .

(1)

In reactive processes between primary collision partners, other forms of matter such as molecules, ionized particles or photons may be produced, e.g., A + B −→ AB + hν

(2)

A + B −→ An+ + Bm+ + (m + n)e− ,

(3)

or

or even exotic atoms, in a special reaction e+ + H −→ p + Ps.

(4)

Examples of decay reactions will be discussed in the following sections in various ways. In the above classifications we have assumed that the magnetic components mJ of the quantum number J of the states of the particles A and B are equally distributed statistically. In accordance with this distribution, the collision processes are characterized by averaging over all quantum states of the components mJ . However, because of recent progress in experimental techniques, studies of atomic collision processes can be carried out and analyzed in which the colliding partners A and B are initially in pure quantum-mechanical states. Initial proposals for such types of experiments, called complete or perfect, can be traced back to the pioneering papers by Fano (1957) and Bederson (1969a, 1969b). It follows from these papers, for example, that particles A and B may be in the quantum states |nA JA mA and |nB JB mB with the corresponding quantum numbers n, J , and m for the two particles. The interaction process of the collision with the particles in pure quantum states can, at least in principle, be described by a quantum-mechanical Hamilton operator H int , which is determined by the interaction potential between the colliding partners. As a consequence of the linearity of the Schrödinger equation, the total system of the particles after the collision will

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also be in pure quantum states. In other words, we can represent atomic collisions between atomic particles in pure quantum states as follows: |ψin = |A|B

−−−−−−−−−−−−−→ linear operator Hint

|ψout = |C|D · · · |K · · · .

(5)

Before the collision, the colliding particles are in the joint quantum state |ψin , after the collision the collisional products are in the state |ψout . If the state vector |ψout after the collision has been extracted from an appropriate experiment, it may be described by applying the quantum-mechanical superposition principle in the form fm |ψm , |ψout = (6) m

where |ψm are wave functions of possible substates of the state vector |ψout and fm are complex amplitudes associated with the collision process. The extraction of the state vector |ψout represents the maximum of information and knowledge that can be extracted from the experimental analysis of the collision process. Measurements of all the amplitudes associated with the collision process is equivalent to performing a complete experiment. On one hand, complete experiments on atomic and molecular collision processes require a high degree of experimental effort and special methods, which have only recently been successfully applied. On the other hand, they also require advanced theoretical models to describe the dynamics in a most detailed way.

2.2. E XAMPLES OF A PPROACHES TO C OMPLETE /P ERFECT S CATTERING E XPERIMENTS Correlation and coincidence experiments in electronic, atomic and molecular collisions (including photoionization and spin effects) require to determine fundamental quantum scattering amplitudes and phases or alternatively, irreducible tensor operators, state multipoles or statistical tensors and expectation values of angular momenta. The state of the art was comprehensively and efficiently reviewed by Andersen and Bartschat (2000) and by Williams (2000) relating the experimental data to atomic and molecular collision processes and theories. A typical density matrix ρ (Fig. 1) illustrates the state of the art for the specific case of the n = 2 excitation of hydrogen atoms from the ground state (Blum and Kleinpoppen, 1979): 0 = |n0 MS0 , p0 m0 −→ 1 = |n1 LML MS1 , p1 m1 ;

(7)

the corresponding scattering amplitude is f (1 , 0 ) while n0 and n1 denote the initial and final principal quantum numbers with the usual meaning of the quantum numbers; σ (LML ) = |fLML |2 is the differential cross section of levels

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APPROACHES TO SCATTERING EXPERIMENTS 

σ (00)

 ∗  ρ =  f11 f00  f10 f ∗ 00 ∗ f1−1 f00

∗ f00 f11

∗ f00 f10

σ (11) ∗ f10 f11 ∗ f1−1 f11

∗ f11 f10 σ (10) ∗ f1−1 f10

∗ f00 f1−1

477



∗   f11 f1−1  ∗  f10 f1−1 σ (1 − 1)

λ, χ , P , µ, β, O Col , ACol , TKQ F IG . 1. Density matrix for excitation of atomic hydrogen from the ground state to the n = 2 state. The quantities below the matrix are data extracted from a complete/perfect collision experiment, (Blum and Kleinpoppen, 1979; Kleinpoppen and Williams, 1980, see text).

with the quantum numbers LML averaged over all spins (the dependence on n1 , p1 , n0 , and p0 is suppressed). The quantities below the matrix are a selection of the typical quantities determined by complete experiments which are described in further subsections. The matrix is divided into four submatrices. The upper one consists only of one element, the differential cross section for the 2s state excitation. The quadratic submatrix contains all elements characterizing the 2p excitation with the definite angular momentum L = 1 (2p submatrix). Its diagonal elements are the differential cross sections for exciting the various magnetic sublevels, its off-diagonal elements characterize the coherence between the different magnetic substates. The remaining elements of the matrix characterize the interference between the 2s and 2p states. This fundamental matrix has been applied particularly in connection with beam–foil and atomic collision experiments.

2.3. A NALYSIS OF S PIN AND C OINCIDENCE E XPERIMENTS I NCLUDING P HOTON P OLARIZATION D ETECTION IN ATOMIC C OLLISIONS One of the most general scheme for a complete/perfect scattering experiment in atomic collision physics is indicated in Fig. 2: A polarized particle, e.g., an electron, collides with another polarized particle, e.g., an atom. In general, coincidence, spin and polarized photon analyzes will be required to approach a complete experiment. Since around the 60’s of the last century the ideal complete experiment has only been approached in certain steps. However, the way to the basic ideas of complete/perfect experiments for atomic collisions developed through the pioneering research from the Franck–Hertz experiment (Franck and Hertz, 1914), the Ramsauer–Townsend effect (Ramsauer, 1921; Townsend and Bailey, 1922), the Hanle effect (Hanle, 1924), electron scattering interference from electron angular distributions by Bullard and Massey (1931), Mott scattering (see, e.g., Mott and Massey, 1965), direct, exchange (Bederson,

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F IG . 2. A polarized beam of electrons (spin up) is colliding with a polarized beam of one-electron atoms (spin down). An electron–photon angu lar correlation experiment consists of the coincident detection of the scattered electron (including up–down spin analysis) and the polarization of the coincident photon emitted from an excited atom; it represents a complete/perfect experiment (Becker and Crowe, 2001; Kleinpoppen, 1971).

1969a, 1969b, 1970) and resonance scattering (Schulz, 1963). As pointed out by Andersen and Bartschat (2000), the pioneering quantum theories of electronimpact excitation of atoms by Oppenheimer (1927a, 1927b, 1928) and Penney (1932a, 1932b) in the 20’s and 30’s were restricted to the calculation of first-order cross sections; polarization effects of impact line radiation were seen already in experimental investigations (even the magnetic depolarization, called Hanle effect, discovered in 1924) the theoretical understanding of all these effects started only with the theory by Percival and Seaton (1958) and its first experimental verification on alkali resonance lines by Haffner et al. (1965) and Haffner and Kleinpoppen (1967). Historically first experimental investigations on the detailed analysis of scattering amplitudes started around the end of the 60’s and the beginning of the seventies. It also involved the applications of various fundamental papers by Bederson (1969a, 1969b, 1970), Blum and Kleinpoppen (1974), Fano (1957), Fano and Macek (1973), Kessler (1985), Kleinpoppen (1971). Comments on direct measurements of phases of quantum mechanical excitation amplitudes and state parameters in atomic collisions were published toward the second half and end of the 70’s (Kleinpoppen, 1976, 1980). On the theoretical side, appropriate formalisms for describing angular correlation and polarization phenomena in collisions were developed in the 50’s, for the needs of the theory of nuclear reactions, and were reviewed for example by

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Biedenharn and Rose (1953), Blatt and Biedenharn (1952), Devons and Goldfarb (1957), Fano (1957), Fano and Racah (1959) and Fergusson (1965). It included the development of such concepts as density matrix and statistical tensors of the angular momentum, efficiency tensors, irreducible tensor operators and amplitudes, and many others, which in the seventies started to be widely applied to atomic and molecular processes, as summarized in review papers and monographs such as those by Andersen et al. (1988, 1997), Andersen and Bartschat (2000), Balashov et al. (2000), Blum (1996), Fano and Macek (1973), Hertel and Stoll (1977), Kessler (1985), Zare (1988) and many others. While applications of the most sophisticated scheme in Fig. 2 require still major efforts for complete/perfect scattering experiments we list partial solutions to it. The special case of the 1 P1 state excitation can be analyzed in a most straightforward way (for helium and light two-electron atoms; Eminyan et al., 1973; Kleinpoppen, 1967, 1971). The in- and out-going state vectors are

|ψin = He 1 1 S0 |pe ,

1 1

|ψout = He P1 pe = ψ P1 pe = fML |n, L = 1, ML pe , (8) ML

0

12

1

= ψ P1

3

with the representation

1

ψ P1 = fML |n, L = 1, ML = f0 |1 0 + f1 |1 1 + f−1 |1 −1, (9) ML

for the amplitudes fML of the magnetic substates; expressing the relevant differential cross section σ (θe , E) in terms of the bracket 1 1 σ (θe , E) = ψ P1 |ψ P1 = |f0 |2 + 2|f1 |2 , (10) and the ratios σ0 |f0 |2 = , σ σ σ1 |f1 |2 =2 , 1−λ=2 σ σ λ=

and

f0 χ = arg , f1

(11) (12)

(13)

as the relative phase between f0 and f1 ; the angular correlation coincidence count rate for observing the photon in the scattering plane at an angle θγ and fixed

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F IG . 3. Electron–photon angular correlations from 81.2 eV electron-impact excitation of He(2 1 P1 ) at various electron scattering angles. The full curve is a least squares of the theoretical representation to the data; the broken curves are first Born approximation (data by Hollywood et al., 1979).

electron scattering angle becomes

Neγ = λ sin2 θγ + (1 − λ) cos2 θγ − 2 λ(1 − λ) cos θγ sin θγ cos χ. (14)

The last term represents an interference effect for the excitation of the magnetic substates. Figure 3 demonstrates measured electron–photon angular correlations from the He(2 1 P1 ) excitation at various scattering angles (Hollywood et al., 1979). Born’s approximation is only close to the experimental data at small scattering angles. Bethe (1933) already predicted that Born’s approximation remains finite only for a ML = 0 with respect to the momentum transfer vector as a selection rule; this means for the excitation/de-excitation 1 S0 → 1 P1 → 1 S0 that only the ML = 0 substate of the 1 P1 state is excited and the emitted photon radiation is that of a dipole oscillating parallel to the momentum transfer p = pi − pf .

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F IG . 4. Magnitude of the orientation vector of helium atoms excited to the 2 1 P1 state as function of the incident electron energy for various electron scattering angles. These data were obtained from the measurement of the angular correlation parameters λ and |χ| (Eminyan et al., 1973, 1974). The figure on the right-hand side shows experimental data at 80 eV electron impact energy; − − − theoretical results calculated from the distorted wave approximation of Madison and Shelton (1973) at 78 eV.

An alternative, equivalent interpretation to λ and χ data is based on the concept of atomic orientation and alignment from the electron–photon coincidences, e.g. the He(1 P1 ) excitation (Eminyan et al., 1974, 1975; Fano and Macek, 1973). The non-vanishing components of alignment and orientation are determined by the parameters λ and χ as follows: 3L2z − L2 = 12 (1 − 3λ), L(L + 1) 1/2 Lx Lz + Lz Lx = cos χ, = λ(1 − λ) L(L + 1)

Acol 0 = Acol 1+

Acol 2+ = col O1−

L2x − L2y

= λ − 1, L(L + 1) 1/2 Ly = sin χ. = − λ(1 − λ) L(L + 1)

(15)

Figures 4 and 5 demonstrate magnitudes of orientation and alignment data of the 2 1 P1 state as extracted from electron–photon coincidence data. We mention in addition that alignment quantities have also been determined in non-coincidence experiments: by the anisotropic angular emission of Auger electrons or the linear polarization of electromagnetic radiation (including x- and γ -rays, Mehlhorn, 1983). After the excitation of an atomic target its state can be represented by various magnetic substates of quantum numbers ML . The relevant distribution of these

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[2

F IG . 5. Alignment parameters as calculated from the experimental angular correlation parameters λ and |χ| of the 2 1 P1 state of helium for various electron scattering angles and an electron impact energy of 80 eV (Eminyan et al., 1975). Dashed curves are predictions according to the distorted wave approximation by Madison and Shelton (1973).

substates can be described by state multi poles (Blum and Kleinpoppen, 1979), statistical tensors (Balashov et al., 1984), or alignment tensors and an orientation vector (Fano and Macek, 1973). Excitation of P states can be particularly simply expressed by orbital angular momentum transfer which is related to the orientation as follows: During the collisional excitation the impinging electron can only transfer orbital angular momentum perpendicular to the scattering plane. The expectation value Ly = L⊥ of this transfer is the relative difference between the number of atoms with positive and negative orbital momentum and can be calculated from the λ and χ data for the 1 P1 excitation process: L⊥ = Ly =

N (Ly ) − N (Ly ) = −2 λ(1 − λ) sin χ. N (Ly ) + N (Ly )

(16)

According to Fano and Macek (1973) this orbital momentum transfer can be recol = L /L(L + 1) and the circular polarization lated to the orientation vector O1− y P3 emitted from the P → S photon emission, Ly = −P3 (the sign difference follows from traditional definitions in classical optics and atomic physics).

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F IG . 6. Electron charge distribution of a coherently excited 1 P1 state. The coordinate indices c, n and a refer to Andersen et al. (1988). In the following the amplitudes without indices correspond to the collision frame (see, e.g., Andersen et al., 1988, p. 9).

The angular and polarization correlation data of the electron–photon coincidences can also determine the electron angular distribution of excited states (socalled electron clouds): The relative charge distribution in the volume element dV √ ∗ is −eψ ψ dV , with a state vector ψ = f0 ψz − 2f1 ψx for 1 P1 excitation. Here, √ ψz = |1 0 and ψx = −1/ 2(|1 1 − |1 −1) denote the corresponding wavefunctions of the pz and px orbitals, respectively. The electron charge distribution depends on amplitudes f0 and f1 or λ and χ which can be calculated and graphically represented for a given electron impact energy and scattering angle. Figure 6 shows a typical picture of the shape of an electron cloud of the excited 1 P1 state; it corresponds to an alignment angle γ = θmin = 45◦ and L⊥ = 0.75h¯ (in h¯ units). The picture also demonstrates the possible repulsive and attractive interactions for atomic orientation by the collision process which has been discussed in detail by Kohmoto and Fano (1981). Figure 7 shows experimental data of the orbital angular momentum transfer L⊥ to the excited 2 1 P1 state and the corresponding alignment angle γ of Fig. 6. Beyer et al. (1982) introduced a concept of attractive (fA ) and repulsive (fR ) scattering and linked these to the amplitudes fML =0 and fML =±1 , and λ, χ, and

484


[2

F IG . 7. Top: Orbital angular momentum L⊥ transferred to the excited 2 1 P1 state of helium at 80 eV impact energy versus scattering angle θcol = θ . Data from Eminyan et al. (1973, 1974) and Hollywood et al. (1979). Broken line guides the eye, only. Bottom: The corresponding alignment angle γ = θmin of the excited electron cloud (see Fig. 6).

σ = |f0 |2 + 2|f1 |2 for light atoms as follows: √ 1 fR = √ f0 + i 2f1 , 2 √ 1 fA = √ f0 − i 2f1 , 2

(17)

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485

F IG . 8. Right–left electron scattering asymmetry as a function of the scattering angle for detecting right (•) or left (◦) hand circularly coincident polarized photons of the helium transition 3 1 P1 −→ 2 1 S0 at 80 eV. The full lines are fitted to the experimental data (Silim, 1985).

with

|fA |2 = 12 σ 1 + 2 λ(1 − λ) sin χ, |fR |2 = 12 σ 1 − λ(1 − λ) sin χ,

and

(18)

√ 2 λ(1 − λ) cos χ. (19) 2λ − 1 Based upon the scattering with repulsive/attractive dynamics for excitation of light atoms a simple classical grazing model by Kohmoto and Fano (1981) and Madison and Winters (1981) lead to the formulation of orienta tion propensity rules. Of particular interest with regard to the classical grazing model and the repulsive and attractive potentials in the electron–atom scattering is a dynamical right–left electron scattering asymmetry for detecting (Fig. 8) right- or tan(δR − δA ) =

486


[2

left-hand circularly polarized photons of the 3 1 P1 → 2 1 S0 helium transitions (Kleinpoppen, 1983, 1988; Silim et al., 1987) in coincidence with the scattered electrons. As pointed out (Herting et al., 2002; Herting and Hanne, 2003) the orienta tion propensity rules should hold for light atoms described in the LS coupling scheme. However, for atomic orientation by polarized electron impact excitation of heavy atoms (such as Hg(6s2 ) 1 S0 → (6s6p) 3 P1 ) a non-classical interference due to intermediate coupling within the excited state obstructs the interpretation of the orientation propensity rule. The above correlation or deviation from the amplitude relations fR /fA and fML =0 /fML =1 may help further clarification of the problems with regard to the validity of the orientation propensity rules. The He(1 S0 → 1 P1 ) electron impact process at a typical energy of 50 eV appears to be well understood in connection with detailed electron-impact coherence parameters; they serve as most sensitive tests of theoretical models (see, e.g., Fig. 7.13 in Andersen and Bartschat, 2000). The agreement between various experimental data sets, such as differential cross section, orbital momentum transfer, photon polarizations, and charge cloud orientation, on the one hand, and theoretical predictions from R-matrix theory with pseudo-states (Bartschat et al., 1996) and the convergent close-coupling (CCC) theory (Fursa and Bray, 1995), on the other hand, are most impressive. We like to draw attention to various reference books (e.g., Andersen and Bartschat, 2000; Balashov et al., 2000) and papers (e.g., Andersen et al., 1988) for theoretical and experimental λ, χ, and σ data, they represented so called complete data in atomic collision physics for the first time (Eminyan et al., 1973). The obvious move to investigate the electron impact excitation of atomic hydrogen is complicated by the appearance of two sets of amplitudes for singlet (f s ) and triplet (f t ) scattering for anti-parallel and parallel spins of the projectile electron and the atomic hydrogen electron. Neglecting an overall phase, we need to determine 7 independent parameters for a complete experiment with polarized electrons and polarized hydrogen atoms for the 1s → 2p → 1s excitation/deexcitation. For such experiments we refer to the partial complete experiments by Yalim et al. (1999) on the 2P state excitation of atomic hydrogen using the scattered electron decay photon angular correlation technique at 54.4 eV with the parameters λ = σ0 /σ and R = Re(a1c a0c )/σ as defined by Morgan and McDowell (1975) with the collision frame amplitudes (z in the incident beam direction, see Fig. 6). Data of Yalim et al. (1999) are shown in Fig. 9 in comparison to other experimental measurements and theories. By using an optical pumping process in laser excited atoms details of the scattering dynamics for electron scattering can also be obtained as first demonstrated by Hertel and Stoll (1974a, 1974b). This leads to a comparison of the above electron–photon coincidence experiment with the inverse de-excitation of a laser excited atom (Fig. 10). Many reviews and papers with this alternative method

2]


487

F IG . 9. Experimental parameters λ and R: • Yalim et al. (1999); Kleinpoppen and Williams (1980); O’Neil et al. (1998); Hood et al. (1979); theories: · · · van Wyngaarden and Walters (1986); − · − Madison et al. (1991); − − − Scholz et al. (1991); — Bray and Stelbovics (1992); − · ·− Wang et al. (1994).

have been published (e.g., Campbell et al., 1988): we mention the successful experiments for the super elastic scattering from alkali atoms where laser radiation was available (Karaganov et al., 2001).

488


[2

F IG . 10. Comparison of the electron–photon coincidence experiment (left) with the inverse deexcitation of a laser excited atom (right) (Hertel and Stoll, 1974a, 1974b; Hertel, 1976).

2.4. S PIN E FFECTS IN ATOMIC C OLLISIONS Spin effects in atomic collisions have been applied to first approaches in perfect scattering experiments (Bederson, 1969a, 1969b). The basic spin reactions are Coulomb direct and exchange processes for elastic electron–atom collisions including an interference effect: e(↑) + A(↓) −→ e(↑) + A(↓),

direct amplitude f, quantity observed |f |2 ,

e(↑) + A(↓) −→ e(↓) + A(↑),

exchange amplitude g, quantity observed |g|2 ,

e(↑) + A(↑) −→ e(↑) + A(↑),

interference amplitude f − g, quantity observed |f − g|2 .

(20)

Calibrating these reactions with the equivalent opposite spin directions gives the relevant differential cross section to σ (θ, E) =

1 2

|f |2 + |g|2 + |f − g|2 .

(21)

In a pioneering experiment by Collins et al. (1967) the atomic beam is spinpolarized (polarization PA ) and velocity selected by a Stern–Gerlach magnet and spin-analyzed after electron scattering with unpola rized electrons by a E–H gradient balance magnet. Neglecting any other types of interactions the spin polarization of the atom can only vary by an exchange process (see Fig. 11). The ratio of the detected beams of atoms with and without spin polarization of the

2]


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F IG . 11. Experimental data for σEX (E, θ)/σ (E, θ) of Collins et al. (1967) for electron scattering on polarized potassium atoms at an energy of 0.5 eV. Horizontal error bars indicate uncertainties in cos θ at three different angles.

atoms resulted in σEX (θ ) = σ (θ)

|g|2 1 2 2 (|f |

+ |g|2 + |f − g|2 )

=1−

PA . PA

(22)

A few years later a paper was published by Hils et al. (1972) on the measurement of the direct differential elastic scattering of unpolarized electrons on polarized potassium atoms (PA ). The scattering process polarizes the electrons and their degree of polarization (Pe ) was measured by Mott scattering. The relation between the scattering amplitude f (θ ) and the measured electron polarization is given by |f (θ )|2 P =1− e. σ (θ) PA

(23)

The data clearly demonstrated the direct scattering process (Fig. 12), although the accuracy of the measurement was limited. Measurements of exchange amplitudes have also been carried out by the usual electron–photon coincidence experiment with electron impact excitation of the 3 3 P state of helium (Silim, 1985; Silim et al., 1987) with emission of light at wavelength of 388.9 nm and the following process: e + He 1 1 S0 −→ He∗ 3 3 P0,1,2 + e −→ He∗ 2 3 S1 + e + hν. (24)

490


[2

F IG . 12. Experimental and theoretical data for |f (θ)|2 /σ (θ) at an energy of 3.3 eV for electron scattering by polarized potassium atoms versus the scattering angle. Experiment by Hils et al. (1972), theoretical values from the work of Karule and Peterkop (1973) (two-state close-coupling approximation) at 3 eV (continuous curve) and 4 eV (dashed curve). But also see Karule and Peterkop (1972).

Only exchange amplitudes g0,1 contribute to the excitation of the 3 P states so that a parameter λ=

|g0 |2 |g0 |2 , = σ |g0 |2 + 2|g1 |2

(25)

and a phase χ can be defined by g1 = |g1 | exp(iχ), g0 = |g0 | which is similar to the 1 P excitation of helium. Experimental data for excitation energy of 60 eV showed total polarization and coherence for the 3 3 P excitation by electron impact. In electron scattering by more heavy atoms spin–orbit interaction between the projectile electron and the target atom has to be taken into account; therefore many more amplitudes are necessary for describing the scattering. The complication due to a large number of amplitudes is reduced by using target atoms without a resulting total spin (spin-less atoms), as for example with rare gas atoms or two-electron atoms with opposite spins. Consequently, two spin reactions can be defined for the scattering of polarized electrons on spin-less atoms A: e(↑) + A −→ A + e(↑),

with amplitude h,

e(↑) + A −→ A + e(↓),

with amplitude k;

(26)

we denote, as before, the first process as a direct one with amplitude h and the second one as a spin-flip process with a phase difference γ1 − γ2 between them and refer to Fig. 13 for experimental and theoretical data. The modulus |h| of the

2]


491

F IG . 13. Moduli of amplitudes |h| and |k| and their phase differences for elastic scattering of polarized electrons on xenon atoms as a function of the scattering angle θ at 100 eV; experimental data points of Berger and Kessler (1986) as compared to various theories: − − − Haberland et al. (1986); · · · McEachran and Stauffer (1987); — Awe et al. (1983). The data for |h| and |k| are in units of Bohr’s radius a0 and calibrated to the differential cross section σ = |h|2 + |k|2 .

492


[2

direct scattering amplitude shows a distinctive diffraction structure due to dipole and exchange interactions and the superposition of partial waves of the scattered electrons of various angular momenta; the spin-flip amplitude k originates from spin–orbit interactions which are considerably smaller and primarily determined by the ( = 1) partial wave, a diffraction structure is hardly discernible. The aim in extending these experiments was applying spin polarized electrons in collisions with spin polarized atoms. First approaches at the end of the 70’s aimed at spin-asymmetry effects of ionization cross sections with spin polarized electrons and polarized target atoms (Alguard et al., 1977; Hils and Kleinpoppen, 1978; Hils et al., 1981, 1982). More recent experiments with spin polarized electrons (|Pe | = 0.65), and polarized cesium atoms (|PA | = 0.90) showed dramatic asymmetry effects in the differential cross section σ0 with unpolarized collision partners; based upon the theory of Burke and Mitchell (1974) the differential cross section for scattering of the spin polarized beams is given by ˆ + A2 (Pe n) ˆ − Ann (PA n)(P ˆ e n) ˆ σ = σ0 1 + A1 (PA n) (27) (Baum et al., 1999, 2002). Figure 14 shows impressive experimental and theoretical data for the differential elastic cross sections and the various spin asymmetry parameters A1 , A2 , and Ann of low-energy electron scattering from cesium atoms. The physical meaning, with respect to the reaction plane, of these spin-asymmetries are as follows: A1 and A2 correspond to spin-up, spin-down asymmetries in the differential cross section for scattering of unpolarized electrons on unpolarized atoms (A1 ) or polarized electrons on unpolarized atoms (A2 ); Ann represents an anti-parallel– parallel asymmetry. Data on differential cross sections and spin asymmetries have also been reported for spin polarized electron impact excitation of spin polarized cesium atoms (Baum et al., 2004). The optically allowed excitation (6s) 2 S1/2 −→ (6p) 2 P1/2,3/2 ,

(28)

and the optically forbidden transition (6s) 2 S1/2 −→ (5d) 2 D3/2,5/2 ,

(29)

were measured and data are compared to theories of the non-relativistic convergent close-coupling (CCC) and semi-relativistic R-matrix with pseudo-states (RMPS). Satisfactory agreements between experiments and theories were notable. By applying both, spin polarized projectile electrons and spin polarized hydrogen atoms, Flechter et al. (1985) were able to fully determine the various scattering processes for direct, exchange and interference, and also triplet- and singlet interactions. As before, with electron–potassium scattering we have σ = σdir + σint = 14 3|T |2 + |S|2 = 14 (3σT + σS ) = 12 σ ↑↑ + σ ↑↓ . (30)

2]


493

F IG . 14. (a) Differential cross section σ0 for scattering of electrons by cesium atoms (normalized to theory at θ = 90◦ , impact energy of 3 eV; • Baum et al. (1986); ◦ Gehenn and Reichert (1977)); and the asymmetries Ann (b), A2 (c), and A1 (d) are compared to theoretical predictions: Breit–Pauli R-matrix approaches BP8 (including after convolution with the experimental angular resolution of θ = 8.5◦ ), a relativistic Dirac 8-state R-matrix model (Dirac 8) and a non-relativistic convergent close coupling calculation (CCC). Many more such data for projectile energies from 5 to 25 eV by Baum et al. (2002) have been published.

The so-called spin asymmetry A is linked to the following expressions A=

σ ↑↓ − σ ↑↑ σint 1 = ↑↓ = , ↑↑ Pe PA σ +σ σ

(31)

with σ ↑↑ = σT = |f − g|2 ,

(32)

σ ↑↓ = 12 (σT + σS ) = |f |2 + |g|2 ,

(33)

and

and T = f − g and S = f + g as triplet and singlet scattering amplitudes, respectively; σint = |f ||g| cos φ, with the phase φ between f and g.

(34)

494


[2

F IG . 15. Polarization spin asymmetry A for the elastic 90◦ scattering on atomic hydrogen. Error bars: — experimental results of the Yale group (Flechter et al., 1985); crosses, solid and dashed curves: theoretical results from van Wyngaarden and Walters (1986).

Figure 15 gives an example of the spin asymmetry A for the elastic 90◦ scattering of polarized electrons on polarized hydrogen atoms (Flechter et al., 1985). Similar measurements, including theoretical predictions, were reported on different atoms by the Bielefeld group on lithium (Baum et al., 1986) and the NBS group on sodium atoms (McClelland et al., 1987). Since the first relevant experiment by Ehrhardt et al. (1969) the physics of lowenergy electron impact ionization with completely determined kinematics has had an outstanding development both, experimentally and theoretically, in atomic, molecular, cluster and surface physics (see, e.g., Ehrhardt and Morgan, 1994; Neudatchin et al., 1999; Weigold and McCarthy, 1999). The last two AIP Conference Proceedings (2001 and 2003) on (e, 2e) and double photoionization link the process of electron impact and photodouble ionization together, respectively

2]


495

F IG . 16. Kinematical and quantum mechanical variables of the (e, 2e) process: momenta of the continuum electrons p0 , p1 , p2 , their spin projections ms and angular momentum states of the atomic target and the residual ion (after Lower et al., 2004).

(Hanne et al., 2003; Madison and Schulz, 2001): e + A −→ A+ + 2e, hν + A −→ A++ + 2e.

(35)

In a recent paper by Lower et al. (2004), schematically, a quantum mechanically complete experiment, in addition to the kinematically complete experiment on the (e, 2e) process would require to determine a scheme of measurements as illustrated in Fig. 16. As pointed out by these authors the ultimate objective to determine experimentally these physical variables lies beyond the present technologies although significant progress has been made recently on the quantum states of the projectile electron and the target atom. Another type of quantum mechanically complete experiment is the analysis of photoionization of polarized atoms according to the reaction |ψin = |A|hν −→ Hlin. operator −→ |ψout = (36) ci A+ |e− i ; i

the photoprocess transfers the initial state into a pure, final state described by amplitudes and phases (Klar and Kleinpoppen, 1982). We particularly refer to the pioneer experiment by Siegel et al. (1983) with polarized metastable neon atoms. We also mention the photoionization of polarized atomic oxygen by Plotzke et al. (1996) and by Prümper (1998). Godehusen et al. (1998) reported a complete photoionization experiment with laser polarized (aligned or oriented) europium

496


[3

atoms; a review on such experimental data has been published by Sonntag and Zimmermann (1995). Quantum mechanically complete collision experiments with ions, atoms and projectiles have been approached; but the large amount of possibilities goes far beyond electron impact and photoabsorption processes. Accordingly, the large selectivity of possibilities is by far more with heavy particle impact. We restrict ourselves here, however, to the charge transfer process p + He −→ H(2p) + He+ −→ L + H(1s), α

(37)

by which the Lyman-α photons and the hydrogen atoms in the ground state are measured in coincidence; the impact directions of the protons and the detected hydrogen atoms define the reaction plane. The analysis of the coincident Lα photons and the hydrogen atoms can again be carried out by means of the amplitudes f0 and f1 for the magnetic substates with ML = 0 and ML = ±1; the fine structure splitting of the 2 2 P1/2,3/2 states leads to a depolarization of the Lα emission which can be accounted for. Accordingly, the description of the coincidence analysis can be carried out with the quantities σ , λ = |f0 |2 /σ , and the phase χ between f0 and f1 similar to the He(2p) excitation by electron impact (Hippler et al., 1987). We are pleased also to refer to the method of Momentum Imaging in Atomic Collisions which has dramatically been applied for many kinematically complete studies of three- and more particle atomic collision systems. The technique emerged form the work of Schmidt-Böcking and his students and colleagues at Frankfurt a. M. and is also named as COLTRIMS, i.e. Cold Target Ion Momentum Spectroscopy, (Cocke, 2004). The technique first applied to ion–atom collisions is now frequently used for a wide range of experimental areas including, both, charged particles and photonic collisions. Most recent and important reviews about COLTRIMS are published by: Dörner et al. (2001); Schmidt-Böcking et al. (2002); Ullrich et al. (1998, 2003).

3. Angle and Spin Resolved Analysis of Resonantly Excited Auger Decay 3.1. G ENERAL C ONSIDERATIONS During the last decade of the bygone century and still continuing in the present one, a number of theoretical and experimental investigations have been focused on the Auger emission after photoionization and photoexcitation, respectively (Armen et al., 2000; Grum-Grzhimailo et al., 1999; Hergenhahn et al., 1991; Kabachnik and Ueda, 2004; Kleiman et al., 1999a; Kuntze et al., 1993; Lagutin et al., 2003; Lohmann et al., 1993; Lohmann, 1999a; Mehlhorn, 2000;

3]


497

Schmidtke et al., 2001; Snell et al., 1999; Ueda et al., 1999; West et al., 1998, and refs. therein). For the profound and still ongoing work by the group of Helena and Seppo Aksela in this field, we just mention Aksela et al. (1996) and Aksela and Mursu (1996), related directly to this paper. This successful and exciting progress has been, as a matter of fact, mainly due to the availability of third generation synchrotron beam sources. Only recently, even more advances have been made toward complete experiments (e.g., Grum-Grzhimailo et al., 2001; Kabachnik, 2004; Lohmann et al., 2003; Meyer et al., 2001; O’Keeffe et al., 2003, 2004; Schmidtke et al., 2000, 2001; Ueda et al., 1999), where the data on the angular distribution and spin polarization of the Auger emission are of major importance; we refer to the subsequent Section 4 for a more detailed discussion. While this type of research, on one hand, improved the understanding of Auger emission, it, on the other hand, disregarded the fact that, in the beginning of angle and spin resolved experiments of Auger emission electron or even proton beam sources have been used. We refer here, for instance, to the first angle resolved experiments by Cleff and Mehlhorn (1971, 1974a, 1974b) and to the first angle and spin resolved Auger emission experiments by Hahn et al. (1985) and by Merz and Semke (1990). The advantage of observing the angular distribution and spin polarization of the emitted Auger electrons using photon beam techniques is that the number of parameters to determine is generally restricted by the dipole approximation (Kleiman et al., 1999a). Moreover, for the case of photoexcitation and applying the well established two-step model (Mehlhorn, 1990), alignment and orientation of the intermediate excited state become constant numbers. Such experiments and its predictions with respect to the spin polarization vector have been discussed extensively (Lohmann, 1999a, 1999b). On the other hand, the alignment and orientation parameters after electron impact excitation have been investigated in somewhat more detailed manner in the light of coincidence and non-coincidence experiments (e.g., Andersen et al., 1997; Srivastava et al., 1996a, 1996b) but in a rather limited manner in the context of studying the subsequent Auger emission processes (see, e.g., Feuerstein et al., 1999; Kaur and Srivastava, 1999; Mehlhorn, 1990; Theodosiou, 1987). However, even such studies have been confined to electron excitation from the more outer subvalence atomic shells, only and their subsequent decay via Auger emission has been to a final state with total angular momentum Jf = 0 in which case the angular distribution parameter of Auger emission becomes a constant number independent of the matrix elements. For both, the electron impact excitation, as well as the photoexcitation case, we will consider the showcase of the resonantly excited Ar∗ (2p−1 3/2 4s1/2 )J =1 L3 M2,3 M2,3 Auger decay. For example, the photoexcited resonant Auger emission is shown in Fig. 17 in contrast to the normal Auger decay.

498


[3

F IG . 17. Normal vs. resonant Auger process; after Lohmann et al. (2003).

In particular, we will investigate the angle and spin resolved resonant Auger emission of argon atoms after electron impact excitation of a deep inner shell which has been investigated by Lohmann et al. (2002), 1 e + Ar S0 −→Ar∗ 2p−1 3/2 4s1/2 J =1 + e −→ Ar+∗ 3p−2 (38) 4s +e . 1/2,3/2

1/2 Jf

Auger

In contrast to the deep inner shell photoexcitation case (see further down), the parameters of alignment and orientation are not independent of the excitation matrix elements (amplitudes). Moreover, they become functions of the electron impact energy. Even more important is the fact that, in some sense, the emission of an Auger electron after electron impact excitation can be seen as a special case of present (e, 2e) experiments (Balashov and Bodrenko, 1999, 2000; Paripás et al., 1997; Taouil et al., 1999), and particularly the work by Birgit Lohmann (1996). Thus, within the limit of the two-step model, the investigation of Auger emission experiments after electron impact excitation can yield complementary information on (e, 2e) experiments, too. We also discuss the photoexcited resonant argon Auger decay which has been recently investigated, both, experimentally and numerically by Lohmann et al. (2003),

3]


499

F IG . 18. The reaction plane and coordinate frame defined for synchrotron beam excitation (Lohmann et al., 2003).

1 γ + Ar S0 −→Ar∗ 2p−1 3/2 , 4s1/2 J =1 −→ Ar+∗ 3p−2 , 4s 3/2

1/2

LJ2S+1=2,4 =1/2,...,5/2 + eAuger .

(39)

The resonant Auger decay (39) has been studied extensively in the past by Aksela and Mursu (1996), Aksela et al. (1997), Chen (1993), de Gouw et al. (1995), Farhat et al. (1997), Hergenhahn et al. (1991), Tulkki et al. (1993), von Raven et al. (1990). Here, we particularly consider the spin polarization of the emitted Auger electrons which manifests itself via two different physical processes. The components of the spin polarization vector of Auger emission in the reaction plane, defined by the incoming synchrotron beam axis and the direction of Auger emission (see Fig. 18), can show a non-zero spin polarization due to spin polarization transfer by circularly polarized light of the incoming synchrotron beam (Blum et al., 1986; Kabachnik and Lee, 1989; Klar, 1980; Lohmann et al., 1993). The transferred spin polarization is generally large because of the asymmetric m-sublevel population generated by the circularly polarized light (Hergenhahn et al., 1999; Lohmann, 1999b; Lohmann and Kleiman, 2001; Lohmann et al., 1993; Müller et al., 1995; Schmidtke et al., 2001; Snell et al., 1996, 1999). On the other hand, the component of the spin polarization vector perpendicular to the reaction plane can have a non-vanishing spin polarization generated via dynamic effects during the Auger emission, i.e. interference between different partial waves emitted, induced by linearly polarized light. In principle, such a dynamic spin polarization should be possible even if the target and the photon beam are unpolarized (Blum et al., 1986; Kabachnik, 1981; Klar, 1980). The dynamic spin polarization shows higher values only if certain conditions concerning the number of contributing partial waves and their relative phase shifts

500


[3

are fulfilled (Hergenhahn et al., 1999; Lohmann, 1999a). Propensity rules for specific types of Auger transitions where a large dynamic spin polarization can be expected have been derived by Lohmann (1999a). We will discuss the theoretical and numerical aspects and the experimental setup for the measurement. We will compare the numerical to the experimental data for the case of the transferred spin polarization. For the dynamic spin polarization we will give a detailed analysis in terms of phase shift differences and partial decay widths. 3.2. T HEORY OF A NGLE AND S PIN R ESOLVED AUGER P ROCESSES 3.2.1. Auger decay following electron impact excitation Let us consider the following two-step process, e + A(J0 ) −→A∗ (J ) + e −→ A+∗ (J ) + e f Auger .

(40)

In the first step, after a primary electron impact excitation the exciting electron is not detected while another electron is excited from a deep inner shell into a Rydberg state. After a certain lifetime, this Rydberg state decays via resonant Auger decay and the emitted Auger electron is eventually detected. The validity of this approach has been proved in a variety of experiments (e.g., see the review by Mehlhorn (1990) and refs. therein). Due to the application of the two-step model, the set of dynamic parameters describing the excitation/scattering and the subsequent emission process, can be factorized into parameters of orientation and alignment of the total angular momentum J of the intermediate excited state A∗ , and angular distribution and spin polarization parameters of the Auger electrons, respectively. The alignment and orientation parameters contain the information about the electron impact excitation while the latter yield information about the Auger decay dynamics. In contrast to the case of photoexcitation the total set of parameters AKQ , describing the excitation–emission process, is not limited by the dipole approximation. Therefore, the maximum rank of irreducible statistical tensors, i.e. state multipoles, or in other words, measurable quantities is given by the general restriction K ≤ 2J , only (Blum, 1996), while the magnetic components obey |Q| ≤ K. For our specific case of electron impact excitation, or more generally for excitation processes with spin 1/2 particles, it has been shown that general selection rules further restrict the values of the magnetic components to |Q| ≤ 1 (Lohmann, 1984, 1998). However, even with this restriction, the general equations of angular distribution and spin polarization after electron impact excitation still remain rather complicated. Explicit expressions have been given by Lohmann (1998). Similar expressions for related cases of interest have been given by Balashov et al. (2000).

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501

Electron impact excitation can populate intermediate states with J > 1 which, within the limits of the dipole approximation, cannot be accessed via photoexcitation. For the considered case of deep inner shell excitation, we are focusing on J = 1 intermediate excited states in order to present the formalism. With respect to closed shell atoms, J0 = 0 → J = 1 excitations are the general case for resonant Auger processes. Therefore, quantities of a maximum rank of K = 2 can occur, only. In order to simplify the discussion we assume an excitation process with either longitudinally or unpolarized electrons for the remainder of this work, only. This simplification yields the advantage that the expressions for angular distribution and spin polarization remain the same as has been derived for the case of photoexcitation with circularly polarized light (Lohmann, 1999b, see further down). In particular, we obtain the angular distribution as I (θ) =

I0 1 + α2 A20 P2 (cos θ ) , 4π

(41)

where I0 denotes the total intensity, and P2 (cos θ ) is the second Legendre polynomial. The Cartesian components of the spin polarization vector, with respect to the helicity system of the emitted Auger electrons, i.e. z-axis kAuger (see Fig. 18), can be written as: px (θ ) =

ξ1 A10 sin θ , 1 + α2 A20 P2 (cos θ )

(42)

py (θ ) =

− 32 ξ2 A20 sin(2θ ) , 1 + α2 A20 P2 (cos θ )

(43)

pz (θ ) =

δ1 A10 cos θ . 1 + α2 A20 P2 (cos θ )

(44)

and

The dynamics of the electron impact excitation into the intermediate excited A∗ state can be generally described by a set of state multipoles or statistical tensors AKQ (see above). Particularly, A10 and A20 are known as orientation and alignment parameters. The Auger decay dynamics is described by the angular distribution parameter α2 and spin polarization parameters δ1 and ξ1 , referring to the transferred spin polarization, and the parameter ξ2 related to the dynamic spin polarization. From their structure Eqs. (41)–(44) are similar to relations obtained for the emission of photoelectrons. The orientation parameters A10 can be non-zero, only, if a longitudinally polarized electron beam is used for the primary excitation process whereas the alignment parameters A20 can be different from zero even for an unpolarized electron beam.

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[3

3.2.2. Photoexcited Auger decay The two-step model can be also applied for the case of Auger decay following a primary photoexcitation process. γ + A(J0 ) −→A∗ (J ) −→ A+∗ (J ) + e f Auger .

(45)

Again, the two-step model allows for a factorization of the angular distribution and spin polarization parameters while, additionally, the dipole approximation restricts the number of parameters, i.e. only parameters with rank K ≤ 2 contribute. This yields as general expression for the angular distribution: I0 1 + α2 A20 P2 (cos θ ) + 32 Re A22 sin2 θ . I (θ) = (46) 4π Note, that in contrast to the electron excitation case an additional tensor parameter A22 occurs which can be related to the deformation of the electronic charge cloud (see Fig. 6). Using the same coordinate frame as in the previous section, i.e. z-axis kAuger (see Fig. 18), we obtain the Cartesian components of the spin polarization vector within the helicity system of the emitted Auger electrons as: √ (ξ1 A10 + 6ξ2 Im A22 ) sin θ , px (θ ) = (47) √ 1 + α2 (A20 P2 (cos θ ) + 3/2 Re A22 sin2 θ ) √ − 32 ξ2 (A20 − 2/3 Re A22 ) sin 2θ , py (θ ) = (48) √ 1 + α2 (A20 P2 (cos θ ) + 3/2 Re A22 sin2 θ ) and δ1 A10 cos θ . pz (θ ) = (49) √ 1 + α2 (A20 P2 (cos θ ) + 3/2 Re A22 sin2 θ ) As discussed in the previous part, the set of parameters describing the dynamics of photoexcitation into the intermediate excited A∗ state is given by a set of state multipoles AKQ , and the Auger decay dynamics is described by the angular distribution parameter α2 and spin polarization parameters δ1 , ξ1 , and ξ2 referring to the transferred and dynamic spin polarization, respectively. Now, let us consider three cases of experimental interest, assuming a primary photoexcitation with a polarized synchrotron beam where the alignment and orientation parameters become constant numbers (Lohmann, 1998, 1999b). Defining the degree of polarization in the system of propagation of the photon beam, i.e. z-axis kphot , we express the photon polarization in terms of Stokes parameters Pi with i = 1, 2, 3. Assuming a 100% circularly polarized light, i.e. P3 = ±1 we get: A10 = ± 32 , (50) A20 = 12 and A22 = 0.

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Thus, the general equations for angular distribution and spin polarization can be reduced: α2 I0 1 + √ P2 (cos θ ) , I (θ ) = (51) 4π 2 √ ± 3ξ1 sin θ , px (θ ) = √ (52) 2 + α2 P2 (cos θ ) − 3 ξ2 sin 2θ py (θ ) = √ 2 (53) , 2 + α2 P2 (cos θ ) and

√ ± 3δ1 cos θ

. pz (θ ) = √ 2 + α2 P2 (cos θ )

(54)

For the second case, we assume a 100% linearly polarized light, i.e. P1 = 1, which yields for the alignment and orientation parameters: √ 3 1 A20 = 2 , Im A22 = 0 and Re A22 = − A10 = 0, . (55) 2 In this case, the general expressions for angular distribution and spin polarization are significantly reduced. We obtain: α2 I0 1 + √ 2P2 (cos θ ) − 1 , I (θ ) = (56) 4π 2 px (θ ) = pz (θ ) = 0, (57) and −3ξ2 sin 2θ py (θ ) = √ . 2 + α2 (2P2 (cos θ ) − 1)

(58)

At first sight, a surprising result yields our last case of discussion. Again, assuming a 100% linearly polarized light, but P1 = −1, the alignment and orientation parameters are: √ 3 1 . (59) A20 = 2 , Im A22 = 0 and Re A22 = A10 = 0, 2 In this case, the angular distribution becomes isotropic, i.e. independent of the emission angle: I0 α2 I (θ ) = (60) 1+ √ . 4π 2 Moreover, all components of the spin polarization vector disappear: px (θ ) = py (θ ) = pz (θ ) = 0.

(61)

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As has been shown by Kleiman et al. (1999b), this result can be explained by the highly symmetric excitation process involving three planes of symmetry and is a direct outcome of the dipole approximation. Here, any information about the origin of the incident photons vanish. Therefore, we only have to consider the direction of the electric field vector which oscillates perpendicular to the reaction plane. Thus, the system is invariant under reflection within all three planes, the x–y, the y–z and the x–z plane. Hence, all components of the spin polarization vector vanish. Note, that for the angular distribution, this case is different to the two previous ones with respect to the integrated intensity over the solid angle. Here, we still have a dependency on the total intensity I0 and the angular distribution parameter α2 . The effect of vanishing polarization of emitted electrons is well-known for the case of photoionization and has been widely used for the production of unpolarized electrons (Kleinpoppen, 1997). With respect to Auger emission, it could be used as a tool for producing 100% unpolarized, mono-energetic electrons. However, for the Auger emission, this effect has not been experimentally demonstrated until today and, only recently, has been theoretically investigated in more detail (Kleiman et al., 1999b).

3.3. N UMERICAL C ALCULATION M ETHODS An important point in obtaining numerical data is to apply a consistent model for the calculation of all parameters. For obtaining the numerical data we employ a relativistic distorted wave Born approximation (RDWA). Detailed information about the calculation of alignment and orientation parameters may be found e.g. in Andersen et al. (1997); Kaur and Srivastava (1999); Srivastava et al. (1996a, 1996b), and for the calculation of angular distribution and spin polarization parameters e.g. in Lohmann (1999b, 1998). We briefly review the methods used for the calculation of the relevant parameters for the resonant Auger decay, i.e. reactions (38) and (39). The calculation has been performed applying a relativistic distorted wave approach (RDWA) employing a relaxed orbital method. In particular, the bound state wavefunctions of the excited intermediate, and the ionized final state are constructed using the multi-configurational Dirac–Fock (MCDF) computer code developed by Grant and co-workers (Grant, 1970; Grant et al., 1980). Intermediate coupling has been taken into account. Eight configu ration state functions (CSF) have been included in the calculation of the final ionic state whereas a single configuration approach has been used for the intermediate state. The mixing coefficients have been calculated applying the average level calculation mode. For the calculation of the continuum wavefunctions electron exchange with the continuum has been accounted for. With this, the electron excitation, the Auger transition matrix elements, and the scattering phases are obtained for calculating the relevant angular distribution and spin polarization parameters, respectively.

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3.4. E XPERIMENTAL D ETAILS AND S ET- UP A detailed description of the experiment and the analysis for the resonant Auger decay!after photoexcitation, reaction (39), has been given by Snell et al. (2002). Here, we will give a short review of the experimental methods and set-up. The experiment has been performed at the Advanced Light Source (ALS) at Berkeley operating in the two bunch mode utilizing the elliptically polarizing undulator (EPU) at beam-line 4.0.2 (Young et al., 2001). The EPU has been set to deliver either 100% circularly or linearly polarized light at all photon energies used. The electron energy analysis has been performed using a time-of-flight (TOF) spectrometer, collecting electrons emitted at 45◦ with respect to the storage ring plane in the plane perpendicular to the photon propagation. A spherical Mott polarimeter of the Rice type, operated at 25 kV, mounted at the end of the TOF has been used to carry out the electron spin polarization analysis (Burnett et al., 1994; Snell et al., 2000). Figure 19 shows the experimental set-up for measuring the dynamic spin polarization; i.e. MCP1 and MCP2 perpendicular to the reaction plane. Measurement of the transferred spin polarization is achieved by rotating the MCP’s by 90◦ around the TOF-axis.

F IG . 19. Experimental set-up with undulator beam-line, TOF electron analyzer and Mott polarimeter (Lohmann et al., 2003).

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3.5. A NALYSIS AND C OMPARISON OF T HEORETICAL AND E XPERIMENTAL DATA 3.5.1. Angular distribution and spin polarization parameters The numerical results of Lohmann et al. (2003) for the relative intensities, angular distribution and spin polarization parameters are shown in Table I for the Ar∗ (2p−1 3/2 4s1/2 )J =1 L3 M2,3 M2,3 Auger transitions together with the numerical and experimental data for the transferred spin polarization. As can be seen, a large transferred spin polarization has been found in the experiment which is in good accordance with the numerical data. The numerical predictions for the resolved spin-up and spin-down partial intensities for excitation with circularly polarized light are shown in Fig. 20(a) illustrating the high degree of transferred spin polarization for the eight lines of the calculated Auger spectrum (Lohmann et al., 2003). Here, a Lorentz profile with a FWHM = 0.1 eV has been assumed for all lines. On the other hand, only small dynamic spin polarization has been found in accordance with the theoretical predictions in terms of propensity rules (Lohmann, 1999a). Therefore, the Auger decay of the Ar(2p → 4s) excited state can be considered as a show case for large transferred but almost vanishing dynamic spin polarization. The calculated spin-up and spin-down partial intensities are plotted in Fig. 20(b) illustrating the almost vanishing dynamic spin polarization. Table I The relative intensities, angular distribution and spin polarization parameters for the Ar∗ (2p−1 3/2 4s1/2 )J =1 L3 M2,3 M2,3 Auger multiplets (Lohmann et al., 2003). Comparison of calculational to experimental results. The transferred spin polarization ptrans has been obtained for the angle of detection θexp = 90◦ . Final statesa  4P 5/2  4 4P 3/2  P 4P 1/2

Int. I0 b

α2

213.74 213.63 213.56

5.8 11.3 6.2

−0.18 0.63 −1.12

−0.26 −1.00 0.15

−0.48 −0.09 −0.57

−0.012 −0.002 −0.005

−0.38

−0.30

213.20 213.07

19.4 7.9

0.70 0.62

−0.46 1.17

0.06 −0.35

0.000 −0.004

−0.10

−0.22

2D

211.48 211.47

6.5 32.7

−0.93 −0.22

−0.04 0.65

0.27 0.84

−0.228 0.048

0.82

0.78

2S

208.44

10.2

−0.71

0.41

0.82

0.000

0.80

0.80

2P 3/2 2 P 2P 1/2 2D 2D

5/2 3/2

2S 1/2

Ang. and spin pol. par. δ1 ξ1

ptrans Theor. Exp.

Energy (eV)

ξ2

a The states have been identified in the LSJ coupling according to (Tulkki et al., 1993). b The total intensity has been normalized to 100.

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F IG . 20. The calculated spin resolved intensities for excitation with (a) circularly and (b) linearly polarized light showing high degree of transferred spin polarization and almost no dynamic spin polarization for any of the Auger lines (Lohmann et al., 2003). The data are calculated for the experimental geometry, θexp = 90◦ , φexp = 45◦ (see Fig. 19).

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Table II The calculated relative phase shifts of the partial waves of the Ar∗ (2p3/2 → 4s1/2 ) L3 M2,3 M2,3 Auger transitions (Lohmann et al., 2003). Final state 4P 5/2 4P 3/2 4P 1/2 2P 3/2 2P 1/2 2D 5/2 2D 3/2 2S 1/2

Relative phases εp3/2 −εp1/2

εf5/2 −εp3/2

εf7/2 −εp3/2

εf7/2 −εf5/2

– −0.011 −0.011 −0.011 −0.011 – −0.011 −0.011

0.658 0.658 – 0.656 – 0.649 0.649 –

0.658 – – – – 0.649 – –

−0.0005 – – – – −0.0005 – –

Table III The calculated partial widths of the Ar∗ (2p3/2 → 4s1/2 ) L3 M2,3 M2,3 Auger transitions; after Lohmann et al. (2003). Relative partial widths (%) Final state

3 4

P 4s P

3 4 5/2

P 4s P

3 4 3/2

P 4s P1/2

3 2

P 4s P3/2

3 2

P 4s P1/2

1 2

D 4s D5/2

1 2

D 4s D

1 2 3/2

S 4s S1/2

εp1/2

εp3/2

εf5/2

εf7/2

– 44.75 68.66 10.40 84.30 – 60.13 0.00

99.93 55.25 31.34 89.59 15.70 50.62 39.18 100.00

0.01 0.00 – 0.00 – 3.95 0.69 –

0.06 – – – – 45.43 – –

However, a large ξ2 parameter for one of the fine structure components of the line has been found. This has been explained by Lohmann et al. (2003) inspecting two features of the dynamic spin polarization. First, the spin polarization parameter ξ2 is a function of the sine of the phase shift difference of the emitted partial waves, ξ2 ∼ f (sin ). I.e., a small relative phase shift between the partial waves automatically results in an almost vanishing dynamic spin polarization. The calculated relative phases of the emitted partial waves are given in Table II. Their analysis yields comparatively large phase shifts between the emitted εf and εp partial waves whereas the phase shift between partial waves having the same orbital angular momentum, e.g. εp3/2 −εp1/2 , has been found almost zero. This is because the latter can be generated via relativistic effects, only. Second, a large ξ2 parameter requires the Auger decay to proceed via at least two partial waves. Otherwise, we have ξ2 = 0 and thus zero dynamic spin polarization. This requires information about the partial decay widths of the fine structure states of the Auger lines (Table III). 2D

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Almost equal intensities for the εf and εp partial waves for the 2 D5/2 fine structure component have been found. This feature, together with the comparatively large phase shift eventually yields the calculated large dynamic spin polarization parameter ξ2 for the fine structure component. On the other hand, the 2 D3/2 fine structure component decays almost solely via the εp1/2,3/2 partial waves, having a small relativistic phase shift, only. This results in an almost vanishing dynamic spin polarization for the 2 D3/2 fine structure state. Unfortunately, the fine structure of the 2 D line is not fully resolved. The preliminary data of Lohmann et al. (2003), however, yield experimental evidence for a non-zero dynamic spin polarization which seemed to stem from the Auger decay to the (2p−2 3d) 2 D final state. Such excited states have not been included in their numerical approach. Using larger basis sets might be able to numerically reproduce the experimental values for the dynamic spin polarization. 3.5.2. Alignment and orientation parameters For the case of electron impact excitation, both, alignment and orientation parameters are functions of the electron impact transition matrix elements and therefore become dependent of the electron impact energy. Lohmann et al. (2002) calculated the primary excitation cross section as a function of the electron impact energy in order to identify energy ranges where a comparatively high cross section coincides with large values for the alignment and orientation parameters, respectively. Results for the cross section of the electronically excited (2p−1 3/2 4s1/2 )J =1 state of argon are plotted in Fig. 21(a) as a function of the electron impact energy. As can be seen, the cross section is comparatively large close to threshold (248.63 eV) from where it rapidly decreases to its minimum at an electron impact energy of ∼300 eV. Then, it continuously increases to a maximum even higher than its threshold value at an energy of ∼800 eV. For larger energies we find the cross section slowly decreasing. Orientation and alignment parameters A10 and A20 are shown in Figs. 21(b) and 21(c) (Lohmann et al., 2002). Considering the alignment parameter, for the √ case of photoexcitation its value remains constant at A = 1/ 2 whereas the 20 √ orientation parameter takes a value of A10 = 3/2. As can be seen from Fig. 21 this is no longer valid for the case of electron impact excitation. Moreover, both parameters vary over a wide range of electron impact energy. Though the orientation parameter varies as a function of electron impact energy and does even change sign at ∼300 eV, its magnitude does hardly exceed values of 10%. Close to threshold negative values can be found whereas at ∼400 eV the orientation takes its maximum of ∼11% after changing sign. For larger energies the orientation smoothly decreases to small values. For the calculation a 100% longitudinally polarized electron beam has been assumed. In contrast to the photoexcitation case, the alignment has always negative magnitude but shows a dramatic behavior. Close to threshold an alignment of ∼ −0.5

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F IG . 21. The cross section σ (a), the orientation parameters A10 (b), and alignment parameters A20 (c) for the electronically excited Ar∗ (2p−1 3/2 4s1/2 )J =1 state; data after Lohmann et al. (2002).

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has been found which slightly decreases to ∼ −0.4 at 300 eV electron impact energy. Then, it increases to a maximum value of ∼ −0.69 at ∼550 eV, from where it smoothly decreases to smaller numbers. The numerical results for the angular distribution and spin polarization parameters for the resonant Ar∗ (4s1/2 ) L3 M2,3 M2,3 Auger transition have been already shown in Table I and discussed in Section 3.5.1. The resonant Auger spectra have been discussed, e.g., by Aksela and Mursu (1996). Here we only mention the large angular distribution parameter α2 for the 4 P1/2 and the comparatively large dynamic spin polarization parameter ξ2 for the 2 D5/2 final states, respectively. For the considered type of experiment the dynamic spin polarization parameter can be accessed via observation of the py -component of the spin polarization vector, only. Inserting α2 and ξ2 values for the 2 D5/2 state into Eq. (43) one needs to identify a range of electron impact energy which yields a large alignment A20 in coincidence with a comparatively large cross section. The cross section gets its maximum around 800 eV where we still have an alignment of ∼ −0.60. At an energy of 500–600 eV the cross section is still large but we are closer to the maximum of the alignment value of −0.69. Inserting both values of A20 into Eq. (43) the py -component of the spin polarization vector has been plotted as a function of the Auger emission angle. As can be seen from Fig. 22, a maximum

F IG . 22. The py -component of the spin polarization vector for the resonant Ar∗ (4s1/2 ) L3 M2,3 M2,3 2 D5/2 final state Auger transi tion as a function of the Auger emission angle θ . Solid line: electron impact energy 550 eV; dotted line: electron impact energy 800 eV; from Lohmann et al. (2002).

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degree of spin polarization of ∼19% has been obtained for the first, and ∼22% for the latter alignment of the (2p−1 3/2 4s1/2 )J =1 excited argon state, respectively. 3.6. F INAL C OMMENTS We reviewed the photoexcited resonant Auger decay, using the example of the Ar∗ (2p−1 3/2 , 4s1/2 )J =1 L3 M2,3 M2,3 Auger state, which has been investigated, both, experimentally and numerically. Emphasis has been laid on the analysis of the spin polarization of the emitted Auger electrons, where a large transferred spin polarization has been numerically predicted for most lines of the spectrum. This has been well confirmed in the experiment. Only small dynamic spin polarization has been calculated which has been found in accordance with derived propensity rules. The discussed preliminary experimental data point towards large dynamic spin polarization for some lines of the spectrum where, however, more extended calculations utilizing larger basis sets are necessary in order to explain this and further features of the spectrum. We also discussed angle and spin resolved Auger emission after electron impact excitation where we focused on the theoretical results for alignment and orientation. In contrast to the previous case of photoexcitation, these parameters became energy dependent. Similar to the photoexcitation, the investigated Auger transition to the 2 D5/2 final sta te shows a dynamic spin polarization of ∼20% over a broad range of energy. The exact knowledge of these parameters is important for the interpretation of possible (e, 2e) experiments. This is due to the fact that the scattered and the Auger electrons are hardly to distinguish if emitted with almost equal energy. Therefore, such experiments should be carried out either as (e, 2e) experiments covering the full solid angle for obtaining the full set of parameters of alignment and orientation, or at certain angles and energies with an accuracy which allows for a secure deduction of the required parameters.

4. Complete Experiments for Half-Collision; Auger Decay 4.1. G ENERAL C ONSIDERATIONS Atomic or molecular processes of disintegration are sometimes referred to as halfcollision, because they represent only a final stage of an atomic reaction. The typical examples are Auger decay, A −→ A+ + eAuger ,

(62)

and molecular dissociation, (AB)+ −→ A+ + B.

(63)

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Similar to the full collision discussed in Section 2, one can set a task of complete experiment for the half-collision. Such a task is meaningful provided the disintegration of a system can be treated independently from its preceding excitation. In the following we consider the atomic Auger decay. The conventional model treats the Auger decay as a part of a two-step process (Fig. 17), as discussed by Mehlhorn (1990): Step 1: Excitation of the Auger state by photon or particle impact γ (e) + A(α0 J0 ) −→ A+ (αJ ) + e(2e);

(64)

Step 2: Decay of the Auger state A+ (αJ ) −→ A2+ (αf Jf ) + eAuger .

(65)

Here J0 , J , and Jf are the total angular momenta of the initial atomic state, the Auger state, and the final ionic state, respectively, with the corresponding sets of other quantum numbers α0 , α, and αf , specifying the states uniquely. Within this model the complete experiment on the Auger decay relates to step 2 and the purpose of the corresponding measurements is to find the absolute ratios and relative phases of the decay amplitudes into different channels in addition to the total decay width. Reactions (64) and (65) represent the normal Auger decay, where the hole is created after ionization by particle or photon impact in an inner atomic shell, with the subsequent Auger decay leading to a doubly charged ion. Similar, a resonant Auger decay can be considered (Armen et al., 2000; Piancastelli, 2000), which has been discussed in Section 3. Here, a hole is, for example, created by photoexcitation of the inner-shell electron with the energy tuned to the resonance: γ + A(α0 J0 ) −→ A∗ (αJ ) −→ A+ (αf Jf ) + eAuger .

(66)

The resonant Auger decay leads to singly charged ions. The two-step approximation is often appropriate especially if the data are integrated over the line in the Auger electron spectrum. On the far wings of the Auger line, the cross section of the two-step process drops down and the contribution from the direct branch leading to the same final state (direct double ionization for the normal Auger decay and direct single photoionization for the resonant Auger decay) cannot be neglected. Post-collision interaction (PCI) between the knocked out electron in reaction (64) and the final state of the system in reaction (65) is neglected in the two-step model. In the full collision the total angular momentum of the combined system of the atom and the scattering particle is not fixed, because the particle is usually characterized by a large number of partial waves. This makes the partialwave representation of the scattering amplitudes not practical for the purpose of complete scattering experiments and the representation of projections of angular momenta is used. The objects of complete scattering experiment therefore are

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angle-dependent scattering amplitudes of transitions between different magnetic substates in the combined atom and scattering particle system. Such an approach was a basis of the analysis in Section 2. In contrast, for the Auger decay (see reaction (65)), the total angular momentum J of the system is fixed. Then, the partial-wave representation of the Auger electron is convenient and complex decay amplitudes, VαJ →αf Jf ,j ≡ αf Jf , j ; Ji |H − E|αJ ≡ Vj exp(iδj ),

(67)

are becoming the objects of interest for complete experiments on the Auger decay. We denoted the orbital and total angular momenta of the Auger electron eAuger as and j . In this respect the complete experiment for Auger decay is close to complete experiments in photoionization (see, e.g., Becker, 1998; Cherepkov, 1979; Heinzmann, 1980; Kessler, 1981), where the incoming dipole photon carries only one unit of angular momentum. In Eq. (67), the transition operator H –E, where H and E are the total atomic Hamiltonian and the total atomic energy, in practical calculations often reduces to the Coulomb interaction between the atomic electrons. Assuming sharp states αJ and αf Jf , we introduced brief notations for the absolute values, Vj (Vj ≥ 0), and phases, δj , of the decay amplitudes. The number of independent amplitudes (67) governing the Auger decay is restricted and determined by the triangle rules, J + J f + j = 0,

+

1 + j = 0, 2

(68)

and the conservation of parity, π = (−1) πf ,

(69)

where π (πf ) is the parity of the αJ (αf Jf ) states, respectively. Introducing further approximations can reduce the number of the independent amplitudes, which have to be found in the complete experiment. Although studies of the Auger process have a long history, recently reviewed by Mehlhorn (2000), complete experiments for Auger decay as a task of determining the decay amplitudes (67) is much younger than the concepts of complete scattering and complete photoionization experiments, respectively. It was first formulated and analyzed theoretically by Kabachnik and Sazhina (1990), while first experimental results were achieved at the end of the 90’s for both normal (Grum-Grzhimailo et al., 1999) and resonant (Hergenhahn et al., 1999; Ueda et al., 1998, 1999; West et al., 1998) Auger processes. Recent advances in the problem of complete experiments for Auger decay were reviewed by Kabachnik (2004).

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4.2. A PPROACHES TO C OMPLETE E XPERIMENT FOR AUGER D ECAY Since the absolute Auger decay width can be measured independently, the main efforts in the complete experiment are directed to measurements of dimensionless parameters of the Auger decay (so-called intrinsic parameters) with subsequent extraction of the absolute ratios, Vj /V j , and the relative phases, δj − δ j , of the decay amplitudes (67). Measurable quantities include relative strength of Auger lines; anisotropy parameters of the angular distribution of Auger electrons and components of their spin polarization (for example, the parameters α2 , ξ1 , ξ2 , δ1 from Section 3.2); polarization and anisotropy of the angular distribution of secondary products of decay of the residual ionic state αf Jf , which carry information on polarization of the residual ion; angular correlations between the Auger electron and the secondary products. Expressions for the observable quantities contain bilinear combinations of the decay amplitudes (67). The general formulas as well as expressions for states with particular values of Ji , πi and Jf , πf can be found in the literature, for example, Balashov et al. (2000), Berezhko and Kabachnik (1977), Kabachnik and Sazhina (1984), and many others given in the references. Unlike the parameters of the angular distribution and spin-polarization of Auger electrons, the angle-integrated quantities are not related to the phase differences between the decay amplitudes into different channels and therefore allow for an extraction of the relative partial widths j / j , where j = 2πVj2 .

(70)

Despite the large variety of measurable quantities, a few factors make the complete experiment for the Auger decay very difficult. As has been found recently, the angular distribution of Auger electrons and their spin components are dependent (Kabachnik and Grum-Grzhimailo, 2001; Kabachnik and Sazhina, 2002; Schmidtke et al., 2000, 2001), which reduces the number of measurable independent parameters. For illustration, the intrinsic parameters α2 , ξ1 , ξ2 , δ1 , describing angular distribution and spin-polarization of the resonant Auger electrons, which have been discussed in Section 3.2 for the decay of J = 1 states, are interrelated by √ √ √ 1 2 + α2 + 3δ1 . 3(ξ1 )2 + (3ξ2 )2 = √ 1 − 2α2 (71) 2 One can check that the numbers in the four corresponding columns of Table I for a particular Auger transition in Ar up to numerical accuracy satisfy Eq. (71). As a result, investigations which solely rely on the detection of the Auger electron can constitute the complete experiment in the simplest case of two decay channels, only. Furthermore, in practice, parts of the necessary experiments are feasible and hard to perform, and/or parts of the quantities to be measured show a very small

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magnitude. For instance, measurements of spin polarization of Auger electrons and different kinds of coincidence studies are difficult because of the low counting rate; the spin component of the Auger electron perpendicular to the reaction plane, which is related to the dynamic spin polarization, is usually small (e.g., see Section 3.5.1); the spin polarization of the Auger state A+ (αJ ) in the normal Auger process (reactions (64)–(65)) is usually also small, but needs to be determined in order to find the intrinsic Auger decay parameters. Polarization of the residual ions is often disturbed by external depolarization effects due to radiative cascades and intra-atomic hyperfine interactions, which should be taken into account in the analysis. To enhance the counting rate, the resonant Auger process is often utilized for performing the complete experiment (Hergenhahn et al., 1999; O’Keeffe et al., 2003, 2004; Ueda et al., 1998, 1999; West et al., 1998). Figure 4 from O’Keeffe et al. (2003) gives the examples of accuracy, which can be reached in the complete experiment by modern facilities.

4.3. E XAMPLES OF C OMPLETE E XPERIMENTS The simplest case for complete experiment, two-channel Auger decay, occurs either if min(J, Jf ) = 1/2, is fulfilled, see Eq. (68), or if additional restrictions are imposed ad hoc on the dynamics of the Auger decay. Figure 23 shows results of complete experiments by O’Keeffe (2003, 2004) for the resonant Auger decay of the photoexcited Xe∗ (4d−1 5/2 6p; J = 1) d-hole state into different ionic fine+ 4 structure Xe (5p 6p; Jf = 1/2) states. The decay proceeds via two channels determined by the conservation of angular momentum and parity with ejection of

F IG . 23. Results of a complete experiment for the resonant Auger decay Xe∗ (4d−1 5/2 6p; J = 1)

→ Xe+ (5p4 (2S+1 LJ )6p[K]1/2 ) + eAug (s1/2 , d3/2 ) for three different final ionic states (Racah notation is used for the Xe+ ionic levels). Allowed values of the absolute ratio of the decay amplitudes into the s1/2 and d3/2 channels and their phase differences are represented by the black areas (see text). Results of the intermediate-coupling multi-configurational Dirac–Fock calculation by Fritzsche (O’Keeffe et al., 2004) are indicated by circles.

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F IG . 24. Results of a complete experiment for the Auger decay Na+ (2s2p6 4p 3 P) → 2+ Na (2s2 2p5 2 P) +eAug (s, d) (see text). Calculations by Zatsarinny (1995): Hartree–Fock ion core (triangle); correlated ionic core (cross); correlated ionic core with polarization potential (circle).

s1/2 and d3/2 Auger electrons. The absolute ratio of the decay amplitudes (area between two vertical lines) was determined from circular polarization of the final ion fluorescence induced by circularly polarized synchrotron radiation, while the angular anisotropy parameter α2 of the angular distribution, measured by Langer et al. (1996) and Aksela et al. (1996), provides relationship between the cosine of the relative phases and the absolute ratio of the amplitudes (area between the two other curves). The set of equations expressing the measured quantities in terms of the decay amplitudes for the resonant Auger decay of the Xe∗ (4d−1 5/2 6p; J = 1) state are presented by O’Keeffe et al. (2004). Further improvement of the theoretical model describing the decay amplitudes is however needed to achieve better agreement with the complete experiment. Displayed in Fig. 24 are results for the normal Auger decay of the 2s-hole state Na+ (2s2p6 4p 3 P), created by fast electron impact from laser-excited sodium atom (Grum-Grzhimailo et al., 1999, 2001). Here, the Auger decay reduces to the twochannel case with emission of s and d electrons, only, after turning to the LSJ approximation, which is valid for the sodium case with high accuracy. The absolute ratio of the amplitudes, Vs /Vd , was deduced from the relative line strength in the Auger fine-structure multiplet, while the angular anisotropy parameter α2 , as in the case of Xe, gave the second relation between the decay amplitudes. The calculations converge to the experiment when improving the theoretical description of the ionic core. The various approximations of wave-functions have almost no effect on the relative phase of the decay amplitudes. From this, it seems that the determination of the relative phases less stringently tests its theoretical value

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F IG . 25. The reduced parameter space for the relative phases of the Auger decay amplitudes for + 4 3 the resonant Auger decay Xe∗ (4d−1 5/2 6p; J = 1) → Xe (5p ( P2 )6p[1]3/2 )+eAug (s1/2 , d3/2 , d5/2 ) from O’Keeffe et al. (2004, see text). The result of the intermediate-coupling multi-configurational Dirac–Fock calculation is indicated by circle.

than the absolute ratio of the amplitudes. A possible reason for this weak dependence on the wave-function approximation is that the relative phase is a property integrated from the inner part of the potential of the atom to infinity whereas the moduli depend directly on the quality of wave-functions being active in the Auger decay. Note that more sophisticated approximations were used for calculations of the Auger decay amplitudes in sodium (Fig. 24) than of those in xenon (Fig. 23). In particular, close-coupling in the continuum and relaxation of the discrete wavefunctions were accounted for sodium. The complete experiment is becoming much more complicated if already three decay channels are involved and not less than four independent dynamical parameters are needed to extract two absolute ratios and two relative phases of the decay amplitudes. An example is the resonant Auger process in Xe discussed above, but with the final ionic fine-structure state with Jf = 3/2 (Fig. 25). The two absolute amplitude ratios, Vs1/2 : Vd3/2 : Vd5/2 = 6(3) : 16(3) : 78(5), were determined from circular and linear fluorescence polarization, induced by circularly and linearly polarized synchrotron radiation (O’Keeffe et al., 2004). This can be compared to the calculated ratios 0.3 : 19.3 : 80.4. The experimental ratios substituted into the formula for the angular anisotropy parameter α2 , measured by Langer et al. (1996) and Aksela et al. (1996), yield the reduced parameter space of the two relative phases, as displayed in Fig. 25. In order to reduce this space into a particular solution, one more independent experimental quantity is needed,

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e.g. one of the spin components of the Auger electron. Nevertheless, a further step can be performed already with existing data assuming that the relativistic splitting of the Auger electron d3/2 and d5/2 wave-functions is negligible. This assumption is confirmed by the negligible relative phase δd3/2 − δd5/2 obtained in the calculations (Fig. 25). Then, the parameter space in Fig. 25 further reduces to two points (with experimental inaccuracies), lying on the horizontal axis δd3/2 − δd5/2 = 0 with only the sign of the (non-relativistic) phase difference δs − δd being undetermined. The negative value of the relative phase is in excellent agreement with the theory. The above examples show that complete experiments for (half-collision) Auger decay processes are feasible. Such experiments can provide the most detailed information about the mechanisms of the Auger decay and the most comprehensive, up to the degree allowed by quantum mechanics, test of theory. Theoretical descriptions of the complete experiments require highly sophisticated models of the Auger decay.

5. Analysis of Molecular Collisions 5.1. BASIC C ONCEPTS In the so-called Born–Oppenheimer approximation movements of the electrons and nuclei in molecules are considered as independent from each other and the total wavefunction is (Fink, 1997) |ψmol = |ψel |ψvibr |ψrot .

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In analogy to the classification of atomic collisions (see Section 2.1) we approach molecular perfect/complete collisions by the transitions between pure quantum states |ψin → |ψout , i.e. |ψmol in = |ψel in |ψvibr in |ψrot in −−−−−−−−−−−−−→ |ψmol out interaction operator

= |ψel out |ψvibr out |ψrot out .

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While atomic collisions have, in part, already reached a state for satisfactory completeness (Section 2.2) we are far away from such requirements in molecular collision processes. However, investigations on polarization, alignment, and orientation studies in molecular collisions have already revealed important information (see, e.g., Andersen and Bartschat, 2000; Becker and Crowe, 2001) for future developments towards various fields of complete molecular collisions. Here, we like to restrict ourselves to photoionization of molecules.

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Collisions with molecules, surfaces and foils have recently resulted in important investigations on orientation and alignment which have been initially discussed in theoretical frameworks by Fano and Macek (1973), Greene and Zare (1983), Jakubovicz and Blum (1978), and Blum (1996). For example, the Stokes parameters, P1 and P2 describing the degree of linear polarization, of electron-impact excitation of X 1 g+ → c 1 u transitions in H2 and D2 molecules were measured by Becker et al. (1984) and Malcolm and McConkey (1979). Hegemann et al. (1993) determined the ratio of electron spin polarizations after and before electron collisions with a beam of NO molecules. A depolarization of the initial electron spin polarization was observed which is apparently much smaller in molecules compared to an atomic Na target. The reason for this is that the experiment averages over all directions of the molecular axis and sums over various rotational states; accordingly bigger effects may be expected with oriented molecules (Hanne, 1997). Although many examples on the analysis of molecular collisions in fundamental and applied physics should be quoted we restrict ourselves on a recent study of photoelectron–photoion angular correlations which provide a complete description of the 4σ −1 photoionization dynamics of the NO molecule. 5.2. P HOTOIONIZATION DYNAMICS : 4σ −1 P HOTOEMISSION OF NO In contrast to atoms, where the study of polarized species makes it possible to perform a complete photoionization experiment, (e.g., Godehusen et al., 1998; Klar and Kleinpoppen, 1982; Plotzke et al., 1996), in molecules this is instead of the orientation or more general the fixation of the molecule in space, a method which became known as photoionization of fixed-in-space molecules. It is based on angle resolved photoelectron–photoion coincidence methods (Golovin, 1997; Heiser et al., 1997; Lafosse et al., 2000; Landers et al., 2001; Motoki et al., 2000; Takahashi et al., 2000) exploiting the so called axial-recoil-approximation, that means if the dissociation of a molecule after photoionization is fast compared to its rotational time scale, one can derive the molecular axis orientation in space by measuring the momentum vectors of the molecular fragments (Zare, 1967). The formulas to derive a complete description of the molecular photoionization process, by the analysis of fixed-in-space molecules photoelectron angular distributions, were given almost 30 years ago by Dill (1976). Several publications consider the feasibility of complete experiments on the basis of these formulas, e.g., Cherepkov et al. (2000), and experimental approaches to implement this idea (Geßner et al., 1999; Hikosaka and Eland, 2000; Jahnke et al., 2002; Motoki et al., 2000, 2002; Shigemasa et al., 1998). The most recent articles (Jahnke et al., 2002; Motoki et al., 2002) emphasize the necessity of using linearly and circularly polarized light in order to derive all matrix elements and phase shifts including their

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signs. For the special case of rotationally resolved (1 + 1 ) resonance-enhanced multi-photon ionization (REMPI) measurements, the feasibility of a complete experiment was shown by Reid et al. (1992) and discussed by Leahy et al. (1992). The 4σ −1 photoemission of NO molecules results in the population of the c 3

state of NO+ which subsequently dissociates into a N+ (3 P) ion state and a neutral O(3 P) atom (Geßner et al., 2002): hν + NO X 2 −→NO+ 4σ −1 , c 3 ; ν = 0 + e −→ N+ 3 P + O3 P. (74) Angle resolved photoelectron–photoion coincidence experiments were carried out using linearly and circularly polarized light of opposite helicities at a photon radiation of 40.8 eV energy falling onto the NO molecules. Electrons and ions resulting from the photoionization process were either extracted in opposite directions by constant electric fields and detected by using time and position sensitive detectors (Takahashi et al., 2000) or alternatively by five independent time-offlight (TOF) analyzers operated coincidently with the ion detector in a pulsed mode (Becker, 2000). The 4σ −1 photoelectron angular distributions were measured for all orientations of the molecular axis of NO relative to the electric vector of the synchrotron radiation. The molecular axis and the electron emission direction were selected co-planar with the electric vector of the incoming light and its propagation direction. The N+ photoion angular distributions were measured relative to the emission direction of the 4σ −1 photoelectrons in the plane perpendicular to the light propagation direction. Figure 26 shows the results by Geßner et al. (2002) for the case of excitation using linearly polarized light. The excellent agreement between the data and the fit is shown by three showcase examples for excitation angles of 0◦ , 45◦ , and 90◦ . Figure 27 shows the N+ ion angular distributions relative to the photoelectron emission direction for the two different light helicities, and the circular dichroism in the angular distribution (CDAD) which is the difference between the intensities (Geßner et al., 2002). The solid curve in the CDAD graph represents the same set of dipole transition moments and phase shifts as the solid curves in Fig. 26. The dotted curve is the output of the multichannel Schwinger configuration interaction calculation (MCSCI) which is in qualitatively good agreement with the measurements. For the data analysis a parameterization was used for the angular correlations of the molecular orientation, the electron emission, and the state of the incoming light (linearly or circularly polarized). A unique set of complex dipole matrix elements with 13 independent parameters was obtained, including 7 dipole transition moments and 6 phase differences including their relative signs. These combined measurements with linearly and circularly polarized incoming light provided, for the first time, such a unique set of complex dipole matrix elements for the

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F IG . 26. 4σ −1 photoelectron angular distribution for a series of excitation geometries using linearly polarized light (Geßner et al., 2002). The maps show the data (left) and the fit (right), respectively. For further details, please refer to the colour figure on the colour plate insert.

F IG . 27. Left: N+ ion angular distributions relative to the 4σ −1 photoelectron emission direction using circularly polarized light of two opposite helicities. Solid line: E-vector rotates counterclockwise (positive helicity); dashed line: E-vector rotates clockwise (negative helicity) when facing the light beam. Right: CDAD of the N+ ion emission. Solid line: fit; dotted line: MCSCI (convoluted with experimental angular resolution), after Geßner et al. (2002).

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photoionization dynamics of a molecule. The measured data and derived dipole transition parameters show good agreement compared to the MCSCI calculations (Stratmann and Lucchese, 1995; Stratmann et al., 1996). A follow up experiment by the group of Danielle Dowek from LURE (Lebech et al., 2003) at the same photoemission line of NO at different photon energies revealed the strong variations in the dynamical behavior of the molecular photoemission process with photon energy. The comparison with theory shows again good agreement of the measured molecular frame photoelectron angular distributions (MFPADs) with the calculated ones shown for three different photon energies in Fig. 28. This example shows that complete experiments are possible even in such intriguing cases as molecular photoionization where in principal an infinite number of outgoing partial waves is possible due to the anisotropic potential seen by the photoelectron. This high number of partial waves is, however, in many cases limited to three or four partial waves which makes it possible to achieve a complete description by fitting the MFPAD’s to a superposition of such a limited number of partial waves including all their relative phase shifts with signs. The 4σ −1 photoemission of NO is a showcase example for this possibility in both respects, a thorough comparison with theory and the unique insight into the molecular photoionization dynamics.

6. Concluding Remarks Selectively, only various parts of the physics indicated in the title of the paper have illustratively been described. While in particle and nuclear physics coincidence experiments leading to correlations of certain physical quantities had already strongly been developed throughout the first part of the 20th century (Frauenfelder and Steffen, 1965) typical correlation and coincidence experiments in atomic and molecular physics began only in the second part of the last century. In particle and nuclear physics usually target densities are that of solid states, i.e. ρ ≈ 1023 /cm3 , while in atomic and molecular physics they have to be low enough to guarantee that the free path lengths of atoms and molecules are large compared to the dimensions of the apparatus; i.e. ρ 1010 /cm3 . However, in particle and nuclear physics typical cross sections for collisional interactions are generally smaller by orders of magnitudes compared to those in atomic and molecular physics. We regret to admit that we are not in the position to mention important subparts of our article: First and most we did not refer to Bederson’s W -parameter (Andersen et al., 1997; Goldstein et al., 1972) by which, in early pioneering experiments, fractional depolarizations of polarized excited atomic beams, resulting from decaying back to the ground state, were measured.

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F IG . 28. Computed molecular frame photoelectron angular distributions (MFPADs) utilizing a 17-channel close-coupling expansion for ionization leading to the c 3 state of NO+ at photon energies of 25.0 eV (a), 30.0 eV (b), and 40.0 eV (c) by Lebech et al. (2003). In all representations the molecular axis is vertical along z. The three geometries correspond to linearly polarized light parallel to the molecular axis (left); linearly polarized light perpendicular to the molecular axis (center); and left-hand circularly polarized light of positive helicity with the direction of propagation of light perpendicular to the molecular axis in the positive x direction (right).

As a selected example we also did not refer to Stepwise Electron and Laser Excitation of Atoms (Mac Gillivray and Standage, 1988; Wang et al., 1995) by which a combination with the electron–photon coincidence techniques was established. Other areas we missed out, e.g., there are Inner Shell Vacancy Productions (Wille and Hippler, 1986); Symmetry Properties and Conservations Laws for Collisional Excitation with Planar Symmetry (Andersen and Hertel, 1986); first de-

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termination of Rank 4 Multipoles by a polarized photon–photon coincidence technique (Mikosza et al., 1993); recent experimental and theoretical results showed that Electron Dichroism exists for Chiral Molecules containing Heavy Atoms (see, e.g., Hanne, 2000; Kessler, 1996); in this connection we draw attention to an electron spin effect detected in solid state physics, namely the effect observed by spin polarized electron energy loss spectroscopy with ultra-thin Co films on copper (001)-crystals (Vollmer et al., 2003); the connection of this solid state effect to electron spin effects in collision physics has still to be worked out. To interpret the different types of radiation spectra observed from NASA, ESA and the Hubble space telescope provides a database on cross sections for electron impact excitation of highly charged ions, electron–ion recombination, photoionization, charge transfers, and X-ray emission (Chutjian, 2004, see Fig. 29); alignment effects after autoionization decay of atomic states have been observed in ions (see, e.g., Zimmermann et al., 2000), furthermore, interference effects in the alignment and orientation in Kr II ion states have been analyzed (Lagutin et al., 2003); interestingly, the direction of electron emission from photoionized atoms can be controlled by varying the phase of the ionizing laser field (Paulus et al., 2003). We may sum up the field by mentioning the bi-annual XII International Symposium on Polarization and Correlation in Electronic and Atomic Collisions held together with the (e, 2e), Double Photoionization and Related Topics Symposium (Hanne et al., 2003) and the XXIII International Conference on Photonic, Electronic and Atomic Collisions (previously called ICPEAC, Schuch et al., 2004). While the first two ICPEAC conferences took place in the USA (1958 in New York, 1961 in Boulder, Colorado), the III ICPEAC conference was held at the University College, London in 1963 with Ben Bederson as member of the Organizing Committee. Approximately 400 scientists from more than 20 countries attended this meeting and about 130 papers were presented (McDowell, 1964). The Present State of the Study of Atomic Collisions was the Opening Talk by Sir Harrie Massey while P.G. Burke arranged an informal session on theoretical and W.L. Fite and M.F. Harrison on experimental techniques. At the London ICPEAC conference an international committee under the chairmanship of W.L. Fite had been set up to arrange the subsequent meetings 1965 and 1967. The V ICPEAC in Leningrad, 1967 had already about 320 papers listed in the conference proceedings. At the last ICPEAC conference 2003, in Stockholm, the XXIII one, the number of accepted Abstracts of Papers went up to 1025. It justifies the statement by H.A. Bethe in the APS-Book More Things in Heaven and Earth (edited by Bederson, 1999a, 1999b) that nearly all physics beyond spectroscopy depends on the analysis of collisions. This tendency of development for atomic collision physics could also be noticed at the Arnold Sommerfeld Centennial Memorial Meeting (Munich, 1968; Bopp and Kleinpoppen, 1969), by which first the Commemoration Speakers Bethe, Bopp, Ewald, Heisenberg, Hermann, Kastler, Laport, van der Weerden, and Welker praised Sommerfeld’s life and academic

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(a)

(b) F IG . 29. (a) A 19th century woodcut from “Civilization in Transition”, Princeton University (Jung, 1970); the title of this picture is The Spiritual Pilgrim Discovering Another World. (b) A part of the view of the universe by the Hubble Deep Field Space Telescope; hundreds of new galaxies are seen, some only formed 109 years after the Big Bang (kindly provided by Chutjian, 2004).

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highlights. A Symposium followed on Physics of the One- and Two-Electron Atoms. It was recognized that a kind of renaissance of research on atomic physics had taken place during about the second part of the 20th century. 29 invited papers were presented in sessions on Atomic Constants and Spectroscopy, 20 invited papers in sessions on Atomic Collision Processes, and 4 papers in the session on Applications on Astrophysics, i.e. partly both on atomic spectroscopy and collision physics. Ben Bederson (Bopp and Kleinpoppen, 1969, p. 642) talked about Summary on Recent Spin-Analysis of Electron Potassium Differential Cross Section Measurements. In 1975 at the Stirling Symposium on Electron and Photon Interactions with Atoms Bederson and Miller (1976) presented an updated summary on Spin Polarization in Electron–Atom Scattering; the Stirling Symposium was dedicated to honor Ugo Fano and a Festschrift for him was published (Kleinpoppen and McDowell, 1976). A large part of invited papers at the Stirling meeting were devoted to angular correlations, (e, 2e) and electron-spin effects, accordingly the meeting can be considered as a predecessor of the present bi-annual Symposia on Polarization and Correlation in Electronic and Atomic Collisions (XII one in 2003, Hanne et al., 2003). We also draw attention to complete experiments in intense laser fields, chemical effects in (e, 2e) processes, as well as (e, 2e) experiments from atomic solid state physics, determination of 16-pole moments, spin exchange in spin polarized atoms by ion impact, non-dipole vector correlations, Auger processes, spin asymmetry measurements, relativistic effects in polarized electron–atom collisions, quantum chaos, dissociative recombination, controlling wave packets, spin polarized metastable atomic hydrogen, and Paul trap resonances in He+ –He collisions. Nearly all these collision processes can be traced in the last two ICPEAC conference proceedings or their satellite meetings. We like to finish our remarks by referring to the impressive beginnings in approaches to complete experiments in molecular collision physics.

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REFLECTIONS ON TEACHING RICHARD E. COLLINS The University of Sydney, NSW 2006, Australia Dedication . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . 2. Recollections . . . . . . . . . . . 3. Characteristics of Great Teachers . 4. Characteristics of Great Teaching 5. Rewards of Teaching . . . . . . . 6. Who Should Teach? . . . . . . . . 7. Recognition of Excellent Teaching 8. Evaluation of Teaching . . . . . . 9. Assessment of Students . . . . . . 10. Conclusion . . . . . . . . . . . . . 11. Acknowledgement . . . . . . . . . 12. References . . . . . . . . . . . . .

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Abstract This paper discusses several issues relating to teaching, including the characteristics of outstanding teachers and excellent teaching, the rewards from teaching, the need for the best people to teach, the recognition of good teaching, and methods of assessment of teaching and of students. The paper is based on the author’s personal experiences at the hands of his teachers, and as a teacher himself.

Dedication Some years ago, my former student Zhou Xian sent me a letter containing the following: A Chinese phrase said “If your friend gave you a drop of water for helping you in your difficult time, you should pay back a river afterwards”. You gave me a river, and I have just given back a drop of water.

It is in the same spirit that I dedicate this paper to my own teachers, who taught me the joys of understanding. I owe them a debt of gratitude that I can never repay, except in small part by trying to pass on these joys to my own students. 535


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1. Introduction Much of our lives are consumed by teaching and learning. Everyone is a learner— we have learned everything that we know. Babies are learning machines, capable of absorbing vast amounts of information, apparently effortlessly. It almost appears that they learn automatically, without having to be taught. Yet even in their earliest years, the learning of children can be extended and enriched by giving them opportunities for new experiences, and exposing them to different stimuli. Obvious examples are the inherent ability of most children to become almost seamlessly multilingual, and to develop near-perfect musical pitch. All that it takes is the provision of the appropriate environment—everyone is also a teacher. Upon entering school, the teaching process becomes more formalized, regimented and obvious. This structured, institutionalized teaching continues, often for up to two or three decades, and is broadened through the extension of social interactions and a myriad of other experiences, both within and beyond school. Our subsequent lives continue to be enriched by the ongoing learning, often directly assisted by teachers and teaching aids, that enables us to develop and adapt as the world around us, and we ourselves, inevitably change. Like many other aspects of our lives, teaching is ubiquitous, and so much a part of our existence that it is often unnoticed, unappreciated, and unacknowledged. However, teaching plays such an essential role in the development of our intellectual and physical skills, and in the maintenance and evolution of our society, that it is arguably one of the most important, if not the most important, of all human endeavors. As pupils, we have all recognized that some teachers are very effective in getting their message across—learning under them seems easy, enjoyable, sometimes even almost unavoidable. A few teachers stick in our minds as truly excellent— people who inspired us, who touched us deeply, who perhaps changed our lives. What is it about these great teachers that cause them to stand above all the others? Why are these few remembered with such affection and gratitude? How would the education of our children be enhanced if many more of our undoubtedly good teachers were like the few that we recall as truly outstanding? Could society aspire to the ideal that all of our teachers should be great? In this paper, I touch lightly on a few of these questions. I make no pretense that my analysis, and the conclusions that I draw, are based on scholarly research, involving a well-planned and executed set of experiments. I have been passionate about teaching for most of my life—even before it became part of my job over three decades ago. My thoughts about this subject started to find coherent form during the recent writing of my life’s story [1] in which I gave considerable attention to my experiences as a student and a teacher. These ideas and opinions developed further as this paper took shape. Although I recognize that there is much of value and relevance to be found in the contributions of others in this field, I have deliberately not referenced any such work because I wanted the views

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that I express here to be personal. And that is what they are—unashamedly and unapologetically, no more and no less. Their justification rests only on my own experiences and observations, formed over a lifetime as a pupil and a teacher, and through countless discussions and interactions with my colleagues and my own students.

2. Recollections In this section I share with you some of my own experiences at the hands of my teachers, and during my own teaching. For me, all of my teachers that I mention here are truly outstanding. Even though I have also included some stories about my own teaching, I make no claim that this places me in the same class. It is up to others to judge that. All that I can say is that I tried. Although most of the interactions with my outstanding teachers were shared with fellow pupils, taken together my experiences are unique. Every individual has a similarly unique set of recollections about the excellent teaching that they have received. I hope that you who read about mine will take some time to reflect on your own. While the details will inevitably be quite different, I suspect that you will find much in common between the attributes of my excellent teachers and yours. My first science teacher in high school, Mr. Cullen, was one of those largerthan-life characters. His classes were spiced with enthusiasm, examples of the application of science, practical demonstrations, slightly off-color stories and religious homilies. Cullen taught us physics and chemistry in the context of the potential for these disciplines to benefit society. I think that my own views about such matters have their roots in his classes. Cullen was also responsible for one of the most effective pieces of learning that I have ever experienced. We had been given a homework problem that required the use of log tables for its solution. When our answers were handed back he had penalized us all by 2 marks for having presented the results to 4 significant figures—the number automatically generated by the log tables. It wasn’t that our answers were wrong—there were just too many numbers in them. To add insult to injury, Cullen had also scrawled on each script: Insane accuracy. None of this had been preceded by any discussion about errors, or significant figures—it came completely out of the blue. We argued bitterly with him. After all, what was the matter with a couple of extra figures? Cullen was completely unmoved and simply kept repeating: Insane accuracy. Modern educational philosophy would hold that positive reinforcement is the best way of getting your teaching message across. Cullen’s technique was anything but that. At least for me, however, his lesson could hardly have been more effective—I don’t think that I have ever since knowingly quoted the result of a calculation to other than the correct number of significant figures.

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I was very fortunate to have several more teachers in high school who stimulated my interest in mathematics and science and extended me to the limit of my abilities. Vince Durack, who taught me mathematics in my final two years at high school, was such a great teacher, although I remember him much more in the process that he caused to occur in our classes than for his presentation techniques. My mathematics classes included several talented boys and we were very competitive. We learned calculus, series, trigonometry, algebra, coordinate geometry and a whole range of clever manipulative skills that I have found useful throughout my professional life. In a typical lesson, Vince introduced a new mathematical procedure to us—perhaps a method of integration, or a technique for summing a series. We then went to the textbook, and commenced work on the set of problems based on this topic. We solved a few of these problems in class, and then competed with each other to come back the next day with the whole set finished. This went on for two years—so much so that, when I finally left high school, I felt that I could integrate anything that was integrable! My fondest memories of all my teachers are reserved for Percy Moss who taught me physics in my last two years at high school. Percy was at the end of his career but he remained enormously enthusiastic about his subject. Over his teaching life, he had built up an extensive stock of wonderful demonstrations with which he illustrated his lessons. I can still see him, a little sparrow of a man, standing on the bench at the front of our class, his arms outstretched wide, shouting, “What a beautiful experiment, boys!” I found it impossible to avoid being excited in such an environment. Percy’s enthusiasm for science, and physics in particular, touched me deeply and helped to shape my future life. It would be nice to be able to give many examples of excellent teaching during my undergraduate years at the University of Sydney. Alas, few of my lecturers stood out in this regard. In those days, most academics appeared to have little concern for their undergraduate students. Indeed, at a welcoming talk during our Orientation Week before we commenced our First Year studies, we were told that our lecturers wouldn’t care whether we attended their classes, that about a third of us would not make it into Second Year, and that even more would eventually fail to graduate. What subsequently occurred was consistent with these predictions. The lecturers delivered their lectures, usually in an uninterested and mind-numbingly boring way, and then disappeared. In fact, I don’t recall having a single, meaningful, one-on-one conversation with a member of academic staff during my first three years at University. Within this sea of mediocrity, one undergraduate teacher stood out for me— Mr. Broe, our First Year chemistry lecturer, who reminded me of my high school physics teacher, Percy Moss. Broe was also right at the end of his career, but he retained his enthusiasm for his subject and he really cared about our progress. As it happened, he had lectured to both of my parents at the University of Queensland a generation before. It is a measure of his sustained excellence that they also

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remembered him with affection as an outstanding young lecturer. Broe put on a chemistry spectacular for us in his last lecture of the year. I still recall my excitement as he dazzled us with a series of experiments—explosions, solutions that changed color in a seemingly random way, liquids that cascaded between different flasks, and more. This lecture served as the inspiration for my own Physics is Fun lectures that I developed many years later after I returned to academia, and gave for over two decades throughout Australia. I should also mention my brief interaction as an undergraduate student with Professor Harry Messel—one of those characters who leaves his mark on you, whether you like it or not. In 1954 at the tender age of 29, Harry had been appointed as a full Professor, and Head of the School of Physics, at the University of Sydney. In 1980 when I returned to the University as an academic, he became my boss until his retirement some years later. Harry almost single-handedly raised the profile of the whole science scene in Australia, from the way it was taught in schools, to Government policy. He influenced most things that he touched, including me. As an undergraduate, Harry gave me just one lecture, but I remember it clearly. This occurred without warning one day in my Second Year. Instead of our regular physics lecturer, Harry walked in and proceeded to harangue us at high volume and speed on many matters. The principal subject of his talk, however, or at least the bit that stuck in my mind, was Enthusiasm, which he spoke about with his characteristic Canadian accent. His message was simple, and was well illustrated by his actions. He held up a small piece of chalk between two fingers, and said, “Students, if your entooosiasm is this big, this is how far you will go”. He then spread his arms out wide, and continued, “But if your entooosiasm is this big. . .” And on and on. After I completed my first degree in physics, I left the university environment for a job in industry. I was fortunate to work for an extraordinary man, Dr. Lou Davies, who became my boss, my mentor, my teacher and my friend. Lou was one of the few physicists in Australia at that time to have left a secure job in government science to work in the commercial environment. Lou started a research laboratory in an Australian electronics company, and I was the first physicist that he employed in this lab. More than any other person, Lou touched my life, including many aspects of my intellectual and professional development. I wasn’t alone in this regard. Lou had a genuine concern and affection for all with whom he interacted, and he was always prepared to take time to listen and to offer encouragement and advice. His door was never closed. In addition, he had an encyclopedic knowledge of matters relating to physics, science in general, and the world at large, and he was happy to share what he knew with anybody who was prepared to listen. Although my interactions with Lou were in an environment far removed from the formality of the classroom, he too was a great teacher. The things I learned from Lou, and his care, concern and encouragement for me, were pivotal for my own professional and personal development.

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In the mid-1960s, I took leave of absence from my company to undertake a Ph.D. degree at New York University under the supervision of the man whom we honor in this volume—Benjamin Bederson. I consider Ben to be one of the very best teachers that I have known. I was fortunate to interact with him both in the context of my own research, and as a student in his graduate classes. As a supervisor, Ben had great concern for the welfare of all of his graduate students, and we knew that we could depend on him for guidance and advice whenever we needed it. Having said that, however, he did not shield us from the rigors and hardships that inevitably occur when undertaking cutting-edge research. He let us make our own mistakes and gave us the opportunity to fix them, but he was always there if we lost our way to give a gentle shove that would put us back on track. Toward the end of my time at NYU, and after an extraordinarily intense and extended period of work, I had managed to extract sufficient data from a recalcitrant experiment that, Ben assured me, would guarantee that my Ph.D. would ultimately be granted. However, another group then published data from a related experiment that were so much better than mine that Ben was forced to revise his opinion. He told me that I would have to go back and do much better. This involved a complete rebuild of the experiment, and essentially starting the data acquisition from scratch again. This must have been a very hard thing for him to do. However, it was exactly the right advice, and the rebuilt experiment delivered, to the benefit of science, and me personally. From my interactions with Ben, I learned the importance of the light and caring, but firm touch of a supervisor that enables his or her students to develop to the fullest of their potential, while making sure that they do not get lost or damaged on the way. By the time that I completed my Ph.D., Ben had ensured that I knew what was inside me. In addition to my interactions with Ben Bederson as my supervisor, I also took an evening Plasma Physics course that he taught. This was a really outstanding set of lectures. Ben presented virtually all of the material in two ways. Firstly, he derived the various results conventionally from the relevant equations with approximations where needed. After each derivation, he would then say something like, “Now let’s imagine what an electron would see in this situation”. He would then use physically insightful arguments to develop the same functional relationships between the various parameters that had been obtained from the more formal approach. In short, he taught us about the physics of the problems. Throughout my career, I have found this a very insightful and productive way of thinking. During my time as a graduate student, I held a part-time Instructorship position at NYU in order to earn enough money to support my family. Because I was also taking the normal graduate courses associated with my Ph.D. program, I often found myself simultaneously teaching and learning essentially the same material, albeit at different levels. This was enormously beneficial to my own development, and I would commend such an approach to any young person contemplating a career in science.

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Viewing the teacher–student interaction from the other side was a quite eye opening experience for me. For example, I clearly remember being astonished at how poorly my students did in their examinations. My initial reaction to their dismal efforts was one of scorn—I felt that these guys must be really dumb if they couldn’t understand what I was saying. My views mellowed rapidly, however. After all, it was my job to ensure that they did understand. Also, I started to wonder what my own exam scripts had been like. I believe that it would be a powerful learning experience for all teachers to be able to go back and see how they performed as a student. My own efforts were probably not much better than those of my students. Following the completion of my Ph.D., I spent a decade or so back in industry. I then returned to the world of teaching when I joined the New South Wales Institute of Technology in 1978. NSWIT required all new lecturers to undertake some training in teaching methods and techniques. I felt very negative about having to do this, and I even remember remarking to a colleague, “These people can’t tell me anything about teaching”. I was completely wrong, and was very surprised at how much I learned. For example, the simple idea of first determining Aims for a course, and then structuring the content to meet these Aims, was not something that I had ever thought about before, and it made a strong impact on me. I even continued to go to a series of optional sessions after the completion of the compulsory ones. The lesson, and there is always a lesson, is obvious. In 1980 I returned to the School of Physics at the University of Sydney as a member of academic staff. I was not too surprised to find that many of the attitudes toward students that I had previously experienced as an undergraduate were still there. However, there were also some important and obvious differences. A few of my colleagues were now carrying the torch for teaching—something that had not been common before. One of these teaching enthusiasts in my School was Ian Johnston, and I learned a great deal from him. Ian is truly passionate about his teaching, and has been responsible for many teaching innovations. In the 1970s, he was one of the first people within the university system in Australia to introduce course evaluations by students. These are now standard practice in virtually all university classes. Ian received one of the inaugural Awards for Excellence in Teaching at the University of Sydney, and justifiably so. One of Ian’s key teaching philosophies is simplicity, and his views on this matter greatly influenced the way that my own teaching developed over the subsequent years. Ian’s argument is quite straightforward—if the students don’t understand even the basic concepts in their lecture classes, there is little point in exposing them to much more complex things. In my teaching of several subjects over the subsequent two decades, I progressively simplified my approach, weeding out nonessential mathematics, replacing abstruse derivations with more insightful physical arguments, and concentrating on the physics of the situation rather than the analysis.

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This emphasis on simplicity is, I believe, enormously important in effective teaching. It had also been my experience that most of the physics that I have used as a professional research scientist, both in industry and in the academic environment, is at this basic level. I have actually formed the view that one can be a very competent and successful physicist even if the only physics that you know is what we teach in our First Year courses. These thoughts were undoubtedly also influenced by my experiences in Ben Bederson’s physically insightful Plasma Physics course. They led me to develop what I believe was my most successful teaching initiative—a senior level course that I called Energy Physics. The basic aim of this course was to show how elementary physics could be used to obtain at least a partial understanding of quite complex systems. Because of my personal research interests, I chose energy-related topics for discussion. The same type of course could be taught with virtually any scientific theme. To the best of my knowledge, Energy Physics was unlike any course that had been offered before within the School of Physics at the University of Sydney. Indeed, I have not heard anything like it described elsewhere. Initially, both the students and the academic staff regarded the course with some suspicion. If it just used simple physics, they argued, then surely it would only be suitable for the less highly achieving students. In fact, the first time that I taught Energy Physics, only one or two of my students came from the upper levels of the cohort. The turning point came when we had an extraordinarily talented group of senior students who decided to do the course. Students like that will very quickly vote with their feet if they are not being challenged, but they stayed with me. We had a most stimulating time together, and afterwords several of them told me that Energy Physics was the most demanding and interesting course of their degree program. I must also include in these recollections a few words about one of the greatest physics communicators that I have known—Professor Julius Sumner Miller. Julius had been around the University of Sydney for many years before I rejoined it in 1980. He was, and still is, internationally famous for his television lectures and his provocative Why is it so? In front of a class, Julius was perhaps the best presenter of physics that I ever saw. His technique was to provoke, embarrass and stimulate, rather than to encourage and explain. My fondest recollection of Julius was the last time that I ever saw him lecture. He was well over 70 years old, could stand unaided only with difficulty, and had lost much of his sight. Nevertheless, I watched spellbound as he held a packed lecture theater in his hand, presenting conundrums in physics, provoking responses, and insulting and teasing us all. The finale of his performance was the collapse of a 200 liter drum that had been boiling away on a stove in the corner throughout the lecture. (This dramatic demonstration of the strength of the forces due to atmospheric pressure is now banned because of the danger of the drum exploding!) Julius directed the lecture demonstration staff to seal the drum and then pour cold water over it, resulting in a spectacular and impressive collapse. This prompted a prolonged standing ovation

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from the staff and students who had been fortunate enough to witness the whole performance. I can still see Julius propped up against the bench at the front of the lecture theater, tears pouring down his cheeks in gratitude for this acknowledgment of his extraordinary talent. In hindsight, I think he probably knew that this was the last lecture that he would ever give. He was very unwell at the time, and died soon afterwords. I treasure having known Julius Sumner Miller. All of the people that I have discussed so far are, or were, well known to me. They taught me, or I interacted with them closely. I conclude these recollections with a few words about someone that I do not know personally, but whom I admire greatly—the great Australian jazz musician, James Morrison, who is famous around the world for his extraordinary talents. He is truly one of the musical giants of our time. What is much less well known about him is his passion for teaching, and for helping young musicians develop their skills. James spends a great deal of time participating in workshops and master classes with talented young players, guiding them along the path that he has walked with such distinction. I understand that he does this without payment—his rewards are in the achievements of his pupils. I have the highest admiration for such a person who, at the very pinnacle of his profession, is happy to spend time passing on to young people his experiences and skills. It is the mark of a really great teacher. So there are my data—a few memories about a handful of people who stimulated and excited me, who caused me to think about the way that I did things, and who changed my life. I will now use these data, supplemented with a little more detail where appropriate, to develop some opinions and ideas about teachers and teaching. In the following discussion, I frequently need to refer to the way that scientific and technical arguments are developed. In order to be brief and specific, I mostly do this in the context of my own discipline of physics. Obviously, the essential features of the arguments that I present can be translated into any discipline.

3. Characteristics of Great Teachers What are the attributes of my great teachers that set them apart from the others? What enabled them to have such an influence on me that I remember them with affection decades later? How were they able to impact so strongly on my mind that they changed my life? If I review, as a group, the great teachers who influenced me during my career, I conclude that they all possess several important attributes. Perhaps the most obvious common feature of my great teachers is their enthusiasm and passion. They are all very intense—about their work, their teaching, and life in general. Being with them is not relaxing. These are people who are excited about what they are doing with their lives, and they are keen to share that excitement with others. They also want their students to experience the beauty that they

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see in their subject, and to share the joy that they have found from developing a deep understanding of it. My friend and colleague, Ian Johnston, is a good example. Speaking about physics with him is like standing under a high-pressure water jet. His eyes are wide, his voice is raised, and he talks like a carpenter hammers in nails. In the presence of such a person, it is difficult to avoid having some of this enthusiasm rub off on you. You cannot help feeling that it would be good to be where he is. Exposure to people like this, both in the classroom and in our lives, almost forces us to develop some of that enthusiasm and passion. Nobody goes to sleep in the presence of such teachers! All of the teachers that I remember as outstanding also have a great love for what they are doing. They love their subject, and they love imparting knowledge about it to others. Teaching for them is not a chore—it is one of the most important, if not the most important thing that they do. They do not seek to minimize their teaching load when the allocations are being made—in fact they often ask for more. These people look forward to the teaching term starting, and are itching to be back in the classroom interacting with their students. And the students know it. They are the experts in telling when a teacher doesn’t want to be standing in front of them, and they react accordingly. But when they see a teacher who is obviously pleased to be there, they too feel comfortable about sharing their valuable time by participating actively in the learning experience. In addition to their love of teaching, I have observed that great teachers also have a strong affection for their students. They respect them as individuals and equals in the learning process, and they care about their progress. These teachers are always willing to listen to students’ problems, and pleased to hear about their successes. In dealing with the inevitably large classes in undergraduate university teaching, it is often all too easy to forget that the group is not a collective noun, but is actually made up of individuals who are at a very formative stage of their intellectual, emotional and physical development. Students need to be, and are entitled to be, treated with care, respect and courtesy. Great teachers show empathy with their students—they try to develop eye contact with each one of them, even in big groups. They make it clear that they are talking to each student as an individual, and that they care about every student’s development. Consistent with this care that they display for their students, great teachers also seem to have an inherent appreciation of the difficulties that students experience in their learning. It has been said that the very brightest people may not make the best teachers, because they found learning much easier than the average student. Certainly, having had to struggle during the learning process provides a good basis for understanding the learning difficulties experienced by others. It is my observation, however, that possessing an outstanding intellect does not preclude a person from being a great teacher. Ben Bederson is an obvious example. Another well known such person in my discipline is the late and great Richard Feynman. A Nobel laureate, and a man of the highest intellectual capabilities, Feynman was

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also famous for his innovative approaches to teaching, particularly with students in the early years at university. I like to recall a most revealing personal experience that was very useful in helping me to comprehend better the difficulties that students experience in learning physics. In the early 1980s, I tried to learn the Japanese language. I was quite unsuccessful. Not everybody’s mind is built the same way. The problems that my students experience in learning physics are directly analogous to my own inability to cram all those unfamiliar Japanese words into my uncooperative brain. My failure to obtain even a basic facility in this language was a most humbling learning exercise for me. I try to think about it whenever I have a tendency to become impatient with my students’ difficulties in comprehending an argument that I am attempting to get across, or their poor performances in examinations. Teaching is so much a part of our lives that it is very easy to lose sight of its importance. Particularly in the university environment, the need to fit in one’s teaching responsibilities with research, administration and all the other facets of an academic job often results in many things getting done only superficially. Yet the teaching part of our academic jobs carries with it an awesome responsibility. As teachers, we are charged with developing the minds of our young people and, through this, we play a key role in shaping the future of our society. This responsibility requires us to do our teaching, not just well, but to the absolute best of our ability. Great teachers never seem to lose sight of this. For them, the responsibility becomes a privilege, as indeed it is.

4. Characteristics of Great Teaching At its most fundamental level, teaching is an exercise in communication. Teachers have something that they wish to give to their students—information, an idea, a way of thinking. The first task for teachers is to ensure that their students are receptive to this knowledge. In order to do this, they need to get their attention, to engage them, and to encourage them to participate in the communications dialog. Viewed in this way, there are many similarities between the teaching process and the work of a stage performer. Indeed, I have often likened the role of a teacher to that of a comedian. As soon as a comedian comes on stage, he/she must engage with the audience and get their attention, then retain it for the duration of the performance so that they appreciate the punch lines of the stories. I have always admired the techniques that comedians use to grab an audience and hold them. In my teaching over the years, I have unashamedly borrowed many of their methods, and developed a few of my own. Having said that, however, the challenge for teachers is to go a lot further than the simple act of communication. Teachers seek an engagement with their students that extends far beyond the duration of the class, or the end-of-course

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examination. A good teacher tries to inspire his/her students to the point that they themselves want to learn more about the subject at their own initiative. Here the analogy with the comedian breaks down. Good teaching is much more than grabbing the attention of the students and communicating with them—it involves getting them to think, taking charge of their own learning and, ultimately, expanding their minds. Perhaps the most obvious characteristic of good teaching is that it involves the students in a true two-way interaction. We have all been the victims of teaching in which such involvement was non-existent—classes that consisted of little more than the teacher providing information and the students writing it into their notes. Attendance by students at such classes is basically a waste of their time, and they know it. The same outcomes can be achieved simply by consulting the textbook. Very little learning takes place in an environment like this. I have to say that most of my undergraduate classes were of this type. To be brutally honest with myself, so was much of my own early teaching! There are many ways of engaging students. Julius Sumner Miller was a master at this: “How many of you think it will go up?” And he would hold one hand high to encourage you to move. “How many of you think it will go down?” And his other hand would go up. “What’s the matter with the rest of you? You’re not thinking!” And we would squirm, and make sure we responded the next time he asked a question! Julius was certainly not warm and cuddly, and he never tried to present himself as Mr. Nice Guy, but everybody in his classes was turned on and participating! A further important characteristic of good teaching is that the students find it enjoyable. At the most basic level, if students are having a good time in a class, they will be more likely to attend the next one. If their classes are not a pleasant experience, there will be a tendency for the other undoubted attractions of their world to take precedence. Again, the analogy with the comedian is a good one. An audience that is enjoying a comedian’s performance will hang on to his/her words in order to make sure that they get the next joke. Students who are having fun will also be more attentive and engaged so that they can experience more of the same. In addition, without wanting to appear self-indulgent, if the students are enjoying themselves in your class, it is much more likely that you as a teacher will do so also. Teaching in this way is fun! Moreover, the positive feedback that you receive from your students will almost certainly help you to teach better. Another characteristic that I believe is very conducive to good teaching is simplicity, as practiced and preached so effectively by my colleague Ian Johnston. I well recall my experiences as a new graduate in the early 1960s at the monthly evening meetings of the Australian Institute of Physics. I used to attend these functions because they gave me the opportunity to meet with other physicists and discuss a range of issues. However, I found the after-dinner talks almost uniformly impenetrable. In hindsight, I formed the view that many speakers felt that

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their status would be raised if their presentations were so complex and abstruse that they could not be understood by most of the audience. Of course this was as absurd then as it is now, although the message never seemed to get through. In fact, the inverse is true. Truly great teachers can take complex and difficult issues and communicate them in a way that permits understanding up to the capabilities of their audience. Teaching undergraduate classes is no different. Particularly in physics, it is all too easy to hide behind the mathematics and to present material in the context of formal derivations. I know—I have done it too many times. It takes much more work, and is much better teaching, to bypass the mathematics and to explain the physics. My experiences as a graduate student in Ben Bederson’s Plasma Physics class showed me how effective this can be if it is done well. In this context, it is important not to equate simplicity with lack of challenge. As a teacher, it takes a lot of work to extract the essential features from a complex physical situation, and to apply elementary physics in a way that develops a useful understanding of the system. As a student, there is usually much more challenge involved in mastering a subtle conceptual issue than simply memorizing a derivation. However, it is precisely the process of understanding the physics, uncluttered by mathematical analysis, which often leads to new insights and concepts. My Energy Physics students confirmed to me the value of this approach to teaching when they wrote in their course assessments, “You made me think”. Most of the discussion in this section is concerned with strategies and approaches that make for good teaching, rather than detailed methods and techniques. There is one aspect of good teaching that I must mention, however, which borders on technique, even though it also has strategic implications—repetition. All too often, we present our lectures, and even design our courses and overall degree programs, on the basis of a logical development of the material: A leads to B, and C follows from this. We now use C to derive D . . . However formally correct such an approach may be, this is not the way that people learn. The first time that students meet a new word, or are exposed to an unfamiliar concept, they may not understand it at all. They may not even remember hearing it. On many occasions I sat in an undergraduate lecture and suddenly realized that an unfamiliar word was being repeated, but that I did not have any idea what that word meant. I later found that the lecturer had introduced and defined the word earlier, because I had written it in my lecture notes. However, it had not stuck in my mind on the way through. My participation in the class had thus been reduced to mere scribe. How beneficial it would have been if the lecturer had not erased that earlier bit from the board—a simple technique, but one that is hardly ever used. At a more global level, we usually design and teach our entire degree programs on the basis that the material in the prerequisite subjects is perfectly understood, without revision or repetition. One simply has to reflect on the students’ performances in examinations to see the flaws in this approach. I have observed that good teachers continually revisit new concepts and ideas during their individual lectures, and

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throughout their overall courses, so that they may be progressively reinforced in the students’ minds. Good teaching involves repetition. I should also make comment here on the necessity of tailoring the methods and content of one’s teaching to the audience. For example, Vince Durack’s way of teaching mathematics to my senior high school class would not have been at all effective with a less motivated and (dare I say it) less competent group of students. A good teacher is aware of the capabilities of his/her students, and of what they might reasonably be able to achieve, and presents the lessons accordingly. Good teaching is a two-way communications exercise. Another important feature of good teaching is that it impacts on the students— it affects them directly in a way that is relevant to them. In hindsight, it is clear to me that this was what my high school teacher Mr. Cullen was doing when he wrote Insane accuracy on our homework assignments and docked us 2 marks. We felt the loss of the marks keenly, even though this penalty was insignificant in the context of our overall course assessment. Nevertheless his methods forced home the lesson. Of course the impact does not need to have negative connotations— indeed it is probably better if it does not. (The elimination of physical impact, which seemed to be a core teaching technique when I was a boy, can only be regarded as a positive development!) Students will be similarly receptive to ideas if they relate to matters that are important to them, or if they lead to new insights that explain things with which they are familiar. The challenge for teachers is to identify those things that will excite the interest of their students: “What we are discussing here is important because . . .”. In this context, we physicists are particularly fortunate in the teaching of our subject, because so much of it can be illustrated by practical experiments. The use of demonstrations in physics lectures is an extremely effective way of helping students to visualize concepts, and to relate the science to a situation with which they are familiar from past experience. Indeed, so much of physics is concerned with the everyday world as we know it that it is often very easy to identify the linkages between the new science that is being taught, and the experiences that the students have previously had. Lecture demonstrations are a very powerful technique that aids in students’ understanding and learning of physics, and they should be used whenever possible. Having said this, however, it is important to reflect on whether it is the lecture demonstrations, the excellent overheads, or whatever, that makes the teaching great. Of course it isn’t. They are just part of the information stream—props, if you like. It is how the information is presented that makes it quality teaching. Some years ago, I gave two lectures to 140 talented high school students from many countries at one of the Professor Harry Messel International Science Schools. These Schools are held every two years by the Science Foundation for Physics within the University of Sydney. The first of these lectures was a continuous stream of interactive demonstrations, experiments, student involvement,

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laughter, barely controlled chaos, with the message almost subliminal—and they thought it was great! In the second lecture, I just stood there and talked. Not an overhead, not a chalk mark on the board, not a sound, except for one voice—and they thought it was great! Today’s obsession with technology as a pathway to good teaching, with developing “innovative teaching methods”, and “new ways of learning” is simply a distraction. Good teaching is not about the props—it’s not even all that much about the content. It’s about what it does to the students. To state that the adoption of new technology is an essential criterion for good teaching is plain bunk. As a final comment on the characteristics of great teaching, it is worth remarking that those who teach extremely well may not always have been able to do so. Certainly, some people possess a natural aptitude for teaching, although most teachers probably start out in much the same way that they themselves were taught. However, a little training and advice, particularly in the early days, will be enormously helpful in developing good teaching skills (as I learned to my chagrin). In addition, I have observed that those who achieve excellence as teachers evolve their methods and techniques. This evolution, which should occur throughout one’s entire teaching career, may be as simple as adopting the discipline after every teaching experience of noting the things that went well, or that were not so good, so that next time around these parts are respectively reinforced, or improved. It is also extremely useful to observe the teaching of others, both good and bad and, where relevant, to adopt or avoid their styles and methods in one’s own teaching. The importance of evolution of methods and content in achieving excellence in teaching cannot be over-emphasized.

5. Rewards of Teaching It is a sad fact of life, and perhaps an indictment of the society in which we live, that teaching is not generally regarded as a high status profession, and that teachers are usually not particularly well paid. Having said this, however, I believe that teaching offers rewards that are more rich and diverse than in almost any other calling. Although the discrepancy between the financial and non-financial rewards for teachers reflects very positively on the altruism of most teachers, I am sure that society would be better served if this difference were not so great. Perhaps the greatest rewards that I personally have received as a teacher are from observing the achievements of my students. It is a wonderful experience to watch your students grappling with challenging concepts, and to see their understanding grow. Some of my most satisfying moments as a teacher have occurred when a student who has been puzzling over something that I said has come back with a question that I could not answer. You can almost see the neurons in their brains linking up as they reason, comprehend, and then challenge. I used to quip

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that my Ph.D. students are ready to finish up when they finally shout at me, “No, Dick! You’re wrong!” To have been given the opportunity of participating in my students’ intellectual growth is, for me, a great joy and privilege. I also find it most satisfying to observe the ongoing development of my students long after they have left my care. This is particularly the case with my former graduate students, because I became very close to all of them, having worked with each for periods of some years. I maintain contact with many of these students, and it gives me great pleasure to observe how the flicker of interest that I helped to ignite has, in many cases, grown into a fire of passion and achievement. The rewards in this process are all the sweeter when, as happens more often than you might imagine, one of my former students comes back to me, reflects on my contribution to their development, and thanks me for it. Such moments are priceless. Another widely acknowledged reward from teaching relates to the benefits that accrue to one’s own understanding. For me personally, this first became evident during my teaching and learning experiences at NYU. The need to clarify one’s thinking about a subject to the point where it is possible to explain it to others is perhaps the most effective way of developing true understanding. I also find it interesting and humbling to observe that my understanding continues to be enhanced in so-called elementary physics courses, even after many years of teaching them. Over my long life, I have frequently thought about the things that lead to true happiness. I have formed the view that this is best achieved through the continuing attainment of challenging goals—understanding a piece of science, the successful completion of some research or a complex paper, the acquisition of a physical or mental skill. I feel very fortunate that I have also found similar happiness and satisfaction from the achievements of those whom I have helped in some way along the same pathways—my students. Such riches are the true rewards from teaching.

6. Who Should Teach? I have already remarked that teaching is one of the most important, if not the most important of all human endeavors. It follows that society as a whole should strive to ensure that teaching is done as well as possible. In order to do this, the best people should teach. The distinguished former Chancellor of the University of Sydney, Sir Hermann Black, used to make the same point in a slightly different way. I recall him saying that universities exist for three purposes: to accumulate knowledge, to create knowledge, and to impart knowledge. His thesis was that there should not be an “elite” within universities that specializes in research, and a “lesser” group that shoulders the bulk of the teaching responsibilities, particularly

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to students in their early years. He believed that all academics should be active in all three of these intellectual endeavors. I wholeheartedly agree with him. Regrettably, the situation in many universities is very different from this. It is a commonly held view among many academics that research is the most important and challenging part of their work, and that teaching is a lower priority task— just a necessary chore, to be disposed of with a minimum of time and effort. The widespread tendency for successful researchers to buyout from their teaching is evidence of this. Past promotions and employment policies in universities have served to reinforce this trend. Even today, despite the much greater importance given to teaching compared with when I was a student, these attitudes still remain strong. Even very worthwhile initiatives can diminish the quality of teaching within our universities. A few years ago, the Australian Government introduced the Federation Fellowship scheme to raise the research performance of the tertiary education sector. Federation Fellows are research-only appointments, and receive a salary that is about twice that of a full Professor in Australia. Needless to say, these Fellowships have become highly prestigious, and are much sought after. One of the aims of the scheme is to attract back to Australia some of our country’s brightest young scientists who are currently working overseas, and this has occurred. Obviously, many good people who are already teaching within the Australian university system also apply and, quite appropriately, some of these have received Fellowships. The research responsibilities of Federation Fellows clearly include some teaching through the supervision of graduate students. In general, however, they do not teach students who are in their earlier years. An adverse effect of the Federation Fellowship Scheme has thus been to remove some of the best young minds from the teaching of undergraduate students at our universities. This is a most unfortunate outcome from a very well intended initiative. In this context, I believe that there are real benefits to be gained if all researchonly staff also undertake some teaching. Post-docs are a good case in point. Although the jobs of these people are usually defined as being devoted entirely to research, it is very much in their interests to develop their teaching skills. Having had the responsibility of teaching undergraduate courses will be beneficial to their cases when they apply for full academic positions in the near future. Also, all scientists need good communication skills, even if they stay entirely within the research environment. In addition, the contributions of such people can only enrich the teaching program offered by university departments through the greater diversity of offerings to the students, and the easing of the often excessive teaching load of full-time academics. I personally disagree vehemently with the concept of giving research a higher priority than teaching, either deliberately through the planned buyout of teaching, or unintentionally as has occurred in the Federation Fellowship scheme. It seems to me that the very best minds are precisely the ones to which our stu-

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dents should be exposed—they deserve nothing less. Think about the benefits to those students who were fortunate to attend Richard Feynman’s lectures; or to the young folk who have had one-on-one personal interactions with my musical hero, James Morrison. It is also worth noting that having students attend recorded lectures from great teachers is a very poor substitute for the real thing. Such an environment lacks that most essential feature of quality learning—interaction. I am perfectly prepared to accept that recorded lectures, or equivalently computer-aided teaching, have their place in the scheme of things. I really believe, however, that the best learning takes place in an environment where minds meet. We must always offer our students such experiences, to the maximum extent possible. It should be emphasized that the benefits from having the best people do the teaching are not just to the students. When I rejoined the University of Sydney in 1980, student numbers in our final undergraduate (honors) classes were low, and had been that way for a long time. Another recently appointed Professor, Max Brennan, and I observed that it had been many years since anybody at our level had taught one of our First Year classes. Without reflecting negatively in any way on the quality of the staff who had been undertaking this responsibility, or on their teaching, we felt that this practice was sending the wrong message to our students. Max and I therefore asked to be given the opportunity to teach some of these classes. Three years later, when the first cohort of students that we taught reached their honors year, student numbers in that year increased by a factor of three. These higher numbers were maintained in subsequent years. Obviously this could have been a statistical fluctuation. However, a decade or so later professorial participation in First Year teaching had again declined, and we repeated the experiment with almost identical results. Ben Bederson told me that you should always do any experiment three times to be sure. In this case, however, twice seems to be enough. Students understand when they are being treated as important, and they respond accordingly.

7. Recognition of Excellent Teaching As noted above, the last two decades or so have seen a substantial improvement in recognition of the importance of teaching, particularly within universities. In some universities a part, albeit small, of budget allocations to individual departments is now determined on the basis of the quality of their teaching. I am frankly not all that keen on such a “stick” approach to improving the quality of teaching— I much prefer the “carrot” of positive recognition and celebration of excellence in teaching. Nevertheless, if this increases the quality of our teaching, I cannot object. (At my own University, the component of the budget allocation that supposedly rewards teaching quality is calculated on the basis of each department’s

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“scholarly activities” relating to teaching. Most such activities are defined as things like teaching qualifications, attendance at teaching courses, or publications on teaching. Very little of the so-called “Scholarship Index” at the University of Sydney is related to the quality of teaching. I regard this as a shameful cop-out. In the next section, I present my views on appropriate methods of evaluating the quality of teaching.) My northern hemisphere alma mater, New York University, introduced its Great Teacher Awards way back in 1959, a time when such things seemed to me as a student to be far removed from the consciousness of most teachers in my first University! The NYU Great Teacher Awards were made to teachers who embody “the spirit of excellence in teaching, as recognized by their students and alumni”. (There must have been some very far-sighted people at NYU in those days!) In 2001, the Great Teacher Awards were rolled into the NYU Distinguished Teaching Awards that had been established in the early 1980s. In a way, I regret the loss of the Great Teacher name—for me it epitomizes exactly what we should be recognizing in our teachers. The University of Sydney also established its own Awards for Excellence in Teaching in the 1980s. I am more proud of my own Award than of anything else in my professional career. One of the nice features of the University of Sydney scheme is that each year the awarders give presentations to which all academics are invited. I always enjoyed attending the performances of these teachers, and I learned a lot from what they said and did. In the late 1990s when I was Head of the School of Physics at this University, I had the pleasure of nominating two of my staff for their successful applications for these Awards. I believe that it is extremely important to establish explicit policies and procedures that acknowledge and reward excellence in teaching. In recent years, however, I have noticed a very regrettable trend, at least in the University of Sydney Awards for Excellence in Teaching. These days in the consideration of nominations for these Awards, much more emphasis seems to be put on the development of teaching process and techniques, rather than to the quality of the actual teaching. While not in any way wishing to diminish the importance of scholarly research on teaching or the necessity to develop new and better ways of teaching, I am strongly of the view that Awards for Excellence in Teaching, or Great Teacher Awards, call them what you will, should be for just that—great teaching.

8. Evaluation of Teaching There has been a lot of debate on the evaluation of the quality of teaching. I have written just one paper in the teaching field and it was on just this subject, in the context of teaching within universities [2]. (I am happy to concede that the

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arguments that I put there, and that I repeat here, may not be applicable for evaluation of the quality of teaching to much younger students, although I think that they are probably relevant to most high school teaching as well.) I believe that it is quite straightforward to determine whether university teaching is good or bad—just ask the students. In my opinion, the key to evaluation of teaching quality at universities is properly structured and administered course evaluation surveys. My arguments are really very simple. Students have spent their whole lives being taught. They have been subjected to a continuous stream of teaching and teachers since before they can remember. They have been variously enthused and inspired, or bored out of their brains, with all shades in between. They know when a teacher cares about their learning and welfare. They recognize that, in a class where the discipline is poor, this is usually caused by the teacher’s inability to gain the students’ attention, interest and respect. They are the experts about determining if they are getting good value from their teaching. They have seen it all, over and over again. And students have a vested interest in the quality of the teaching that they receive. With very few exceptions, they do not want to waste their time in classes that are not interesting to them, or are not productive for their development. They have better things to do with their lives. Of course, there is the old argument that some teaching may only be seen as being good in hindsight—that the students may not know enough to recognize whether teaching is good or not. Frankly, I think that this is rubbish, at least in the context of the reasonably mature students about whom I am writing. When I was a student, I had many teachers that I thought were good—and now I still think that they were good. And vice versa. And my current colleagues whom I think are great teachers—because they care, because they return assignments on time, because their doors are always open to their students, because they talk about teaching, because they love to teach—the students also think that they are great teachers. And vice versa. Those who try to complicate this issue also ask whether it is reasonable to give to the students the task of determining who are the good teachers. Will they tell the truth? No more and no less than you or me. They are, after all, responsible citizens. The students that I am talking about are adults by law. They have the vote. They are accountable for their own actions. I actually believe that we demean them by not giving credibility to their views. The response to my lone sortie into the teaching literature was quite interesting. Some of my colleagues who, like me, are passionate about teaching, loved it. The teaching theorists simply ignored it. I am intrigued that those who were responsible for the establishment of the New York University Great Teacher Awards nearly half a century ago had no problems about using the opinions of students evaluate the quality of teaching!

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9. Assessment of Students It will be obvious to anyone who reads this that I love to teach. There was one part of this activity, however, that I did not enjoy—marking examination scripts. I always found this a tedious, and at times depressing, chore. Particularly with large classes, marking exam scripts feels like wading through treacle (or how I imagine that feels!). The size of the pile of unmarked scripts seems to decrease far too slowly. To be sure, there are all sorts of dodges for speeding up the process, particularly in a quantitative discipline like physics. If the number in the answer is correct and the working looks OK, one can award full marks, and move on quickly to the next question. In addition, the students can usually make only a limited number of mistakes (although they never cease to surprise me!). Thus a short distance down into a mountain of scripts, an astute marker will have established a one-to-one correspondence between the numerical value of most wrong answers and the mistake that the student has made, so that a reduced mark can be allotted rapidly. I only ever taught one course in which I enjoyed marking the assessment papers—this was my Energy Physics offering. As I mentioned above, in this unconventional course I explored with the students ways of developing an understanding of complex systems using elementary physics. Consistent with the unusual nature of the course, I also made the assessments different from the norm. The way in which they were done is perhaps best illustrated by an assignment question that I used to set: Professor Snilloc retired from the University many years ago and lives in a mansion with his wife. He is quite mad in almost every way—the only thing that he always gets correct is the physics of what he is talking about. He likes to have a mug of tea (milk, no sugar) in bed each night before going to sleep in his room in the attic. He insists that the tea should be scalding hot, and his wife is always in trouble because, by the time that she brings it up from the kitchen downstairs, it has cooled off too much . . . What instructions does the mad professor give his wife to ensure that the tea gets to him as hot as possible? Explain the physics associated with each one, as quantitatively as you can. Estimate the temperature of the tea that he would drink if she does everything correctly.

I still remember the first time that I handed this assignment question to the students. They expressed consternation, because it was not what they had been used to—the problem was ill defined, the material needed to tackle it had not previously been covered in the lectures, and evidently there is no “correct” answer. As I pointed out to them, however, this is just what will occur when they work as practicing physicists. They accepted the challenge with great enthusiasm. Reading the students’ responses to my Professor Snilloc problem was an absolute joy. Over the years, I have received many memorable submissions, a couple of bits of which are recorded elsewhere [1].

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An added bonus from this type of assessment question is that it requires the students to present their answers in the form of prose. It is very beneficial for physics students to be required to structure their thoughts into logical arguments, with complete sentences and correct grammar. Most courses in our discipline do not pay much attention to the development of these skills. This contrasts sharply with the offerings in many other fields, particularly in the Arts faculties. We really should place more emphasis on such matters in our Science courses, because good writing and verbal skills are as essential for success in the scientific field as they are in any other. Despite my distaste for marking conventional examination papers, I always resisted the temptation to move toward multiple-choice examinations in my courses. Even though I admittedly used all sorts of short cuts in my marking, I actually believe that it is quite important to go through the tedious discipline of reading what students write. The students’ words provide a window to their minds. They enable us to understand better the difficulties that they are having in their learning so that we, in turn, can help them do better next time. I would have loved to find a better way of assessing the performance of my students, because this would have eliminated the only aspect of teaching that I disliked. Regrettably, I never did. However, it seems a small price to pay for having so much fun.

10. Conclusion I was delighted to be invited to contribute to this volume that honors Ben Bederson—a person who played a pivotal part in the development of my life. I chose to write about teaching for several reasons. Firstly, my scientific research interests over the past two decades are so far removed from Ben’s that I felt that a technical contribution on them would be out of place. Secondly, I am passionate about teaching, and there were some things about this subject that I wanted to say, and that I had not previously managed to get off my chest, let alone articulate to my satisfaction. Finally, writing the paper has given me the opportunity to honor all of my teachers, including Ben, for the things that they have given to me that enabled my life to be what it is. For me, this alone justifies my decision to write this paper. As I said at the beginning, the thoughts that I have presented here are based on my own observations and experiences—no more and no less. That I have not referenced any of the prior teaching literature, other than my own tiny contribution, is deliberate—I wanted this paper to be about how I personally feel about teaching. If it is criticized for this, so be it. Indeed, I would welcome such criticism, because it would at least mean that my words have been read, and reflected upon. I really believe, however, that others will find much in common between their experiences at the hands of excellent teachers, and mine. At the very least,

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my thoughts might strike a chord with some who are as passionate about teaching as I am, and who want to ensure that all students are offered the best possible learning experiences. Perhaps this paper may even stimulate rigorous and wellplanned research to validate or negate my views. If it does nothing more than promote debate, I will be content.

11. Acknowledgement I thank all of my teachers for their enrichment of my life.

12. References [1] Collins R.E., “Lots of Scars”, Lexington Avenue Press, Copacabana, NSW, Australia, 2004. [2] Collins R.E., Recognizing excellent teaching, Synergy 9 (November 1998) 3–4.


Index Note: in this index, “BB” stands for “Benjamin Bederson”; suffix “n” indicates a footnote, italic numbers an illustration (figure), and boldface numbers a table. Aberdeen, University of (UK), 35–37 Abraham Pais Prize for the History of Physics, 73, 74 Absorbing sphere model, 418 AC Stark effect, 245 Actinides – atomic spectroscopy, 275, 276 – database site, 276 – radioactivity, 274 Activation energy, in electron attachment experiments, 317–319 Advances in Atomic, Molecular and Optical Physics, co-editors, 9, 10, 12, 409 Alignment angle, of excited electron cloud, 483, 484 Alignment effects, after autoionization decay of atomic states, 525 Alignment parameters, 482, 500, 503, 509– 512 Alignment tensors, 482 Alkali series, 291 – cesium, 296 – – 134 Cs, 4, 5, 277 – – 135 Cs, 277 – – 137 Cs, 277 – – 137m Cs, 277 – – isotope production, 280, 281 – – polarizability, 352, 354 – francium, 274, 283, 284 – – atomic spectroscopy, 283–286 – – Bohr–Weisskopf effect, 296 – – discovery, 274, 283 – – high-lying states, 293 – – parity non-conservation, 285 – – Perey’s work, 274, 283 – – polarizability, 354 – – Yagoda’s work, 274, 283, 284 – potassium, 278 – – 40 K, 277, 278 – sodium, 282

– – electron-impact excitation cross sections, 385–409 – – 31 Na deformation, 289 – see also Cesium; Francium; Potassium; Sodium Ambipolar diffusion, 305, 317, 318 American Institute of Physics, 60 American Physical Society (APS) – BB as Editor-in-Chief, 9, 11, 44, 45, 49–55 – court cases, 61, 62 – Forum on the History of Physics – – BB as Chair, 73, 74 – – BB as Editor of Newsletter, 13, 45, 74, 334 – Physical Review A: Atomic, Molecular, and Optical Physics, BB as Editor, 12, 44, 57–62 – Physical Review OnLine Archive, 54 – Task Force on Journal Growth, 52–54 – Task Force on Publication Policy, 50, 51 Angle-resolved photoelectron–photoion coincidence experiments, 520, 521 Angular anisotropy parameter, 518 Angular correlation coincidence count rate, 479 Angular correlation in collisions, 478 Angular correlation parameters, 481 – alignment parameters calculated from, 482 Angular distribution – measurements, 414 – of Auger emission, 497, 501–504, 506, 515 Angular distribution parameter – argon Auger transitions, comparison of calculational to experimental results, 506– 509 – for resonant Ar* (4s1/2 )L3 M2,3 M2,3 Auger transition, 506, 511 Anisotropic angular emission of Auger electrons, 481 559

560 Anisotropy, 89 Anti-parallel–parallel asymmetry, 492 Arnold Sommerfeld Centennial Memorial Meeting, 525 Astrophysics – applications, 416, 434 – laboratory, 417 Asymptotic condition, 127, 128, 134 Atmospheric physics, applications, 416, 423 Atom Based Metrology, 381 Atom dynamics, 369–379 Atom manipulation, 368, 369, 380, 381 – STM images, 372, 375 Atom–molecule collisions, 424 – see also Collisions Atom motion, STM observations, 369–374 Atom tunneling, 378 Atomic Auger decay, 513 Atomic beam magnetic resonance (ABMR), 277 – flop-in techniques, 277 – focussing magnets, 278 – radioactive experiments, 277 Atomic beam recoil, cross sections for sodium, 398, 399, 403, 404 Atomic collision physics, 525 Atomic collision processes, 472 – applications, 473 – classification, 474–476, 519 Atomic collisions, spin effects in, 488–496 Atomic cross section, 473 Atomic number density, measurement, sodium, 389, 390 Atomic number fluctuations, 245, 261 Atomic orbitals, 102, 164 Atomic orientation and alignment, 481 Atomic polarizability see Polarizability Atomic reactions, 512 Atomic spectroscopy, 472 – actinides, 275, 276 – francium, 283–286 Atomic structure, 274 Atomic theory, 364 Atomic/molecular collision experiments, 414, 473 Attractive scattering, 483 Auger decay, 512 – after electron impact excitation, 500, 501 – after photoexcitation, 502–504 – Ar(2p → 4s) excited state, 506

Index – complete experiment for, 474, 513–519 – – examples, 516–519 – Na+ , 517 – in two-step model, 513 Auger decay amplitudes, 514 – for resonant Auger decay Xe+ , 518 – sodium, 518 – xenon, 518 Auger decay dynamics, 500, 501, 509, 516 Auger decay parameters, intrinsic, 515, 516 Auger electron, 417, 443 – anisotropic angular emission, 481 – detection of, 515 – partial-wave representation, 514 – spin components, 516, 519 Auger emission, 474, 496 Auger emission angle, 511 Auger fine-structure multiplet, 517 Auger process – normal – – compared with resonant Auger process, 497, 498 – – spin polarization of Auger state, 516 – – two-step model, 497, 498, 500, 502, 513 Auger spectrum, 506, 507, 513 Auger state – decay of, 513 – excitation by photon or particle impact, 513 Auger transition matrix elements, 504 Auger transitions – angular distribution parameters, 501, 502 – – theoretical compared with experimental data, 506–509 – spin polarization parameters, 501, 502 – – theoretical compared with experimental data, 506–509 Aurora, 414, 423, 424, 426, 441 Auroral optical emissions, 422, 423 Australian Institute of Physics, 546 Autonomous Atom Assembler (AAA), 381 Awards for Excellence in Teaching (University of Sydney), 541, 553 Axial-recoil-approximation, 520 Back, Ernst E.A., 282 Bates, Sir David, 9 Bederson, Benjamin – as Chair of Forum on the History of Physics, 73, 74

Index – as Co-Editor of Advances in Atomic, Molecular and Optical Physics, 9, 10, 12 – compared with Benjamin Franklin, 42–47 – as conference organizer, 5, 15, 44, 525 – consulting and advising positions, 14, 15, 45 – curriculum vitae, 11–16 – doggerel in homage to, 23–27 – as Editor-in-Chief, American Physical Society, 9, 11, 44, 45, 49–55 – as Editor of Forum on the History of Physics Newsletter (APS), 13, 45, 74, 334 – as Editor of Physical Review A: Atomic, Molecular, and Optical Physics (APS), 12, 44, 57–62 – other positions, 12 – other publications, 21, 22 – patents, 16, 42 – PhD students, 13, 345, 346, 350 – as physicist–historian, 65–74 – “popular” physics courses, 16, 47 – professional memberships, awards, 13, 14, 42 – research interests, 13 – research publications, 17–21 – staff positions, 4, 5, 11 – as teacher, 15, 16, 47, 540, 542, 544, 547 – wartime army service, 11, 29–34, 45, 66–69 Binding energy, 102, 417 Bismuth, isotonic shifts, 288 Black, Hermann, 550 Bohr–Hartree–Fock approximation, 184, 186 Bohr model, 96, 97, 99 – correlation energy treated, 100, 101 – Hartree–Fock results improved using, 184, 185 – hydrogen molecule, 96, 97, 99, 107, 110, 188 – interpolated, 108–111 – recent progress based on, 107–111 Bohr model energy, 97 Bohr molecule, 96–99 Bohr radius, 97 Bohr–Weisskopf effect, 277, 285, 294–296 Born, Max, 70 Born approximation, 480 Born–Oppenheimer separation/approximation, 111–113, 162, 519 BrCH2 CN, 313, 320

561 Breit–Pauli R-matrix approach, 493 Brennan, Max, 552 Broe (chemistry lecturer), 538, 539 Brookhaven Conference, 5, 414 Bureaucracy, 40 C5+ ions, 416 Carbonyl hydrides, transition-metal, 310, 311, 312 Cascade processes, 416 Cascading transitions, 417 Casimir effects, 76–81 Cavalleri diffusion cell, 301, 336 Celotta, Robert J., 13, 350 Center-of-mass system, 208–210, 415 Centre de Spectroscopie Nucléaire et de Spectroscopie de Masses, 283, 289 Centrifugal force, 98 Cesium, 296 – 134 Cs, 4, 5, 277 – 135 Cs, 277 – 137 Cs, 277 – 137m Cs, 277 – isotope production, 280, 281 – polarizability, 352, 354 – spin polarized atoms, 492 Charge cloud orientation, 486 Charge transfers, 525 Chlorine nitrate (ClONO2 ), electron attachment studies, 314, 321, 322 Chromium (Cr), 350 Circular dichroism in angular distribution (CDAD) graphs, 521, 522 Circular polarization, 482 Circularly polarized light – photoexcitation by, 501, 507, 522 – spin polarization transfer by, 499 City College, New York (CCNY), 11, 71 City University of New York (CUNY), 71 Civic scientists, 41–47 – attributes – – communication skills, 43, 44 – – consensus-building ability, 44 – – policy-influencing skills, 44, 45 – – scientific ability, 42 – – wisdom, 43 Classical (over the barrier) model (for electron capture), 418, 419, 421 Classification of atomic collisions, 474–476, 519

562 ClCH2 CN, 313, 320 Clueless: Gould and Miller, 357 Co/Cu(111) system, 367, 368 – atom moving impedance, 369 – binding site image, 372–374, 375 – STM imaging impedance, 369 Coalescence wave function, 105, 120–123 Coalescent approach, 129–130 Coherence, electrons in nanostructures, 364 Coincidence experiments, 415, 422, 473, 476, 497 – analysis of, 477–487 – in particle and nuclear physics, 523 Cold Target Ion Momentum Spectroscopy (COLTRIMS), 496 Colliding atomic particles, 472 Collinear laser spectroscopy, 291–293 – advantages, 292 – line-width reduction, 291, 292 Collision frame, 483 Collision frame amplitudes, 486 Collision theories, 473 Collisional products, 476 Collisional stabilization, 310 Collisions – Ar17+ + Ag(s), 443 – electron–atom – – atomic beam recoil method, 398, 399 – – emission measurements, 390–394 – – laser-excited targets, 401–407 – electronically elastic, 415, 424, 425, 432, 433 – electronically inelastic, 424, 425, 433 – experimental approaches, 414, 415 – H+ + SO2 , 413, 441 – H0 + Ar (energy reference) – – electron capture collisions, 430, 437, 439 – – stripping collisions, 427, 435 – H0 + H2 , 432–434, 435, 440 – – direct scattering, 432–434 – – electron capture (H− ), 434, 437 – – stripping (H+ ), 434, 435, 436 – H0 + N2 , 424, 440 – – direct scattering, 424, 425 – – electron capture (H− ), 424, 426, 430 – – stripping (H+ ), 424, 426, 427–429 – H0 + O2 , 434–439, 440 – – direct scattering, 434, 435 – – electron capture (H− ), 438, 439, 439, 440 – – stripping (H+ ), 435–437, 438

Index – H0 projectiles, 422–441 – He+ with CO and NO, 424 – heavy solar wind ions, 416–422 – Io plasma torus, 441 – multi-electron process, 421 – O5+ + CO, 417, 421, 422 – O6+ + CO, 413, 418, 419, 421 – roles, 414 – stripping, 424, 426 Combustion plasmas, 319 Comet Hyakutake, 416 Cometary problem, of collision systems, 418 Cometary X-ray emissions, 413, 416, 417, 421 – electron capture model, 417 Comets, composition, 417 Complete data, in atomic collision physics, 486 Complete experiment, 474, 476 – for Auger decay, 474, 513–519 – – approaches, 515, 516 – – examples, 516–519 – – Na+ , 517 – for half-collision Auger decay, 513, 519 – for molecular photoionization, 523 – for other collision processes, 527 Complete photoionization experiment, 495, 514 Complete/perfect scattering experiments, 473, 514 – approaches, 476, 477 – objects, 513, 514 Complex decay amplitudes, 514 COMPLIS system, 293, 294 Configuration interaction, 175, 179 Configuration state functions (CSFs), 504 Confinement, electrons in nanostructures, 364 Confluent hypergeometric function, 153 Confocal Fabry–Perot interferometer, 248 Conservation of momentum, 415 Conservation of parity, 514 Convergent close coupling (CCC) theory, 397, 408, 455, 486, 492, 493 Cooper, Peter, 71, 73 Cooper Union for the Advancement of Science and Art, 71 Coordination, atoms in nanostructures, 364 Correction energy, 101 Correlation diagram, 174

Index Correlation energy, 100, 101 Correlation experiments, 476, 523 Correlation function, 123–127, 130, 147– 155, 176 – Le Sech form, 125, 126, 151 – Padé–Jastrow form, 125, 126 – Patil form, 125, 126, 148 Coulomb interaction energy, 168 Coulomb potential energy, 97 Crompton, Robert W., 336 Cross sections – apparent excitation, 387 – Born-approximation normalization, 395– 397 – cascade, 387 – differential, 400, 405, 414, 428, 476, 479, 486, 488 – for collisional interactions, 523 – for electronically excited argon, 510 – integral, 405–407 – measurements with trapped atoms, 409 – optical emission, 387 – partial, 399–401 – relations between, 387 – summation relations, 387, 394, 395 – total, energy dependence, 414 – total ionization, 456, 459 Crossed-beam technique, 303, 474 Cullen (high school science teacher), 537, 548 Cusp condition, 118, 119, 127, 128, 133, 134, 145–147 – derivation of, 203–208 – verification of, 210–215 1,3,5,7-Cyclooctatetraene (COT, C8 H8 ), 313, 325, 328 Cylindrical coordinates, 97 Davies, Lou, 539 De Broglie wavelength, 97, 310 Decay amplitudes see Auger decay amplitudes Deep inner shell photoexcitation, 498 De-excitation of laser-excited atom, 486, 487 – compared with electron–photon coincidence experiment, 488 Deflection, beam, 347 Demkov process, 424 Density functional theory (DFT), 319, 320, 329

563 – calculations, 317, 325 – time-dependent, 85, 344 Density matrix, 479 – for excitation of atomic hydrogen, 476, 477 Detector efficiency, 416 Determinations of emitting states, 415 Deuterium molecule – collision with He+ , 415 – polarizability, 355 Diatomic molecules – modelling, 155–184 – – Hartree–Fock self-consistent method, 162–166 – – Heitler–London method, 155–158, 185, 186 – – Hund–Mulliken method, 158–161 – – James–Coolidge wave functions used, 166–174 – – two-centered orbitals used, 174–184 Diatomic orbitals, 105, 106 Dielectric constant, 344 Differential cross section, 414, 428, 476, 479, 486, 488 – at very small angles, 424 – cesium, 492, 493 – reduced, 414, 429, 431 – sodium, 400, 405 Diffusion cooling, 309 Dimensional scaling (D-scaling), 98, 99, 185–190 – in spherical coordinates, 232–235 Dipole approximation, 497, 500, 501, 502 Dipole interactions, 492 Dipole moment, electric, 346, 347 Dirac 8-state R-matrix model, 493 Dirac–Fock calculations, 504, 516, 518 Direct differential elastic scattering, 489 Direct double ionization, for normal Auger decay, 513 Direct process, spin reactions, 490 Direct scattering, 477, 478, 488 – H0 collisions, 424, 425, 432–435 Direct scattering amplitude, 488, 492 Direct single photoionization, for resonant Auger decay, 513 Dirichlet boundary conditions, 77, 80 Dissociation of molecule, 512 – after photoionization, 520 Distinguished Teaching Awards (New York University), 553

564 Distorted wave approximation, 481, 482, 504 Distribution of nuclear magnetism, 277 – Bohr–Weisskopf effect, 294–296 – – hydrogenic bismuth, 295 – – muonic bismuth, 295 – – neutron wave functions, 295 – – PNC, 295 Divergent integral, 198 Dots, light force method, 350 Double photoionization, 494 Doubly charged ion, Auger decay leading to, 513 Drift tube measurements, 301, 321 Durack, Vince (high school mathematics teacher), 538, 548 Dynamic effects, during Auger emission, 516 Dynamic spin polarization, 499–502, 508, 509, 512, 516 – measurement of, 505 (e, 2e) experiments, 494, 498, 512 (e, 2e) process, 495 E–H gradient balance, 334, 335, 346, 452, 488 Eckart wave function, 122, 123 Education systems, problems facing, 36, 37, 47 Effective charge, 100, 101 Efficiency tensors, 479 Einstein, Albert, 5, 35, 70 Ekacesium see Francium Elastic collisions, 415, 424, 425, 432, 433, 475 Elastic scattering, 252 Electric dipole moment, 346, 347 Electric dipole polarizability see Polarizability Electric field induced diffusion, 365 Electron affinity (EA), 324, 329, 330, 344 Electron–atom collisions – atomic beam recoil method, 398, 399 – emission measurements, 390–394 – laser-excited targets, 401–407 – sodium excitation cross sections, 385–409 Electron attachment, 305, 306, 309–323, 336 – nitrogen trifluoride, 316, 318 – phosphorus fluorides, 317, 318 – single-center hexafluorides, 316 – sulfur-fluoride compounds, 314–316

Index – transition-metal trifluorophosphines and carbonyl hydrides, 311–314 Electron beam ion trap (EBIT), 443, 444 Electron beam sources, 497 Electron capture – H+ , 441, 443 – H0 , 424, 426, 430, 434, 437 – O6+ , 417 – single, 417 Electron cascade front (in shock wave), 463, 464 Electron charge distribution – of coherently excited 1 P1 , 483 – to collision, natural, and atomic frames, 483 Electron cloud, 483 – of He(21 P1 ), orbital angular momentum transferred to, 484 Electron cyclotron resonance (ECR) ion source, 418, 443 Electron detachment, 323–329, 336 Electron dichroism, for chiral molecules, 525 Electron–electron correlation function see Correlation function Electron–electron cusp condition see Interelectronic cusp condition Electron impact energy, orientation parameter affected by, 509 Electron impact excitation – of atomic hydrogen, 486 – of atoms, 478 – Auger emission after, 497, 498, 500, 501 – – alignment and orientation parameters, 497 – of He(21 P1 ), 480 – of highly charged ions, 525 – of sodium atom, 517 – X 1 g+ → c 1 u transitions in H2 and D2 molecules, 520 Electron impact excitation cross sections, optical method for measuring, 387–389 Electron impact ionization, 452, 454–457, 464 Electron–ion recombination, 309, 316, 321, 333, 525 Electron–nucleus cusp condition, 146, 147 Electron number density distribution, 464 Electron–photon angular correlation experiment, 478, 480 Electron–photon coincidence, atomic orientation and alignment based on, 481

Index Electron–photon coincidence experiment, 473, 524 – compared with inverse de-excitation of laser-excited atom, 486, 487, 488 – exchange amplitudes measured using, 489 Electron–potassium scattering, 492 Electron scattering interference, 477 Electron spin effect, solid state, 525 Electron spin polarization – depolarization after collision with NO molecular beam, 520 – emitted Auger electrons, 497, 499, 503, 512, 516 Electron storage ring, 505 Electron temperature, 306, 309, 333 Electronically elastic collisions, 415, 424, 425, 432, 433 Electronically inelastic collisions, 424, 425, 433 Electrostatic energy analysis, 425 Ellipsoidal coordinates, 138, 139 Elliptically polarizing undulator (EPU), 505 Emitted radiation, measurements, 415 Emitting state, lifetime, 416 Energy and angular distribution measurements, 414 Energy corrected binary-encounter approximation (ECBEA), 455 Energy difference, collisions, 415 Energy function, 108 Energy loss measurements, 415, 416 Energy loss spectra – H0 + H2 collision, 432, 435, 437 – H0 + N2 collision, 425, 427, 430 – H0 + O2 collision, 436, 439 Energy Physics course (Dick Collins), 542, 547, 555 Energy pooling ionization coefficient, neon, 459 Energy pooling processes, 454, 457–462 – “directional”, 460 – “thermal”, 460 Energy reference spectrum – electron capture collisions, 430, 437, 439 – stripping collisions, 427, 435, 436 Energy resonance model, 418 Entropy of attachment, 329, 330 Ephebic Oath, 73 Equilibrium constant, 329 Euler constant, 171, 217

565 Europium, laser-polarized (aligned/oriented), 495–496 Evans, Nigel G., 303, 316, 336 Exchange amplitude, 488, 490 – measurement of, 489 Exchange interactions, 492 Exchange scattering, 477, 478, 488 Excitation, of Auger state by photon or particle impact, 513 Excitation energy (Q), 415 Excitation/de-excitation 1S1 → 1P1 → 1S0 , 480 Excited state fraction, 401–403 Expectation values of angular momenta, 476 Fabry–Perot interferometer, 248, 249 FCH2 CN, 313, 320 Federation Fellowship Scheme (Australia), 551 Fermi, Enrico, 67 Feynman, Richard, 544, 545, 552 Feynman–Hellman theorem, 116 First-generation type experiments, 473 Fisher, Leon Harold, 3, 69, 300n Fission isomers, shape isomer shift, 294 Flowing afterglow experiments, 301, 314, 321, 334 Flowing-afterglow Langmuir-probe (FALP) method, 301, 331 – apparatus, 303–306 – Langmuir probe operation, 306–309 Fluctuation in number of atoms, 245, 261 Fluorescence polarization, 518 Fluorescent light – frequency distribution, 248 – spectrum, 247–251 Fluorine (F2 ) molecule, 316 Fluxionality, 317 Fock operator, 163 Forbidden transitions, 417 Force equation, 98 Fordham University, 71 Forum on the History of Physics (APS) – BB as Chair, 73–74 – BB as Editor of Newsletter, 13, 45, 74, 334 Fountain clock, 5 Fractional depolarizations, of polarized excited atomic beams, 523 Francium, 274, 283, 284 – atomic spectroscopy, 283–286

566 – Bohr–Weisskopf effect, 296 – discovery, 274, 283 – high-lying states, 293 – parity non-conservation, 285 – Perey’s work, 274, 283 – polarizability, 354 – Yagoda’s work, 274, 283, 284 Franck–Condon transitions, 424 Franck–Hertz experiment, 477 Franklin, Benjamin, 41 – comparison with BB, 42–47 – scientific achievements, 42 Frequency distribution of fluorescent light, 248 Full collision, 513 – see also Atomic collisions; Molecular collisions G2 theory, 322 G3 theory, 317, 319, 322 G3(MP2) theory, 325 G3(MP2)/B3LYP theory, 320 Galileo spacecraft, 441 Gaussian-type orbitals (GTOs), 165 Gegenbauer functions, 201 Gerade states, 159 Goudsmit, Samuel A., 25n, 282 Grazing model, 485 Great Teacher Awards (New York University), 553, 554 Greenglass, David, 30, 68 Grotarian diagram, 419, 420 Guillemin–Zener wave function, 127–129, 134, 153 H0 projectiles, collisions involving, 422–441 H2 + -like molecular ion, 136, 138–143, 192, 193 H3 , 426 Half-collision, 512 – complete experiments for, 512–519 – examples, 512 Half-collision processes, 474 Hanle effect, 247, 477 Harris, Townsend, 71, 73 – see also Townsend Harris High School; Townsend Harris Medal Hartree–Fock approximation, 100 Hartree–Fock equation, 163, 164 – results improved using Bohr model, 184, 185

Index Hartree–Fock self-consistent method, 162– 166 Hartree–Fock-like wave function, 129 HCo(CO)4 , 312 HCo(PF3 )4 , 310, 311, 312 He+ collisions, with CO and NO, 424 Heavy solar wind ions, collisions involving, 416–422 HeH molecule, ground state potential curve, 108, 109 Heitler–London method, 104, 107, 155–158, 185, 186 Heitler–London trial function, 185 Heitler–London wave function, 129, 216, 217 Heitler–Weisskopf effect, 265 Helicity system, of emitted Auger electrons, 501, 502 Helium atom, correlation diagram, 174 Helium-like ions, 100, 101 Heterodyne detection technique, 254–259, 270 Hexafluorides, 311, 316 High-pressure mass spectrometry, 301 Highly oriented pyrolytic graphite (HOPG), ions scattered from, 444, 445, 446 Historian, BB as, 65–74 HMn(CO)5 , 310, 312 “Hollow” atom, 443 Homodyne detection technique, 269 HRe(CO)5 , 310 HRh(PF3 )4 , 310, 311, 312 Hubble space telescope, 441, 525, 526 Hund–Mulliken method, 104, 158–161 Hydrogen atom – cusp condition, 146 – wave mechanical solution, 137, 138, 235 Hydrogen molecular ion, 127 Hydrogen molecule – Bohr model for, 96, 97, 99, 107, 110, 188 – correlation diagram, 174 – ground-state potential curves, 110, 132, 184, 188, 190 – polarizability, 355 Hylleraas correlation factor, 102, 103 Hyperfine structure, 274 – Breit–Rosenthal correction, 274, 275 – information provided by, 282 – multiple moments, 282 – sodium, 282 Hyperpolarizability, 347

Index ICH2 CN, 320 Ideal two-state fluctuator, 374 Impact line radiation, polarization effects, 478 Inelastic collisions, 424, 425, 433, 475 Inelastic processes, 414 In-going state vector, 479 Inner shell photoexcitation, 498, 513 Inner shell vacancy production, 524 Institut de Physique Nucléaire (Orsay), 289 Intensity correlation, 244 – experiments, 260–266 Interelectronic cusp condition, 146, 147 Interference amplitude, 488 Interference effects, 480, 488 International Conference on Atomic Physics (ICAP), 5, 15 International Conference on the Physics of Electronic and Atomic Collisions (ICPEAC), 5, 15, 44, 414, 525, 527 Interpolated Bohr model, 108–111 Intra-atomic–molecular scattering, of photoelectron, 474 Intra-atomic hyperfine interactions, 516 Io plasma torus, 441 Ion–ion recombination, 309 Ion mobility, 344 Ion–molecule collisions, 424 – see also Collisions Ion sources, 443, 498 Ionic state, 441, 513 – intermediate, 441 Ionization–recombination model, 454, 462– 465 Ionizing shock waves, 453–455 – neon, 454, 462–466 Irreducible statistical tensors, 500 Irreducible tensor operators, 415, 479 ISOL, 289 ISOLDE, 280 – early experiments, 283 – isotope production, 280 – – cesium, 280 – – francium, 283 – – laser resonance ionization used, 280 – – long chains, 281 – – yields, 280, 281 Isomer shifts, 289 – fission isomers, 294 – 178 Hf, 293

567 – shape isomer shift, 240m Am, 294 Isotone shifts, 289 – examples, 288 Isotope production, 280 – long chains – – cesium, 281 – – mercury, 286 – – sodium, 282 Isotope shifts, 274, 275 – field shift, 275 – mass shift, 275 – odd–even staggering, 286, 289 – stellar chemistry, 296 – variation of fundamental constants, 296 James–Coolidge wave function, 143, 166– 174, 196–198 Japanese language, trying to learn, 545 Jastrow, Robert, 23n Johnston, Ian, 541, 544, 546 Kinematically complete experiment, (e, 2e) process, 495 Kistiakowsky, George, 31, 32, 67 Kraner, Carol (at APS), 58, 60, 62 Laboratoire Aimé Cotton, 274–276, 283, 289 – single mode tunable pulsed dye laser system, 293 Laboratoire René Bernas, 274, 280 Laboratory astrophysics, 417 Lamb–Dicke limit, 258 Langevin equation-of-motion approach, 242 Langmuir probe, 304, 306–309, 316, 334 Langmuir probe hysteresis, 308 Laser absorption spectroscopy, 461 Laser-cooled atoms, 352–354 Laser-excited atoms, 486 – de-excitation of, 486, 487, 488 Laurent expansions, 194 Lazarus, David, 25n, 50 Lead, isotonic shifts, 288 Leadership, 38–39 Lecture demonstrations, use in teaching, 542, 548 Lennard-Jones potential, 117, 120 Level-crossing spectroscopy, 279 Levine, Judah, 13, 414 Lewis structure, 187 Lifetime – atmospheric, 316

568 – emitting state, 416 LiH molecule, ground state potential curve, 109–111, 111 Line widths, 276, 280 – Doppler broadening of, 276 – Doppler-free, 279 – reduction in accelerated ion beam, 291, 292 Linear combination of atomic orbitals (LCAO) method, 159, 164 Linearly polarized light, photoexcitation by, 507, 522 Loeb, Leonard Benedict, 3, 300 Los Alamos Laboratory (Manhattan Project), BB at, 11, 29–34, 66–69 Low-energy electron scattering, of cesium atoms, 492 Mach–Zender atom interferometer, 351 Magic angle, measurements at, 416 Magnetic substates, 481, 482 – amplitudes, 479, 514 Malloy, Margaret (at APS), 58–60, 62 Manipulated atom image, 372, 375 Many-centred one-electron problem, 143, 144, 198–202 Markov approximation, 243 Markovian master equation approach, 242 Maxwell, James Clerk, 35, 36 McMillan correlation function, 118 Mercury, 278, 279 – isotonic shifts, 288 – nuclear shape transition, 286, 288 Messel, Harry, 539, 548 Microwave spectroscopy, 355, 356 Miller, Julius Sumner, 542, 543, 546 Millikan, Robert A., 3, 301 Modified effective range theory (MERT), 345 MoF6 , 311, 316 Molecular collision processes, 472, 519 – applications, 473 Molecular collisions, analysis of, 519–523 Molecular dissociation, 512 – after photoionization, 520 Molecular frame photoelectron angular distributions (MFPADs), 523, 524 Molecular orbital approach, 129, 130 Molecular orbital theory, 103, 158 Molecular perfect/complete collisions, 519 Molecular photoionization, 474 Molecular photoionization dynamics, 523

Index Molecular spectroscopy, 472 Momentum, conservation of, 415 Momentum Imaging in Atomic Collisions method, 496 Momentum transfer, 480 Morrison, James (musician), 543 Morse potential, 120 Moss, Percy (high school physics teacher), 538 Mott polarimeter, 505 Mott scattering, 477, 489 Multi-channel Schwinger configuration interaction (MCSCI) calculations, 521, 522 Multi-charged ions, in solar wind, 413 Multi-configurational Dirac–Fock (MCDF) calculations, 504, 516, 518 Multiple-choice examinations, 556 Mutual neutralization, 309 N+ ion angular distributions, 521, 522 N5+ ions, 416 Nanofabrication, 350, 380, 381 Nanostructures, 364, 381 Nanotechnology, 364 Nanotransducer, 381 Ne8+ ions, 416 Negative ion production, 444, 445 Neon – electron impact ionization in, 455–457 – energy pooling processes in, 457–462 – ionizing shock waves in, 454, 462–466 – metastable state, 458 – polarizability, 355, 458, 459 Neutralization, probability at surface, 446 Neutralized ions, 355, 356 New South Wales Institute of Technology, 541 New York Academy of Sciences, 71 New York City, and physics, 69–73 New York high schools, 70, 71 New York Public Library, 73 New York University (NYU), 5, 11, 72, 345, 346, 540 – Atomic Beams and Plasma Physics Laboratory, 302, 303 – Dean of Faculty of Arts and Sciences, 38 – Dean of Graduate School of Arts and Sciences, 11, 57 – Distinguished Teaching Awards, 553

Index – Great Teacher Awards, 553, 554 Night air-glow phenomena, 423 Ni(PF3 )4 , 310, 312 NIST wavelength tables, 421 Nitric oxide (NO) molecule, 4σ −1 photoemission, 520–523 Nitrogen molecule, 301 Nitrogen trifluoride (NF3 ), 312, 316, 318 Non-coincidence experiments, 497 Nuclear isomers, 279 Nuclear magnetic moments, 279 Nuclear magnetic resonance, medical application, 37 Nuclear shape transition, 286–288 O5+ ions, 418, 419 – collisions involving, 421, 422 O6+ ions, 416, 418 – collisions involving, 413, 418, 419, 421 O7+ ions, 416 Off-line experiments (radioisotopes), 275– 280 One-electron excitation, in collisions, 432 One-electron homonuclear wave function, 127, 128 One-electron many-centered problem, 143, 144 One-electron molecules, 135–145 One-electron two-center problem, 182 Optical Bloch equations, 241, 242 Optical double resonance, 278, 279 Optical emission spectrum, H+ + SO2 , 441, 442 Optical experiments, collisions studied using, 415 Optical pumping, 279, 486 Optical spectroscopy – collisions, 421, 422 – radioisotopes, 274 Optically forbidden transitions, 492 Orbital angular momentum transfer, 482, 483, 484 Orbital momentum transfer, 486 Orientation parameters, 500–503, 509–512 Orientation propensity rules, 485, 486 Orientation vector, 482 – of He(21 P1 ), 481 Oscillator strengths, 394, 458 Out-going state vector, 479

569 Over-the-barrier model (for electron capture), 418–421, 443 Oxygen molecule, 301 Ozone (O3 ), electron attachment studies, 309, 310, 314, 321 1 P excitation, 479, 482 1

Pais, Abraham, Prize for the History of Physics, 73, 74 Parameters of collision processes, 472 Parity, conservation of, 514 Parity non-conservation (PNC) effects, 285, 295, 296 Partial waves, of argon Auger transitions, relative phase shifts, 508 Partial-wave representation – of Auger electron, 514 – of scattering amplitudes, 513 Partial widths (Auger decay), 515 Pauli, Wolfgang, 282 Pauli exclusion principle, 108 PCl3 , 312, 318, 319 Percival–Seaton theory, 478 Perfect scattering experiments, 473, 488 – approaches, 476, 477 PF3 , 312, 317–319 PF5 , 312, 317, 318 Phase diffusion model (PDM), 245 Phase shift, 351, 352 Phase shift difference, argon Auger transitions, 508 Phosphorus compounds, electron attachment studies, 312, 317, 318 Photodetachment threshold, 328, 344 Photoelectron angular distributions, 521 Photoelectron–photoion angular correlations, 520 Photoelectron spectroscopy, 301, 324, 328 Photoemission of NO, 520–523 Photoexcitation, 474 – Auger emission after, 496–498 – of inner-shell electron, 498, 513 Photoexcited Auger decay, 502–504 Photoexcited resonant argon Auger decay, 497, 506–512 Photoionization, 474, 525 – Auger emission after, 496, 504 – double, 494 – of fixed-in-space molecules, 520 – of polarized atoms, 495

570 Photoionization dynamics, 4σ −1 photoemission of NO, 520–523 Photoionization/excitation experiment, 474 Photon antibunching, 244, 245, 261 Photon beam techniques, 497 Photon bunching, 244, 260 Photon polarization, 486 Physical Review A: Atomic, Molecular, and Optical Physics (APS), BB as Editor, 12, 44 Pirani gauge, 4 Plasma aerodynamics, 453 Plasma effects, 330–332 Plasma electron spectroscopy, 461 POCl2 , 319 POCl3 , 312, 319 Poissonian fluctuations, 262 Polarizability, 344 – alkali halide dimers, 346, 349 – alkali metal, 354 – anisotropy, 347 – barium, 355 – cesium, 352, 354 – clusters, 349 – core electrons, 354, 355 – deuterium molecule, 355 – dynamic, 84–86 – – TDDFT formalism, 85, 344 – electric deflection experiments, 346–349 – francium, 354 – helium, 355 – hydrogen molecule, 355 – indium, 346 – interferometry experiments, 350–352 – – uranium, 350 – ions, 349, 354 – light force method, 349, 350 – measurement of – – E–H gradient balance method, 346, 349, 452 – – electric deflection method, 346, 452 – metastable atoms, 350 – metastable mercury, 346, 350 – metastable noble gas, 346, 458 – neon, 355, 458, 459 – nitrogen, 355 – quadrupole, 355 – radiative corrections, 344 – radium, 354 – relativistic corrections, 345

Index – scalar, 85, 347 – Si+ core, 356 – silicon ions, 355 – sodium, 350 – sodium-like Mg+ , 354–356 – sodium-like Si+ core, 356 – spherical component, 347 – static – – isotropic, 88, 89 – – tensor, 89 – tensor, 85, 349 – valence electron, 354 Polarization of electromagnetic radiation, 481 Polarization energy, 355 Polarization phenomena in collisions, 478 Polarization spin asymmetry, 494 Polarized hydrogen atoms, electron scattering on, 494 Polarized light, 416 – circularly polarized – – photoexcitation by, 501, 507, 522 – – spin polarization transfer by, 499 – linearly polarized, photoexcitation by, 507, 522 Polarized potassium atoms, electron scattering on, 489 Polytechnic University, 72 Post-collision interaction (PCI), 513 Postdocs – BB’s, 13 – teaching by, 551 Potassium, 278 – 40 K, 277, 278 Potential energy surfaces, 415, 427, 428 Primordial matter, 417 Professor Harry Messel International Science Schools (University of Sydney), 548, 549 Prolate spheroidal coordinates, 105, 138, 139, 143 Propensity rules – for Auger transitions, 500 – orientation propensity rules, 485, 486 Proton beam sources, 497 Proximal condition, 120 Proximal limit, 117 PSCl3 , 312, 319 Pseudorotation, 317 Pt(PF3 )4 , 310, 312

Index Pulse-radiolysis microwave-cavity method, 301 Pulsed dye laser systems, 293, 406 Pure quantum states, collisions between particles in, 475, 476, 519 Q (excitation energy), 415 Quadrature squeezing, 267, 268 Quantum electrodynamics (QED), 76, 241, 242 Quantum scattering amplitudes and phases, 476 Quantum-mechanical collision dynamics, 473 Quantum-mechanical superposition principle, 476 Quantum-mechanically complete experiments, 495, 496 R-matrix with pseudo-states (RMPS) theory, 486, 492 Rabi frequency, 241, 249 – factors affecting, 248 Racah notation, 516 Radiation–atom interactions, 239 Radiation trapping, 455 Radiative cascades, 516 Radioactivity – actinides, 274 – discovery, 274 – off-line experiments, 275–280 – on-line experiments, 280–294 Radioisotopes – artificially produced, 274 – natural, 274 RADOP method, 286–288, 292 Ramsauer–Townsend effect, 477 Ramsauer–Townsend minimum, 309 Rank-4 multipoles, 525 Rayleigh–Jeans law, 76 Rayleigh scattering, 344 Reactive processes, 475 Recurrence relations, 169, 222, 223, 225, 227, 229 – 3-term, 179 – 5-term, 180, 231, 232 Recursion relations, 169, 171, 172, 222–231 Reduced parameter spaces, resonant Auger decay Xe+ , 518 Reduced scattering angles, 414

571 – stripping collisions, 426, 428, 429, 438 ReF6 , 311, 316 Regular singular point, 195 Relative phase shifts, partial waves of argon Auger transitions, 508 Relativistic contraction, 354 Relativistic distorted wave Born approximation (RDWA), 504 Relaxed orbital method, 504 Renaissance of research on atomic physics, 527 Repetition, use in good teaching, 547, 548 Repulsive scattering, 483 Repulsive/attractive dynamics, 485 Residence time distribution, 376 Resonance enhanced multi-photon ionization, rotationally resolved, 521 Resonance fluorescence, 239–270 – theory, 240–244 Resonance ionization spectroscopy, 293, 294 – isotope-shift data, 295 – refractory elements, 293 Resonance scattering, 478 Resonant Auger decay, 474, 499 – analysis and comparison of theoretical and experimental data, 506–512 – compared with normal Auger decay, 497, 498 – experimental details and setup, 505 – numerical calculation methods, 504 – Xe* , 517 Resonant radiative transfer, 464 Resonant state, 309 Resonant-electron-capture model, 419 Resonantly excited Auger decay, 474 – angle and spin resolved analysis, 496–512 Riccati equation, 117, 130 Riccati function, 119, 124, 128 Right–left electron scattering asymmetry, 485, 486 Robinson, Ed, 13, 414 Rood, Ogden Nicolas, 3 ROSAT (Röntgen satellite), 416 Rubin, Ken, 13, 346 Rydberg atoms, 406, 409 Rydberg electron, 355 Rydberg–Ritz combination principle, 275, 276 Rydberg state, 500

572 Saha–Langmuir equation, 444 Salop, Arthur, 13, 345, 414 Santilli, Ruggero Maria, 26n Saturation parameter, 255 Scanning tunneling microscopy (STM), 364 – atom manipulation using, 368, 369, 380, 381 – experimental system, 366 – tip–adatom interaction, 364, 369–371, 373, 380 Scattering amplitudes, 473 – analysis of, 478 Scattering dynamics, for electron scattering, 486 Scotland, university physics departments, 37 Second-generation type experiments, 473 Second-order correlation function, 260 Selective field ionization (SFI), 406 Self-consistent field (SCF) method, 162, 163 – see also Hartree–Fock self-consistent method SF4 , 311, 314, 315 SF5 − , 314, 316 SF6 , 311, 314–316, 327, 334 SF5 C2 H3 , 311, 315, 315 SF5 C6 H5 , 311, 315, 315 SF5 CF3 , 305, 307, 311, 315, 315, 316 SF5 Cl, 311 Shielded diatomic orbitals, 105 Shock tube experiments, 462 – regions in shock structure, 462, 463 – see also Ionizing shock waves Simplicity, as characteristic of good teaching, 546 Single electron capture cross sections, 418 Single-atom fluorescence, 266 Singlet interactions, 492 Singlet scattering, 486 Singlet scattering amplitude, 493 Singly charged ion, resonant Auger decay leading to, 513 Slater determinant, 162 Slater–Kirkwood approximation, 458, 460 Slater orbitals, 156 Slater-type orbitals (STOs), 165 Slow atoms, 352 Smith, David, 303, 316, 336 Sodium, 282 – cross sections – – comparison to theory, 397, 398, 407

Index – – differential, 400, 405 – – direct excitation, 391–393, 456 – – from laser-excited states, 404–406 – – integral, 405–407 – – partial, 399–401 – – superelastic, 403–405 – electron-impact excitation cross sections, 385–409 – electronic structure, 385 – energy level diagram, 388 – experimental difficulties, 385, 386, 389, 390 – measurement of atomic number density, 389, 390, 391 – 31 Na deformation, 289 – oscillator strength, 394 – production of excited atoms, 401, 402 Solar wind – composition, 417, 441 – major ionic components, 416 – multi-charged components, 413, 416, 418 – O6+ in, 418 Solid state effect to electron spin effect in solid state physics, 525 Sommerfeld, Arnold, Centennial Memorial Meeting, 525 Spectral wavelength resolution, 418, 419 Spectrum, fluorescent light, 247–251 Spherical coordinates, dimensional scaling in, 232–235 Spin asymmetry, 492–494 Spin components, of Auger electron, 516, 519 Spin effects, in atomic collisions, 488–496 Spin experiments, 473 – analysis of, 477–487 Spin–orbit interaction, 490, 492 Spin–orbital wave function, 156 Spin polarization – of atoms, 489 – of emitted Auger electrons, 497, 499, 503, 512, 516 Spin polarization parameters, 501, 502, 506, 508 – comparison of calculational to experimental results, 506–509 – for resonant Ar* (4s1/2 )L3 M2,3 M2,3 Auger transition, 506, 511 Spin polarization transfer, by circularly polarized light, 499

Index Spin polarization vector, 499, 503, 504 – Cartesian components, 501, 502 – for resonant Ar* (4s1/2 )L3 M2,3 M2,3 2 D5/2 final state Auger transition, 511 Spin polarized atoms, 492 Spin polarized electrons, 492 Spin resolved Auger emission, 500–504, 512 Spin-flip amplitude, 492 Spin-flip process, 490 Spinless atoms, 490 Spins – odd–odd nuclei, 277 – zero-moment method, 277 Squeezing, 267, 268 – observation of, 269, 270 Standing wave, 349 Stapp, Henry P., 26n Stark effect, 245, 347 Stark ionization, 356 State multipoles, 476, 482, 500, 501 State University of New York (SUNY, Stony Brook), 285, 296 Statistical tensors, 476, 482, 501 – irreducible, 500 – of angular momentum, 479 Stepwise electron and laser excitation of atoms, 524 Stevenson, Edward C., 68 Stirling Symposium, 527 STM see Scanning tunneling microscopy STM topograph, 368, 372 Stokes parameters, 502, 520 Stripping collisions – H0 , 424, 426, 427–429, 434–437, 435, 436, 438 – with reduced scattering angles, 426, 428– 429 Structural analysis – atoms, 472 – molecules, 472 Students – assessment of, 555, 556 – exam performance, 541 – quality of teaching evaluated by, 554 – two-way involvement with, 546 Sub-Poissonian statistics, 262 Subvalence atomic shells, electron excitation from, 497 Sulfur fluorides, electron attachment studies, 311, 314–316, 334

573 Sulfur trioxide (SO3 ), electron attachment studies, 310, 314, 322, 323 Superacids, gas-phase, 311, 336 Super-elastic scattering, 487 Surface collisions, highly charged ions, 443– 446 Susceptibility, electric, 349 Sydney, University of, 538, 539, 541, 552 – Awards for Excellence in Teaching, 541, 553 – Chancellor, 550 – Scholarship Index, 553 – Science Foundation for Physics, 548 Symmetric excitation process, 504 Symmetry Properties and Conservation Laws for Collisional Excitation, 524 Synchrotron beam excitation, 499 – reaction plane and coordinate frame defined, 499 Synchrotron beam sources, 497 Synchrotron radiation, fluorescence polarization induced by, 518 Target molecules, dissociation of, 421 Teachers – analogy with comedians, 545, 546 – characteristics of great, 543–545 – recollections, 537–543 Teaching – characteristics of great, 545–549 – evaluation of, 553, 554 – personal reflections on, 535–557 – recognition of excellent, 552, 553 – rewards, 549, 550 – who should do, 550–552 Teaching process, formalization of, 536 Teaching skills, developing, 541, 549 Terrorist attacks, 69 – effects, 46 Thallium, isotonic shifts, 288 Theoretical models, tests of, 486 Third-generation type experiments, 473 Time-dependent density functional theory (TDDFT), 84, 344 Time-of-flight (TOF) experimental techniques – collisions studied using, 423, 425, 434 – resonant Auger decay studied using, 505 Townsend Harris High School (CCNY), 70, 73

574 Townsend Harris Medal, award to BB, 14, 42 Transferred spin polarization, 499, 501, 502, 507, 512 – measurement of, 505 Transition metal trifluorophosphines and carbonyls, 310, 311–314 Transition rate, cobalt atom dynamics, 375, 376–379 Traps – electron beam ion, 443, 444 – francium in, 285, 354 – magneto-optical, 285, 352, 452 – Paul trap, 252, 279 – Penning trap, 279 – radium in, 354 – ytterbium in, 354 Triatomic molecules, 415, 426 Trifluoromethylbenzonitrile see C8 H4 F3 N Trifluorophosphines, transition-metal, 310, 312, 314 Triplet interactions, 492 Triplet scattering, 486 Triplet scattering amplitude, 493 Tunable chemical bond, 380 Tunable laser, 276, 280 – pulsed, single mode, 293 Two-center one-electron problem, 182 Two-center orbitals, 101–103, 174–184 – cusp conditions, 210–215 Two-channel Auger decay, 515–517 Two-electron bond, 104 Two-electron homonuclear wave function, 129–131 Two-electron molecules, 145–155 Two-electron transfer process, 424 Two-level atom, resonance fluorescence, 240 Two-state fluctuators, 376 Uhlenbeck, George, 25n Ultraviolet catastrophe, 76

Index Ungerade states, 159 United Nations, International Year of Physics (2005), 35 Unmanned space missions, 413 Upper atmospheric processes, 423 Uracil, 313, 321 US Army, Special Engineering Detachment (SED), 11, 29–34, 45, 66–69 Valence-bond approach, 155 van der Waals dispersion coefficients, 89, 90, 354, 458 van der Waals interaction, 344, 364 Variational parameters, 152, 158, 167, 179 Variational wave function, 100 Vibrational excitation, 378, 380 Vibro-rotational excitation, 415 Virial theorem, 115, 116 Vogt–Wannier theory, 310 Voyager mission, 441 W parameter, 523 Weakly ionized gas (WIG), 452 – energy pooling processes, 460 – interactions, 451–469 – in shock wave, 462, 463 WF6 , 311, 316 Wilkins, John, 26n Writing skills, 556 Xenon atoms, electron scattering on, 491 X-ray emissions, 414, 525 – from comets, 413, 416, 417 X-ray spectrum, O6+ + CO collisions, 419 Yukawa potential, 120 Zeeman states, 248 Zhou, Xian, 535

CONTENTS OF VOLUMES IN THIS SERIAL Volume 1 Molecular Orbital Theory of the Spin Properties of Conjugated Molecules, G.G. Hall and A.T. Amos Electron Affinities of Atoms and Molecules, B.L. Moiseiwitsch Atomic Rearrangement Collisions, B.H. Bransden The Production of Rotational and Vibrational Transitions in Encounters between Molecules, K. Takayanagi The Study of Intermolecular Potentials with Molecular Beams at Thermal Energies, H. Pauly and J.P. Toennies High-Intensity and High-Energy Molecular Beams, J.B. Anderson, R.P. Anders and J.B. Fen

Volume 2 The Calculation of van der Waals Interactions, A. Dalgarno and W.D. Davison Thermal Diffusion in Gases, E.A. Mason, R.J. Munn and Francis J. Smith Spectroscopy in the Vacuum Ultraviolet, W.R.S. Garton The Measurement of the Photoionization Cross Sections of the Atomic Gases, James A.R. Samson The Theory of Electron–Atom Collisions, R. Peterkop and V. Veldre Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, F.J. de Heer Mass Spectrometry of Free Radicals, S.N. Foner

Volume 3 The Quantal Calculation of Photoionization Cross Sections, A.L. Stewart Radiofrequency Spectroscopy of Stored Ions I: Storage, H.G. Dehmelt Optical Pumping Methods in Atomic Spectroscopy, B. Budick Energy Transfer in Organic Molecular Crystals: A Survey of Experiments, H.C. Wolf Atomic and Molecular Scattering from Solid Surfaces, Robert E. Stickney Quantum, Mechanics in Gas Crystal-Surface van der Waals Scattering, E. Chanoch Beder Reactive Collisions between Gas and Surface Atoms, Henry Wise and Bernard J. Wood

Volume 4 H.S.W. Massey—A Sixtieth Birthday Tribute, E.H.S. Burhop Electronic Eigenenergies of the Hydrogen Molecular Ion, D.R. Bates and R.H.G. Reid Applications of Quantum Theory to the Viscosity of Dilute Gases, R.A. Buckingham and E. Gal Positrons and Positronium in Gases, P.A. Fraser Classical Theory of Atomic Scattering, A. Burgess and I.C. Percival Born Expansions, A.R. Holt and B.L. Moiseiwitsch Resonances in Electron Scattering by Atoms and Molecules, P.G. Burke 575

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Contents of Volumes in This Serial

Relativistic Inner Shell Ionizations, C.B.O. Mohr Recent Measurements on Charge Transfer, J.B. Hasted Measurements of Electron Excitation Functions, D.W.O. Heddle and R.G.W. Keesing Some New Experimental Methods in Collision Physics, R.F. Stebbings Atomic Collision Processes in Gaseous Nebulae, M.J. Seaton Collisions in the Ionosphere, A. Dalgarno The Direct Study of Ionization in Space, R.L.F. Boyd Volume 5 Flowing Afterglow Measurements of Ion-Neutral Reactions, E.E. Ferguson, F.C. Fehsenfeld and A.L. Schmeltekopf Experiments with Merging Beams, Roy H. Neynaber Radiofrequency Spectroscopy of Stored Ions II: Spectroscopy, H.G. Dehmelt The Spectra of Molecular Solids, O. Schnepp The Meaning of Collision Broadening of Spectral Lines: The Classical Oscillator Analog, A. Ben-Reuven The Calculation of Atomic Transition Probabilities, R.J.S. Crossley Tables of One- and Two-Particle Coefficients of Fractional Parentage for Configurations s λ s tu p q , C.D.H. Chisholm, A. Dalgarno and F.R. Innes Relativistic Z-Dependent Corrections to Atomic Energy Levels, Holly Thomis Doyle Volume 6 Dissociative Recombination, J.N. Bardsley and M.A. Biondi Analysis of the Velocity Field in Plasmas from the Doppler Broadening of Spectral Emission Lines, A.S. Kaufman

The Rotational Excitation of Molecules by Slow Electrons, Kazuo Takayanagi and Yukikazu Itikawa The Diffusion of Atoms and Molecules, E.A. Mason and T.R. Marrero Theory and Application of Sturmian Functions, Manuel Rotenberg Use of Classical Mechanics in the Treatment of Collisions between Massive Systems, D.R. Bates and A.E. Kingston

Volume 7 Physics of the Hydrogen Maser, C. Audoin, J.P. Schermann and P. Grivet Molecular Wave Functions: Calculations and Use in Atomic and Molecular Process, J.C. Browne Localized Molecular Orbitals, Harel Weinstein, Ruben Pauncz and Maurice Cohen General Theory of Spin-Coupled Wave Functions for Atoms and Molecules, J. Gerratt Diabatic States of Molecules—QuasiStationary Electronic States, Thomas F. O’Malley Selection Rules within Atomic Shells, B.R. Judd Green’s Function Technique in Atomic and Molecular Physics, Gy. Csanak, H.S. Taylor and Robert Yaris A Review of Pseudo-Potentials with Emphasis on Their Application to Liquid Metals, Nathan Wiser and A.J. Greenfield

Volume 8 Interstellar Molecules: Their Formation and Destruction, D. McNally Monte Carlo Trajectory Calculations of Atomic and Molecular Excitation in Thermal Systems, James C. Keck

Contents of Volumes in This Serial Nonrelativistic Off-Shell Two-Body Coulomb Amplitudes, Joseph C.Y. Chen and Augustine C. Chen Photoionization with Molecular Beams, R.B. Cairns, Halstead Harrison and R.I. Schoen The Auger Effect, E.H.S. Burhop and W.N. Asaad Volume 9 Correlation in Excited States of Atoms, A.W. Weiss The Calculation of Electron–Atom Excitation Cross Section, M.R.H. Rudge Collision-Induced Transitions between Rotational Levels, Takeshi Oka The Differential Cross Section of Low-Energy Electron–Atom Collisions, D. Andrick Molecular Beam Electric Resonance Spectroscopy, Jens C. Zorn and Thomas C. English Atomic and Molecular Processes in the Martian Atmosphere, Michael B. McElroy Volume 10 Relativistic Effects in the Many-Electron Atom, Lloyd Armstrong Jr. and Serge Feneuille The First Born Approximation, K.L. Bell and A.E. Kingston Photoelectron Spectroscopy, W.C. Price Dye Lasers in Atomic Spectroscopy, W. Lange, J. Luther and A. Steudel Recent Progress in the Classification of the Spectra of Highly Ionized Atoms, B.C. Fawcett A Review of Jovian Ionospheric Chemistry, Wesley T. Huntress Jr. Volume 11 The Theory of Collisions between Charged Particles and Highly Excited Atoms, I.C. Percival and D. Richards

577

Electron Impact Excitation of Positive Ions, M.J. Seaton The R-Matrix Theory of Atomic Process, P.G. Burke and W.D. Robb Role of Energy in Reactive Molecular Scattering: An Information-Theoretic Approach, R.B. Bernstein and R.D. Levine Inner Shell Ionization by Incident Nuclei, Johannes M. Hansteen Stark Broadening, Hans R. Griem Chemiluminescence in Gases, M.F. Golde and B.A. Thrush

Volume 12 Nonadiabatic Transitions between Ionic and Covalent States, R.K. Janev Recent Progress in the Theory of Atomic Isotope Shift, J. Bauche and R.-J. Champeau Topics on Multiphoton Processes in Atoms, P. Lambropoulos Optical Pumping of Molecules, M. Broyer, G. Goudedard, J.C. Lehmann and J. Vigué Highly Ionized Ions, Ivan A. Sellin Time-of-Flight Scattering Spectroscopy, Wilhelm Raith Ion Chemistry in the D Region, George C. Reid

Volume 13 Atomic and Molecular Polarizabilities— Review of Recent Advances, Thomas M. Miller and Benjamin Bederson Study of Collisions by Laser Spectroscopy, Paul R. Berman Collision Experiments with Laser-Excited Atoms in Crossed Beams, I.V. Hertel and W. Stoll Scattering Studies of Rotational and Vibrational Excitation of Molecules, Manfred Faubel and J. Peter Toennies

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Low-Energy Electron Scattering by Complex Atoms: Theory and Calculations, R.K. Nesbet Microwave Transitions of Interstellar Atoms and Molecules, W.B. Somerville Volume 14 Resonances in Electron Atom and Molecule Scattering, D.E. Golden The Accurate Calculation of Atomic Properties by Numerical Methods, Brain C. Webster, Michael J. Jamieson and Ronald F. Stewart (e, 2e) Collisions, Erich Weigold and Ian E. McCarthy Forbidden Transitions in One- and Two-Electron Atoms, Richard Marrus and Peter J. Mohr Semiclassical Effects in Heavy-Particle Collisions, M.S. Child Atomic Physics Tests of the Basic Concepts in Quantum Mechanics, Francies M. Pipkin Quasi-Molecular Interference Effects in Ion–Atom Collisions, S.V. Bobashev Rydberg Atoms, S.A. Edelstein and T.F. Gallagher UV and X-Ray Spectroscopy in Astrophysics, A.K. Dupree Volume 15 Negative Ions, H.S.W. Massey Atomic Physics from Atmospheric and Astrophysical, A. Dalgarno Collisions of Highly Excited Atoms, R.F. Stebbings Theoretical Aspects of Positron Collisions in Gases, J.W. Humberston Experimental Aspects of Positron Collisions in Gases, T.C. Griffith Reactive Scattering: Recent Advances in Theory and Experiment, Richard B. Bernstein Ion–Atom Charge Transfer Collisions at Low Energies, J.B. Hasted

Aspects of Recombination, D.R. Bates The Theory of Fast Heavy Particle Collisions, B.H. Bransden Atomic Collision Processes in Controlled Thermonuclear Fusion Research, H.B. Gilbody Inner-Shell Ionization, E.H.S. Burhop Excitation of Atoms by Electron Impact, D.W.O. Heddle Coherence and Correlation in Atomic Collisions, H. Kleinpoppen Theory of Low Energy Electron–Molecule Collisions, P.O. Burke Volume 16 Atomic Hartree–Fock Theory, M. Cohen and R.P. McEachran Experiments and Model Calculations to Determine Interatomic Potentials, R. Düren Sources of Polarized Electrons, R.J. Celotta and D.T. Pierce Theory of Atomic Processes in Strong Resonant Electromagnetic Fields, S. Swain Spectroscopy of Laser-Produced Plasmas, M.H. Key and R.J. Hutcheon Relativistic Effects in Atomic Collisions Theory, B.L. Moiseiwitsch Parity Nonconservation in Atoms: Status of Theory and Experiment, E.N. Fortson and L. Wilets Volume 17 Collective Effects in Photoionization of Atoms, M.Ya. Amusia Nonadiabatic Charge Transfer, D.S.F. Crothers Atomic Rydberg States, Serge Feneuille and Pierre Jacquinot Superfluorescence, M.F.H. Schuurmans, Q.H.F. Vrehen, D. Polder and H.M. Gibbs Applications of Resonance Ionization Spectroscopy in Atomic and Molecular

Contents of Volumes in This Serial Physics, M.G. Payne, C.H. Chen, G.S. Hurst and G.W. Foltz Inner-Shell Vacancy Production in Ion–Atom Collisions, C.D. Lin and Patrick Richard Atomic Processes in the Sun, P.L. Dufton and A.E. Kingston Volume 18 Theory of Electron–Atom Scattering in a Radiation Field, Leonard Rosenberg Positron–Gas Scattering Experiments, Talbert S. Stein and Walter E. Kaupplia Nonresonant Multiphoton Ionization of Atoms, J. Morellec, D. Normand and G. Petite Classical and Semiclassical Methods in Inelastic Heavy-Particle Collisions, A.S. Dickinson and D. Richards Recent Computational Developments in the Use of Complex Scaling in Resonance Phenomena, B.R. Junker Direct Excitation in Atomic Collisions: Studies of Quasi-One-Electron Systems, N. Andersen and S.E. Nielsen Model Potentials in Atomic Structure, A. Hibbert Recent Developments in the Theory of Electron Scattering by Highly Polar Molecules, D.W. Norcross and L.A. Collins Quantum Electrodynamic Effects in Few-Electron Atomic Systems, G.W.F. Drake Volume 19 Electron Capture in Collisions of Hydrogen Atoms with Fully Stripped Ions, B.H. Bransden and R.K. Janev Interactions of Simple Ion Atom Systems, J.T. Park High-Resolution Spectroscopy of Stored Ions, D.J. Wineland, Wayne M. Itano and R.S. Van Dyck Jr.

579

Spin-Dependent Phenomena in Inelastic Electron–Atom Collisions, K. Blum and H. Kleinpoppen The Reduced Potential Curve Method for Diatomic Molecules and Its Applications, F. Jenˇc The Vibrational Excitation of Molecules by Electron Impact, D.G. Thompson Vibrational and Rotational Excitation in Molecular Collisions, Manfred Faubel Spin Polarization of Atomic and Molecular Photoelectrons, N.A. Cherepkov Volume 20 Ion–Ion Recombination in an Ambient Gas, D.R. Bates Atomic Charges within Molecules, G.G. Hall Experimental Studies on Cluster Ions, T.D. Mark and A.W. Castleman Jr. Nuclear Reaction Effects on Atomic Inner-Shell Ionization, W.E. Meyerhof and J.-F. Chemin Numerical Calculations on Electron-Impact Ionization, Christopher Bottcher Electron and Ion Mobilities, Gordon R. Freeman and David A. Armstrong On the Problem of Extreme UV and X-Ray Lasers, I.I. Sobel’man and A.V. Vinogradov Radiative Properties of Rydberg States in Resonant Cavities, S. Haroche and J.M. Raimond Rydberg Atoms: High-Resolution Spectroscopy and Radiation Interaction—Rydberg Molecules, J.A.C. Gallas, G. Leuchs, H. Walther, and H. Figger Volume 21 Subnatural Linewidths in Atomic Spectroscopy, Dennis P. O’Brien, Pierre Meystre and Herbert Walther Molecular Applications of Quantum Defect Theory, Chris H. Greene and Ch. Jungen

580


Theory of Dielectronic Recombination, Yukap Hahn Recent Developments in Semiclassical Floquet Theories for Intense-Field Multiphoton Processes, Shih-I Chu Scattering in Strong Magnetic Fields, M.R.C. McDowell and M. Zarcone Pressure Ionization, Resonances and the Continuity of Bound and Free States, R.M. More Volume 22 Positronium—Its Formation and Interaction with Simple Systems, J.W. Humberston Experimental Aspects of Positron and Positronium Physics, T.C. Griffith Doubly Excited States, Including New Classification Schemes, C.D. Lin Measurements of Charge Transfer and Ionization in Collisions Involving Hydrogen Atoms, H.B. Gilbody Electron Ion and Ion–Ion Collisions with Intersecting Beams, K. Dolder and B. Peart Electron Capture by Simple Ions, Edward Pollack and Yukap Hahn Relativistic Heavy-Ion–Atom Collisions, R. Anholt and Harvey Gould Continued-Fraction Methods in Atomic Physics, S. Swain Volume 23 Vacuum Ultraviolet Laser Spectroscopy of Small Molecules, C.R. Vidal Foundations of the Relativistic Theory of Atomic and Molecular Structure, Ian P. Grant and Harry M. Quiney Point-Charge Models for Molecules Derived from Least-Squares Fitting of the Electric Potential, D.E. Williams and Ji-Min Yan Transition Arrays in the Spectra of Ionized Atoms, J. Bauche, C. Bauche-Arnoult and M. Klapisch

Photoionization and Collisional Ionization of Excited Atoms Using Synchrotron and Laser Radiation, F.J. Wuilleumier, D.L. Ederer and J.L. Picqué Volume 24 The Selected Ion Flow Tube (SIDT): Studies of Ion-Neutral Reactions, D. Smith and N.G. Adams Near-Threshold Electron–Molecule Scattering, Michael A. Morrison Angular Correlation in Multiphoton Ionization of Atoms, S.J. Smith and G. Leuchs Optical Pumping and Spin Exchange in Gas Cells, R.J. Knize, Z. Wu and W. Happer Correlations in Electron–Atom Scattering, A. Crowe Volume 25 Alexander Dalgarno: Life and Personality, David R. Bates and George A. Victor Alexander Dalgarno: Contributions to Atomic and Molecular Physics, Neal Lane Alexander Dalgarno: Contributions to Aeronomy, Michael B. McElroy Alexander Dalgarno: Contributions to Astrophysics, David A. Williams Dipole Polarizability Measurements, Thomas M. Miller and Benjamin Bederson Flow Tube Studies of Ion–Molecule Reactions, Eldon Ferguson Differential Scattering in He–He and He+ –He Collisions at keV Energies, R.F. Stebbings Atomic Excitation in Dense Plasmas, Jon C. Weisheit Pressure Broadening and Laser-Induced Spectral Line Shapes, Kenneth M. Sando and Shih-I. Chu Model-Potential Methods, C. Laughlin and G.A. Victor

Contents of Volumes in This Serial Z-Expansion Methods, M. Cohen Schwinger Variational Methods, Deborah Kay Watson Fine-Structure Transitions in Proton–Ion Collisions, R.H.G. Reid Electron Impact Excitation, R.J.W. Henry and A.E. Kingston Recent Advances in the Numerical Calculation of Ionization Amplitudes, Christopher Bottcher The Numerical Solution of the Equations of Molecular Scattering, A.C. Allison High Energy Charge Transfer, B.H. Bransden and D.P. Dewangan Relativistic Random-Phase Approximation, W.R. Johnson Relativistic Sturmian and Finite Basis Set Methods in Atomic Physics, G.W.F. Drake and S.P. Goldman Dissociation Dynamics of Polyatomic Molecules, T. Uzer Photodissociation Processes in Diatomic Molecules of Astrophysical Interest, Kate P. Kirby and Ewine F. van Dishoeck The Abundances and Excitation of Interstellar Molecules, John H. Black Volume 26 Comparisons of Positrons and Electron Scattering by Gases, Walter E. Kauppila and Talbert S. Stein Electron Capture at Relativistic Energies, B.L. Moiseiwitsch The Low-Energy, Heavy Particle Collisions—A Close-Coupling Treatment, Mineo Kimura and Neal F. Lane Vibronic Phenomena in Collisions of Atomic and Molecular Species, V. Sidis Associative Ionization: Experiments, Potentials and Dynamics, John Weiner Françoise Masnou-Seeuws and Annick Giusti-Suzor On the β Decay of 187 Re: An Interface of Atomic and Nuclear Physics and

581

Cosmochronology, Zonghau Chen, Leonard Rosenberg and Larry Spruch Progress in Low Pressure Mercury-Rare Gas Discharge Research, J. Maya and R. Lagushenko Volume 27 Negative Ions: Structure and Spectra, David R. Bates Electron Polarization Phenomena in Electron–Atom Collisions, Joachim Kessler Electron–Atom Scattering, I.E. McCarthy and E. Weigold Electron–Atom Ionization, I.E. McCarthy and E. Weigold Role of Autoionizing States in Multiphoton Ionization of Complex Atoms, V.I. Lengyel and M.I. Haysak Multiphoton Ionization of Atomic Hydrogen Using Perturbation Theory, E. Karule Volume 28 The Theory of Fast Ion–Atom Collisions, J.S. Briggs and J.H. Macek Some Recent Developments in the Fundamental Theory of Light, Peter W. Milonni and Surendra Singh Squeezed States of the Radiation Field, Khalid Zaheer and M. Suhail Zubairy Cavity Quantum Electrodynamics, E.A. Hinds Volume 29 Studies of Electron Excitation of Rare-Gas Atoms into and out of Metastable Levels Using Optical and Laser Techniques, Chun C. Lin and L.W. Anderson Cross Sections for Direct Multiphoton Ionization of Atoms, M.V. Ammosov, N.B. Delone, M.Yu. Ivanov, I.I. Bandar and A.V. Masalov Collision-Induced Coherences in Optical Physics, G.S. Agarwal

582


Muon-Catalyzed Fusion, Johann Rafelski and Helga E. Rafelski Cooperative Effects in Atomic Physics, J.P. Connerade Multiple Electron Excitation, Ionization, and Transfer in High-Velocity Atomic and Molecular Collisions, J.H. McGuire Volume 30 Differential Cross Sections for Excitation of Helium Atoms and Helium-Like Ions by Electron Impact, Shinobu Nakazaki Cross-Section Measurements for Electron Impact on Excited Atomic Species, S. Trajmar and J.C. Nickel The Dissociative Ionization of Simple Molecules by Fast Ions, Colin J. Latimer Theory of Collisions between Laser Cooled Atoms, P.S. Julienne, A.M. Smith and K. Burnett Light-Induced Drift, E.R. Eliel Continuum Distorted Wave Methods in Ion–Atom Collisions, Derrick S.F. Crothers and Louis J. Dube Volume 31 Energies and Asymptotic Analysis for Helium Rydberg States, G.W.F. Drake Spectroscopy of Trapped Ions, R.C. Thompson Phase Transitions of Stored Laser-Cooled Ions, H. Walther Selection of Electronic States in Atomic Beams with Lasers, Jacques Baudon, Rudalf Dülren and Jacques Robert Atomic Physics and Non-Maxwellian Plasmas, Michèle Lamoureux Volume 32 Photoionization of Atomic Oxygen and Atomic Nitrogen, K.L. Bell and A.E. Kingston Positronium Formation by Positron Impact on Atoms at Intermediate Energies, B.H. Bransden and C.J. Noble

Electron–Atom Scattering Theory and Calculations, P.G. Burke Terrestrial and Extraterrestrial H+ 3, Alexander Dalgarno Indirect Ionization of Positive Atomic Ions, K. Dolder Quantum Defect Theory and Analysis of High-Precision Helium Term Energies, G.W.F. Drake Electron–Ion and Ion–Ion Recombination Processes, M.R. Flannery Studies of State-Selective Electron Capture in Atomic Hydrogen by Translational Energy Spectroscopy, H.B. Gilbody Relativistic Electronic Structure of Atoms and Molecules, I.P. Grant The Chemistry of Stellar Environments, D.A. Howe, J.M.C. Rawlings and D.A. Williams Positron and Positronium Scattering at Low Energies, J.W. Humberston How Perfect are Complete Atomic Collision Experiments?, H. Kleinpoppen and H. Handy Adiabatic Expansions and Nonadiabatic Effects, R. McCarroll and D.S.F. Crothers Electron Capture to the Continuum, B.L. Moiseiwitsch How Opaque Is a Star?, M.T. Seaton Studies of Electron Attachment at Thermal Energies Using the Flowing Afterglow–Langmuir Technique, David Smith and Patrik Španˇel Exact and Approximate Rate Equations in Atom–Field Interactions, S. Swain Atoms in Cavities and Traps, H. Walther Some Recent Advances in Electron-Impact Excitation of n = 3 States of Atomic Hydrogen and Helium, J.F. Williams and J.B. Wang Volume 33 Principles and Methods for Measurement of Electron Impact Excitation Cross

Contents of Volumes in This Serial Sections for Atoms and Molecules by Optical Techniques, A.R. Filippelli, Chun C. Lin, L.W. Andersen and J.W. McConkey Benchmark Measurements of Cross Sections for Electron Collisions: Analysis of Scattered Electrons, S. Trajmar and J.W. McConkey Benchmark Measurements of Cross Sections for Electron Collisions: Electron Swarm Methods, R.W. Crompton Some Benchmark Measurements of Cross Sections for Collisions of Simple Heavy Particles, H.B. Gilbody The Role of Theory in the Evaluation and Interpretation of Cross-Section Data, Barry I. Schneider Analytic Representation of Cross-Section Data, Mitio Inokuti, Mineo Kimura, M.A. Dillon, Isao Shimamura Electron Collisions with N2 , O2 and O: What We Do and Do Not Know, Yukikazu Itikawa Need for Cross Sections in Fusion Plasma Research, Hugh P. Summers Need for Cross Sections in Plasma Chemistry, M. Capitelli, R. Celiberto and M. Cacciatore Guide for Users of Data Resources, Jean W. Gallagher Guide to Bibliographies, Books, Reviews and Compendia of Data on Atomic Collisions, E.W. McDaniel and E.J. Mansky Volume 34 Atom Interferometry, C.S. Adams, O. Carnal and J. Mlynek Optical Tests of Quantum Mechanics, R.Y. Chiao, P.G. Kwiat and A.M. Steinberg Classical and Quantum Chaos in Atomic Systems, Dominique Delande and Andreas Buchleitner

583

Measurements of Collisions between Laser-Cooled Atoms, Thad Walker and Paul Feng The Measurement and Analysis of Electric Fields in Glow Discharge Plasmas, J.E. Lawler and D.A. Doughty Polarization and Orientation Phenomena in Photoionization of Molecules, N.A. Cherepkov Role of Two-Center Electron–Electron Interaction in Projectile Electron Excitation and Loss, E.C. Montenegro, W.E. Meyerhof and J.H. McGuire Indirect Processes in Electron Impact Ionization of Positive Ions, D.L. Moores and K.J. Reed Dissociative Recombination: Crossing and Tunneling Modes, David R. Bates Volume 35 Laser Manipulation of Atoms, K. Sengstock and W. Ertmer Advances in Ultracold Collisions: Experiment and Theory, J. Weiner Ionization Dynamics in Strong Laser Fields, L.F. DiMauro and P. Agostini Infrared Spectroscopy of Size Selected Molecular Clusters, U. Buck Fermosecond Spectroscopy of Molecules and Clusters, T. Baumer and G. Gerber Calculation of Electron Scattering on Hydrogenic Targets, I. Bray and A.T. Stelbovics Relativistic Calculations of Transition Amplitudes in the Helium Isoelectronic Sequence, W.R. Johnson, D.R. Plante and J. Sapirstein Rotational Energy Transfer in Small Polyatomic Molecules, H.O. Everitt and F.C. De Lucia Volume 36 Complete Experiments in Electron–Atom Collisions, Nils Overgaard Andersen and Klaus Bartschat

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Stimulated Rayleigh Resonances and Recoil-Induced Effects, J.-Y. Courtois and G. Grynberg Precision Laser Spectroscopy Using Acousto-Optic Modulators, W.A. van Mijngaanden Highly Parallel Computational Techniques for Electron–Molecule Collisions, Carl Winstead and Vincent McKoy Quantum Field Theory of Atoms and Photons, Maciej Lewenstein and Li You Volume 37 Evanescent Light-Wave Atom Mirrors, Resonators, Waveguides, and Traps, Jonathan P. Dowling and Julio Gea-Banacloche Optical Lattices, P.S. Jessen and I.H. Deutsch Channeling Heavy Ions through Crystalline Lattices, Herbert F. Krause and Sheldon Datz Evaporative Cooling of Trapped Atoms, Wolfgang Ketterle and N.J. van Druten Nonclassical States of Motion in Ion Traps, J.I. Cirac, A.S. Parkins, R. Blatt and P. Zoller The Physics of Highly-Charged Heavy Ions Revealed by Storage/Cooler Rings, P.H. Mokler and Th. Stöhlker Volume 38 Electronic Wavepackets, Robert R. Jones and L.D. Noordam Chiral Effects in Electron Scattering by Molecules, K. Blum and D.G. Thompson Optical and Magneto-Optical Spectroscopy of Point Defects in Condensed Helium, Serguei I. Kanorsky and Antoine Weis Rydberg Ionization: From Field to Photon, G.M. Lankhuijzen and L.D. Noordam Studies of Negative Ions in Storage Rings, L.H. Andersen, T. Andersen and P. Hvelplund

Single-Molecule Spectroscopy and Quantum Optics in Solids, W.E. Moerner, R.M. Dickson and D.J. Norris

Volume 39 Author and Subject Cumulative Index Volumes 1–38 Author Index Subject Index Appendix: Tables of Contents of Volumes 1–38 and Supplements

Volume 40 Electric Dipole Moments of Leptons, Eugene D. Commins High-Precision Calculations for the Ground and Excited States of the Lithium Atom, Frederick W. King Storage Ring Laser Spectroscopy, Thomas U. Kühl Laser Cooling of Solids, Carl E. Mangan and Timothy R. Gosnell Optical Pattern Formation, L.A. Lugiato, M. Brambilla and A. Gatti

Volume 41 Two-Photon Entanglement and Quantum Reality, Yanhua Shih Quantum Chaos with Cold Atoms, Mark G. Raizen Study of the Spatial and Temporal Coherence of High-Order Harmonics, Pascal Salières, Ann L’Huillier, Philippe Antoine and Maciej Lewenstein Atom Optics in Quantized Light Fields, Matthias Freyburger, Alois M. Herkommer, Daniel S. Krähmer, Erwin Mayr and Wolfgang P. Schleich Atom Waveguides, Victor I. Balykin Atomic Matter Wave Amplification by Optical Pumping, Ulf Janicke and Martin Wikens

Contents of Volumes in This Serial Volume 42 Fundamental Tests of Quantum Mechanics, Edward S. Fry and Thomas Walther Wave-Particle Duality in an Atom Interferometer, Stephan Dürr and Gerhard Rempe Atom Holography, Fujio Shimizu Optical Dipole Traps for Neutral Atoms, Rudolf Grimm, Matthias Weidemüller and Yurii B. Ovchinnikov Formation of Cold (T ≤ 1 K) Molecules, J.T. Bahns, P.L. Gould and W.C. Stwalley High-Intensity Laser-Atom Physics, C.J. Joachain, M. Dorr and N.J. Kylstra Coherent Control of Atomic, Molecular and Electronic Processes, Moshe Shapiro and Paul Brumer Resonant Nonlinear Optics in Phase Coherent Media, M.D. Lukin, P. Hemmer and M.O. Scully The Characterization of Liquid and Solid Surfaces with Metastable Helium Atoms, H. Morgner Quantum Communication with Entangled Photons, Herald Weinfurter

Volume 43 Plasma Processing of Materials and Atomic, Molecular, and Optical Physics: An Introduction, Hiroshi Tanaka and Mitio Inokuti The Boltzmann Equation and Transport Coefficients of Electrons in Weakly Ionized Plasmas, R. Winkler Electron Collision Data for Plasma Chemistry Modeling, W.L. Morgan Electron–Molecule Collisions in Low-Temperature Plasmas: The Role of Theory, Carl Winstead and Vincent McKoy Electron Impact Ionization of Organic Silicon Compounds, Ralf Basner, Kurt Becker, Hans Deutsch and Martin Schmidt

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Kinetic Energy Dependence of Ion–Molecule Reactions Related to Plasma Chemistry, P.B. Armentrout Physicochemical Aspects of Atomic and Molecular Processes in Reactive Plasmas, Yoshihiko Hatano Ion–Molecule Reactions, Werner Lindinger, Armin Hansel and Zdenek Herman Uses of High-Sensitivity White-Light Absorption Spectroscopy in Chemical Vapor Deposition and Plasma Processing, L.W. Anderson, A.N. Goyette and J.E. Lawler Fundamental Processes of Plasma–Surface Interactions, Rainer Hippler Recent Applications of Gaseous Discharges: Dusty Plasmas and Upward-Directed Lightning, Ara Chutjian Opportunities and Challenges for Atomic, Molecular and Optical Physics in Plasma Chemistry, Kurl Becker Hans Deutsch and Mitio Inokuti

Volume 44 Mechanisms of Electron Transport in Electrical Discharges and Electron Collision Cross Sections, Hiroshi Tanaka and Osamu Sueoka Theoretical Consideration of Plasma-Processing Processes, Mineo Kimura Electron Collision Data for Plasma-Processing Gases, Loucas G. Christophorou and James K. Olthoff Radical Measurements in Plasma Processing, Toshio Goto Radio-Frequency Plasma Modeling for Low-Temperature Processing, Toshiaki Makabe Electron Interactions with Excited Atoms and Molecules, Loucas G. Christophorou and James K. Olthoff

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Volume 45 Comparing the Antiproton and Proton, and Opening the Way to Cold Antihydrogen, G. Gabrielse Medical Imaging with Laser-Polarized Noble Gases, Timothy Chupp and Scott Swanson Polarization and Coherence Analysis of the Optical Two-Photon Radiation from the Metastable 22 Si1/2 State of Atomic Hydrogen, Alan J. Duncan, Hans Kleinpoppen and Marian O. Scully Laser Spectroscopy of Small Molecules, W. Demtröder, M. Keil and H. Wenz Coulomb Explosion Imaging of Molecules, Z. Vager Volume 46 Femtosecond Quantum Control, T. Brixner, N.H. Damrauer and G. Gerber Coherent Manipulation of Atoms and Molecules by Sequential Laser Pulses, N.V. Vitanov, M. Fleischhauer, B.W. Shore and K. Bergmann Slow, Ultraslow, Stored, and Frozen Light, Andrey B. Matsko, Olga Kocharovskaya, Yuri Rostovtsev George R. Welch, Alexander S. Zibrov and Marlan O. Scully Longitudinal Interferometry with Atomic Beams, S. Gupta, D.A. Kokorowski, R.A. Rubenstein, and W.W. Smith Volume 47 Nonlinear Optics of de Broglie Waves, P. Meystre Formation of Ultracold Molecules (T ≤ 200 µK) via Photoassociation in a Gas of Laser-Cooled Atoms, Françoise Masnou-Seeuws and Pierre Pillet Molecular Emissions from the Atmospheres of Giant Planets and Comets: Needs for Spectroscopic and Collision Data, Yukikazu Itikawa, Sang Joon Kim, Yong Ha Kim and Y.C. Minh

Studies of Electron-Excited Targets Using Recoil Momentum Spectroscopy with Laser Probing of the Excited State, Andrew James Murray and Peter Hammond Quantum Noise of Small Lasers, J.P. Woerdman, N.J. van Druten and M.P. van Exter Volume 48 Multiple Ionization in Strong Laser Fields, R. Dörner Th. Weber, M. Weckenbrock, A. Staudte, M. Hattass, R. Moshammer, J. Ullrich and H. Schmidt-Böcking Above-Threshold Ionization: From Classical Features to Quantum Effects, W. Becker, F. Grasbon, R. Kapold, D.B. Milošević, G.G. Paulus and H. Walther Dark Optical Traps for Cold Atoms, Nir Friedman, Ariel Kaplan and Nir Davidson Manipulation of Cold Atoms in Hollow Laser Beams, Heung-Ryoul Noh, Xenye Xu and Wonho Jhe Continuous Stern–Gerlach Effect on Atomic Ions, Günther Werth, Hartmut Haffner and Wolfgang Quint The Chirality of Biomolecules, Robert N. Compton and Richard M. Pagni Microscopic Atom Optics: From Wires to an Atom Chip, Ron Folman, Peter Krüger, Jörg Schmiedmayer, Johannes Denschlag and Carsten Henkel Methods of Measuring Electron–Atom Collision Cross Sections with an Atom Trap, R.S. Schappe, M.L. Keeler, T.A. Zimmerman, M. Larsen, P. Feng, R.C. Nesnidal, J.B. Boffard, T.G. Walker, L.W. Anderson and C.C. Lin Volume 49 Applications of Optical Cavities in Modern Atomic, Molecular, and Optical Physics, Jun Ye and Theresa W. Lynn

Contents of Volumes in This Serial Resonance and Threshold Phenomena in Low-Energy Electron Collisions with Molecules and Clusters, H. Hotop, M.-W. Ruf, M. Allan and I.I. Fabrikant Coherence Analysis and Tensor Polarization Parameters of (γ , eγ ) Photoionization Processes in Atomic Coincidence Measurements, B. Lohmann, B. Zimmermann, H. Kleinpoppen and U. Becker Quantum Measurements and New Concepts for Experiments with Trapped Ions, Ch. Wunderlich and Ch. Balzer Scattering and Reaction Processes in Powerful Laser Fields, Dejan B. Milošević and Fritz Ehlotzky Hot Atoms in the Terrestrial Atmosphere, Vijay Kumar and E. Krishnakumar

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Volume 50 Assessment of the Ozone Isotope Effect, K. Mauersberger, D. Krankowsky, C. Janssen and R. Schinke Atom Optics, Guided Atoms, and Atom Interferometry, J. Arlt, G. Birkl, E. Rasel and W. Ertmet Atom–Wall Interaction, D. Bloch and M. Ducloy Atoms Made Entirely of Antimatter: Two Methods Produce Slow Antihydrogen, G. Gabrielse Ultrafast Excitation, Ionization, and Fragmentation of C60 , I.V. Hertel, T. Laarmann and C.P. Schulz


Advances in Atomic, Molecular, and Optical Physics

Advances in Atomic, Molecular, and Optical Physics, Volume 59 (Advances in Atomic, Molecular and Optical Physics)

Advances in Atomic, Molecular, and Optical Physics

Atomic, Molecular, and Optical Physics

Atomic, Molecular, and Optical Physics