Advunces in
ATOMIC AND MOLECULAR PHYSICS
VOLUME 15
CONTRIBUTORS TO THIS VOLUME D. R . BATES RICHARD B. BERNSTEIN
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Advunces in
ATOMIC AND MOLECULAR PHYSICS
VOLUME 15
CONTRIBUTORS TO THIS VOLUME D. R . BATES RICHARD B. BERNSTEIN
B. H. BRANSDEN E. H. S. BURHOP
P. G. BURKE A. DALGARNO H. B. GILBODY T. C. GRIFFITH
J. B. HASTED D. W. 0. HEDDLE J. W. HUMBERSTON H. KLEINPOPPEN H. S. W. MASSEY
R. F. STEBBINGS
ADVANCES IN
ATOMIC AND MOLECULAR PHYSICS E d i t d hy
Sir David R. Bates DEPARTMENT OF APPLIED MATHEMATICS A N D THEORETICAL PHYSICS THE QUEEN’S UNIVERSlTY OF BELFAST BELFAST. NORTHERN IRELAND
Benjamin Bederson DEPARTMENT OF PHYSICS NEW YORK UNIVERSITY N E W YORK, N E W YORK
VOLUME 15
@
1979
ACADEMIC PRESS A Subuidiurv of Hcrrcourt Brace Joiwnnvich, Piihlishers
New York London Toronto Sydney San Francisco
COPYRIGHT @ 1979, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART O F THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.
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United Kiripdorn Edition Dublished bv ACADEM~CPRESS, rNc. (LONDON) LTD. 24/28 Oval Road, London NW1 7DX
LIBRARY OF CONGRESS CATALOG CARD NUMBER: 65-18423 ISBN 0- 12-003815-3 PRINTED IN THE UNITED STATES OF AMERICA
79 80 81 82
9 8 765 4 3 2 1
Proceedings of a Symposium held September 20-22, 1978 at the Royal Society London in honor of Sir Harrie Massey 's seventieth year
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Contents
ILISI OF C O N T R I B U l O R S SIR H A R R I E M A S S E i : I N T R O D U C T O R Y C O N F E R E N C E A D D R E S S
...
Xlll
xv
Negative Ions H . S. W . Mrrssc.y I. The Ground State of Negative Atomic Ions 11. Excited States of Atomic Negative Ions 111. Electron Affinities and Structures of Negative Molecular Ions
IV. V. VI. VII. VIII.
Dissociative Attachment Photodetachment and Photodissociation Ionic Reactions at Thermal and Epithermal Energies Ionic Reactions at High Energies Negative Ions in Electric Discharge and Breakdown Phenomena References
2 7 9 13 18 23 26 28 33
Atomic Physics from Atmospheric and Astrophysical Studies A . Lkilgrrnio
I. Introduction 11. Dissociative Recombination 111. Ion-Molecule Reactions
1V. V. VI. VI I. VIII. IX. X.
Neutral-Particle Reactions Accidental Resonance Charge Transfer Charge Transfer of Multiply Charged Ions Fine-Structure Transitions Radiative Association Microwave Spectroscopy Oscillator Strengths and Branching Ratios XI. Radiative Recombination X11. Spontaneous Radiative Dissociation of Diatomic Molecules XIII. Relativistic Magnetic Dipole Transitions References vii
37 38 42 44 46 50 53 55 56 59 62 62 67 69
...
CONTENTS
Vlll
Collisions of Highly Excited Atoms
R . F . Stehhings I. lntroduction 11. Thermal Collisions with Heavy Particles
References
77 77 99
Theoretical Aspects of Positron Collisions in Gases J . W . Hirmherston 1. Introduction 11. Positron-Hydrogen Elastic Scattering 111. Positron-Helium Elastic Scattering
IV. V. V1. VII.
Annihilation in Positron-Atom Scattering Angular Correlations in Positron Annihilation Positronium Formation in Positron- Atom Collisions Concluding Remarks References
101 102 105 1 I8 124 126 130 I3 1
Experimental Aspects of Positron Collisions in Gases T . C. Grifjith 1. Introduction 11. Scattering Techniques with Positron Beams
111. Accuracy of the Cross Section Data 1V. Total Cross Section Measurements V. Lifetime Studies References
135
138 142 146 159 164
Reactive Scattering: Recent Advances in Theory and Experiment Richard B . Bernstein 1. 11. 111. IV.
Introduction Potential-Energy Surfaces Classical Trajectory Methods and Results Transition-State Theory: New Developments V. Collisional Ionization: Nonadiabatic Reactions
167 168 171 173 175
CONTENTS
VI. VII. VIIl. IX. X. XI. XII.
Accurate Quanta1 Scattering Calculations Information-Theoretic Approach to Reactive Scattering Molecular-Beam Chemistry Crossed-Beam Chemiluminescence State-to-State Cross Sections Influence of Different Forrns of Energy upon Reactivity Translational Thresholds References
ix 179 i80 181
183 187
189 193 198
Ion- Atom Charge Transfer Collisions at Low Energies
.I. B . Hri s t r d I. 11. Ill. IV .
Introduction Symmetrical Resonance Processes Nonresonant Atomic Chatge Transfer Processes ‘Total Cross Sections of Pseudocrossing Atomic Charge Transfer Processes V. Curve-Crossing Spectroscopy VI. Charge Transfer Processes of Excited Ions References
205 206 21 1 214 22 1 229 23 I
Aspects of Recombination
D . R. Brrtos I . lntroduction 11. Radiative e-O+ Recombination and the Nightglow 111. Complex Ions IV. Recombination in an Ambient Electron Gas V. Recombination in an Ambient Neutral Gas References
235 235 238 245 250 259
The Theory of Fast Heavy Particle Collisions
B . H . Brotisclcti I. 11. 111. IV.
Introduction Excitation of Atoms by Ions Electron Capture from Atoms by Fast Ions Ionization and Charge Exchange into the Continuum References
263 266 214 286 288
CONTENTS
X
Atomic Collision Processes in Controlled Thermonuclear Fusion Research H . B . Cilhody I . Introduction 11. Classification of Relevant Heavy-Particle Collision Processes 111. Experimental Studies
References
293 295 300 326
Inner-Shell Ionization
E . H . S. Birrhop I . Introduction 11. Inner-Shell Ionization by Electrons 111. Inner-Shell Ionization by Atomic Ions
IV. Radiations Following Inner-Shell Ionization References
329 329 335 362 377
Excitation of Atoms by Electron Impact D. W . 0. Hedcilr I. Introduction 11. Secondary Effects 111. Behavior near Threshold IV. Measurements by Different Techniques
V. Time-Resolved Measurements V1. The Determination of Cross Sections in Absolute Terms VII. Miscellaneous Measurements References
38 1 382 391 394 398 40 I 415 419
Coherence and Correlation in Atomic Collisions
H . Kleinpoppon I . Introduction
423
11. Angular Correlation and Spin Experiments as Tools for Studying
Impact Ionization 111. Particle-Photon Angular Correlations
1V. Electron-Ion Angular Correlations from Autoionizing States
425 437 455
CONTENTS
V. Summary and Conclusions Appendix: Coherent Excitation of Degenerate States with Different Angular Momenta References
xi 460 462 464
Theory of Low Energy Electron- Molecule Collisions P . G. Burke I . Introduction 11. Laboratory Frame Representation 111. Molecular-Frame Representation IV. Frame Transformation Theory V . L 2 Methods VI. Vibrational Excitation VII. Conclusions References AUTHOR INDEX SUBJECT INDEX CONTENTS OF PREVIOUS VOLUMES
47 1 473 480 485 488 495 503 504 507 53 1 54 1
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List of Contributors Numbers in parentheses indicate the pages on which the authors’ contributions begin.
D. R. BATES, Department of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Belfast BT7 INN, Northern Ireland (235) RICHARD B. BERNSTEIN, Chemistry Department, Columbia University, New York, New York 10027 (167) B. H . BRANSDEN, University of Durham, Durham, England (263) E . H. S. BURHOP”, CERN, Geneva, Switzerland (329) P. G. BURKE, Department of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Belfast BT7 INN, Northern Ireland, and Science Research Council, Daresbury Laboratory, Daresbury Warington, England (471) A. DALGARNO, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138 (37) H. B. GILBODY, Department of Pure and Applied Physics, The Queen’s University of Belfast, Belfast BT7 1 NN, Northern Ireland (293)
T. C. GRIFFITH, Department of Physics and Astronomy, University College London, Gower Street, London WClE 6BT, England (135) J. B. HASTED, Birkbeck College, University of London, Malet Street, London WC 1, England (205) D. W. 0. HEDDLE, Physics Department, Royal Holloway College, University of London, Egham, Surrey, England (381) J. W. HUMBERSTON, Department of Physics and Astronomy, University College London, Gower Street, London WClE 6BT, England (101)
* Present address: University College London, London. England. xiii
xiv
LIST OF CONTRIBUTORS
H. KLEINPOPPEN?, Fakultat fur Physik, Universitat Bielefeld, Bielefeld, West Germany (423) H. S. W. MASSEY, Department of Physics, University College London, London WClE 6BT, England (1) R. F. STEBBINGS, Department of Space Physics and Astronomy, Rice University, Houston, Texas 77001 (77)
? Present address: Institute of Atomic Physics, University of Stirling, Stirling,
Scotland.
Sir Hurrie Mcrssey Iiztvodirctsry Conferencc Address By Sir David R. Bates
This is a magnificant gathering of far-flung Clan M'Atomic to honor The M'Atomic himself in this, his seventieth year. A scientific clan has obvious differences from and similarities to a blood clan. In it the child is fertilized and the process of fertilization helps to keep the parent young. The age difference between parent and child is less than in a blood clan. For instance, Sir Harrie is my scientific father, yet the difference between our ages is only 8 years. It is by no means exceptional for a scientific child to be about the same age as his parent. Or even to be older! This gathering extends over many scientific generations. In ordinary terms Sir Harrie is a grandparent; but though a stranger would not judge him to be such he is a scientific grand-to-the-power-six-or-moreparent. Clan M'Atomic is fortunate in having such a patriarch. Sir Harrie did not seek the role. It fell to his lot naturally and inevitably because of the unique part he has played in the advancement of the subject. Moreover he has a special place in our hearts owing to his unassuming manner, his ready accessibility, his warmth of personality, and his faculty for preventing time and distance from cooling friendship. I can only speak for myself but I am confident that all of you will endorse my feelings. These I can best express by saying that he has always been so much more than kind that I have come to regard him, and Lady Massey also, as little less than kindred. I will give an example of scientific generosity. It arose from his discovery with E. C. Bullard of diffraction effects in the observed scattering of electrons by argon atoms. Those of you familiar with the history of our field will know that prior to their work in 1930, scattering experiments had been confined to angles less than 60". They extended the range to 120". Almost half a century later Sir Edward Bullard, now of course a geophysicist of great distinction, was good enough to write an account of the discovery of diffraction for me from which it is worth giving this extract: I remember very clearly the day we got the critical results. I was moving the collector round in 5" steps and Massey was reading the electrometer which measured the collected current. The current as in the earlier studies fell rapidly as we went away from the main beam: then suddenly Massey said "You've turned it the wrong way the xv
xvi
INTRODUCTORY CONFERENCE ADDRESS current has gone up.” I said “I haven’t,’’ and we checked the measurements. We had found a peak in the scattered current at an angle around 90” from the main beam.
Paying tribute to the value of Massey’s already deep knowledge of wave mechanics, then only a 4-year-old infant, he added, “It is rare to be able to recognize a significant new observation as it is made. I can only think of one other example in my own experience.” It was a very exciting discovery and the two gave an account of it in a paper that Rutherford submitted to the Royal Society (Bullard and Massey, 1931). About a month afterwards Rutherford came along and announced “Arnot says he got results like yours in mercury vapor some time ago. I asked him why he hadn’t written them up at once and he said he thought they were wrong and hadn’t told anyone.’ Now he says he knows they are all right because they are similar to your results and has written a paper! What would you like me to do? Hold his paper until your paper has come out‘?” Back came the generous reply. “No! Have it published as soon as possible .” In his fine appreciation of Sir Harrie published in Adi~zncvsin Atomic rrrzd Molecrilar Physics, Eric Burhop has included a summary of his old friend’s career and achievements up to 1968 (Burhop, 1968). Originally 1 had intended to add to this appreciation by discussing at length several of Sir Harrie’s research papers in order to attempt to convey the savor of some of his qualities, for instance, his quite exceptional ability at mathematical analysis illustrated by his treatment with C . B. 0. Mohr (Massey and Mohr, 1933) of the problem of ionization of atomic hydrogen by electron impact. And again his penetrating insight into collision theory illustrated by his work with R. A . Smith (Massey and Smith, 1933) on the passage of positive ions through gases. But on reflection I realized that for me to make such an attempt would be presumptuous. And moreover that it would be superfluous for the majority of you. The minority should try and repeat the analysis of Massey and Mohr even with their paper open in front of them-as well as with a pile of foolscap sheets handy. And they should ponder too over the concepts in the paper by Massey and Smith bearing their novelty and timeliness in mind. Since 1968 Sir Harrie has not decelerated. I shall mention briefly a few aspects of his work as a scientist and as a man of affairs during the past decade. He has written the huge third edition of “Negative Ions“ (Massey, 1976), the forthcoming “Atomic and Molecular Collisions” (Massey, Arnot’s attitude may well have been influenced by a recent announcement of results on diffraction, which proved to be spurious. In his 1930 paper he stated that the slight hump he had noticed earlier but not reported was ‘.just about the size of the experimental error” (Arnot, 1931). He acknowledged (1933) that it was the careful tests carried out by Bullard and Massey that first showed the diffraction effects to be genuine.
INTRODUCTORY CONFERENCE ADDRESS
xvii
1979), and with Eric Burhop and Brian Gilbody the five-volume second edition of “Electronic and Ionic lmpact Phenomena” (Massey, 1969, 1971; Massey and Burhop, 1969: Massey and Gilbody, 1974; Masseyet u / . , 1974). At University College, London he initiated and negotiated t h e mutually beneficial amalgamation of the Physics and Astronomy Departments; he was responsible for the important new infrared observational program using ground-based and balloon-borne telescopes for the study of planetary nebulae, HI1 regions, and other objects: and he arranged with the Appleton Laboratory for collaborative instrumental research in connection with the International Ultra Violet Explorer Satellite. Again he has played a vigorous part in the multifarious activities of the Royal Society as its Physical Secretary. This prestigious office entails many duties. They range from attending numerous formal receptions, lunches, and dinners, through having editorial responsibility for Series A of the Philosophicwl Ti-tinscrctiotis and of the Proceedings, to being a member of about 50 committees and currently being Chairman of the Ordnance Survey Committee and the British National Committee on Space Research. For a reason that will be apparent to you all later, I shall concentrate your attention on one early feat. As has been wisely said it is given only to the immortals of a subject to write the first book on it, or the last. Sir Harrie has surely scored a double with the slim first (Massey, 1938) and thick third (1976) editions of “Negative Ions”--my own favorite. It is to be doubted too if a comprehensive treatise covering the field of “Electronic and lonic Impact Phenomena” could ever again be written. There is no question regarding the place in the literature of the classic monograph “The Theory of Atomic Collisions,“ the first edition of which he wrote with N . F. Mott, for whose originality he has often expressed admiration (Mott and Massey, 1933). On several occasions Sir Harrie has referred to “the remarkable good nature” of his senior by 3 years, now Sir Neville Mott, Nobel Laureate, in inviting him to collaborate in the project so soon after he had entered the field. We must be thankful that Sir Neville was so percipient. Let me recall a few facts. In 1926 when Himself was plain Harrie Massey a brilliant student of 18, Schrodinger published the four communications in which he developed wave mechanics. In the same year Born published the paper in which he presented the approximation named after him and in which also, when discussing scattering, he proposed the probability interpretation of the wavefunction. Collision theory was then developed rapidly. To participate in writing the first coherent account of this fastgrowing subject ready for publication in 1933 was a truly formidable task. To help bring the work to a successful conclusion was a quite extraordinary achievement for a young man of 24 as Dr. H. S . W. Massey was
xviii
INTRODUCTORY CONFERENCE ADDRESS
when the book was being written. Just how extraordinary each of us can easily judge by asking ourselves one question-and answering it candidly. What had we achieved when we were 24 years old? “The Theory of Atomic Collisions” is much more than a summary of the contents of research papers. It is itself an original work and exemplifies to perfection the difficult art of synthesis. The difficulty of the art is not always fully appreciated. When the first edition of the great monograph appeared, P. M. S. Blackett commented that he was sorry that it had so much mathematics but he supposed that this was inevitable. Much mathematics was certainly inevitable in “The Theory of Atomic Collisions.” Considerably more than mathematics is, however, contained. And a reader, as distinct from a looker-up-of-equations, will recognize an attitude of mind that all who have been privileged to have a personal association with Sir Harrie will be very familiar: that while Mathematics may be the Queen of Sciences she is but a Servant of Thought. Few receive the satisfaction of being honored by an occasion such as this in their seventieth year. And most of those who are so honored are prudently not allowed a speaking part. But Sir Harrie is of course exceptional. Happily he continues with undiminished zest to merit star rating at any conference he chooses to address. REFERENCES Arnot, F. L. (1931). P ~ ~ JR. c , Soc. . Londori. S e r . A 130, 655. Arnot, F. L. (1933). “Collision Process in Gases,” p. 59. Methuen, London. . A 130, 579. Bullard, E. C . , and Massey, H. S. W. (1931). P r o c . R. S i r . L o n d ( ~ / iSer. Burhop, E. H. S. (1968). Adv. A t . Mol. P h y s . 4, I . Massey, H. S. W. (1938). “Negative Ions,” 1st ed. Cambridge Univ. Press, London and New York. Massey, H. S. W. (1969). “Electronic and Ionic Impact Phenomena,” Vol. 2. Oxford Univ. Press (Clarendon), London and New York. Massey. H. S. W. (1971). “Electronic and Ionic Impact Phenomena,” Vol. 2. Oxford Univ. Press (Clarendon), London and New York. Massey, H . S. W. (1976). ”Negative Ions,’‘ 3rd ed. Cambridge Univ. Press, London and New York. Massey. H. S. W. (1979). “Atomic and Molecular Collisions.” Taylor Francis, London. Massey. H. S. W., and Burhop, E. H. S. (1969). “Electronic and Ionic Impact Phenomena,” Vol. 1. Oxford Univ. Press (Clarendon), London and New York. Massey, H. S. W., and Gilbody, H. B. (1974). “Electronic and Ionic Impact Phenomena,” Vol. 4. Oxford Univ. Press (Clarendon), London and New York. Massey, H. S. W., and Mohr, C. B. 0. (1933). Proc. R. Soc. L i m h i , Ser. A 140, 613. Massey. H. S. W., and Smith, R. A. (1933). Proc,. R. Soc. London. Ser. A 142, 142. Massey, H. S. W., Burhop, E. H. S., and Gilbody, H. B. (1974). “Electronicand Ionic Impact Phenomena,” Vol. 5 . Oxford Univ. Press (Clarendon), London and New York. Mott, N. F., and Massey, H. S. W. (1933). “The Theory of Atomic Collisions,” 1st ed. Oxford Univ. Press (Clarendon), London and New York.
Atl\*nticc.v in
ATOMIC A N D MOLECULAR PHYSICS
VOLUME I5
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ADVANCES IN ATOMIC A N D MOLECULAR PHYSICS, VOL. I5
NEGATIVE IONS H. S . W . MASSEY Depurtment vf Physics and Astronomy University College London London, England
I. The Ground State of Negative Atomic I o n s . . . . . . . . . . . . . . . . . . . . A. Determination of Atomic Electron Affinities . . . . . . . . . ......................... B. Calculations for Li- . . . . . . . . C. B, C, N, 0, and FAtorns . . . . . . ...................... D. A I , S i , P , S , a n d CI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. New Methods for Calculating Electron Affinities . . . . . . . . . . . . . . . . .
B. States Metastable toward Aut
s ... . ... .
111. Electron Affinities and Structures
.......... B. Polyatomic Ions A. Measurements of Total Dissociation Cross Sections
............ . .
C. Attachment to H N 0 3 . . . D. Angular Distributions of ...................... E. Effect of Vibrational Ex sociative DetachmentH, and . . . . . . . . . . . . . . . . . . . . . . . F. SF6 ......................................................... V. Photodetachment and Photodissociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Measurement of Photodissociation Cross Sections. . . . . . . . B. Experimental Results-Photodetachment from NO- . . . . . . . . . . . . . c . c0,-. . . . . . . . . . . ...... . .. VI. lonic Reactions at Th A. Observations of Infrared Luminescence in Ionic Reactions . . . . . . . . B. The Reactions 0: + CO, COT + 0, . ......... C. Energy Dependence of 0 - Reaction Rates.. . . . . . . . . . . . . . . . . . . . . . , D. Negative-Ion Reactions with Atmospheric Trace Constituents . . . . . VII. Ionic Reactions at High Energies . . . . . . . . . . . . . . . . . . . . . . . A. Collisional Detachment from H - . . . . , . . . . . . . . . . . . . , . . . . . . . . . . . . . B. Ion Pair Production in Dissociation of H, and H i . C. Collisions lnvolving Other Negative Ions . . . . . . . . . VlII. Negative Ions in Electric Discharge and Breakdown Phenomena . . . . . . A. Equilibrium Conditions in Discharges Containing Negative Ions B. Stability of the Discharge.. . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . ..... ..............................
2 2 3 4 6 7 7 7 8 9 9 12 13 13 14 15 15 16 17 18 18 20 21 23 23 23 25 25 26 26 21 27 28 28 30 33
I Copyright 0 1979 by Academic Press. Inc. All rights of reproduction in any form reserved.
ISBN 0- 12-W3815-3
2
H . S. W . Massey
In attempting to review the present position of knowledge of such a large subject as negative ions, involving as it does almost all branches of atomic and molecular physics, it is convenient to refer to the most recent extensive publication on the subject and limit consideration to advances made since that time. For this purpose the third edition of the author’s book “Negative Ions,” published in 1976, which reviews the subject up to about the end of 1974 is a convenient reference base. However, this volume contained 716 pages and in this chapter we are limited to about 25. The new results obtained in the last four years are substantial and so we must here be very selective as well as very brief. For this reason reference will frequently be made to “Negative Ions” for background material. We shall usually follow the same order of subject matter as in that book.
I. The Ground States of Negative Atomic Ions A. DETERMINATION OF ATOMICELECTRON AFFINITIES During the last few years the accurate measurement of the electron affinities of atoms has become possible through the availability of highresolution laser light sources for photodetachment measurements, including photoelectron spectroscopy. The methods used and many of the earlier results obtained have been discussed in “Negative Ions” and an excellent review has since been written by Hotop and Lineberger (1979, which covers most of the applications to atoms. We shall therefore refer the reader to these sources. In 1974 the only negative atomic ion whose electron affinity could be calculated to an accuracy well beyond that attainable experimentally was H-. In many ways this is an instructive case for it showed how important is the effect of correlation between the motions of the atomic electrons in determining whether or not a negative ion is stable. This renders difficult the extension of the theory to other atomic ions. Methods were available for calculating the self-consistent or Hartree-Fock wavefunctions for all atoms and negative ions (when stable) but in this approximation electron correlation is neglected, the wavefunction being built up by properly antisymmetrized sums of products of single-electron orbitals. However, even by 1974, progress had been made in calculating correlation energies. In principle these depended on the use of a variational wavefunction in the form of a configuration expansion
3
NEGATIVE IONS
Here (Do is the Hartree-Fock wavefunction, while (Dg:::.'is a wavefunction for a configuration in which occupied orbitals + a , &, . . . are replaced by excited orbitals Jla, +o, J l V , . . . of a suitably chosen basis set. To make such calculations practicable various approximations have been resorted to. Thus it is to be expected that the major contributions come in the configuration sum from single and double excitations so that even in the most elaborate calculations carried out up to the present no account has been taken of excitations of higher order than quadruple. A further useful approximation is that of independent pairs based on the Bethe-Goldstone method in nuclear physics. This means that to calculate the contribution to the correlation energy from the ij pair one need only work with a function
+,
To a good approximation the total pair correlation energy may then be obtained by adding the individual contributions. This procedure may be extended to calculate higher-order correlation terms involving three or more orbitals. The orbitals used are usually expanded in terms of Slater-type functions are also though Gaussian functions either of the form, P"Wrzor e--p(r-rl)z employed. In Chapter 2 of "Negative Ions" (Massey, 1976), an account is given of the earlier independent pair calculations foi atoms and ions of the first main row of the periodic table. By present standards these calculations were of comparatively small scale but achieved quite good results. Since then very much more elaborate calculations have been carried out that take advantage of the greatly increased power of the latest computers. Nevertheless, as we shall see, it is remarkably difficult to calculate electron affinities of even the first row atoms to an accuracy of better than 0.1 eV.
B. CALCULATIONS FOR LrAs Li- is the next stable negative ion in order of simplicity from H-, it has received special attention. As early as 1968 Weiss had calculated the electron affinity to quite good accuracy on the assumption that the correlation energy between the two K electrons and between the K and L electrons contributes very little. Using a ten-configuration expansion he calculated the LL correlation energy and then the electron affinity. He obtained 0.614 eV, quite close to the observed value 0.620 & 0.007 eV. Sims et al. (1976) have since carried out two very much more elaborate
H . S. W . Massey
4
calculations. The first took the form of a general configuration expansion using a basis set of 9s, 7p, 4d, and 3f Slater-type orbitals transformed to natural orbitals (Lowdin, 1955). Multiple excitations up to quadruple were included so that the expansion included no less than 234 terms. Nevertheless the electron affinity obtained by comparison of the resulting energy with that obtained in a similar calculation for Li (Larsson, 1%8) was found to be 0.589 eV, less satisfactory than in the much less elaborate calculations of Weiss (1968). The same authors also calculated the electron affinity using a wavefunction that included explicitly positive integral powers of the interelectronic separations, so it could then be written
with
where A is an antisymmetrizing operator and O(L2)an orbital angular momentum projection operator; uAranged from 0 to 2. Using a basis set of 8s and 3p functions, the overall expansion included 147 terms. This gives, by comparison with results of a similar calculation for Li (Sims and Hagstrom, 1971) an electron affinity of 0.604 eV, a quite good result but not perhaps as might be expected from a calculation of such a magnitude. C. B, C, N , 0,
AND
F ATOMS
Sasaki and Yoshimine used an IBM 360/195 computer to make a thorough study of the use of a configuration expansion to calculate correlation energies and electron affinities of the remaining first row atoms (1974a) and the corresponding negative ions (1974b). 8s, 7p, 6d, 5f, 4g, 3h, and 2i type orbitals were used, expanded in terms of 47 Slater-type functions. All configurations involving single and double excitations were included. This already required 672 configurations for B and 1571 for F. To reduce the number of triply and quadruply excited configurations to a manageable number, only those obtained from combining the most important double excitations were included. Nevertheless this left as many as 798 configurations for B and 2649 for F (see Table I). Table I1 shows the results obained for the correlation energy analyzed into contributions from different classes of configuration. It will be seen at once that, as assumed by Weiss (1968) in his calculations for Li- referred to earlier, almost all the contribution to the electron affinity comes from
5
NEGATIVE IONS
TABLE I CORRELATION
ENERGYIN ATOMICU N I T S , WITH S I G N REVERSED, FOR ATOMSOF FIRSTR o w OF T H E PERIODIC TABLE( E X C L U D I N G Li)"
THE
K L intershell
K shell
BB CC NN 00
FF
L shell
SDb
SD
TQb
SDTQb
SD
TQ
SDTQ
0.0420 0.0420 0.0417 0.0418 0.0414 0.0414 0.0410 0.041 I 0.0407 0.0407
0.0085 0.0090 0.0121 0.0126 0.0154 0.0165 0.0189 0.0197 0.0226 0.023 1
0.0020 0.002 1 0.0016 0.0015 0.0017 0.0010 0.0016 0.001 1 0.0012 0.0010
0.0105 0.01 I I 0.0137 0.0141 0.0171 0.0175 0.0205 0.0208 0.0238 0.0241
0.0856 0.0703 0. I168 0.0961 0.1807 0.1225 0.2409 0.1828 0.3012 0.2409
0.0027 0.0017 0.003 1 0.0023 0.0059 0.0019 0.0082 0.0035 0.0104 0.4049
0.0883 0.0720 0.1199 0.0984 0. I866 0. I244 0.2491 0.1863 0.31 14 0.2458
Calculated by Sasaki and Yoshimine (1974a,b). So refers to contributions from single and double excitations, TQ from triple and quadruple, SDTQ to the sum of all. a
correlations between the outermost L shell electrons. In fact, to obtain results of high accuracy there should be more concentration on expanding the orbital basis so as to improve the calculation of the L shell contribution while keeping the scale of the operation within reasonable bounds by eliminating all configurations that contribute only to the K and KL correlation energies. Table I1 compares the calculated electron affinities with those calculated by independent pair methods with much smaller orbital basis sets and with the observed. Weiss et a / . ( 1971) used symmetry-adapted pairs. TABLE I I
ELECTRONA F F I N I T I E( SI N eV)
~~~~~
~
~
a
See Section IV.
B . C, N , 0,
AND
F
B
C
N
0
F
0.15 0.223 0.243
1.11
1.29 1.322
-0.52 -0.12 -0.299
1.13 1.47 1.43 1.505
3.12 3.47 3.37 3.621
0.28
1.25
-0.07"
1.465
3.448
~
Calculated Sasaki and Yoshimine (1974b) Weiss (1971). Marchetti er a / . (1972) Moser and Nesbet (1971) Staemmler and Jungen (1975) Observed Hotop and Lineberger (1975) ~~~
OF
-
6
H . S . W . Mussey
Moser and Nesbet (1971) did not but did include triple excitations. Staemmler and Jungen (1975) used orbitals of Gaussian type and calculated L shell correlations only, but otherwise their calculation is roughly comparable in scale to the other two. It is remarkable that the very elaborate calculations of Sasaki and Yoshimine yield less satisfactory results than do the much smaller ones. The explanation seems to be that a cancellation of errors occurs in the latter, neglect of triple and quadrupole excitations being balanced by truncation errors due to use of incomplete basic sets. Evidence in support of this was obtained by Sasaki and Yoshimine (1974b) who examined the effect of reducing these separate errors systematically. The magnitude of the problem of accurate calculation of the electron affinity may be gauged from the fact that Sasaki and Yoshimine obtained 94-95% of the true correlation for the neutral atoms but only 83% of the true contribution to the electron affinity.
D. AL, SI, P, S,
AND
CL
The fact that almost the entire contribution to the electron affinity from correlation effects for the atoms of the first row arises from correlations between outer-shell electrons suggests that calculations of electron affinities for second row atoms should not be appreciably more complex. Thus only M shell correlations need be considered. Moser and Nesbet (1975) took advantage of this by carrying out Bethe-Goldstone-type calculations for Al, Si, P, S , and C1 of similar complexity to those referred to in Table I1 for the first row atoms. Their results for the electron affinities are given in Table 111. Comparison with observed values shows good agreement. Apparently the cancellation of errors remains effective in these cases also. Moser and Nesbet also calculated the energy separation of the ground term and the first excited term for Si- and P- as 0.91 and 0.85 eV, respectively, agreeing closely with values 0.94 and 0.84 eV obtained by quadratic extrapolation from isoelectronic sequences. TABLE 111
ELECTRON AFFINITIES(eV) FOR SECOND Row ATOMS ~
Calculated Moser and Nesbet (1975) Observed Hotop and Lineberger (1975)
~~
P
S
CI
1.52
0.74
2.18
3.79
I .39
0.743
2.0772
3.715
Al
Si
0.49
-
NEGATIVE IONS
7
E. N E W METHODSFOR CALCULATING ELECTRON AFFINITIES All the methods discussed above depend on the determination of the small difference between two directly calculated large numbers. In principle at least it would be an advantage if the small difference could be calculated directly. Several methods have been proposed for doing this. Le Dourneuf and Vo Ky Lan (1977) have adapted the eigenfunction expansion method of collision theory to the situation in which all the scattering channels are closed. Solution of the resulting eigenvalue problem gives the attachment energy of the additional electron. An important part of their method, known as the polarized frozen core approximation, is the inclusion into the expansion of atomic pseudostates (Vo Ky Lan et a l . , 1976), which lead to the correct atomic polarizability. This is necessary because core polarization provides an important additional attraction for the additional electron. With a comparatively simple expansion they obtained quite good results for 0 and C but. just as with the configuration interaction method, there are convergence difficulties (Cornille el al., 1978)-the best results are probably obtained with not too large an expansion! Other methods calculate the energy difference by what is effectively third-order perturbation theory based on the wavefunction of the ion or, less usually, the neutral atom. Different procedures have been used for doing this, one (Cederbaum and von Niessen, 1974) based on the Dyson equation for the Green's function as in many-body theory, one using a variation-perturbation technique (Cederbaum el rrl., 1977a), and one (Simons and Smith, 1973) based on the equations of motion for an operator that adds or subtract an electron from the system. While developed primarily for application to molecules, tests have been made by applying them to atoms. Thus Cederbaum et (11. (1977b) obtained electron affinities for Li, Na, F, and CI that are generally of comparable accuracy to those obtained by Weiss (see Table 11) and involve calculations of comparable magnitude. Once again the convergence question remains uncertain.
11. Excited States of Atomic Negative Ions A. RESONANCESTATES Considerable effort continues to be devoted to the study of excited states of negative ions, with lifetimes toward autodetachment of order sec or less, by the observation of resonances in scattering or photodetachment processes. Measurement of the resonance width is important
H . S. W . Massey as from it the lifetime of the state may be obtained. As the widths are only of the order of a few millielectron volts this makes heavy demands on experimental technique to obtain sufficiently monochromatic electron beams. Gallagher and York (1974) have developed an ultrahigh-energy resolution electron source based on the photoionization near threshold of a beam of metastable barium atoms by light from a cw ultraviolet laser. With such a source van Brunt and Gallagher (1977) have observed narrow resonances in the differential scattering of electrons with an energy resolution of about 2 meV. Used in conjunction with a target beam of neutral atoms from a supersonic source to reduce Doppler broadening due to motion of the atoms, measurements have been made of the structure of resonance scattering in rare gases that yield considerably more accurate determinations of width than have been possible hitherto. B. STATESMETASTABLETOWARD AUTODETACHMENT The ls2s2p4P state of He- is metastable toward both autodetachment and radiation and its properties have been thoroughly investigated. The ls2p2 2P state, while metastable toward autodetachment, may make an allowed radiative transition to ls2s2p2P that suffers fast autodetachment (compare with the 2s2p2 state of H-; Drake, 1973). Saponova and Senaskenko (1976) calculated the radiative lifetime as 2 x lo-' sec, which is marginally too short for He- 2p to be readily observed in beam experiments. However, Baragiola and Salvatelli (1975) observed the production of He- ions, by charge transfer collisions between a beam of 40 keV He('S) atoms and the vapors of lead and magnesium, to take place with comparable cross sections. For Pb the He- will be produced in a 4P state: He('S)
+ Pb(SP)-+ He-('P) + Pb+(*P)
(5)
but for magnesium if the target is left in the Mg+(%) state, only He-(2P) will be produced in significant quantities. Arguments based on the absence of curve crossing suggest that it would be improbable to leave the target either in a quartet state of Mg+ or a triplet state of Mgz+.The observations therefore pointed toward the existence of a comparatively longlived *P state. Confirmation of this has been provided by the work of Dunn et a f . (1978) who applied the same technique used by Gilbody et af. (1969) for studying He-(4P). In this earlier work charge transfer collisions of the mainly 23S metastable atoms produced from the neutralization of 200 keV He+ ions in gases were studied. For the recent experiments they obtained the metastable atoms from neutralization of He2+ions in single collisions in Hz. These will be exclusively in the 2IS state, following
NEGATIVE IONS
HeZ++ H,
He(2'S)
-+
+ H + + H+
9 (6)
2% atoms can only capture electrons to produce He- in doublet states, but nevertheless these latter ions were observed when the metastable beam was passed through Hz: He(2IS) + H,
-P
H e - ( l ~ 2 p ~ )+~ H P:
(7)
He-(4P) could not be produced if spin were conserved. The positive results of these experiments require that the lifetime of He-T2P) exceed sec, considerably longer than the calculated value.
111. Electron Affinities and Structures of Negative Molecular Ions A.
DIATOMIC
IONS
From the discussion of Section I it is clear that the problem of calculating uh initio the electron affinity of even the first row atoms is a difficult one, making great demands on computing time. It is not surprising that much progress has not yet been made with the corresponding problem for molecules, which is naturally much more difficult. Even the calculation of the SCF energy is a major task and from experience with atoms this approximation is quite inadequate for the calculation of electron affinities. It seems more promising, however, for the determination of structural properties such as the form of the potential energy curve. There has been some progress beyond the SCF approximation using the methods outlined in Section I. in which third-order perturbation theory is applied in some form to calculate the electron affinity directly from the wavefunction of the negative ions. In judging the success of this work similar convergence questions arise as with the atomic calculations referred to in Section I. Quite a good value for the electron affinity may be obtained with a relatively limited basis set of orbitals. Expanding the basis set may, at least at first, yield poorer results. Cederbaum et al. (1 977b) have applied a variation-perturbation method to obtain a third-order approximation to the electron affinity of OH. The basis set for the calculation was built up from Cartesian gaussian functions and included 7s and 4p functions contracted to 4s and 2p. They obtained an electron affinity of 2.07 eV, much closer to the observed value 1.825 eV than the earlier configuration interaction calculations of Meyer (1974) which gave I .265 eV. Cederbaum et a / . (1977a) have applied the Green's function formalism to obtain a third-order approximation to the electron affinity of C2 using
I0
H . S. W . Mussey
three different basis sets, again built up from gaussian functions. These gave good results, the last agreeing within experimental error with the observed value. This was obtained with 12s8pld functions contracted to 6s5pld and including diffuse functions of s and p type. Vibrational constants were also computed. In a more ambitious calculation the same authors studied Pz, predicting an electron affinity of 0.30 eV-the best experimental value (Bennett et al., 1974) is 0.24 5 0.23 eV! The equation of motion method has been used to obtain third-order approximations to the electron affinities to a number of diatomic molecules [BH (Griffing and Simons, 1974); BeH (Kenney and Simons, 1974); CN and BO (Griffing and Simons, 1975)]but there seems to be some doubt as to the convergence of their results (Liu, 1977). There has been considerable interest in the scattering of slow electrons by F, aroused by the development of hydrogen fluoride and rare gas fluoride lasers. Calculations (Rescigno et al., 1976; Schneider and Hay, 1976)of the scattering found it to be dominated by a “C: shape resonance of F; at an energy of around 2 eV. However, the sensitivity of the results to the basis functions used was such as to make it uncertain whether, with a more complete set, the resonance would move to such lower energies as to become in reality a bound state. To examine this Rescigno and Bender (1976) calculated the potential energy curve of F; using a configuration interaction method with a molecular basis set comprising l l c g , llu,, ST,, 5 ~ , ,16, and 16, orbitals. 256 spin eigenfunctions were involved for F, and 169 for F;. Results were obtained at 20 internuclear separations from 2.0 to 1 0 . 0 ~ Despite ~. the scale of the calculation the electron affinity came out to be only 1.82 eV compared with the observed 3.44 eV, but the equilibrium separation for F2 and the dissociation energies for F, and F; were in good agreement with observation. It seems likely from this work that the potential energy curve for F, intersects that for Fzvery close to the equilibrium separation for the neutral molecule, a conclusion that is in agreement with experimental data on dissociative attachment of electrons to F, (see Section IV). Since 1974 accurate electron affinities have been measured for further diatomic ions by photoelectron spectroscopy. These include the hydrides CH (1.238 & 0.008 eV) and Si H (1.277 -C 0.009 eV) (Kasdan et al., 1975b) as well as NH (0.381 f 0.014 eV) (Engelking and Lineberger, 1976), PH (0.905 & 0.030 eV), and PH2 (1.250 +- 0.030 eV) (Zittel and Lineberger, 1976). For Si H- two bound excited states were found. FeO has also been studied by Engelking and Lineberger (1977) who found an electron affinity of 1.492 f 0.020 eV.
NEGATIVE IONS
I1
Crawford and Garrett ( 1977) have pointed out some important general conclusions applying to electron affinities of dipolar molecules. In the approximation in which the nuclei are stationary the electron affinity must be positive if the dipole moment exceeds 1.625 Debye. When nuclear motion is included they produce evidence suggesting that the electron affinity, though smaller, will remain > O if that given in the stationary approximation is greater thanfB, where B is the rotational constant h2/8?r21of the molecule and f is between 0.05 and 0.10. Turning now to excited states of diatomic ions, strong experimental evisec has been obtained dence of a state of H; with a lifetime of at least by Aberth et al. (1975). The existence of relatively long-lived H; was first reported in mass spectra by Khvostenko and Dukel'skii (1958) and much later by Hurley (1974) but no observations with deuterium were made in confirmation. Aberth et al. used a tandem mass spectrometer in which the ions extracted from a duoplasmatron source were focused by an einzel lens and then passed successively through a Wien velocity filter and a 90"sector magnet to a suitable detector. They observed H;, HD-, and D; ions as a function of the pressure ratio of H2 to D2in the duoplasmotron source, the total pressure being maintained constant. Comparison was also made between the HD- and D; lines and those of 3He- and 4He-, respectively, when 3He and 4He were introduced into the source. Observations made with different flight times showed that the half-life of the D; exceeds sec. It is possible that the ions are in a quantum state analogous to the long-lived 4P state of He-. Theoretical analyses of the electron scattering data concerned with the ,IIg resonance of 0; (Linder and Schmidt, 1971) and the 32-and 'A resonances of NO- (Tronc et d . , 1975) have been carried out by Parlant and Fiquet-Fayard (1976) and by Teillet-Billy and Fiquet-Fayard (1977), respectively. For the former the equilibrium separation R , was found to be 1.355 A, which differs appreciably from that 1.341 A, obtained from photodetachment data by Celotta er al. (1972). The agreement is found to be much closer for NO(32-), 1.267 1 g ~ ~from 8 , electron scattering, 1.258 ~fr. 0.010 8, from photodetachment (Siege1 et a / . , 1972). In the NO analysis allowance was made for the fact that the lifetime of the audodetaching levels does not greatly exceed the vibrational period, in constrast to 0;. The high-resolution electron spectroscopic observations by Grestau et al. (1977) have revealed excitation of high vibrational levels of the ground states of NO and 0, up to v = 27 for NO and 20 for 0,. This arises through the intermediate resonance state of the corresponding negative ion suffering autodetachment when the nuclear separation is much larger than the equilibrium value for the neutral molecule.
H . S. W . Massey
12 B. POLYATOMIC IONS
Since 1974 the electron affinities of a number of polyatomic molecules have been measured with some accuracy. Herbst et al. (1974) obtained 2.36 f 0.10 eV for NOz by photoelectron spectroscopy. By the same technique Kasdan el al. (1975b) found 1.124 k 0.02 eV for SiH2. For NO,, a case of major atmospheric interest, Refaey and Franklin (1976) obtained 3.77 0.25 eV from measurement of the threshold energy for the reaction
*
I-
+ HNO, + NO; + H I
(8)
From threshold measurements for production of ion pairs in collisions with alkali metal atoms, Rothe et al. (1975) found that the electron affinity of SO, exceeds 1.70 ? 0.15 eV while Compton et al. (1975) found values of 0.46 ? 0.2 and 1.0 5 0.2 eV for COS and CSz, respectively. The ion CO;, also of major atmospheric interest, has been extensively studied through photodetachment and photodissociation techniques (see Section V). The electron affinity has been measured as 2.69 t:!: eV (Hong et ul., 1977) but certain difficulties remain in the interpretation of observed data on photodissociation and on ionic reaction rates (see Sections V and VI). Direct experimental evidence of the existence of H; (and also D;) ions with lifetimes in excess of sec has been obtained by Aberth et al. (1975) using the same tandem mass spectrometer as that with which they observed H, and D,. Cooper et af. (1975) have reported observations of the production of ion pairs in collisions at thermal energies between Cs atoms and the hexafluorides of W, In, Re, and Mo. K atoms were also found to produce pairs on collision with the first two of these molecules, under similar conditions. These observations imply that the electron affinities of WF, and IrF, exceed 4.34 eV and of MoF6 and ReF, exceed 3.89 eV. In experiments with a sodium atom beam of variable homogeneous energy a threshold energy for the production of pairs in collision with WF, was found that yielded an electron affinity of 4.5 eV, a remarkably high value. Studies of ionization in flames have shown that certain oxides and acids of boron (Jensen, 1969), tungsten (Jensen and Miller, 1970), molybdenum (Jensen and Miller, 1971), and chromium (Miller, 1972), in the gas phase, also have quite large electron affinities. Gould and Miller (1975) have extended this work to H2/OZ/Nzflames containing potassium with added rhenium-containing compounds and find that the electron affinities of gaseous ReO, and ReOl are approximately 3.01 and 4.5 eV, respectively. An interesting, elaborate calculation of the NzO and NzO- potential energy surfaces in the neighborhood of the potential minima has been car-
NEGATIVE IONS
13
ried out by Hopper et (11. (1976) with a view to interpreting the remarkable experimental data on electron attachment to N,O. They constructed the molecular orbitals in their configuration analysis from Gaussian basis sets optimized for the individual atoms-9s5p sets contracted to 4s3p (Dunning, 1971). Only valence orbitals were included. While inadequate to obtain a good value for the electron affinity these bases should be much more satisfactory for determining equilibrium parameters. Quite good results were indeed obtained for the atomic equilibrium separations, the bond force constants, the vibrational frequencies, and the dipole moment of the neutral molecule. Similar satisfactory results would be expected for the negative ion. The calculated surface for N,O near equilibrium is found to be stable with respect to both dissociation and detachment, in agreement with experiment. Many observed features of attachment and detachment reactions can also be understood from the forms of the surfaces.
IV. Dissociative Attachment The study of dissociative attachment has continued to engage the attention of many research groups both experimental and theoretical. On the experimental side the work can be classified into three main categories: (a) measurements, at room temperature, of total dissociation cross sections as a function of electron energy for further molecules as well as, with increased energy resolution, for molecules previously investigated; (b) measurement of differential cross sections, including the angular distributions of the product ions; (c) further studies of the effect of initial vibrational or rotational energy excitation on the rate of dissociative attachment. We can refer here to only a few of the major new results obtained
A.
M E A S U R E M E N T S OF
TOTALDISSOCIATION CROSS SECTIONS
Dissociative attachment to halogen molecules has been observed in a number of experiments but because of the difficulty of working with such highly reactive substances much of the earlier work was of doubtful reliability. Recently, the development of the fluorine gas laser has drawn attention to the importance of having good data available on dissociative attachment to F,. The most recent measurements are those of Chen and Chantry (1978), who observed the variation of the attachment cross section at room temperature with electron energy over an energy range extending from very
H . S. W . Massey
14
low energies up to 9 eV. They used the electron beam technique with high-energy resolution. When allowance is made for the small electron energy spread, the cross section has a sharp maximum value of 8 x 10-14 cm2at zero energy, declines rapidly to 8 X cm2as the energy rises to 0.1 eV, and then falls rather more gradually to a minimum close to 4 x cm2 at 4 eV. It then rises to a broad maximum of 8 x cmz at 6 ev. Hall (1978) has carried out semiempirical calculations that show that these results are consistent with the disposition of ground-state F, and F; potential curves calculated by Rescigno and Bender (1976). The best fit is obtained if the latter curve intersects the former about 0.1 a.u. inside the equilibrium separation of F,. In his calculations Hall also took into account the measurements of Chen et al. (1977) and Sides et al. (1976), using swarm techniques, which yield rate coefficients directly rather than cross sections.
B. PRODUCTION OF N -
FROM
N2 A N D NO
N- ions in the ‘D term of the ground configuration are expected to be metastable with lifetimes of order sec or larger (“Negative Ions,” Massey, 1976, pp. 84, 144,254) so that they could survive passage through a mass spectrograph. Hiraoka et al. (1977) have observed N- ions in a quadrupole mass spectrometer when the ion source contained either NO or N,. As a check on the nature of the ions they used 14N15Nas the source gas and found peaks of equal height at mass numbers 14 and 15. The appearance energy for the ions from N 2 was found to be close to 16 eV as compared with 8-9 eV for the 0- ions resulting from an N 2 - 0 2 mixture. On the other hand for NO the appearance energies for the N- and 0- ions were nearly equal. When allowance is made for the fact that the values are only very approximate, they are consistent with the N - ions arising from N, via the reaction N2(X1Si) + e + N(*D”) + N-(ID)
(9)
and from NO via NO(XTI)
+e
4
N-(’D)
+ Of3P)
(10)
the respective appearance energies for ions of zero kinetic energy being 13.58 and 7.97 eV, respectively. For 0- from N, and NO the expected values are 4.53 and 7.39 eV. From theoretical considerations the ground state of N - was known to be almost certainly unstable with an electron affinity close to - 0.1 eV, indicating the existence of a shape resonance in electron collisions with N atoms at 0.1 eV. Strong evidence now exists of the formation of
15
NEGATIVE IONS
ground-state N - in dissociative attachment collisions of electrons with N2. As long ago as 1970 Hall et uf. observed the threshold excitation of N, by electron impact in the energy range 6-13 eV, using the trappedelectron method. Most of the features were identified but at 9.6 eV a process appeared with a sharp onset followed by a continuum falling off gradually in intensity up to 1 1 eV. Spence et al. (1978). using a modified trapped-electron technique that enabled them to measure the intensity of scattered electrons at energies other than zero, found that the feature had a vertical onset within 0.5 eV of the dissociation limit of N, and disappeared for scattered electrons with energies greater than zero to within the energy resolution of the incident electron beam (0.08 eV). The identification of the process concerned was made by Mazeau et ul. (1978). They observed the energy distribution of the electrons resulting at a fixed scattering angle (135") from collisions of an electron beam with a crossed beam of N , It was found that, if the scattered electron energy was fixed at 0.07 eV and the energy loss spectrum obtained by scanning the incident electron energy, a loss process appeared with a sharp onset near 9.7 eV followed by a continuum with intensity falling off up to 11.7 eV. This feature was no longer present at scattered electron energies of 0.1 eV or greater. These are the characteristics expected if the reaction involved is N 2 + e - . N,*
-
N
+ N--
N
+N +e
( 1 1)
From the observed energy distribution of the electrons emitted at incident energies near 10.5 eV the energy of the state of N - was determined as 0.07 0.02 eV above that of the ground state of N . quite close to the best theoretical prediction (see Table 11). The width of the distribution was 16 -+ 5 meV. Similar features were observed when the N2 was replaced by NO.
*
C. ATTACHMENT TO H N 0 3
In the course of an investigation of the ion chemistry of HNO, at room temperature, using the flowing-afterglow technique, Fehsenfeld rt ul. (1975) (see also Section IV) found that the dissociative attachment reaction HNO,
+ e + OH + NO;
is very fast, with a rate constant of 5
2
3 x
(12)
cm3 sec-I.
D. ANGULAR DISTRIBUTIONS OF PRODUCT IONS Van Brunt and Kieffer have extended their measurements of the angular distribution of ions resulting from dissociative attachment, first carried
16
H . S. W . Massey
out for 0, (1970), to NO (1974). For ion energies between 8 and I 1 eV they fou,nd that the distribution was given closely by sin3 8, where 8 is the angle between the direction of the motion of the ions and of the incident electron beam. This indicates that the repulsive state of NO- into which the electron is initially captured is either of C or A type. Hall ef al. (1977) have carried out measurements of the energy and angular distributions of the 0- ions resulting from dissociative attachment in CO. Electrons and negative ions emerging from the collision region in a well-defined direction were energy-analyzed electrostatically and then separated by a magnetic momentum filter. From analysis of the data in terms of the theory developed by O'Malley and Taylor (1968). strong evidence was obtained that both the process that leads to 0- and ground ?P) state C and that in which the C is formed in the excited ('D) state result from capture into II states of CO-. E. EFFECTOF VIBRATIONAL EXCITATION ON ASSOCIATIVEDETACHMENT-H, A N D D, At room temperature the cross section for dissociative attachment in H2 and D, is effectively zero at low electron energies but rises to a first maximum at an energy of 3.8 eV. Even at this maximum the magnitude is very small (1.6 x cm2 for H, and 8 x cm2 for D,). Allen and Wong (1978) have used a modified form of the electron impact spectrometer designed by Stamatovic and Schulz (1970) to observe dissociative attachment by electrons in the energy range 2-4 eV in H, and D, over a gas temperature range from 300 to 1400 K, with such highenergy resolution that contributions from separate initial vibrational and rotational states could be distinguished. A very great sensitivity of the attachment cross section to the initial internal state was observed. Thus for H, the cross sections for initial vibrational quantum numbers 0, 1 , 2, 3, 4, respectively, are in the ratio 1 : 3 x 10, :5 x lo3:4 x lo4:3 x los. The corresponding ratios for D, are 1:4 x 102:8 x 103:104:4 x los. For H, with u = 0 the cross section for an initial rotational quantum numberj = 7 is about 40 times larger than f o r j = 0. It follows that in hot hydrogen dissociative attachment in this electron energy range can be quite rapid in contrast with the very low cross section at 300 K. In this connection it is of interest that Nikotopoulou ef al. (1977) found it difficult to account for the high rate of H- production they observed in an H, plasma at high pressure. The enhanced rate of dissociative attachment to vibrationally excited H, may remove this difficulty. Remarkable as these results are, Bardsley and Wadehra (1978) have
NEGATIVE IONS
17
shown that they may be understood theoretically in terms of a reasonable model for the 'X, potential energy curve of H,.
F. SF, Chen and Chantry (1978) in their experiments on the effect of temperature on dissociative attachment processes found that the rate of production of SF; through the reaction SF,
+e
-
SF; + F.
(13)
which occurs with very low energy electrons, increases rapidly with the gas temperature. This shows that the cross section for (13) depends very much on the initial vibrational state of the SF,. From a plot of the relative SF; signal as a function of inverse gas temperature it appeared that an activation energy of 0.20 eV is required for the reaction to proceed effectively. This is close to twice the quantum of the u3 vibrations of SF,. Light from a Cot laser may be tuned to resonance with these vibrations so that it should be possible to enhance the rate of production of SF; at room temperature by irradiation of the gas with this light. Moreover, the laser may be tuned to resonance with the ug vibration of the 34SF6isotope so that selective enhancement of 34SF; production as compared with 32SF; should be achievable. The first evidence of enhancement of SF; production by laser irradiation was obtained by Schermann et ul. (1977) but Chen and Chantry (1978) were the first to demonstrate the effect quantitatively and verify the isotope selectivity. They used an apparatus similar to that in their earlier extensive experiments on dissociative attachment (the so-called hightemperature apparatus). The focused laser beam was introduced along the axis of the electron beam. Because the laser beam heated up the collision chamber and the filament of the electron gun it was not sufficient merely to take measurements with the laser beam on and off. The effect of collision chamber heating was avoided by chopping the laser beam with a period short compared with the thermal time constant of the chamber. On the other hand modulation of the filament emission occurred in phase with that of the laser beam and so was allowed for by monitoring the electron current both during the on and off parts of the laser modulation cycle. Results are conveniently expressed in terms of a photoenhancement factor E ( E ) ,which is the ratio of the SF; currents observed with and without irradiation, at the same electron current, and depends on the electron energy E . It was found that with the P(28) CO, laser line at 936.86 cm-', E ( E )for 32SF;had a sharp peak for E = 0 while 34SF; was not en-
18
H . S . W . Massey
hanced. On the other hand with the P(44) C 0 2laser line at 920.8 1 cm-’ the reverse situation occurred, 34SF; being enhanced but 32SF; unaffected. It was verified that E ( E )was proportional to the gas pressure and electron beam intensity. Surprisingly it was also proportional to the laser power even though two-quantum excitation is involved. A further observed feature that is not easy to explain is the fact that the largest peak value of Q E ) , considered as a function of frequency, occurs about 8 cm-I away on the red side from the peak of the absorption curve for SF, for small radiant intensities.
V. Photodetachment and Photodissociation The availability of laser light sources with photon energies especially appropriate for photodetachment has led to major experimental developments in this field. Electron affinities have been determined with great precision for many atoms (see Section I) and molecules (see Section III), fine-structure separations in the ground states of atomic ions and vibrational energies in molecular ions determined, and in some cases, excited electronic states identified. An account of the techniques involved has been given in Chapter 2 of “Negative Ions” (Massey, 1976), as well as in the excellent review published by Hotop and Lineberger (1975). We shall call attention here to some of the more recent developments. A . MEASUREMENT OF PHOTODlSSOCIATlON
CROSS SECTIONS
In the past three years attention has been paid to the measurement of photodissociation cross sections for molecular ions as distinct from those for photodetachment. Moseley et al. (1975a) have adapted for this purpose the combined drift tube and mass spectrometer technique developed (McDaniel et al., 1962) for the study of ion drift velocities, diffusion coefficients and reaction rates in different gases. A pulse of ions under study, from a suitable source, drifts a distance in the gas at pressure p under the motion of a uniform electric field F . Ions reaching an orifice at the end of the drift space pass out in a jet from which the core is extracted by a suitable skimmer and passed into a quadrupole mass spectrometer. Ions of a selected mass are then detected by an electron multiplier. By repeated pulse operation, using a time-of-flight analyzer, the time spectrum of arrival of different ions at the exit aperture may be obtained. From such spectra taken at different gas pressures and drift distances the required quantities may be measured. To adapt this arrangement for photodissociation measurements Moseley ef al. crossed
19
NEGATIVE IONS
the ion stream close to the exit aperture with a laser beam and operated the source in a direct current mode. Measurements were made by chopping the laser beam at 100 Hz and counting the flux of chosen ions with and without the laser flux, typical count rates ranging from lo3 to lo5ions per second. If I and I , are the numbers of ions counted with the laser beam respectively on and off, the photodestruction cross section QPdis given by
+
where is the photon flux, t the mean time an ion spends in the photon beam, and k a geometrical factor determined by the overlap between the photon beam and the ion stream. To eliminate the need for the difficult measurements of k and t , comparison was made with observations for a standard ion such as 0- for which Qpdis known. Thus if measurements for 0- are distinguished by the suffix s,
Pv -ps us where P denotes the laser power output and u the speed of the drifting ions. In general QPdincludes contributions from both photodissociation and photodetachment. To help in the identification of the processes involved measurements could be made of the flux of different secondary ions produced by the laser beam. One of the major complications in studying photodissociation of molecular ions is the need to ensure that the ions in the region of interaction with the laser beam are devoid of internal excitation. To examine whether this is achieved or not, measurements can be made for different drift distances and gas pressures, thereby changing the number of collisions suffered by the ions in passing from the source to the interaction region, and at different values ofF/p, thereby changing the mean energies of the ions and hence the chance of excitation in the drift region. A different arrangement for measuring photodestruction cross sections has been used by Vestal and Mauclaire (1977a). Ions from a source, which are mass selected by a quadrupole analyzer, enter a reaction region with quadrupole ion trapping, near the center of which it is crossed by a laser beam. The resulting ions in the beam pass out of this region into a second quadrupole analyzer with an electron multiplier detector. In this arrangement the interaction region is maintained at a high vacuum. On the other hand the degree of internal excitation of the ions may be varied by changing the pressure in the ion source-under the experimental conditions the residence time of an ion in the source is proportional to this pressure.
20
H . S . W . Massey
With the well-defined beam geometry absolute measurements of cross sections may be made. One factor that must not be overlooked in both sets of experiments is the effect of the polarization of the laser beam. In general, for a given polarization, transitions in which the fragment ions are ejected in directions either parallel or perpendicular to the direction of flow of the primary ions will be favored. Since the detection of secondary ions will be more effective in the former case, there is a need to carry out observations with the laser beam polarized both parallel and perpendicular to the direction of primary ion flow.
B. EXPERIMENTAL RESULTS-PHOTODETACHMENT FROM NO, Reference has already been made in Sections I and I1 to the latest accurate measurements of electron affinities of atoms and molecules made using laser photodetachment techniques, particularly photoelectron spectroscopy. These experiments have also provided other data concerning the properties of the negative ions such as the fine-structure separations of the ground states for atomic ions or the vibrational energy levels for molecular ions. In some cases the existence of bound excited states of the ions has also been revealed. Herbst el al. (1974) measured the absolute apparent photodetachment cross section for NO, over the quantum energy range 2.0-2.7 e V using a tunable dye laser as light source. With NO2 as the gas in the plasma ion source the beam of ions of mass corresponding to NO; did not suffer photodetachment until the photon energy exceeded 2.2 eV. However, ions of the same mass generated from a plasma in which the source gas was O2with a trace of N 2contained a component that could suffer photodetachment at photon energies greater than 1.8 eV. In view of earlier measurements of the electron affinity of NO2, which were close to 2.2 eV, it was assumed that the ions obtained with NO2 as source gas were normal NO,. Analysis of the photodetachment cross section as a function of frequency then yielded an electron affinity of 2.36 -t 0.10 eV. It was then suggested that the second component was the peroxy isomer (N-0-0)-. More recently Huber et al. (1977) studied the photodestruction of NO; ions at photon energies of 1.97 and 2.34 eV, below the electron affinity of NO2, using the drift-tube mass spectrometer technique. Most of the measurements were made in O2 containing small amounts of NO2. It was found that the apparent photodestruction cross section decreased rapidly with increasing drift distance, total gas pressure, and relative NO2 concentration, the decrease being faster at the lower photon energy. Huber et al. showed that their results are consistent with the assumption that the
NEGATIVE IONS
21
process concerned is one of photodetachment from vibrationally excited NO, and derived effective rate constants for deactivation of vibration, in cm3 sec-', of 8.5 x by 0, and 5.9 x by NO, from observations at the lower photon energy, 1.4 x and 9.7 x at the higher. The much greater rate in NO, is readily understood because of the probability of resonant charge and vibrational transfer in that case. c.
co,
This ion is of considerable importance in the D region of the ionosphere (see "Negative Ions," Massey, 1976, p. 666) and its photodestruction has been extensively studied in the last few years. Hong et ui. (1977), using the drift tube technique first developed by Woo et al. (1969) in which the electrons produced by photodetachment from a drifting ion source are observed, measured the photodetachment cross section as a function of frequency. They established the identity of the ions in a separate massanalyzing system. At the relatively high working pressure there is no doubt that the ions would be in their ground vibrational states. A threshold energy of 2.69 ?::I eV was obtained that, because of the comparable geometry of the ion and neutral molecule, should be very close to the adiabatic electron affinity. The rate of photodetachment by sunlight derived from the cross section measurements is 0.022 -t- 0.003 sec-*. Moseley and his collaborators (1975a, 1976; Cosby and Moseley, 1975; Cosby ef al., 1976) have used their drift-tube mass spectrometer to measure the photodestruction cross section over the wavelength range from 457.9 to 694 nm (photon energies from 1.84 to 2.7 eV). Preliminary measurements were made of the dependence of the apparent cross section on drift distance, gas pressure, and F / p . It was found that over a considerable range of these variables the cross section was constant, suggesting that under these conditions the ions entered the interaction region in their ground states. All cross-section data were then taken under these conditions. From comparison with the yield curve for secondary 0- ions it is clear that the process involved is one of photodissociation. Between 1.80 and 2.3 eV the cross section exhibited a number of maxima and minima, the largest observed value being 4 x I O-l8 cml at 1.9 eV. There is evidence of a threshold close to 1.8 eV, suggesting that I .8 eV is an upper limit to the energy D(CO,-O-) of dissociation into C 0 2 + 0-. This is not inconsistent with the observed electron affinity E,(CO,), which is related to it by
22
H . S . W . Massey
Thus with E,(C03) = 2.69 eV, E,(O) = 1.462 eV, D ( C 0 2 - O ) comes out to be less than 0.57 eV as compared with the accepted value 0.4 2 0.2 eV. However, as will be discussed further in Section VI, an upper limit of 1.8 eV for D ( C 0 2 - O - ) is not consistent with the lower limit 1.97 eV derived by Dotan et al. (1977) from their experimental studies of the reactions 0; + coz z co; + 0 2 . Moseley et al. (1976) analyzed their photodissociation spectrum on the assumption that the process occurred through excitation of a predissociating bound electronic state, in which case the minima corresponded to particular vibrational levels of this state. They identified it as 12A, 1.52 eV above the ground state, with three vibrational modes with energies of 990, 1470, and 880 cm-l. The subject is still somewhat confused nevertheless, because of the measurements made recently by Vestal and Mauclaire (1977aj using the apparatus described earlier. They found that at ion source pressures so high that the CO; ions in the beam should possess no internal excitation, the photodissociation cross section over the wavelength regions from 620 to 585 nm varied with frequency in much the same way as that observed by Moseley et al. (1976j, but the amplitude of the oscillations was somewhat smaller and the absolute cross sections about eight times smaller. Measurements at much lower pressures showed greater amplitudes of oscillation but the absolute cross section did not change greatly. On the other hand it appears that at wavelengths between 435 and 488 nm the absolute cross sections measured by the two methods come into reasonable agreement. The reasons for these marked discrepancies are not yet clear. It may be that in the drift tube experiments the ions are not free of internal excitation but this in itself is insufficient to explain all the differences. Again, the fact that Vestal and Mauclaire made all their measurements with the laser beam polarized parallel to the ion beam may be significant. Moseley ef al. (1976) found experimentally that this sense of polarization gives a smaller yield of 0- fragment ions than does the perpendicular sense-the factor might be markedly smaller still when the primary ions are in a high vacuum. Further experiments are clearly necessary to resolve these various problems. Measurements have also been made by Vestal and Mauclaire (1977b) of the photodissociation cross sections of 0; and 0;. For the former the threshold was found to be greater than 3.5 eV, consistent with the bond energy 4.1 eV of 0; derived from the measured electron affinity. The results for O;, measured at wavelengths between 587 and 620 nm and a low source pressure (0.1 tom) agree with earlier results obtained by Cosby et al. (1976) but at these pressures the apparent cross section decreases with pressure-a similar problem of inconsistency seems to be present to that
NEGATIVE IONS
23
for COT. Cosby et ul. (1976) have also measured photodestruction cross sections for O;, O;.H,O, CO;, and CO;.H20 over the range from 695.0 to 457.9 nm (1.78-2.71 eV). They observed no photodestruction of HCO;.H,O over this range.
VI. Ionic Reactions at Thermal and Epithermal Energies A . OBSERVATIONS REACTIONS
OF I N F R A R E D
LUMINESCENCE I N IONIC
In the last few years further measurements of ionic reaction rates have been carried out, many of which have been concerned with ions of importance in the Earth’s atmosphere. Before discussing some of this work we draw attention to the new experimental possibilities opened up by the successful observation of infrared radiation emitted from vibrationally excited CO, produced in associative detachment collisions of CO with 0-. Bierbaum et a!. (1977) produced the reaction in a flowing-afterglow system and observed infrared emission at a port 5 5 m downstream from the ion source. A narrow-band interference filter was used to isolate emission at 4.3 pm arising from antisymmetric stretch vibrations of the CO, product molecules. Signals of strength as great as 10 times background were observed and many checks were made to verify that they really arose through the production of vibrationally excited molecules in the ionic reaction. These results open the way for detailed study of the distribution of internal energy in ionic reaction products.
B. THE REACTIONS0;
+ CO,
* COT + 0,
Dotan et ul. (1977) have applied flowing-afterglow techniques to measure the rates of these reactions as functions of relative kinetic energy and thence derive limits for certain dissociation energies and electron affinities. The 0; ions were formed by adding a mixture of O2and a small quantity of 0, into the helium buffer gas at a point past the ion source. 0 - a n d 0; were produced by attachment of slow electrons and these were rapidly . the 0; produced converted to 0; by charge transfer reactions with 0 3 As in this way may be initially in an excited state, check measurements were also made in which 0; was produced by charge transfer from OH-, a reaction that is considerably less exothermic. N o difference was observed between the results obtained with these separate sources. For study of the reverse reaction, COT ions were produced by adding a small flow of CO,
24
H. S. W. Massey
together with the O2 and 0 3 .The forward reaction converted the 0; to CO; before leaving the ion production region. Measurements were first made with a flow drift tube over a range of values of F / p corresponding to kinetic energies in the c.m. system for O;-CO, collisions up to 1 eV. The rate constant for the forward reaction fell from 5.5 x lO-'O cm3 sec-' at room temperature to about 1.6 x 10-lo cm3 sec-I at 1 eV. On the other hand no evidence was obtained of the reverse reaction, suggesting that D(C02-O-) > D ( 0 2 - O - ) . Further support for this came from observations of the relative rates of collisional dissociation from CO; and 0; in collisions with Ar at mean kinetic energies in the c.m. system between 0.9 and 2 eV. At the same energy the rates for 0; were about 10 times larger than for COT. From observations taken at high temperatures with variabletemperature flowing-afterglow equipment, a lower limit to the equilibrium rate constant K for the reactions at 595 K of 5 x lo4 was obtained. From the formula RT In K = -AG
(16)
where R is the molar gas constant and AG the change in the molar free energy due to the forward reaction, we have -AG > 12.8 kcal mole-'
(17)
Now -AG
=
-AH
+ TAS
(18)
where -AH is the difference in the dissociation energies D(C02-0-) -
D(02-O-) and AS is the change in molar entropy due to the forward reaction. Assuming that the entropies of COT and 0; are nearly equal to those of NO, and 0 3 ,respectively, T AS at 600 K = - 0.66 kcal mole-', so (17)
gives D (C0 2- O- )
-
D(Oz-O-)
2
13.5
kcal mole-'
=
0.58
eV
(19)
Now D(Oz-O-)
=
EJO,) + D ( 0 , - 0 ) - E,(O)
=
E,(0,)-0.41
eV
(20)
Since the charge transfer reaction of OH- with 0, is exothermic E , ( 0 3 ) 2 E(OH) = 1.825 +- 0.002 eV. Hence from (20), D(0,-0-) 3 1.39 eV and so from (19) D(C0,-0-)
B
1.97 eV
(21)
which, as mentioned in Section V, is inconsistent with the lower limit 1.85 eV obtained from the photodissociation work of Cosby et al. The incon-
N E G A T I V E IONS
25
sistency is probably even greater because E,(OJ is closer to 2 eV (Byerly and Beaty, 1971; Wong ef al., 1972). It is important that this discrepancy be resolved by further experiments.
c. ENERGYDEPENDENCE OF 0- REACTIONRATES Lindinger ef al. (1975) have used a flow drift tube to measure the variation with kinetic energy in the c.m. system of the rates of reaction of 0with N,. NzO, S O z , NH,, CHI, and CzH4. For O--Nz the rate constant was found to be < lo-', cm3 sec-* over the energy range from thermal to about I eV, whereas Comer and Schulz (1974) found values > cm3 sec-' for energies >0.2 eV. D. NEGATIVE-ION REACTIONS WITH ATMOSPHERIC TRACECONSTITUENTS
Fehsenfeld and Ferguson ( 1974). using the flowing-afterglow system, measured the rates of three-body association reactions of 0-, OH-, 0;. Og, CI-, COT, OH-(H,O), and O,(H,O) with H,O and, for several ions, with CO, and SO2. They also measured the rates of a number of binary reactions of these ions and their hydrates with HzO, COP,SO,, NO,, O,, and NO. From these and certain other observations they note the following general effects of hydration or clustering on reaction rates: (a) With simple loosely bound clusters new reaction channels will be opened. (b) Binary reaction rates are reduced. (c) Eventually, with increased clustering, certain reactions are terminated. (d) Negative ions may occur in different isomeric forms with different reactivities. This has been observed both with NO; and with SO;. Fehsenfeld ef (~1.(1975) have also studied the ion chemistry of HNO, at room temperature. Rapid reactions of HNO, occur with NO;, CI-, and COT so that, in experiments with NO,, it is important to ensure that HNO, is not present as an impurity even to as much as 0.1%. HNO, forms such a strong bond to NO; (21.2 kcal mole-' as compared with 12.1 for NO, . HzO) that it must partially displace H 2 0from clustered NO; in the lower atmosphere. Fehsenfeld (1975) has measured rates of reactions of CO;, NO;, and CO; with H and finds that the two isomeric forms of NO; react quite differently. There is no doubt of the profound effect that trace constituents must have .on the negative-ion composition of the lower atmosphere.
26
H. S . W. Massey
VII. Ionic Reactions at High Energies A . COLLISIONAL DETACHMENT FROM H-
Gillespie (1977) has calculated the asymptotic form of the cross sections for detachment of electrons from H-in collisions with H and He, by extending the method used by Inokuti and Kim (1968) to calculate the corresponding cross sections for detachment by electron impact. This consisted in applying Bethe's method for evaluating the first Born approximation and applying closure formulas. Gillespie finds that the total detachment cross section is given for high (but nonrelativistic) impact velocities by Qd
=
(22)
8~a~(az//Pz)fd
where cy is the fine-structure content and p the ratio of the impact velocity u of the ions to that ( c )of light:f, = 2.42 2 0.01 for H and 2.81 ? 0.01 for
He collisions. The formula is valid when d / p " 4 x what larger region is found for discharges in which the CO, is replaced by the same concentration of O,, falling between values off > 2 x lo-, and 6 3 x 10-4. To take account of cluster formation semiquantitatively we note that, for the CO, mixture, the relative concentration of CO required to produce a given effective detachment rate and hence a given value of A will need to be a few hundred times larger than it would be in the absence of the cluster reaction (see the discussion above about the relation between T , andf). On the other hand, for the 0, mixture detachment will be dominant for values off > which falls within the instability range calculated neglecting cluster formation. We would therefore expect the latter calculations to give good results in practice. Nighan and Wiegand carried out experiments to check these conclusions. They worked with a cylindrical convection-dominated discharge, 3.8 cm in diameter and 70 cm long. Although appearing uniform to the eye, observations carried out with electrostatic probes and with photomultipliers observing the light emitted sideways showed that, in fact, the discharge in He-N, mixtures with CO, and with 0, exhibited striations. This is in agreement with the fact, apparent from (31), that the instability condition is most readily satisfied when 6 = 0. When CO was added in increasing amounts, the fluctuations decreased gradually toward zero. Also no striations were observed with the discharge operating in a mixture of He-N, with CO alone, presumably because of the fast rate of reaction ( 2 3 . To put these experiments on a more quantitative basis the minimum fraction fmin of CO present to damp out instability was determined by observing with a spectrum analyzer the magnitude of the voltage fluctuation at the striation frequency (about 5 Hz) as the fraction increased. The value o f f when the voltage fluctuation became smaller than the noise background was taken to be Jmjn. Measurements were made offminfor the mixtures of CO, and of 0, with N,-He already referred to above. For the 0, mixturefmlnwas close to
NEGATIVE IONS
33
2 x and hence not far from that predicted on the assumption of no clustering, whereas for the C 0 2 mixture it was more than 100 times larger as expected from the discussion above. There is no doubt that this work has greatly increased our understanding of the role of negative ions in producing instability in collisiondominated discharges. We have confined the discussion above to selfsustaining discharges, but the analysis may be extended without difficulty to discharges maintained through an external ionizing source.
ACKNOWLEDGMENTS
I am most indebted to Dr. P. J . Chantry for providing me with information on recent experimental studies of dissociative attachment and to Dr. W. C. Lineberger for valuable discussion of a number of matters concerned with the interpretation of expenmental data.
REFERENCES Aberth, W.. Schnitzer, R.,and Anbar. M. (1975). Phy.s. Rev. Lett. 34, 1600. Allen, M.. and Wong, S . F. (1978). Phys. Re\,. A [3] (to be published). Baragiola. R. A.. and Salvatelli. E . R. (1975).J. Phys. B [ I ] 8 , 382. Bardsley. J . W., and Wadehra, J. M. (1978). Phys. R e v . A [3] (to be published). Bates. D. R . . and Walker, J . C. G . (1967). Proc. Phys. Soc., London 90, 3 3 3 . Bennett. S. L., Margrave. J . L., and Franklin. J . L . (1974). J. Chem. f h y s . 61, 1947. Berkner, K. H . , Kaplan. S. N., and Pyle. R . V. (1964). Phys. Rev. [2] 134, A1461. Bierbaum. V. M., Ellison, G. B., Futrell, J . H.. and Leone, S. R. (1977). J. Chem. Phys. 67, 2376. Brouillard. F.. Claeys. W., and Delfosse, J. M . (1975). J . Phys. B [ I ] 8, 1149. Byerly, R.. and Beaty, E . C. (1971). J . Geophys. Res. 76, 4596. Cederbaum, L. S . , and von Niessen. W. (1974). Phys. Lett. A 47, 199. Cederbaum. L . S . , Domcke. W.. and von Niessen, W. (1977a).J. Phys. B [ I ] 10, 2963. Cederbaum, L. S., Schonhammer, K., and von Niessen, W. (1977b). Phys. Re\!. A [3] 15, 833. Celotta. R. H.. Bennett, R. A., Hall, J. L., Siegel. M. W., and Levine, J. (1972)Phvs. Re\,. A [3] 6, 63 1 . Chen. C. L.. and Chantry. P. J . (1978). To be published. . Lert. 30,99. Chen. H. L., Center, R. F., Trainor, D. W., and Fyfe. W. I . ( 1 9 7 7 ~ A p p lPhys. Comer, J . . and Schulz. G. J . (1974). Phys. Re,*.A [3] 10, 2100. Compton, R. N., Reinhardt, P. W.. and Cooper, C . D. (1975). J . Chem. Phys. 63, 3821. Cooper, C. D., Compton, R. N., and Reinhardt, P. W. (1975). Pror. f n r . Conf Phys. Elcciron. Ar. Collisions. %\I, 1975, p. 922. Cornille, M., Hibbert, A., Moser. C . , and Nesbet, R. K . (1978). Phys. R e v . A [3] 17, 1245. Cosby. P. C.. and Moseley. J . T. (1975). Phys. Rev. Lert. 34, 1603. Cosby. P. C.. Ling, J . H.. Peterson, J . R., and Moseley. J . T. (1976). J. Chem. Phys. 65, 5267. Crawford. 0. H . , and Garrett. W. R. (1977). J . Chem. Phys. 66, 4968.
34
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Dimov, G. I., and Dudnikov, V. G . (1966).Z. Tekh. Fiz. 36, 1239;Sov. Phys. --Tech. Phys. (Engl. Trans/.)11, 919 (1967). Dotan, I., Davidson, J. A., Streit, G. E., Albritton, D. L., and Fehsenfeld, T. C. (1977). J . Chem. Phys. 67, 2874. Doverspike, L. D., Smith, B. T., and Champion, R. L. (1977).Proc. Int. Conf. Phys. Electron. A t . Collisions, loth, 1977 Abstracts, p. 1254. Drake, G . W. J. (1973). Asrrophys. J . 184, 145. Dunn, K. F., Gilmore, B. J,, Simpson, F. R., and Gilbody, H. B. (1978).J. Phys. B [I] 11, 1797. Dunning, T. H. (1971).J. Chem. Phys. 55, 3958. Engelking, P. C., and Lineberger, W. C. (1976). J. Chem. Phys. 65, 4323. Engelking, P. C., and Lineberger, W. C. (1977).J. Chem. Phys. 66,5054. Esaulov. V., Dhuicq, D., and Gauyacq, J . P. (1978). J . Phys. B [I] 11, 1049. Fayeton, J., Dhuicq, D.. and Barat, M. (1978).J. Phys. B [I] 11, 1267. Fehsenfeld, F. C. (1975). J. Chem. Phys. 63, 1686. Fehsenfeld, F. C., and Ferguson, E. E. (1974). J. Chem. Phys. 61, 3181. Fehsenfeld, F. C., Howard, C. J . , and Schmeltekopf, A. L. (1975).J. Chem. Phys. 63,2835. Gallagher, A. C., and York, G. (1974). R e v . Sci. Instrum. 45, 662. Gilbody, H. B., Browning, R., Dunn. K . F., and McIntosh, A. (1969).J. Phys. B [I] 2,465. Gillespie, G. H. (1977). Phys. Rev. A [3] 15, 563. Could, R. K., and Miller, W.J. (1975). J . Chem. Phys. 62, 644. Grestau, F., Hall, R. I., Mazeau, J., and Vichon, D. (1977).Proc. Int. Conf. Phys. Electron. At. Collisions, loth, 1977 Abstracts, p. 148. Griffing, K. M., and Sirnons, J. (1974).J. Chem. Phys. 62, 535. Griffing, K. M., and Sirnons, J. (1975). J. Chem. Phys.64, 3610. Haas, R. A. (1973).Phys. Rev. A [3] 8, 1017. Hall, R. I. (1978). J. Chem. Phys. 68, 1803. Hall, R. I., Vazeau, J., Reinhardt. J., and Schermann, C. (1970). J . Phys. B [1] 3, 991. Hall, R. I., Cadei, I., Schermann, C., and Tronc, M. (1977). Phys. Rev. A [3] 15, 599. Harnois, M., Risley, J. S., and Geballe, R. (1975).Proc. Inr. Conf.Phys. Electron. A t . Collisions, 9th, 1975 Abstracts, p, 35. Harnois, M., Falk, R. A., Geballe, R., and Risberg, J. S. (1977).Phys. Rev. A [3] 16,2256. Hayward, T. D., and Tesmer, J. (1977). Private communication to G. H . Gillespie. Heinemeier, I . , Hvelplund, P., and Simpson, F. R. (1976).J. Phys. B [I] 9, 2669. Herbst. E.. Patterson, T. A., and Lineberger, W. C. (1974).J . Chem. Phys. 61, 300. Herbst, E., Mulholland, K. A., Champion, R. L., and Doverspike, L. D. (1977).J. C e m . Phys. 61, 5074. Hiraoka, H., Nesbet, R. K., and Welsh, J . R. (1977). Phys. Rev. Lett. 39, 130. Hong, S. P., Woo, S. B., and Helmy, E. M.(1977). Phys. Rev. A [3] 15, 1563. Hopper, D. G., Wahl, A. C., Wu, R. L . C., and Tiernan, T. 0. (1976).J . Chem. Phys. 65, 5474. Hotop, H., and Lineberger, W. C. (1975).J. Phys. Chem. Re$ Data 4. Huber, B. A.. Cosby, P. C., Peterson, J . R., and Moseley, J. T. (1977).J. Chem. Phys. 66, 4820. Hurley, R. E. (1974). Nucl. Insfrum. & Methods 118, 307. Inokuti, M., and Kim, Y . K . (1968). Phys. R e v . [2] 173, 154. Jensen, D. E. (1969). Trans. Faruday SOC. 65, 2123. Jensen. D. E., and Miller, W. J. (1970).J . Chem. Phys. 53, 3287. Jensen, D. E., and Miller, W. S. (1971). Symp. ( I n r . ) Combust. [Proc.] 13, 363. Kasdan, A., Herbst, E., and Lineberger, W. C. (1975a). Chem. Phys. Lett. 31, 78.
NEGATIVE IONS
35
Kasdan, A., Herbst. E . , and Lineberger, W. C. (197%). J. Chem. Phys. 62, 541. Kenney, J . . and Simons, J. (1974).J. C h e m . Phys. 62, 592. Khvostenko, V. I . , and Dukel'skii, V . M. (1958). Z h . Eksp. Teor. Fie. 34, 1026; SOV. Phys.-JETP (Engl. Transl.) 7, 709 (1958). Larsson. S. (1968). Phys. R e v . [2] 169, 49. Le Dourneuf. M.. and Vo Ky Lan (1977). J . Phy.s. B [ I ] 10, L97. Linder. F., and Schmidt, H . (1971). Z. Naturforsch., Teil A 26, 1617. Lindinger. W.. Albritton. D. L.. Fehsenfeld. F. C . , and Ferguson. E. E. (1975). J . Chem. Phys. 63, 3238. Liu, B. (1977). J . C h e m . Phys. 67, 373. Lowdin. P. 0. (1955).P h y s . R e v . [2] 97, 1474. McDaniel, E. W.. Martin, D. W.. and Barnes. W. S. (1962). R e ) . . Sci. Instrum. 33, 2. Marchetti, M. A., Krauss, M., and Weiss, A. W. (1972) Phys. R e v . A 5, 2387. Massey, H. S . W . (1976). "Negative Ions," 3rd ed. Cambridge Univ. Press, London and New York. Mathis, R. F.. and Snow, W. R. (1974). J. Chem. Phys. 61, 4274. Mazeau, J . , Gresteau, F.. Hall, R. 1.. and Hurtz, A. (1978).J . P h y s . B [ I ] (to be published). Meyer, W. (1974). Theor. Chim. Actii 11, 441. Miller, W. J . (1972). J. C h e m . P h y s . 57, 2354. Moseley, J . T., Cosby, P. C., Bennett, R. A,, and Peterson, J. R. (1975a).1. f h m i . Phys. 62, 4826. Moseley, J. T., Olson, R. E., and Peterson, J. R. (1975b). Case Stud. At. Phys. 5, 1-45. Moseley, J . T., Cosby, P. C., and Peterson, J. R. (1976). J. C h e m . Phys. 65, 2512. Moser, C. M.. and Nesbet, R. K. (1971). Phys. R e v . A [3] 4, 1336. Moser, C. M., and Nesbet. R. K . (1972). P h y s . R e v . A [3] 7, 1710. Moser, C. M., and Nesbet, R. K. (1975). Phys. R e v . A [3] 11, 1157. Nicolopoulou, E . , Bacal, M., and Doucet, H . J. (1977). J . Phys. (Paris) 38, 1399. Nighan. W. L., and Wiegand, W. J . (1974). Phps. R e v . A [3] 10, 922. Oliver, A , , Brouillard, F., Claeys, W., and Poulaert. G. (1976). J. Phys. B [ I ] 9, 3295. O'Malley, T. F., and Taylor, H. S. (1968). Phys. R e v . [2] 176, 207. Oparin, V. A , , Il'in, R. N., Serenkov, I. T . , Solov'ev, E. S . , and Fedorenko, N. V. (1970). Z h . Eksp. Teor. Fiz. Pis'ma R e d . 13, 351; JETP Lett. (Engl. Transl.) 13, 249 (1970). Parlant, G.. and Fiquet-Fayard. F. (1976). 1.Phys. B [ I ] 9, 1617. Peart, B., Grey, R., and Dolder. K. T. (1976a).J . Phys. B [ I ] 9, 3047. Peart, B., Grey, R., and Dolder, K. T. (1976b) J . Phys. B [I] 9, L373. Refaey. K. M. A.. and Franklin, J . L. (1976). J. C h e m . Phys. 64, 4810. Rescigno, T. N . , and Bender. C. F . (1976). J. Phys. B [ I ] 9, L329. Rescigno, T. N.. Bender, C. F.. McCurdy, C . W., and McKoy. B. V. (1976). J. Phys. B [ I ] 9, 2141. Risley, J. S.. d e Heer, F. J., and Kerkdijk, C. B. (1978). J . Phys. B [ I ] 11, 1783. Rose, P. H., Connor, R. J., and Bastide, R. P. (1958). Bull. Am. Phys. SOC. [2] 3, 40. Rothe. E. W., Tang, S. Y., and Reck, G . P. (1975).J.Chetn. Phys. 62,3829. Saponova. U. I . , and Senaskenko, V. S. (1976). Phys. Lett. A 55, 401. Sasaki. F.. and Yoshimine, M. (1974a). P h y s . R e v . A [3] 9, 17. Sasaki, F.. and Yoshirnine, M. (1974b). Phys. R e v . A [3] 9, 26. Schermann. J . P., Barbe. R., and Avrillier. S . (1977). Proc. Int. Conf. P h y s . Electron. At. Collisions. 10th. I977 Abstracts, p. 842. Schneider, B. I., and Hay, P. J. (1976). Phys. R e v . A [3] 13, 2049. Serenkov, I . T . , Il'in R. N., Oparin, V . A., and Solov'ev, E. S. (1975). Z h . Eksp. Teor. Fiz. 68, 1686; Sov. Phys.-JETP (Engl. Transl.) 41, 845 (1976).
36
H . S. W . Massey
Sides, G. D., Tieman, T . O., and Hanrahan, R. I. (1976). J . Chem. Phys. 65, 1966. Siegel. M. W., Celotta, R. J., Hall, J. M., Levine, J., and Bennett, R. A. (1972). Phys. Rev. A [3] 6, 607. Simons, J., and Smith, W. D. (1973). J. Chem. Phys. 58, 4899. Sims, J. S., and Hagstrom, S. A. (1971). Phys. Rev. A [3] 4, 908. Sims, J. S., Hagstrom, S. A., Munch, D., and Bunge, C. F. (1976). Phys. Rev. A [3] 13,560. Smith, K.andThomson,R. M.(1978). PlenumPress,NewYork. Smythe, R., and Toevs, J. W. (1965). Phys. Rev. [2] 139, A15. Snyder, R. (1973). J. Phys. B [ I ] 6, L8. Spence, D., Huebner, R. H., and Burrow, P. D. (1978). Bull. Am. Phys. SOC. [2] 23, 143. Staemmler, V., and Jungen, M. (1975). Theor. Chim. A c f a 38, 303. Stamatovic. A., and Schulz, G. J . (1970). J . Chem. Phys. 53, 2663. Teillet-Billy, D.,and Fiquet-Fayard, F. (1977). J. Phys. B [ I ] 10, L111. Tronc, M., Huetz. A., Landau, M., Pichon, F., and Reinhardt, I. (1975). J . Phys. B [ I ] 8, 1160. van Brunt, R. J., and Gallagher, A. C. (1977). Pruc. Inr. Con$ Phys. Electron. At. Collisions, lOrh, 1977 Abstracts, p. 940. van Brunt, R. J., and Kieffer, L . J. (1970). Phys. Rev. A [3] 2 , 1899. van Brunt, R. J., and Kieffer, L. J. (1974). Phys. Rev. A [3] 10, 1633. Vestal, M. L.. and Mauclaire, G . H. (1977a). J. Chem. Phys. 67, 3758. Vestal, M. L., and Mauclaire. G. H. (1977b). J . Chem. Phys. 67, 3767. Vo Ky Lan, Le Doumeuf, L., and Burke, P. G. (1976). J . Phys. B [ I ] 9, 1065. Weiss, A. W. (1968). Phys. Rev. [2] 166, 70. Weiss, A. W. (1971). Phys. Rev. A [3] 3, 126. Wung, $. F., Vorburger, T. V., and Woo, S. B. (1972). Phys. Rev. A [3] 5, 2598. Woo, S . B., Branscomb, L. M., and Beaty, E. C. (1969). J . Ceophys. Res. 74, 2933. Zittel, P. R . , and Lineberger, W. C. (1976). J. Chem. Phys. 65, 1236.
/I
ADVANCES IN ATOMIC A N D MOLECULAR PHYSICS. VOL. 15
ATOMIC PHYSICS FROM ATMOSPHERIC AND ASTROPHYSICAL STUDIES A . DALGARNO Hurvard-Smithsonian Center for Astrophysic.! Cumbridge. Massachuserts
I . Introduction . . . . . . . . . . . ................. 11. Dissociative Recornbinatio .................. 111. Ion-Molecule Reactions . ........................ IV. Neutral-Particle Reacti V. Accidental Resonance VI. Charge Transfer of Multiply Charged Ions .......................... VII. Fine-Structure Transiti VIII. Radiative Association . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX. Microwave Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X. Oscillator Strengths and Branching Ratios .......................... XI. Radiative Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XII. Spontaneous Radiative Dissociation of Diatomic Molecule XIII. Relativistic Magnetic Dipole Transitions.. . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38 42
50
55 56 59 62
I. Introduction Interpretations of atmospheric and astrophysical phenomena have frequently led to the identification of processes of general significance in atomic and molecular physics, and the precision of atmospheric and astrophysical data is often sufficient that quantitative conclusions can be drawn about the efficiencies of specific atomic and molecular processes. Ionospheric data on the electron content and distribution in the terrestrial atmosphere were discussed in the classic papers of Massey (1937) and Bates and Massey (1946, 1947) in which an extensive description was presented of processes that affect the ionization balance and a framework was established for the qualitative and quantitative analysis of the properties of weakly ionized diffuse plasmas in terms of the detailed atomic and molecular processes that occur. Essentially the same theoretical approach has been used subsequently to interpret the experimental data on the atmospheres of the planets, obtained particularly by instrumentation carried aboard spacecraft. 37 Copyright 0 1979 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-12-W3815-3
38
A . Dalgurno
For this review the author has selected a number of topics that display the intimate interactions and connections between atomic and molecular physics and atmospheric and astrophysical phenomena. The selection is limited and largely arbitrary though with some emphasis on recent developments.
11. Dissociative Recombination The early ionospheric analyses of Bates and Massey demonstrated that radiative recombination O+ + e -+ 0
+ hu
proceeds too slowly and concluded that the probable sequence was charge transfer of O+ to form a molecular ion followed by dissociative recombination XY+
+e+X +Y
Because dissociative recombination does not involve an interaction with the radiation field, it might well proceed rapidly. However, at the time Bates and Massey (1947) introduced the process, there was no quantitative evidence on its efficiency. Later, a laboratory investigation of the decay of ionization in a helium afterglow revealed the existence of an anomalously rapid recombination rate (Biondi and Brown, 1949). Although an extensive series of laboratory experiments was to show that the original measurements of Biondi and Brown (1949) referred not to dissociative recombination He:
+
e
+
He
+ He
but to the three-body process He+te+e+He+e
the discovery of a fast recombination stimulated a theoretical analysis of dissociative recombination by Bates (1950), which established that the process often proceeds with a rate coefficient as large or larger than cm3 sec-I. Extensive laboratory measurements have since been performed in which the rate coefficients for the dissociative recombination of the atmospheric ions NO+, N:, and 0; were determined (cf. Bardsley and Biondi, 1970). In the case of NO+, two values of the rate coefficient with a different dependence on electron temperature have been obtained. The value of Walls and Dunn (1974) is appropriate to NO+ ions in the u = 0 vi-
39
ATMOSPHERIC A N D ASTROPHYSICAL STUDIES
brational level, whereas that of Huang et al. (1975) corresponds to a mixture of vibrationally excited and ground-state ions. Further information can be supplied by analysis of atmospheric data. During the past four years an extensive data base of atmospheric parameters has been gathered by instruments aboard the Atmospheric Explorer satellites AE-C, -D, and -E (Dalgarno et al., 1973b). A sample of the data on NO+ concentrations is shown in Fig. 1, which includes the results of a detailed theoretical model of the terrestrial ion chemistry (Oppenheimer et al., 1977a) corresponding to the different rate coefficients for dissociative recombination. The value of Walls and Dunn (1974) is clearly to be preferred, presumably because the experiment involves NO+ ions in the u = 0 vibrational level. The radiative lifetimes of vibrationally excited NO+ ions are of the order of sec (Billingsley, 1973) and in the atmosphere vibrationally excited NO+ ions decay by infrared emission in a time short compared to the mean time for dissociative recombination (Dalgarno, 1963). An analysis of nocturnal data on NO+, when fewer mechanisms exist for populating excited vibrational levels, is also consistent with the coefficient measured by Walls and Dunn (D. G . Torr e t al., 1976a). 400.
+ x .
+ 360.
X.
+
X .
+
Y
-
+
K
t
E 3201
X.
x
+
w l 2 280
m
+
m
u
+
k c
U
X .
+
o(
.X .X
+
.X
U
240/
1
I
I
I , 1 1 1 1 1
(02
I
I
I
I
1
,,,I
I
I
1
lo3
1
1
1
1
1
l
1
I
, 1 1 1 ,
lo4
DENSITY ( ~ r n - ~ )
FIG.1. Densities of NO+ ions are shown as a function of altitude. 0 , Experimental data: x, results of calculations using the recombination coefficient of Walls and Dunn (1974); + results of calculations using the recombination coefficients of Huang p t al. (1975). (Oppenheimer e t u / . , J . Grophy.~.Res. 82, 191, 1977, copyrighted by the American Geophysical Union).
40
A . Dalgarno
An earlier analysis (M. R. Torr et al., 1975) of a more limited set of Atmospheric Explorer data had supported the laboratory value of Huang et al. (1975) but there may have occurred some atmospheric disturbance leading to an enhanced vibrational population (D. G. Torr et a f . , 1976b). Some dependence of the rate coefficient of dissociative recombination on the vibrational population is expected on theoretical grounds (Bates, 1950; Bardsley, 1968; O'Malley, 1969; Lee, 1977) so that it appears the laboratory and atmospheric data can be easily reconciled. However, recent merged electron-ion beam measurements by Mu1 and McGowan (19791, which involve NO+ ions in an unknown distribution of vibrational levels, give a rate coefficient in harmony with that measured for NO+ in the ZI = 0 vibrational level. The atmospheric analysis, though persuasive, is not entirely convincing. Uncertainties in the rates of reactions in the ion chemistry may become amplified to the point where measurement errors are significant. The chemistry itself may still be incomplete. In particular, Zipf (1978) pointed out that associative ionization N(2D) + 0 -+ NO+ + e
- 0.4
eV
is a source of NO+ that has not been included in the Atmospheric Explorer analyses. For 0: the laboratory values of Mehr and Biondi (1%9) and of Walls and Dunn (1974) are consistent with each other but somewhat higher than those of Mu1 and McGowan (1979). Rate coefficients have also been derived from Atmospheric Explorer data (D. G. T o n et al., 1976b; Oppenheimer et al., 1977b) and the values presented by Torret ul. are shown in Fig. 2 with the laboratory data. The agreement is better than ? 30%. Vibrationally excited 0: ions are probably quenched efficiently in the atmosphere by an atom interchange mechanism (Bates, 1955a) O$(V) + 0 4 O$(IJ'
and O+(2P)ions of atomic oxygen and to obtain estimates of their rate coefficients. The O+(2P)metastable species decays by the emission of a doublet near 7320 and 7330 A, and its observation provides valuable additional data on chemical reactions in the ionosphere (Dalgarno and McElroy, 1963, 1965). It is measured routinely in the Atmospheric Explorer satellite (Hays e l a / . , 1973; Rusch et al., 1977) and it has also been detected by groundbased instruments (Carlson and Suzuki, 1974; Meriwether et a / . , 1974, 1978). There remain considerable uncertainties in the metastable rate coefficients (cf. Oppenheimer et al., 1976; Rusch et al., 1977; Orsini et a / . , 1977b; Torr and Orsini, 1977, 1978; Torr and Torr, 1978). Table I compares some of the results for reactions of 0+(2D).Further analyses of Atmospheric Explorer data taken over a wide range of atmospheric conditions should help to resolve the discrepancies and refine the estimates. Basic laboratory and theoretical studies are also needed.
A . Dulgarno
44
TABLE I
RATE COEFFICIENTS OF REACTIONSOF O+(*D) W I T H Nz A N D 0 (cm'sec-') Of(2D) + N,-+ 0 + N J Oppenheimer ef a / . , (1976) Torr and Orsini (1977) Torr and Orsini (1978)
1 x 10-8 5 x 10-10 1 x 10-10 1 x 10-10
2 x 10-11 2 make an insignificant contribution to the total cross section in the energy range 0- 14 eV. The theoretical results for helium models H5 and H14 are in very good
THEORETICAL ASPECTS OF POSITRON COLLISIONS IN GASES
115
0 4
03
I
T
0
02
04
06
Positron wavenumber
1 4 +
08
10
k (ao-’]
FIG.6. Total cross sections for positron-helium scattering. Exp. results: 0 ,Cantervr d. (1972, 1973); A , Jaduszliwer and Paul (1973): A, Brenton el trl. (1976); x, Burciaga P I a / . (1977); 0.Stein et a / . (1978); f , Wilson (1978). Theoretical results: ---, model HI; -, model HS; ---, model H14.
agreement with the experimental results of Canter rf ul. (1973) down to 2 eV, the lowest energy reached in this experiment. They are also in very good agreement with an extrapolation of these experimental results below 2 eV (Bransden er nl., 1974; Humberston, 1974) based on a least-squares fit to the functional form (+
=
A.
+ A,k + A , k 2 In k + A3k2 + A,k4
(10)
As an example of this agreement, the total cross section at zero positron whereas the value calcuenergy, the value of A . in Eq. ( l o ) , is 0.883~~120, 2 0 helium model H 14 and 0.89 lated from the scattering length is 0 . 9 2 ~ ~ for for model HS. Above 6 eV the results of Canter et al. (1973), Burciaga rt NI. (19771, and Stein et LJI. (1978) are all in reasonably good agreement with each other and with the theoretical values for models H5 and H14. Below 6 eV, however, and particularly in the vicinity of 2 eV where all the results display a Ramsauer-type minimum, the results of Stein er u l . and Burciaga e f
J . W. Humberston
116
al. fall significantly below the theoretical values. (Unless defined otherwise the term theoretical values will imply the results for models H5 or H14.) But, because the values of the p- and d-wave phase shifts are rather insensitive to the precise form of the helium wavefunction at an energy of 2 eV (see Figs. 4 and 5 at k2 = 0.147), provided the dipole polarizability has the correct value, a rather firm lower limit on the total cross section at this energy can be set at 0.066~~20 (Humberston, 1978). This is signifi~ ~0 ~ . 0 5 5 ~ ~measured : by Burciaga cantly larger than the values 0 . 0 4and et al. and Stein e f al., respectively, and it is possible that there are systematic errors in these experimental results associated with the neglect of scattering through small angles. As can be seen from Fig. 7, which con-
UI * .-
4 9OeV
C
2
I * .-
2-
2.18 eV
Q)
t .-
e UI
b
0.54 eV
0 136eV
0
30
60
90
I20
150
I80
0 (degrees) FIG.7. Angular distributions of positrons scattered by helium atoms at various incident energies. The model H5 phase shifts were used to obtain these results.
THEORETICAL ASPECTS OF POSITRON COLLISIONS I N GASES
I I7
tains plots of the angular distributions of scattered positrons at various energies, the neglect of small-angle scattering is more serious at energies close to 2 eV than elsewhere in the elastic-scattering energy region. One might have expected similar systematic errors in the results of Canter et ul. yet the good agreement with the accurate theoretical values suggests that this is not so to any significant extent, unless there is some fortuitous cancellation of the effects of this and some other systematic error. Further support for the accuracy of the results of Canter et a/. and the theoretical values for helium model H5 has recently been provided by the measurements of Wilson (1978). This has prompted Griffith et ul. (1978) to reexamine the original experimental apparatus of Canter et ul., and they have found that positrons scattered through small angles in collisions with helium atoms will be further deflected through much larger angles because of variations along the flight tube in the strength of the axial magnetic field. Such positrons are then clearly identified as having been scattered. This additional deflection explains the rather mysterious absence of evidence for small-angle scattering in the time of flight spectra of Canter et a / . at positron energies less than 4 eV. Griffith et al. conclude that the errors in the measurements of the total cross section of Canter et ul. due to the neglect of small-angle scattering are less than the statistical errors. Other rather less direct comparisons between experimental and theoretical results for scattering parameters have been made by Bransden et ul. (1974) and Bransden and Hutt (1975) using techniques based on forward dispersion relations (Gerjuoy and Kroll, 1960). The dispersion relation can be written as
wherefk(0) andfB(0)are the exact and the first Born approximation to the forward scattering amplitude, respectively. For zero-energy scattering ( k = 0 ) this equation reduces to
where N is the scattering length. This equation has been used to check the consistency of the measured values of the total cross section with the calculated values of the scattering length and the Born amplitude. Using the data of Coleman et ul. (1976) up to 800 eV and the Born approximation thereafter, the right-hand side of Eq. (12) is calculated to be ( - 1.24 -t 0.05ja0. A very similar result is obtained with the data of Brenton et al. (1976). The left-hand side of Eq. (12) has the value - 1 . 2 8 ~for 1 ~ models HI4 and H5 and - 1 . 3 1 for ~ ~ model H I . In calculating these results the
I18
J . W . Hurnherston
values offB(0)appropriate to each helium model have been used, whereas Bransden et ul. used the value -0.791u0, which was obtained by Bransden and McDowell (1969) with a very accurate helium function. Comparison between the theoretical values of the real part of the forward scattering amplitude, which is given in terms of the phase shifts by
and the “experimental” values, derived from the total cross-section measurements according to a modified form of Eq. (1 l), has been made by Bransden and Hutt at a number of positron energies below the positronium formation threshold (17.8 eV). The agreement between the experimental results derived from the data of Coleman et al. (1976) and Jaduszliwer et af. (1975) and the theoretical results from Eq. (13) for helium model H5 is good up to a positron energy of approximately 6 eV, but thereafter the theoretical value falls slightly below the experimental value. If the lowest set of Drachman’s (1966) p-wave phase shifts are used in these calculations instead of those for model H5, good agreement with the experimental values of Refk(0) is obtained up to an energy of 9 eV, which suggests that the p-wave phase shifts for model H5 plotted in Fig. 4 are rather low at energies above 6 eV. This was perhaps to be expected in view of the rather poor convergence of the p-wave phase shifts in this energy region referred to earlier.
IV. Annihilation in Positron- Atom Scattering The probability per unit time of annihilation of a positron and an electron, which are in a singlet spin state, into two y rays is h = nrkn
(14)
where ro ( = e 2 / m c 2 )is the classical radius of the electron, c the velocity of light, and n the electron density in the immediate vicinity of the positron. Three-quarters of the electrons form triplet spin states with the positron and annihilation is then into three y rays. But the rate of annihilation into three y rays is only 1/370 of the rate into two y rays, so that Eq. (14) is a very good approximation to the overall annihilation rate. If the density of atoms in the vicinity of the positron is N per unit volume, the annihilation rate can be written as h = micNZeff
(15)
where Zerris the effective number of electrons per atom. The value of Zeff will in general vary with the positron velocity. The annihilation cross sec-
THEORETICAL ASPECTS OF POSITRON COLLISIONS IN GASES
tion
(T,
119
is defined as
A
(Ta
= -=
Nu
1 - 7rY:cZ,[f(u) L]
(16)
where u is the positron velocity. Except in the limit of zero velocity where it becomes infinite (although the annihilation rate remains finite), the annihilation cross section is several thousand times smaller than the cross sections for elastic and inelastic scattering. Therefore, although the annihilation channel is always open, its coupling to the elastic and inelastic scattering channels is so weak that it can be neglected when calculating the scattering parameters. If the total wavefunction describing the scattering of a positron, with position vector r l , by an atom containing N electrons with position vectors r2, . . . , rN+lis P ( r l , r z , . . . rN+l),then
.
where P i s normalized to unit positron density as Y, -+ =. The calculation of Zeffis therefore a relatively trivial extra calculation once the scattering wavefunction has been obtained. Unfortunately, however, because Eq. (17) does not constitute a variational principle, the error in Zeffis of first order in the error in the wavefunction, whereas errors in phase shifts are usually of second order. Consequently the accuracy of Zeff calculated from a particular wavefunction is likely to be significantly less than the accuracy of the variationally determined phase shift. At high positron energies, where the total wavefunction is accurately represented by the Born approximation, the value of Zeffis the electron number Z , but at low positron energies it can rise to several times this value due to the polarization of the target atoms by the incident positron. Indeed, there is quite a strong correlation between the dipole polarizability P of an atom and its value of Zeffat very low positron energies, which is represented reasonably accurately by (Osman, 1965) Zeff
P1.25
(18)
In positron-hydrogen scattering the most accurate values of the s- and p-wave contributions to Zeff have been calculated by Humberston and Wallace (1972) and Bhatia et al. (1974b, 19771, but calculations have also been made by Chan and Fraser (1973) and Chan and McEachran (1976). The sum of these values and the results from the Born approximation for the contribution from all higher partial waves is plotted in Fig. 8. The corresponding results for positron-helium scattering from a number of calculations are plotted in Fig. 9. Although, for the reason mentioned above, the accuracy of these results is not expected to be as
J . W . Humberston
120
01
02
03
04
05
06
07
k (a,,-')
8. ZeHfor positron-hydrogen scattering: these values were obtained by adding the accurate s- and p-wave contributions of Humberston and Wallace (1972) and Bhatia er a / . (1971, 1977) to the results given by the Born approximation for the contribution from all higher partial waves. FIG.
good as that of the corresponding phase shifts, the value of 3.86 for Zefffor helium model H5 at an energy of 40 meV (thermal energy at room temperature) is in very satisfactory agreement with the equilibrium value 3.94 ? 0.02 obtained by Coleman el nl. (1975a) in experimental studies of the lifetime spectrum of positrons diffusing through helium gas. For accounts of the general features of the lifetime spectra of positrons in gases the reader is referred to the articles by Massey (1971) and Griffith and Heyland (1978). Studies of the nonequilibrium part of the lifetime spectrum enable experimental information to be obtained about the value of Zeffat energies other than thermal energy, but the experimentally measured quantity is Zeff(t), the value of Zeffaveraged over the velocity distribution of the positrons. Thus, if y ( u , t) is the velocity distribution of the positrons, i.e., y ( u , t ) du is the number density of positrons with velocities in the interval v to (v + dv) at time I , then
IOm lom ~ ( 0 t)zefXv) ,
zedt) =
du
(19)
Y ( W ) dv
The most interesting part of the lifetime spectrum and the one most readily investigated theoretically is the so-called shoulder region, where the
THEORETICAL ASPECTS OF POSITRON COLLISIONS IN GASES
12 1
42 II .O
38
36 Zeff
32
0
02
04
06
08
10
k (a;') FIG.9. Z , , for positron-helium scattering. Exp. results: X , Roellig and Kelly (Fraser, 1968) at T = 77 K; 0 , Coleman et ol. (1975a). Theoretical results: -, Campeanu and Humberston H5, Campeanu and Humberston HI; McEachran r i ul. (1977).
positrons are slowing down below the threshold energy for positronium formation by purely elastic collisions, and the measured annihilation rate A(t) = &,cN&(r) is changing with time because of the variation of Zeff(u) with positron velocity. Eventually the positron velocity distribution acquires its equilibrium form and the annihilation rate becomes constant. Before direct comparisons can be made between experimental and theoretical annihilation rates the velocity distribution of the positrons must be calculated. Its developement in time from an initial velocity distribution y(u.0) in a gas at temperature T and in a uniform electric field of strength E is governed, to a good approximation, by the diffusion equation (see Orth and Jones, 1969)
as
-(u,t) = ac at a
[(
3 d L e2F ~vu,(U)
T )( u , 1 ) + L J NMU , ( L . ) ~2 au 2e'F 3m2L~'NCr~j( U)
M (20)
122
J . W . Humberston
where ( T ~ ( uis) the momentum transfer cross section, k Boltzmann’s constant, and M the mass of each gas atom. All other symbols have been defined previously. The momentum transfer cross section is given in terms of the phase shifts by
and plots of uM(k)for models H1 and H5 are given in Fig. 10. Campeanu and Humberston (1977b) have solved Eq. (20) for several plausible forms of the initial velocity distribution, and used the solution to calculate Zeff(t)from Eq. (19). They found that the results do not depend very strongly on the form of y(v, r = O ) , provided it is reasonable, except within a short interval of time close to I = 0. Significant differences were
1 .o
\
m -
03
N
Y
b’
0.1
0.03
0 01 00
02
06
06
0.8
1.0
k (ad’)
FIG. 10. The momentum transfer cross section for positron-helium scattering: -, H5: __. HI.
123
THEORETICAL ASPECTS OF POSITRON COLLISIONS I N GASES
found between the results for model HI and H5,as can be seen from Fig. 11, but the results for model H5 are in quite good agreement with the experimental values of Coleman ef a / . (1975a) at all values o f t other than close to t = 0. There is even better agreement between the H5 results and recent, previously unpublished results of Griffith and Heyland, which are also plotted in Fig. 11. The differences between theoretical and experimental values of Zeff(t)close to t = 0 arise because the experimental annihilation rate then includes contributions from processes other than the annihilation of free positrons with helium atoms. Campeanu and Humberston also investigated the effect on the lifetime spectrum of varying the temperature of the gas and' the magnitude of the applied electric field. The results for the equilibrium annihilation rate obtained with the H5 data only are presented in Fig. 12, and they agree quite well with the results of a number of experimental measurements. More accurate experimental results for varying electric field strength are, however, required. Confirmation that the diffusion of positrons through helium gas is accurately described by the diffusion equation, Eq. (20) has been provided by Farazdel and Epstein (1977, 1978). These authors used a Monte Carlo technique with the phase shifts and values of Zefffor model H5 and reproduced the theoretical results given in Figs. I 1 and 12 very closely indeed. Investigations have also been made recently of positron diffusion in the other inert gases (Hara and Fraser, 1975; Campeanu, 1977) and in mixtures of two gases (Massey r f NI., 1972; Grover, 1978), but the lack of
I
0
I
MO
I
moo
I
I
I
I
1500
axil
2500
3000
time Insec amaqatsl
FIG.1 1 . The time dependence of Zetffor positrons in helium at room temperature and zero electric field: ---, exp. measurement of T. C. Griffith and G . R . Heyland (unpublished); _.._, exp. measurement of Coleman ei at. (1975a); -calculated from the data for model calculated from the data for model HI. H5:
J . W . Humberston
124
TIK)
0
500 1
lo00
1500
I
-
10
20
E I v cm-' arnogat-'1
----
30
2m
-i LO
FIG.12. The dependence of the equilibrium value of Zeffon temperature and electric field for positrons in helium. Exp. results: 0, Lee et a / . (1969) at T = 293 K ; 0 , Leung and Paul (1969) with h2 = 8.6 pet-' (see reference for explanation) at T = 77 K: +, Leung and Paul (1969) with hz = 8.1 psec-l at T = 77 K ; A, Roellig and Kelly (Fraser, 1968) at T = 77 K and zero electric field; x, Coleman er a / . (1975a) at T = 293 K and zero electric field. Theoretical results: -, variation of Zenwith electric field strength at T = 77 K: -, variation of Z,,, with temperature with zero electric field. All the theoretical results are for model H5.
accuracy of the input data for the diffusion equation, uM and Zeff, for any gas except helium prevents accurate lifetime spectra from being obtained.
V. Angular Correlations in Positron Annihilation In the annihilation of a positron and an electron that form a singlet spin state, two 0.51 MeV y rays are produced, which in the center-of-mass coordinate system of the electron-positron pair are emitted at 180" to each other. Owing to the motion of the center of mass relative to the laboratory frame of reference, however, the angle between the two y rays, as measured in the laboratory frame, is 7~ - 8, where 8 = p / m c with p the momentum of the electron-positron system at the moment of annihilation and c the velocity of light. Measurement of the angular correlation of the y rays therefore enables information to be obtained about the momentum distribution of the annihilating electron-positron pairs. This is the basis of a technique that is widely used to obtain the momentum distribution of electrons in solids and liquids. In the usual angular correlation experiment only the projection of the angle between the two y rays onto one particular plane is considered and the measured angular distribution is therefore
THEORETICAL ASPECTS O F POSITRON C O L L I S I O N S I N GASES
w) =
+m
j-mr(ps.pu.pz
I25
+m
=
(22)
1nc.o) dp, dpu
where T ( p ) is the momentum distribution function of the annihilating electron-positron system. For positron-helium scattering this has the form up)
=
I Ij dr3
dr, exp(-ip
1'
rl)*(rlrr2 = r l , r 3 )
(23)
c a w
-
Q)
U
0.2
5
10
0 imradl FIG. 13. Angular distribution of y rays from positron annihilation in helium. The black circles are the experimental results of Briscoe er (11. (1968). Theoretical results: -, model H5;--, model HI; ..., McEachraner a/.(1977). Drachman (1969) obtained results for model HI very similar to those plotted here.
126
J . W . Humber-ston
Theoretical angular distributions calculated from a number of zero-energy positron-helium scattering wavefunctions are plotted in Fig. 13 together with the experimental results of Briscoe et al. (1968). Rather surprisingly, the agreement with experiment is less good for the model H5 results than it is for the other theoretical results. This is probably due to the fact that the theoretical results refer to zero-energy scattering by isolated helium atoms, whereas the experimental results were obtained in liquid helium, where the proximity of other helium atoms distorts the wave function and also prevents the complete thermalization of the positrons before annihilation.
VI. Positronium Formation in Positron- Atom Collisions Positronium is the bound state of a positron and an electron. Its energy level structure is that of a hydrogenic atom with a reduced mass which is half the electron mass, and the ground-state energy is therefore -6.803 eV. It exists in two spin states, which are referred to as parapositronium, with total spin S = 0, and ortho-positronium, with S = lh. The lifetimes of the ground states of para- and ortho-positronium for annihilation into two and three yrays are 1.25 x 10-lo and 1.4 x lo-' sec, respectively. Positronium formation is the process in which an incident positron picks up an electron from the target atom and forms positronium, leaving behind a singly ionized atom. The threshold energy for positronium formation is the difference between the first ionization energy of the target atom and 6.803 eV. For positron collisions with hydrogen and helium atoms the threshold energies are 6.8 and 17.8 eV, respectively. A. POSITRONIUM FORMATION I N POSITRON-HYDROGEN COLLISIONS
This is the simplest example of a rearrangement process, yet it is considerably more complicated than elastic scattering, and the state of theoretical calculations is not very satisfactory. At a formal level a major difficulty, but one that has been overcome (Hahn, 1966), is that the wavefunctions of the hydrogen atom and the positronium, although known exactly, are not orthogonal to each other. More practical problems are related to the fact that the coordinate of the center of mass of the positronium is not one of the interparticle distances. Also, the rather high polarizability of the hydrogen atom and the even higher polarizability of the positronium (eight times that of the hydrogen atom) means that a very accurate description of the polarization of the system is required if accurate values of the positronium formation cross section are to be obtained, particularly just above threshold.
THEORETICAL ASPECTS OF POSITRON C O L L I S I O N S I N G A S E S
127
The first calculation of the positronium formation cross section was carried out by Massey and Mohr (1954) using the first Born approximation. The results rose quite steeply from zero at the threshold to a maximum of 4 . 5 ~ u at i 12 eV and then decreased more slowly, so that at 24 eV the . authors also used a form of the distorted-wave value was 2 ~ 4 These approximation in which the s-wave component of the plane wave approximation to the positron wavefunction was replaced by the function given by the static approximation. The latter method gave results approximately 50% lower than the results from the Born approximation. There have since been a number of calculations of the various partial-wave contributions to the cross section. Bransden and Jundi (1967) solved the coupled-channel problem with various forms of approximation for the polarization effects. They found that the addition of polarization potentials to the static potentials in the coupled equations for the radial scattering functions produced results that were very different from those obtained in the coupled static approximation. This can be seen in Fig. 14, where the results of various calculations of the s-wave contribution to the positronium formation cross section are given. The results have a very
c
N O
m
Y
0.005
C
0 ”
; 0.OOL ln v)
e u
0.003
0 .”
2
p
0.002
E,
.-
E-
0.001
4-
In
a“
05
06 kZ (a;‘
07 )
F I G .14. The s-wave cross sections for positroniurn formation in positron-hydrogen scattering: A . Chan and Fraser (1973); B , coupled static approximation: C, Fels and Mittleman (1967) model V; D, Dirks and Hahn (1971) x 10: E, Wakid and Labahn (1972) approximation t’ x 10; F , Bransden and Jundi (1967) approximation B(1) x lo-*: G , Stein and Sternlich (1972). The positroniurn threshold is at L L = 0.5.
J . W . Humberston
128
sensitive dependence on the details of the approximations being used, so that very different results are obtained even from methods that might be expected to give similar results. As an example the formulation of Fels and Mittleman (1967) is basically similar to that of Bransden and Jundi, the only difference being the use of a different coordinate system, and yet the cross sections are more than two orders of magnitude smaller than those of Bransden and Jundi. Two other rather similar methods of calculation were used by Stein and Sternlich (1972) and Chan and Fraser (1973), but here also one set of results is consistently almost 50% higher than the other. Stein and Sternlich extended the elastic s-wave calculations of Schwartz (1961) above the positronium formation threshold with two-channel versions of the Kohn and Hulthen variational methods and trial functions containing up to 84 Hylleraas correlation functions. Chan and Fraser used the formulation of the coupled static approximation with the addition of 26 correlation functions. This method gives rigorous lower bounds on the diagonal elements of the R-matrix, but not on the positronium formation cross section, in the energy region up to the lowest eigenvalue of Q H Q (see Section 11). Both calculations reproduce the s-wave phase shifts of Schwartz quite well, and both give s-wave elastic cross sections above the positronium threshold that are smooth continuations of the values below. The values of the diagonal elements of the R-matrix for both calculations are above the rigorous lower bounds of Dirks and Hahn (1971), with the values of Stein and Sternlich being slightly more positive than those of Chan and Fraser. One is therefore inclined to believe that the results of Stein and Sternlich are the more accurate, although the absence of a bound principle in the calculation makes this conclusion uncertain. Calculations of the p-wave contribution to the positronium formation cross section have been performed by Bransden and Jundi (19671, Fels TABLE 1V POSlTRONlUM
k(a;') 0.71 0.75 0.80 0.85 0.866
FORMATION CROSS OD(/=
O)*
0.0015 0.0029 0.003 1 0.0032 0.0033
SECTIONS'
IN
up(/=
POSITRON-HYDROGEN SCATTERING I)C
0.0147 0.357 0.505 0.584 0.667
The cross sections are in units of sa', . Chan and McEachran (1976).
* Chan and Fraser (1973).
OD(/=
0) +
OD(/=
0.0162 0.360 0.508 0.587 0.670
1)
THEORETICAL ASPECTS OF POSlTRON COLLISIONS I N GASES
129
and Mittleman (1967), and Chan and McEachran (1976). Again there is a wide spread in the results, although those of Chan and McEachran are probably quite accurate. The method used by Chan and McEachran is essentially the same as that used by Chan and Fraser in the s-wave calculations referred to above, but with up to 50 correlation terms in the trial functions. The results of these calculations are given in Table IV, where it is seen that the p-wave contribution to the cross section is much larger than the s-wave contribution. Higher partial-wave contributions may also be important but no such elaborate calculations have yet been performed.
B. POSITRONIUM FORMATION IN POSITRON-HELIUM COLLISIONS Considering the unsatisfactory state of calculations of the positronium formation cross sections in positron- hydrogen scattering it is hardly surprising that in positron-helium scattering the theoretical situation is if anything worse. There is, however, some experimental information on the magnitude of the cross sections, although it is not of a very precise nature. In the energy range 17.8-20.5 eV, between the threshold for positronium formation and the lowest threshold for excitation of the helium atom, only elastic scattering and positronium formation can occur. The total cross section is therefore the sum of these two partial cross sections, and by making assumptions about the behavior of the elastic cross section, the positronium formation cross section can be subtracted out. The assumption made by Coleman r t al. (1975b) is that the elastic cross section in this energy range is a smooth and almost constant continuation of the elastic cross section immediately below the positronium formation threshold. According to this assumption, the positronium formation cross section rises linearly from zero at the threshold to (0.07 2 O.O3).rru2,at 20 eV. Several of the techniques employed in calculations of positronium formation cross sections in positron- hydrogen scattering have also been used in such calculations for positron- helium scattering, although nothing equivalent to the elaborate variational calculations of Stein and Sternlich and Chan and Fraser has yet been attempted. Massey and Moussa (1961) used the Born approximation to calculate the total positronium formation cross section and obtained results that rose to a maximum value of 0 . 4 4 at 27 eV, in fairly good agreement with the “experimental” values. Similar results were obtained by Kraidy and Fraser ( 1967) with the coupled static approximation, but the inclusion of a polarization potential in the positronium channel raised the maximum value of the cross section to 1.4~rcr$at 23 eV, somewhat in excess of the experimental value of the total
130
J . W . Humberston
cross section. According to these calculations the positronium formation cross section is dominated by the d-wave contribution. Fels and Mittleman (1969), however, using essentially the same method as they had used previously for positron-hydrogen scattering, found that the s-wave contribution was dominant. Their results are again very much smaller than those of any other authors, with a maximum value of only 0 . 0 0 3 ~ ~ 2 , at 20 eV. More recent calculations have been carried out by Mandal et ul. (1975, 1976) using integral forms of the close-coupling and polarizedorbital methods. The two methods give quite different sets of results, neither of which is in very good agreement with the experimental values. Calculations of positronium formation cross sections for neon and argon have been made by Gillespie and Thompson (1977) using the Born and distorted-wave approximations, but both sets of results are in rather poor agreement with the experimental data.
VII. Concluding Remarks The theoretical situation regarding low-energy positron- helium elastic scattering now seems to be quite satisfactory, even if it is not yet quite as good as that for positron-hydrogen scattering. Good agreement has been obtained between the theoretical predictions for helium model H5 and the results of a number of disparate experiments, and the phase shifts and values of Zefffor this model are almost certainly very close to the exact values. Unfortunately, the results of calculations of inelastic processes and resonances in the positron-hydrogen and positron-helium systems are in a far less satisfactory state, and a considerable effort will be required if accurate results are to be obtained. There is a particularly urgent need for accurate theoretical values of the positronium formation cross sections in positron-helium scattering because direct experimental measurements of this quantity may soon be made. In this chapter we have concentrated almost exclusively on positron scattering by hydrogen and helium atoms, the two systems that have attracted most theoretical interest, but investigations have also been made of positron scattering by heavier atoms and molecules. The complexity of the target wavefunction then precludes the use of variational methods with very elaborate wavefunctions, as have been used for scattering by hydrogen and helium, and simpler methods of approximation must be used. The most satisfactory of these seems to be the polarized-orbital method. In one form or another it has been used quite successfully by Massey et al. (1966), Montgomery and Labahn (1970), Gillespie and
THEORETICAL ASPECTS OF POSITRON COLLISIONS I N GASES
f31
Thompson (1973, and McEachran et a / . (1978) to calculate the parameters for the elastic scattering of positrons by neon and argon. The particular form of the method used by McEachran et al. has produced elasticscattering cross sections for neon in excellent agreement with the experimental values. Unfortunately however, the shoulder in the experimental positron-neon lifetime spectrum is not reproduced very accurately (Campeanu, 19771, indicating that the momentum transfer cross sections, and hence the phase shifts, are not as accurate as might have been thought from the values of the scattering cross sections. Also the value 6.99 for Zeffat an energy of 40 meV is somewhat higher than the experimental equilibrium value of 5.99 2 0.06 (Coleman er u / . , 1975a). Calculations of cross sections for low-energy positron scattering by the molecules H2 and N2 have been performed by Hara ( 1974), Darewych and Baille (19741, and Gillespie and Thompson (1975). The results of Hara for H, and Gillespie and Thompson for N, agree fairly well with the experimental values up to the dissociation energies of the molecules. A considerable quantity of experimental data is also available on positron collisions with other atoms and molecules, and further complementary theoretical work on these systems is now required. ACKNOWLEDGMENTS The author wishes to thank Professor Sir Harrie Massey for introducing him to the subject of positron-atom collisions, and for his continued interest and guidance. He is also grateful to Drs. T. C. Griffith, G. R. Heyland, and T. R. Twomey, and other members, past and present, of the Gaseous Positronics Group at University College London for many useful discussions.
REFERENCES Amusia, M. Y.,Cherepkov, N. A,, Chernysheva, L. V., and Shapiro, S. G. (1976).J.f h y s . B [ I ] 9, L531. Armstead. R. L. (1968). f h y s . R e v . [2] 171, 91. Aulenkamp, H., Heiss, P., and Wichmann, E. (1974). Z. Phys. 268, 213. Bhatia, A. K . , Temkin, A , , Drachman, R. J.. and Eiserike, H. (1971). Phys. Rev A [ 3 ] 3 , 1328. Bhatia. A. K., Temkin, A . , and Eiserike, H. (1974a). f h y s . Re\,. A [3] 9, 219. Bhatia. A. K., Drachman, R. J . , and Temkin, A. (1974b). fhys. Rev. A [3] 9, 223. Bhatia, A. K., Drachman, R. J., and Temkin, A. (1977). Phys. Rev. A [3] 16, 1719. Bransden, B. H. (1969). Case Stud. A r . Collision f h y s . 1, 171. Bransden. B. H., and Hutt, P. K. (1975).J . f h y s . B [ I ] 8, 603. Bransden. B. H . , and Jundi, Z. (1967). Prur. fhys. Snc. London 92, 880. Bransden. B. H., and McDowell, M. R. C. (1%9). J . f h y s . B [ l ] 2 , 1187. Bransden, B. H., Hutt, P. K., and Winters, K. H . (1974).J . f h y s . B [ I ] 7 , L129.
132
J . W . Humherston
Brenton, A. G., Dutton, J . , and Harris, F. M. (1976). P m c . I n t . Conf PositronAn,zihilation, 4th, 1976 Abstract A16. Briscoe, C. V . , Choi, S.-I., and Stewart, A. T. (1968). Phys. Rev. Lett. 20, 493. Burciaga, J. R., Coleman, P. G., Diana, L . M., and McNutt, J . D. (1977). J . Phys. B [ I ] 10, 569. Campeanu, R. I. (1977). Ph.D. Thesis, University of London. Campeanu, R. I., and Humberston, J. W. (1975). J . Phys. E [ I ] 8, L244. Campeanu, R. I., and Humberston, J. W. (1977a). J . Phys. B [ I ] 10, L153. Campeanu, R. I., and Humberston, J. W. (1977b). J . Phys. B [I] 10, 239. Canter, K. F., Coleman, P. G . , Griffith, T. C., and Heyland, G. R. (1972). J . Phys. B [1]5, L167. Canter, K. F., Coleman, P. G.. Griffith, T. C . , and Heyland, G. R. (1973).J . PhyJ. B [1]6, L201. Chan, Y. F . , and Fraser, P. A. (1973). J . Phys. B [ I ] & 2504. Chan, Y . F., and McEachran, R. P. (1976). J . Phys. B [ I ] 9,2869. Coleman, P. G., Griffith, T. C., Heyland, G. R., and Killeen, T. L. (197Sa).J . Phys. B [I]& 1734. Coleman, P. G., Grifith, T. C., Heyland, G. R., and Killeen, T. L. (1975b).A t . Phys., 4th, 1974 p. 355. Coleman, P. G., Griffith, T. C., Heyland, G . R., and Killeen, T. C. (1976). "Electron and Photon Interactions with Atoms" (H. Kleinpoppen and M. R. C. McDowell, eds.), p. 181. Plenum, New York. Dalgarno, A., and Kingston, A . E. (1960). Proc. R. Soc. Londan, Ser. A 259, 424. Darewych, J. W., and Baille, P. (1974). J . Phys. B [137, L I . Dirks, J . F., and Hahn, Y . (1971). Phys. Rev. A [3]3, 310. Doolen, G. D., Nuttall, J . , and Wherry, C. J. (1978). Phys. Rev. Lett. 40, 313. Drachman, R. J. (1965). Phys. Re\'. [2] 138,AlS82. Drachman, R. J . (1966). Phys. Rev. [2] 144, 25. Drachman, R. J. (1968). Phys. Rev. [2] 173, 190. Drachman, R. J. (1969). Phys. Rev. [2] 179,237. Drachman, R. J. (1972a). Pror. I n t . Cot$ Phys. Electron. A . Collisions, 7th. I971 Invited Talks and Progress Reports, p. 277. Drachman, R. J. (1972b). J . Phys. B [ I ] 5 , L30. Drachman, R. J . (1975). Phys. Rev. A [3] 12, 340. Farazdel, A., and Epstein, I . R. (1977). Phys. Rev. A [3] 16, 518. Farazdel, A . , and Epstein, 1. R. (1978). Phys. Rev. A [3] 17, 577. Fels, M. F., and Mittleman, M. H. (1967). Phys. Rev. [2] 163, 129. Fels, M. F., and Mittleman, M. H. (1969). Phys. Rev. [2] 182,77. Feshbach, H. (1962). Ann. Phys. (Leipzig)19, 287. Fraser, P. A . (1968). Adv. A t . M o l . Phys. 4, 63. Gailitis, M., and Damburg, R. (1963). Pror. Phys. Soc.. London 82, 192. Gerjuoy, E., and Kroll, N . Q. (1960). Phys. Reis. [2] 119, 705. Gillespie, E. S . , and Thompson, D. G. (1975). J . Phys. B [ I ] 8, 2858. Gillespie, E. S . , and Thompson, D. G. (1977). J. Phys. B [ I ] 10,3543. Griffith, T. C . , and Heyland. G . R. (1978). Phys. Rep. 39c, 169. Griffith, T. C., Heyland, G. R., Lines, K. S., and Twomey, T. R. (1978).J. Phys. B [ I ] 11, L635. Grover, P. S. (1978). J . Phys. B [ I ] 11, 2555. Hahn, Y. (1966). Phys. Rev. [2] 142, 603. Hara. S. (1974). J . Phys. B . [ I ] 7, 1748. Hara, S.,and Fraser, P. A. (1975). J . Phys. I3 [ I ] 8, 219.
THEORETICAL ASPECTS OF POSITRON COLLISIONS IN GASES
133
Hazi, A . U., and Taylor, H. S. (1970). Phys. R e ) . . A [3] 1, 1109. Ho, Y. K . , and Fraser, P. A. (1976).J. Phys. B [1]9, 3213. Ho, Y. K . , Fraser, P. A., and Kraidy, M. (1975). J. Phvs. B [ I ] 8 , 1289. Houston. S . K. (1973).J. Phys. B [ I ] 5, 136. Houston, S. K., and Drachman, R. J. (1971). P h y s . R e v . A [3]3, 1335. Humberston, J. W. (1973). J . Pliys. B [ l ] 6 , L305. Humberston, J. W. (1974). J . Phys. B [ I ] 7, L286. Humberston, J. W. (1978). J. Phys. B [ I ] 11, L343. Humberston, J. W . , and Wallace, J. B. G. (1972). J. Phvs. B [I] 5, 1138. Jaduszliwer. B., and Paul, D. A. L. (1973). Con. J . Phys. 51, 1565. Jaduszliwer, B., Nakashima. A., and Paul, D. A. L . (1975). Cun. J. Phys. 53, 962. Kohn, W. (1948). Phys. Re\,. [2] 74, 1763. Kraidy. M., and Fraser, P. A. (1967). Proc. I n t . Conf. P h y s . Electron. A t . Collisions. Sth, 1967 p. 110. Lebeda, C. F., and Schrader, D. M. (1969). Phys. Rev. [2] 178, 24. Lee, G. F., Orth, P. H. R., and Jones. G. (1969). Phys. Lett. A 28, 674. Leung, C. Y., and Paul, D.A.L. (1969). J. Phys. B [1]2, 1278. McEachran, R. P., Morgan, D. L . . Ryrnan, A. G . . and Stauffer. A. D. (1977).J. Phys. B [ I ] 10,663: corrigendum: 11, 951. McEachran, R. P., Ryman, A. G., and Stauffer, A. D. (19781.J. P h y s . B [l] 11, 551. Mandal, P., Ghosh, A. S., and Sil, N. C. (1975). J . Phys. B [I] 8 , 2377. Mandal, P., Basu, D., and Ghosh, A. S. (1976).J. Phys. B [1]9, 2633. Massey, H. S. W. (1971). A t . Phys.. 2nd 1970 p . 307. Massey, H. S. W., and Mohr, C. B. 0. (1954). Proc. Phys. Soc.. London 54,695. Massey, H. S. W., and Moussa, A. H. (1961). Proc. Phys. Soc.. London 77, 811. Massey, H. S. W., Lawson, J., and Thompson, D. G. (1966). In "Quantum Theory of Atoms. Molecules, and the Solid State ( P . - 0 . Lowdin. ed.), p. 203. Academic Press, New York. Massey, H . S. W., Lawson, J., and Hara, S . (1972). J. P h y s . B [I] 5 , 599. Massey. H. S. W., Burhop, E . H. S., and Gilbody, H. B. (1974). "Electronic and Ionic Impact Phenomena," Vol. 5 . Oxford Univ. Press, London and New York. Mittleman, M. H. (1966). Phys. R e v . [2] 152, 76. Montgomery, R. E., and Labahn, R. W. (1970). C o n . J. Phys. 48, 1288. O'Malley. T . F., Rosenberg, L., and Spruch. L. (1962). P h y s . Rev. [2] 125, 1300. Orth, P. H . R., and Jones, G. (1969). Phys. Rev. [2] 183, 16. Osman, P. E. (1965). Phys. R e v . [2] 140, A8. Page, B. A. P. (19751.J. Phys. B [ l ] 8 , 2486. Page, B. A. P. (1976). J. Phys. B [ l ] 9, 2221. Peterkop. R.. and Rabik, L. (1971). J. Phys. B [ I ] 4, 1440. Schwartz, C. (1961). Phys. Re\,. [2] 124, 1468. Seiler. G. J., Oberoi. R. S . , and Callaway, J. (1971). P h y s . R e v . [3] 3, 2006. Shimamura. I. (1975).J. Phys. B [ I ] 8 , 2352. Stein, J . , and Sternlich, R. (1972). Phys. R e ~ sA. [3]6, 2165. Stein, T. S . , Kauppila, W. E., Pol, V., Smart, J. H.. and Jesion, G. (1978). P h y s . Rev. A [3] 17, 1600. Thomas. M. A., and Humberston, J. W. (1972). J. Phys. B [ I ] 5, L229. Wakid, S. E. A. (1975). Phys. Lert. A 54, 103. Wakid. S. E. A,, and Labahn, R. W. (1972). Plrys. ReLj. A [3] 6 , 2039. Wardle, C. E. (1973). J. Phys. B [ I ] 6 , 2310. Wilson, W. G. (1978). J. Phys. B [ I ] 11, L629.
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Ii EXPERIMENTAL ASPECTS OF ADVANCES IN ATOMIC A N D MOLECULAR PHYSICS, VOL. 15
POSITRON COLLISIONS IN GASES T. C . GRIFFITH Department of Physics and Astronomy Unii*ersiiyt o l f e g e Uniwrsily of London London, England
I. 11. 111. IV.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scattering Techniques with Positron Beams Accuracy of the Cross Section Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total Cross Section Measurements . . . . A. Positron-Helium Scattering at Low ....................... B. Neon and Other Inert Gases at Low C. Inert Gases-Intermediate Energies D. Inert Gases-Positronium Formation Cross Section . . . . . . . . . . . . . . . . E. Cross Sections for Molecular Gases .......................... F. Inelastic Collisions of Positrons in Helium above the Ionization Threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
135 142 146
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A. Experimental Methods . . . . . . . . . . . . . . C. Lifetime Parameters for Molecular Gases D. Vacuum Lifetime of Ortho-Positronium ........................... 163 References.. . .................. 164
I. Introduction It was exactly ten years ago that Sir Harrie Massey inspired the formation of a group to study the interactions of positrons in gases at University College. At that time the thought that positron beams would be available for this work seemed a remote and fanciful dream. The research program embarked upon was therefore designed to extract as much information as possible from the study of the annihilation time spectrum of positrons moderated to thermal energies in the inert gases and mixtures of small amounts of molecular gases with certain inert gases. As a pivot for the discussion of the progress in this field over the past decade a brief discussion of the liftime spectrum of positrons in argon, shown in Fig. I , will show the nature of the information that can be 135
Copyright 0 1979 by Academic Press. Inc All rights of reproduction in any form reserved.
ISBN 0-12-003815-3
T . C . Grijjith
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FIG. 1. Typical lifetime spectrum for argon at 297 K and 6.3 amagats with channel widths of 1.92 nsec taken by Coleman ef n l . (197Sa). (a) Raw data, (b) restored signal, (c) free-position component, (d) fitted orthopositronium.
gleaned from this work. The fast positrons from a radioactive source are moderated in the gas to less than 100 e V in about 2 nsec. At energies close to the inelastic thresholds some collisions lead to the formation of ortho- and para-positronium (Ps) as well as to excitation, ionization, and elastic scattering. Consequently the lifetime spectrum divides into three distinct regions: (a) the prompt peak due to the decay of para-Ps and positron annihilations in the source holder and chamber walls, (b) the shoulder region that results from the slowing down of positrons by elastic collisions from energies near the inelastic thresholds to thermal energies, (c) a composite region containing two overlapping exponential decay curves, one due to thermalized positrons and the other due to the decay of ortho-Ps. The shoulder width and the decay constant for the thermalized positrons are both dependent on the positron-atom interaction and both can be calculated from the phase shifts derived from theory. Ten years ago this was the only way of testing the theory and at that time neither experi-
EXPERIMENTAL ASPECTS OF POSITRON COLLISIONS IN GASES
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ment nor theory could claim much accuracy. The shoulder (and other features) were clearly visible in argon and in helium at low temperature but in general the statistical accuracy based on runs that lasted several weeks was poor and often unreliable. Studies of the change in shoulder widths with the addition of small amounts of N, to argon had also been reported. The quality of the lifetime experiments have now been considerably improved, with the result that shoulders have been observed and measured in all the rare gases and in N S . Much more precise values of the decay rates have also been taken. Two interdependent factors have determined these improvements, namely, improved statistics resulting from much more rapid accumulation of data and the formulation of an exact method of evaluating and removing the background due to randomly generated events. These improvements have also led to precise determinations of the vacuum lifetime of ortho-Ps. Useful data on the cross sections for formation of positronium in various gases have also been obtained from the annihilation spectra. When Groce et nl. (1968, 1969) announced that beams of monoenergetic positrons had become a reality it was immediately realized that, instead of using expensive accelerators for the purpose, such beams might also be produced using radioactive sources. The development of the techniques used for this purpose has been reviewed in detail by Griffith and Heyland (1978). A weak source in a time-of-flight (tof) technique, similar in principle to that used for lifetime studies, was developed by Coleman et a l . (1973) to produce monoenergetic positrons. Other groups, notably Jaduszliwer and Paul (1973) and Brenton ct a / . (1976) have used static monochromators for the same purpose. The yield of positrons depends critically on the moderator employed-fine magnesium oxide powder being one of the favored substances. Kauppila et n l . (1976) have recently reverted to the use of an accelerator to produce a strong positron source in sitir from a proton beam. They have one of the strongest beams of lowenergy positrons currently in use for cross section measurements. Progress has been rapid and accurate total cross sections for positron interactions in a variety of gases at energies between 0.4 and 1000 eV have already been measured. Positron beams have also been used for other purposes, notably by Canter et n / . (1974a) for the production and study of ortho-positronium from which the Lyman-a radiation corresponding to the 2P-IS transition was detected for the first time. Another important experiment has been that of Gidley P I N / . (1976b), where a direct determination of the vacuum lifetime of ortho-Ps formed by a low-energy positron beam was made. As well as improving the accuracy of the total cross section measurements, efforts are now being concentrated on differential cross section measurements and a determination of the partial cross sec-
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tions for some of the inelastic processes. Twomey et u l . (1977) have recently incorporated a localized target in their time of flight system for this purpose. Along with the advance of experimental techniques there has also been considerable activity on the theoretical side. At energies below the inelastic thresholds (elastic-scattering region) accurate calculations on the positron-helium system have been performed by Campeanu and Humberston (1977). and the results are very close to the measured values. At higher energies (up to 1000 eV) several groups of theorists, whose contributions are reviewed by Byron and Joachain (1977a), have used different approximations to evaluate the cross sections. Dispersion relations have been applied to the e+-He problem by Bransdenet a l . (1974) and the e+-Ne system by Inokuti and McDowell (1974). They derive a sum rule relating the zero-energy scattering length and the forward Born elasticscattering amplitude to the integral of the total cross sections over all energies between zero and infinity. Dewangan and Walters (1977)and Byron and Joachain (1977b) with their EBS model have both considered modifications of the second Born approximation for e+-He and e+-Ne scattering and the agreement with experiment is good for helium but rather poor for neon. The ab inirio optical model of Byron and Joachain (1977~) has been evolved to deal with more complex atoms such as argon. Some success, assessed in terms of agreement with experiment, has been achieved at lower energies for e+-He and e+-Ne scattering by McEachran et a / . (1978) using refinements of the polarized orbital method. In the scattering of positrons by molecular gases at low energies extensive approximations are necessary but there are indications, from the work of Darewych and Baille (1974) and of Gillespie and Thompson (1975) for the e+-N, system and of Baillie et al. (1974) and Hara (1974) for e+-Hz, that results in acceptable agreement with experiment are being obtained. Positron beam techniques constitute the direct means of studying positron interactions, so that it is appropriate to start with an account of some of these methods and the results obtained with them and then conclude with a survey of some of the new lifetime data.
11. Scattering Techniques with Positron Beams Only a brief account of two of the systems used for producing positron beams will be given here. A detailed survey of the principles behind these techniques has already been presented by Griffith and Heyland ( 1 978). The accuracy of the data obtained by Stein et al. (1978) at Detroit appears
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to be superior to most of the earlier data and their system, illustrated in Fig. 2, will be considered first. They use an intense beam of low-energy positrons obtained by moderation of the fast positrons from a "C source. This source is produced in sitir in the reaction l'B(p,n)llC by bombardment of a boron target with 4.75 MeV protons from a Van der Graaff accelerator. Positrons of energy determined by the positive bias on this boron target are extracted and focused by an electrostatic lens system and then, after a system of vertical and horizontal deflector plates, pass through a "filter lens" type of retarding field energy analyzer. The beam is then confined by an axial magnetic field of 10 G in a solenoid wound around the stainless steel scattering chamber of length 102 cm and curved on a radius of 91.5 cm (45" bend). This chamber can be evacuated to 3 X lo-' torr. with a mercury diffusion pump or flushed with gas at a steady rate with pressure differentials of 220: 1 and 90: 1 between the scattering
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FIG. 2. Schematic diagram of the system used by Kauppila
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a / . (1976).
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region, respectively, and the source and the detector vacuum chambers. A Bendix spiraltron electron multiplier with its cone maintained at - 180 V is used to detect the positrons at the exit end. About one in lo6 of the fast positrons from the source gets detected as a low-energy positron and the energy is defined to better than -+ 0.1 eV. Total cross sections for positrons scattered in various gases have been determined from transmission measurements involving the counting of the total number of positrons detected by the channeltron with and without gas flowing through the scattering chamber. In the gas runs, discrimination against positrons scattered through small angles from the forward direction was effected using a retarding potential element between the end of the scattering chamber and the channeltron detector. At positron energies above 1 eV the signal-to-noise (background) was better than 100: 1 as determined on application of appropriate potentials to the retarding element. Many features of their method are similar to those used by other authors but they have a stronger beam and better energy resolution extending to lower energies than for the earlier systems. The maximum possible error in their cross sections due to omission of small angle scattering can be deduced from the upper limits of 13" at 1 eV and 7" at 30 eV, which they estimate for the angular discrimination of their system for elastically scattered positrons. The error due to the change of path length in the scattering chamber due to the spiraling of the positrons in the axial magnetic field was estimated to be less than 1% at 2 eV. A baratron capacitance gauge was used for the measurement of their target gas pressures and corrections were applied for the thermal transpiration due to the difference in temperature between the gas in the scattering chamber and at the baratron manometer head. The second positron beam system that merits discussion was briefly described by Coleman et 41. (1975a) and Twomey et al. (1977) and consists of a modification of the earlier tof method of Coleman e? af. (1973) where scattering occurred along a column of gas 100 cm long. This method has been much improved by the introduction of a gas cell into the flight path of the positrons, thus localizing the scattering and allowing the moderator region to be evacuated for both the vacuum and gas runs. At the end of the flight path the positrons are now being detected with a channeltron electron multiplier instead of the sodium iodide well counter used for the earlier tof work. With a 100 pCi source up to 7 slow positrons per second can be timed and detected from a moderator in the form of a fine tungsten grid with the wires coated with fine MgO powder. An axial magnetic field of magnitude up to 600 G is applied so that positrons of energy 1000 eV at angles of up to 60" to the axis can be transported between the moderator and the detector. The flight path, including the gas cell of
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length 8 cm, has a total length of 130 cm and is evacuated with three SO mm and one 100 mm diffusion pumps, which. as can be seen in Fig. 3 , constitutes a differential pumping arrangement around the gas cell. The pressure differential between the cell and the flight path and the moderator regions was 1000: 1 and at a pressure of 3 x torr in the flight path 96% of the scattering occurs in the gas cell. Localization of the scattering ensures maximum tof for all the scattered positrons without the smearing experienced in the earlier work due to scattering from all points on the flight path. Fast positrons entering the moderator in the evacuated flight tube traverse the thin plastic scintillator, with the source deposited centrally on it. to provide a start pulse for the timing sequence. The sequence is stopped by a pulse from the channeltron and the data acquisition system is that of conventional tof electronics. The fast counting rates of lo6 sec from the fast plastic scintillator as compared with -20 sec from the channeltron means that the latter. suitably delayed, have to be used as start pulses and the faster rates as stop pulses. An important aspect of this work has been the signal restoration procedure developed by Coleman P I (11. ( 1 974c) to allow for random background events and for the conversion of signal events by unrelated stop pulses. The background decreases from zero time, t = 0. on the analyzer according to an exponential form, exp(- 4 r ) , determined by the stop rate n , . The true signal distribution has to be derived from the tof spectra recorded on the analyzer. The technique used to deduce the total cross sections mT from the tof data involves detailed examination of the low-energy positron peaks ob-
-
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Flc. 3 . Schematic diagram of the time-of-flight system of Twomey calization of the scattering region.
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tained at a given energy with and without gas flowing through the gas cell. Scattered positrons in the gas run have longer tof than those unscattered and will either appear as a tail on the longer time side of the peak observed in the vacuum run or, when there is an appreciable amount of small-angle scattering, will cause “distortion” of the unscattered peak. Coleman et a / . (1975d, 1976) have described in detail their method of analysis to obtain mT from these spectra. The tail of the gas spectrum also contains information on the energy loss and angular distributions of the scattered positrons. The two methods described have only been used for total cross section measurements. Other methods, namely those of Jaduszliwer et a l . (1972), Dutton er a / . (1979, Canter et al. (1972, 1973), Burciaga et a / . (1977), and Wilson (1978) as well as the early system of Costello et a / . (1972), have used slow positron beams for the same purpose. Variants and modifications of some of these methods have also utilized low-energy positron beams for other important experiments briefly mentioned in Section I.
111. Accuracy of the Cross Section Data Since the main emphasis of this chapter lies in those aspects of atomic physics that can be investigated with positron beams it is of interest to assess the accuracy of the cross section data so acquired. As can be seen in Section IV there are significant disagreements between some of the measurements from different laboratories. The total cross section mT at a given energy is given by In S, - In S,
=
uTI
(1)
where S, and S, represent the total number of positrons detected at the target in the vacuum and gas runs, respectively, and I = Jk n(s) ds with n(s) the gas molecule number density, deduced from the pressure and temperature, at a point s on the flight path of total length L . The slope of a graph of In S, against I gives uTdirectly, but it is generally sufficient to deduce uTfrom a vacuum run and one gas run taken at a pressure for which the attenuation A = S , / S , = 3. In the tof work, instead of obtaining A directly from the counting rates, the attenuation ai in each corresponding channel of the gas and vacuum spectra has to be calculated from the relation
EXPERIMENTAL ASPECTS OF POSITRON COLLISIONS I N GASES
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where (Nv)jand (N,Ij are the counts in channel j , respectively, in vacuum and gas runs of the same duration. Channel 1 is chosen near the leading (short time) edge of the peaks in the spectra and the asymptotic value of ui at this leading edge of the peak is taken as the true attenuation A . Subtraction of the vacuum spectrum, scaled by a factor of 1/A, from the gas spectrum yields the spectrum of scattered positrons as illustrated in Fig. 4. The statistical errors on the data are generally quite small so that dis-
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FIG. 4. Time-of-flight spectra for 100 eV positrons taken with the apparatus illustrated in Fig. 3 (400 nsec range. 1.62 nsec/channel). In (a) the crosses represent the spectrum (after correction for random background) with helium in the scattering cell and the open circles the vacuum spectrum scaled down by the attenuation factor A - I . The spectrum of scattered positrons, obtained from the difference between the spectra in (a). is shown in (b).
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crepancies in the results from different laboratories must be attributed to unidentified systematic errors. One obvious source of error ensues from the evaluation of I , and Tsai et 01. (1976). for example, have found it necessary to perform a subsidiary experiment to determine their “effective path length”; by varying the position of their target along the flight path they deduced an “end correction,” which was applied to L. Such effects were, however, not considered to be significant in the experiments of Canter pt al. (1972, 1973), Stein rf a / . (l978), or Brenton et a / . (1977). In the localized scattering system described by Twomey et ul. (1977) the gas pressure measured at the center of the cell was used to normalize their measurements to accurately known cross sections, because of uncertainties in the evaluation of I by calculation. Both the pressure and temperature measurements may be subject to errors. Pressures are usually measured using McLeod and/or Baratron capacitance gauges with appropriate corrections for thermal transpiration, pressure gradients, and other effects. In flowing-gas systems, time-dependent drifts can be monitored with a pen recorder operating from the output of a Pirani gauge. Fluctuations of both pressure and temperature are generally found to be less than the statistical errors. Another source of error encountered with those systems using axial magnetic fields arises from the spiraling of the positrons along the axis so that the true path length becomes L sec a , with a the pitch angle of the positron on the helical trajectory. Stein et al. (1978) estimate that the error due to this effect is less than 1%, while Canter et a / . (1973) consider this error to be small at energies greater than 4 eV. Errors due to spiraling would tend to overestimate the cross sections. The lack of adequate compensation for the failure to detect small-angle scattering is probably the most serious source of error in this work. It leads to cross sections that are smaller than the true values, and a number of workers have attempted to quantify the error for elastic scattering in terms of a minimum cutoff angle. Several methods have been used to reduce this error. In the tof work, for example, the method of analysis compensates for small-angle scattering by deducing the asymptotic value of A from spectra accumulated on the shortest possible time range. The limit is reached for scattering which corresponds to delaying the positron through energy loss or angular deflection for a time within the width of one channel when using the optimum time range to record the spectra. Localization of the scattering in a region close to the moderator has significantly decreased the magnitude of the elastic scattering cutoff angles compared with those for the earlier system of Canter et ul. (1973) and the errors due to the omission of small-angle scattering are estimated to be
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small at energies below 800 eV. The recent experiments of Stein et [ I / . (1978). Tsai rt a / . (1976), and Brenton p t [ I / . (1977) have minimized the error by using baffles along the flight path to prevent scattered positrons reaching the target. They have also used retarding potential elements in front of the final detector to reject positrons with reduced axial velocities resulting from both angular deflection and energy loss. Such retarding elements are effective for inelastic scattering up to about 1000 eV but for elastic scattering it is still necessary to define a minimum cutoff angle. A recent reassessment of the low-energy positron-helium data of Canter et (11. (1973) by Griffith et al. (1978a) has revealed that the axial magnetic field distribution has played a crucial role in compensating for small-angle scattering at low energies. It had been noted by Coleman et a / . (1975b) that, over the energy range 2-20 eV, no positrons scattered into the forward hemisphere could be identified on the tof spectra. There is a simple explanation of this observation to be found in the fact that a charged particle moving in a slowly varying axial magnetic field of strength B has. as discussed by Chandrasekhar (1975) and Twomey (1977), to satisfy the condition that sin2aj'B = const.
(3)
where cy is the pitch angle. Changes in the field strength along the trajectory cause corresponding changes in a,and it is evident that when the field strength has a value corresponding to cy = 90" the positrons have zero axial velocity and will fail to reach the target. Canter et ul. (1973) had a weak field along the main flight path and stronger fields at the source and target ends of the system. The initial pitch angle was therefore reduced over the main flight path and the effects of spiraling were small. Positrons scattered into the forward hemisphere with resulting increase in pitch angle would, on approaching the target, experience a further increase in their pitch angle such that all those scattered at angles greater than 23" would fail to reach the target. Instead of an angle of 35" suggested by Canter et id. (1973) the actual cutoff angle in their experiment is now believed to have been less than 10". Corrections due to both small-angle scattering and to spiraling, which in any case act in opposite directions, were therefore relatively small in this experiment. Similar considerations should be applied to all the other experiments involving an axial magnetic field but they do not apply to the system of Brenton et al. (1977), which operates with a transverse magnetic field. or to the experiment recently reported by Wilson (1978). which does not use an axial magnetic guiding field.
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IV. Total Cross Section Measurements The amount of data on positron total cross sections that has been accumulated using positron beams over the last five years is quite extensive. It covers a range of atomic and molecular gases at positron energies between 0.5 and 1000 eV. In general there is reasonable agreement between the measurements from different laboratories but, as suggested in Section 111, there are also significant discrepancies in one or two important regions. It is convenient to deal with different gases and energy regions separately.
A . POSITRON-HELIUM SCATTERING AT Low ENERGY There are more data for this interaction than any of the others because the theory is considered to be of greater reliability here than for more complex systems. The theoretical background to the problem is considered in detail by Hurnberston (this volume, Chapter 4). Five sets of experimental data from different groups have been reported covering the energy range from 0.5 eV through the elastic scattering region to 17.8 eV, the positronium formation threshold, and to higher energies. The results are shown in Fig. 5 and, with the exception of the data of Jaduszliwer and Paul (1974b), the different experiments are in agreement to within about 10% at energies above 6 eV. Stein et al. (1978) and Wilson (1978) have clearly verified the existence of the Ramsauer minimum in the cross sections. Three of the e+-He sets of data were taken with a tof technique, the experiment of Burciaga et ul. (1977) only differing from that of Canter et al. (1973) in having a flight path of 3 1.8 cm instead of 96 crn. Wilson (1978) has also used a tof system similar to that of Coleman et ul. (1973) but with a straight flight path and a beam geometry that does not have an axial magnetic field. He actually determines the ratio of cr,(e+)/cr,(e-) as a function of energy below 6 eV and then normalizes to the absolute e--He cross sections determined by Kennerly and Bonham (1978). In Fig. 5 only the recent theoretical calculations of Campeanu and Humberston (1977) have been shown. Other calculations were reviewed by Griffith and Heyland (1978) and are also discussed in this volume by Humberston (Chapter 4). According to Humberston (1978) the curve in Fig. 5 represents the lowest values that are acceptable by the theory. The differences of up to 20% between the data of Stein et ul. (1978) and Burciaga et al. (1977) on the one hand and those of Canter et al. (1973) and Wilson (1978) on the other, at energies below 5 eV, is therefore of particular significance. The theoretical curve lies close to the higher values. The recent considerations by Griffith et al. (1978a) regarding small-angle scat-
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i
HELIUM
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FIG. S. Total cross sections for e+-He scattering at low energies. Experimental points: A- Costellovf d.(1972); 0. Jaduszliwer and Paul (1974b); 0. Canteret a / . (1973); 0. Stein ct a / . (1978): A, Burciagaer a / . (1977); +. Wilson (1978). The theoretical calculation of Campeanu and Humberston (1977) is represented by the solid curve.
terjng and spiraling. discussed in Section 111 suggest that the data of Canter er al. (1973) are not subject to serious systematic errors. The agreement between these results and those of Wilson (1978) supports this view because the method used by Wilson is reputed to be free from the usual sources of errors. Burciaga et d.( 1977) have not assessed the influence of the axial magnetic field on their result. Stein et al. (1977, 1978) have used a retarding potential element and have a convincing argument that they have effectively discriminated against forward scattering. Thus, despite the proliferation of data on this problem, there is still the need for more accurate measurements to resolve the present discrepancy.
B. NEON A N D OTHERINERT GASESAT Low ENERGY All the measurements and theoretical curves that have been reported are shown in Figs. 6, 7, and 8. Both neon and argon have been investigated theoretically using the polarized orbital approximation applied initially by Massey ef al. (1966) and then by Montgomery and LaBahn
T . C . Griffith
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FIG. 6. Positron-neon total cross sections at low energies. Experimental points: A, Jaduszliwer and Paul (1974b); 0, Steinef a / . (1978); 0, Canteref a / . (1973, 1974b). Theoretical curves: -, Massey e l al. (1966); --, Gillespie and Thompson (1975); --, Montgomery and LaBahn (1970), (2p-d) “un-norm”; ..., Montgomery and LaBahn (1970). (2s-p), (2p-d) “norm”; -, McEachran et a / . (1978). Thresholds for positronium formation and for excitation and ionization of neon atoms are indicated by the arrows.
(1970). Refinements of the same method have recently been applied to the e+-Ne problem by McEachran et a / . (1978), who have also shown that these approximations give results in helium that are in close agreement with those of Campeanu and Humberston (1977). Other calculations using different approximations have been performed for neon and argon by Gillespie and Thompson ( I 975). These theoretical curves are compared with experiment in Figs. 6 and 7. The experimental data for both neon and argon are not in very close agreement. Stein et a / . (1977, 1978) and Kauppila et al. (1976) show a broad minimum in argon and a deep narrow minimum in neon. They also show the sharp rise in the cross section at the positronium formation threshold and this feature is also clearly visible at the same energy in the data of Canter et a l . (1973, 1974b). It is not possible at this stage to form any meaningful conclusions about the agreement or otherwise between
EXPERIMENTAL ASPECTS OF POSITRON COLLISIONS IN GASES
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FIG. 7 . Positron-argon total cross sections at low energies. Experimental points: 0. Jaduszliwer and Paul (1974b); V. Canter rt n / . (1974b): 0. Kauppilarf n / . (1976). Theoretical curves: ..., Massey r f a / . (1966);-..-. Gillespie and Thompson (1975); -, Montgomery and LaBahn (1970), (3p-d) norm; -'-, Montgomery and LaBahn (1970). (3p-d), (3p-s), (3p-p); Montgomery and LaBahn ( 1970). (3p-d). The arrows indicate positronium formation and excitation and ionization of argon atoms.
---.
the various theoretical curves and the experimental data. There is probably much more small-angle scattering for these systems than for helium and it is worth noting that the Compensation for forward scattering by the axial magnetic field distribution occurs for the data of Canter et crl. (1973, 1974b) in all gases. It is not known whether the compensation is as effective for the data of Steinet c d . (1977, 1978)in these gases as it was for their helium results. Some new results in neon have recently been reported by Coleman et nl. (1979). At energies between 5 and 25 eV they lie between the lower two sets of data in Fig. 6 but below 5 eV they are in agreement with the results of Stein et c i l . (1978). In krypton and xenon there are no theoretical calculations for comparison. The two sets of data in Fig. 8 are broadly in agreement above 6 eV
T . C . Grv$th
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FIG. 8. Total cross sections for e+-Kr and e+-Xe at energies up to 50 eV. Experimental points: 0, Steineral. (1977)forKr; A, Steinet al. (1977)forXe; A,Canteretal. (1974b)for Kr; 0, Canter er al. (1974b) for Xe.
EXPERIMENTAL ASPECTS OF POSITRON COLLISIONS IN GASES
15 1
for krypton but for xenon the data of Stein et al. (1977) are significantly higher than that of Canter et al. (1974b). In both gases the data of Stein et al. (1977) show the minimum in the cross sections quite clearly and also the sharp rise in the cross sections at the positronium formation thresholds of 7.2 eV in krypton and 5.3 eV in xenon. The results of Canteret al. (1973, 1974b) have much larger statistical errors and increase steadily to a maximum value from their lowest point at 2 eV. C. INERT GASES-INTERMEDIATEENERGIES
The energies considered in this section lie in the range between 100 and 1000 eV. Over 80% of the collisions are inelastic and the scattered positrons tend to be strongly peaked in the forward direction for both elastic and inelastic collisions. I t is therefore essential to have sensitive means of discriminating against small-angle scattering. especially above 200 eV. The system of Brenton et (11. (1977), based on the original total cross section experiment of Ramsauer (19211, has been used for positrons of energies up to 1000 eV in helium, neon, and argon. Small-angle scattering is discriminated against by means of a retarding potential element. The new localized scattering system of Twomey et al. (1977) described in Section 11, coupled with the method of analysis outlined in Section I11 to allow for small-angle scattering, has also been used to measure reliable cross sections over the same energy range in the same gases. The new measurements were normalized to the data of Canter et al. (1973, 1974b)at energies below 100eV. The earlier work of Canter ef al. (1973, 1974b) and of Coleman et a l . (1 976) had failed to achieve full compensation for small-angle scattering above 200 eV. It was noted by Bransden et al. (1974), who applied dispersion relations to the problem, that the e+-He data should satisfy the sum rule and that even with reasonable extrapolation procedure this could not be achieved with the data of Canter et al. (1973, 1974b). Similar arguments were applied by Inokuti and McDowell (1974) to e+-Ne scattering but for argon and the other inert gases there is not sufficient information to define and apply the sum rule. The cross sections at energies up to 1000 eV are shown in Figs. 9, 10, and 1 I , where a comparison is made with the recent calculations of Dewangan and Walters (1977) and of Byron and Joachain (1977a.b.c) for helium and neon and with Joachain ef al. (1977) for argon. Within the stated errors all the measurements in helium and neon satisfy the sum rule. In helium the values of the e+-He cross sections appear to be converging to the e--He values above 600eV. Although the new values of Brenton et al. (1977) for helium are slightly higher than those of Twomey et al. (1977), in some regions the disagree-
T. C . Grjffith
I52
0301
OZ 0 I
'
0 1
01
I
0
50
L
I
I
I
I
I
I
I
I
100
200
300
LOO
5w
600
700
Bw
900
Y
POSITRON ENERGY lev1
FIG. 9. Positron-helium total cross sections at intermediate energies. Experimental Jaduszliwerei a / . (1975); 0, Coleman ef d . (1976); 0, points: +, Brenton ef a/. (1977h 0. Twomey r t a / . (1977). Theoretical curves: -, Byron and Joachain (1977~)for their optical model theory; ---, Dewangan and Walters (1977). The e--He total cross-sections of de Heer and Jansen (1975) fall almost exactly on the solid curve of Byron and Joachain (1977b3 for e+-He scattering.
ment can not be regarded as serious. In neon the results from these two groups are in very good agreement, both being substantially lower than the theoretical curves, but both satisfy the sum rule on extrapolating smoothly to join the first Born estimate of the total cross section at 6 keV. It is only for argon that there appears to be a significant disagreement between the two sets of data, Those of Brenton et al. (1978) are close to the theoretical curve but the data of Griffith et al. (1979) are well below the theory. In the range up to 270 eV covered by their data Tsai et al. (1976) are in agreement with the other data for neon and argon.
I53
E X P E R I M E N T A L ASPECTS OF POSITRON COLLISIONS IN GASES
30
-
e' - Neon
*m a
-c
\'k
.
6 \\'\ '
b
'
z
p 20V
m w
m
-
U _I
2 102
+ 0
D
O
1
1
700
800
900
I
-
0
t
0
I
3
0
200
I
1
I
I
300
LOO
500
600
POSITRON ENERGY f eV)
FIG. 10. Positron-neon total cross sections at intermediate energies. Experimental points: 0 , Griffith et N/. (1979); +, Brenton e t t i / . (1978): 0, Tsai e t ( I / . (1976); 0 , Coleman er ti/. (1976); V, electron results of de Heer and Jansen (1975). Theoretical curves: -.-, Dewangan and Walters (1977): -. Byron and Joachain f 1977~).
D. I N E R I GASES-POSITRONI U M FORMATION CROSSSECTION I t is of much interest to examine the rise in the cross sections at the positronium formation threshold in the inert gases. In helium, Massey and Moussa (19611, Fels and Mittleman (1969), Kraidy and Fraser (19671, and Mandal cf t i l . (1976) have calculated the positronium formation cross section upsas a function of energy. The agreement with experiment is poor but there is hope that, following the successful application of the theory to elastic scattering, improved calculations of ul,scan be performed. On the experimental side. if it is assumed that the elastic cross section in helium can be extrapolated linearly from a region below the threshold at 17.8 e V to t h e threshold for the first electronic excitation of the helium atom at 20.6 eV, the excess cross section between 17.8 and 20.6 eV can then be attributed to ups. Similar arguments can be applied to the other inert gases. The data of Canter ef a / . (1973, 1974b) for helium, neon and argon show a definite increase at their respective thresholds and this rise is even more sharply defined in the data of Stein e t a / . (1977, 19781 and is.
I54
T . C . Griffith
0
l o t 1
0
I
50
I
I
I
I
I
700 200 300 LOO POSITRON ENERGY lev1
I
I
1
500
600
700
I
800
900
FIG. 1 1 . Total cross sections for e+-Ar scattering at intermediate energies. ExperiTsai et a / . (1976); 0 , mental points: 0, Griffith et ( I / . (1979); f , Brenton et a/. (1978); 0, Coleman et a / . (1976); 0.electron results of de Heer and Jansen (1975). The solid curve represents the calculations of Joachain et a / . (1977).
in addition, present in their data for krypton and xenon. Coleman et u l . (1975~) assigned a value of 0.07 0 . 0 3 ~ to ~ ;up,in helium at 20 eV, while Stein et al. (1978) quote 0 . 0 8 ~ ~at: 20.6 eV. In neon the corresponding 2 , 16.7 eV by Griffith and Heyland (1978) values are quoted as 0 . 1 5 ~ ~at and 0 . 2 3 ~ ~by 2 ,Stein et al. (1978). In argon the value at 11.5 eV is about
*
2TU2.
There is an important connection with lifetime data to be considered for ups.Coleman et al. (1975b) have shown that, by measuring the fraction of
positrons that form positronium in helium, neon, and argon in lifetime experiments, lower limits on upscan be set as 0 . 1 6 ~ ~at2 ,24.5 eV in helium, 0 . 2 5 ~ ~ at; 21.6 eV in neon, and m$at 15.7 eV in argon. The energies cited here are as defined in Figs. 6, 7 , and 8, and in all cases upsis assumed to increase linearly from zero at threshold. At the same energy the values of upsdeduced from the beam and lifetime measurements are in good agreement.
EXPERIMENTAL ASPECTS OF POSITRON COLLISIONS I N GASES
I55
E. CROSSSECTIONS FOR MOLECULARGASES Total cross sections for positron interactions in H,, D,, N,. O,, CO, and CO, have been measured by Coleman et al. (1974a, 1975~)and were discussed by Griffith and Heyland ( 1978). Kauppila el n l . (1977) have recently taken measurements for some of these gases and a comparison between these new results and the earlier data is made in Figs. 12 and 13. There are large differences, which are as yet unexplained, between the data for CO,. There is very good agreement between the two experiments for e+-H, and tolerable agreement for e+-Nz. The results of Kauppila et al. (1977) show a broad minimum in the cross sections for all three gases
80-
i OL
.'\..,..'. 0
2
I
L
1
I
6
8
, ,
I
I0 12 1.4
I
I
I
I
I
1
1
16 18 20 LO 60 80 I00
POSITRON ENERGY
I
I
250
LOO
lev1
FIG. 12. Comparison of the measured total cross sections for e+-H, and e+-N, scattering with theory. Experimentalpoints: V, Colemanrr a / . (1974a)forN,; x. KauppilaPrul. D. (1977) for N,; 0. Griffith and Heyland (1978) for H,; 0. Kauppila rr a / . (1977) foi H,; 0, Davies. J . Dutton. D. Everett. and F. M . Harris (personal communication, 1978) for H,. Theoretical curves: Gillespie and Thompson (1975) for N,; Darewych and Baille (1974) for N,; -.-, Hara (1974) for H,; .... the lowest of the calculated curves of Baille rr NI. (1974) for H,. The thresholds for positronium formation, for molecular dissociation. and for ionization are indicated by the arrows.
-.
---.
T . C . Grxtith
156
I
1 I
CARBON DIOXIDE
I I
I I
if
I I I
Vt€€
I
I I I I
I I
I
P I
I
I
I I
I I
I
I
I 2-
I 0-
I
1
I
I
I
I I
P I
Q I
but it is only for hydrogen that the data of Coleman e? ul. (reported by Griffith and Heyland, 1978) are of sufficient accuracy to show the minimum. In fact, both sets of data are in excellent agreement in the region of the minimum for ef-H,. Some new data for e+-H, scattering at energies up to 1000 eV have recently been reported by D. E. Davies, J. Dutton. D. Everett, and F. M. Harris (private communication, 1978). Their results, at energies up to 500 eV, have been included in Fig. 12. The minimum for e+-H, scattering lies at about 4 eV, which is close to the dissociation energy of 4.48 eV for H,. At energies below 8.8 eV, however, the cross section for H, dissociation is very small indeed, and the gradual rise in cross section up to 8.8 eV has to be attributed to elastic scattering, and rotational and vibrational excitations. Calculations have only been attempted for the e+-Hz and e+-N, systems and, considering the nature of the approximations that have been used, the agreement with experiment is surprisingly good at energies below 6 eV. The calculations only take account of elastic scattering so that comparisons at energies above 6 eV are not meaningful. The calcula-
E X P E R I M E N T A L ASPECTS OF POSITRON C O L L I S I O N S I N GASES
157
tions of Hara (1974) for hydrogen and of Gillespie and Thompson (1975) for nitrogen are quite close to the experimental results below 6 eV. The approximations used by Baille r t d.( 1974) and by Darewych and Baille (1974) do not appear to be in such good accord with experiment. F.
INELASTIC C O L L l S l O N S OF POSITRONS IN ABOVE THE
HELIUM
IONIZATION THRESHOLD
The localized scattering system of Twomey e t (11. ( 1977) has been used for a preliminary analysis of the partial cross sections for some of the inelastic processes in helium at incident positron energies above 40 eV. The system is operated with an axial magnetic field of sufficient magnitude to transport the majority of the positrons scattered into the forward hemisphere to the target. The nature of the information that can be deduced can be assessed by reference to the typical tof spectrum of the scattered positrons given in Fig. 4. In the region on the high-energy (short time) side of the secondary peak only the positrons that have been scattered elastically are recorded. The secondary peak and its tail is due to large-angle elastic scattering and positrons involved in excitation and ionizing collisions. Positrons that have formed positronium will not appear on the spectrum. At energies above 40 eV the elastic scattering becomes increasingly more peaked in the forward direction so that its contribution to the secondary peak is small. The secondary peak is therefore mostly due to positrons involved in excitation and ionizing collisions. The kinematics of the two processes are very different since excitation has two bodies in the final state while ionization has three. Thus the energy and angular distribution of the outgoing positrons can be expected to be dissimilar for the two processes. In excitation collisons above 40 eV the energy loss of the positrons will not vary greatly and the tof spectrum is largely due to the positron angular distribution. If the difference in the excitation energies for different levels is ignored, then as the energy increases above 40 eV, since the angular distributions are expected to become increasingly more peaked in the forward direction, the apparent energy separation A E between the position of the unscattered positron peak and the mean position of the secondary peak should decrease as the energy of the incident positron increases. Ionizing collisions for electrons in helium have been extensively studied by Ehrhardt rt ul. (1972). They show that at energies above 50 eV there are two distinct groups of electrons-a high-energy group sharply peaked in the forward direction attributed to the scattered incoming electrons and a low-energy group at angles close to 90" due to the ejected atomic electrons. This assignment for the two groups is supported by the
T . C . Grijjjth
158
fact that the electron exchange and capture amplitude is very small when compared with the direct scattering amplitude. It is therefore reasonable to expect that positrons that produce ionization should behave in a manner similar to the small-angle high-energy group of electrons discussed above. By analogy with the angular distribution of the scattered fast electrons the tof spectrum of positrons involved in ionizing collisions will have a secondary peak spread that is predominantly due to energy loss rather than to angular deflections, as would be the case for excitation. The energy loss will increase with the incident energy and so, therefore, will the separation energy A E . This behavior is consistent with the socalled binary encounter model of Ehrhardt et ul. (1972), where as the incident energy increases the collision becomes more of a two-body process in which an increasing amount of energy is transferred to the ejected electron. The behavior of A E with energy is thus quite different for the two inelastic processes under consideration. Table I , compiled by Twomey (1977), shows that A E increases with incident positron energy, confirming that, as for electrons (Ehrhardt et ul., 1972), ionization is easily the dominant process over the energy range considered. The above observations would be consistent with assuming that cross sections for the excitation of helium by positrons are roughly equal to those for electrons. The e+-He elastic cross sections predicted by Dewangan and Walters (1977) may be added to the excitation cross sections for e--He and the sum subtracted from the e+-He total cross sections at the same energies. If this is done it is found that the ionization cross sections for positrons at energies in the range 100-300 eV are appreciably greater than the corresponding values for electrons. At higher energies the positron and electron ionization cross sections converge to the same values. It is quite interesting t o ' note that the classical positron-atomic hydrogen ionization calculation of Percival and Valentine (1967), when scaled to correspond to e+-He, accounts almost exactly for the difference between e+-He and e--He ionization. Examination of the spectra in Fig. 4 shows that the energy resolution in the peaks is poor and that this severely limits the interpretation of the data. Upper limits to the cross sections for excitation plus ionization can be set by evaluating the integral under the difference spectrum at different TABLE I ENERGYDISPLACEMENT OF THE INELASTIC PEAKS 75 100 150 E," (eV) 50 200 250 300 400 500 A E (eV) 22 ? 4 23 2 2 25 2 30.5 2 2 32 2 2 37 t 8 34 * 6 38 2 9 47 2 12
*
Incident positron energy.
EXPERIMENTAL ASPECTS OF POSITRON COLLISIONS IN GASES
159
energies but this procedure is of doubtful value at this stage. At present the only firm conclusion that can be drawn is that most of the collisions at energies above 40 eV are ionizing collisions. The independence of the shape of the secondary peak and its tail with respect to the magnetic field configuration (see Section 111) supports the above conclusion since it suggests that the tof distributions are reflecting energy loss rather than angular deflections.
V. Lifetime Studies A. EXPERIMENTAL METHODS
Recent advances in the design of electronic equipment used in fast counting techniques have contributed significantly to the improvement in the quality of the measurements of the positron lifetime spectra briefly described in Section 1. The basic principles of the measurement techniques are similar to those used for the tof experiments with positron beams. Full details of recent improvements in technique have been given by Griffith and Heyland (1978) so that only a brief summary of the salient points need be given here. Start and stop pulses for the timing sequence have been derived, respectively, from the prompt and annihilation y rays associated with the positron. By containing the positrons in the gas within a gold-plated vessel of small dimensions and using large scintillation counters to detect the y rays (Coleman et a l . , 1974b) it has been possible to realize large solid angles and increase the data accumulation rate by a factor of at least 10 compared with the earlier work. The same objective has also been achieved using a thin plastic scintillator for direct detection of the positron entering the gas, contained in a larger vessel, as an alternative to detection of the prompt y rays (Coleman r t d.. 1972; Gidley rt u l . , 1976a.l Another important advance has been the formulation of the exact method of evaluating the random coincidence background and number of signal events converted by unrelated stop pulses (see Section 11) described by Colemanet a l . (1974~). The results shown in Fig. I , taken with the system of Coleman ef c i l . (1975a). are representative of the quality of present data.
B. LIFETIME PARAMETERS FOR INERT GASES A lifetime spectrum for the inert gases furnishes a number of convenient parameters that lend themselves to comparison with theory. After
T . C . Cr$3th
160
the subtraction of the random coincidence background events and signal restoration (Coleman rt d.,1974c) the spectrum in the equilibrium region can be described by Y ( t ) = A exp(- A f t )
+ B exp( - A,t)
(4)
where Af and A, are the decay constants for free positron and ortho-Ps annihilations, respectively; A and B are constants that give the relative amplitudes of these components; the parameters Af and A, are density dependent and are related to two further parameters, Zeff and ,Zeff, through the relations Af
= wpZefr9
A,
=
OX, + 4wpiZeff
(5)
the vacuum decay rate of where p is the gas density in amagats, ortho-positronium and o p = .rrr',cn has already been defined by Humberston (this volume, Chapter 4). Humberston (this volume, Chapter 4) has also shown that the shoulder or nonexponential region of the spectrum can be described in terms of a diffusion equation that, among other factors, involves the momentum transfer cross sections predicted by the theory. The shoulder width, discussed by Griffith and Heyland (1978), is characteristic of the gas and is a parameter related to the thermalization time of the positrons. The fraction F of positrons that form positronium is another important parameter. It can be deduced from the decay constants Af and A, and the amplitude terms A and B and, as discussed by Coleman et ul. (3975d), can be compared with the predictions of the simple Ore model. An extension of this model has been used to predict the positronium formation cross sections mentioned in Section IV,D. The present state of the data represented by these parameters is summarized in Table 11. In helium, neon, and argon at moderate densities Zeff, TABLE 11 MEASURED VALUES
OF THE
Gas
Density range (amagats)
Zeir
He Ne Ar Kr Xe
2-60 7-40 0-280 0- I17 1-6
3.94 2 0.02 5.99 -t 0.08 27- 18" 66.8-39.5" 320 ? 10
LIFETIME PARAMETERS
FOR THE I N E R T
GASES
,Z,,
Shoulder width (nsec amagat)
F
0.125 t 0.002 0.235 2 0.008 0.314 t 0.003 0.445 t 0.005 1.03 2 0.08
1700 t 50 1700 t 200 362 2 5 296 -t 6 200 t 20
0.23 0.26 0.33 0.16-0.28" 0.05- 0.07"
Where there is variation with density the two values correspond to the extremes of the cited density range.
EXPERIMENTAL ASPECTS OF POSITRON COLLISIONS I N GASES
16 1
,Zeff,F, and shoulder width are sensibly constant but this is not the case for krypton and xenon, or for argon at high density. The results in Table I1 are based on recent measurements at University College, London. Other data have been listed and discussed by Griffith and Heyland (1978). Comparison with theory is mostly restricted to helium, where as shown by Humberston (this volume, Chapter 4) the
0.25 U
0.20
0.15
DENSITY
p
AMAGATS
FIG. 14. Z p ~,Zeff, . and Ffor krypton at 297 K and densities up to 120 amagats from Griffith and Heyland (1978).
T . C. GrifJith
I62
shoulder region is well reproduced by the calculated phase shifts. The calculated value of 3.88 for Zeffat room temperature is also in satisfactory agreement with the measurements. The recent calculations of Zeff for neon by McEachran et al. (1978) give a value of 6.99, which is significantly higher than the experimental values. F can be compared with the extreme values predicted by the Ore model and the measurements are consistent with this model for helium, neon, and argon but much too low for krypton and xenon. Speculations about these discrepancies are presented in the review article by Griffith and Heyland (1978). There are in
7
I
I
I
I
I
L
h B
25-
f-\rP' '\
N
'L
\
=-4 20L
7
I
I
1
L
,--I----*-,
0.46-
-%-
'Y,
0.30-
/
!L
Ld/A -/ ?/
NITROGEN 1 = 297K
// / i
0.20 0-20
I 00
I
I
I
I
50
100
150
200 200
DENSITY p
and h'forNzat ZY7 K
250
(AMAGATS)
and densities up to 240 amagats. Upper curve shows that lZ,nis constant at 0.26. Lower curve shows that F has a maximum value at 140 amagats (see Griffith and Heyland, 1978, for details). F I G . 13.
Len. &w.
I63
EXPERIMENTAL ASPECTS OF POSITRON COLLISIONS I N GASES
fact many interesting new features that have been revealed in the lifetime data for the heavy inert gases but all that can be done within the confines of this chapter is to illustrate the observed variations of Z,,, F, and ,Zeff with density in the case of krypton, as given in Fig. 14.
c. LIFETIMEPARAMETERS
FOR
MOLECULARGASES
Nearly all the data for molecular gases are as complicated as for krypton and xenon. The case of nitrogen is illustrated in Fig. 15. With the exception of nitrogen, no shoulders are observed and the interpretation of most of the data has hardly commenced. A great deal of useful information, which may help in understanding the phenomena, is being obtained by studying mixtures of some of the inert gases among themselves and with small amounts of molecular gases. Table 111 gives a summary of the main parameters for the gases that have been studied in some detail. D. VACUUMLIFETIME OF ORTHO-POSITRONIUM
An account of lifetime studies would not be complete without a brief that have reference to the important measurements and calculations of been performed over the last few years. The discrepancies between theory and experiment and between the various measured values have been discussed by Griffith and Heyland (1977, 1978). It is now accepted that gas extrapolation procedures involving a series of measurements of A, at different densities in a suitable gas or mixture of gases are more reliable than experiments using magnesium oxide powders (Gidley et al., 1976a,b). Two such experiments have recently been completed, one by Gidley et al. (1978) and the other by Griffith et ai. (1978b). Both groups TABLE 111 MEASURED VALUESOF
Gas D2 H2
N2
co
co2 0 2
CCIZF2
THE
LIFETIME PARAMETERS FOR MOLECULAR GASES
Density range (amagats)
ZeK
12-39 19- I70 0-234 0- 172 0-48 7-215 0.3-5
13.6 t 0.1 13.6-12.6 30.6- 18.6 38.5- 24 so- 120 26- 19 750- 1500
~zeff
0.186 t 0.004 0.186 0.260 t 0.005 0.285 ? 0.01 0.48 t 0.02 80- 14 0.57 2 0.02
F
-0.45 0.19-0.36 0.28-0.55 0.50-0.60 0.32-0.56 0.4s
164
T . C. GrifJith
have given careful attention to correct evaluation of the virial coefficients for the gas and to estimation of the “wall effects” as well as other considerations. It can now be asserted that the earlier discrepancies between theory and experiment have almost been resolved. The theoretical value reported by Caswell et al. (1977) was 7.0379 k 0.0012 psec-I while the new experimental values are 7.056 k 0.007 psec-’ by Gidley et al. (1978) and 7.045 f 0.006 psec-l by Griffith et al. (1978b). ACKNOWLEDGMENTS
I am greatly indebted to my colleagues Drs. G. R. Heyland. J. W. Humberston, T. R. Twomey, and Mr. K. S . Lines for many helpful suggestions and comments during the preparation of this chapter. I am also grateful to Miss Una Campbell for preparing the illustrations, the Photographic Section for photographs, and Mrs. Pat Crowther for typing the manuscript.
REFERENCES Baille. P., Darewych. J. W., and Lodge, J. G . (1974). C a n . J . Phys. 52, 667. Bransden, B. H., Hutt, P. K . , and Winters, K. H. (1974). J . Phys. B [ I ] 7 , L129. Brenton, A. G., Dutton, J . . and Harris, F . M. (1976). Prac. Int. C m f Positron Annihilation, 4rh, 1976 Abstracts A16. Brenton, A. G.. Dutton. J . . Harris, F. M., Jones, R. A., and Lewis, D. M. (1977).J. Phys. B [ I ] 10, 2699. Brenton, A . G.. Dutton. J . , and Harris, F. M. (1978). J . Phys. B [I] 11, L15. Burciaga. J . R., Coleman, P. G . , Diana, L . M . , and McNutt. J . D. (1977).J . P h y . B [ I ] 10, L.569. Byron, F. W., and Joachain. C. J. (1977a). Phys. R e p . 34, 234. Byron. F. W., and Joachain, C. J . (1977b). J . Phys. B [ I ] 10, 207. Byron, F. W., and Joachain, C. J. (1977~).Phys. Rev. A [ 3 ] 15, 128. Campeanu, R. I., and Humberston. J. W. (1977).J . Phys B [ I ] 10, L153. Canter. K. F., Coleman, P. G . , Griffith, T. C.. and Heyland, G . R. (1972). J . Phys. B [ I ] 5, L167. Canter, K . F., Coleman. P. G.. Griffith, T. C.. and Heyland, G . R. (1973). J . Phys. B [ I ] 6 , L201. Canter, K. F., Mills, A. P.. and Berko, S. (1974a). Phys. Rev. Lett. 33, 7. Canter, K. F., Coleman. P. G., Griffith, T. C., and Heyland, G . R. (1974b).Appl. P h y . 3 , 249. Caswell. W . E., LePage, G. P., and Sapirstein. J . (1977). Phys. Rev. Lett. 38, 488. Chandrasekhar. S . (1975). “Plasma Physics.” p. 24. Univ. of Chicago Press. Chicago, Illinois. Coleman. P. G., Griffith, T. C.. and Heyland, G. R. (1972). J . Phys. E 2 , 376. Coleman, P. G., Griffith, T. C., and Heyland, G . R. (1973). Proc. R . Soc. London, S e r . A 331, 561. Coleman, P. G., Griffith. T. C.. and Heyland. G . R . (1974a). A p p l . Phys. 3, 89.
E X P E R I M E N T A L ASPECTS OF POSITRON COLLISIONS I N GASES
I65
Coleman. P. G.. Griffith. T. C., Heyland. G . R . . and Killeen, T . L. (1974b). A p p l . P h y s . 3, 271. Coleman. P. G.. Griffith, T. C . , and Heyland, G . R. ( 1 9 7 4 ~ )A. p p l . P h y s . 5, 223. Coleman. P. G.. Griffith. T. C.. Heyland. G . R.. and Killeen. T. L. (1975a).J. P h y s . B [ I ] 8, 1734. Coleman. P. G.. Griffith. T . C.. Heyland. G . R.. and Killeen. T. L. (1975b).J. Phps. B [ I ] & L185. Coleman. P. G.. Griffith. T . C.. Heyland, G . R.. and Killeen. T. L. ( 1 9 7 5 ~ J) .. P h y s . B [ l ] 8, L454. Coleman. P. G . , Griffith. T . C., Heyland. G . R.. and Killeen. T. L. (1975d).A t . P h y s . 4,355. Coleman. P. G.. Griffith. T. C.. Heyland. G. R.. and Twomey, T. R. (1976).Appl. P h y s . 11, 321. Coleman. P. G., McNutt. J . D.. Diana, L. M . . and Burciaga, J. R. (1979). Phys. R e v . A [3] (in press). Costello. D. G.. Groce, D. E . , Herring, D. F., and McGowan. J . W. (1972).P h y s . R e v . B [3] 5, 1433. Darewych, J . W., and Baille. P. (1974). J . Phys. B [ l ] 7, L1. de Heer. F. J.. and Jansen, R. H. J . (1975). Reports Nos. 37173 and 37174. FOM Inst. At. Mol. Phys., Amsterdam. Dewangan. D. P . . and Walters. H. R. J . (1977). J . P h y s . B [I] 10, 637. Dutton. J . . Harris. F. M., and Jones, R. A. (1975).J . P h y s . B [ I ] 8, L65. Ehrhardt. H.. Hasselbacker, K. H.. Jung, K.. and Williamson, K. (1972). J . P h y s . B [ I ] 5, 1.559. Fels, M. F.. and Mittleman, M. H. (1969). P h y . ~ Rev. . [2] 182, 77. Gidley, D. W.. Marko. K. A,. and Rich, A. (1976a). Phys. Reit. Leif. 36,395. Gidley. D. W.. Zitzewitz. P. W.. Marko. K. A . . and Rich, A. (1976b). P h y s . Rev. Loti. 37, 729. Gidley. D. W.. Rich. A . , Zitzewitz. P. W.. and Paul. D. A . L. (1978). P h y s . Rerr. L e f t . 40, 737. Gillespie, E. S . . and Thompson. D. G. (197.5).J . Phys. B [ I ] 8, 2858. Griffith. T. C.. and Heyland, G. R. (1977). N a t r t w (London) 269, 109. . 39, 169. Griffith. T. C . . and Heyland. G. R. (1978). P h y . ~Rep. Griffith, T. C., Heyland, G. R., Lines, K. S. , and Twomey, T. R. (1978a).J . P h y s . B [I] 11, L635. Griffith, T. C . , Heyland, G . R.. Lines, K. S . , and Twomey, T. R. (1978b). J . Phys. B [I] 11, L743. Griffith, T. C., Heyland, G. R., Lines, K . S . , and Twomey, T . R. (1979). A p p l . P h y s . 19, 43 1. Groce. D. E., Costello, D. G., McGowan, J. W . , and Herring, D. F. (1968).Bull. A m . Phvs. Soc. [2] 13, 1397. Groce, D. E.. Costello. D. G., McGowan, J. W.. and Herring, D. F. (1969). Proi.. f n r . Conf: Phyr. Elrctron. A t . Collisions, 6 i h . 1969 p. 757. Hara. S . (1974). .I. Phys. B [ I ] 7, 1748. Humberston. J . W. (1978)J. P h y s . B [ I ] 11, L343. Inokuti. M . . and McDowell, M. R. C. (1974).J . P h y s . B [ I ] 7 , 2382. Jaduszliwer. B.. and Paul. D. A. L. (1973). Con. .I. P h y s . 51, 1565. Jaduszliwer. B.. and Paul. D. A. L. (l974a). Can. J . P h y s . 52, 272. Jaduszliwer, B., and Paul. D. A . L. (1974b). Ccrn. J . Phys. 52, 1047. Jaduszliwer. B., Keever, W. C., and Paul, D. A. L. (1972). Can. J . Phps. 50, 1414. Jaduszliwer. B.. Nakashima, A , . and Paul. I). A. L. (1975). Cun. J . P h y s . 53, 962.
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Joachain, C. J . , Vandespoorten, R., Winters, K. H., and Byron, F. W., Jr. (1977). J . Phys. B [ I ] 10, 227. Kauppila, W. E., Stein, T. S., Pol, V., and Jesion, G. (1976). Phys. Rev. Left. 36, 580. Kauppila, W. E.. Stein. T. S., Smart, J . H., and Pol, V. (1977). Proc. f n t . Conf. Phys. Electron. At. Collisions. loth, 1977. Abstracts, p. 826. Kennerly. R. E.. and Bonham, R. A. (1978). Phys. Rev. A [3] 17, 1844. Kraidy, M., and Fraser, P. A. (1967). Proc. Int. Conf. Phys. Electron. A t . Collisions, 5th, 1967. Abstracts, p. 110. McEachran, R. P., Ryman, A. G., and Stauffer, A. D. (1978). J. Phys. B [I] 11, 551. Mandal, P., Basu, D.,and Gash, A. S. (1976). J . Phys. B [I] 9, 2633. Massey. H. S . W., and Moussa, A. H . A. (1961). Proc. Phys. Soc., London 77, 811. Massey, H. S. W., Lawson, J . , and Thompson, D. G. (1966). In “Quantum Theory of Atoms, Molecules and the Solid State” (P.-0. Lowdin, ed.), p. 203. Academic Press, New York. Montgomery, R. E . , and LaBahn, R. W. (1970). Can. J . Phys. 48, 1288. Percival, I . C., and Valentine, N . (1967). Proc. Int. Conf. Phys. Electron. A t . Collision.,, 5th, 1967. Abstracts, p. 121. Ramsauer, C. (1921). Ann. Phys. (Leipzig) 66,546. Stein, T. S . . Kauppila, W. E.. Pol, V . , Jesion, G . , and Smart, J. H. (1977). Proc. f n r . Conf. Phys. Electron. At. Collisions. loth, 1977. Abstracts, p. 804. Stein, T. S . , Kauppila, W. E., Pol, V., Smart, J. H., and Jesion, G. (1978). Phys. Rev. [3] 17, 1600. Tsai, J . S . , Lebow, L., and Paul, D. A. L . (1976). Cun. J . Phys. 54, 17. Twomey, T. R. (1977). Ph.D. Thesis, University of London. Twomey, T. R., Griffith. T. C., and Heyland, G. R. (1977). Proc. Int. Conf. Phys. Electron. At. Collisions, loth, 1977. Abstracts, p. 808. Wilson, W . G. (1978). J . Phys. B [ I ] 11, L629.
ADVANCES IN ATOMIC AND MOLECULAR PHYSICS. VOL. IS
REACTIVE SCATTERING: RECENT ADVANCES IN THEORY A N D EXPERIMENT RICHARD B . BERNSTEIN Chemistry Deparrment Columbia University N e n York, New York
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Potential-Energy Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Classical Trajectory Methods and Results ..........................
167 168 171 173
Transition-State Theory: New Developments ....................... Collisional Ionization: Nonadiabatic Accurate Quantal Scattering Calculations .................... Information-Theoretic Approach to Molecular-Beam Chemistry . . . . . . . Crossed-Beam Chemihminescence State-to-State Cross Sections . . . . . . Influence of Different Forms of Ene A. Electronically Excited Reagents.. . ........ 189 B. Vibrational Enhancement. ..... C. Rotationally Excited Reagents . . . . . . . . . . . . 190 D. Translational Excitation of Reac XII. Translational Thresholds . . . . . . . . . . . . . References ...............................................
IV. V. VI. VII. VIII. IX. X. XI.
I. Introduction The purpose of this chapter is to report on some of the significant recent advances, both theoretical and experimental, in the field of reactive molecular scattering. Most of the attention will be devoted to systems of atoms and small molecules. Consideration will be limited to research published since the appearance of Sir Harrie Massey’s monumental treatise “Slow Collisions of Heavy Particles” (Massey, 1971). In that volume most of the significant results of research in the field have been lucidly presented and thoroughly discussed, leaving little to be added here for the pre-1970 period. Thus the reference list of the present chapter contains only references dated since 1970. There are several recent reviews that deal quite extensively with 167 Copyright @ 1979 by Academic Press. Inc All nghfs of reproduction in any form reserved
ISBN 0-12403815-3
168
Richard B . Bernstein
various aspects of the reactive scattering of neutral particles. A partial list follows. Those concerned primarily with the theoretical and interpretive aspects include books by Child (1974), Levine and Bernstein (1974a), Nikitin (1974), Miller (1976a) and Bernstein (1979), plus review articles on various subtopics by Levine (1972, 1978), Polanyi (1972), Polanyi and Schreiber (1974), Micha (1979, Bernstein and Levine (1975), Kuntz (1976, 1979), Pechukas (1976), Tully (1976), Child (1979), Diestler (1979), Levine and Kinsey (1979), Light (1979), Schaefer (1979), Truhlar and Dixon (1979), Truhlar and Muckerman (1979), and Wyatt (1979). On the more experimental side are books by McClure and Peek (1972), Fluendy and Lawley (1973), Lawley (1979, Brooks and Hayes (1977), and reviews on various specific aspects of the subject by Kinsey (1972), Lee (19721, Herschbach (1973, 1976), Los (1973), Polanyi (1973), Wexler (1973), Farrar and Lee (1974), Toennies (1974), Baede (1975), Grice (1973, Brooks (1976), Bernstein (1977, 1978a), Bauer (1978), Herm (19791, and Los and Kleyn (1979). The impact of theory upon experiment in the field of molecular reaction dynamics has been discussed by Bernstein (1975). It should be pointed out that the above listing of (post-1970)review literature is not intended to be complete; it is rather a reflection of the writer’s own interests in reactive scattering. In what follows, an attempt is made to illustrate, largely by example, some of the recent significant developments in both theory and experiment in the field of reactive molecular collisions.
11. Potential-Energy Surfaces Much of the progress in this field has been reported in the Faraday Discussion of the Chemical Society (1977) of this same title. There have also been a number of reviews, e.g., that by Schaefer (1979) on ab initio computational methods and results, and by Kuntz (1979) on semiempirical methods, especially the diatomics-in-molecules (DIM) procedure (see also Kuntz and Roach, 1972, 1973). Baht-Kurti and Yardley (1977) have reported ab inirio (CI) valence bond calculations of the LiFH potential surface, a presumed prototype for all the alkali-hydrogen halide systems. Since there are a number of molecular beam experiments on both M + HX reactions (cf. Herm, 1979) and H + MX (cf. Herschbach, 1977), it is important to have at least a qualitative understanding of the main topological features of the surfaces. The above-calculated surface shows a substantial potential barrier in
REACTIVE SCATTERING: RECENT ADVANCES
169
the ”exit” valley (MX + H), as anticipated from an earlier, lesssophisticated computation on KBrH by Roach (1970, 1977). This is consistent with experimental dynamical results by Blackwell rt a / . (1977, 1978), who found a substantial “vibrational enhancement” in the reactivity of hydrogen halides (HF and HCl) in collision with Na, i.e., HFt + Na- NaF + H This is a well-known characteristic of a reaction with an activation barrier in the exit valley. Semiempirical potential surfaces for LiFH and LiFz have been published recently by Zeiri and Shapiro (1978). One of the most thoroughly studied elementary reactions is that of atomic fluorine with HZ. HD. and D2,e.g., F
+ H2+
HF
+H
which serves as a case study in reaction dynamics (Polanyi and Schreiber, 1977; Berry, 1973; Polanyi and Woodall, 1972; Levine and Bernstein, 1974a). Considerable effort has gone into the determination of the FHp potential surface. The best ab iizitio computations (Benderef N / . , 1972a.b)are in fair agreement with the experimentally favored semiempirical potentials (e.g., Muckerman, 1972a,b). A “most-preferred” semiempirical surface (Polanyi and Schreiber, 1977) is an extended LEPS functionality with parameters adjusted such that trajectory calculations yielded best agreement with the experimental values of the activation energy und the mean product vibrational energy. Figure 1 shows this potential surface for the collinear FHH configuration, in the usual skewed and scaled (mass-weighted) representation. The
1.00
/
/
/
/
1.25
1.50
1.75
2.00
IF-,,
I\\‘\lo
20
(A)
FIG. 1 . Preferred semiempirical potential energy surface for FH2. collinear configuration: skewed and scaled (mass-weighted) representation (Polanyi and Schreiber. 1977).
I70
Richard B . Brrnstein
minimum energy path for the F + H, reaction has an entrance valley barrier of 2.16 kcal mol-', yielding a computed activation energy of 1.9 kcal mol-I [vs. an experimental value of 1.6 kcal mol-l, Homann et al. (1970)], a mean fractional vibration energy .& = 0.66, and a mean translational release &, = 0.24, compared to the experimental values (Polanyi and Woodall, 1972) of 0.66 and 0.26, respectively. It is encouraging that the best semiempirical surface is so similar to the ub initio surface for FH2, but such "chemical accuracy" has not been obtained routinely by existing CI computational procedures, except when applied to few-electron systems. The question of the sensitivity of the reactive scattering cross sections to the assumed potential energy surface has recently been studied by Connor et al. (1978), who carried out exact quanta1 calculations (on the collinear geometry) for the F + H, reaction using three different potential surfaces. Purely empirical methods for determining the main topological features
:
O0
2.0
4.0
RHg-I'
taPO
8.0
FIG.2. Empirical potential surface for Hg + I2 system (Mayer er al., 1977b). Left por,, collinear configuration; right, CZvinsertion approach, separated by oblique line. tion, C Note different coordinates for the two configurations. Solid contours based on observations, dashed contours estimated.
REACTIVE SCATTERING: RECENT ADVANCES
L
171
I HgI
( - I .45eV)
FIG.3. Minimum energy path for the Hg + I, reaction (Mayer C I al., 1977a). Note van der Waals minimum, then barrier preceding deep IHgl potential well responsible for complex-mode dynamicat behavior.
of a potential surface are sometimes useful. From a crossed-molecularbeam study of the endoergic reaction Hg+I,+Hgl+I
Mayer rf (11. (1977b) were able to deduce an approximate ground-state potential surface, shown in Fig. 2. Figure 3 is a plot of the minimum energy path, which is somewhat more directly determined from the experiments. The atom exchange reactions of H2and its isotopes with H+and with H are the most “fundamental” of chemical reactions. For some time accurate Lib inifio potential surfaces have been available [Csizmadia ef al. (1970) and Bauschlicher ef nl, (1973) for H t and Liu (1973) for H3]. Classical, semiclassical, and even exact 3-D quantal-scattering calculations have been carried out for these systems. Agreement with experiment is, in most cases, remarkable, as will be discussed later.
111. Classical Trajectory Methods and Results Polanyi and Schreiber (1977, 1978) and Truhlar and Muckerman (1979) have discussed optimal classical trajectory procedures for simulation of the detailed reactive-scattering behavior of model systems on (assumed) potential surfaces. Methods for improving efficiency, such as importance sampling, window function methods, moment inversion, and quasiclas-
172
Richirrd B . Bernstein
sical trajectory reverse histogram technique, are described, and results compared with more primitive Monte Carlo, quasi-classical histogrammic methods. The problem of the reliability of the classical mechanical method per se to deal with a basically quantal scattering problem is still a matter of some concern. With the current availability of accurate 1-D quantal calculations for a number of model systems and 3-D quantal computations for a few, it is now possible to make meaninghl appraisals of each of the various approximation schemes, including the classical and quasi-classical methods. In general it has been found that, except for interference effects and resonances, the quasi-classical trajectory method is remarkably successful in mimicking the major dynamical features provided by the quantal calculations (see, e.g., Truhlar and Kuppermann, 1972; Bowman and Kuppermann, 1973; Truhlar et al., 1973; Baer, 1974; Essen et a l . , 1976; Gray rf al., 1978). An ingenious approximation scheme to deal with the problem of multiple intersecting potential surfaces was introduced by Tully and Preston (1971), namely, the trajectory surface-hopping (TSH) method (see also Preston and Tully, 1971). For the important H i system extensive TSH computations were carried out by Tully and Preston (1971). They were based on DIM potential surfaces known to be in good agreement with the ab initio potential of Csizmadia e f al. (1970). Hopping locations were chosen to be the avoided intersections between the two lowest singlet surfaces of H3+. Nonadiabatic couplings were computed by the DIM program itself. Transition probabilities were calculated using a modified Landau-Zener (LZ) approximation. The calculations gave remarkably good predictions of the experimental cross sections of Ochs and Teloy (1974) for the reactions H*+D,
0
2
4
6
0
REACTIONS
2
-
4
6
0
2
4
6
8
E+,(eV)
FIG. 4. Absolute cross sections for H + + D2 reactions. yielding (a) D+ + HD, ( b ) HD+ + D, (c) DJ + H. Points: calculated via TSH method by Tully and Preston (1971). Curves: experimental measurements by Ochs and Teloy (1974).
REACTIVE SCATTERING: RECENT ADVANCES
I 0 0
5.5
2 4 Eir ( e v )
173
1
6
FIG.5 . Product translational energy (recoil) distributions for the reverse reaction of Fig. 4a. at collision energies of 3.0, 4.0. and 5 . 5 eV. Histograms via TSH method by Tully and Preston (1971 ). Curves: experimental measurements by Krenos ('I ( I / . (1974).
H++Dt+
I
D+ + H D HD++ D D: + H
as shown in Fig. 4. Figure S shows the agreement with experimental (Krenos rr ul., 1974) translational recoil energy distributions for the reverse of the first of these reactions. Unfortunately no fully quantal close-coupled numerical computations on this system are yet available. For a recent review of classical trajectory studies of a whole class of reactions involving alkali halide molecules and their status with regard to experiment, see Kwei (1979).
IV. Transition-State Theory: New Developments The now-familiar transition-state theory (TST) approach to elementary reaction rates has been tested by comparison with exact quantalscattering (and also classical trajectory) calculations for several model systems over a wide range of conditions. For reactions with an appreciable activation barrier Eh and at energies not too much in excess of Eh,
174
Richard B . Bernstein
TST is moderately successful (e.g., see Muckerman, 1971; Truhlar and Wyatt, 1976) in providing good estimates of the averaged quantities such as the overall reaction rate coefficient and the Arrhenius activation energy. For exoergic reactions without an activation barrier (such as most ion-molecule reactions) the standard TST must be modified. This has now been carried out explicitly by Bates (1978) to deal with bimolecular ion-molecule reactions, termolecular recombination processes, and radiative association reactions. Comparison with several experimental rate coefficients was favorable in most cases. An important new development in the field has been introduced by Miller (1976b) in the form of a unified statistical model that interpolates between the TST formalism, intended to deal with direct reactions, and the statistical theory, applicable to reactions involving a long-lived complex. Pollak and Pechukas (1979) have tested the Miller statistical theory numerically on the collinear H + H2exchange reaction, treated classically. Figure 6 shows the results: TST vs. the unified statistical theory (UST), vs. the accurate dynamical calculations. Clearly the UST corrects for the main defect of the TST, i.e., transition probabilities greater than unity,
STATE THEORY
UNIFIED STATISTICAL THEORY
0 0
a
0.5 POINTS: EXACT CALC.
0.0 0
1.0
2.0
3
E (eV)
FIG. 6 . Reaction probability vs. energy for the collinear H + H, reaction calculated using classical trajectories; comparison of results of transition-state theory (TST), the Miller unified statistical theory (UST), and the numerically exact (classical dynamical) results (points connected by dashed lines). From Pollak and Pechukas (1979).
REACTIVE SCATTERING: RECENT ADVANCES
I75
due to ignoring “recrossing” trajectories. In the UST a trajectory contributes to reaction if and only if it crosses a critical dividing surface an odd number of times. Apparently the effort involved in the computational procedure is no more than that required to apply TST in its variational form (where one calculates many dividing surfaces and chooses the surface of minimum flux) (Pollak and Pechukas, 1978). Unfortunately, the UST results are not really in quantitative accord with the exact dynamical calculations, and so the UST can only be regarded as an improvement over TST (see also Sverdlik and Koeppl, 1978). The question of the agreement of the classical dynamical with exact quantal results is a separate one (see Schatz and Kuppermann, 1976).
V. Collisional Ionization: Nonadiabatic Reactions The theory of atom-atom collisions at hyperthermal energies is well developed (see, e.g., Massey, 1971). depending for its execution upon reliable diabatic potential curves and interaction matrix elements H,,(R) in the vicinity of the relevant curve crossings at separations R , and an efficient close-coupling quantal computational program leading to the Smatrix at each collision energy of interest. Faist ef al. (1975) were the first to carry out a reasonably complete computation of the ionization cross sections for alkali- halogen atom collisions and compare with LandauZener-Stueckelberg (LZS) approximation calculations [see also Faist and Levine (1976) and Faist and Bernstein (1976) for more extensive comparisons between close-coupled and LZS calculations for electronically inelastic atom-atom scattering]. Some time before, Delvigne and Los (1973) had showed the remarkable ability of the LZS approximation to mimic the angular distribution of the collisional ionization cross section for Na
+1
4 Na+ + I-
Results on this and other systems, including halogen molecules, are discussed by Baede (1975) and Los and Kleyn (1979). In order to deal with the three-atom systems and account for the experimental cross sections for reactions such as (KBr
K
+ Br2+
I
+ Br
low E
K+ + Br,
t K + + Br-
+ Br
high E
176
Richard B . Brrnstein
the Tully and Preston (197 1) trajectory surface-hopping (TSH) method has been employed. Here one follows a classical trajectory until it encounters a surface intersection and then applies the Landau-Zener formula for the transition probability for a jump from one surface to the other. Evers and de Vries (1976), Evers (1977), Atenet al. (1977), and Everser al. (1978) have very successfully applied the TSH method to the alkali atom-halogen molecule systems. For the K + Br, system experimental results and TSH calculations are now available for collision energies over the entire range from lo-’ to lo4 eV, as reviewed by Evers (1978). Figure 7 shows the cross section for the reaction forming neutral KBr; Fig. 8 (upper) shows that for K+ formation and Fig. 8 (lower) the branching fraction F of Br- formation (with respect to total negative-ion yield). Figure 9 presents a graphic overview of the essential results of both experiment and theory on the nonelastic cross sections for K + Br, collisions. As an outgrowth of an extensive series of experimental studies and TSH calculations on a variety of M + XY reactions (M = Na, K; X, Y = I , C1, Br) the group of Los, de Vries, and their co-workers has evolved a unified description of this important class of “harpoon” reactions. The model is based on a two-state approximation; the electron transfer is assumed to occur at the diabatic crossing “seam” of the intersecting potential surfaces. The probability of an electron jump is calculated by the LZ formula, modified to allow for an orientation dependence 200
I
I
I
K + Br2
-
E + ~( K J MOL-’1
FIG.7. Total cross section for the reaction K + Br, + KBr + Br vs. collision energy. Points, experimental (Van der Meulen et ( J / . , 1975); dashed curve, calculated via TSH method by Evers (1978).
REACTIVE SCATTERING: RECENT ADVANCES
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__________---------0 100
10’
102 -
103
1’04
Etr (eV)
FIG.8. Total cross section for the collisional ionization reaction of K + Br,. Upper: K + formation; lower. branching fraction F for Br- formation [i.e.. Br-/(Br- + Brc)]. Solid curves, experimental (Baede c f ~ l . 1973; , Hubers ~t ai.. 1976); dashed curves. calculated as in Fig. 7 (Evers, 1978).
of the coupling matrix element H l z . The model takes into account the stretching of the XY bond during the approach of the M atom, important only at low collision energies. The model was extensively tested by comparison with TSH calculations. Values of HI2were adjusted to achieve best fit to all experimental data. 200 I
I
-
E+,(eV)
FIG.9. Overall summary plot of experimentally deduced total cross sections for different nonelastic processes in K + Br, collisions vs. translational energy, from E < lo-’ to lo4 eV (Los and Kleyn, 1979).
Richard B . Bernsrein
178
It was then found possible to make a semiempirical correlation of the important parameters in the model for all the systems studied (see, e.g., Aten and Los, 1977). A “reduced” LZ matrix element H& was found to be a smooth function of a reduced crossing separation R,*,with a “universal” experimental functionality much like that of Olson et al. (1971; cf. also Grice and Herschbach, 1974):
H:,
=
1.73R,* exp(-0.875R,*)
where R: = &(21M)1’2,with Re the curve-crossing separation, and I M the ionization potential of the alkali metal. The coupling matrix element H,,(R,) is expressed in terms of Hf, by
HI2
=
HTZ(ZMEAXY)1‘2
where E A x yis the vertical electron affinity of the halogen molecule. Figure 10 shows this correlation, i.e., a semilog plot of H&/R,* vs. R,*,with a “best straight line” drawn through the data. This universal function makes possible calculation of ionization cross sections for these many alkali-halogen systems over a wide energy range. Furthermore, most of the experimental behavior at low energies (i.e., the neutral product formation) can be accounted for semiquantitatively by the same model.
.
*-Na+XY A-K
+XY
- cs + XY I
I
4
I
1
5
I
1
I
6
Rc.
FIG. 10. Empirical “reduced” Landau-Zener coupling matrix element HT, vs. reduced crossing distance R,* (Hubers et al., 1976; Los and Kleyn, 1979). Points: experimental, for the indicated systems. Line: best exponential fit.
REACTIVE SCATTERING: RECENT ADVANCES
I79
VI. Accurate Quantal Scattering Calculations Only since 1975 has it been possible to execute a numerically exact 3-D close-coupled quantal scattering computation for even the simplest chemical reaction, H
+ H,+
H,
+H
This was accomplished through the independent efforts of Kuppermann and Schatz (1975) and Elkowitz and Wyatt (1975). [Further details on the methodology and results were published by Schatz and Kuppermann (1976) and Wyatt (1977, 1979).] Figure 1 1 shows a comparison of a calculated with an observed product angular distribution, that of Geddes r t al. (1972). The detailed quantal results are of great value not only because they are essentially predictive of experiment, but because they serve as benchmarks for testing various approximation schemes, such as the semiclassically calculated angular distributions of Doll ef a / . (1973), sudden approximation calculations by Bowman and Lee (1978), and of course, “conventional” quasi-classical trajectory calculations. As adapted by Wyatt (1977) the so-called Jzconserving approximation, originally developed for nonreactive collisions by Pack (1974) and McGuire and Kouri (1974), has been found to be remarkably successful in dealing with the reactive scattering problem. Applications have been made to both the H + H, and the F + H, systems by Redmon and Wyatt (1975). There has been considerable interest in a modified Franck-Condon
4 ai
a a
----
D + H ~ - - D H + H (EXP’T’L)
1
z
0
;0.5 I-
v) cn
0 a V
z
0 t-
-
K)
FIG.1 1 . Differential reaction cross section for D + H, DH + H , from experiments of Geddes ei nl. (1972). solid curve with hatched zone of uncertainty. Dashed curve. from ab initio quantal scattering computation for H + H2exchange reaction by Kuppermann and Schatz (197.5) and Schatz and Kupperrnann (1976).
180
Richard B . Bernstein
(golden rule) overlap approximation to the quanta1 reactive-scattering problem, proposed by Berry (1974a,b) and further developed by Halavee and Shapiro (1976), Schatz and Ross (1977), Mukamel and Ross (1977), and Fung and Freed (1978). Although not yet fully tested, it appears to be a very promising approximation scheme, capable even of dealing with nonadiabatic collisions (Zvijac and Ross, 1978). Another potentially useful approximation method for electronically nonadiabatic collision processes is the generalized Stueckelberg model of Miller and George (1972). As pointed out by Miller (1978), since this procedure incorporates interference and tunneling within the framework of classical S-matrix theory, it properly describes resonance effects, due to different avoided crossings being highly correlated with one another. However, the theory is difficult to apply in practice. Certain simplifications introduced by Komornicki et al. (1976) make the application of the Miller-George theory somewhat more convenient. In this form it may be superior to the TSH approximation, which is inherently unable to provide interference or resonance behavior.
VII. Information-Theoretic Approach to Reactive Scattering The information-theoretic (IT) approach to molecular reaction dynamics was introduced by Bernstein and Levine (1972), and applied to the codification, compaction, and correlation of a considerable body of experimental (and computer-generated) reactive-scattering data by Ben-Shaul et af. (1972). Making use of the principle of maximal entropy subject to (a limited number of) dynamic constraints, it was possible to develop a number of useful procedures for the worker in molecular reaction dynamics, such as surprisal analysis and synthesis, and entropy deficiency (i.e., information content) of disequilibrium product state distributions. The theoretical foundation of the subject has recently been consolidated by Alhassid and Levine (1977, 1978) and reviewed by Levine (1979). An interesting computational finding by Wyatt (1975) and Schatz and Kuppermann (1976) was a linear dependence of the surprisal of the final rotational state distribution for the Hzproduct of the , H + H2exchange reaction (cf. Levine et al., 1973). Nesbet (1976) and Clary and Nesbet (1978) have found that the same is true throughout the course of the collision! A number of general reviews on various aspects of the IT method have by now appeared, e.g., those by Levine and Bernstein (1974b, 1976),
REACTIVE SCATTERING: RECENT ADVANCES
181
Bernstein and Levine (1975). and Levine and Ben-Shad (1977). Levine (1978) has summarized the extensive literature on the subject for the 1972- 1978 period. and Levine and Kinsey (1979) have presented an upto-date “handbook” for IT practitioners in the field. Because of these extensive reviews, no further discussion of the IT approach will be presented here, despite its wide application to state-to-state processes of great experimental interest (cf. Levine and Manz, 1975; Kaplan et d., 1976; Bernstein, 1977: Levine, 1977; Polanyi and Schreiber, 1978).
VIII. Molecular-Beam Chemistry The decade of the 1970s saw the beginning of the end of the “alkali age” of molecular-beam scattering. The crossed-beam study by Gillen et al. (1971) of the detailed differential reactive cross section for the reaction of K + Iz was soon followed by research by McDonald et al. (1972) of comparable detail on the analogous reaction of D + I, (and other halogens). Soon thereafter came similar studies of reactions of CI atoms with several molecules (e.g., Cheung et ul., 1973), and of related F atom reactions (e.g., Parson et al., 1973), as well as reactions of 0 atoms (e.g., Parrish and Herschbach, 1973; see also Kingrt al., 1974; Dixon and Herschbach, 1975). A pioneering, landmark crossed-beam study was the very early one of Schafer et nl. (1970) on the important F + D, reaction. There has gradually been a trend from “physics” toward “chemistry” slowly transforming the field of molecular-beam scattering (cf. Faraday Discussion of The Chemical Society, 1973) to “molecular-beam chemistry,” as exemplified by the work of Herschbach and collaborators (see, e.g., Herschbach, 1973, 1976). This trend can also be perceived in the work of Lee and co-workers (1971; Valentini et al., 1977), Herm and his collaborators (see, e.g., Freedman et al., 1978; Parr el al., 1978), Grice and co-workers (see, e.g., Whitehead et al., 1973; Mascord et al., 19771, Zare and his colleagues (see, e.g., Zare and Dagdigian, 1974; Engelke et al., 1977), and the writer and co-workers (see, e.g., Bernstein, 1978b). Reagent species have included many reactive atoms hitherto unavailable in beams. Examples include reactions of rare earth and heavy-metal atom beams (e.g., Ba, Ca, Sr, etc., reviewed by Herm, 1979, Al, Zare, 1974; Sn, Parret al., 1978; Hg, Mayerer al., 1977a), N atom beams (Loveet af., 1977), F atom beams (Farrar and Lee, 1975a,b), and many others. Coreagents include halogens, inorganic and organic halides, oxygen and various oxides, and cyanides. A partial list of metal atom reactions has been presented by Herm ( 1 978). Recently a thorough crossed-beam study on a family of elementary
182
Richard B . Bernstein
reactions of considerable significance has been reported by Bauer et a / . (1978), namely, the D + HX exchange reactions
HI
+F:
+D+H
DI
They measured the angular distribution of the H product, which could then be compared with previous measurements by McDonald and Herschbach (1975) of the “complementary” DX products. Figure 12 shows the consistency of the results of these independent studies (by comparing at complementary angles). This confirms that the molecular product is strongly back-scattered with respect to the incident atom in the center-of-mass system, a characteristic of “rebound” reactions. Figure 13 shows a comparison of the new experimental data (for the H product of the D + HCI reaction) with previously published classical trajectory calculations by Raff et al. (1975) based on a semiempirical potential surface. Unfortunately the limited angular range of the experimental results precluded observation of a possible bimodal distribution.
IX. Crossed-Beam Chemiluminescence Although for decades it has been customary to follow certain exothermic chemical reactions in the bulk phase by monitoring their spectral emission, it is only fairly recently that single-collision experiments (via crossed molecular beams) have been carried out in which uv or visible radiation from nascent reaction products is observed. For reactions involving electronically excited reagents, it is perhaps not surprising that chemiluminescence is often observed. However, for many exoergic reactions of ground-state reagents, products are formed in excited electronic states and emission of visible or uv radiation is readily observable, providing considerable information on the reaction dynamics and potential surfaces involved. Only a few examples will be mentioned here. One is the reaction of metastable Hg atoms with halogens, e.g., Hg(63P,) + Br2(’Z:)
-*
HgBr(B*Z+)+ Br(*P,,*) hu
I
5tNO 8,
HgBr(X*Z+)
This reaction was one of a series studied by Krause et a f . (1975). These authors measured a value for the total chemiluminescence cross section
REACTIVE SCATTERING: RECENT A D V A N C E S
I83
%.m.
FIG.12. Differential reaction cross section (arbitrary units) for the reactions of D + HCI. HBr and HI vs. scattering angle in the center-of-mass system with respect to the incident D atom. Points: experiments of Bauer r? a/. (1978). via measurements of H atom product, forward scattered. Shaded zones: data band of McDonald and Herschbach (1975). via measurements of DX product, back scattered.
3n -
w
-
0
Richard B . Bernstein
184
uCL of -160 A, for this reaction, and -90 h2for the corresponding one with Clz. They also pointed out the possibility of the existence of a strong, unexpected spin-orbit effect, since their measurements provided upper limits for ucLof 40 and 20 A,, respectively, for the reactions (with Br, and ClJ of Hg(63Po) reagent atoms. This was confirmed experimentally by = 3 Azfor the reaction Hayashi et al. (1978), who measured a value of cCL of Hg(63Po)with Br,. The theory of this spin-orbit effect is not yet developed, but a correlation diagram approach to the problem of the entire potential energy surface manifold for the Hg + x, systems has been undertaken by Muckerman (1979). The chemiluminescence cross sections for the reactions of metastable Ca* and Sr* atoms with N,O have been measured by Dagdigian (1978) and Wilcomb and Dagdigian (1978). A number of molecular beam reactions of ground-state reagents have been investigated, including that of NO
+ Oa+
NO$
+
0 2
This reaction produces both visible and infrared emission from the electronically and vibrationally excited nascent NO, product (see, e.g., Redpath and Menzinger, 1971, 1975; Golde and Kaufman, 1974; Redpath et a / . , 1978). The reaction rate is known to be strongly increased by vibrational excitation (via laser irradiation) of the ozone (Gordon and Lin, 1973, 1976; Stephenson and Freund, 1976) and by translational excitation, the latter from accelerated beam studies of Menzinger and co-workers (see Redpath et al., 1978). Figure 14 shows the experimentally derived translational excitation
I
o*Mp I I
I
NO + 0,
-
I
No;
I
I
+ 0,
0.06
0.02 02
3
4
5
1 . -
0
2
-
I
4
6
I
0
1
0
E , ~( KCAL MOL-')
FIG. 14. Cherniluminescence cross section function for the reaction NO NO,* + 0, (Redpath er u l . , 1978).
+ 03+
REACTIVE SCATTERING: RECENT ADVANCES
I85
function for the visible chemiluminescence (CL) cross section for the reaction N0(zI13,2)with 03.The translational threshold for CL was estimated to be 3.0 t 0.3 kcal mol-I, consistent with the known thermal activation energy of the bulk reaction. In another CL study, namely, N,O + Ba + BaO* + N2 Wren and Menzinger (1974) found smaller cross sections, which declined the energy. The analogous reactions with O z , NOz, Clz, etc. have been studied. For the beam reaction Ba + C12 -,BaCl* + CI the chemiluminescence first observed by Ottinger and Zare (1970) accounts for only a very small fraction of the total reactive cross section (from product angular distributions) observed directly by Haberman ef al. (1972) and Lin et al. (1973). The translational energy dependence of rcL for the reaction NEO
+ Pb + PbO* + N2
has been measured by Wicke et al. (1978); the results are shown in Fig. 15. The data were consistent with a statistical model of Krenos and Tully (1975). Crossed-beam CL reactions of La, Y. and Sc with 0, were studied by
FIG. 15. Cherniiuminescence cross section function for the reaction Pb PbO*
+ N, (Wicke er a l . . 1978).
+ N20+
I86
Ritlzard 3. 3rrnstvin
Manos and Parson (1978). Figure 16 shows the CL translational energy dependence for the three concurrent reactions:
1
(CZrIl,2)
La
+ O2 + 0 + La0 (B22) (Azn,,2)
The information-theoretic ‘‘prior expectation” (density of states, statistical) model is seen to be a good representation of the observed energy dependence. Laser-induced fluorescence experiments on the reactions of Sc and Y with 02,NO, and SO2 by Liu and Parson (1977) showed that product vibrational and rotational distributions were usually quite close to the ‘‘prior” statistical predictions. A crossed-beam measurement of the CL cross section was reported by Alben et al. (1978) for several of the so-called dioxetane reactions, the simplest of which is O f ( ’ A 3+ HZC=CHz
+
HZCO*
+ HZCO
Figure 17 shows crCL(&) for the reaction of the electronically excited oxygen molecule with methyl vinyl ether. Lo 10
1
+ 02
-
a
LOO + 0
Lao
i . *
FIG.16. Relative rates of formation, as a function of collision energy, of the three indicated excited electronic states of La0 from the reaction of La + 02.Points: experimental. Solid curves: information-theoretic“prior expectation,” on density of states considerations (Manos and Parson, 1978).
REACTIVE SCATTERING: RECENT A D V A N C E S
I87
FIG. 17. Chemiluminescence cross section for the dioxetane reaction of O,(’AJ with 1978). methyl vinyl ether (Alban pr d.,
A very interesting CL reaction was recently observed and characterized in a flow system by Sridharan er ul. (1978): PbO
+ C + CO + Pb*
The total cross section for the production of excited states was estimated The ground-state reaction is exoergic by 7.2 eV, and light to be - 5 emission from Pb atoms excited up to this limit was observed. A related reaction,
xz.
N20
+ B + BO* + N ,
has been studied by Tang et 01. (1976) and Brzychcy et al. (1979). Clearly one must take cognizance of the many excited-state potential surfaces in dealing with the dynamics of chemiluminescent reactions.
X. State-to-State Cross Sections Recent advances have led to the development of a new subfield of chemical dynamics known as “state-to-state chemistry” (see Brooks and Hayes, 1977). There has been a great deal of current experimental effort directed toward the measurement not only of state-to-state reaction rates,
Richard B . Bernstrin
188
i.e., k,-,f(T), but also the more fundamental quantities, state-to-state cross as a function of collision energy. Here i refers to a sections, i.e., specific set of initial quantum numbers of the reagent molecules and f a final set, for the products. A recent overview of the experimental and theoretical advances (Bernstein, 1977) obviates the need for extensive discussion here. From an experimental viewpoint the key new tool is the laser, being used in two different ways: (a) product internal state analysis, via laserinduced fluorescence (LIF) (Cruse et al., 1973; Zare and Dagdigian, 1974; Smith and Zare, 1976; Kinsey, 1977) and (b) reagent internal state selection, via laser excitation (Odiorne et al., 1971; Pruett and Zare, 1976; Engelke et af., 1977). As an illustration of the power of the new experimental method, Fig. 18 shows a comparison of the LIF spectra of the SrF product from the reactions HF(u) + Sr -+ SrF
+H
for u = 1 vs. u = 0, from a recent laser-molecular-beam experiment of Karny and Zare (1978). The cross section for the reaction of HF(u = 1) is at least four orders of magnitude greater than for that of ground-state HF. This is not entirely unexpected since the ground-state reaction is endothermic by some 6.4 kcal mol-1 while that with HF(v = 1) becomes exothermic by 4.9 kcal rnol-'. For an information-theoretic treatment of the role of internal excitation on reactivity, see Levine and Manz (1975).
Sr
+ HF(v)
-+
SrF t H
(0.0)
I
I
I
649
650
651
I
652 Xnrn
1
I
653
654
FIG. IS. Laser-induced fluorescence spectra of the nascent SrF product from the crossed-beam reaction of HF with Sr. Upper, HF ( u = 1); lower, HF ( u = 0).
REACTIVE SCATTERING: RECENT ADVANCES
189
XI. Influence of Different Forms of Energy upon Reactivity This is an area of intense experimental interest and activity, closely related to the subject of the previous section. Space permits inclusion of only a few examples of such studies. A. ELECTRONICALLY EXCITED REAGENTS The crossed-beam reaction of I,*(B) with F, has been studied by Engelke et al. (1977):
Excited iodine molecules were prepared by irradiation of I, with an Ar+ ion laser ( h = 514.5 nm), yielding I,[B311(O:), u = 431, which then reacted with F, to produce chemiluminescence from the IF* products. An energy level diagram is shown in Fig. 19. Analysis of the IF* emission spectrum 0
I + I + F + F
514.5 nm laser
-\
IF(X)+IF(X) (-132)
FIG. 19. Energetics of the 1 ; + F, reaction. Energy units: kcal mol-'. The processes indicated by the arrows were observed by Engelke ct ~ r l .(1977).
Richard B. Bernstein
190
showed that the IF*(B) product was vibrationally hot, with the vibrational distribution extending to the predissociation limit. The cross section for the reaction was estimated to exceed 10 hA2. The direct four-center reaction of the electronically ground-state reagent molecules does not proceed under single-collision conditions (Valentini et a f . , 1976, 1977), as discussed in Section XI.
B . VIBRATIONAL ENHANCEMENT For endoergic reactions or reactions with an appreciable activation barrier, vibrational energy can often be efficiently utilized to overcome the energetic requirements (subject, of course, to dynamical constraints imposed by the topology of the potential surface). An example is provided by data on the influence of vibrational excitation on the rate of the endoergic reaction (A& = 16.5 kcal mol-’) Br
+ HCI(u) + HBr + CI
Douglas et al. (1973) found that the ground-state thermal reaction rate is very small but that if the HCl is excited to u 5 2 (Evlbz= 16 kcal mol-’f the reaction proceeds with near gas-kinetic rate. From this study and further work by Leone e f al. (1975), Arnoldi et al. (1975), and Arnoldi and Wolfrum (1976), it is found that the rate constant (at 298 K) for HCI(u = 2) is -1Og M-’ sec-’ compared to that for HCl(u = 0) of M-’ sec-* (a “vibrational enhancement” of 1 1 orders of magnitude!). These and other related results are discussed by Bernstein (1977). C. ROTATIONALLY EXCITEDREAGENTS The first direct determination of the rotational energy effect on a reaction rate was reported by Polanyi and co-workers (Ding et al., 1973) for the reaction F
+ HCI(J = 0-10)
---*
HF
+ CI
Further work by Blackwell et al. (1978) has dealt with the reactions HF(J = 0-14) HCI(J = 0-19)
] + Na+
H
+
E:Fl
They found that the rate constant at first decreased (by about a factor of 2) as J increased, passed through a minimum and then increased with 1.The results are only partially explained; one must consider the separate effects of rotational angular momentum and rotational energy, and the topology of the potential surface.
REACTIVE SCATTERING: RECENT ADVANCES
191
The only crossed-beam measurements of the rotational effect on reactivity come from the author’s laboratory. Studies were carried out on the complex-mode alkali halide-alkali exchange reaction
using rotationally state-selected beams of CsF (Stolte et d.,1975, 1977) and RbF (Zandee and Bernstein, 1978). The influence of the rotational energy of the alkali fluoride molecule upon its reactivity was measured as a function of the collision energy ,TI,. The reactions are known to proceed 1976). The measurements were of via a long-lived complex (Stolte et d., where mR is the the so-called reactive branching fraction, i.e., (T~/(T,-, reaction cross section and (T,- the complex formation (capture) cross section. Figure 20 shows the directly observed rotational energy effect, i.e., the ratio of the reactive branching fraction for a “low-J” beam (,Fro,< 0.3) vs. a “thermal” (Boltzmann) beam (E,,, > 2 kcal mol-I). For the CsF reaction, which is endoergic, rotational energy enhances reaction, while for the exoergic RbF reaction the opposite is found. These results are in qualitative accord with expectation on simple theoretical (phase space) grounds.
I
I
3
1
4
5
I
6
I
Etrlhcol mol-’)
FIG. 20. Observed rotational energy effect for the reactions of RbF + K + Rb + KF (Zandee and Bernstein. 1978) and CsF + K 4 Cs + KF (Stolte c r a / . , 1977). The ordinate is the ratio of the reactivity of “low-J” reagent molecules ( & s 0.2 kcal mol-I) to that of 2.3 kcal rnol-I); the abscissa is the average collision energy “thermal” reagents (&,
192
Richard B . Bernstrin
D. TRANSLATIONAL EXCITATION OF REACTION Molecular-beam scattering measurements of the translational energy dependence of the reaction cross section have provided valuable information of three types. First, on the energy threshold Eth,or Eo (i.e., the endoergicity AEo and/or activation barrier Eb)(for an overview, see Bernstein, 1978a). Second, on the form of the immediate postthreshold functionality of the cross section aR(Etr- Eo). Various theoretical postthreshold laws were discussed by Levine and Bernstein (1971, 1972), Eu (1974a,b), Eu and Liu (1975), Krenos and Tully (1973, Ureiia and Aoiz (1977), Yokozeki and Menzinger (1977), and others. A fairly common form is E,'(E
UR
-
Eo)",
0
=S n
s 5/2
where E is the total (translational plus internal) energy. Third, information on the overall shape (and magnitude) of the cross section function aR(Etr) over a wide energy range. The first crossed-beam measurements of the translational excitation function for a reaction involving neutral reagents and products were for the exoergic reactions
I
II
II
11
II
I I
II
11
II
I
I
I
!
C+,(eV) FIG.21. Comparison of experimental excitation functions for the indicated CHJ + K , Rb reactions. Dashed portion (determined indirectly) is uncertain. From Wu et al. (1978).
REACTIVE SCATTERING: RECENT ADVANCES
0 0
I93
I
5
10 -Etr (KCAL
15
MOC'I
FIG.22. Experimental excitation function for the HCI + K reaction. Circles, data of Geis et a / . (1977); squares and solid curve (theoretical extrapolation) from Pruett et c d . ( I975 ).
(Gersh and Bernstein, 1971, 1972; Litvak et a / . , 1974; Pace et a / . , 1977). Figure 21 shows a comparison of the K and Rb cross sections (Wu et al., 1978). Figure 22 shows the excitation function for the reaction HCI+K-+KCI+H
measured at low energies by Pruett er ul. (1975) and at higher energies by Geis et a / . (1977); the data points are compared with a semitheoretical curve based on the low-energy behavior (Pruett er af., 1975). A significant study of the energy dependence of a reaction of considerable theoretical interest has been reported by Hepburn et al. (1978): H(D)
+ Br2+
HBr(DBr)
+ Br.
The excitation function declines monotonically with &r over the experimental range from 1 to 8 kcal mol-'. At a given Gr, the cross section for the D atom reaction is larger than for H atom attack, but when the data are compared as a function of relative velocity the isotopic reaction cross sections are the same.
XII. Translational Thresholds For an exoerzic reaction with an activation barrier, a molecular-beam threshold measurement provides an essentially direct determination of the barrier energy E b . For an endoergic reaction the threshold yields the endoergicity plus the energy required to overcome any intrinsic activation barrier. For the endoergic reaction Hg
+ 12+
HgI
+I
Rich urd B . Be rns rein
194
studied by Wilcomb et al. (1976), the observed translational threshold was 1.14 k 0.03 eV, as shown in Fig. 23. Assuming the average internal energy of the I2 to be 0.04 eV, and given the bond dissociation energies of I2 (1.54 eV) and HgI (0.39 eV), one calculates the intrinsic activation barrier to be Eb = 0.39 - 1.54 + 1.14 + 0.04 = 0.03 & 0.03 eV, essentially zero (see also Mayer e? a / . , 1977a). For the ion-pair formation reactions Xe
+ CsCl+
C1-
Xe + Cst + {CsXet
Sheen et a/. (1978) measured the thresholds for each reaction and determined the postthreshold energy dependence of the cross sections, as shown in Fig. 24. Thermochemical data (i.e., product dissociation energies) from threshold energies were obtained by Batalli-Cosmovici and Michel (1972) for SrO, and Dirscherl and Michel (1976), for EuO. One of the recurring problems in extracting accurate thresholds is the well-known deleterious influence of the reagent’s relative translation energy distribution upon the data near threshold, requiring deconvolution to yield the true cross section (excitation) function (see Chantry, 1971; Chupka et al., 1971; BernI
I
Hg+12-
I
I
HgI+I
I
I
-
r” L
t
0 -I
w>
U CI,
I
-
E+,(eV 1
+ I, reaction vs. average collision energy. Solid curve: calculated, best fit of convoluted line-of-centers cross section function, = 1.14 f 0.03 eV). From Wilcomb rt a / . with indicated translational threshold energy (/it,, ( I 976). FIG.23. In-plane yield of HgI product from the Hg
I95
REACTIVE SCATTERING: RECENT ADVANCES
csxe+l
LCS+
Etr ( e V )
FIG.24. Experimental excitation function for Cst and CsXe+ formation from Xe + CsCl collisions vs. average collision energy. Points (solid curves) experimental: dashed curves. deconvoluted cross section functions with indicated thresholds. From Sheen et al. (1978).
''m ''@I----K+CH3Br-KBr
-._2 c
+CH3
Rb
+
CH3Br
RbBr
+
CH,
3
0.75-
* /
II
0.8
FIG. 25. Left: in-plane yield of KBr product from the CHJ3r + K reaction vs. average collision energy (same presentation as Fig. 23). Best experimental translational threshold indicated (El,, = 0.24 t 0.06 eV). Right: same, for RbBr from the CH,Br + Rb reaction (Elh= 0.20 2 0.06 eV). From Pang ct al. (1978).
196
Richard B . Bernstein
stein, 1973). This introduces nonnegligible uncertainties in the experimentally derived Ethand the postthreshold functionality, even after optimal deconvolution. There are few examples of an activation barrier measurement for an exoergic reaction, one of which [for the dioxetane reaction (Alben et al., 1978)] was shown in Fig. 17. Presented in Fig. 25 are results for the exoergic reactions
(Pang et al., 1978), which show significant barriers (cf. Fig. 21 for the related CHJ reactions). The crossed-beam reaction of I, with F, has been thoroughly studied by Valentini et al. (1976, 1977). Threshold experiments revealed that no reaction occured at energies below 4 kcal mol-', above which a new trihalogen molecule IIF was detected. Upon increasing the translational energy to 7 kcal mol-', the IF molecule itself was observed. The direct four-center reaction I,
+ Fz --* IF + IF
does not occur under single-collision conditions (as predicted by Woodward-Hoffman rules). Figure 26 shows the threshold determina-
-
E+,(kcal mol-'1
FIG.26. Experimental cross section functions for the reactions of F, with I,. ICI, and H I to produce IIF, CIIF, and HIF. respectively; thresholds indicated. From Valentini er a / . (1976).
REACTIVE SCATTERING: RECENT ADVANCES 0
197
I + I + F + F
(- 36)
I
*x \ \
\ (-73!,-;?’ I2+F2
(-66)
\-3--F
IIF+F
FIG.27. Energetics of the Iz + F, reaction. Energy units: kcal mol-’. The processes indicated by the arrows have been observed (Wong and Lee, 1973: Valentini et a / . , 1976, 1977).
tions for the three related reactions F,
+
11
CII+ F
+
IIF CllF /HIF
(Valentini et a / . , 1976). Taken together with knowledge gained from a previous study by Wong and Lee (1973) on the F + I2 reaction (which proceeds via a long-lived IIF complex), and via a detailed crossed-beam study, Valentini et a/. (1977) have provided a significant, self-consistent body of data on the 12-F2 system. (The energetics are summarized in Fig. 27.) Their results are of great interest not only to workers in the field of chemical dynamics but also for inorganic and structural chemists, as well as researchers in quantum chemistry (i.e., chemical bonding). Clearly there have been great advances, both experimental and theoretical, in the field of reactive molecular scattering during the 1970s. It is hoped that this brief presentation has offered a glimpse of these developments, thus serving as a 1979 footnote (or postscript) to Sir Harrie Massey’s great treatise on slow collisions of heavy particles. ACKN 0W LEDG M E NT The research program of the writer and his co-workers has received financial support from the National Science Foundation Grant CHE 77-1 1384 A01. hereby acknowledged with thanks.
198
Richard B . Bernstein REFERENCES
Alben, K. T., Auerbach, A., Ollison, W. M.,Weiner, J . , and Cross, R. J., Jr. (1978). J . A m . Chem. Soc. 100, 3274. Alhassid, Y., and Levine, R. D. (1977).J. Chem. Phys. 67, 4321. Alhassid, Y . . and Levine, R. D. (1978). Phys. Rev. A [3] 18, 89. Arnoldi, D., and Wolfrum, J. (1976). Ber. Bunsenges. Phys. Chem. 80, 892. Arnoldi, D.. Kaufmann, K., and Wolfrum, J. (1975). Phys. Rev. Lett. 34, 1597. Aten, J . A., and Los, J. (1977). Chem. Phys. 25, 47. Aten, J . A., Evers, C . W. A , , de Vries. A. E., and Los, J. (1977). Chem. Phys. 23, 125. Baede. A. P. M . (1975). Adv. Chem. Phys. 30,463. Baede, A. P. M.,Auerbach, D. J., and Los, J. (1973). Physica (Cltrecht) 64, 134. Baer, M . (1974). J. Chem. Phys. 60, 1057. Balint-Kurti, G. G., and Yardley, R. N. (1977). Faraday Discuss. Chem. Soc. 62, 77. Batalli-Cosmovici, C., and Michel, K. W. (1972). Chem. Phys. Lett. 16, 77. Bates, D. R. (1978). Proc. R. Soc. London, Ser A 360, 1. Bauer, S. H. (1978). Chem. Rev. 78, 147. Bauer, W . . Rusin, L. Y., and Toennies, J. P. (1978). J . Chem. Phys. 68, 4490. Bauschlicher, C. W., O’Neil, S. V.. Preston, R. K., Schaefer, H . F., and Bender, C. F. (1973). J . Chem. Phys. 59, 1286. Bender, C. F.. O’Neil, S. V., Pearson, P. K., and Schaefer, H. F. (1972a). Science 176, 1412. Bender, C. F., Pearson, P. K., O’Neil, S. V., and Schaefer, H. F. (1972b). J . Chem. Phys. 56, 4626. Ben-Shad, A., Levine, R. D., and Bernstein, R. B. (1972). J . Chem. Phys. 57, 5427. Bernstein, R. B. (1973). Comments A t . Mol. Phys. D 4, 43. Bernstein, R. B. (1975). Isr. J . Chem. 14, 79. Bernstein, R. B. (1977). In “State-to-State Chemistry” (P. R. Brooks and E. F. Hayes, eds.), ACS Symp. Ser. No. 56, p. 3. Am. Chem. SOC.,Washington, D.C. Bernstein, R. B. (1978a). In “Advances in Laser Chemistry” (A. H. Zewail, ed.). p. 384. Springer-Verlag Berlin and New York. Bernstein, R. B. (1978bf. Pure Appl. Chem. 50, 781. Bernstein, R. B., ed. (1979). “Atom-Molecule Collision Theory: A Guide for the Experimentalist.” Plenum, New York. Bernstein, R. B., and Levine, R. D. (1972). J . Chem. Phys. 57, 434. Bernstein. R. B.. and Levine, R. D. (1975). Adv. At. Mol. Phys. 11, 215. Berry, M . J . (1973). J . Chem. Phys. 59, 6229. Berry, M. J . (1974a). Chem. Phys. Lett. 27, 73. Berry, M . J . (1974b). Chem. Phys. Lett. 29, 323 and 329. Blackwell, B. A., Polanyi, J. C., and Sloan, J. J. (1977). Faraday Discuss. Chem. Soc. 62, 147. Blackwell, B. A., Polanyi, J. C., and Sloan, J . J. (1978). Chem. Phys. 30, 299. Bowman, J . M . , and Kuppermann, A. (1973). Chem. Phys. Lett. 19, 166. Bowman. J. M., and Lee, K, T. (1978). J . Chem. Phys. 68, 3940. Brooks, P. R. (1976). Science 193, 11. Brooks, P. R.. and Hayes, E. F., eds. (1977). “State-to-State Chemistry,’’ ACS Symp. Ser. No. 56. Am. Chem. SOC.,Washington, D.C. Brzychcy, A., DeHaven, J., Prengel, A. T., and Davidovits, P. (1979). Chem. Phys. Lett. 60, 102. C h a n t y , P. J. (1971).J. Chem. Phys. 55, 2746.
REACTIVE SCATTERING: RECENT ADVANCES
199
Cheung. J. T . , McDonald, J. D., and Herschbach. D. R. (1973).J.A m . Chern. SOL..95,7889. Child. M. S. (1974). “Molecular Collision Theory.” Academic Press, New York. Child. M. S. (1979).In ”Atom-Molecule Collision Theory: A Guide for the Experimentalist” (R. B. Bernstein, ed.), p. 427. Plenum, New York. Chupka, W. A., Berkowitz, J . , and Gutman, D. (1971). J. Chem. Phys. 55, 2724. Clary, D. C., and Nesbet, R. K. (1978). Chern. Ph Connor, J. N. L., Jakubetz, W., and Manz, J. (1978). M o l . Phys. 35, 1301. Cruse, H. W.. Dagdigian. P. J., and Zare. R. N . (1973). F ~ r ~ d Discuss. ay Chern. Soc. 55, 277. Csizmadia. 1. G., Karl, R. E., Polanyi, J. C., Roach, A. C., and Robb, M. A. (1970). J. Chem. Ph.vs. 52, 6205. Dagdigian, P. J. (1978). Cham. Phys. Letr. 55, 239. Delvigne. G. A., and Los. J. (1973). Physicu (Urrechf)67, 166. Diestler. D. I . (1979). In “Atom-Molecule Collision Theory: A Guide for the Experimentalist” (R. B. Bernstein, ed.), p. 655. Plenum, New York. Ding. A. M., Kirsch, L. J.. Perry, D. S., Polanyi, J. C., and Schreiber. J . L. (1973).Furuduy Discuss. Chem. Soc. 55, 252. Dirscherl, R., and Michel. K. W. (1976). Chern. Phys. Lrrt. 43, 547. Dixon. D. A . , and Herschbach. D. R. (1975). J . A m . Chem. SOC. 98,6268. Doll. J. D.. George, T. F., and Miller. W. H. (1973). J. Chrm. Phys. 58, 1343. Douglas, D. J . , Polanyi, J. C., and Sloan, J. J. (1973). J. Chem. Phys. 59, 6679. Elkowitz. A,, and Wyatt, R. E . (1975). J . Chem. Phys. 62, 2504. Engelke, F.. Whitehead. J . C.. and Zare. R. N. (1977). Furuday Discuss. Chem. S O C . 62, 222. Essen, M.. Billing. G. D.. and Baer, M. (1976). Chem. Phys. 17, 443. Eu. B. C. (1974a). J . Chem. Phys. 60, 1178. Eu, B. C. (1974b). Chem. Phys. 5, 95. Eu. B. C., and Liu. W. S. (1975). J. Chem. Phys. 63, 592. Evers, C. W. A. (1977). Chem. Phys. 21, 355. Evers, C. W. A. (1978). Chem. Phys. 30, 27. Evers, C. W. A,, and de Vries, A. E. (1976). Chern. Phys. 15, 201. Evers, C. W. A., de Vries, A. E.. and Los, J. (1978). Chem. Phys. 29, 399. Faist, M. B., and Bernstein. R. B. (1976). J. Chem. Phys. 64, 2971. Faist, M. B.. and Levine. R. D. (1976). J. Chem. Phys. 64, 2953. Faist. M. B., Johnson, B. R., and Levine, R. D. (1975). Chem. Phys. Lerr. 32, 1. Faraday Discussion of the Chemical Society (1973). “Molecular Beam Scattering,” Vol. 55. FDCS, London. Faraday Discussion of the Chemical Society (1977). “Potential Energy Surfaces,’’ Yo!. 62. FDCS , London. Farrar. J. M., and Lee, Y. T. (1974). Annu. Rev. Phys. Chem. 25, 357. Farrar, J . M., and Lee, Y. T. (1975a). J. Chem. Phys. 63, 3639. Farrar. J. M . . and Lee, Y . T. (1975b).J. A m . Chem. SOL..96,7570. Fluendy, M. A. D.. and Lawley, K . P. (1973). ”Chemical Applications of Molecular Beam Scattering.” Chapman & Hall, London. Freedman. A., Behrens, R . , Parr, T. P., and Herm. R. R. (1978). J. Chem. Phys. 68,4368. Fung, K. H.. and Freed, K. F. (1978). Chem. Phys. 30, 249. Geddes, J., Krause, H. F., and Fite, W. L. (1972). J . Chem. Phys. 56, 3298. Geis, M. W., Dispert, H.. Budzynski, T. L., and Brooks, P. R. (1977). In “State-to-State Chemistry” (P. R. Brooks and E. F. Hayes. eds.), ACS Symp. Ser. No. 56, p. 103. Am. Chem. SOC.,Washington, D.C.
200
Richurd B . Bernstein
Gersh, M. E.. and Bernstein, R. B. (1971). J . Chem. Phys. 55, 4661. Gersh, M. E.. and Bernstein, R. B. (1972). J. Chem. Phys. 56, 6131. Gillen, K. T., Rulis, A. M., and Bernstein, R. B. (1971). J . Chem. Phys. 54, 2831. Gordon, R. J., and Lin, M. C. (1973). Chem. Phys. Leu. 22, 262. Gordon, R. J . , and Lin, M. C. (1976). J . Chem. Phys. 64, 1058. Gray, J . C . , Truhlar, D. G., Clemens, L., Duff, J . W., Chapman, F. M., Morrell. G. O., and Hayes, E. F. (1978). J . Chem. Phys. 69, 240. Grice, R. (1975). Adv. Chem. Phys. 30, 247. Grice, R., and Herschbach, D. R . (1974). Mol. Phys. 27, 159. Haberman, J. A., Anlauf, K. G., Bernstein, R. B., and Van Itallie, F. J. (1972). Chem. Phys. L e t / . 16, 442. Halavee, U., and Shapiro, M. (1976). J . Chem. Phys. 64, 2826. Hayashi, S., Mayer. T. M., and Bernstein, R. B. (1978). Chem. Phys. Left. 53, 419. Hepburn. J . W., Klimek, D.. Liu, K., Polanyi. J . C.. and Wallace, S . C. (1978). J. Chem. Phys. 69, 43 1 1 . Herm, R. R. (1979). In “The Alkali Halide Vapors” (P. Davidovits and D. L. McFadden, eds.), p. 189. Academic Press, New York. Herschbach, D. R. (1973). Furuduy Discuss. Chem. Soc. 55, 233. Herschbach, D. R. (1976). Pure Appl. Chem. 47, 61. Herschbach, D. R. (1977). Furuduy Discuss. Chem. Soc. 62, 162. Homann, K. H . , Solomon, W. C., Warnatz, J., Wagner, H. G., and Zetzsch, C. (1970). Ber. Bunsenges. Phys. Chrm. 74, 585. Hubers, M. M., Kleyn, A. W., and Los. J . (1976). Chem. Phys. 17, 303. Kaplan, H., Levine, R. D., and Manz. J . (1976). Chem. Phys. 12,447. Karny. Z . , and Zare, R. N. (1978). J . Chem. Phys. 68, 3360. King. D. L., Dixon, D. A., and Herschbach, D. R. (1974). J. Am. Chem. Soc. 96,3328. Kinsey. J . L. (1972). Phys. Chem., Ser. One, 9, 173. Kinsey, J . L. (1977). Annu. Rev. Phys. Chem. 28, 349. Komornicki. A.. George, T. F., and Morokuma, K. (1976). J . Chem. Phys. 65, 48. Krause, H. F., Johnson, S . G., Datz, S., and Schmidt-Bleek, F. K. (1975). Chem. Phys. L e / / . 31, 577. Krenos, J. R.. and Tully, J. C. (1975). J. ,Chem. Phys. 62, 420. Krenos, J. R., Preston, R. K., Wolfgang, R.. and Tully, J. C. (1974). J. Chem. Phvs. 60, 1634. Kuntz, P. J . (1976). In “Dynamics of Molecular Collisions” (W. H. Miller, ed.), Part B, p. 53. Plenum. New York. Kuntz, P. J. (1979). In “Atom-Molecule Collision Theory: A Guide for the Experimentalist” (R. B. Bernstein, ed.), p. 79. Plenum, New York. Kuntz. P. J.. and Roach, A . C. (1972). J. Chem. Soc.. Furuduy Truns. 2 68, 259. Kuntz, P. J., and Roach, A. C. (1973). J. Chem. Soc., Furuduy Truns. 2 69, 926. Kuppermann, A., and Schatz, G. C. (1975). J. Chem. Phys. 62, 2502. Kwei, G. H. (1979). In “The Alakli Halide Vapors” (P. Davidovits and D. L. McFadden, eds.), p. 441. Academic Press, New York. Lawley, K. P., ed. (1975). “Molecular Scattering: Physical and Chemical Applications,” Adv. Chem. Phys., Vol. 30. Wiley, New York. Lee, Y. T. (1972). In “Physics of Electronic and Atomic Collisions” (T. R. Govers and F. J . deHeer, eds.), p. 357. North-Holland Publ.. Amsterdam. Lee, Y. T.. Gordon, R. J., and Herschbach, D. R. (1971). J. Chem. Phys. 54, 2410. Leone, S. R., McDonald, R. G., and Moore. C. B. (1975). J . Chem. Phys. 63,4735. Levine, R. D. (1972). Theor. Chem., Ser. One 1, 229.
REACTIVE SCATTERING: RECENT ADVANCES
20 I
Levine, R . D. (1977). In "State-to-State Chemistry" (P. R. Brooks and E . F. Hayes, eds.). ACS Symp. Ser. No. 56. p. 226. Am. Chem. SOC..Washington. D.C. Levine, R. D. (1978). Annu. Rev. f h y s . Chrm. 29, 59. Levine, R. D. (1979). In "Maximum Entropy Formalism" (R. D. Levine and M. Tribus, eds.). MIT Press, Cambridge, Massachusetts. Levine, R. D., and Ben-Shaul. A. (1977). In "Chemical and Biochemical Applications of Lasers" (C. B. Moore, ed.), Vol. 2. p. 145. Academic Press. New York. Levine. R. D.. and Bernstein. R. B. (1971). Chem. fhys. Lett. 11, 552. Levine, R. D., and Bernstein. R. B. (1972). J. Chetn. f h y s . 56, 2281. Levine, R. D., and Bernstein, R. B. (1974a). "Molecular Reaction Dynamics." Oxford Univ. Press (Clarendon). London and New York. Levine. R. D.. and Bernstein, R. B. (1974b). A r c . Chum. Res. 7, 393. Levine, R. D.. and Bernstein, R. B. (1976). I n "Dynamics of Molecular Collisions" (W. H. Miller. ed.), Part B, p. 323. Plenum, New York. Levine, R . D.. and Kinsey, J. L . (1979). In "Atomic-Molecule Collision Theory: A Guide for the Experimentalist" (R.B. Bernstein, ed.), p. 693. Plenum, New York. Levine, R . D.. and Manz. J . (1975). j? Chem. f h y s . 63,4280. Levine. R. D., Johnson, B. R.,and Bernstein. R. B. (1973). Chem. f h y s . L e u . 19, 1. Light. J . C. (1979). In "Atom-Molecule Collision Theory: A Guide for the Experimentalist" (R.B. Bernstein, ed.), p. 467. Plenum, New York. Lin. S.-M.. Mims, C . A . , and Herm, R. R. (1973). J . Chem. f h y s . 58, 327. Litvak, H. E.. Ureha, A. G.. and Bernstein, R. B. (1974). J. Chem. f h y s . 61, 738. Liu. B. (1973).J. Chem. f h y s . 58, 1925. Liu. K.. and Parson, J. M. (1977). J . Chem. Phys. 67, 1814. Los,J. (1973). f r o c . In!. Cottf: f h y s . Elcctvon. A t . Cnllisions, Brh, 1973. Invited Lectures. Los, J . , and Kleyn, A. W. (1979). In "Alkali Halide Vapors" (P. Davidovits and D. L. McFadden, eds.), p. 275. Academic Press, New York. Love. R. L., Herrmann, J. M.. Bickes, R. W., Jr., and Bernstein, R. B. (1977). J . A m . Chem. Soc. 99, 8316. McClure, G . W., and Peek, J. M. (1972). "Dissociation in Heavy Particle Collisions." Wiley (Interscience), New York. McDonald. J . D.. and Herschbach, D. R . (1975). J . ChrJm.Phys. 62, 4740. McDonald, J. D., LeBreton, P. R., Lee, Y. T., and Herschbach, D. R. (1972). 1.Chem. f h y s . 56, 769. McGuire, P., and Kouri, D. J. (1974). J . ChtJtn.f h y s . 60, 2488. Manos, D. M.. and Parson, J. M. (1978). J. Chein. fhys. 69, 231. Mascord, D. J., Gorry, P. A., and Grice. R. (1977). Fnwday Discuss. Chem. Soc. 62,255. Massey. H. S. W. (1971). In "Electronic and Ionic Impact Phenomena" (H. S. W. Massey, E. H. S . Burhop, and H. B. Gilbody. eds.), 2nd ed., Vol. 3, "Slow Collisions of Heavy Particles." Oxford Univ. Press (Clarendon), London and New York. Mayer, T. M., Wilcomb, B. E . . and Bernstein. R . B. (1977a). J . Chem. f h y s . 67, 3507. Mayer. T.M.. Muckerman. J. T.. Wilcomb, B. E . . and Bernstein. R. B. (1977b). J. Chem. f h y s . 67, 3522. Micha, D. A. (1975). A h . Chem. Phys. 30,l. Miller, W. H., ed. (1976a). "Dynamics of Molecular Collisions," Parts A and B. Plenum. New York. Miller. W. H. (1976b). J . Chem. Phys. 65, 2216. Miller. W. H. (1978). J. Chem. f h y s . 68,4431. Miller. W. H., and George. T . F. (1972). J. Chrtn. Phys. 56, 5637. Muckerman. J. T . (1971). J. Chem. f h y s . 54, 1155.
202
Richard B . Bernstein
Muckerman, J . T. (1972a). J. Chem. Phys. 56, 2997. Muckerman, J . T. (1972b). J. Chem. Phys. 57, 3388. Muckennan, J . T. (1979). In preparation. Mukamel, S .. and Ross. J. (1977). J. Chem. Phys. 66, 3759. Nesbet, R. K. (1976). Chem. Phys. Lett. 42, 197. Nikitin, E. E. (1974). “Theory of Elementary Atomic and Molecular Processes in Gases.” Oxford Univ. Press (Clarendon). London and New York. Ochs. G.. and Teloy, E. (1974). J. Chem. Phys. 61,4930. Odiorne. T. J . , Brooks, P. R., and Kasper, J. V. (1971). J . Chem. Phys. 55, 1980. Olson. R. E.. Smith, F . T., and Bauer, E. (1971). Appl. O p t . 10, 1848. Ottinger, C., and Zare, R. N. (1970). Chem. Phys. L e f t . 5, 243. Pace, S. A., Pang, H. F., and Bernstein, R. B. (1977). J. Chem. Phys. 66, 3635. Pack, R. T. (1974). J. Chem. Phys. 60, 633. Pang, H. F., Wu, K. T., and Bernstein, R. B. (1978). J . Chem. Phys. 69, 5267. Parr, T . P., Behrens, R., Freedman, A.. and Herm, R. R. (1978). Chem. Phys. L e f f .56.71. Parrish. D. D., and Herschbach, D. R. (1973). J. Am. Chem. Soc. 95, 6133. Parson. J. M . . Shobatake, K., Lee, Y. T., and Rice, S. A. (1973). Faraday Discuss. Chem. Soc. 55, 344. Pechukas, P. (1976). In “Dynamics of Molecular Collisions” (W. H. Miller, ed.), Part B. p. 269. Plenum, New York. Polanyi. J. C. (1972). Acc. Chem. Res. 5, 161. Polanyi. J . C. (1973). Faraday Discuss. Chem. Soc. 55, 389. Polanyi, J. C . , and Schreiber, J. L. (1974). Phys. Chem. 6A, 383. Polanyi, J. C., and Schreiber, J. L. (1977). Furaduy Discuss. Chem. Soc. 62, 267. Polanyi. J . C . , and Schreiber, J. L. (1978). Chem. Phys. 31, 113. Polanyi, J. C . , and Woodall. K. B. (1972).J. Chem. Phys. 57, 1574. Pollak, E.. and Pechukas. P. (1978). J . Cheni. Phys. 69, 1218. Pollak, E., and Pechukas, P. (1979). J . Chem. Phys. 70, 325. Preston. R. K . . and Tully, J. C. (1971). J . Chem. Phys. 54, 4297. Pruett. J . G.. and Zare. R. N. (1976).J. Chem. Phys. 64, 1774. Pruett, J . G., Grabiner, F. R.. and Brooks, P. R. (1975). J. Chem. Phys. 63, 1173. Raf€, L. M., Suzukawa, H. H., and Thompson, D. L . (1975). J . Chern. Phys. 62, 3743. Redmon. M. J . , and Wyatt, R. E. (1975). I n t . J . Quantum Chem., Symp. 9,403. Redpath. A. E., and Menzinger, M. (1971). Can. J , Chem. 49, 3063. Redpath. A. E.. and Menzinger. M. (1975). 1.Chem. Phys. 62, 1987. Redpath. A . E., Menzinger, M., and Carrington. T. (1978). Chem. Phys. 27,409. Roach, A. C. (1970). Chem. Phys. L e f f .6, 389. Roach, A . C . (1977). Faraday Discuss. Chem. Soc. 62, 151. Schaefer, H. F. (1979). In “Atom-Molecule Collision Theory: A Guide for the Experimentalist” (R. B. Bernstein, ed.), p. 45. Plenum, New York. Schafer, T. P., Siska, P. E., Parson, J. M.. Tully. F. P., Wong, Y . C.. and Lee, Y. T . (1970). J. Chem. Phys. 53, 3385. Schatz. G. C . . and Kuppermann, A. (1976). J. Chem. Phys. 65,4642 and 4668. Schatz, G. C . . and Ross, J. (1977). J. Chem. Phys. 66, 1021 and 1037. Sheen. S. H . , Dimoplon, G., Parks, E. K., and Wexler, S. (1978). J. Chem. Phys. 68,4950. Smith, G . P.. and Zare, R. N . (1976). J. Chem. Phys. 64, 2632. Sridharan, U. C., DiGiuseppe, T. G . , McFadden, D. L., and Davidovits. P. (1978). Chem. Phys. Lett. 59, 43. Stolte, S . , Proctor, A. E., and Bernstein, R. B. (1975). J. Chem. Phys. 62, 2506. Stolte, S . , Proctor, A. E., and Bernstein, R. B. (1976). J. Chem. Phys. 65, 4990.
REACTIVE SCATTERING: RECENT ADVANCES
203
Stolte, S.. Proctor. A. E.. Pope, W. M.. and Bernstein, R. B. (1977). J . Chem. Phys. 66, 3468. Sverdlik, D. I., and Koeppl, G. W. (1978). Chem. Phys. Lett. 59, 449. Tang, S. P.. Utterback, N. G.. and Friichtenicht. J. F. (1976). J . Chem. Phys. 64, 3833. Toennies. J . P. (1974). Phys. Chem. 6A, 228. Truhlar, D. G.. and Dixon, D. A. (1979).In "Atom Molecule Collision Theory: A Guide for the Experimentalist" (R. B. Bernstein, ed.), p. 597. Plenum, New York. Truhlar. D. G., and Kuppermann, A. (1972). J. Chem. Phys. 56, 2232. Truhlar, D. G., and Muckerman, J. T . (1979). I n Atom-Molecular Collision Theory: A Guide for the Experimentalist" (R. B. Bemstein, ed.), p. 505. Plenum, New York. Truhlar. D. G., and Wyatt, R. E. (1976). Annu. R r v . Phys. Chem. 27, I . Truhlar, D. G.. Kuppermann, A., and Adams. J . T. (1973). J. Chem. Phys. 59, 395. Tully, J. C . (1976). In "Dynamics of Molecular Collisions'' (W. H. Miller, ed.). Part B, p. 217. Plenum, New York. Tully, J . C. (1977). In "State-to-State Chemistry" (P. R. Brooks and E. F. Hayes. eds.). ACS Symp. Ser. No. 56, p. 206. Am. Chem. SOC..Washington, D.C. Tully. J. C.. and Preston, R. K. (1971). J. Chern. Phys. 55, 562. Ureiia, A. G., and Aoiz, F. J . (1977). Chem. Phys. L e f t . 51,281. Valentini, J. J., Coggiola, M. J., and Lee, Y. T. (1976). J. A m . Chem. Soc. 98, 853. Valentini. J. J., Coggiola, M. J., and Lee, Y. T. (1977). Faraday Discuss. Chrm. Soc. 62, 232. Van der Meulen, A., Rulis, A. M., and DeVries, A. E. (1975). Chrm. Phys. 7, 1. Wexler. S. (1973). Bur. Bunsenges. Phys. Chivn. 77,606. Whitehead, J. C., Hardin, D. R., and Grice, R. (1973). M o l . Phys. 25, 515. Wicke, B. G., Tang, S. P., and Friichtenicht, J. F. (1978). Chem. Phys. Lett. 53, 304. Wilcomb. B. E., and Dagdigian. P. J . (1978). J . Chem. Phys. 68, 3990. Wilcomb. B. E.. Mayer, T. M., Bemstein. R. B.. and Bickes. R. W.. Jr. (1976). J. A m . Chem. Soc. 98, 4676. Wong, Y. C.. and Lee, Y. T. (1973). Furadoy Discuss.Chem. Soc. 55, 383. Wren, D. J., and Menzinger, M. (1974). Chrrn. Phys. Lett. 25, 378. Wu, K. T.. Pang. H. F.. and Bernstein. R. B. (1978). J . Chem. Phys. 68, 1064. Wyatt, R. E. (197.5).Chem. Phys. Lett. 34, 167. Wyatt, R . E . (1977). In "State-to-State Chemistry" (P.R. Brooks and E. F. Hayes, eds.), ACS Symp. Ser. No. 56, p. 185. Am. Chern. Soc.. Washington, D.C. Wyatt, R. E. (1979). In "Atom-Molecule Collision Theory: A Guide for the Experimentalist" (R. B. Bernstein, ed.). p. 567. Plenum, New York. Yokozeki. A , , and Menzinger, M. (1977). Chrm. Pliys. 22, 273. Zandee. L..and Bernstein, R. B. (1978). J. Chem. Phys. 68, 3760. Zare, R. N. (1974). Ber. Bunsenges. Phys. C h r m . 78, 153. Zare. R. N.. and Dagdigian. P. J . (1974). Science 185,739. Zein. Y.,and Shapiro, M. (1978). Chem. Phys. 31, 217. Zvijac. D. J.. and Ross. J . (1978). J. Chem. Phys. 68, 4468. "
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A D V A N C E S IN ATOMIC A N D MOI.FCUL.AR PHYSICS. V O L
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ION-ATOM CHARGE TRANSFER COLLISIONS AT LOW ENERGIES J . B . HASTED Birhbech College Univcnitj of' L o t h n
London, Eriglond
I . Introduction.. . . . . . . . . . . . . .
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11. Symmetrical Resonance Proc ............................. 111. Nonresonant Atomic Charge Transfer Processes ... IV. Total Cross Sections of Pseudocrossing Atomic Charge Transfer
205 206 .
211
Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 I4 V . Curve-Crossing Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 V1. Charge Transfer Processes of Excited Ions . . . . . . . . . . . . . . . . . . . . . . . 229 ........................................... 231 References
I. Introduction At impact energies not in excess of one atomic unit of velocity. ionatom charge transfer collisions are a powerful technique for study of ion-atom interaction energies. Our increasing understanding of the collision processes has come about through increasingly sophisticated beam-gas and crossed-beam experiments, the data from which are compared with fully quanta1 and semiclassical impact parameter calculations. It is on comparisons with the latter that we shall concentrate in this chapter. Semiquantitative agreement with measurements can be obtained with the use of simple wavefunctions and interaction energies; parametric representation of the cross-section functions is sometimes possible. Distinctions can be drawn between symmetrical resonance processes, nonresonance processes in which transitions proceed at pseudocrossings of potential energy curves. and nonresonance processes in which this is not the case. In the interference oscillations present in the scattering functions there is much information about both adiabatic and nonadiabatic potential energy curves. 205 CoDrrlehl . . - ti) - I979 hk Academic R e \ \ In' All right, of reproduction in an) fi,rrn re\erved ISBN 0- I?-OO3XI?-3
206
J . B. Husted
11. Symmetrical Resonance Processes The symmetrical resonance charge transfer process has always been regarded as a particularly simple and fortunate example of how quantum theory can be applied with good precision to an inelastic collision process, ever since Massey and Smith (1933) first showed that total cross sections are largely controlled by the long-range interactions between the species. Not only can an impact parameter formulation be used without appreciable loss of accuracy, but the interactions can be computed simply, using asymptotic theories. The probability of transition P(u, 6 ) at impact velocity u and impact parameter h can be expressed in terms of the phase shifts r), and r), for scattering along the gerade and ungerade potential curves of the ion-atom system:
P(u, b)
=
sin2(r), -
vg)
(1)
Using the JWKB approximation the phase can be expressed in the form
(cf. Massey and Smith, 1933). Here R,, is the closest distance of approach of the nuclei, E the impact energy, V ( R )the interaction potential, and p the reduced mass. The initial development of computations of total cross sections involved a number of approximations that avoided the necessity of using the entire collision path for the calculation. The probability of transition oscillates between zero and unity for impact parameter values less than a value bo, outside of which it falls off exponentially. Without great loss of accuracy, P ( b , v ) can be replaced by its mean value of 0.5 for b =sb,, and taken to be zero for b > b,; then the cross section c = &ab2,,where ho is defined to be the highest impact parameter at which P assumes a certain value P o ; this has been taken to be either 0.1 (Firsov, 1961), 0.25 (Rapp and Francis, 1%2), or 0.075 (Smirnov, 1964, 1965a,b). When collision trajectories are taken to be rectilinear, the calculated cross section is found to depend upon impact velocity u in a very simple fashion: (+11’
= A - B In u
(3)
where A and B are constants both of which show a strong dependenceLon the ionization potential of the atom.
ION-ATOM
CHARGE TRANSFER COLLISIONS AT LOW ENERGIES
207
Although the existence of fast computers might lead us to suppose that = hbZ, is today an unnecessarily simple approximation, yet the comparisons with experimental data have shown a surprising degree of success (Hasted, 1%8) when the ionization potential is not much smaller than 1 R. But with decreasing ionization potential the older calculations increasingly underestimate the constant A , although in view of discrepancies between different measurements no such unqualified statement can be made about the constant B . The principal reason for this has recently been shown by Hodgkinson and Briggs (1976) to be not so much the neglect of the region outside h, as the use in earlier calculations of simple exponential wavefunctions. When these are replaced by functions suggested by quantum defect theory (Seaton, 1958), the agreement with alkali metal experiments is very much improved. as can be seen from the example in Fig. 1. Another important factor is the neglect in the earlier calculations of atomic electrons of higher angular momentum than I = 0. These give rise to 2 , n, A, . . . , etc., molecular states, and the energy splitting and charge transfer between such states as ngand nuis different from that between C, and Xu.When the calculations are generalized and averaged over the approu
-
v ( c m /sed FIG. 1. Symmetrical resonance charge transfer cross-section function ti) for Cs+Cs. Full line, Hodgkinson and Briggs (1976) calculations; lower full line. Perel et a / . (1970) experiments; dashed line, Marino el a / . (1962) experiments and Marino (1966) experiments; dash-dot line, Chkuaseli er ol. (1960) experiments; dotted line. Rapp and Francis (1962) calculations; 0, Gentry et a / . (1968) experiments; Kushnir et a / . (1959) experiments and Kushnir and Buchma (1960); A, Perel ct a / . (1965) experiments.
.,
priate values of magnetic quantum number rn, the cross sections are found to be smaller than those for the single m = 0 companent. The effect of rotational coupling between states of different values of tn is not considered, since most of the total cross section is made up of contributions from large nuclear separations where the angular velocity is small, so that the rotation of the internuclear vector has little effect (Johnson, 1970). Figure 2 shows a comparison of various calculations for a rare gas, Kr+Kr, chosen for two reasons. First, the ion cyclotron resonance data (Smith and Futrell, 1973) have usually been considered to be anomalously high. But recent injected ion drift tube data (Okuno ('I d., 1978) are in reasonable agreement with the calculations, as is the case for other modern rare gas data. The second interesting feature of the krypton drift tube data is that despite their extension down to thermal energies the effects of nonrectilinear collision trajectories are still not apparent. Departure from rectilinear collision trajectory is to be expected in thermal and suprathermal energy collisions. It has been realized for more than a decade that sufficiently slow charge transfer collisions should be limited by the Langevin orbit rather than by the hb2, limit, and an attempt was made by Wolf and Turner (1967) to interpret this idea quantitatively. Detailed solution of the
12
t
-
lo "5 1 8 b
6
-
4
lo6 v (cmkec)
lo7
FIG.2. Symmetrical resonance charge transfer cross section function U ' ' ~ ( I Jfor ) Kr+Kr. Hodgkinson and Briggs (1976) calculations: Rapp and Francis (1962) calculations: 0.Smith and Futrell (1973) ion cyclotron resonance experiments: +. Okuno P I a / . (1978) Flaks and Solovev (1958) experiments; experiments; A , Dillon P I ul. (1955) experiments; 0, H, Kushnir P I n l . (1959) experiments; ..., Langevin orbiting limit.
-,
---.
ION-ATOM
CHARGE TRANSFER COLLISIONS AT LOW ENERGIES
209
phase shift equations over nonrectilinear trajectories (Sinha and Bardsley, 1976) confirm the prediction that there is an intermediate region ~ and the Langevin limit of energy in which both the ( A - B in v ) behavior are exceeded. But because the laboratory impact energies are too high, the experimental data fail to demonstrate this behavior. Because symmetrical resonance charge transfer cross sections are very large and rare gas polarizabilities are not, the impact energy for which Langevin orbiting cross sections exceed them is unexpectedly small, and is in fact well below thermal energy for Kr+Kr. For some years it has been known that there are cases of oscillations in total symmetrical resonance cross-section functions (Perel et a / ., 1965). When the cross section is plotted as a function of inverse impact velocity, these oscillations are regular, as appears in Fig. 3. The oscillations were shown by Smith (1966) to arise from a failure of the energy difference between the two interaction potentials V , and V , to increase monotonically with decreasing nuclear separation. When there is a maximum in the energy difference function, a region of stationary phase difference and oscillations is found. However, the oscillations are not always sufficiently large, nor are they always at sufficiently high impact velocity to be detectable by experiment; on occasion (Latypov ef a / . , 1970) spurious structure has been reported. Alkali metal ion collisions are the most important instances of oscillating behavior, and the Li+Li collision illustrated in Fig. 3 I
I
I
I
I
120
b
100
8
12
..
-
16
v-'(a.u.)
i 20
FIG.3 . Li+Li charge transfer cross section as a function of inverse velocity; U, calculations of Sinha and Bardsley (1976); calculations of McMillan (1971); A , experimental data of Perel et crl. (1965).
210
J . B. Hasted
shows regularly spaced oscillations both in experiment (Perel er af., 1970) and calculation (McMillan, 1971; Sinha and Bardsley, 1976). But even the most accurate LCAO calculations (Hodgkinson and Briggs, 1976) fail to reproduce the frequency of the oscillations at all well. According to Smith, accounting for the constructive or destructive interference in the region of the maximum separation, where the random phase approximation breaks down, modifies the total cross section by superimposing an oscillatory structure, according to the formula
where b, is the position of the stationary point. The He+He collision, for which oscillations were predicted at extremely low impact velocity, demonstrates only the recognized (A - B In v ) behavior ~ in the latest beam experiments at higher energies (Hegerbergef af., 1978), as is shown in Fig. 4. The difficulties inherent in E (keV) 1
2
3.4
7
4
10
3.2
3.0
2.8 \ N
-b
2.6
2.5
I 3
I 4
I
I
1
5
6
7x10'
v ( c m sec-')FIG.4. He+ He symmetrical resonance charge transfer cross sections m*'* as a function of impact velocity. Experimental data, points fitted by a line, of Hegerberger (11. (1978); D. calculated from theory of Rapp and Francis (1962). as corrected by Dewangan. B. calculated using interaction potentials of Bardsley er a / . (1975); MS,calculated using interaction potentials of Marchi and Smith (1965); ---, calculations of Hodgkinson and Briggs (1976).
ION-ATOM
CHARGE TRANSFER COLLlSlONS AT LOW ENERGIES
21 I
accurate gas pressure determination have been reflected in the scatter that prevails in the many single-collision condition measurements of this cross section; but there is now a difference of only 2% between the two most recent (Hegerberg et al., 1978; Eisele and Nagy, 1976) sets of experimental data. Two sets of interaction potentials (Marchi and Smith, 1965; Sinha and Bardsley, 1976) bracket the data, and the calculations of Hodgkinson and Briggs (1976) lie closer still. Part of the difficulty in calculating interaction energies lies in the computation of the exchange splitting (Bardsley er a / . , 1975), which contributes an appreciable term to the long-range polarization interaction. The variation of sign of this term has presented difficulty (Bardsley et a / . , 1975; Duman and Smirnov, 1970; Johnson, 1970). For rare gas symmetrical resonance charge transfer, the important issue of the fine-structure splitting has recently been investigated using ionic mobility measurements. Mobility measurements are less subject to systematic errors than beam charge transfer measurement, because the Torr) determination as well as difficulties of accurate low-pressure (those of total charge collection are avoided. Accurate measurement of length, time, and high pressures (-- I Torr), together with analysis of diffusion, are all that is necessary for a drift velocity measurement. The momentum transfer cross section, which determines the zero-field limit of the mobility, can be taken to be very close to twice the charge transfer cross section, since the momenta transferred in inelastic and elastic scattering are almost identical, although the charge locations are contrasted. It has been reported (Helm, 1975) that the mobility of the metastable Krf2P1,2ion in krypton is 3.3 rt 0.2%, higher than that of the Kr+2P3,2 ground-state ion. This is smaller than the 10% difference reported both in beam measurements (Hishinuma, 1972) and calculations (Kimura and Watanabe, 1971; Johnson, 1972) at higher energies. In the low limit of energy the Langevin limiting cross section depends only on the atomic polarizability, so that the difference should be zero. Calculations (Sinha and Bardsley. 1976) yield a difference of 1.8%, which may not be significant. For Xe+ both experiment and calculation yield about 6% difference, although the individual cross sections differ by about the same amount.
111. Nonresonant Atomic Charge Transfer Processes During the 1970s further experimental data on total cross-section functions for nonsymmetric charge transfer have been accumulated. It is possible to classify these as ( 1 ) processes in which the input and output
J . B . Hasted
212
channel nonadiabatic potential energy curves are separated from each other in energy, and (2) pseudocrossing potential energy curve processes. The latter are the most commonly occurring, and we shall see in the next section that one simple assumption is that of a linear crossing of the two curves, together with a constant coupling-matrix element H 1 2 . In the former processes the simplest approximation is that the two potential energy curves are parallel to each other, with the coupling matrix element having the exponential form H 1 2 ( R )= exp(- AR)
(5)
with A adjustable. Models of noncrossing situations were considered by Stuckelberg (1932), Rapp and Francis (1962), and Nikitin (1961; also see Nikitin and Ovchinnikova, 1970). Bykhovskii and Nikitin’s (1965) formulation covers both noncrossing and pseudocrossing processes. One of the best known noncrossing formulations has been that of Demkov (1964), following Rosen and Zener (1932), who supposed that the transfer of charge takes place at a nuclear separation R, when the coupling matrix equals one-half of the difference between the intermolecular potentials: IHdRII
!ilVi(Rc)
=
-
VARAI = !dAV(Rc)I
(6)
This leads to an expression for the transition probability P at impact parameter b: P = sech2
(+V(RC)l 2AY(R,) ) sin2 (!-n 1-H12dr) --m
(7)
but see also Melius and Goddard, 1974). One can approximate the sin2 term by 4, which is quite accurate for b < R , , but neglects the region b 2 R,. Typical calculations of the oscillating transition probabilities are illustrated in Fig. 5; it becomes clear from inspection just how great the h R , 2 underestimation of the total cross section is likely to be. Some comparisons of measured total cross sections at maximum with Demkov approximation are made in Table I. For impact parameter formulations such as this, it is necessary to compute the collision paths by a classical scattering equation, using previously assumed interaction energies, taken to be of the form 11
-
22
-
H
c1 . -hEm - .2. +
Aa ? C+ 2 ~ 4 R=
...
(9)
ION-ATOM
C H A R G E TRANSFER C O L LI S I O NS AT LOW ENERGIES
b (a,)
2 I3
-
FIG.5. Transition probability P as a function of impact parameter calculated (Olson, 1972) for the following situation: H I 2 = e-m, 4, = 0.30;' V ( R , ) = 0.0 a.u.. R, = 15.350,
-.
Calculated from equation; ---, calculated with the sinz term approximated by
1.
where C1,2are van der Waals constants and A a is the difference between the polarizabilities a l , a2of the two atoms. Additional R-I2 terms may be added, and where both systems carry ionic charge there is a Coulomb R-' term and also a term - 2 a l a 2 / R 7 . It is of interest to note that the Demkov approximation can be used to derive algebraically (Olson, 1970) an adiabatic maximum rule of form rather similar to that originally used (Hasted, 1951, 1%0) as a test of the Massey adiabatic criterion (Massey, 1949). The equation is derived in the form u,,,
=
alPE,J/E/'2
(10)
TABLE I CALCULATED A N D MEASURED TOTALCROSSSEcrioNs
NaLi+ S Na'Li RbK+ S Rb+K Li+K + Li(2p)
cmZ) AT M A X I M U M
Calc.
Exp.
Reference
5.5
6.4 9.5 5.4
Daley and Perel (1969) Perel and Yahiku (1967) Daley and Perel (1969)
7.5 3.1
214
J . B . Hasted
I A E I / E i ' (eV)'
-
FIG.6. Variation of impact velocity of maximum cross section with IhE,I/E,"2 (Perel and Daley. 1971). I . Ar+Rb: 2. K+Rb. Rb+K;3 . Hg+Cs;4 . Na+Li; 5 . Li+Na; 6. He+Cs;7 . Cs+Rb, Rb+Cs; 8 . Li+Cs; 9. K+Cs; 10. H+Cs; 1 1 . Li+Rb; 12. K+Na; 13. Na+K; 14. Li+K.
with energy defect AE, and mean ionization potential Ei both in electron volts; the impact velocity om,, at which the cross section has maximum value in cm/sec; and the adiabatic parameter a numerically equal to 14.5 x 107. In this derivation it is assumed that
AE,
= AV(R,J
X = (Ei/13.6)1/2
(1 1 )
(12)
An exacting test of the adiabatic maximum rule is with the collected experimental data for alkali metal charge transfer. This rule is illustrated in Fig. 6, from which it is seen that there are two asymptotic slopes of the ~ I , , , ~ ~ ( ~ A Egraph ~ ~ / Egiving ~ ' ~ )two values of a, equal to 22 x lo7 and 8 x lo7. Another useful feature of the Demkov two-state approximation is that it can be used to interpret quantitatively the near-thermal energy minima, which are observed in many atomic ion charge transfer functions with diatomic molecules (Birkinshaw and Hasted, 1971).
IV. Total Cross Sections of Pseudocrossing Atomic Charge Transfer Processes The role of avoided crossings in charge transfer processes is now reasonably well understood quantitatively in the simplest cases where a single avoided crossing of potential energy curves makes a dominant con-
I O N - A T O M CHARGE TRANSFER COLLISIONS AT LOW ENERGIES
2 I5
tribution to the cross section. The more usual situation, when several or even many crossings must be taken into account, together with possible contributions from noncrossing regions, is less well understood. The most powerful method of studying crossings is by means of the double differential measurements now termed “curve-crossing spectroscopy” (Section V). But it is also important to know how far total charge transfer cross section functions can be interpreted quantitatively on the avoidedcrossing model. The earliest application of the Landau-Zener-Stuckelberg model to single-electron capture processes resulting in Coulomb-dominated products was made by Bates and collaborators more than 20 years ago (Bates and Moiseiwitsch, 1954; Dalgarno, 1954; Bates and Lewis, 1955; Bates and Boyd, 1956). The probability P, of its occurrence is proportional to the exponent w, as follows: P , = exp(-w) 7r
H1Z2(Rx)
4R,)
Iv; - v;1
w’-
(13) R=R,
(14)
where p is the reduced mass of the system, V’ denotes aV/aR, and R , is the separation at the avoided crossing. The total cross section is (T
= 47rR:I(r))
where
with p in amu and the energies, including the impact energy E, in eV. Since the integral I(?) is known to maximize at 0.113 when r) = 0.424 (Boyd and Moiseiwitsch, 1957). Eqs. (15)-(17) demonstrate that the cross section function m(E) passes through a maximum; the energy at which this maximum occurs can be used to calculate an empirical HI2.Quanta1 calculation of ~
1
2
14iv+f
(18)
using initial and final atomic eigenfunctions C$ is straight-forward for simple atomic systems (see, however, Janev, 1976), but unsatisfactory in such cases as heavy-metal vapors, for which the empirical H I 2 is useful.
.I. B . Hasted
216
An asympotically exact determination of H l z can be made by the Landau-Herring method (Landau and Lifschitz, 1963; Herring, 1962), which has been applied both to molecular systems and to two-electron exchange. An attempt was made by Hasted and Chong (1962) to parametrize the dependence of the empirical H12upon R , using experimental data and computations. The most recent empirical parametrization, due to Olson er a / . (1971) is
HT2 = 0.662 exp( - 0.858R:)
(19)
where
R:
( a - y)Rx
(20)
H12 = 2Hlz/aYRX
(21)
=
with a2/2 the lower ionization potential of the reactants and y 2 / 2 the lower ionization potential of the products. A comparison of HTz values with the empirical Eq. (16) is given in Fig. 7. The failure of nonresonance thermal energy cross sections to increase monotonically with decreasing A E m is clearly shown in the experiments and calculations of Turner-Smith e f a / . (1973), illustrated in Fig. 8 . The "near-resonance'' principle, which dates from the 1930s, is finally dead.
'....
\
i
..
,
1 IX
\ \\
,5p'FIx
'..
I
J
'..'.... i , J .I....
1
2
3
I It
m5flS! 5f
7d,
,...'..
1x
\
4
j
I
5
6
5!
I
%ASS
7
8
9
-
AV ( 1 o 3 C ~ ' )
m5dFI):
j
I 10
11
FIG.7. Variation of statistically weighted thermal energy cross section with energy defect (AC,wavenurnbers) for metal-ion laser charge transfer processes (Turner-Smithet al., 1973). .-.,He+Cd, Ne+Mg,calculated using Landau-Zener approximation with empirical H , 2 .
---.
ION-ATOM
CHARGE TRANSFER COLLISIONS AT LOW E N E R G I F S
,
10"
1
"
'
'
~
1
2 I7
'
\
10'
1
-
104
?
0
Y
* N
1-
loe
lo-'" 0
4
8
12
16
20
24
28
Rf (au.)
FIG.8 . Empirical variation (Olson er l i / . , 1971) with H: o f values ofHT, calculated either from the maxima in cross-section functions using Eqs. (13)-(15) or from Eq. (19).
An important problem is that of assessing the accuracy of the Landau-Zener approximation in calculating total cross sections. Not too much data are available for making this assessment, partly because it is assumed in the theory that only a single s electron makes the transition; in many collision processes other possibilities of transition must also be considered. Helium or atomic hydrogen are the most suitable targets for this assessment; single-electron capture by a doubly charged ion presents Coulomb-dominated product potential energy curves that ensure an avoided crossing at a suitable nuclear separation. The process B3+ + He
-
B2+ + He+
'
(22)
is one in which the excited-state channels can be distinguished clearly from the ground-state channel: it has been studied by Zwally and Cable (1971) and the comparison of the experiment and Landau-Zener caiculation is given in Fig. 9. Only two channels contribute to the process, and H I 2calculations were made using Morse wavefunctions. It is seen that the measured cross section function, for which the error is given rather conservatively as *7%, exceeds the calculated values by a factor of almost two. Other tests of Landau-Zener calculations show a similar overesti-
J . B . Hcisted
218
-
(EXP) v i ~ ~ ' c r n / s e c ) 1
2
I 8 1 1 1
I
0 4 06 I
I
,
6 8 10
4 I
1
1
1
l
1
20
1
,
60
40 '
(
I
30
t
I
3
25
2o 15
I
N O 0
!.
10
b
5 0
2
3
4 (BJS) log E (eV)
-
5
FIG. 9. Comparison of experimental (Zwally and Cable, 1971) ground-state B3+He single-electron capture cross sections (...) with Landau-Zener (LZ) calculations (--4.Also shown is a comparison for the ground-state BeZf H process (-) of Landau-Zener calculations (LZBJS) with Bates er al. (1964) calculations (BJS) in which full acount is taken of the oscillations from beyond the crossing.
mate (Olson et al., 1971). However, in a recent study (Huber and Wiesemann, 1978) of ArZ+Neand Ar2+Arthere is exact correspondence between experiment and Landau-Zener calculations. More than a single s electron transfer is involved. A reason for the possible underestimation of cross sections by the Landau-Zener approximation is to be found in the pattern of the oscillations of probability of transition, considered either as a function of time, or of nuclear separation, or of impact parameter. Figure 5 illustrates such oscillations for a noncrossing case, and Fig. 10 shows similar calculations and measurements for an avoided crossing. There is an appreciable contribution from regions outside the avoided crossing. The oscillating probabilities of transition can be calculated by solution of equations of type (Bates et al., 1964)
where Ic1,2(2are the probabilities Pl,2of the system being on potential energy curves 1,2. For the ground-state Be2+Hcollision both types of calculation were made, and are illustrated in Fig. 9.
ION-ATOM
C H A R G E T R A N S F E R C O L L I S I O N S A T LOW E N E R G I E S
C"
2 I9
He
2 keV r.=4 35
i
P
o
0
I
1
3.0 H,,(01)
0
1
1
I
I
I
I
-
4
law 2
3
b (a.u.)
FIG. 10. Comparison of measured (-) and calculated (...)oscillatingtransition probabilities P ( h ) at 2 keV impact energy for CZ+Heground-state channel single-electron capture (Makhdis et NI.. 1976). Values of H,,(01) and 3H,,(OI)are calculated from empirical Eq. (16).
However, it is possible that some underestimation by the LandauZener approximation might arise from other factors (Janev, 1976). such as an underestimation of the dynamical width of the nonadiabatic region, or the neglect of tunneling transitions (Ovchinnikova. 1973) and rotational coupling. A difficulty of this and indeed all LCAO impact parameter formulations arises from the fact that the transition probability is calculated over a single collision trajectory, which is itself calculated on the basis of the impacting ion-atom interaction energy [Eqs. (8) and (9)]. But this is not the actual collision trajectory, since the probability of this interaction energy being the appropriate one is itself time dependent, in oscillatory fashion. Oscillations due to this cause, first predicted by Stuckelberg (1932), arise in elastic-scattering cross sections. This situation is most serious at small scattering angles, which correspond to large impact parameters. It is not yet possible to estimate the magnitude of errors in charge transfer cross sections arising from this cause alone. The role of avoided crossings is of particular significance in electron capture processes that either start or terminate or both start and terminate in states for which Coulomb repulsion or attraction acts between the two species:
Such collision processes are difficult to measure (most suitably by crossed- or inclined-beam techniques), but are of importance for esti-
J . B . Hrrstrd
220
mating the cooling effects of impurities in hot plasma machines built for thermonuclear energy research. In the course of becoming completely stripped the impurity atom of atomic number Z effectively removes Z deuterons from the electrical heating action. Programs of calculations of cross sections for stripping processes of this type have been reported (e.g., Greenland, 1978). It must be stressed that it is only in the low-impact-energy region (impact velocities less than one atomic unit) that transitions at the avoided crossings dominate the electron transfer. At higher energies the collisions are dominated by Rosen-Zener transitions taking place at other nuclear separations, and a different energy dependence of the cross sections is to be expected. If both energy regions are taken into account, then the cross section function for a Coulomb-dominated charge transfer process can show two distinct regions. Figure 1 1 displays such an effect, comparing theory (Presnyakov and Ulantsev, 1974) and experiment. Bykhovskii and Nikitin (1965) have considered a model that generates both Landau-Zener and Rosen-Zener types of transition; for certain values of the parameters, two total cross-section function maxima are expected, and also a sharper increase of cross section with velocity in the adiabatic region than that predicted by Landau-Zener approximation alone. Other models for calculating probability of transition in curve-crossing collisions have been given by Child (1971) and by Crothers (1971).
10
8 6 4
2 0 0.1
1
-
10
E(keV)
*
10
FIG. 1 1 . Comparison of calculations (-) (Presnyakov and Ulantsev. 1974) for the single-electron electron capture in Kr3+Ne collisions with experiments (---). 1, Flaks and Filippenko (1959); 2 . Hasted and Chong (1962).
ION-ATOM
C H A R G E T R A N S F E R COLI.ISIONS A T L O W E N E R G I E S
221
V. Curve-Crossing Spectroscopy Curve-crossing spectroscopy (Cooks. 1978) is aimed at deriving details of the interaction potentials of an ion-atom or ion-molecule system from a study of the double differential inelastic cross sections. The oscillations corresponding to an identified avoided crossing of potential curves yield information about the form of these curves and also about the interaction matrix element H I 2 . An ion beam is scattered from a gas or crossed-beam target and is detected after traveling over a collimated path at a certain polar scattering angle. The kinetic energies before and after collision are simultaneously measured, so that the ingoing and outgoing channels of the collision process can be inferred. The kinetic energy difference measurement can be made either for the total cross section or for the differential cross section in angle. The length of the postcollision collimation must ensure that an angular resolution of better than 0.3"is obtained, for impact energies in the region of 1 keV. A range of scattering angles between 0 and 10" is covered either by traversing the detecting system together with its energy analyzer in angle, or by electrical deflection. Use of tof velocity analysis reduces the size of the movable part of the apparatus, and allows an energy resolution of 0.001 to be obtained at the expense of intensity loss due to low duty cycle. Alternatively, conventional electrostatic velocity analysers can be used; variable electrical deflection between fixed collision chamber and analyzer entrance can be used to avoid mechanical difficulties of movement of the entire analyzer and detection system. Neither electrical deflection nor electrostatic velocity analyzer can be used when a scattered neutral component is required to be investigated: but tof analysis is still possible, using a flat-surface particle multiplier. Low kinetic energies are used, so that the scattering angles for given impact parameter are as large as possible, and the intensities for a given scattering angle are also large. The product of impact energy E and scattering angle 0 is approximately invariant and is known as the reduced angle 7 = EO. Relative reduced cross sections p = Ocr(0) sin O are measured as a function of T for the appropriate channel. Typical small-angle energy loss spectra are illustrated in Figs. 12- 14 together with their assignments (Chen. 1972; Chen et ( i l . , 1974; Kubach c t r r l . , 1976). Angular distribution charge transfer spectra show one or more sets of oscillations. depending on the number of pseudocrossings. The first experiments showing oscillations in differential cross sections were carried out without energy analysis by Ziemba and Everhart (1959) and
222
J . B . Hasted 2600 200 eV
1300
0
I
128
0
$ c
?! c
C
3
8 +a,
I
I ,
, , ,
I
,Ill1, I
, ,
,I>
1
I
,
, I
I
,
,
1 ,
11,
I
j
Lockwood and Everhart (1962). The first such experiments with doubly charged ions were due to Hasted r t a / . (1971). In many modern experiments the emphasis is more on energy analysis at zero angle; the relevance of this approach to the study of processes involving excited ions (Section VI) is clear. In the discussion up to this point, only atomic orbitals have been considered, although the nature of the collision process demands that correlations between the atomic electrons on each of the two systems be taken into account. This can be done by making use of the molecular orbitals in
+ C
$15
10
20 10 19
8 4
18
,-.
I l l I I I I I I I I I I I I I 1IIII11lIIIIIII11111 I A E (eV)----FIG.13. Energy loss spectra at different scattering angles for single-electron capture in He2+Ar(Chen, 1972; Chen el al., 1974).
J . B. Hustrd
224 A
0
0
5
AE (eV)
10
15
Fic. 14. Energy loss spectrum at E = 500 e V , 0 = 0.75" for single-electron capture in H+Xe(Kubacheta/.,1 9 7 6 ) . ( A ) H ( n = I ) + X e + 5 p 5 z P , , z , ( B ) H ( = n I ) + XeSps2P,,,.(C) H ( n = 2) + Xe+ 5p5 2P,,2, (D) H ( n = 2) + Xe+ 5p5 and H ( n = 1 ) + Xe+ 5s 5p6 zS,,2.
common use in theoretical chemistry, whose energies can be calculated by techniques well developed and tested by molecular spectroscopy. Such orbitals differ from the combined atomic orbitals in that they do not correspond to adiabatic adjustment of the atomic wavefunctions to movement of the nuclei relative to each other. They are nonadiabatic, or diabatic orbitals, and as such they are not forbidden by the Wigner rule to cross each other, since they correlate to different levels in the unitedatom limit. In the first consideration of their applications to heavy-particle collision processes, by Lichten (1967), diabatic orbitals were represented on a logarithmic energy scale, terminating in the two-atom and unitedatom energy levels; but in this representation it is less easy to compare them with the more familiar adiabatic orbitals. We therefore show in Fig. 15 in the familiar linear representation V ( r )an example of either type, calculated for (H-Cs)+ by Sidis and Kubach (1978). The avoided crossing situation with adiabatic orbitals can be contrasted with the crossing diabatic orbitals. These crossings are responsible for the promotion of electrons to higher orbitals, from which radiation often accompanies their decay. Oscillations in the elastic differential cross section accompany these promotions; these are to be distinguished from the oscillations in the charge transfer channels, which we considered earlier. As examples of curve-crossing spectroscopy of charge transfer collisions, we select the following single-electron capture processes: He2+Ne,
ION-ATOM
CHARGE TRANSFER COLLISIONS AT LOW ENERGIES
I \ \
225
i
i
01
0
-0-35
H CS' 5
10
15
20
25
R(au)-
FIG.IS. Adiabatic and diabatic interactions calculated for (H-Cs)+ (Sidis and Kubach. 1978).
Ar (Chen, 1972; Kubach et al., 1976) H+ Xe (Kubach et al., 1976) and C2+, 02+He(Makhdis et ul., 1976). A more complete list of references is given in Tabie 11, but it should be emphasized that these do not include studies
of excitation without charge transfer. Figure 12 shows the variation with angle of the He+ energy loss specT A B L E I1
PUBLISHED CURVE-CROSSING SPECTRA H+He, Ne, Ar, Kr, Xe H+Xe He+Ne He+He He+Ne. Ar, K r , Xe Ne+He
I
He2+Ar
C2+He 1 C2+Ne
Abignoli el a / . (1972) Kubach et 11/. (1976) Barat e f a / . (1976) Rrenot e t ( I / . (1975) Baudon
PI
a / . (1970)
Chen et
ti/.
(1974): Chen (1972)
Makhdis et cr/. (1976) Sat0 and Moore ( 1978)
OZ+He APHe Ar2+Ne Ar2+Ar
Makhdis el a / . (1976) Huber and Wiesernann (1978)
226
J . B . Husted
trum in the electron capture collisions of He2+ with Ne. The zero-angle spectrum contains signals corresponding to one channel, but as the angle is increased, other channels successively start to contribute. At the smallest angles, which are subtended by the resolution of the apparatus when it is set to zero angle, only the largest impact parameter collisions play a part. Thus the avoided crossings at small nuclear separation cannot be reached, and make no contribution. Only the crossings at very large nuclear separation, corresponding to a positive energy defect of a few electron volts, can contribute at larger angles. This behavior is also found in the contribution of endothermic noncrossing transitions to a spectrum. As the angle is increased, contributions from successively smaller nuclear separation regions make their appearance. The appropriate assignments of the spectral bands have been marked on Fig. 12. When the separation of the first two peaks is plotted against T, it is found that there is a critical range of T at which the separation changes from the separation of the S channels to that of the lZ+ molecular states; this can be shown to imply that spin-orbit coupling of the outer electron with the open shell of the Ne+ plays a role. The He2+Arspectra (Fig. 13) (Chen, 1972) show additional features of interest, among which is the possible contribution of the continuum process Hez+ + Ar + He+ + Ar+
+e
(26)
Such processes have previously been studied in total cross section, using coincidence technique (Flaks et ( I / . , 1967). This continuum band is much broader than individual channel bands, since a large number of unresolved channels contribute. But the He+ ( n = 2) band is also unexpectedly broad; as 7 increases (decreasing impact parameter) the band grows and broadens, the maximum shifting toward greater endothermicity. There is no crossing, and the system is weakly coupled to a variety of excited states of Ar+ and He+. It relaxes into a statistical distribution of these states, which contribute in varying amounts to the band. Similar contributions are found with the He2+Krspectrum. The proton-rare gas collision processes are of some simplicity because the excited-state processes lie nearly 10 eV above the near-resonant ground-state channels, as can be seen from the H+Xe calculated potential energy curves in Fig. 16. For all except He there is the complication of the J = 3/2, 1/2 sublevels. The H+ Xe potential energy curves (Kubach and Sidis, 1973) are calculated and adjusted to obtain the spectroscopic en) ergies at large separations, so that the crossings appear as in the ~ ( bdiagram in Fig. 17, at 10ao and 6u0.
F I G .16. Calculated diabatic potential energy curves (Kubach and Sidis, 1973) for H+Xe. Labeling of curves refers to the usage in Figs. 14 and 17.
I
loo0 500
-2
200
7 Q) 3
0 -200 -400
0
2
4
6 b (a,)
8
1
0
F I G . 17. ~ ( hfunction ) for production of Xe+ *PI,, from H+Xe. Numbering refers to the paths labeled in Fig. 16.
f. B . Hasted
228
In this situation there is the possibility not only of X-A and X-B curve crossings, but also of virtual transitions between A and B. An energy loss spectrum corresponding to channels A and B is displayed in Fig. 14 and two other channels C and D are also observed, although not discussed here. However, the experimental data have insufficient angular resolution to detect the calculated oscillations in angle (Fig. 18), and convolution is necessary in order to make comparison. When this is done, however, the improvement obtained by taking the noncrossing transitions into account can be clearly seen. The complicated forms of the oscillations arise from the fact that there are interferences between waves scattered by different branches of the potentials inside the crossings. Figure 17 shows the re. 7 s 417 eV deg, four impact duced classical deflection functions ~ ( b )For parameters are related to the same angle, and a mixture of rainbow and Stuckelberg oscillations is produced. Single-electron capture by doubly charged ions shows less complication, because the Coulomb repulsion between products ensures that there is a suitable avoided crossing even though the energy defect is large, and transitions elsewhere can safely be neglected. In the 02+Heangular spectrum presented in Fig. 19 the contributions form three different crossings to a single energy loss channel can be discerned (Makhdis et a/., 1976), and in the simpler case C2+He the oscillations are compared in Fig. 10 with calculations, following Bates et a / . (1964) in the form P ( b ) , where P i s the probability of transition. Equations of the type (8) and (9) are used for the interactions from which the functions b(0) are deduced.
0
200
400
600
T (eVdeg)FIG. 18. Reduced differential cross sections p(7) measured for H+Xe (Kubach et a / . . (1976). 0 , Process A; A, process B (Fig. 14); convoluted calculations: process A; _ _ _ , process B .
ION-ATOM
CHARGE TRANSFER COLLISIONS AT LOW ENERGIES
229
10 5
2
t
-
\
I
v)
5
e :
3
o'2P
Ir2D
10
-E
02'He
2 10
5 2 1
1000
0
z
2000
(eV deg)-
FIG.19. Differential scattering function with singles electron capture in the channel 02+ 3P0,1,p+ He's, + O+ 2P~,2,,,2+ He2S1,*(Makhdiser d.,1976). Arrows indicate angles corresponding to impact parameters equal to values of R , calculated from interaction energies of Eqs. (8) and (9).
VI. Charge Transfer Processes of Excited Ions The most common applications for which charge transfer cross sections are required are situations in which the ions can be present in a number of different long lifetime states, and it is required to know at what rates each one of these is destroyed, or alternatively how effective each one is in transferring its charge to a gas. Ion beams from sources usually contain a mixture of excited states, and while it is sometimes possible to infer the relative populations at least qualitatively, a precise estimate can only be obtained by indirect means. The most refined method of studying excited ion processes is by the method of curve-crossing or translational spectroscopy discussed in the previous section. The differential cross sections are measured for characteristic energy defects, of either sign; momentum analysis is included on either side of the collision. When total cross sections are desired integration over angle must be carried out. When it is possible to produce a pure ground-state ion beam, comparison of its spectrum with that of the
230
J . B . Hasted
mixed-state beam enables the proportions of excited states and the cross sections to be infererred. However, other methods of studying mixed-state ion beams have become popular. The most usual derives from the early analyses of charge-changing collisions in gases over a wide range of pressure (Allison, 1958). One can achieve the elimination of one component of a mixed state beam by utilization of its larger attenuation coefficient in a suitable gas. The attenuation technique in its simplest form makes use of the relation for ion beam intensity I(L) after transversing gas path length L, at gas density n:
wherefis the fraction of the beam whose attenuation cross section is (+*, the fraction I - f having attenuation cross section (+. In the case of a beam of ions attenuated by charge transfer conversion to neutrals, a collisional separation is carried out in a region of magnetic field. In this way the regeneration of positive ions by gas collisions of the neutrals diverges from the ion beam path. If only these two cross sections contribute, then both, together with the quantity f , can be determined by analyzing the attenuation with varying n. Only when Eq. (27) is found
E (eV)
-
FIG. 20. Total cross-section function for O+N, and O+Ar charge transfer (Matic ef a / . , 1977). Data points are all experimental with smooth lines drawn through them: 0.0, Matic ef a/. (1977); V, Hughes and Tiernan (1971); A, Rutherford and Vroom (1971); f , Lo and Fite (1970); * , Ormrod and Michel (1971).
ION-ATOM
CHARGE TRANSFER COLLISIONS AT LOW ENERGlES
23 1
experimentally to be satisfied can the simple analysis be applied to derive
f. Allowance can then be made for the collisional interconversion of the two ionic species by excitation and deactivation. The attenuation method allows only the two sums of charge-changing cross sections to be deduced (i.e., single and multiple electron capture and loss), but it is possible to deduce the individual cross sections by measuring the rate of attenuation in gas at very low pressures, and also by the standard “condenser” technique (Hasted, 1960). An important advance is to build all these facilities into a single beam-gas apparatus, so that the excited-state composition of the beam issuing from a given source can be monitored in different ways. Some recent cross sections (Matic cf a / . , 1977) for excited state and ground state ions are shown in Fig. 20. Such measurements must be checked (Sidis and Matic, 1977) by means of quantum theory, for example, by the semiempirical techniques of Sections Ill and IV. It would also be desirable for checks to be made using the translational spectroscopy technique of Section V .
REFERENCES Abignoli. M.. Barat, M..Baudon, J . , Fayeton, J . , and Houver, J.-C. (1972).J.Phys. E [1]5, 1533. Allison, S. K. (1958). Rat,. Mod. Phys. 30, 1137. Barat. M.. Brenot, J . C.. Dhuicq, D.. Pommier, J., Sidis. V. Olson. R. E.. Shipsey, E. J . , and Browne. J . C. (1976). J. Phvs. B [ I ] 9, 269. Bardsley. J. N . , Holstein, T.. Junken, B. R., and Sinha, S. (1975).Phys. Rev. A [3] 11, 191 I . Bates. D. R.. and Boyd, T. J. M. (1956).Proc. Phys. Soc., London, S e c t . A 69, 910. Bates. D. R.. and Lewis. J . T. (1955). Proc. Phvs. S n c . . London, Sect. A 68, 173. Bates, D. R., and Moiseiwitsch, B. L. (1954).Proc. Phys. Soc., London, S e c t . A 67, 805. Bates. D. R.. Johnson, H. C., and Stewart, I. (1964). Proc. Phys. S o c . . London 84, 517. Baudon. J.. Barat, M.,and Abignoli, M. (1970). J . Phys. E [I] 3, 207. Birkinshaw, K . , and Hasted, J . B. (1971). Proc. fnl. Conf. Phys. foniz. G a s e s , 10th. 1971 p. 10.
Boyd. T. J. M., and Moisewitsch, B. L. (1957). Proc. Phys. Soc., London. S e c t . A 70,809. Brenot, J. C., Pommier. J . . Dhuicq. D., and Barat. M. (1975). J. Phys. E [ I ] 8, 448. Bykhovskii, V. K..and Nikitin. E. E. (1965). SOP.Phy.7.-JETP (EngI. Trunsl.) 21, 1003. Chen, Y. H. (1972). Ph.D. Thesis, University of Virginia. Charlottesville. Chen, Y. H . , Johnson, R. E., Humphries, R. R.. Siegel, M. W., and Boring. J . W. (1974).J . Phys. E [ I ] 8, 1527. Child, M. S . (1971). Mol. Phys. 20, 171. Chkuaseli, D., Nikoleishvili, V., and Gouldamashvili, A. (1960). Bull. Acud. Sci. USSR, Phys. Ser. (Engl. Trunsl.) 24, 972. Cooks. R. G . . ed. (1978). “Collision Spectroscopy.” Plenum, New York. Crothers. D. S. F. (1971). A d v . Phys. 20, 405. Daley. H. L., and Perel, J. (1969). Proc. Int. Conf Phys. Electron. A t . Collisions, 6rh. 1969 p. 1051.
232
J . B . Hasted
Dalgarno. A. (1954). Proc. Phys. SOL..,London, Sect. A 67, 1010. Demkov, U . H. (1964). A t . Collision Processes. Proc. Inr. Conf, Phys. Electron. A t . Collisions. 1963 p. 831. Dillon, J . A , , Sheridan, W. F., Edwards, H. D., and Ghosh, S . N . ( 1 9 5 9 . J .Chem. Phys. 23, 776. Duman, E. L., and Smirnov, B. M. (1970). Sov. Phys.-Tech. Phys. ( B i g / . Trunsl.) IS, 16. Eisele, F. L., and Nagy, S . W . (1976). 1. Chem. Phys. 65, 752. Firsov, 0. B. (1961). Z h . Exp. Teor. Fyz. 21, 1001. Flaks, I. P., and Filippenko, P. G. (1959). Zh. Exp. Teor. Fyz. 17, 67. Flaks, I. P., and Solovev. E. S . (1958). Sov. Phys.-Tech. Phys. (Engl. Trunsl.) 3, 564. Flaks, 1. P., Afrosimov, V. V . , Manaev, Yu. A , , Panov, M. N . , and Fedorenko, N . V. (1967). Proc. Int. Conf. Phys. Electron. A t . Collisions, 5th, 1967 p. 210. Gentry, W. R..Lee, Y. T.. and Mahan, B. H. (1968). J . Chem. Phys. 49, 1758. Greenland, P. T. (1978). J . Phys. B [l] 11, L191. Hasted, J. B. (1951). Proc. R . Soc. London, S e r . A 205, 421. Hasted, J. B. (1960). A d v . Electron. 13, 1. Hasted, J. B. (1968). Adti. At. Mol. Phys. 4, 237. Hasted. J. B., and Chong, A . T. J. (1962). Proc. Phys. S o c . , London 80, 441. Hasted, J. B., Iqbal, S . M . , and Yousaf. M. M . (1971). J . Phys. B [ l ] 4, 343. Hegerberg. R., Stefansson, T.. and Elford, M . T. (1978). J . Phys. B [ l ] 11, 133. Helm, H. (1975). Chern. Phys. Lett. 36, 97. Herring, C. (1962). R e v . Mod. Phys. 34, 631. Hishinuma. N . (1972). J . Phys. Soc. Jpri. 32, 227 and 1452. Hodgkinson, D. P., and Briggs. J. S . (1976). J . Phys. B [ l ] 9, 255. Huber, B. A., and Wiesemann, K. (1978). “Frunjahrstagung DPG Fachausschuss Atomphysik,” No. 2, p. 439. Verhandlung der DPG, Munchen. Hughes, B. M., and Tiernan. T. 0. (1971).J. Chem. Phys. 55, 3419. Janev, R . K . (1976). A&. A t . Mol. Phys. 12, 1 . Johnson, R. E. (1970). J . Phys. B [ I ] 3, 539. Johnson, R . E. (1972). J . Phys. Soc. Jpn. 32, 1612. Kirnura, M., and Watanabe, T. (1971). J. Phys. Soc. Jpn. 31, 1600. Kubach, C.. and Sidis, V. (1973). J . Phys. B [ I ] 6 , L289. Kubach, C.. Benoit, C., Sidis, V . , Pommier, J., and Barat. M . (1976). J . Phys. B [ I ] 8, 2073. Kushnir. R . , and Buchma, I. (1960). Bull. Acud. Sci. U S S R , Phys. Ser. (Engl. Transl.) 24, 989. Kushnir, R., Palyukh, B. M., and Sena. L. A . (1959). Bull. Acad. Sci. U S S R , Phys. Ser. (EngI. Transl.) 23, 995 Landau, L. D., and Litschitz, E. M. (1963). “Quantum Mechanics.” Pergamon, Oxford. Latypov. Z. A.. Fedorenko, N . V . , Flaks, I. P., and Shaporenko. A . A . (1970). JETP Lett. (EngI. Transl.) 11, 116. Lichten, W. (1967). Phys. Rev. 164, 131. Lo, H. H., and Fite. W. L. (1970). A t . Duru 1, 312. Lockwood, G. J.. and Everhart, E. (1962). Phys. R e v . 124, 567. McMillan, W. (1971). Phys. Rev. A [3] 4, 69. Makhdis. Y . Y., Birkinshaw, K . , and Hasted, J . B. (1976). J . Phys. B [ I ] 9, 1 1 1 . Marchi, R. P., and Smith, F. T. (1965). Phys. Rev. A [2] 139, 1025. Marino, L. L. (1966). Phys. Rev. [2] 152, 46. Marino, L. L., Smith, A . C. H.. and Caplinger, E. (1962). Phys. R e v . [2] 128, 2243. Massey, H. S. W. (1949). Rep. Prog. Phys. 12, 248. Massey, H. S. W., and Smith, R . A . (1933). Proc. R . S o c . London. Ser. A 142, 142.
ION-ATOM
CHARGE TRANSFER COLLISIONS AT LOW ENERGIES
233
Matic. H.. Cobic. B.. and Vujovic. M . (1977). I n / . Con,f: Phys. N c ~ t r c ~ An t.. Collisions, 10th. I977 p. 882. Melius. C. F.. and Goddard. W. A. (1974). Phy.s. H e r . A [3] 10, 1541. Nikitin. E. E. (1961). Opr. Spektrosk. 12, 452. Nikitin. E. E.. and Ovchinnikova. M. Ya. (1970). Usp. Fiz. N m k 104, 379. Okuno, Y.. Kobayashi, N., and Kaneko, Y. (1978). Phys. Reis. Lurr. 40, 170B. Olson. R. E. (1970). Phys. Rev. A [3] 2, 121. Olson. R. E. (1972). Phys. R e v . A [3] 6 , 1822. Olson, R. E.. Smith, F. T., and Bauer. E. (1971). Appl. Opt. 10, 1848. Ormrod, J . H.. and Michel. W. L. (1971). Can. J. Phys. 49, 606. Ovchinnikova, M. Ya. (1973). Zh. Eksp. Tror. Fiz. 64, 129. Perel. J . , and Daley, H. L. (1971). Phys. R e v . A [3] 4. 162. Perel. J . . and Yahiku. A. Y. (1967). Proc. h i t . Coqf: Phys. EIec~tron.A t . Collisions, 5 t h . 1967 p. 400. Perel, J . , Vernon, R. H., and Daley. H. L. (1965). Phys. Rev. A [3] 138, 937. Daley, H. L.. and Smith. F. J . (19701. Phys. R e v . A [3] 1, 1626. Perel. .I., Presnyakov. L. P.. and Ulantsev, A. D. (1974). SOI.. Qimntiim Ekcrrctn. (EngI. Trcrnsl.) 1, 2377. Rapp. D.. and Francis, W. E. (1962). J. Chem. Phys. 37, 2631. Rosen. N .. and Zener. C. (1932). P h y s . R e r . 40,502. Rutherford. J . A., and Vroom, D. A. (1971). J . C h m . Phys. 55, 5622. Sato. Y..and Moore. J. H. (1978). J. Chcm. Phys. (in press). Seaton. M. J. (1958). M o n . Not. R. Astron. SOC. 118, 504. Sidis. V.. and Kubach, C . (1978). J. Plzys. B [ I ] 11, 2687. Sidis, V., and Matic, M. (1977). 10th In/. Cot!/: Phys. Electron. A t . Collisions, 10th I977 p. 720. Sinha, S . . and Bardsley, J . N. (1976). Phys. Rrr,. [3] 14, 104. Smirnov, B. M. (1964). So\'. Phys.-JETP (Engl. Trcrnsl.) 19, 692. Smirnov, B. M. (1965a). Sov. Phys.-JETP (Engl. T r m s l . )20, 345. Smirnov. B. M. (1965b). Soit. Phys.-Tech. Phys. (Engl. T r a n s / . )10, 88. Smith, D. L.. and Futrell, J . H. (1973). J. Chcwi. Phys. 59,463. Smith, F. T. (1966). P;iys. Left. 20, 271. Stuckelberg. E. C. G. (1932). Helrz. Phys. Actu 5, 369. Turner-Smith, A. R.. Green, J. M., and Webb. C. E. (1973).J . Phys. B [I] 6 , 114. Wolf, F. A , , and Turner. B. R. (1967). Rep. 7919, Gulf General Atomic. Ziemba, F. P.. and Everhart. E. (1959). Phys. R e i , . Lett. 2, 299. Zwally, H. J.. and Cable. P. G . (1971). Phvs. Rri.. A [3]4, 2301.
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A D V 4 N C E S IN ATOMIC A N D MOLECULAR PHYSICS, VOL
I5
ASPECTS OF RECOMBINATION D . R . BATES Depurrmenr of Applied Muthrnrcitic.c utid Theoretical Phvsic.r Queen's University of Beljiisr B e l j h t , Norrhenr Irelurid
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radiative e - O + Recombination a Complex Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Ion-Ion Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . ........ B. Electron-Ion Recombination .............. .......... Recombination in an Ambient Ele Recombination in an Ambient Neutral Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Electron-Ion Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Ion-Ion Recombination . . .......... ........ References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....
235 238 238 241 250 250 255 259
I. Introduction Recombination is not a coherent subject but instead consists of a set of widely diverse processes. Thus it cannot be surveyed in a strictly logical manner, nor can it be surveyed thoroughly in the space available here. In selecting the material to be included, account was taken of the articles by Bardsley and Biondi (19701, Flannery (1972, 1976), Bates (l975a) Janev (1976), and Moseley e? al. (1976) and of the book by Massey and Gilbody (1974) but some overlap could not be avoided.
11. Radiative e- Of Recombination and the Nightglow It is fitting on this occasion to commence with radiative e-O+ recombination, because in 1937 Massey asked me and a fellow graduate, J. J. Unwin, to carry out quanta1 calculations on it and to investigate whether it could be responsible for the rate of loss of free electrons in the iono235 Copyright 0 1979 hy Academic Precs. Inc. 411 right.. of reproduction in any form reserved ISBN 0-12-003815-3
236
D. R. Butes
sphere reported by radio scientists. We (Bates et al., 1939) found it to be far too slow to be of any importance in this connection. Interest of aeronomers in the process O+ + e
+
0'
+ hv
(1)
has been rekindled during the past decade because of the discovery of 0 1 lines originating from levels having principal quantum number 3 or greater in the tropical nightglow. The earliest identification of such a line, 4368 A, 4p3P -+ 3s3S, was made by Ingram (1962) from a station in the Bolivian Andes at low geomagnetic latitude ( - 3") but the significance of its presence was not appreciated at the time. Appreciation only came when the Ogo 4 polar orbiting satellite allowed the luminosity to be viewed from above. The lines 1304 A, 3s3S + 2p3P, and 1356 A, 3s5S + 2p3P, were then discovered by Hicks and Chubb (1970) using photometers, and by Barth and Schaffner (1970) using a spectrometer. They were found to be emitted from bands around 12- 15"on either side of the geomagnetic equator (that is, in the region of the Appleton anomaly peaks). Shortly afterward Weill and Joseph (1970), working in Israel, recorded 7774 A, 3p5P + 3s5S,from the direction of an intertropical arc. Brune et al. (1978) have recently reported a rocket ,xperiment that showed that the lines 1304 and 1356 8, are emitted feebly at midlatitudes. A conference report of some of the Ogo 4 results led Hanson (1969) to suggest that radiative recombination ( 1 ) is responsible. This is now generally accepted and attention will here be confined to the process though mutual neutralization
o++ 0 -
4
0'
+ 0"
(2)
proposed by Knudsen (1970), may make an appreciable contribution (cf. Hanson, 1970; Olson el NI., 1971). Guided by hydrogenic results quoted by Bates and Dalgarno (19621, Hanson (1969) estimated that at 1000 K the radiative recombination processes leading to 1304 and to 1356 8, each have a rate coefficient a! (1304 or 1356) of about 5 x cm3 sec-'. On integrating the calculated volume emission through the F layer, the deduced intensities were found to be very much below those which had been reported (Barth and Schaffner, 1970; Hanson, 1970; Hicks and Chubb, 1970; Anderson, 1972). Kirkpatrick (197 1 ) carried out calculations on direct recombination into nl configurations of neutral oxygen with the 4S parent using the scaled Thomas-Fermi approximation for n s 10, I s 4, and the hydrogenic approximation otherwise. Adopting his results, assuming that the triplet to quintet population rates are in the ratio 3 : 5 and paying some attention to how cascading proceeds, Tinsley et 01. (1973) found (~(1304)and
ASPECTS OF RECOMBINATION
237
a(1356) to be 4.0 and 8.1 X cm3 sec-I, respectively. The accord with Hanson's value of 5 x cm3 sec-' was encouraging. It suggested that the discrepancy noted in the preceding paragraph should not be ascribed to the calculated rate coefficients. Meier and Opal (1973) traced it to two sources: the absolute calibration of the Ogo 4 sensors and the failure to appreciate that in making comparisons the ionosonde data and nightglow intensities should be obtained nearly simultaneously. Working at a site in Brazil approximately under the Appleton anomaly, Tinsley et a / . (1973) made measurements on the absolute intensity of 7774 8, and also got ionosonde data from a neighboring station with the optical line of view and the radio beam intersecting at 300 km. Taking 47774) to be as calculated, cm3 sec-' at 1000 K, they found close agreement between the 5.8 x measured and deduced intensities. The tropical nightglow appeared to be explained satisfactorily. However, as Massey has often warned one should not rest content merely because theory and observation agree; and indeed a striking change was soon effected in the problem under discussion. An elaborate investigation of (1) was undertaken by Julienne et a / . (1974). Direct recombination intc; :he levels n d 20, / S 3, was treated by the quantum defect method. Cascading was allowed for using transition probabilities tabulated by Wiese et a / . (1966) or obtained from the Coulomb approximation. Throughout, the experimental energy for each level was adopted not, as in the earlier work of Kirkpatrick (1971), an average over the angular momentum sublevels. Julienne et ul. (1974) found that the refinement is of major importance for the triplets but of only minor importance for the quintets: thus it reduces a(1304) at 1000 K by a factor of cm3 sec-lbut reduces a(1356) and 47774) by factors (7.5)-.'to 5.3 x and 5.2 x of ( 1 3 - l and ( I . l ) - ' to 5.3 x cm3 sec-', respectively. The good agreement that Tinsley et a / . had reported no longer existed for the triplets. It was immediately recovered. Julienne et a / . (1974) pointed out that the atmosphere is optically thick toward all transitions terminating on the ground term of oxygen and that the transitions should therefore be ignored in treating casading. When this is done the effective a(1304) at 1000 K is raised to 3.0 x cm3 sec-', which is close to the value of Tinsley et a / . (1973). Solution of the radiation transfer equation (Strickland and Anderson, 1977) shows that owing to the multiple scattering of the triplet the ratio of the nadir photon intensities I( 1304): I( 1356) in the tropical nightglow should not be a( 1304):a(1356) but instead should be around three times higher. This brings about satisfactory agreement with the most recent measurements of Carruthers and Page (1976). Certain real or apparent difficulties relating to the cascading spectrum predicted by Julienne et d.(1974) have been resolved by Zipf et rrl. (1979).
238
D. R . Bates
111. Complex Ions Interest in complex ions stems partly from their presence in the atmospheres of the Earth and other planets and in interstellar clouds. Important recombination measurements have been made in recent years but the relevant theory has not yet been developed properly. A . ION-ION RECOMBINATION For most simple species the transfer of an electron from the negative to the positive ion releases energy and consequently potential energy surfaces of the ionic reactants and neutral products cross. An electronic transition leading to mutual neutralization may occur near a crossing (cf. Janev, 1976; Moseley et al., 1976). Bennett et al. (1974) pointed out that the position may be quite different if either or both ions form clusters with polar molecules like H 2 0 because such clustering reduces their energy relative to that of the neutrals. Should a sufficient number of polar molecules be clustered the ions would be more stable than the neutrals. In this circumstance mutual neutralization through transitions near crossings cannot take place. To derive a lower limit a, to the rate coefficient, Bennett et al. (1974) considered contacf collisions between oppositely charged clusters moving under the influence of their mutual Coulomb attraction. On contact the clusters would coalesce, their energy of relative motion would dissipate, and they would decompose into neutrals. Integrating the cross section over a Maxwellian distribution it may readily be shown that
where M is the reduced mass of the clusters, R, the sum of their effective radii, and the other symbols follow standard practice. In general e2/kT >> R,
(4)
so that the R: term may be omitted from (3), which then may conveniently be written
6)(y) 112
a, = 9.8 x 10-9 R,
300
112
cm3sec-1
with R, expressed in angstroms M on the chemical scale. This is naturally the same as the corresponding formula derived by Olson (1972) for his semiempirical absorbing-sphere model of mutual neutralization with R, replacing RT,the electron transfer distance that is the internuclear dis-
239
ASPECTS OF RECOMBINATION
tance at the effective crossing. Utilizing an ion-ion flowing afterglow plasma combined with Langmuir probe diagnostics, measurements have recently been made on some clustered ion recombination coefficients (Smith and Church, 1976; Smith et al., 1978). From the results, which are given in Table I, and formula (5) values of R , or RT were obtained and are also given. As may be seen, these values are larger than is likely for R , (which would be expected to be only about 4 A) but are within the range of RT that Olson (1972) encountered. Mutual neutralization may account for all the results in Table I with the possible exception of the first and second. For these it is, on present knowledge, conceivable that the reactant ion pairs are more stable than the neutral products: mutual neutralization through transitions near crossings could not then occur and some other process that is more rapid than contact neutralization would have to be invoked. A novel cluster ion process was suggested by Smith et al. (1973). They judged it to be rapid with a rate coefficient at 300 K of from 1 to 5 x cm3 sec-l or possibly larger. This is an overestimate. The process is recombination due to the action of the Coulomb field increasing the energy in the internal modes of the ions. The initial increase causes the relative motion of the ions to become bound. This is the rate-limiting step. Subsequent increases ultimately lead to recombination. The process takes place irrespective of whether or not the ionic reactants are more stable than the neutral products, and therefore the measurements of Smith and his associates (Table I) set upper limits to the values of the recombination coefficients aIfor the cases treated. Bennett et a / . (1974) have investigated the problem. They reached conclusions (some of which will be used here) on certain aspects of it. TABLE I CLUSTERED ION-ION
RECOMBINATION
Reactant ions
Reduced mass (am)
Rate coefficient at 300 K cm3sec-l)
H30+.(HZO), + NO,*HNO, NO+.(NO,), + NO,*(HNO,), H@+.(H,O)3 + NOT NH:.(NH3), + CINH:*(NH,), + NO, NH:*(NH& + NO, H,O+'(H&)3 + CI-
46.1 82.0 33.6" 20.9 24.5 27.6 23.6
5.7 3.5 5.5 7.9 4.9 6.7 4.8
Deduced Rc or RT Ref.
('4)
h h h
8.8 7.2 7.3 8.2 5.5 8.0 5.3
C C
C
C
" The measurement was made at 182 K and a T-", power law assumed. Smith and Church (1976).
Smith ct ul. (1978).
D. R. Bates
240
If RI is the closest distance of approach of the two ions the maximum energy that a cluster molecule of dipole moment p may gain from the Coulomb field is clearly about E
=
2pe/R:
( 6)
Proceeding as for (3) with the cross section corresponding to RI and noting that the strong inequality analogous to (4) is valid so that this cross section is approximately cr = rre2R,/E
(7)
it is found that the recombination coefficient, which is assumed to be the same as the ion-binding rate coefficient, is
aI = 4ne2(pe/M)1'2F/kT
(8)
where F is a factor that takes into account that the optimum collision conditions required for (6) may not be satisfied. If p is in debye units ( esu cm) and M is again on the chemical scale, (8) may be written a,
=
2.6 x lO-'(2O p/M)1'2(300/T)F cm3 sec-'
( 9)
Some values of p are given in Table 11. For (9) to be consistent with the results in Table I it would suffice for F to be 0.15 or less. Bennett et al. (1974) have performed some calculations on energy transfer between an ion and an oppositely charged dipole rotor. These demonstrate that the rotor gains energy only if its angular momentum vector has a positive component along the orbital angular momentum vector and only if the initial phase angle of the rotor relative to that of the orbital motion is within a restricted range. In order that the gain should be near the maximum, each condition would be more stringent and each might well reduce the rate coefficient based on (6) by a factor 0.4 or indeed by a smaller factor, leaving the process not more than about twice as fast as contact recomTABLE I 1 DIPOLEMOMENTS" Species
Dipole moment (debye units) 1.9
1.3 0.3 2.2 a
Hodgman (1951).
24 1
ASPECTS OF RECOMBINATION
bination. Understanding the rather low rate coefficients measured (Table I) thus presents no difficulty. In the absence of experimental evidence on cluster ion-ion recombination brought about by internal excitation or any definite application requiring information on the process, there is little inducement to make the very complicated theory quantitative.
B. ELECTRON-ION RECOMBINATION Using a microwave afterflow-mass-spectrometer apparatus with the electron temperature T, raised above the neutral and ion temperatures by controlled microwave heating, Biondi and his associates (Huang ef al., 1976, 1978) have obtained some quite unexpected results, summarized in Table 111, on the rate coefficients a[NH:.(NH,),] and a[H,O+*(H,O),] for NH,+*(NH,),
+e
H,O+-(H,O),
+e
-
-
neutrals
(10)
neutrals
( 1 1)
The great rapidity with which electron polar-cluster ion recombination takes place had been discovered earlier (Leu r t al., 1973). The rate coefficient is typically around ten times that for diatomic ions. The later investigations revealed that whereas a[NH:*(NH,),] does not change appreciably with s , a[H30+.(H20),] increases monotonically; and that the rate TABLE I 1 1 ELECTRON CLUSTERED-ION RECOMBINATION COEFFICIENTSa
S
NH:.(NH,),s family 0 1 L
3 4 H,o+-(H,O)~family 0
Recombination coefficient ( cm3sec-')
Electron temperature
1 .5(300/Te,0.5 2.8(300/ T,)O."' 2.7(300/ T,)o.050 3 3
300-410 300-3000 300-3000 200 200
1
2 3 4 5 Leu er a / . (1973), Huang er al. (1976, 1978)
(K)
540 300-8000 300-6000 300-6000 about as above about as above
242
D. R . Butes
coefficients are remarkably insensitive to T , instead of approximately following the common dependence. The first of these features, which is not very remarkable, may be connected with some facet of the structures of the two families of clusters, perhaps due to NH: being methane-like and H30+ being ammonia-like, and is not necessarily of fundamental significance; the second should be explicable in general terms as should the great rapidity of the recombination. Huang et al. (1976, 1978) postulated that the initial rate-limiting step is the internal excitation of the cluster ion accompanied by capture of the electron into a high Rydberg level. Owing to the many degrees of freedom possessed by the excited cluster they supposed that the lifetime of the system toward autoionization is long enough to allow a transition to a repulsive level and that stabilization of the recombination by dissociation then takes place. The sequence is a version of the indirect dissociative recombination process introduced by Bardsley (1968). The conflict, which is acute for diatomic ions, between electron capture having to proceed very quickly and yet predissociation of the Rydberg level entered having to compete with the inverse of the capture autoionization is resolved by appeal to the many internal modes of the complex. Huang et ul. (1976, 1978) noted that the electron capture cross section would have to be nearly proportional to T;112 in order to reproduce the weak T, dependence of the measured recombination coefficients. However, they did not attempt to explain such proportionality, which is the key part of the problem. It is necessary to consider what would be entailed. Let the cross section for the excitation of a molecular ion to a vibrational-rotational level with energy Eiabove the initial level by electrons of energy E(>Ei) be represented approximately by Qx(ilE) = CE-”
(12)
C and p being constants. The corresponding cross section for capture of electrons into the Rydberg level n is then
where
(cf. Stabler, 1963). Integration of the product of the velocity of the electron and the cross section (13) over the Maxwellian distribution gives the partial rate coefficient to be
ASPECTS O F RECOMBINATION
243
ELIkT >> 1
(16)
Provided the total rate coefficient, obtained by integrating over n, is simply
and in particular q 2 ( i ) = C(2/m)'/'
(18)
As an indication of one aspect of what is required it may be noted from (18) that for the temperature-independent a,,&) to be, say, 3 x lop6cm3 sec-', the constant C would have to be such as to make the excitation cross section at 0.1 eV about 1600 A2. This is so large that to proceed without questioning it would be imprudent. Almost half a century ago Massey (1932) treated a problem central to the issue: the rotational excitation of molecules by slow electrons. He pointed out that the cross section for the process is large in the case of a neutral molecule having a permanent dipole moment because the effective interaction then falls off quite slowly with distance. The cross section is naturally enhanced if the molecule is positively charged. Chu and Dalgarno (1974) have developed the relevant Coulomb-Born theory and applied it to C H + ( J = 0)+ e 4 C H + ( J = 1)
+e
(19)
At the energy, 0.1 eV chosen for illustrative purposes they found the cross section for (19) to be 370 A2,which is of the order of magnitude needed. Semiclassical calculations by Dickinson and Muiioz (1977) show that many /-partial waves contribute. Discussing the recombination mechanism, Huang et al. (1976) observed that electron capture by pure rotational excitation has a high probability. They observed also, without discussing the consequences, that the energy charge Ei is small. In view of the high moments of inertia of NH:-(NH,), and H30+*(H20)s, it is indeed evident that in the permitted J + J + 1 transitions E , / k T , > 1
(64)
ASPECTS OF RECOMBINATION
257
where hl,z are the mean-free paths of the ions in the ambient gas (cf. Bates, 1975b); and although P(rec, =) cannot be obtained directly from computer-simulated experiments of practicable duration it may be obtained from a combination of two sets of such experiments. To achieve this Bates and Mendas (1978b) introduced the concept of apartial-parting of the ions, which is reached when their separation R exceeds Ro and certain clearly specified conditions, ensuring randomness, are satisfied. Computer-simulated experiments allow the determination of the probability P(rec, pp) of recombination preceding partial-parting. After a partialparting there is a nonzero probability P(Ro) that the ions again approach within a distance Ro of each other. It may be shown that P(rec, m ) of (63) may be expressed in terms of P(rec, pp) and P(R,). Thus
where P(0,pp) = 1
-
P(rec,pp)
(66)
is the probability of a partial-parting. Provided condition (64) is satisfied, P(Ro) and hence a may be found from further computer-simulated experiments with the region R < Ro made field-free and with recombination ignored. Equilibrium requires that the mean time T(a) an ion pair reaching the field-free region resides within it before separating completely is such that
The corresponding mean residence time T(pp) before partial-parting occurs may readily be evaluated by computer-simulated experiments. As may be verified without difficulty, the two times are related through the probability P(Ro) sought:
Except at the higher gas densities P(rec,pp) is very small compared with unity. This allows the procedure to be simplified and moreover obviates the need for condition (64) to be satisfied provided Ro >> e2/kT Introducing the rate coefficient defined by
( 69)
258
D . R . Bates
it is apparent from (63) and (65)-(70) that a = p(R,)P(rec,pp)
The program may be arranged to distinguish between the contributions to (Y from idealized Thomson recombination (in which the relative energy of the ions diminishes indefinitely) and from mutual neutralization (for this purpose regarded as taking place only if it is not followed by Thomson recombination). Bates and Mendas (1978b) carried out illustrative computer-simulated experiments on 0:
+ 0, + o*+ [O,] + o2
(72)
0:
+ 0, + o*+ [O,] + o*
(73)
and
in which the square brackets indicate that the contents are in some neutral state. Their objective was to test a semiempirical formula that Natanson (1959) had constructed to cover all ambient gas densities and that Bates and Flannery (1969) had slightly adjusted to take into account results obtained by the quasi-equilibrium statistical method and to incorporate mutual neutralization (then presumed to have simply an additive effect in the low-density region). Because of their limited objective they were content to take the ion-neutral interactions to be of the hard-sphere type (which is rather a crude approximation in that, for example, it ignores rotational excitation) and to represent mutual neutralization by the absorbing sphere model of Olson (1972). Since the paper of Bates and Mendas (1978b) was written, computer output data have been accumulated to reduce the statistical errors. These confirm that the adjusted Natanson formula meets with only modest success in bridging the gap between the lowand high-gas-density regions. As the density is raised isothermally, the ~ the total recombination coefficient from the binary contribution ( Y to process of mutual neutralization increases sharply, passes through a maximum, and then declines. The enhancement occurs because ion-neutral collisions may render the internal energy of an ion pair transitorily negative and may make it more likely that the minimum distance of approach is less than the radius of the absorbing sphere introduced by Olson (1972) in his treatment of mutual neutralization. The initial increase in the binary recombination coefficient is naturally manifest in the isobaric variation of aM with temperature. This is of interest in connection with ionic recombination in flames and is therefore being investigated (Bates, 1979~).Table VI gives preliminary illustrative results on process (73) for a selection of assumed values of aM for the isolated
259
ASPECTS OF RECOMBINATION TABLE VI
RATE COEFFICIENTS Temperature (K)
CKM
FOR
BINARY RECOMBINATION"
Rate coefficient (cm3secc1)
Assumed value for isolated ions 300 4-7 2-7 I -7 5 -8 Computed value in molecular oxygen at 1.09 atm (0: + 0; recombination) 4.1-7 300 6.6-7 2.5-' 1.4-7 600 7.7-7 4.6-7 2.7-7 1.5-7 lo00 6.7-7 3.7-7 .2.1-7 1.2-7 2.1-7 1500 4.0-7 I .2-7 0.6-' I .5-7 2000 3.0-7 0.8-7 0.4-7
2.5-8 0.8-7 0.8-7 0.7-7 0.4-' 0.2-7
The indices give the power of 10 by which the entries must be multiplied.
reactants. It is apparent that the presence of the ambient gas enhances the value of aMand markedly changes its temperature dependence from the standard T-1'2 inverse pow, law-in the flame region it makes the fall off more rapid. The absorbing-sphere model adopted is not valid for a process like K+
+ Cl-+
K
+ C1(2P3,2,,,2)+ 0.72 or 0.61
eV
(74)
for which the internuclear distances of the crossings are 20 and 23 8, where the interaction is very weak so that there is only a very small probability that traversals of the crossings in a collision lead to mutual neutralization. Measurements by Burdett and Hayhurst (1977) show that the rate of (74) in N2]021H2flames at atmospheric pressure is 4 x 10-11(2000/T)*'2 cm3 sec-' through the temperature range 1820-2400 K. Calculations have been done (Bates, 1979c) using a model which allows for the probability being small. They demonstrate that an ambient gas may bring about a great enhancement of the mutual neutralization coefficient. ACKNOWLEDGMENTS
I thank both Dr. N . F. Bardsley and Professor A . Dalgarno for helpful discussions.
REFERENCES Anderson, D. N . (1972). J . Geophys. Res. 77, 4782. Bardsley, J . N . (1%8). J . Phys. B 1, 365. Bardsley, J. N . , and Biondi, M . A. (1970). Adv. At. Mol. Phys. 6 , 1.
260
D.R . Bates
Barth, C . A., and Schaffner, S. (1970). J. Geophys. R e s . 75, 4299. Bates, D. R. (1955). Proc. Phys. Soc. London, Sect. A 68, 344. Bates, D. R. (1975a). Case Stud. At. Phys. 4, 59. Bates, D. R. (1975b). J . Phys. B 8, 2722. Bates, D. R. (1979a). “Phys. Ionized Gases,’’ Invited Papers to SPIG-78 (R. K. Janey ed.). Bates D. R. (1979b). J . Phys. B 12, L35. Bates, D. R. (1979~).In preparation. Bates, D. R., and Dalgarno, A. (1962). I n “Atomic and Molecular Processes” (D. R. Bates ed.), p . 177. Academic Press, New York. Bates, D. R., and Flannery, M. R. (1968). Proc. R . Soc. London, Ser. A 302, 367. Bates, D. R., and Flannery, M. R. (1969). J. Phys. B 2, 184. Bates, D . R., and Khare, S. P. (1965). Proc. Phys. Soc. London 85, 232. Bates, D. R., and Mendas, I. (1975). J. Phys. B 8, 1770. Bates, D. R., and Mendas, I. (1978a). Proc. R . Soc. London, Ser. A 359, 275. Bates, D. R., and Mendas, I. (1978b). Proc. R . Soc. London, Ser. A 359, 287. Bates, D. R.,and Moffett, R. J. (1966). Proc. R . Soc. London, Ser. A 291, 1.. Bates, D. R., Buckingharn, R.A., Massey, H. S. W., and Unwin, J. J. (1939). Proc. R . Soc. London, Ser. A 170, 322. Bates, D. R., Kingston, A. E., and McWhirter, R. W. P. (1962a). Proc. R. Soc. London, Ser. A 267, 297. Bates, D. R., Kingston, A. E., and McWhirter, R. W. P. (1962b). Proc. R . Soc. London, Ser. A 270, 155. Bates, D. R., Hays, P. B., and Sprevak, D. (1971a). J . Phys. B 4, 962. Bates, D. R., Malaviya, V., and Young, N . A. (1971b). Proc. R . Soc. London, Ser. A 320, 437. Bennett, R. A., Huestis, D. L., Moseley, J . T., Mukherjee, D., Olson, R. E., Benson, S . W., Peterson, J. R., and Smith, F. T. (1974). Report TR-74-0417. Air Force Res. Lab., Cambridge, Massachusetts. Bottcher, C. (1978). J . Phys. B. 11, 3887. Boulmer, J., Devos, F., Stevefelt. J., and Delpech, J. F. (1977). Phys. R e v . A 15, 1502. Bmne, W. H., Feldman, P. D., Anderson, R. C., Fastie, W. G., and Henry, R. C. (1978). Geophys. Res. Lett. 5, 383. Burdett, N . A., and Hayhurst, A. N. (1977). Chem. Phys. Lett. 48, 95. Byron, S., Stabler, R. C., and Bortz, P. I. (1962). Phys. Rev. Lett. 8, 376. Carruthers, G. R., and Page, T. (1976). J. Geophys. Res. 81, 1683. Chu, S.-I., and Dalgarno, A. (1974). Phys. Rev. A 10, 788. Cohen, J. S. (1976a). Phys. Rev. A 13, 86. Cohen, J. S. (1976b). Phys. Rev. A 13, 99. Collins, C. B. (1965). Phys. Rev. 140, A1850. Deloche, R.,Monchicourt, P., Cheret, M., and Lambert, F. (1976). Phys. R e v . A 13, 1140. Dickinson, A. S . . and Mufioz, 3. M. (1977). J. Phys. B 10, 3151. Drawin, H. W., Emard, F . , Dubreuil, B., and Chappelle, J. (1974). Beitr. Plasmuphys. 14, 103. Flannery, M. R. (1972). Case Stud. At. Collision Phys. 2 , 1. Flannery, M. R. (1976). I n “Atomic Processes and Applications” (P. G. Burke and B. L. Moiseiwitsch, eds.), p. 407. North-Holland Pub]., Amsterdam. Gousset. G., Sayer, B., and Berlande, J. (1977). Phys. Rev. A 16, 1070. Gryzinski, M. (1959). Phys. Rev. 115, 374. Hanson, W. B. (1969). J. Geophys. Res. 74, 3720. Hanson, W. B. (1970). J. Geophys. Res. 75,4343.
ASPECTS OF RECOMBINATION
26 1
Harper, W. R. (1932). Proc. Cambridge Philos. Soc. 28, 219. Harper, W. R. (1935) P r v c . Cambridge Philos. Soc. 31, 429. Hayhurst, A. N., and Sugden, T. M. (1965). IUPAC M e e t . Plusincis. Moscow. Hicks, G . T., and Chubb, T. A . (1970). J. Geophys. R e s . 75, 6233. Hodgman, C. D.. ed. (1951). “Handbook of Chemistry and Physics,“ 41st ed., p. 2541. Chem. Rubber, Publ. Co., Cleveland, Ohio. Huang. C.-M., Biondi., M. A.. and Johnsen, R. (1976). Phys. R e v . A 14, 984. Huang, C.-M., Whitaker, M., Biondi, M. A., and Johnsen, R. (1978). Phys. R e v . A 18, 64. lngram, M. F. (1962). Mort. Nor. R . Astrvn. Soc. 124, 505. Jaffe, G. (1940). Phys. Rev. 58, 968. Janev, R. K., (1976). Adv. A t . Mol. P h y s . 12, I . Johnson, A. W.. and Gerardo, J. B. (1972a). Phys. R e v . A 5, 1410. Johnson, A. W., and Gerardo, J. B. (1972b). Phys. Reij. L e t t . 28, 1096. Johnson, A . W., and Gerardo, J. B. (1976). Report SAND-76-5498. Sandia Laboratories. Albuquerque, New Mexico. Johnson, L. C. (1972). Astrophys. J. 88, 52. Johnson, L. C., and Hinnov. E. (1973). J. Qrtunt. Spectrosc. 8 Radiat. Transfer 13, 333. Julienne, P. S., Davis, J., and Oran, E. (1974). J. G e o p h y s . R e s . 79, 2540. Kebarle, P., Searles. S. K.,Zolla, A., Scarborough, J., and Arshade, M. (1967). J. A m . Cheni. Soc. 89, 6939. Kirkpatrick, R. C. (1971). Unpublished work. quoted by Tinsley et a / . (1973) and described further by Julienne et a / . (1974). Knudsen, W. C. (1970). J . Geophys. Res. 75, 3862. Langevin. P. (1903). Ann. Chirn. P h y s . 28, 433. Leu, M. T., Biondi. M. A., and Johnsen, R. (1973). Phys. R e v . 7, 292. Liu. B. (1971). Phys. R e v . L e t t . 27, 1251. Mansbach, P.. and Keck. J. (1969). P h y s . R e v . 181, 275. Mason, E. A.. and Schamp, H. W. (1958). Anti. Phys. ( N . Y . ) 4, 233. Massey, H. S. W. (1932). Proc. Cambridge Philos. Soc. 28, 99. Massey. H. S. W., and Burhop. E. H. S. (1952). “Electronic and Ionic Impact Phenomena,” 1st ed. Oxford Univ. Press (Clarendon), London and New York. Massey. H. S. W., and Gilbody, H. B. (1974). ”Electronic and Ionic Impact Phenomena” (by H. S. W. Massey, E. H. S. Burhop, and H. B. Gilbody), Vol. 4. Oxford Univ. Press (Clarendon), London and New York. Meier. R. R., and Opal, C. B. (1973). J. Geophys. R e s . 78, 3189. Moiseiwitsch, B. L. (1962). In “Atomic and Molecular Processes” (D. R. Bates ed.), p. 280. Academic Press, New York. Moseley, J. T., Olson, R. E., and Peterson, J . R . (1976). Case Stud. A t . Phys. 5, I . Mulliken, R. S. (1964). P h y s . R e v . 136, A962. Natanson. G . L. (1959). Sov. Phys.-Tech. Phys. (Engl. T r a n s / . )4, 1263. Olson, R. E. (1972). J. Chem. Phys. 56, 2979. Olson, R. E . , Peterson, J. R.. and Moseley. J. (1971). J. Geophys. R e s . 76, 2516. F’itaevskii, L. P. (1962). Sov. Phys.-JETP ( D i g / . Trrrn.d.) 15, 919. Smirnov. B. M. (1977). Sov. Phys. -Usp 20, 119. Smith, D.. and Church, M. J. (1976). f n t . J . Mrrss Specfrom. Ion P h y s . 19, 185. Smith. D.. Church. M. J., and Miller, T. M. (1978). J. C h e m . Phys. 68, 1224. Smith, F. T . , Huestis, D. L., and Benson, S. W. (1973). I n “Electronic and Atomic Collisions” (B. C. CobiC and M. V. Kurepa, eds.). p. 895. Inst. Phys., Belgrade, Yugoslavia. Stabler, R. C. (1963). Phys. R e v . 131, 1578. Stevefelt, J . , Boulmer, J., and Delpech, J. F. (1975). Phys. R e v . A 12, 1246.
262
D.R . Bates
Strickland, D. J . , and Anderson, D. E. (1977). J . Geophys. Res. 82, 1013. Thomson, J . J. (1924). Philos. M u g . [8] 47, 337. Tinsley, B. A . , Christensen, A . B., Bittencourt, J . , Gouveia, H . , Angreji, P. D . , and Takahashi, H. (1973). J . Geophys. Res. 78, 1174. Warman, J. M., Sennhauser, E. S., and Armstrong, D. A. (1979). J . Chem. Phys. 70, 995. Weill. G . , and Joseph, J. (1970). C . R . Hebd. Seances Acad. Sci. 271, 1013. Wiese, W. L., Smith, M . W., and Glennon, B. M. (1%6). “Atomic Transition Probabilities,” Vol. I . Nat. Bur. Stand., Washington, D.C. Zipf, E., McLaughlin, R . W., and Gorman, M. R . (1979). Plant. Space Sci. 27,719.
11
ADVANCES IN ATOMIC A N D MOLECULAR PHYSICS. VOL. IS
I
I I
THE THEORY OF FAST HEAVY PARTICLE COLLISIONS B . H . BRANSDEN University of Durham D u r h a m , England
I . Introduction 11. Excitation o A. The Born and Clos
B. Higher Order Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III. Electron Capture from A. Intermediate Energies. . . . . . . . . . . . . . . . B. High-Energy Charge Transfer. ............................ IV. Ionization and Charge Exchange into the Continuum . . . . . . . . . . . A. Total Cross Sections B. Differential Cross Se References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
288
I. Introduction The subject of ion-atom collisions is a very large one and only a few selected topics will be discussed in this review. For an overall view the reader may be referred to Vol. 4 of the monograph by Massey and Gilbody (1974). Very many important and interesting phenomena have been studied experimentally, but comparatively few topics have received more than preliminary theoretical investigation, and the theory has been most fully developed, as might be expected, for systems with only a few electrons. Apart from specifically many-body phenomena, such as multiple excitation or ionization, most theoretical methods can be discussed conveniently, and tested, in terms of a one-electron system in which a bare nucleus A, of charge ZAeis scattered by a hydrogenic ion containing a nucleus B of charge ZBe. We shall examine the one-electron theory and comment, at appropriate places, when the extension to a system containing several electrons contains new points of principle. Certain many-electron systems can be described accurately in terms of a one-electron model. This applies to inner-shell processes in heavy atoms, because the large nuclear Coulomb field experienced by an inner-shell electron is much more important than the field due to other electrons. An interesting re263 Copyright 0 1979 by Academic Ress. Inc All rights of reproduction in any form reserved ISBN 0-12-003815-3
B . H . Bransden
264
view of this topic has been given by Briggs (1976). In a one-electron system, various processes can occur: elastic scattering; discrete excitation and ionization of the ion ( B + e-); charge exchange, in which the electron transfers from the nucleus B to the nucleus A A
+ (B + e-) + (A + e-)* + B
(1)
and reactions involving the emission, or absorption, of radiation. An interesting example of a radiative process is radiative electron capture: A
+ (B + e-) + (A + e-)* + B + y
(2)
This process becomes relatively more important at high energies and is the dominant charge exchange mechanism at energies greater than about 9 MeV per nucleon (Briggs and Dettmann, 1974, 1977; Lee, 1978). Except at very low energies ( E A < 1 eV), the relative motion of the nuclei A and B can be treated classically.' The trajectory is determined by assuming an effective potential W ( R ) between the colliding systems and solving the classical equations of motion. The internuclear vector R (using the coordinate system of Fig. 1) is then known as a given function of the time
R
=
R(t)
(3)
The theoretical problem then reduces to finding solutions of the timedependent Schrodinger equation (in atomic units2),
subject to suitable boundary conditions as t
+----t
---*( Z B )
+ ? m.
Z
FIG. 1 . A coordinate system for two nuclei and one electron. Introductory accounts of the theory have been given by McDowell and Coleman (1970) and Bransden (1970). ' Except where stated, atomic units are used throughout.
THE THEORY O F FAST HEAVY PARTICLE COLLISIONS
265
The parameter that determines the nature of the solution is A = u / u , , where u is the velocity of the incident ion A, and u, the Bohr velocity of the electron in the target ion. For a K shell electron, u, = ZBa.u. The energy of the incident ion in the laboratory coordinate system is E = 25 v2MA keV, where u is expressed in atomic units, and MA in atomic mass units. A < I defines the low-velocity region, for which a molecular orbital description is appropriate. In this case q(rB, t ) can be expanded in terms of electronic wavefunctions, calculated by solving the Schrodinger equation at fixed values of R. This is the method followed in the perturbed stationary-state model, introduced by Massey and Smith (1933), which has formed the basis of all modern work in the low-velocity region. At very low energies, the electron is shared between the two nuclei and excitation and charge exchange occur with comparable probabilities (see Briggs, 1976; Bransden, 1972). For protons incident on hydrogen, the upper limit of the low-energy region is -25 keV; but for heavy ions, the corresponding energies are much higher, extending into the MeV region, and these energies have been reached only comparatively recently with the development of heavy-ion accelerators. The fast collisions with which we are concerned here are those for A b 1. If A exceeds 1 by a large margin, charge exchange is small, compared with excitation. This leads to some simplification, in that the two processes can be treated independently. In the high-velocity region, it is usually sufficient to put W = 0 and take a rectilinear trajectory for the nuclear motion R(t)
=
b
+ vt,
v .b
=
0
(5)
where b is a two-dimensional impact parameter vector. Exceptional cases occur if very small values of the impact parameter are important, or if ZAZB is large, and then some allowance for the deflection of the incident ion must be made. A general point of importance arises because the internuclear interaction can be removed from the Schrodinger equation (4) by the transformation
This phase transformation does not alter the computed transition probabilities and total cross sections. In contrast, since differential cross sections depend on the magnitudes and phases of the probability amplitudes, the correct phase must be restored when calculating angular distributions. An interesting discussion of this point has been given recently, in connec-
B . H . Brnnsden
266
tion with the differential cross section for charge exchange, by Belkic and Salin (1978). The mathematical details of the derivation of the impact parameter model from the many-body Schrodinger equation may be consulted in the recent paper of Campos and Kruger (1978), which may be compared with the standard treatment given by McCarroll and Salin (1968).
11. Excitation of Atoms by Ions Taking as a basis the eigenfunctions &(rB) of the Hamiltonian for the target ion (B + e-), and expanding NrB,t ) as
WrB,t ) =
2 an@, f)+n(re)e-i~c
(7)
n
the Schrodinger equation (4) can be reexpressed in a coupled-channel representation as iin(b, t ) =
C Vnm(R)e*(C-'n'U,(b
9
I)
(8)
m
where
where en is the energy of the nth level of the target. If the target is initially in the level i, the boundary conditions are lim an(b,t ) = tini
t+-m
The differential cross section for excitation of the level f is (McCarroll and Salin, 1968) f(O,+)
=
2 lom
h db[6if - af(b,~ ) ] J A ( 2 psin v 8/2)
(10)
where p is the reduced mass of A and B, and A the change in the magnetic quantum number of the target. The total cross section is
Qf, = 27r
lom
l ~ r ( h , m ) )db ~h
( 1 1)
No single center expansion, such as (7), truncated to a j n i t e number of basic states, can represent the situation in which the electron is found bound to nucleus A as t + 03, and so methods based on truncating the infinite set of Eq. (8) are only suitable if the charge transfer probability is
T H E THEORY OF FAST HEAVY PARTICLE COLLISIONS
267
small. If several electrons are centered on nucleus B, the generalization is straightforward. If some electrons are centered on A and some on B, the generalization is again straightforward, provided electron exchange is neglected; 4, exp(-e,r) being replaced, in the expansion ( 7 ) , by 4,(rB)xn(rA) exp[-i(c, + q m ) t ] , where xm is the eigenfunction and r ) , the energy associated with the ion A. To include the exchange, these basis functions must be antisymmetrized in the electron coordinates, and this modifies Eq. (8) by the addition of exchange interaction terms. For values of A b 1 , exchange effects are small, and have usually been n e g l e ~ t e d , ~ except of course in discussion of reactions that can occur only through exchange, such as H(ls) + He(1'S) 4 H(n/) + He(n3L)
(12)
A. THE BORNA N D CLOSE-COUPLING APPROXIMATIONS
For sufficiently high energies, the first Born approximation for total cross section for excitation becomes accurate, and many calculations4 have been carried out for simple systems, such as H + H, H + He, H + He+, p + H, p + He. Some of these calculations have not used the impact parameter approach, but it has been known for a long time that the Born approximation in the impact parameter form is equivalent to that in the wave form, for heavy-particle excitation (McDowell and Coleman, 1970), and further clarification of the connection has been given recently by Taulbjerg (1977). Comparison with the results of higher-order calculations suggests that, for total cross sections, the Born approximation is accurate above energies of 200 keV per nucleon for excitation of low-lying, strongly coupled states. This is not so for differential cross sections at large angles, for which it is necessary to allow for the nuclear deflection, either by solving the coupled equations or by using a distorted wave approximation. This is seen rather clearly in the results of a calculation, of the excitation of the 2lS state of helium by proton impact, by Flannery and McCann (1974a), which are shown in Fig. 2. The importance of higher-order terms in determining the high-energy limit of the differential cross section at fixed angle depends on the masses of the colliding particles and has been discussed generally by Shakeshaft (1977).
-
Coupled-channel calculations for H(1s) + H(1s) H(1s) + H(2s or 2p) by Bottcher and Flannery (1971) and Ritchie (1971) demonstrate the importance of exchange in the region A c 1 . ' For details of Born approximation calculations, reference should be made to the recent reviews by Bell and Kingston (1974, 1976).
B . H . Bransden
268
--
-
\
\
lo’
‘
’
\
’
I
’
I
I
’
To go beyond the first Born approximation, expansion (7) can be truncated to include a few discrete target levels and the resulting finite set of coupled equations solved n~merically.~ In many cases, this “closecoupling” procedure is not satisfactory, because it does not allow for the continuum states of the target, which second Born approximation calculations suggest can be important. Using a closure approximation, to evaluate the sum over continuum intermediate states Holt and Moiseiwitsch (1%8) and Holt (1969) showed that for excitation of hydrogen by proton impact, the continuum contributions were particularly important in the excitation of s states. For example, at 100 keV, the continuum contribution increased the cross section for excitation of the 2s level from 0,027 to -O.O36(.rra2,).
-
The close-coupling approximation has been used to calculate cross sections for p + H, H + H, p iHe, H + He, He+ + H , . . . in a series of papers by Flannery (1%9a-e) and Flannery and McCann (1973, 1974a,b,c) and also by Bell and Kingston (1978) and Bell et al. (1973, 1974).
THE THEORY OF FAST HEAVY PARTICLE COLLISIONS
269
B. HIGHERORDERMETHODS Apart from higher Born approximations, a number of methods have been proposed that take into account both discrete and continuum terms in the basic expansion (7). 1 . The Second-Order Potential Model The second-order potential model of Bransden and Coleman (1972; see also Sullivan et al., 1972; Berrington et al.. 1973; Begum et al., 1973) starts from the coupled-channel method, by retaining a finite number of equations of the form (8), but then adds to the potential matrix V,, further nonlocal potentiaIs &,(t, 1’). which take account of the channels not explicitly represented to second order. As in the simplified second Born approximation, the continuum contributions are calculated using closure, and this at present is one of the principal limitations of the model. 2 . Glauber Approximation If all the energy differences (a, - em) are ignored in Eq. (8), an exact solution of the resulting equations can be obtained (Byron, 1971) using closure, in the form
where
Using this approximation to a r , we obtain the Glauber approximation (Glauber, 1959) to the cross section. At least as far as some of the important low-lying intermediate states are concerned, neglecting the energy differences (a, - E,) is not a good approximation, and the overall accuracy of the method can be questioned. However, Dewangan (1978) has shown how to correct the results of a coupled-channel calculation, by igam) in all channels other than those noring the energy differences ( a , treated explicitly. He obtained ~
ut(b,w) =
NF +
a:‘ -
N F O ~
(15)
where uf is the amplitude calculated from the coupled-channel calculation, ufGthe Glauber amplitude, and UFO” a correction term to avoid double counting:
270
B . H . Bransden
This procedure offers a way of removing the disadvantages of the Glauber model. Not surprisingly, numerical results for p + H are close to those of the second-order potential model.
3. Pseudostate Expansions A general method that avoids the difficulty of the target continuum, which can be used in conjunction with the coupled-channel approach, or to calculate the second Born approximation or the optical potential, is to use as a basis members of some complete, but discrete, set of functions, termed pseudostates.s A recent example of this approach is the work of Reading and his collaborators (1976, 1977; Fitchard et al., 1977; Swafford et al., 1977), who have used, for the one-electron system, the set of functions (Zimmerman, 1972) N
The constants are determined by diagonalizing the target Hamiltonian on a finite set of these functions, having chosen the complex numbers c i , so that the more important bound states of the target are represented accurately. Working with a finite basis it is possible to compute the second Born approximation and also to generate the U-matrix approximation. Reading and his collaborators have shown that this technique is often numerically preferable to solving the coupled differential equations formed from the same basic set. It may by remarked that to the extent that a pseudostate overlaps the target continuum, it is possible to calculate ionization cross sections as well as those for discrete excitation. The ionization cross section is just the difference between the total cross section (into all states) and the sum of the cross sections for discrete excitation and elastic scattering . 4 . The Cheshire-Sullivan Model
Several procedures have been proposed that take continuum intermediate states into account but that are not based on Eq. (8). Cheshire and Sullivan (1967) expanded both the interaction and the wavefunction q ( r B ,t ) in spherical harmonics. This leads to a set of coupled partial differential The coupled-channel calculations of Bell et al. (1973, 1974) for the H clude pseudostates to represent the continuum.
+ He system in-
27 1
T H E THEORY OF FAST HEAVY PARTICLE COLLISIONS
equations in the variables rB and t . The solution of these equations, retaining spherical harmonics of up to order L , is equivalent to employing the truncated eigenfunction expansion including all states, both discrete and continuous, with angular momentum up to L. The method was applied to the excitation of the n = 2 levels of hydrogen, by proton impact, in the approximation in which L s 1, which is equivalent to retaining all s and p target states. This procedure is complicated numerically, and it is impractical to apply it widely. However, the results provide a benchmark against which to test other approximations that retain s and p target levels.
5 . Excitation of n
=
2 and n = 3 Levels of Hydrogen
All of the methods discussed have been applied to 2s and 2p excitation of hydrogen and results are also available in some cases for 3s, 3p, and 3d excitation. This is an excellent testing ground as there are no uncertainties due to the complexities of many electron wavefunctions. None of the models can be expected to be satisfactory below -60 keV, where charge exchange is important and methods based on two center expansions are required (Bransden, 1972). The qualitative effects of higher-order terms differ from transition to transition. This can be seen from Table I, where first Born approximation cross sections are compared with the UTABLE I CROSS
SECTIONS FOR
EXCITATION OF HYDROGENATOMS B Y
E(keV)
Cross sections in mi:
Is
1s-2s 60 105 200
B 0.161 0.0975 0.0530
c4 0.329 0.174 0.080 Is-
60 105 200
PROTON IMPACTn
B 0.032 0.019 0.011
CIO 0.301 0.158 0.074
S 0.29 0.16 0.075
3s
CIO 0.089 0.043 0.018
F 0.182 0.108 0.057 Is-
F 0.041 0.024 0.012
B 0.239 0.178 0.120
B 1.36 1.02 0.70
C4 1.23 1.03 0.72
-
S 0.88 0.86 0.67
C10 1.16 1.00 0.72
Is
3p
CIO 0.237 0.182 0.124
2p
F 0.168 0.147 0.109
B 0.019 0.103 0.0071
---f
F 0.95 0.85 0.65
3d
C10 0.083 0.043 0.016
F 0.040 0.023 0.010
B , First born approximation; C4, (Is-2s-2p) close coupling ( I am indebted to Dr. C. J. Noble for calculating these numbers and the next): CIO, (ls-2s-2p-3s-3p-3d) close coupling; S, second-order potential model (Sullivan et a/.. 1972); F, fourth-order calculation with a large pseudostate basis (Fitchard et a / . , 1977).
272
B . H . Brunsden
matrix pseudostate calculations by Fitchard et al. (1977). The pseudostate sets employed were of the form (17) and were rather large with up to lOs, 22p, and 21d states, and it is expected the results will be accurate. In fact, if the d states are omitted, the calculated cross sections agree well with those of Cheshire and Sullivan, which tests the adequacy of the s and p members of the pseudostate set. It is interesting that the d state contributions are significant, but the second Born calculations of Holt and Moiseiwitch (1968, 1969) suggest that f state, and higher, contributions can be ignored. To obtain accurate cross sections, it was necessary to calculate to fourth order;' the second Born approximation is not adequate, except at energies a200 keV. The calculated cross sections agree with the measurements of the total n = 2 and n = 3 excitation cross sections by Park et al. (1975, 1977) (see Fig. 3) down to -60 keV. We can now compare these results with those of the other methods listed earlier, and we shall take the case of (2s) excitation first, because it is sensitive to the choice of approximation, compared with (2p) excitation. From Table I , we see that the accurate cross section (F) is greater than that given by the first Born approximation (B), by about 13% at 60 keV and by 8% at 200 keV. This does not imply that higher-order corrections to the Born cross section are individually small. They are not, and for example at 60 keV the second Born cross section (evaluated without closure by Bransden and Dewangan, 1979) is about twice the first Born cross section (see Table I), and this increase is almost totally cancelled by higher terms. Few of the approximations lead to even a qualitatively correct cross section for this transition. The 2-state 1s-2s close-coupled equations lead to a cross section that is considerably smaller than the first Born cross section and so do the Glauber approximation (Ghosh and Sil, 1971; Franco and Thomas, 1971) as well as the distorted-wave method of Joachain and Vanderpoorten (1973). This is to be expected, as in the case of comparatively weak transitions such as s + s or s + d, we expect several intermediate states to be significant (Bransden and Issa, 1975), up to energies of the order of 600 keV per nucleon. If the important (2p) intermediate state is taken into account as in the 1s-2s-2p close-coupling approximation (Flannery, 1969a-e) or in the work of Sullivan et al. (1972) or of Baye and Heenan (1973), the calculated cross section is greater than the first Born approximation, but greatly exceeds the presumably accurate results of Fitchard et al. From this,
' B. H. Bransden and C. J . Noble (unpublished, 1978) have found that it is sometimes necessary to go as far as sixth-order terms to obtain convergence, and have verified this by comparing with the results of the coupled-channel method, using the same basis set.
THE THEORY O F FAST HEAVY PARTICLE COLLISIONS
273
FIG.3 . Total cross sections for (a) II = 2 and ( b ) I I = 3 excitation. Data points, data of Born approximation: -, U-matrix calculations of Fitchard Park e l a / . (1975, 1977); et a / . (1977) using a large basis set.
---.
it is clear that, for accurate results, a large basis set is necessary. The two-center expansion methods of Shakeshaft (1976) and of Cheshire rt a / . (1970) are on the other hand in fair agreement with the results of Fitchard et al. (1977). For the strong s -+ p transition, the accurate cross section is smaller than the Born cross section and in this case, the second-order potential
B . H . Bransden
274
calculations agree reasonably well with the accurate results. The fourstate close-coupling model (1s-2s-2p) is again rather poor, providing a cross section greater than the Born at the higher energies. The conclusions are perhaps rather unpalatable. It would seem that to be sure of improving on the first Born cross section, it is necessary to both employ a fairly large basis set, and either to solve the coupled equations or to calculate to at least fourth order, and both of these requirements are expensive for systems with more than one electron. At lower velocities ( 6 6 0 keV), it is essential to use a two-center expansion, and even small basis sets will provide cross sections of about the correct magnitude for the more important transitions.
111. Electron Capture from Atoms by Fast lons To calculate electron capture cross sections the basis set must be enlarged so that the rearranged system can be represented asymptotically as t + ? m . At low velocities, molecular orbitals can be taken as the basis functions, and this approach, introduced by Massey and Smith (1933), has been outstandingly successful (Briggs, 1976; Bransden, 1972). For higher velocities (A 3 l ) , it is usually better to make an atomic two-center expansion, and to replace Eq. (7) by WrB, t )
an(b, r)4(rB)exp( - k n t )
= n
+
Cj(b, f)Xj(rA) eXp(-iVjf
iV
rg - 3 i U 2 f )
(18)
1
where X,(rA)is an atomic wavefunction (or pseudostate function) centered on A, and is the corresponding energy. The factor A h , t ) = exp[i(v * rB - 3 u 2 t ) ] allows for the change in momentum of the captured electron, and is included to preserve Galilean invariance (Bates and McCarroll, 1958). Strictly the factor A(rB,t) is only required to take this form in the asymptotic region It1 -+ m, and attempts have been made to impart more flexibility to trial wavefunctions by using other forms (Schneiderman and Russek, 1%9; Levy and Thorson, 1%9; Riley and Green, 1971), but often at the expense of unacceptable numerical complication. The probability amplitudes an,cj satisfy the system of coupled equations8 iH + N; = Ha + Kc, iN+H + I! = Ka + Rc (19) A derivation can be found in Bransden (1972).
THE THEORY OF FAST HEAVY PARTICLE COLLISIONS
275
with
H,
=
(F, IH
A
- I -
)
Fj ,
Hnj = (G,, IH
-
i t 1 Gj)
(20)
where Fn
&(rB) eXp(-ie,t) Gj
E
Xj(r,)A(rB,t) exp(-ir)jt)
(21)
To obtain the cross section for capture these equations are solved, with the boundary conditions lim a,(b, t ) =
t+-m
a,,,
lim c j ( b ,t )
t+-
m
=
0
(22)
and then
At energies such that the velocity of the incident ion is comparable or greater than the orbital velocity of the electron in the target atom, the momentum transfer of the captured electron cannot be ignored. The introduction of the momentum transfer factors A ( r B ,t ) causes a large reduction in the size of the matrix elements K and K compared with H and H [see Eq. (9)], rapidly reducing the importance of the charge exchange cross sections at high velocities compared with those for excitation. However, the evaluation of K and K becomes very time consuming, and may seriously limit the number of terms used in the expansions. A. INTERMEDIATEENERGIES 1 . Two-Center Expansions Although in particular cases, such as capture into the ground state, the expansion method can sometimes be used quite successfully using a limited number of atomic eigenfunctions centered about A and B, contributions from the continuum states in the expansion will be generally of importance. The atomic expansion is particularly slowly convergent if the two nuclei are close, when the wavefunction resembles that of the united atom. To represent the continuum states, pseudostates can be introduced centered on A and B. An alternative and promising approach is to expand the wavefunction about three, rather than two, centers, for example,
276
B . H . Bransden
about A and B and the center of charge C (Anderson et al., 1974; Antal et al., 1975). In this approach, the probability amplitude for capture into the fictitious system centered on C remains finite as t + m. This loss of flux from the genuine excitation and rearrangement channels can be identified with the ionization flux. The most extensive work has been for the proton-hydrogen atom system. In the pioneering work of Cheshire et al. (1970) the pseudostate wavefunctions were taken to be Slater orbitals, chosen to be orthogonal to the hydrogenic Is, 2s, and 2p wavefunctions, which were represented explicitly in the expansion. The parameters in the were chosen to pseudostate orbitals, which were designated % and maximize the overlap with the He+ united-atom intermediate states. Another choice, introduced by Gallaher and Willets (1968), and recently thoroughly investigated by Shakeshaft (1976), is to make the expansion in terms of Sturmian functions, which form a complete, discrete set of which the radial functions satisfy the equation
p,
1 d2 / ( I + 1) [-zp+-2;"---
r
Snl(r) = -
1
2(/
~
+ 1) S n l ( r )
(24)
where the parameter aAlis treated as an eigenvalue. The most extensive results of Shakeshaft include six s and six p states centered about each proton, giving rise to a large computing problem. A rather different approach has recently been introduced by Morrison and Opik (1978). This is too complicated to be described in detail here. First, a modified system of elliptical coordinates is introduced. The wavefunction is then expanded in terms of sets of orthogonal polynomials, which are functions of the new coordinates, and which are chosen so that the asymptotic wavefunctions, as t + k w , can be expressed exactly using a few terms. The method is rather economical and large basis sets of up to 30 functions can be used. The general impression is that the results for the combined (2s + 2p) excitation cross section agree reasonably with the data (Park et al., 1975), and with the pseudostate calculations of Shakeshaft (1976) and Cheshire et al. (1970). Atomic expansion methods have also been applied extensively to the case of the (He2+ + H) system. The elaborate calculations of Morrison and Opik (1978) agree with experiment about as well as those of the atomic expansion of Rapp and Dinwiddie (1972), but presumably are more accurate. The two-electron system (p + He) was studied in detail a few years ago by Winter and Lin (1974) using an atomic expansion with up to 1 1 states. Cross sections for capture into the ground state and the n = 2 and n = 3 levels of hydrogen were computed for energies up to 1 MeV. In general, good agreement was obtained with the experimental data. If many states are required in the expansion method, the calculations
THE THEORY OF FAST HEAVY PARTICLE COLLISIONS
277
become extremely lengthy: however, as Lin (1978) has recently emphasized, there are circumstances in which it is possible to obtain useful accuracy with a two-state coupled system, retaining just the initial atomic state centered on B and the final state of interest centered on A. Except for the case or resonance, the cross section for capture into the ground state exhibits a maximum, near incident velocities u , which are close to the orbital velocity of the target electron c. At these velocities, capture occurs at large impact parameters, for which the wavefunctions can be approximated by the two-state expansion. For ground-state capture by protons from helium the result of the two-state calculations and 1I-state calculations agree closely above the cross section maximum. This does not imply that the two-state model is adequate at high energies ( A >> 1 ) because second-order terms become important for u >> u, and these require continuum intermediate states to be represented in the expansion. For example, in the case of helium the experimental and calculated cross sections agree reasonably up to 100 keV, but by 500 keV the coupled-state values are about twice those given by the data. Making use of these ideas, Lin (1978) has calculated a series of cross sections for capture into the K shell, by fully stripped ions from the K shell of the inert gases. His results are shown in Table 11, compared with the data. The agreement is remarkable. Lin et af. (1978) have used the two-state approximation to calculate the capture into the ground state by protons incident on C, N , 0, Ne, and Ar, and by applying a correction factor to account for capture into excited states (-20%), they compare these results with the experimental data. The agreement is very good, considering the simplicity of the model in the region A = 1 . The case of neon is illustrated in Fig. 4. First-order perturbation methods have also been used for some of these cross sections (Omidvar et al., 1976; Halpern and Law, 1975; Lapicki and Losonsky, 1977) and fail badly; Born and Brinkman-Kramers (BK) cross sections9 often disagreeing with the data by factors of tens or hundreds. The reason for the gross failure of the Born and BK approximations in this instance is not because of the importance of those second-order terms with continuum intermediate states, which are known to dominate the cross section in the high-energy limit (see below), but because of the failure to orthogonalize the initial and final states. This defect is removed in the distorted-wave model of Bates (1958; see also Bassel and Gerjuoy, 19601, which can be derived from the two-state coupled equations and which 9 The Brinkman-Kramers model uses (.ZA/rA)as the perturbation, while the Born approximation uses (ZA/rA)- ( Z , Z , / R ) . The large contribution arising from the internuclear potential in the latter case in some cases leads to cross sections of the correct order of magnitude, but this is spurious since the internuclear potential cannot contribute to the charge exchange probability (see Bransden, 1972).
278
B . H . Brunsden TABLE I I
ELECTRON CAPTUREFROM T H E K SHELLI N T O THE K SHELL,PER TARGET ELECTRONI N T H E TWO-STATE APPROXIMATION^ Cross-sections in 10-20cm2 Process
Energy (MeV)
ulue
qth
Uexp
N7+ + Ne
14 19 24 30 35 20 25 30 12.6 19.0 22.6 14.7 26.3 20 30 46 76
0.79 0.92 0.96
368 342 368 269 195 516 440 360 1.30 2.52 5.40
355b 350 435 330 300 440 430 410 1.79' 2.34 5.83 3.2 13.2 9.7d 29.0 47.7 44.3
OR++ Ne
FQ++ Ne C6+ + Ar N7+ + Ar
Fet
+ Ar
1.08
1.17 0.8 I 0.90
1 .oo
0.42 0.52 0.57 0.42 0.56 0.42 0.52 0.64 0.82
5.1
12.6 25.2 32.4 52.8 39.6
From Lin (1978). Woods ef ul. (1976). Winters ef a / . (1973). Typical experimental uncentainly is +20% Hopkins ef ul. (1976).
automatically allows for the nonorthogonality of the initial and final states. This approximation reduces at high velocities to
where hil = (+ilZB/rAl+l).The appearance of hit in the effective perturbation is essential, if cross sections of the correct magnitude are to be obtaine d. 2 . Asymmetric Charge Exchunge An interesting situation, which may have some simplicity, is when the projectile has a charge that is much smaller than the nuclear charge of the target ( Z , 2u,. Assuming the model is valid in a limited velocity range above u = u,, Olson and Salop (1977) have calculated charge exchange and ionization cross sections for fully stripped ions with 1 s Z s 36 colliding with hydrogen atoms, in the range u = 2-7 x lo8 cm/sec. Experimental data are rather limited (Phaneuf et al., 1977) but no large discrepancies appear to occur. A further application by Berkner et al. (1978) has been to electron capture and ionization by iron ions FeQ+,q = 10, 15, 20, 25, incident on atomic hydrogen, at energies in the range 50-1200 keV per nucleon. If the experimental data, obtained for molecular hydrogen, are divided by two, good agreement is found with the calculated cross sections. B. HIGH-ENERGY CHARGETRANSFER Following the early worklo of Drisco (1955), it has been shown by a number of workers (Brodskii et al., 1970; Dettmann, 1971) that the highenergy limit of the cross section for a charge exchange process is not given by the first-order Born approximation, but by the second Born approximation. The second Born cross section for ground-state capture by a particle of charge ZAfrom a target with charge ZBhas the form, in the high-velocity limit, WB2 =
where
uBK
~BK[0.295f 5nV/211(zA+ ZB)] a.u.
(27)
is the Brinkman-Kramers first-order cross section (+BK
=
n218(zAzB)5/5u12 a.u.
(28)
The approach to the high-energy limit is slow and the second term in (27) becomes large only at energies of several terms of MeV per nucleon. For higher energies than about 9 MeV per nucleon, charge exchange takes place predominantly by a radiative process (see Section I), and so the Some details of Drisco's work are given in Bransden (1965).
T H E THEORY O F FAST HEAVY PARTICLE COLLISIONS
28 1
region in which the u - l l behavior dominates is not one of practical importance. Recently, some further developments have taken place. Shakeshaft (1978) has confirmed the work of Drisco (1955), that the cross section for ground-state capture up to third order behaves at high velocities as =
cBK[0.319
+ .5ru/211(ZA + Z B ) ]
(29)
while Briggs and Duke (1978) have obtained the high-velocity form of the second Born approximation for capture from the ground state in the excited (nlrn) state. The general result is of the form
+ U , U + L I ~ U+~ .
cB2 =
. .
+ u ~ ~ - , u ~ ~ - ~(30) )
Since the Brinkman-Kramers cross section behaves like u-12-21, the is always of the form u - l ' , for sufficiently high velocities. behavior of cBz If we take a finite basis set and solve the corresponding coupled equations, the cross sections will approach the Brinkman-Kramers values at high velocities. This means that even if the continuum intermediate states are well represented, at a certain velocity, by a finite number of pseudostates, when we wish to calculate at a higher velocity a larger basis set will be required. Clearly it is desirable to have other methods, which include the effect of higher order Born terms with less labor. About a decade ago, it was proposed to use the impulse approximation, in which the influence of the potential binding the electron to the target was only included to the lowest order. In the impulse approximation, the probability amplitude for charge exchange is cri(b) =
- i J dq [/:mexp(-iqfr)
(
Xf(fA)
IM
- +(q -
v,
) dt
rA,t)
1
+(q) 1 exp(iq b) (31)
where +(k, r, 1 ) is a wavefunction for Coulomb scattering in the potential is the Fourier transform of the initial bound-state function. The results obtained with this approximation were disappointing (Coleman, 1969). The reasons for this have been analyzed recently by Briggs (1977), who showed that a peaking approximation used to simplify the integrals is only valid for ZA>> ZB, whereas the applications had been to systems with ZA= ZB.In addition, the basic approximation can only be expected to be accurate when the (projectile-electron) interaction is much larger than the interaction binding the electron to the target nucleus, which again requires ZA>> ZR.There is some reason to expect the imZ A / R , and
+,
282
B . H . Bransdrn
pulse approximation to be accurate when used in the asymmetrical situation ZA>> ZB, although applications have not yet been published. The momentum wavefunction $q(q) is sharply peaked about q = 0, while the other terms in the integrand of (3 1) vary slowly. If the q dependence of the slowly varying terms is ignored completely, Briggs shows (see also McDowell, l%I) that the cross section is related to the ionization cross section of the final state f, producing an electron of velocity u ,
where +,(O) is the initial wavefunction (in coordinate space), evaluated at rB = 0. The close connection of charge exchange and ionization cross sections is undoubtedly a general feature of the high-velocity region. Making a less severe peaking approximation, in which only the dependence of the Coulomb wavefunction on q is ignored, it is found that Cf(h,rn) =
where v = Z A / u and N ( u ) = r(l - iu)en”12. Apart from a phase factor, which does not affect total cross sections, this expression has also been derived by Belkik (1977), who has called this result the continuum intermediate-state approximation (CIS). Although this approximation would be expected to be more accurate for ZA> ZB, because the ( Z B / r B ) interaction appears only in first order, the CIS model is remarkably successful when applied to the proton-hydrogen system. Some results are shown for capture from the ground state into the 2s and 3s states in Fig. 5 . The agreement with experiment is excellent. In (3 1) and (33), the internuclear potential has been ignored completely, as this potential cannot contribute to the total cross section. To calculate differential cross sections it is necessary to determine the phase of the amplitudes as a function of 6 , and the internuclear potential must be retained (see Belkic and Salin, 1976). A more symmetrical approximation, in which the distortion of the wavefunction by both of the Coulomb fields ( Z A / r A ) and (ZB/rA)appears in the transition amplitude, is the continuum distorted-wave approximation (CDW). It was originally introduced by Cheshire (1%4), and put on a sound theoretical foundation by Gayet (1972), who showed it to be the first term in the multiple scattering series of Dodd and Greider (1966). It takes the form
THE THEORY OF FAST H E A V Y PARTICLE COLLISIONS
30
I
l
50
70
283
l 100 200 300 500 700 1000 IMPACT ENERGY IKrV)
FIG. 5 . Total cross sections for capture into the (2s) and (3s) levels of hydrogen by protons incident on atomic hydrogen in the ground state. The (3s) cross sections are multiplied by 0.1. -.-, Impulse approximation of Coleman and Trelease (1968); ---, CDW model of BelkiC and Gayet (1977); -, CIS model of Belkid (1977). Experimental data for p + H(ls)H(2s) + P: A, Rydinget a / . (1966): V. Andrew er ( I / . + (1%8); 0 , Bayfield (1%9); Bayfield+ (1969); A, Morgan et a/.+(1973): 0, Hughes el a / . + (1971); V, Ryding er a/.+ (1966); +, transformed from data on H, target (see BelkiC, 1977). Experimental data for p + H( Is) + H(3s) + p: 0, Hughes et a/.+ (1970): Ford and Thomas+ (1972); +, transformed from data on H2 target (see BelkiC, 1977). The data of Rydinget a / . have been normalized to those of Bayfield at 44 keV.
.,
+,
C4b.x)
=
-i
I_m_ dr
N(uA)N(uB)
IFi(rB, f ) I F I ( i V A , l ;
v
(Gg(rA9
r A
t ) l F 1 ( i u B , Iv;
+ iUrA))
rB
+ ivrB)IO (34)
with vA = Z J v , yB = ZB/u, and where 0 is a certain differential operator. Recently, BelkiC and Gayet (1977) have applied the CDW model to electron capture from hydrogen by protons and alpha particles for energies above 25 keV and up to 10 MeV per nucleon. Good agreement with
B. H . Bransden
284
the experimental data was found for the total capture cross sections and for capture into the individual s states of hydrogen (see Fig. 5 ) . The data for capture into excited p and d states of hydrogen are rather uncertain and definite conclusions could not be drawn in that case. Similar calculations for electron capture by protons and alpha particles from helium also agree well with the high-energy data. This work has been extended by BelkiC and McCarroll(1977), who have calculated cross sections for capture by highly charged ions (1 s Z , s 30) from atomic hydrogen. For those cases for which experiments have been performed, the theoretical results are in good agreement with the data. The capture cross section into the (2s) state as a function of projectile charge (Z,) is shown in Fig. 6, where it is compared with the Brinkman-Kramers first-order cross sections. The failure of the first-order approximation is seen clearly. For a few systems, at high velocities, not only total cross sections but differential capture cross sections have been measured. First-order theories have been used to analyze the data (Rodgers and McGuire, 1977; Omidvar e? al., 1976), but as we have seen, the simple Brinkman-Kramers or Born models fail to provide cross sections of the correct magnitude. In the impact parameter formulation, the differential cross section is (Belkii and Salin, 1976)
0
4
8
12
16
20
24
28
PROJECTILE CHARGE 2, l a u )
FIG.6. Total cross sections for electron capture into the (2s) level by fully stripped ions of charge ZA from atomic hydrogen in the ground state, at o = 5 a.u. -, CDW model of BelkiC and McCarrol (1977); ---, Brinkman-Kramers first-order approximation.
T H E THEORY OF FAST HEAVY PARTICLE COLLISIONS
da
-=
dR
285
h db exp[i6,(b) + i8f(h)]Jo(2,vhsin 8/2)cf(h,z)12 (35)
/ipv
where p is the reduced mass, and 6 , ( h ) ,af(h) are the phases associated with elastic scattering in the initial and final states due to any potentials omitted in calculating the amplitude cf(b,=). Thus, in expressions (311, (33), or (34), the internuclear potential ( Z , Z , / R ) has been omitted so that to lowest order in the ratio of the electron to proton masses, exp i(6, + &) = b2lu,with u ZAZB/v. Calculating cf(b,E ) from the CDW model, Belkic and Salin (1976) have calculated da/dR for electron capture by 6 MeV protons from argon, and by 293 keV protons from helium. Good agreement is found with the data, in both cases. We illustrate the results for helium in Fig. 7 .
-
I
I
I
I
\
\
\ \ \ \
\
\ \
\ \
0 02
0 0.4
e
0 06
0.08
Ideg)
FIG.7 . Differential cross section (laboratory system) for electron capture by 293 keV protons from helium. CDW model of Belkic and Salin (1978) including the phase due to excluding the internuclear potential; A,experimental data of the internuclear potential: Bratton el u / . (1977).
-.
---.
286
B . H. Bransden
IV. Ionization and Charge Exchange into the Continuum A. TOTALCROSS
SECTIONS
At sufficiently high velocities, total cross section for ionization can be determined from the first Born approximation, either in the quanta1 plane-wave form (Bates and Griffing, 1953)or in the impact parameter formulation (Bang and Hansteen, 1959). Both treatments provide the same total cross sections provided the same atomic wavefunctions are employed, and a review of the results of calculation has been given by Bell and Kingston (1974, 1976). For collisions between light ions, the Born approximation is accurate for total cross sections at velocities such that A = v / v , b 2; but when ZA (To 3 p A r )
- 3 s Ar
-2pAI' - 2sAr+
3P 3s
- 2 p Ar - 2 sAr
-10
> (3
U
W
z W
9 z
2P 2s
- lsAlf - IsAr
0 U F
$
1
W
-100
1
.-.-a
/*=--/ Atomic Levels
Go+ Levels IN T E RNUCL EAR SEPARATION ( A.U.) F I G . 22. Correlation diagram for the asymmetric A]+-Ar system calculated by Larkins (1972). In this case the united atomic system is Ga+.
collision, after the 2pu vacancy has been formed by rotational coupling 2p77 + 2 p a at small R . Writing w for the probability of the lsu-2pa transition, the ratio of K vacancies in the heavy to the light particle emerging from the collision is given, according to the Demkov model, by M'
P,, = -1 --w
e-zx
( 10a)
where U K ( l ) , uK(2) are, respectively, the mean K electron orbital velocities of the collision partners, u the projectile velocity, and m the electron mass. This relation is plotted in Fig. 23 and is seen to represent the experi-
356
E . H . S . Birrhop
2 2x
FIG.23. Ratio of K vacancy cross sections produced in target to projectile, uK(2)/uK(1), as a function of the parameterx [Eq. (lob)] (given by Meyerhofet al., 1977). The beams used were 0,30 MeV Br; 0 , 43 MeV Br; A, 47 MeV I; V, 62 MeV I; 0, 202 MeV Kr; X , 326 MeV Xe; f , 470 MeV Xe.
mental results very well over eight decades. Briggs and Taulbjerg (1975) applied PSS theory to calculate the cross section for the 1s excitation of Ne in the collision of Ne+ with 0 through the 2pw-lsu radial coupling transition and obtained good agreement with the measured values of Stolterfoht (quoted by Briggs and Taulbjerg). Quite generally, for all values of ZH/ZL, we write the following Meyerhof and Taulbjerg (1977) for the cross sections wK(H), uK(L) for K shell vacancy production in each of the two partners
(+K(H)= 4 1 s ~ + ~ )wCT(~PV), vK(L)
=
(1 -
+
W ) ~ ~ P C T ) VK-L
(11)
where 41sw) and wCT(2pu) are cross sections for forming a lsw vacancy by direct ionization and by the radial transition 1sw-2pu, respectively, and CTK-L is the cross section for transfer of the K vacancy of the lighter partner to an L vacancy of the heavier partner. When ZH/ZLis not too large, the terms 4 l s u ) , cK-L are negligible compared to the other terms, so that
~ K ( H+) ~ K ( L=) 4 2 ~ ~ ) This total cross section is shown in Fig. 24 for the data shown in Fig. 15. The full line represents the estimated contribution of multiple collisions in
357
INNER-SHELL IONIZATION
43-MeV Er 5.
L7-MeV I
A)
fl
0.
Y
0
40
60
00
22
FIG.24. Sum of projectile and K vacancy production cross sections, a,(tot) for 43 MeV Br and 47 MeV I plotted against Z of target (figure reproduced from Meyerhof and Taulbjerg, 1977). The solid curves give the computed contribution from multiple collisions.
solid targets (i.e., of ions that enter the collision with L shell vacancies) to the total cross section. The experimental results appear to reflect the detailed structure of the variation with Z arising from multiple collisions. For ZH >> ZL the K vacancy sharing effects become negligible, (+K(H) = dlscr) and (+K(L) = cK-L. The rise in crK(H) + (TK(L)in regions of Fig. 24 marked (A) is attributed to the (+K-L term (Meyerhof rt af., 1977). Vacancies in the M shell of the heavier partner, either when it enters the collision o r when produced in an early stage of the collision, are shared between the 3 u and other MOs. At small separations rotational coupling causes vacancy transfer to the 3dcr orbital. Radial coupling induces transitions to the 2su MO as the partners separate and they emerge from the collision with the vacancies shared between the K shell of ZLand the L shell of Z H .For a given projectile Z the cross section (+K-L will vary as the target Z varies through values where EL(H) = EK(L) when the 3d and 2 s orbitals ~ come close together, thus increasing the 3dcr-2scr transition probability. A similar effect is expected when E d H ) = EK(L), thus interpreting the fluctuations in cross section with Z 2 noted in Fig. 15. The Demkov -Meyerhof model is not successful in representing the results of K - L vacancy sharing (Boving. 1977; Meyerhof, 1976; Stolterfoht er al., 1978). Instead Meyerhof has interpreted the results in terms of the much more complicated Nikitin (1970) formalism. In this formalism an analytic expression for the energy splitting AE(R)between the two MOs involved is given in the form
E. H. S . Burhop
358
Ae(R)
=
A[l - 2 cos 8 exp{-a(R - R,)}
+ exp{-2a(R
-
Rp)}]1/2 12(a)
The four parameters AE, R , , a,8 are determined from a least squares fit of AE(R)to the calculated energy difference of the two states. The sharing ratio is then given by PSR
exp[2rA~cos2(b8 ) / a u ] - 1 = exp[2rA~/au]- exp[2nA~cos2 3O/av]'
(12b)
Woerlee et a l . (1978) have measured the K-L vacancy sharing ratio in Ne-Kr collisions using Ne ions of energy 100- 1100 keV. The DemkovMeyerhof theory gave values that were several orders of magnitude too low. On the other hand the Nikitin formalism with 8 = 38"; a = 4.7, AE = 27.6, R, = 0.35 a.u. represented the results very well, as is shown in Fig. 25. Perhaps with four disposable parameters it could scarcely fail to do so!
7 . Impact Parameter Studies of Heavy-Particle Collisions It has already been pointed out in connection with inner-shell ionization by electrons that differential cross section studies provide a more sensitive test of theoretical models than do measurements of total cross secI
c
3 60 11, \ \
FIG.25. Ratio P S Rof Kr 2p to Ne 1s vacancy production cross sections plotted against l / u taken from Woerlee ef a / . (1978). The experiment points were measured by Woerlee et a / . (1978) for projectile energies 100-1100 keV. The full Line is the adjusted fit to the four-parameter Nikitin (1970) formalism. The broken line is the Demkov- Meyerhof estimate. The insert shows the calculated gap between the 3, and 4u molecular orbitals.
INNER-SHELL 1ONlZATION
359
tions. Since it is legitimate to treat the motion of the heavy particle classically in the velocity region below the velocity of the inner-shell electrons the impact parameter derived from a measurement of the momentum change of the heavy particle is meaningful in such collisions. The impact parameter method of calculating cross sections lends itself readily to comparison with the results of such experiments. In these experiments X-ray or Auger electron emission is measured in coincidence with ions scattered through a given angle. The energy loss of the projectile ion is also measured. The results of recent impact parameter studies of the inner-shell ionization of Au and U by 4 MeV protons are shown in Fig. 26 (Clark et al., 1975). They are compared with the relativistic SCA calculations of Amundsen and Kocbach (1977). The agreement is surprisingly good both with respect to shape of the distribution and also absolute magnitude (within 15% of the experimental values). The nonrelativistic SCA calculations, while giving the shape reasonably well, give values of the cross section about one quarter of those observed. Figure 27 shows measurements
0
200
400
600
lmpoct
porometer.
b (fm)
FIG.26. Measurements of the product bP(b) as a function of b [ P ( b ) = probability of K shell ionization by projectile passing target atom with impact parameter b for proton collisions in Bi and U (taken from Clark er a / . , 1975)]. The measurements are compared with the relativistic (RSCA) calculations of Amunsden and Kocbach (1975) and nonrelativistic (SCA) calculations of Hansteen er a / . (1975). The broken line shows the fit of RSCA when normalized to the experimental value of the total cross section.
3 60
E . H . S . Burhop
00
Proton energy (keV) FIG.27. Measurements of Sackrnann ef a / . (1974) of the probability P x ( b )of K X-ray production in Ne+-Ne collisions as a function of impact parameter 6 for an ion energy of (a) 235, (b) 363 keV. The curves give the theoretical predictions of Briggs and Macek (1972). [Reproduced from Briggs (1976).]
of Sackmann et al. (1974) of the experimental yield of Ne K series X rays as a function of the impact parameter for Ne+-Ne collisions of 235 and 363 keV. This is of particular interest since it is compared with the results of the theory of Briggs and Macek (1972) whose total K shell ionization cross sections were compared with experiment in Fig. 21. The left-hand
INNER-SHELL IONIZATION
36 I
(experimental) scales can be converted to the right-hand (theoretical) scales by multiplying by n/wKwhere n (= 6) is a statistical factor related to the 2pr, MO vacancy fraction and wK is the fluorescence yield. Comparison of the scales at each ion energy implies wK = 0.0228 at 235 keV and wK = 0.0276 at 363 keV. For the higher-energy curve the flat adiabatic maximum occurs at intermediate impact parameters. The narrow maximum at small impact parameters was explained by Bates and Sprevak (1970) as corresponding to a sudden nonadiabatic rotation of the internuclear axis at an impact parameter corresponding to 90" scattering. Since this is one of the few reactions for which a serious theory is available, the agreement with experiment is gratifying. Figure 28 shows the experimental results of Schuch et af. (1977) on the impact parameter dependence of the K shell vacancy-sharing ratio P,,(b) = PK(H)/PK(L) for collisions between 35 MeV CI ions on Ti and Ni. Comparison is made with a simple analytical expression
P,,(b) = expL-2 A E y ( b I l u 1
(13)
derived by Briggs as an approximate solution of the coupled equations describing the interaction. In (13) A E is the energy difference between the two Is energy levels, u the particle velocity, and y ( b ) a function containing the impact parameter dependence. P,,(O) = P,, [Eq. (lo)] of the Demkov-Meyerhof theory.
J-3-07
1000
2000
3000
4000
b Cfml
FIG.28. Measured vacancy-sharing ratios PK(H)/PK(L) as a function of impact parameter b for 35 MeV CF-Ti and CI+-Ni collisions (Schuch P I d . , 1977). The lines represent the expression (13).
E. H. S . Burhop
362
IV. Radiations Following Inner-Shell Ionization A. SURVEY OF STRUCTURE IN
THE
SPECTRA
I . Introduction The existence of satellite lines in X-ray spectra and their interpretation in terms of transitions in atoms multiply ionized in inner shells has been long known (Compton and Allison, 1935) while Auger satellite spectra from such multiply ionized atoms were identified more recently (Mehlhorn, 1965). Recently several new types of fine structure in X-ray and Auger spectra have been observed, both on the low- and high-energy sides of diagram lines. Many of these effects are specifically associated with excitation in heavy particle collisions. 2 . Double-Electron Radiative Transitions We first discuss radiative transitions in which two electrons are involved. The first of these, the radiative Auger effect, discovered by Aberg and Utriainen (1969) appears as a structure on the low-energy side of a K a doublet. It arises from an Auger-type transition in which a K shell vacancy is filled by an L electron but the second electron, instead of being ejected into the continuum with the full available energy as in the normal Auger effect, is promoted to a higher discrete level, or even into a lower-energy state of the continuum. The remaining energy is radiated. The structure, shown in Fig. 29, on the low-energy side of the S K a doublet, excited by electron impact, contains both discrete structure and a continuum. The former type of radiative Auger effect associated with the specific transition (ls)-'(2p)-" += (2~)-~(2p)-"+' + hu and produced by heavy-particle impact has been observed by Jamison et al. (1975) and renamed "radiative electron rearrangement" (RER). A further type of double-electron transition leading to the emission of Kaa radiation is more conveniently discussed in the next paragraph. 3 . Radiative Transitions in Atoms with Doubly Ionized K Shells
There has been much interest in X-ray satellite lines emitted following transitions in an atom doubly ionized in a K shell. These, known as hypersatellites Kcr etc.) seem to have been first observed by Catterall and Trotter (1958) in the X-ray spectra of Li and Be. They have also been observed in the X-ray spectra of isotopes such as 203Hgthat decay by K capture and in which a second K electron may be ejected by internal conversion of accompanying y radiation @berg et a l . , 1976; Cue et al., 1977).
363
INNER-SHELL IONIZATION
2150
2100
2050
2000
1950
ENERGY ( c V )
F I G .29. Structure observed on the low-energy side of the K a doublet of sulfur. The radiative Auger effect is shown in (b). The peaks E , , E2are due to the promotion of an L electron to a higher-energy bound state during emission of the photon (RER).The rest of the structure (peaks ABCD) is due to electrons emitted with the photon [after Aberg and Utriainen (1969)l.
Since double K-shell ionization is copiously produced in collisions of fast heavy atoms, the K& line is seen prominently in such collisions. These double K-shell vacancies can also be filled by a correlated transition of two electrons to the K shell with the emission of a single photon. X radiation due to the transition (ls)-, + ( 2 ~ ) has - ~ been observed by Wolfli et al. (1975) and referred to as Kaa to distinguish it from K&. While the hypersatellite K& is close to the parent K a doublet (with energy some tens or hundreds of eV greater, depending on the atomic number) the K a a is in a different part of the spectrum with energy a little more than double that of Ka. Figure 30 shows the Mn K& line obtained by Cue (1976). It is seen to resolve into a doublet like its parent K a line. However, the intensity ratio K&,:K&, corresponding to decay to the (ls)-l( lp)-y3P1) and
E. H . S . Burhop
3 64
7
a)
Mn 1061
r
-200’
WAVELENGTH 1
I
I
I
265
264
263
2d2
WAVELENGTH
(n’)
1
I
261
2do
(dl
I
Ib9
1
’
FIG.30. K spectrum of Mn showing structure in the K& hypersatellite. (b) Detail of the part of the spectrum within the rectangle shown in (a). [Adapted from Cue (1976).]
(ls)-l(lp)-l(lPJ states, respectively, is found to vary from around 0.1 for elements near Z = 30 to 1.9 near Z = 80 (Briand et a / . , 1976). This provides an interesting test of the onset of intermediate coupling in inner shells with two vacancies, since in pure LS coupling this ratio should be 0 and in j - j coupling, 2. Figure 3 1 given by Stoller et a / . (1977a) shows the Fe K series spectrum given by collisions of 40 MeV Fe projectiles in Fe. The positions of both the Kaa and K& satellites relative to the main diagram lines are seen. Stoller et al. measured the energies and intensities relative to the parent K a line of the two satellites. The energy separations from the parent line
-
3 65
INNER-SHELL IONIZATION
-I
W
z Z
U I V U
W
a Z 3
8
J Z W
z
a U
w
a I-
z
3
8
II
I
6.5
I
7
7.5
J
PHOTON ENERGY (keV) FIG. 31. Fe K a spectrum obtained by Stoller P t ul. (1977a) in 40 MeV Fe+-Fe collisions, showing both the K& and Kaa lines.
agree reasonably with the calculations of Gavrila and Hansen ( 1978). The branching ratios n(K&)/n(Kaa) were found to increase from around 1000 for Z = 13 to 4000 for Z = 26, about double that calculated by Gavrila and Hansen. Some of this discrepancy could come from uncertainty in the fluorescence yields, which would not in general be expected to be the same for the two transitions. Stoller et nl. found the ratio n(Ka)/n(K&) to vary from 60 to 188 over this range of Z . There is considerable evidence for the sharing of double K vacancies in
E. H. S . Burhop
366
ion-atom collisions. In symmetric collisions it is found that the K a a yields are nearly the same for the Doppler-shifted projectile X ray and the unshifted target X ray. This has been observed for Fe-Fe collisions (Stoller et al., 1977a), Ge-Ge and Ni-Ni collisions (Frank et al., 1976), and also in Ni-Ni collisions by Greenberg et a l . (1977). Lennard et a l . (1978) have observed double K vacancy sharing in the bombardment of Fe foils by 50 and 95 MeV Cu beams. Figure 32 shows the X-ray spectrum observed in these experiments. Peaks corresponding to both the Ka a and KaP lines are apparent for both Cu and Fe. MacDonald et al. (1978) in a sophisticated experiment have measured coincidences between K& and ordinary K a satellite emission from successive stages of deexcitation of Ar and S atoms doubly K-ionized in collisions of 32 MeV S ions in gaseous Ar. They found that about 15% of all double K vacancies were to be found in the heavier collision partner, Ar. Both they and Lennard et al. (1978) pointed out the available data on double K vacancy sharing could be understood in terms of a DemkovMeyerhoff type of theory involving radial coupling between the lscr and 2pw MOs (Section III,C,7). 4 . Three-Electron Transitions
Afrosimov et al. (1976) have observed three-electron Auger transition processes closely analogous to the two-electron process involved in K a a
1o6
E
._ ZI
y
lo4
X
Io3 2
10
FIG.32. Cu and Fe Kaa and Ka!p lines are both produced in the impact of 55 MeV Cu+ ions on Fe, thus providing direct evidence for double K vacancy sharing (given by Lennard et a / . . 1978). The Fe and Cu Ka! and KP lines are severely attenuated by appropriate absorbers.
3 67
INNER-SHELL IONIZATION
and K a p emission, the emitted photon being replaced by a third electron. Figure 33 shows the Auger electron spectrum observed from 50 keV Ar+-Ar and CI+-Ar collisions. For double L shell ionization of Ar they estimate the cross section ratio a(LL-MMM)/a(LL) to be 3.8 x lop4.
5 . Radiative Electron Capture ( R E C )b y Fast Highly Stripped Heuvy Ions Schnopper et a l . (1972) observed structure on the high-energy side of the K a spectrum of high-energy highly stripped S and Br atoms (including charged states up to 17+ in S and 23+ in Br) on targets with Z from 5 to 92. It is interpreted as electron capture radiation arising from the capture of electrons, either free or bound in target atoms. The spectra were consistent with this interpretation. In practice in atom-atom collisions it is often difficult to distinguish between electron capture radiation and molecular orbital radiation discussed in Section IV,A,6, especially at high energy, where the projectile velocity is comparable with that of the bound electrons and the REC becomes important.
100
200
300 E,,
400
500
600
ev
FIG.33. Auger electron spectrum from 50 keV Ar+-Ar (0) and CI+-Ar observed by Afrosimov er a / . (1976) due to transition L2,3LZ.3-MMM.
(X)
collisions
368
E . H . S. Burhop
6. X Radiation during Quasi-Molecule Formation
Saris et al. (1972) reported the observation of an X-ray band of radiation on the high-energy side of the characteristic Ar L series radiation when 70-600 keV Ar ions were incident on various targets. They attributed this radiation to transitions to vacant orbitals of the pseudomolecule produced during the collision of the two particles. Figure 34 shows the X-ray spectrum obtained by Kaun et al. (1976) from the impact of 67 MeV Nb on Nb. The two continua marked C1, Cz are attributed to molecular (MO) X rays. The subject of MO X radiation has become so important in relation to heavy-particle collisions during the past few years that we devote the next section to it.
B. MOLECULAR ORBITAL RADIATION Under conditions in which the quasi-molecular picture describes the collision between projectile and target atoms, i.e., ZL = ZH, there is a small but finite probability for a radiative transition to a vacancy in one of the inner molecular orbitals. For example, a characteristic collision time could be taken as tc -- ao/uZUA, where a,, is the Bohr radius, u the projectile velocity, and ZUAthe total charge of the central atom. The lifetime of a K shell vacancy in the united atom can be written T = 5 x 10-'OZ~tsec, so that the chance of filling a lscr vacancy during a collision is lO.Za.,. / u , where v is expressed in cm sec-'. For collision of two atoms with Z = 50, u = log cm sec-' (correspondingly to -50 MeV kinetic energy), this
-
FIG.34. The X-ray spectrum produced by the impact of 67 MeV Nb+ ions on a Nb solid target. The regions C,, Cpare due to molecular orbital radiation. CI is produced by a single collision, C , by a double collision. The Cu absorber greatly reduces the intensity of Nb K series radiation (taken from Kaun er a / . , 1976).
INNER-SHELL IONIZATION
3 69
-
chance is so that the intensity of the MO X radiation is expected to be very weak in comparison with the K series radiation of the separated atoms. The energy of the radiation will depend on the energy separation between the initial and final MOs. A glance at any correlation diagram shows that in general this separation varies markedly with R. The separation generally varies most for small R, where the chance of a radiative transition is greatest. The spectrum of radiation forms a continuum and is not distinguishably characteristic of the particular molecular system involved. This has led to the radiation sometimes being called “noncharacteristic.” But this is a misnomer. This lack of distinguishable frequency characteristics of the radiation has so far rather quenched the exciting prospects of using the MO radiation to determine the spectra and thence the energy levels of the united atom. Since by colliding two U atoms one could obtain a united atom of Z = 184 the phenomena that could emerge from the study of the spectra of such a system would indeed be fascinating. The interpretation of the MO X radiation as reflecting a property of the transient quasi-molecule formed during the collision is strongly supported by the observation that the Doppler shift of the radiation depends on the CM velocity and not on the projectile velocity (Meyerhof et a l . , 1975; Folkmann et al., 1976). The continuous nature of the MO X radiation makes it difficult to separate from various continuous backgrounds such as electron capture radiation and various kinds of bremsstrahlung-nuclear-nuclear bremsstrahlung, bremsstrahlung produced by secondary electrons, and inner bremsstrahlung, which occurs in the process of ionization of an atom and is therefore sometimes referred to as radiative ionization. Fortunately these background effects are of greatest importance under higher-energy conditions, when the projectile velocity is at least of the order of the velocity of the inner-shell electrons. For this reason most of the studies of MO radiation have so far involved conditions in which u < uK and ( Z , Z , ) / ( Z , + Z,J is small. I , Characteristics of the MO Radiations and Their Interpretation
The features of the MO radiation are very clearly demonstrated in the results of the Dubna group shown in Fig. 34 for the collision of two Nb atoms. It is interesting to compare this spectrum with that obtained by the same group for MO X radiation produced in collisions between Kr atoms with a projectile energy of 42 MeV. This is shown in Fig. 35. In Fig. 34 the two continuum regions C1, Czare very prominent. In Fig. 35 only C1appears: CBis completely absent. The difference between the two cases is
E . H . S. Burhop
3 70
2 6 z W 5
5
8
a 4
W
z 3 3 J
1
0'
'
I 10
20
30
40
ENERGY ( K e V )
FIG.35. The X-ray spectrum produced by the impact of 42 MeV Kr+ ions on Kr. In the gas the two-collision process is not possible and only the C , region of MO radiation is seen. [From figure given by Kaun er a / . (1976).]
attributed to the state of the target, solid for Nb, gaseous for Kr. The Cz continuum is ascribed to a two-collision process. In the first collision a K shell vacancy is produced in the projectile. For 50 MeV Nb atoms the vacancy life time corresponds to a flight path -lo-' cm, several times greater than the mean distance between atoms in a solid. The Nb projectile then enters the second collision with a K shell vacancy that branches equally between the lscr and 2pm orbitals of the quasi-molecule (Section III,C,6). The Cz continuum is emitted in radiative transitions to the Ism orbital, which transfer the vacancy to an outer orbital. The C1 continuum on the other hand is ascribed at least partly to a one-collision process. The inner-shell vacancy is produced in say the 2p7r MO at an early stage of the collision. At small R the vacancy is transferred to 2pa by rotational coupling. The C , continuum is emitted by radiative transitions to the 2pu orbital as the atoms separate. C , continuum radiation can obviously also be emitted in the two collision process but the Cz continuum requires two collisions. The mean separation of atoms in Kr gas at normal pressures is greater than lo-' cm so that the two-collision process cannot occur and the Cz continuum is absent in Fig. 35. The situation is illustrated in Fig. 36, which shows the correlation diagram of the Nb-Nb system. The Cz continuum is more energetic than C owing to the dipping of the Ism MO toward the 1s level of the united atom. The dip in the 2pcr orbital at R = 5 x cm ensures that the C1continuum in turn lies on the high-energy side of Nb K series radiation. Further evidence for the two-collision process is provided by the exper-
IN NER-SHELL ION IZATION
lo2
10'
lo4
37 1
lo5 R ( f r n )
6oi I E(keV)
to the dipping of the Isu MO, the transition giving rise to C1is more energetic than that giving rise to C I . [After Kaun rr a / . (1976).] FIG. 36. Correlation diagram for the N b + - N b system. Owing
iments of Saris and Hoogkamer (1977) on MO X radiation produced by 200 keV N + collisions in gaseous Nz and N H 3 . It is seen in Fig. 37 that the MO radiation of the N-N system is seen in N z , not in NH3. A twocollision process can take place with N z , the incident N + ion suffering K ionization in collision with one N atom, and transitions to the vacant I s m or 2pu MOs occur during collision with the second N atom. In NH3, however, the initial K vacancy in N + has been filled long before it reaches an N atom in another NH3 molecule.
2 . Anisotropy of Quasi-Molecular X R a y s The MO X rays are not emitted isotropically. This was first observed by Greenberg et al. (1974). Figure 38 (Wolfli et al., 1976a) shows the measured anisotropies, defined as [1(90")/1(30")]- l , where I ( @ ) is the intensity of the radiation observed to an angle 6 to the beam direction. Figure 38 shows that the anisotropy depends both on the collision partners and
372
E. H. S. Burhop
I
I
1000
2000
I
3000
X - R A Y ENERGY [eVl
FIG.37. 15 MeV N + ions bombarding gases targets of N 2 and NH,. The MO radiation is seen in the former but not in the latter case (taken from Sans and Hoogkamer, 1977).
on the quantum energy of the radiation. A surprising feature of the MO X radiation revealed in this figure is that it extends beyond the energy of the K a transition of the united atom of charge Z , + Z H ,although of course it is very weak in this region. The history of the discovery of anisotropy is of some interest. It was originally predicted by Muller and Greiner (1974) as the result of Coriolis-induced radiative transitions due to the effect of the rotation of the internuclear axis on the electronic motion. The fact that the photon energy extended beyond the united atom limit suggested a dynamical origin of the effect but the theory of Muller and Greiner could not account for the large size of the anisotropy. The importance of the anisotropy is that it appears to be the only property of the MO radiation seen so far that is characteristic of the particular collision and seems capable of providing information about the spectroscopy of superheavy united atoms. In this respect Stoller et al. (1977b) have investigated the main anisotropy peak for systems with a combined atomic number ZUA= ZH + ZLbetween 26 (AI-AI) and 94 (Ag-Ag) and have found a linear relation between the X-ray energy E, at the inflection point of the positive slope in the observed anisotropy peak and the K a
373
INNER-SHELL IONIZATION
20
801
0 +
,-..,:"i" 0
4 20 0[;
0
C F I e- N- I t 4 1 } ~ ' ~ ~ K r - Zr
45
40
20
,
200 MeV
4
30
40
50
60
PHOTON ENERGY (keV)
FIG.38. The ratio [1(90"/1(30")]- I for four symmetric and three asymmetric systems. In each case a marked anisotropy depending both on the systems and the photon energy is apparent (given by Wolfli el a / . , 1976a). The energies of the K a transitions in the united atoms are indicated by arrows.
transition energy of the united atom E # . They find E, = (1.08 & O.O3)E",. However, E; is the nonrelativistic transition energy. This suggests that the effects seen are molecular effects and that true "united atom" effects involving large relativistic corrections to the energy levels have not yet been seen.
3. Possibility of Positron Emission in Collisions between Heavy Atoms The perspective has already been referred to of using energetic collisions of heavy atoms to study the spectroscopy of the united-atom states that are found transiently. This opens up some fascinating possibilities that have been reviewed by Betz et (11. (1976). For a point nucleus, according to Dirac's theory the binding energy of all the j = 4 states become infinite at Z = 137. For an extended nucleus with radius R = 1.2 x 10-'3A#3, different states reach a critical binding energy of 2mc2 when they "dive'' into the negative-energy continuum at Z values of 170 for the Is state,
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3 74
E . H . S. Burhop
184 for the 2p state, and so on. Figure 39 shows the estimated binding energy of these states as a function of Z as estimated by Muller et al. (1972, 1973). For a fully stripped atom with Z greater than 170, the empty atomic level becomes degenerate with the vacuum. Two electron-positron pairs separate out. The electrons are captured into bound states and the positrons are repelled to infinity and can be observed. This process is quite distinct from dynamical electron pair production processes. Betz et al. give a rough estimate of the cross section for positron production in a process such as this in the collision of two U atoms as rising from a threshold at a U+ ion energy of 250 MeV to 5 x cm2 at 800 MeV. The difficulty of separating spontaneous positron emission from the usual dynamical pair production process in a Coulomb field will make it difficult to identify spontaneous positron emission convincingly. To achieve spontaneous sec. positron emission the U ions would need to stay together for
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C. MISCELLANY 1 . Auger Electron Spectra The past 20 years have seen continued refinements in the theory of Auger electron spectra, each improving the agreement between calculation and observation, namely, the introduction of intermediate coupling (Asaad and Burhop, 1958), relativity (Asaad, 1959), and configuration interaction (CI) between residual ion states (Asaad, 1965). Even with these refinements discrepancies remain, even for the K-LL spectrum. Recently Howat et a l . (1978) have introduced two new refinements, orbital relaxation and intercontinuum interactions between the different 2Sfinal states. Since the initial hole is a 2S state the final two-hole-one-continuum state must also be 2S for each of the residual two-hole states 'S, 1,3P, 'D. Howat et al. showed that the usual expression for the transition rate used by Wentzel (1927) had to be supplemented, in lowest order, by an extra term containing the intercontinuum coupling (ICC). Howat (1978) applied the result to calculate Auger rates for Mg K-LL transitions in LS coupling as various interactions are added. The results are given in Table I and are seen to agree with experiment within the stated errors except for the ( 2 ~ ) -'S~ transition, which is still a little too small. Another important contribution to the theory of the Auger effect has been made by Niehaus (1977), who has added postcollision interactions (PCI)to the theory. The conventional Wentzel theory separates the production process of the inner-shell vacancy from its subsequent decay. In most cases the lifetime of the vacancy is long enough to make this a good approximation. If, however, the inner-shell vacancy is produced by a
INNER-SHELL IONIZATION
-loo0
375
-
FIG.39. Electronic binding energies for superheavy nuclei, supposed to have a radius R = 1.2A1%n, illustrating how the bound states successively "dive" into the negativeenergy continuum (Betz er d.,1976). [Figure given by Muller er a / . (1972).]
photon of energy very close to the threshold, the ejected electron will be moving very slowly and may not have got clear of the atomic system before an Auger electron is ejected. In such a case one should treat the vacancy formation and decay as part of the same process and at least take account of the PCI between the two electrons. The copious soft X radiation available in synchrotron radiation sources was used by Schmidt et a l . TABLE I RELATIVETRANSITION RATESFOR K-LL TRANSITIONS (LS COUPLING) I N Mg" Final L L state
Without CI or ICC
With CI
With both CI and ICC
Experiment
(2s)-2 1s (2s)-'(2p)-' 'P (2s)-L(2p)-' 3P (2p)-2 'S (2p)-l ID
10.03 21.94 8.82 4.37 54.83
1.34 22.03 8.80 6.20 55.63
5.41 18.80 8.20 7.82 59.77
6.32 ? 0.67 17.44 ? 1.58 8.40 f 0.88 8.97 -+ 1.37 57.61 f 3.52
a
From Howat (1978).
376
E. H . S. Burhop
(1977) to study the change in shape and energy displacement and energy displacement of the N5-02,302,3(1SO) peak of Xe following N, shell ionization by photons of energy only 0.8 eV above the threshold. They observed this Auger peak to be unsymmetrical with a tail extending toward the high-energy side. The FWHM was increased by -0.2 eV and the maximum displaced -0.1 eV toward high energy. The details of their observed line profile was interpreted by Niehaus in terms of PCI.
2. Natural Width and Energy Measurements of K Series Radiation Finally, it is of interest to refer to two other experiments that introduce refinements in the study of inner-shell ionization as a result of new technical developments. Both concern accurate measurements of X-ray spectra under conditions in which single K vacancies can be produced without contamination by multiple ionization effects. In the first of these experiments (Chevallier et a l . , 1978) the tunable X-ray beam of the Orsay synchrotron radiation facility was used to excite the K a lines of Fe just at the energy threshold when there is insufficient energy available to permit shake-up and shake-off processes. Under these conditions the full-width at half-maximum of Fe K a l was 2.70 f 0.05 eV compared with 2.90 f 0.05 eV when the photon had an energy 500 eV above threshold. The corresponding figures for FeKa2 were 3.4 and 3.7 eV, respectively, showing the effect of multiple ionization. The other experiment (Borchert et a l . , 1978) was made possible by the Isolde facility at CERN, which can obtain adequate concentrations of radioactive neutron-deficient isotopes. The K a spectrum of lQ7Auwas measured using the isotope lQ7Hg(64h), which decays by electron capture to the 77.3 keV level of Au, which is below the K threshold so that only 0.7% of the K X rays result from internal conversion. In the electron capture process, outer electrons do not see any change in the effective nuclear charge, so that a K vacancy unaccompanied by outer-shell vacancies due to shake-up and shake-off effects can be obtained. After making various corrections it was concluded that the energies of the gold K a , , K a 2 and KP, X-ray lines excited by photoionization are -0.6 eV larger than when excited by electron capture, reflecting the effect of unresolved satellite lines arising from the photoionization process.
ACKNOWLEDGMENTS
I am very grateful to Dr. J . Rafelski for valuable discussion. I am also indebted to Miss Muriel King for the preparation and typing of the manuscript and to Mme. Claude Rigoni for preparing the figures.
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REFERENCES Aberg, T., and Utriainen, J: (1969). Phys. Rev. Lett. 22, 1346. Aberg. T., Briand. J. P., Chevallier, P., Chetioui, A., Rozet, J. P., Tavernier, M., and Touati, A. (1976). 1.Phys. B 9, 2815. Afrosimov, V. V., Gordeev, Y. S., Zinoviev, A. N . , Rasulov, D. H . , and Shergin, A. P. (1976).In “Inner Shell Ionization Phenomena” (W. Mehlhorn and R. Brenn. eds.), Invited Paper, p. 258. Freiburg University. Amundsen, P. A. (1977a). J. Phys. B 10, 1097. Amundsen, P. A. (1977b).J. Phys. B 10, 2177. Amundsen, P. A., and Kocbach, L. (1975).J. Phys. B 8, 422. Anholt, R., Nagamiya, S., Rasmussen, J . O., Bowman, H . , Doannou-Yannou. J. G., and Rauscher, E. (1976). Phys. Re\,. A 14, 2103. Arthurs. A. M., and Moiseiwitsch, B. L. (1958). Proc. R . Soc. London. Ser. A 247, 550. Asaad, W. N. (1959). Proc. R . SOC.London, Ser. A 249. 555. Asaad, W. N. (1%5). Niccl. Phys. 66, 494. Asadd, W. N., and Burhop, E. H. S. (1958). Proc. Phys. Soc. London 71, 369. Ashley, J. C . , Ritchie, R. H., and Brandt, W. (1972). Phys. Rev. B 5, 2393. Bang, J., and Hansteen, J. M. (1959). Mut.-Fys. Medd. Danske Vid. Selsk. 31, No. 13. Basbas, G., Brandt, W., and Laubert, R. (1973a).Phys. Rev. A 7, 983. Basbas, C . , Brandt, W., and Ritchie, R. H. (1973b). Phys. Rev. A 7, 1971. Basbas, G., Brandt, W., and Laubert, R. (1978). Phys. Rerj. A 17, 1655. Bates, D. R., and Sprevak, D. (1970). J . Phys. B 3, 1483. Bates, D. R.. and Williams, D. A. (1964). Proc. Phys. Soc. London 83, 425. Berezhko, E . G., and Kabachnik, N. M. (1977). J . Phys. E 10, 2467. Betz, H. D., Delvaille, J. R., Kalata. K., Schnopper, H. W., Sohval, A. R., Jones, K. W., and Wegner, H . E. (1972). In “Inner Shell Ionization Phenomena” (R. W. Fink et ul., eds.), p. 1374. U.S. At. Energy Comm., Oak Ridge, Tennessee. Betz. W., Heiligenthal, G . , Reinhard, T. J., Smith, R. K., and Greiner, W. (1976). Phys. E/ectroti. At Collisions. Proc. Int. c o n j : . 9th. 1975 p. 531. Univ. of Washington Press, Seattle. Bohr. N. (1948). Mat.-Fys. Mcdd. Danshe Vid. Selsk. 18, No. 8. Borchert, C. L., Hansen, P. G., Jonson, B.. Lindgren, I., Ravn, H. L., Schult, 0. W. B., and Tidemand-Petersson, P. (1978). Phys. Lett. A 65, 297. Boving, E. G. (1977). J. f h y s . B 10, 63. Brandt, W. (1973). A t . Phys. Proc. Inr. Conf:. 3rd. 1972 p. 155. Plenum Press, New York. Brandt, W., and Laubert, R. (1969). Phys. Rev. 178, 225. Brandt, W.. Laubert, R., and Sellin, I . (1966). PhJ1.s. Rev. 151, 56. Breuckmann, B., and Schmidt, V. (1974). Z. Phys. 268, 235. Briand, J. P.. Touati, A., Frilley, M.. Chevallier, P., Jonson, A., Rozet, J. P., Tavernier, M.,Shafroth, S ., and Krause, M. 0. (1976). J. Phys. E 9, 1055. Briggs. J. S. (1972). In “Inner Shell Ionization Phenomena” (R.W. Fink et al.. eds.), p. 1209. U . S . At. Energy Comm., Oak Ridge, Tennessee. Briggs, J . S. (1976a). Phys. Electron. At. Colli.sions. Invited Lec.1.. Proc. Int. Conf:. 9th 1975 p. 384. Univ. of Washington Press, Seattle. Briggs, J . S . (1976b). Rep. Prog. Phys. 39, 217. Briggs. J . S., and Macek, J . (1972). J. Phys. B 5 , 579. Briggs. J. S.. and Taulbjerg, K. (1975). J. Phys. E 8, 1909. Brinkman, H. C.. and Kramers, H. A. (19301. Proc. K . Ned. Akad. Wet. 33, 973. Burch, D., Stolterfoht, N., Schneider, D., Wieman, H . , and Risky, S. S. (1975). Nuclear Physics Laboratory Annual Report for 1974. Univ. of Washington Press, Seattle.
378
E. H . S . Burhop
Burhop, E. H. S. (1940). Proc. Cambridge Philos. Soc. 36, 43. Camilloni, R., Giardini-Guidoni, A., Triribelli, R., and Stefani, G. (1972). Phys. Rev. Lett. 29, 618. Camso, E., and Cesati, A. (1977). Phys. Rev. A 15, 432. Catterall, J. A., and Trotter, J. (1958). Philos. Mag. [8] 3, 1424. Chen, J . R. (1977). Phys. Rev. A 15, 487. Chen, M . H., Craseman, B., and Matthews, D. L. (1975). Phys. Rev. L e f t . 34, 1309. Chevallier, P., Tavemier, M., and Briand, J. P. (1978). J. Phys. E 11, L171. Clark, D. L., Li, T. K., Moss, J. M., Greenless, G. W., Cage, M. E., and Broadhurst, J. H. (1975).J . Phys. E 8, L378. Clark, J. C. (1935). Phys. Rev. 48, 30. Cleff, B . , and Mehlhom, W. (1971). Phys. Lett. A 37, 3 . Cleff, B., and Mehlhorn, W. (1974). J . Phys. E 7, 593 and 605. Compton, A. H., and Allison, S. K. (1935). “X-rays in Theory and Experiment.” Macmillan, New York. Cue, N. (1977). Proc. Conf. Sci. Ind. Appl. Small Accd. 4th, 1976, p. 299. IEEE, New York. Cue, N., Scholz, W., and Li-Scholz, A. (1977). Phys. Lett. A 63, 54. Dangerfield, G. R., and Spicer, B. M. (1975). J . Phys. B 8, 1744. Das. J . N. (1972). Nuovo Cirnenfo E 12, 197. Davidovic, D. M., and Moiseiwitsch, B. L. (1975). J . Phys. E 8, 947. Davidovic, D. M., Moiseiwitsch, B. L . , and Norrington, P. H. (1978). J . Phys. E 11, 847. Davis, D. V., Mistry, V. D., and Quarles, C. A. (1972). Phys. Lett. A 38, 169. Deconninck, G., and Longree, M. (1977). Phys. Rev. A 16, 1390. Demkov, Y. N. (1964). Sov. P h y s . - J E T P (Engl. Trans/.) 18, 138. Drisko, R. M. (1955). Thesis, Carnegie Institute of Technology, Pittsburgh, Pennsylvania. Ehrhardt, H., Hesselbacher, K., Jung, K., and Willmann, K. (1971). Case Stud. A t . Collision Phys. 2, 159. Fano, U., and Lichten, W. (1965). Phys. Rev. L e f t . 14, 627. Fischer, D. W., and Baun, W. L. (1965). J . Appl. Phys. 36, 534. Folkmann, F., Armbruster, P., Hagmann, S., Kraft, G., Mokler, P. H., and Stein, H. J. (1976). Z . Phys. A 276, 15. Ford, A. L., Fitchard, E., and Reading, J. F. (1977). Phys. Rev. A 16, 133. Frank, W . , Gippner, P., Kaun, K. H., Manfrass, P., and Tretyakov, Y. P. (1976).Z. Phys. A 277, 333. Gavrila, M., and Hansen, J. E. (1978). J. Phys. E 11, 1353. Genz, H., Hoffmann, D. H. H., and Richter, A. (1976). In “Inner Shell Ionization Phenomena” (W. Mehlhom and R. Brenn, eds.), Abstracts, p. 229. Freiburg University. Green, G. W. (1%2). Thesis, Cambridge University. Greenberg, J. S., Davis, C. K., and Vincent, P. (1974). Phys. Rev. L e f t . 33, 473. Greenberg, J. S ., Vincent, P., and Lichten, W. (1977). Phys. Rev. A 16, 964. Gryzinski, M. (1965). Phys. Rev. A 138, 305. Guffey, J. A., Ellsworth, L. D., and MacDonald, J. R. (1977). Phys. Rev. A 15, 1863. Hansteen, J. M., Johnsen, 0. M., and Kocbach, L. (1975). Ar. Dara Nucl. Dara Tables 15, 3050. Hardt, T. L., and Watson, R. L. (1973). Phys. Rev. A 7, 1917. Helstroom, R., Petty, R. J., and Spicer, B. M. (1977). Phys. Lett. A 62, 146. Hopkins, F., Little, A,, and Cue, N. (1976). Phys. Rev. A 14, 1634. Howat, G. (1978). J . Phys. E 11, 1589. Howat, G., Aberg, T., and Goscinski, 0. (1978). J . Phys. E 11, 1575.
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Ishii, K., Kamiya, M.. Sera, K., Morita, S., Tawara, H., Oyamada, M., and Chu, T. C. (1977). Phys. Rev. A 15, 906. Jamison, K. A., and Richard, P. (1977). Phys. R e v . Lett. 38,484. Jamnik, D., and Zupancic, C. (1957). Mat.-Fys. Medd. Danske Vid. Selsk. 31, No. 2. Jarvis, 0. N . , Whitehead, C., and Shah, M. (1972).Phys. Rev. A 5 , 1198. Kaun, K. H., Frank, W., and Manfrass, P. (1976). In “Inner Shell Ionization Phenomena“ (W. Mehlhom and R. Brenn, eds.), Invited Papers, p. 68. Freiburg University. Khan, J. M., Potter, D. L., and Worley, R. D. (1965). Phys. Rev. 139, A1735. Knudson, A. R., Burkhalter, P. G., and Nagel, D. J. (1972). I n “Inner Shell Ionization Phenomena” (R. w. Fink, ed.). p. 1675. U . S . At. Energy Comm., Oak Ridge, Tennessee. Kolbenstvedt, H. (1967). J. Appl. Phys. 38, 4785. Kubo, H . , Jundt, F. C., and Purser, K. H. (1973). Phys. Rev. Lett. 31, 674. Lapicki, G . , and Losonsky, W. (1977). Phys. R e v . A 15, 8%. Lark, N. L. (1962). Bull. Am. Phys. SOC. [2] 7, 623. Larkins, F. P. (1972). hi “Inner Shell Ionization Phenomena” (R.W. Fink, ed.), p. 1543. U.S. At. Energy Comm., Oak Ridge, Tennessee. Laubert, R., and Losonsky, W. (1976). Phys. Rev. A 14, 2043. Lennard. W. N . , Mitchell, I. V., and Phillips. D. (1978). J. Phys. E 11, 1283. Li, T. K . , Clark, D. L., and Greenlees, G. W. (1976). Phys. Rev. A 14, 2016. McDaniel, F. D., Duggan, J. L., Miller, P. D., and Alton, G. D. (1977).Phys. Rev. A 15,846. MacDonald, J. R., Cocke, C. L., and Eidson, W. W. (1974). Phys. R e v . Lett. 32, 648. MacDonald, J. R., Schule, R., Schuch, R., Schmidt-Bocking, H., and Liesen, D. (1978). Phys. Rev. Lett. 40, 1330. Macek. J . , and Briggs, J. S . (1973). J . Phys. E 6, 841. McFarlane, S. C. (1972). J. Phys. E 5, 1906. McKnight, R. H., and Raines, R. 0. (1976).Phys. Rev. A 14, 1388. Massey, H. S. W., Burhop, E. H. S . , and Gilbody, M. B. (1974). “Electronic and Ionic Impact Phenomena,’’ Vol. 4, p. 2968. Oxford Univ. Press, London. Mehlhorn, W. (1%5). 2. Phys. 187, 21. Merzbacher, E., and Lewis, H. W. (1958). In “Handbuch der Physik” (S. Fliigge, ed.), Vol. 34, p. 166. Springer-Verlag. Berlin and New York. Meyerhof, W. E. (1973). Phys. Rev. Lett. 31, 1341. Meyerhof. W. E . (1976). Science 193, 839. Meyerhof, W. E., and Taulbjerg, K. (1977). Annu. Rev. Nucl. Sci. 27, 279. Meyerhof, W. E.. Saylor, T. K . . and Anholt, R. (1975). Phys. Rev. A 12, 64. Meyerhof. W. E., Anholt. R., and Saylor, T. K . (1977). Phys. Rev. A 16, 169. Middleman, L. M., Ford, R. L.. and Hofstadter, R. (1970). Phys. R e v . A 2, 1429. Morgan, J . F., Sega, R. S., Schraeder, R. J.. Suiter, H. R., and Blatt, S. L. (1977). Phvs. Rev. A 16, 2187. Motz, J. W., and Placious, R. C. (1%4). Phy.5. Rev. 136, A662. Miiller, B . . and Greiner. W. (1974). Phys. Re\.. L e f t . 33, 469. Miiller, B . , Rafelski, J., and Greiner, W. (1972). 2. Phys. A 257, 62. Muller, B., Rafelski, J.. and Greiner, W. (1973). Phys. Lett. E 47, 5 . Needham, P. B., and Sartwell, B. D. (1970). Phvs. Rev. A 2, 27. Niehaus, A. (1977). J. Phys. E 10, 1845. Nikitin. E. E. (1970). Ad\,. Quantum Chem. 5, 135. Nikolaev. V. S. (1966).Zh. Eksp. Teor. Fiz. 51, 1263; Sov.Phys.-JETP (Engl. T r a n s / . )847 (1967). Oppenheimer, J . R. (1928). Phys. Rub,. 31, 349. Perlman. H . S. (1%0). Proc. R . Soc. London. Ser. A 7 6 , 623.
380
E . H . S. Burhop
Pockman, L. T., Webster, D. L., Kirkpatrick, P., and Harworth, K. (1947). Phys. Rev. 71, 330. Quarles, C. A., and Faulk, J. D. (1973). Phys. Rev. Lett. 31, 859. Rester, D. H., and Dance, W. E. (1966). Phys. Rev. 152, 1 . Sackmann, S., Lutz, H. O., and Briggs, J. (1974). Phys. Rev. Lett. 32,805. Sandner, W., and Schmitt, W. (1978). J . Phys. B 11, 1833. Sandner, W., Weber, M., and Mehlhorn, W. (1978). Intl. Workshop on Coherence and Correlation in Atomic Collision 1978 (H. Kleinpoppen, ed.). Sans, F. W.,and Hoogkamer, T. P. (1977). A t . Phys., Proc. Int. Conf., 5th 1976 p. 531. Plenum Press, New York. Sans, F. W., and Onderdelinden, D. (1970). Physica (Utrecht) 49, 441. Saris, F. W., van der Weg, W. F., Tawara, H., and Laubert, R. (1972). Phys. Rev. Lett. 28, 717. Schiebel, U., Bentz, E., Muller, A., Salzbarn, E., and Tawara, H. (1976). Phys. Lett. A 59, 274. Schmidt, V.,Sandner, N., Mehlhorn, W., Adam, M. Y ., and Wuilleumier, F. (1977). Phys. Rev. Lett. 38, 63. Schneider, D., Moore, C. F., and Johnson, B. W. (1976). J . Phys. B 9, L153-6. Schnopper, H. W., Betz, H. D., Dalvaille, J. P., Kalata, K., Sohval, A. R., Jones, K. W., and Wegner, H. E. (1972). Phys. Rev. Lett. 29, 898. Scholer, A., and Bell, F. (1978). J. Phys. A 286, 163. Schuch, R., Schmidt-Bocking. H., Schule, R., and Tserruya, I. (1977). Phys. Rev. Lett. 39, 79. Smick, A. E . , and Kirkpatrick, P. (1945). Phys. Rev. 67, 153. Stoller, C., Wolfli, W., Bonani, G., Stockli, M., and Suter, M. (1977a).Phys. Rev. A 15,990. Stoller, C . , Wolfli, W., Bonani, G., Stockli, M., and Suter, M. (1977b). J. Phys. B 10, L347. Stolterfoht, N. (1977). Proc. Conf. Sci. Ind. Appl. Small Accelerators, 4th. 1976 p. 311. IEEE, New York. Stolterfoht, N.. Schneider, D., and Harrison, K. G. (1973a). Phys. Rev. A 8 , 2363. Stolterfoht, N., Gabler, H., and Leithauser, U. (1973b). Phys. Lett. A 45, 351. Stolterfoht, N., Schneider, D., Burch, D., Aagaard, B., Boving, E., and Fastrup, B. (1975). Phys. Rev. A 12, 1313. Stolterfoht, N., Schneider, D., Mann, R., and Folkmann, F. (1977). J. Phys. B 10, L281. Stolterfoht, N., Schneider, D., and Brandt, D. (1977). Abstr. Conf. Phys. Electron A t . Collisions, loth, 1977 p. 902. Warren, B. E. (1969). “X-ray Diffraction,” p. 334. Addison-Wesley, Reading, Massachusetts. Webster, D. L., Hansen, W. W., and Duveneck, F. B. (1933). Phys. Rev. 43, 839. Wentzel, G. (1927). Z. Phys. 43, 524. Woerlee, P. H., Fortner, R. J., Doorn, S., Hoogkamer, T. P., and Saris, F. W. (1978). J. Phys. B 11, L425. WoHi, W., Stoller, C. H., Bonani, G., Suter, M., and Stockli, M. (1975). Phys. Rev. Lett. 35, 656. WoMi. W., Stoller. C. H.. Bonani. G., Stockli, M., and Suter, M. (1976a). In “Inner Shell Ionization Phenomena” ( W . Mehlhorn and R. Brenn, eds.), Invited Papers, p. 92. Freiburg University.
ADVANCES IN ATOMIC A N D MOLECULAR PHYSICS. VOL. IS
EXCITATION OF ATOMS BY ELECTRON IMPACT D. W . 0 . HEDDLE Physics Department Royal HoNoway College University of’London Egham, Surrey. England
I . Introduction
..................................
B. Collisional Transfer of Excitation . . . . . . . . . . 111. Behavior near Threshold . . . . . . . . . . . . . . . A. Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
391
. . . . . . . . . . . . . . . . . . 392 IV. Measurements by Different Te V. Time-Resolved Measurements
. . . . . . . . . . . . . . . 403
B. Application of the Bethe Approximation A. The Balmer Lines of Hydrogen .
. . . . . . . . . . . . . . 415
C. Photon Labeling . . . . . . . . .
...........................
References . . . . . . . . . .
419
I. Introduction Ten years ago, two reviews of this field (Moiseiwitsch and Smith, 1%8; Heddle and Keesing, 1968) noted that there were serious discrepancies among measurements of the same excitation functions and commented that while the technology was available to obtain reliable measurements of relative excitation functions it was frequently the case that the importance of major secondary processes appeared not to be appreciated. Both reviews concluded hopefully that experimental research in the field was entering a new phase and that a greater degree of consistency would be achieved. While it would not be true to say that the subject is now com381 Copyright 0 1979 by Academic Press, Inc. All rights of reproduction in any form reserved.
ISBN 0-12-003815-3
3 82
D . W . 0 . Heddle
pletely understood, there have been a number of excellent and wefldocumented measurements and the situation is undoubtedly very much better than ten years ago. It is difficult to overemphasize the importance of a clear and detailed description of a measurement and of the reduction of the data. There is no value in making a measurement of high precision unless you can demonstrate that precision and describe in clear and unambiguous terms what uncertainties have contributed to the measurement. You can take nothing for granted, can assume no linearities unless you have verified them, for unless your account is complete in all details it will not be accepted and you will have wasted your time. This is not the occasion for a comprehensive review of the subject, but rather for pointing out some of the major advances that have been made in the past decade. Each of the subsequent sections of this chapter treats a particular area where technical advance or improved understanding has made possible measurements that in many cases must be regarded as definitive. Ten years ago an absolute measurement of the excitation function of a multiply charged ion would have been considered almost impossible, but such measurements for the resonance lines of C3+(Taylor et al., 1977) and N4+( G . H. Dunn, private communication) have now been made. The selection of topics is personal, and therefore arbitrary, and I have made few comparisons between the results of different experiments (except in Section IV where this is central to the topic of that section) or between experiment and detailed theory. The measurements described in Section VII are only “miscellaneous” in the sense that they do not fit readily into the earlier sections. There are also, of course, some experiments that illustrate more than one of the topics I have selected, but they are described in one section only.
11. Secondary Effects A. THE IMPRISONMENT OF RESONANCE RADIATION Consider the processes that populate and depopulate a statej that combines optically with the ground state g . We observe radiation corresponding to the transitionj + I ; i and k denote other states that combine with j and are respectively of higher and lower energy. If we ignore the collisional transfer of excitation we may write the loss rate per unit length of the collision chamber as
EXCITATION OF ATOMS B Y ELECTRON IMPACT
(..+
AJl
+
2k AJk)
1 S
383
Nj dS
where the A,, are radiative transition probabilities, N, are densities of atoms in state n , and S is the cross-sectional area of the excitation chamber. The population rate per unit length is
where Qj is the cross section for election impact excitation of the state j, J is the current density in electrons per second per unit area, and gJis the probability that a resonance photon will reach the wall of the collision chamber and so not be absorbed by an atom. These rates are equal. We define an apparent cross section Q; by Q;
J: N g J dS
=
Aj
I,
Nj dS
where A j is the total radiative transition probability per second from state j . Using similar expressions for the apparent cross sections for the excita-
tion of other states, we can write
where we have written J j to describe the cascade cross section Jj
Qi‘ A d A i
= i
It is clear without detailed calculation that gj will approach unity for vanishingly small pressures and zero for very large pressures and that Q;/(Qj + Jj) takes values between 1 and A j / ( A j - A j J . Phelps (1958) has calculated the fraction of resonance photons that escape from an infinitely long cylinder with axial excitation. He treats two limiting cases. In the first the excited atoms produced by absorption of a resonance photon are considered to reradiate from the axis: this will clearly underestimate the probability of escape, because in reality the excited atoms will reradiate from a position closer to the wall. The second case assumes that the original excitation is distributed across a section of the cylinder in the same way as the ultimately excited atoms: this would be expected to overestimate the probability of escape because the original excitation is really more concentrated around the axis. A mean value has frequently been used combined with an “effective radius” for the collision chamber. Heddle and Samuel (1970) measured the variation of the
3 84
D . W . 0 . Heddle
apparent cross section for excitation of the 3IP + 2 9 line of helium at 501.6 nm in an excitation chamber of radius 4.15 cm. This large value was chosen so that imprisonment effects would be serious at pressures low enough that collisional transfer of excitation would be unimportant. Their results are shown in Fig. 1 for a collision chamber length of 19 cm and it is clear that if the effective radius is taken as the true value there is an excellent fit to g', the second of Phelps' cases. Heddle and Samuel (1970) showed that this remained true for cylinder lengths somewhat greater than the diameter, but that for shorter cylinders the effective radius was rather close to the distance to the nearest part of the electrode surface. Anderson et al. (1969) used a three-sided rectangular cavity around their electron beam in order to provide a sink for resonance photons and hence reduce the concentration of nlP atoms in an experiment (described in Section V) where collisional transfer from atoms in these states could interfere with their measurements. Showalter and Kay (1975) used the mean of Phelps' g values and found, for a long collision chamber, an effective radius some 80% of the true value. This is rather more than the results of Heddle and Samuel (1970) would have shown (75%). Because their collision chamber was of smaller radius (about 7.5 mm) collisional depopulation of the 4IP and 5IP states led to a decrease in apparent cross section at pressures above 20 mTorr. This effect was also observed by van Raan and van Eck (1974). Their collision chamber had a radius of 15 mm and a length of 20 mm and they found the best fit to their observations using g' and an effective radius of 7.5 mm. They give a very useful table of g', g > , and the mean value.
FIG.1. The enhancement of the apparent excitation cross section of the 501.6 nm line of helium (3lP-2IS) as a result of imprisonment of resonance radiation.
EXCITATION OF ATOMS BY ELECTRON IMPACT
385
B. COLLISIONAL TRANSFER OF EXCITATION The excitation state of an atom can be changed in collision with another particle. The process
x, + x, x, + X k c*
has been invoked to explain the observed pressure dependence of excitation cross sections, but there has been considerable debate as to the particular states involved. The basic problem is that, in an electron excitation experiment, all states are excited and while certain states having large excitation cross sections can be identified as the source of additional excitation, the path by which the excitation is transferred from statej to state k is by no means clear. Even in the most-studied case of helium the picture was very confused. An important difficulty lay in the observation that the excitation of the n3D states is very greatly enhanced with increase of pressure, while the source of this additional excitation is known to be the nlP states. This implies a breakdown of the Wigner (1927) spin rule. Models that attribute this to a breakdown of Russell-Saunders coupling in the F states have had some success, but van Raan and Heideman (1974) have pointed out that the product of the singlet-triplet energy difference and the radiative lifetime greatly exceeds Planck's constant, so while the n1F3 states could be considered as a linear combination of n1F3and n3F3,the n3Fz and n3F, states remain pure triplets. With spin-changing collisions only possible via F states, the transfer of excitation to 33D must be via cascade with its source the niP states, where n > 3 . The analysis of the transfer processes observed in electron impact experiments requires such complex models that the results, while perhaps phenomenologically sound, may not be of great physical significance. The problem is made particularly difficult as the concentrations of atoms in the various nlP states are almost independent of n (Gabriel and Heddle, 1960). A more direct experiment would involve the initial excitation of a single state by absorption of light and the observation of the states to which this excitation has been transferred. Figure 2 shows a partial energy level diagram that illustrates the experiment of Shaw and Webster (1976). They used a flowing afterglow to produce a high concentration of atoms in the 2IP state and excited some of these to the 4lD state using a pulsed dye laser. The subsequent decay of the other states was observed either directly or, in the case of the 4F states, by their cascade population of the 3D states. For a variety of reasons only the 4IP-2'S and 43D-23P lines were suitable for analysis. By changing the discharge conditions the electron density could be varied and the cross sections for the associative ion-
3 86
D . W . 0 . Heddle
(23s FIG.2. Partial energy level diagram of helium showing the transitions relevant to the experiment of Shaw and Webster (1976).
ization process He,
+ He, 4 He: + e
could be measured. A similar experiment was done by Wellenstein and Robertson (1972a,b) but in their case they used a helical discharge lamp surrounding a steady discharge. They isolated single lines connecting the n = 2 and n = 3 states of helium with filters, modulated the helical lamp, and observed the modulation of the other lines in the discharge. The results of these two experiments are summarized in Table I.’ There is no singlet-triplet transfer at the n = 3 level, but the 4F states are indeed involved in this transfer. Note that the forward and reverse transitions between pairs of states are in the expected ratio of ( g k / g J )exp(& - Ek)/kT
Provided the pressure is low enough that reverse collisions can be neglected, the apparent excitation cross section varies linearly with pressure. Figure 3 shows results of van Raan and van Eck (1974) for the 43D state of helium. It is only safe to make a linear extrapolation using data taken below 0.8 mTorr. Figure 4 shows the excitation functions of the 43S and 43D states of helium extrapolated to zero effective pressure and with an allowance made for cascade population. The results of van Raan et al. (1974) and of Showalter and Kay (1975) are in good agreement, particu-
387
EXCITATION OF ATOMS B Y ELECTRON IMPACT
TABLE I CROSSSFCTIONS FOR E X C I I A T I O TRANSFER N A N D FOR ASSOCIATIVE IONIZATION IN H ~ L I U M ~
Initial levelb
~
Initial level'
Final level
4'P
4'D
4'F
43P
43D
43F
41P 4'D 4'F 43P 43D 43F He:
170 0 0 0 0 94
80 97 0 0 160 S6
0 71
0 0
38
-
0 81 0 31
0 0 51
~
~
~
0 110
0 40 _
_
_
_
-
0 0 100
140 19
13
150
_
0
-
Final level
3'P
3'D
3'P 3'D 33P 33D Held
32 0 0 3.1
10.6
33P 0 0
-
0.067 1.6
0 E , , but are progressively smaller at lower energies, as in the H and Ca cases, and show substantial effects due to resonances. Both the oscillator strength and the c, parameter can be calculated for excitation of the 3IP, state of helium, and though the calculations are not exact the oscillator strength is probably known to about 1%. In Fig. 18 we show an analysis of the measurements of Moustafa Moussa et al. (1%9), who made absolute measurements and took great care to correct for instrumental polarization and cascade effects. Their measurements show that the polarization passes smoothly through zero at 300 eV. This implies E,, = 300/e3 = 15 eV in contrast to the value E, = 21.3 eV, which corre-
412
D. W . 0. Heddle
Electron E n e r g Y / e V
FIG.18. Excitation and polarization of the 501.6 nm line of helium. 0, Moustafa Moussa et a / . (1969); 0, Heddle and Lucas (1963); t,Standage (1977).
sponds to the value of c3,p= 0.160 calculated by Kim and Inokuti (1968). It is also lower than any published experimental value based on extrapolation from high energies. Figure 18 also shows the experimental results of Heddle and Lucas (1963), which are probably some 10-20% too low because of the depolarizing effect of the imprisonment of resonance radiation, and two points deduced by Standage (1977) from angular correlation measurements and which are free from cascade and other secondary effects. These data give considerable support to the Bethe curve for energies down to some 60 eV. Figure 18 also shows the QE data of Moustafa Moussaet al. (1969). The line has been drawn through the points between 2.5 and 3.5 keV as those at higher energies seem to show the effect of secondary processes. The experimental points are corrected for polarization and cascade, and so Eq. (4) is the appropriate one to use. The slope of the
413
EXCITATION OF ATOMS B Y ELECTRON IMPACT I
10
I
I
,
100 Electron
1
I
I
I
1000 Energy/eV
FIG. 19. Excitation of the 53.7 nm line of helium (3'P-I'S). Note that the origin of the Q8& ordinate is displaced from that of the Q E .
line is 4.20 x m2 eV, which leads to a value of 6.47 x lo-* for the oscillator strength of the l's0-3~p1 transition. The generally accepted value is 7.34 x and it appears that the absolute calibration of the experiment requires correction by multiplication by a factor of 1.13. Westerveld et af. (1978) have recently measured both Q and QsOfor the first two resonance lines of helium and we show their data for the 53.7 nm line (3'P-l'S) in Fig. 19. The intercept of the Q & line is given by ln(E/E,,) = P , = - 1 and for E , = 15 eV is 5.5 eV. The ratio of the slopes of the Qs& and QE lines is in excellent agreement with the value of 0.75 given by Eq. (7a). It is interesting to note that Eq. ( 5 ) seems to be a good approximation a t energies below those for which Eq. (2) is valid and so the values of E, for H, Ca, Ba+, and He are all quite well determined. However, there will always be cases where polarization measurements over a wide range of energies encompassing E, will be difficult and where the linear dependence of QJQ,, on log E, which is only ever valid for Po = 1, cannot be assumed. One example of this situation is the 2s-2p excitation of Be+.
D. W . 0.Heddle
414
This has been studied experimentally by Taylor, Phaneuf, and Dunn and their results, though technically not published have been cited by Hayes et al. (1977) and Kennedy et af. (1978). Equation (5) can be rearranged as
and so a graph off(P) against log E should approach an asymptote having a slope of 2.302 and an intercept a t Eo. Values off(P) are shown in Fig. 20. A line of the correct slope has been drawn through the point at the highest energy and it is clear that this is at least very close to the correct asymptote. For P = 0, f(P) = 3.0, and the line gives E , = 55 eV and Eo = 2.74 eV. A line of slope 1.91 x m2 eV [corresponding tof2, = 0.505 (Weiss, 1963)] has been drawn to represent the asymptotic behavior of Q E . In comparing this line with the experimental points it should be remembered that the latter include a cascade contribution from higher states. 4
I
I
I
I
I
I
I
Electron Energy/eV
FIG.20. Excitation and polarization of the 313 nm line of Be+ (2'P-23). Note that the upper part of the figure shows a function of the polarization and not the polarization itself as in Figs. 15-18.
EXCITATION OF ATOMS BY ELECTRON IMPACT
415
93% of the excitation to the 32Pstate decays directly to the ground state and the total cascade contribution from P states via S or D states will increase the slope of the line by about 1%. Cascade from directly excited S and D states might contribute an additional 10% at low energies, or an inm2 eV. crease in QE by some 0.2 x
VII. Miscellaneous Measurements A. THE BALMERLINESOF HYDROGEN
Mahan et af. (1976) observed the H a line and were able to distinguish excitation to the 3S, 3P, and 3D states by modulating their electron beam over a range of frequencies and measuring the depth of modulation. Knowledge of the radiative lifetimes was the key to the analysis. They calibrated their measurements against a Born total cross section at 500 eV having used calculated values for the polarization of the 3P-2s and 3D-2P components. They show plots of QE vs. log E for the three components and, as with the Lyman a results of Long et al. (1968), their data for the 3P excitation do not appear to have reached the asymptotic region. A recalibration of their QQO data using the known parameters f,, = 7.9 x lo-’, c,, = 0.49 ( E , = 6.92 eV), E,, = 12.09 eV, and Po = 0.43 ( P , = - 0.27) would be of interest. In almost all of the published measurements of electron-hydrogen collisions the source of atoms has been a furnace at a temperature of 2500 K or more. A notable exception is the work of Walker and St. John (1974), who used a Wood’s discharge tube to produce in an excitation chamber an atom number density of 5.6 X rn-, in the presence of molecules at some six times this concentration. This is some lo4 times the atom density in a typical crossed-beam apparatus, but the advantage of chopping the beam is lost. Their excitation functions of the first five members of the Balmer series are shown in Fig. 21. The data have not been corrected for polarization or cascade and have been put onto an absolute scale by comparison with molecular excitation cross sections measured in the same system. The agreement with Born approximation calculations (Vainshtein, 1965; Green et (11.. 1957) at high energies is particularly good fol: the higher members. B. “COMPREHENSIVE” STUDIES
For reasons that are usually of technical origin few measurements of optical excitation functions cover more than a handful of the spectrum
8 00
H
P
6 00
H
d
4 00 N
E N P
200
2
\
Z
0 I-
0
50
5c
m m cn
4c
4c
30
3c
20
2c
10
I0
W
0 U 0
0
1 100
1 200
I 300
1 400
I 500
H €
0 ELECTRON
E N E R G Y (eV)
FIG.21. Apparent excitation functions of five members of the Balmer series of hydrogen. 0,Walker and St. John (1974) (exp.);-, Vainshtein (1%5) and Green et a / . (1957) (th.).
EXCITATION OF ATOMS B Y ELECTRON IMPACT
417
lines of an element. While such measurements can still provide valuable data of sometimes general applicability (such as the difference in form between transitions that are optically allowed or forbidden by various selection rules, or the dependence of the excitation cross section on effective principal quantum number for members of a series) they could not be called comprehensive. Some work in the past decade has been rather more systematic. For example, Gallagher has studied the resonance lines of all the alkalis (Leep and Gallagher, 1974; Enemark and Gallagher, 1972; Chen and Gallagher, 1978) and of four of the alkaline earths (Leep and Gallagher, 1976; Ehlers and Gallagher, 1973; Chen et a / . , 1976; Chen and Gallagher, 1976) and has found great similarity between the members of each group. They suggest, indeed, that in all cases the excitation functions can be expressed as the product of an asymptotic term [similar to which vanishes at the excitation threshEq. (2)] and a term 1 - (Er/E)1’2, old. A set of measurements that is “comprehensive” in a stricter sense is that of Sharpton et a / . (1970) on neon. They observed lines from some 60 states including all transitions from the 2p (Paschen notation) levels, a representative selection of those from the 3p, and most of the 3s, 4s, and 5s as well as those involving excitation of a d electron. They found that the J = 1 states of the 2p5ns and 2p5nd configurations, which combine optically with the ground state, have excitation functions similar to those of helium n’P and in general the excitation functions of other states had a form that could be described in terms of the singlet-triplet mixture of the states. Within a configuration 2p%l they found that states with odd values of J + I had larger excitation cross sections than those with even values, and that within the states of odd J + I those having the smaller value of J have the larger cross sections. C. PHOTONLABELING It sometimes happens that the optical excitation function of a level cannot be measured, because the only spectrum line in an accessible spectrum region is too close in wavelength to some other line of the same element (a number of examples occur in the spectra of the rare gases). A much more common problem occurs in the measurement of inelastic electron scattering because the resolution of electron spectrometers is inEerior to that of optical spectromers and because atomic energy levels converge toward the ionization potentials. Because optical detection of excitation is usually made by observing transitions other than those back to the ground state, the wavelengths of transitions from states that are very close in energy are frequently well separated. It is therefore possible to identify an
D . W . 0. Heddle
418
electron that has excited a specific state by observing the scattered electrons in delayed coincidence with the photons that result from excitation of that state. This technique has been applied by Pochat et al. (1973) to the measurement of the differential cross section for scattering of electrons that have excited the 4lS, 5lS, 4lD, and 5ID states of helium. Figure 22 shows their results for the 5IS state. From their observations they deduce values of the generalized oscillator strengths for the four transitions and compare these with Born approximation calculations by Bell et al. (1969). The agreement is not particularly good, but is best at 200 eV, the highest energy they used. This technique of labeling a scattered electron by an associated photon has great potential for further application. It is not the same as the coherence technique of Eminyan et al. (1974); indeed it is a simpler technique as the photons should be collected over a large solid angle. A detector insensitive to polarization should be used to observe perpendicular to the scattering plane.
D. A SALUTARY EXAMPLE The two reviews cited at the start of this chapter (Moiseiwitsch and Smith, 1968; Heddle and Keesing, 1968) each compared a number of experimental results for the excitation of the 4IS state of helium, a case for which secondary processes should be of little importance and yet for which substantially different shapes of excitation function had been reported within a few years. Both pairs of authors found the comparison rather depressing! The latter authors also noted a serious disagreement between recent measurements of the excitation function of the A+ line at 20 0
50eV
+ 0
+
N
* 10
4
*
-
1OOeV
0
2WeV
+
0
+ 0
*
+
o
'
t
o
+. 0;
10
o
.
+ 20
Scattering Angle/'
FIG.22. Differential cross sections for the excitation of the SIS state of helium.
EXCITATION OF ATOMS BY ELECTRON IMPACT
419
466 nm and measurements made at Jena by Fischer (1933) (Note: Fischer's results are shown in that review ten times too large: the disagreement is still very serious.) Latimer and St. John (1970) and Clout and Heddle (1971) have measured the excitation functions of a number of lines of A+. The agreement in shape between these measurements is good and they also agree well with those of Fischer (1933) in the two cases common to all three experiments. These measurements are arguably more difficult than measurements of the helium 4IS excitation function because the spectrum of argon is much richer and the excitation cross sections are of similar size. These results demonstrate the importance of ensuring that secondary processes are effectively absent from excitation function measurements.
REFERENCES Albert, K . , Christian, C., Heindorff. T., Reichert, E . , and Schon, S. (1977).J . P h y s . B 10, 3733. Anderson, E. M.,and Zilitis. V. A. (1964). Opt. Spectrosc. ( U S S R ) 16, 99. Anderson, R. J., Hughes, R. H . , and Norton, T. G. (1969). Phys. R e v . 181, 198. Bell, K. L., Kennedy, D. J.. and Kingston. A. E . (1969).J . P h y s . B 2 , 26. Bethe, H . (1930).Ann. Phys. (Leipzig) [5] 5, 325. Bogdanova, 1. P., and Marusin, V . D. (1971). O p t . Spectrosc. ( U S S R ) 31, 184. Borst, W. L. (1969). P h y s . R e v . 181, 257. Brunt, J . N. H . , King, G. C., and Read, F. H. (1977a). J . Elecfron. Spectrosc. Relal. Phennm. 12, 221. Brunt, J . N. H., King, G. C.. and Read, F. H. (1977b). J. P h y s . B 10, 3781. Burgess. A., and Sheorey, V. B. (1974).J . P h y s . B 7, 2403. Chen, S . T., and Gallagher, A. (1976). P h y s . Re\.. A 14, 593. Chen, S. T., and Gallagher, A . (1977). P h y s . Reit. A 15, 888. Chen, S . T . , and Gallagher, A . (1978). P h y s . R e , , . A 17, 55 1. Chen, S . T.. Leep, D.. and Gallagher, A. (1976). Phys. R e v . A 13, 947. Clout, P. N., and Heddle, D. W. 0. (1969). J . Opt. Soc. Am. 59,715. Clout, P. N . , and Heddle, D. W. 0. (1971). J . Phys. B 4, 483. Clout, P. N., Haque, M . A., and Heddle, D. W. 0. (1971). J . P h y s . E 4 , 8 9 3 . Crandall, D. H., Taylor, P. O., and Dunn. G. H. (1974). P h y s . R e v . A 10, 141. de Jongh, J . P., and van Eck, J . (1972). Proc. Inr. C'nrif. P h y s . Electron. A t . Collisions, 71h, 1971 Abstracts, p. 701. Donaldson, F. G . . Hender, M. A., and McConkey. J. W. (1972). J . P h y s . B 5 , 1192. Ehlers, V . J . , and Gallagher, A. (1973). P h y s . R e v . A 7, 1573. Eminyan. M., MacAdam, K. B., Slevin, J., and Kleinpoppen, H. (1974).J. P h y s . B 7, 1519. Enemark, E . A., and Gallagher, A. (1972). P h y s . R e v . A 6 , 192. Fano, U. (1954). P h v s . R e v . 95, 1198. Fischer, 0 . (1933). Z. P h y s . 86, 646. Flower, D. R., and Seaton, M. J. (1%7). P r o c . Phys. Soc. London 91, 59. Gabriel, A . H., and Heddle, D. W. 0. (1960). Proc. R . Soc. London, Ser. A 258, 124.
420
D . W . 0. Heddle
Garstang, R. H. (1962).J . Opt. Soc. Am. 52, 845. Green, L. C.,Rush, P. P., and Chandler, C. D. (1957).Astrophys. J . , Suppl. 3, 37. Hayes, M. A., Norcross, D. W., Mann, J. B., and Robb, W. D. (1977).J. Phys. B 10, L429. Heddle, D. W. 0. (1975).J . Phys. B 8, L33. Heddle, D.W.0. (1976).Contemp. Phys. 17, 443. Heddle, D. W. O., and Keesing, R. G. W. (1967).Proc. Phys. SOC.London 91, 520. Heddle, D.W. O., and Keesing, R. G. W. (1%8). Adv. Ar. Mol. Phys. 4, 267. Heddle, D.W. O., and Lucas, C. B. (1963).Proc. R . SOC. London, Ser. A 271, 129. Heddle, D. W. O., and Samuel, M. J. (1970).J. Phys. B 3, 1593. Heddle, D.W. O., Keesing, R. G. W., and Watkin, R. D. (1974).Proc. R . Soc. London, Ser. A 337, 443. Heddle, D. W. 0.. Keesing, R. G. W., and Parkin, A. (1977).Proc. R . SOC.London, Ser. A 352, 419. Inokuti, M. (1971).Rev. Mod. Phys. 43, 297. Inokuti, M., Itikawa, Y., and Turner, J. E. (1978).Rev. Mod. Phys. 50, 23. Jobe, J. D.,and St. John, R. M. (1967a).Phys. Rev. 164, 117. Jobe, J. D., and St. John, R. M. (1%7b). J. Opt. SOC.Am. 57, 1449. Jobe, J. D., and St. John, R. M. (1972).Phys. R e v . A 5, 295. Karstensen, F. (1968).2. Astrophys. 69, 214. Kennedy, J. V.,Myerscough, V. P., and McDowell, M. R. C. (1978).J . Phys. B 11, 1303. Kim, Y.-K., and Bagus, P. S. (1973).Phys. Rev. A 8, 1739. Kim, Y.-K., and Inokuti, M. (1968).Phys. Rev. 175, 176. King, G. C. M., Adams, A., and Read, F. H. (1972).J . Phys. B 5 , L254. Korotkov, A. L.(1970).Opt. Spectrosc. ( U S S R ) 28, 347. Korotkov, A. L.,and Prilezhaeva, N. A. (1969).Sov. Phys. J . 13, 1625. Krause, H. F., Johnson, S. G., Datz, S., and Schmidt-Bleek, F. K. (1975).Chem. Phys. Lett. 31, 577. Krause, H. F., Johnson, S. G., and Datz, S. (1977).Phys. Rev. A 15, 611. Latimer, I. D., and St. John, R. M. (1970).Phys. Rev. A 1, 1612. Lee, E. T. P., and Lin, C. C. (1965).Phys. Rev. 138, A301. Leep, D.,and Gallagher, A. (1974).Phys. Rev. A 10, 1082. Leep, D., and Gallagher, A. (1976).Phys. Rev. A 13, 148. Long, R. L.,Cox, D. M., and Smith, S. J. (1968).J.Res. Nail. Bur. Stand., Sect. A 72,521. McFarland, R. H. (l%7). Phys. Rev. 156, 5 5 . McFarlane, S. C. (1974).J . Phys. B 7, 1756. Mahan, A. H., Gallagher, A., and Smith, S. J. (1976).Phys. Rev. A 13, 156. Moiseiwitsch, B. L.,and Smith, S. J. (1%8). Rev. Mod. Phys. 40, 238. Moustafa Moussa, H. R., de Heer, F. J., and Schutten, J. (1969).Physica (Utrecht)40,517. Mumma, M. J., Misakian, M., Jackson, W. M., and Fans, J. L. (1974).Phys. Rev. A 9,203. Ochkur, V. I. (1964). Sov. Phys.-JETP (Engl. Trans/.) 18, 503. Ott, W. R., Kauppila, W. E., and Fite, W. L. (1970).Phys. Rev. A 1, 1089. Percival, I. C.,and Seaton, M. J. (1958).Philos. Trans. R . SOC. London. Ser. A 251, 113. Phelps, A . V. (1958).Phys. Rev. 110, 1362. Pochat, A., Rozuel, D., and Peresse, J. (1973).J. Phys. (Paris) 34, 701. Read, F. H.(1975).J . Phys. B 8, 1034. Ruthberg, S. (1972).J. Vac. Sci. Technol. 9, 1457. Sharpton, F. A . , St. John, R. M., Lin, C. C., and Fajen, F. E. (1970).Phys. R e v . A 2, 1305. Shaw, M.J., and Webster, M. J. (1976).J. Phys. B 9, 2839. Showalter, J. G., and Kay, R. B. (1975).Phys. Rev. A 11, 1899.
EXCITATION OF ATOMS BY ELECTRON IMPACT
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Shpenik, 0. B. Souter, V. V., Zavilopulo, A. N., Zapesochnyi, I. P., and Kontrosh, E. E. (1976). Sov. Phys.-JETP (Engl. Transl.)42, 23. Stair, R., Fussell, W. B., and Schneider, W. E . (1%5). Appl. Opt. 4, 85. Stamatovic, A., and Schulz, G. J. (1970). R e v . Sci. Instrum. 41, 423. Standage, M. C. (1977). J . Phys. B 10, 2789. Sutton, J. F., and Kay, R. B. (1974). Phys. Rev. A 9, 697. Taylor, P. O., and Dunn. G. H. (1973). Phys. R e v . A 8, 2304. Taylor, P. O., Dolder, K. T., Kauppila, W. E., and Dunn, G. H. (1974). Rev. Sci. Instrum. 45, 538. Taylor, P. 0.. Gregory, D., Dunn, G. H., Phaneuf, R. A., and Crandall, D. H. (1977).Phys. Rev. L e f t . 39, 1256. Vainshtein, L. (1965). Opt. Spectrosc. ( U S S R ) 18, 538. Vainshtein, L., Opykhtin, V., and Presnyakov, L. (1965). Sov. Phys.-JETP (Engl. Transl.) 20, 1542 (corrected in Moiseiwitsch and Smith, 1968). van Raan, A. F. J . (1973). Physica (Urrechr) 65, 566. van Raan, A. F. J., and Heideman, H. G. M. (1974). J. Phys. B 7 , L216. van Raan, A. F. J., and van Eck, J. (1974). J . Phys. B 7, 2003. van Raan, A. F. J., Moll, P. G., and van Eck, J. (1974). J. Phys. B 7, 950. Van Zyl, B., Chamberlain, G. E., Dunn, G. H., and Ruthberg, S. (1976). J . Vac. Sci. Technol. 13, 721. Walker, J. D., and St. John, R. M. (1974). J . Chem. Phys. 61, 2394. Weaver, L. D., and Hughes, R. H. (1967). J. Chem. Phys. 47, 346. Weiss, A. W. (1%3). Asrrophys. J. 138, 1262. Wellenstein, H . F . , and Robertson, W. W. (1972a).J. Chem. Phys. 56, 1072. Wellenstein, H. F., and Robertson, W. W. (1972b).J. Chem. Phys. 56, 1077. Westerveld, W. B . , Heideman, H. G. M., and van Eck. J. (1979), J . Phys. B 12, 1 15. Wigner, E . P. (1927). Nachr. G e s . Wiss. Goettingen Marh.-Phys. KI. p. 375. Williams, J. F. (1976). I n “The Physics of Electronic and Atomic Collisions” (J. S. Risley and R. Geballe, eds.), p. 139. Univ. of Washington Press, Seattle.
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ADVANCES I N ATOMIC A N D MOLECULAR
PHYSICS. VOL.
I5
COHERENCE AND CORRELATION IN ATOMIC COLLISIONS H . KLEINPOPPEN* Fakuhat fur Physik Universirdt Bielefeld Bielefeld, West Germany
I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Angular Correlation and Spin Experiments as Tools for Studying Impact Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . A. (e, 2e) Experiments.. ........................................... B. Interference Effects in Electron Impact loni 111. Particle-Photon Angular Correlations. . . . . . . . . Coherent and Incoherent Impact Excitation ........................... IV. Electron -Ion Angular Correlations from Autoionizing States . . V. Summary and Conclusions.. ............................... Appendix: Coherent Excitation of Degenerate States with Different Angular Momenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . ..............
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I. Introduction The large variety of atomic collision processes is most adequately and comprehensively described and summarized in the monumental volumes of Massey and his collaborators (1969, 1971, 1974; Massey, 1976). Measurements of absolute total and differential cross sections have played a dominant role and will continue to be of great importance for the understanding of many applications of atomic collision processes. Apart from cross section measurements, recent progress in detailed studies of impact polarization, anisotropy, angular correlations, spin effects, resonance process, etc., have provided us with a wealth of new information on atomic collision processes. These new types of information are mostly related to scattering amplitudes and their phases, to state parameters of the atomic target, to a detailed kinematical analysis of the collision products, and to coherence parameters in atomic collisions. * Permanent address: Institute of Atomic Physics, University of Stirling, Stirling FK9 4LA, Scotland. 423 Copyright 0 197Y by Acddemic Re%. Inc All rights of reproduction in any form reserved ISBN 0- I2-OO3815.3
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H . Kleinpoppen
In this review we report on some of the highlights of the new types of investigations indicated above. A more comprehensive summary can be found in the Workshop Proceedings dedicated to Sir Harrie Massey on “Coherence and Correlation in Atomic Collisions” (Kleinpoppen and Williams, 1979). The choice of content of this review is given by its title, which implies that problems of “coherence and correlation” are correlated to each other. Correlations between physical quantities can simply be understood as a physical law between the physical quantities involved. We shall predominantly discuss “angular” correlations between the resulting particles of an atomic collision process. While angular correlations had already been introduced to nuclear physics in the 1940s (Hamilton, 1940; Brady and Deutsch, 1947), angular correlations in atomic physics only recently came into full swing. In the subsequent sections we shall report on recent progress in studies of particle-particle angular correlations (electron-electron coincidence experiments for investigating impact ionization and electron-ion coincidences for the study of autoionization processes) and of particle-photon angular correlations (electron-photon, ion-photon, and atom-photon coincidences for the study of impact excitation). Coherence properties of impact excitation processes are of central importance in connection with angular correlation experiments. We shall also report on the first experiments for the detection of interference effects in electron impact ionization. Furthermore, we shall draw attention to interference effects resulting from coherently excited states with different angular momentum. Y
t
\
FIG. 1. Directional correlation between spin-polarized inelastically scattered electrons [e( t ), e( J. )I and polarized photons [fro( t ), fro( J. )] resulting from excitation of polarized atoms [A( t )] by polarized electrons [e( t )].
COHERENCE A N D CORRELATION I N ATOMIC COLLISIONS
425
Angular correlation experiments can be carried out with various degrees of sophistication, which are determined by both the experimental technology and the detailed analysis. In the simplest way such experiments are restricted to “directional” correlations only where, after the collisional reaction, the particles are observed in coincidence without analyzing the spin state. Most complete information is obtained when both the state of the target atom as well as the angular correlation and the spin states of all particles resulting from the collision are determined. Figure 1 indicates the scheme of an experiment with a complete analysis of an excitation of atoms by electron impact. At least experimentalists have made a successful start on such a task (see Sections 11-IV).
II. Angular Correlation and Spin Experiments as Tools for Studying Impact Ionization Experiments on total ionization cross sections by electron impact were already reported in the 1920s and 1930s (e.g., see Massey and Burhop, 1969; also Kieffer and Dunn, 1966). The theory of ionization by impact processes has been introduced, based on the principles of quantum mechanics by Massey and Mohr (1931) and by WetzeI (1933). Recent progress in basic studies of impact ionization of atoms has particularly been achieved by angular correlation measurements of the two outgoing electrons (e, 2e processes) and by electron spin experiments with polarized electrons and polarized atoms. While directional correlations of outgoing electrons in impact ionization provide information on transfer of momentum and the momenta of the outgoing electrons, spin analysis in impact ionization is the first step into an analysis of amplitudes describing the ionization process. We first report on studies of electron-electron angular correlations studies (Section 11,A). Applications of polarized electrons and polarized atoms had only recently been successful in connection with studies of impact ionization in atomic hydrogen and alkali atoms (Section 11,B). A. (e, 2e) EXPERIMENTS
I . Electron -Electron Angular Correlations from Impact Ionization By replacing the photon by an electron in Fig. 1 we obtain the schematic arrangement for the study of an ionizing electron collision with an atom. So far electron-electron coincidence experiments from ionization
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H . Kleinpoppen
processes have been restricted to those without spin state analysis. After the first electron-electron coincidence experiment by Ehrhardt et al. (1%9) similar measurements were performed for the intermediate energy range by Beaty et al. (1978a,b), who measured absolute coincidence cross sections within and outside the reaction plane for argon and helium at about 100 eV. Other groups (McCarthy and Weigold, 1976; Weigold and McCarthy, 1978; Hood et al., 1977; Camilloni et al., 1978; Coplan et al., 1978) carried out (e, 2e) measurements at 1 keV or higher energies, where the theory of impact ionization has adequately been developed. Such measurements provide information on the structure of target electrons in atoms and molecules (Section II,A,2). In order to simulate photoionization processes van der Wiel and Brion (1974)and also Tan and Brion (1978) applied (e, 2e) measurements at impact energies of several keV and small momentum transfer. Cvejanovit and Read (1974) applied a coincidence time of flight method for two pairs of scattering angles to study the threshold laws of electron impact ionization (Wannier, 1953; Rau, 1976; Peterkop, 1971). The extensive measurements of various experimental groups can be summarized as follows: (1) For sufficiently high (say two to three times ionization threshold) impact energies the ionizing collision proceeds qualitatively like a classical binary interaction that is characterized by a collision between a quasi-free electron and the projectile electron (potential 1/rI2). The amplitude caused by this interaction is forward peaked with a symmetry axis parallel to the momentum transfer direction. The angular correlation may be modified by the influence of the electron-ion interaction (potential 2/r1). Ehrhardt et al. (1969) called the peak in the direction of the momentum transfer the “binary peak” and the peak in the opposite direction the “recoil peak.” In the directions nearly perpendicular to the momentum transfer the amplitudes of the binary and the recoil process cancel each other (see Fig. 2). These terms suggest that in the “direct” interaction between quasi-free electrons the momentum transfer characterizes the electron-electron angular correlation, whereas for interactions leading to the recoil peak electrons have recoiled around the ion so as to leave the scattering center in the direction opposite to that from the binary process. The classical binary encounter theory, however, does not describe the angular correlation of the electrons quantitatively; it provides only a qualitative understanding of the angular correlations especially if the energies of the scattered and ejected electrons are not too small. In this theory the electrons resulting from the ionization process are treated as free and
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t ieoo
FIG.2. Angular correlation of the (e,2e) coincidence cross section for electron impact ionization of helium (Ehrhardt ct a / . , 1969, 1972). EO,initial electron energy, =256.5 eV; 8, = 4", fixed scattering direction of the one electron with an energy E , = 212 eV; the distance of the dots from the center of the polar plot is proportional to the coincidence rate for the second electron with an energy Eb = 20 eV and detected in coincidence at scattering angle Ob;full-line plane-plane approximation (Veldre ct a / . , 1966).
their momenta can be written k,, + k, = k,
+ kt,
(1)
(k,,, k,, k, , kb are the momenta of the initial electron, the target electron, and the two outgoing electrons, respectively). In case no recoil peak occurs, the coincidence cross section is determined by the momentum distribution of the atomic electron (see Section II,A,2). Figure 3 shows, as an example, most recent (e,2e) data from distorted-wave approximations (Bransden P t al., 1978) compared to experimental data. (2) For small impact energies (about twice the ionization threshold or less than 50 eV in case of helium) the coincidence cross sections of the two outgoing electrons show quite different behavior compared to the binary encounter theory. For light and intermediate collision energies most scattering events display values of large E, and small values of 29,, i.e., most scattered electrons are scattered into a cone in the forward direction (E, and 29, are scattering energy and scattering angle of the electron with the larger energy) whereas the ejected electrons are slow and nearly isotropically distributed. Low-energy ionization processes, however, show a distribution of the scattered electron that covers a wide range of values of 29, ; furthermore the energy E, is more evenly spread over values from 0 to Eo - El,, (EiOn is the ionization energy).
428
H . Kle inpoppe n
180“
FIG.3. Coplanar electron-electron angular correlations from electron impact ionization of helium (initial energy 256.6 eV); (a) fixed scattering direction 0 = 4”; 3.0 eV energy of ejected electrons; distorted-wave approximation with second-order nonlocal potential (polarization and exchange); -, asymptotic charge 1; ---, asymptotic charge 0; --, static potential, asymptotic charge 1 (Bransden er a l . , 1978); experimental data points of Ehrhardt ef al. (1972).
Ehrhardt and his collaborators demonstrated a large variety of shapes in angular correlation curves (depending on the various kinematical conditions of the two outgoing electrons) from helium impact ionization. We only present an example in Fig. 4, which exhibits three peaks, a strong one in the forward direction, a broad one in the backward direction, and a less intense one near 90”. Contrary to the correlation curves described by the binary encounter theory the momentum transfer has lost its significance for the symmetry shown in Fig. 4. The measurements of Fig. 4 also demonstrate that the coincidence cross section does not depend much on the ratio E, /Eb.According to threshold predictions for ionization (Peterkop, 1971; Vinkalns and Gailitis, 1968) the cross section should even be independent of E, /Eb close to threshold. Further tests on theory of ionization processes have been reported in connection with (e, 2e) processes from atomic hydrogen by Weigold et al. (1977). Their data for 250 eV incident electrons, 186.4 eV “scattered,”
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429
o,,
t1800
FIG.4. Measured (e, 2e) distribution in helium (Ehrhardt ef a / . , 1979) for three different E , /Eb ratios. The measured data points have been replaced by curves representing averaged experimental data.
and 50 eV "ejected" electrons are in poor agreement with Born and plane-wave Born approximation. The Coulomb-projected Born approximations of Geltman and Hidalgo (1974) appear to give improved agreement. Interesting extensions (first suggested by Balashov et al., 1972) of electron-electron angular correlation measurements have been made in connection with the study of the decay of the autoionizing states of helium (2s2p)'P and (2p2)'D (Weigold et a / . , 1975) and of the 3s3pe4p state of argon by Jung et a / . (1977). It could be shown that the energy distribution of the coincidence count rate with the electrons ejected from the autoionizing states can be fitted to a Fano resonance profile. From the data obtained it was concluded (Ehrhardt et a / . , 1979) that the autoionizing state of argon was aligned. Theoretical predictions of the (e, 2e) angular correlation associated with the above autoionizing state were given by Amusia et a / . 1979b (ref. 5 in Jung et a / . , 1978; random-phase approximation) and by Balashov et a / . (1979)based upon the plane-wave Born approximation. Weigold et a / . (1975) were able to determine the relative contributions of the resonance cross section and the direct cross section within the region of the energy of the autoionizing states of helium. 2 . Atomic Structure from (e, 2e) Experiments Although this review is mainly concerned with applications in atomic collisions it would be inappropriate to exclude the most impressive infor-
H . Kleinpoppen
430
mation on atomic structure obtained from (e, 2e) angular correlations. However, it is sufficient to report only selected examples and to refer to comprehensive recent reviews (McCarthy and Weigold, 1976; Weigold and McCarthy, 1978; McCarthy, 1979). At high initial energy and for geometrical and kinematical symmetry (i.e., with the conditions lkal = lkbl for the momenta of the outgoing electrons and 0, = - 0 b for the scattering angles for coplanar electronelectron coincidences) the electrons interact only weakly with the residual ion. Accordingly, the recoil momentum q = ko - k, - kb can closely be connected to the momentum of the struck electron in the target system (neglecting the recoil energy of the ion and the thermal energy of the atom). This q-profile resulting from (e, 2e) experiments gives direct information on the square-space wavefunction of the struck electron. Further atomic structure information is related to electron binding energies el (which follows from the relation E = E , + Eb = Eo - el, where Eo is the energy incident on the target, and E,, Eb are the energies of the two coincident electrons), identification of the characteristic orbitals of each ion eigenstate formed in the collision, spectroscopic factors (probability that an eigenstate has a configuration consisting of a hole coupled to the ground-state target), and electron correlation effects. Camilloni et al. (1972) first applied (e, 2e) correlation experiments at high incident and outgoing energies (symmetric coplanar geometry) for information on the momentum distribution of target (ejected) electrons in the 1s state of carbon. Weigold et al. (1973) first reported studies on 3s and 3p orbitals of argon by using noncoplanar symmetric geometry (in which the azimuthal angle is variable). These authors found strong correlation effects in the 3s orbitals. The theory of (e, 2e) experiments with regard to structure information has recently been summarized by McCarthy and Weigold (1976) and McCarthy (1979). The theory of (e, 2e) processes involves complete knowledge of the momenta and energies of four bodies, an incident electron (ko),a target atom (or molecule) in its ground state Ig), two outgoing elecThe amplitude Mf trons (k,, kb), and the residual ion in an eigenstate for the (e, 2e) reaction is then written as
If).
kbl (flTlg) IkO) (2) Under both the assumptions (“direct knockout approximation”) that the T operator depends, as a three-body operator, only on the coordinates of the center of mass of the residual ion and of two electrons (one of which is the incident electron and one of which is initially bound) and that the which contains neither electron, commutes with T, the state vector (e, 2e) amplitude becomes Mf(k0
7
If),
ka 7 kb) = (ka
COHERENCE A N D CORRELATION IN ATOMIC COLLISIONS
43 I
Mf = ( k a , kblTl(flg)ko) (31 This amplitude represents a momentum transform of the generalized overlap function (flg), which depends only on the atomic structure of the target and ion. The differential cross section may then be considered as a measure of the overlap function via a profile of recoil momenta q = ko k , - kb. By applying configuration interaction calculations and using we obtain target Hartree-Fock functions as basis sets for 18) and
If),
[coefficient a,(g) for the target ground state and coefficient tj$) for the final state with a single hole in orbitalj]. The overlap function with the ground state 10) (so that ah”’ = 1) is then (flg) = Zjt$Pj and the differential cross section (for the coincident electrons) follows as m = Kn,Slf’((ka, kb(TI*ikO)12
(5)
where K is an appropriate kinematic factor, n, the degeneracy, and the spectroscopic factor S f f )= [t‘#]2. Sif’obeys the sum rule derived from the normalization and closure properties of the wave functions (ZfSif’ = 1). The spectroscopic factor is an important quantity, which can accurately be determined from the (e, 2e) reaction if the assumption leading to the expression for m in the above differential cross section is true. As pointed out by McCarthy (1979) a direct experimental test of Eqs. (3) and ( 5 ) is the measurement of the profile of recoil momenta, which should be correctly described by a Hartree-Fock orbital. Figure 5 shows the (e,2e) momentum profile q for the 29.3 eV (3s) transition in argon at various energies from 400 to 1200 eV. They confirm the validity of the above direct knockout assumption since all the profiles have essentially the same shape (which has also been shown for further argon states at E = 38.6,41, 44, 48, and 51 eV; McCarthy, 1979) and, second, the profile is correctly described by a Hartree-Fock orbital. Similar results in other rare gas atoms confirmed the applicability of the (e, 2e) reaction as a tool for studying atomic structure. As pointed out before, binding energies of individual electronic orbitals could also be obtained from (e, 2e) studies. The (e, 2e) data also confirm the application of the sum rule for spectroscopic factors. Finally, as emphasized by McCarthy and Weigold (1976), in the short time that (e, 2e) experiments have been carried out they have been rather more successful than the corresponding (p, 2p) experiments in nuclear physics. In a similar way proton-proton coincidence experiments have been used to study proton momentum distributions of nuclear states (Jacob and Maris, 1973; Mors et al., 1977).
H . Kleinpoppen
432
0
i 2
1
9(au)
F I G .5. Recoil momentum distribution for the 29.3 eV (3s) transition in argon for 400(A), 800 (x), and 1200 (0)eV initial energy (Weigold et a / . , 1973; Hood er a / . , 1974; Weigold er a / . , ref. 154 in McCarthy and Weigold, 1976). The curve represents the plane-wave theory using Hartree-Fock wavefunctions of Froese-Fischer (1972).
B. INTERFERENCE EFFECTSIN ELECTRON IMPACTIONIZATION Recent progress in the technology of crossed-beam experiments with polarized electrons and polarized atoms has enabled physicists to study interference effects in electron impact ionization of alkali atoms (D. Hils and H. Kleinpoppen, 1977, 1978) and of atomic hydrogen (Alguard et a l . , 1977). The basic idea behind these investigations rests upon the fact that an interference effect is connected with the analysis of the following spin reactions in the ionization process of one-electron atoms:
The ionization rates for parallel and antiparallel spin directions of the electrons and atoms in the initial states before the collision may differ from each other, which may be observable as an “integral” effect in the total number of ions produced (in other words, the difference in the rate of ionization for parallel and antiparallel spin directions may not only be detectable in the differential cross sections but also in the total cross sections).
COHERENCE A N D CORRELATION IN ATOMIC COLLISIONS
433
The ionization process is customarily described in terms of the direct { f ( k ' , k ) } and the exchange { g ( k ' ,k ) } scattering amplitude. By neglecting inner-shell ionization, the spin-averaged total ionization cross section can be written as (in atomic units)
with E as the total energy of the system, 6 the energy of the ejected electron, and k , , k , k' momenta of the incident, ejected, and scattered electron, respectively (Peterkop, 1961b; Rudge, 1968). The ionization cross section can then be written in the form
=
Q'
-
( 8)
Qint
By applying partially polarized electrons (polarization P,) and partially polarized atoms (polarization P A ,one-electron systems) an analysis of the spin dependence in the total ionization cross sections leads to the equation [e.g., see Kleinpoppen (1971) where an equivalent analysis for the scattering of partially polarized electrons on partially polarized electrons has been described] A = PePAQint/Qo
(9)
where A is the asymmetry factor, which represents the ratio of the ionization rate electron and atomic spin initially parallel to that with the spins initially antiparallel. Measurements of A , P,, PAcan be applied to determine the relative total interference cross section Qlnt/Qo. A diagram of the apparatus necessary for such a study is given in Fig. 6. A beam of thermal alkali atoms is polarized by passing through a hexapole magnet. The alkali beam leaving the hexapole magnet has a calculated polarization of 72%. This was verified within 5% by analyzing the beam with a Stern-Gerlach magnet. The atomic beam passes the interaction region where the ionization process takes place in a crossed-beam arrangement with polarized electrons in the presence of a small magnetic field. Hyperfine coupling between the electronic and nuclear spin in the alkali atom reduces the atomic polarization by a factor (21 + l)-'. For potassium and sodium (1 = #) the atomic polarization is reduced to 18%. The alkali beam intensity is monitored with a hot-wire detector. Spin-polarized electrons are produced by elastic scattering of unpolarized electrons on mercury atoms at an energy of 80 eV and a scattering angle of 80" (Kessler, 1%9). A filter lens selects the elastically scattered electrons and retards the electrons to any desired energy in the ionization
0
DYE LASER
KLU C W P L R
f
-0 KCELERATOR
wtc
T
Iw k
rn
V
W K)N
/
DETECTOR
0;:; H)T
WIRE DETECTOR
FIG.6. Scheme of the experiment for the measuremen >frelative cross sections of interference in th electron impact ionization of alkali atoms (Hils and Kleinpoppen, 1977, 1978). The beam of polarized atoms is produced by means of a hexapole magnet; polarized electrons are produced by scattering of unpolarized electrons on a mercury atomic beam. In the intersection region where the polarized electron beam crosses the polarized atomic beam, a small homogeneous magnetic field can be switched into the direction parallel or antiparallel to the atomic beam direction. A laser beam can be focused into the interaction region in order to change the atomic polarization by optical pumping; a Mott detector serves for the measurement of the electron polarization, a Stern-Gerlach magnet combined with a hot-wire detector measures the atomic polarization.
COHERENCE A N D CORRELATION IN ATOMIC COLLISIONS
435
region. The energy scale was calibrated by the onset of the ionization of mercury. The electron spin polarization was measured by Mott scattering after accelerating the electrons to 100 keV. The electron beam current A with a measured polarization of 20%. The relwas typically about ative orientation of the two spin directions is controlled by a longitudinal magnetic field between the hexapole field and the interaction region, which adiabatically rotates the atomic spin either parallel or antiparallel to the electron spin direction (the initial electron spin direction is determined by the scattering of the electrons by the mercury atomic beam; the electron spin is not affected by the direction of the small magnetic field, because the electrons are too fast to adiabatically follow the changes of the magnetic field). As an alternative procedure in the alkali experiment, unpolarized resonance light of a dye laser that induced the resonance transition in sodium was applied in order to destroy the atomic polarization. The difference between the ionization rates with the laser on and off determined Qo and Qint. Figure 7 presents measurements of Qint/Qofor sodium and potassium
0 3
01
0 1
2
3
4
5
6
7
8910
20
E/I
FIG. 7 . Qln,/Qo for electron impact ionization of alkali atoms: experimental data for 0
potassium and 0 sodium (Hils and Kleinpoppen, 1977, 1978: Hils et a / . . 1979); tB and A, Born approximation for Li and Na, respectively, by Peach (1965, 1966).
436
H . Kleinpoppen
(Hils and Kleinpoppen, 1978; Hils ef al., 1979) compared to theoretical predictions based upon the Born approximation (Peach, 1965, 1966) for lithium and sodium. Figure 7 shows fair agreement between the experimental and theoretical data at higher energies. Of course, it is not expected that Born approximation will be valid at lower energies. Recently Klar and Schlecht (1976) predicted a threshold value for the ratio Qlnt/Q = 1 , which disagrees with the measurement in Fig. 7 unless there is a rapid fall of Qlnt/Qfrom unity to just above threshold. Similar results on atomic hydrogen (see Fig. 8) were presented by Alguard et al. (1977). They too, produced polarized hydrogen atoms by means of a hexapole magnet but used the Fano effect for the production of polarized electrons. The atomic hydrogen data also suggest that if the threshold prediction of Qint/Q = 1 is correct it has to be approached rapidly within a small energy range above threshold.
13 15
20
30
40
50 60
80 100
150
200
E NERGY (eV 1 PIG. 8. Experimental and theoretical data of Qm/Q for electron impact ionization of atomic hydrogen: the crosses represent experimental results of Alguard e / a / . (1977). (a-d) Born exchange (BE) calculations from Rudge and Seaton (1965). Peterkop (1961". 1962), Geltman e / a / . (1963), and Goldin and McGuire (l974), respectively; (e) BE calculation with maximum interference taken from Peterkop (]%la, 1962); (f) BE calculation with angledependent potential from Rudge and Schwartz (1%6); @, h) special average exchange calculations allowing for maximum interference according to Rudge and Seaton (1965); (i-k) Glauber exchange (Goldin and McGuire, 1974), modified Born-Oppenhhimer (Ochkur, 1964) and close-coupling calculations (Gallaher, 1974), respectively; (I) BE calculations taken from Rudge (1978).
COHERENCE A N D CORRELATION I N ATOMIC COLLISIONS
437
The Stirling and Yale experiments with both polarized electrons and polarized atoms extend the previous experiments with polarized atoms (Rubin et al., 1969; Campbell e f al., 1971 and 1972; Hils et al., 1972) and the triple electron-scattering experiment of Hanne and Kessler (1 974, 1976) to the next state of sophistication in experimental technology. Most of the original suggestions on the extraction of scattering amplitudes and phases with polarized electrons and polarized atoms can be traced back to the publications on the theoretical analysis of experiments with polarized electrons and polarized atoms (Kleinpoppen, 1967, 1971; Rubin et al., 1969; Bederson, 1969a,b; Blum and Kleinpoppen, 1974, 1975).
111. Particle- Photon Angular Correlations It has only been very recently that angular correlation experiments have been successful with regard to atomic excitation processes. Such experiments have been carried out in connection with impact excitation by electrons, ions, and atoms. It is interesting to note that the analysis of particle-photon angular correlations are particularly dependent on the coherence of the excitation processes involved. This is contrary to the description of collisional excitation by means of quantities like the total and differential cross section or the impact polarization of line radiation in experiments with axial symmetry (traditional type of experiments with unidirectional particles impinging on the atoms and separate detection of observables from collisional excitation); for example, the theoretical description of line polarization in impact experiments with axial symmetry (Percival and Seaton, 1958; Baranger and Gerjuoy, 1958) was based upon the assumption of incoherent excitation of magnetic substates. Macek and Jaecks (1971) first drew attention to the problem of accounting for coherent magnetic sublevel excitation in angular correlation experiments of impact excitation, which had been verified first by Standage and Kleinpoppen (1976). In order to account for the importance of coherence in angular correlation experiments, we first consider a special case of complete coherence in impact excitation in detail; this example should illustrate the important aspects of coherent excitation in a transparent way.
COHERENT A N D INCOHERENT IMPACTEXCITATION We discuss the topics of this section with reference to either completely coherent or completely incoherent excitation. Most of the arguments presented here were originally introduced by Fano (1957) and were recently
H . Kleinpoppen
438
discussed in connection with particle-photon angular correlations by Macek (1976) and Blum and Kleinpoppen (1976).
I . Completely Coherent Impact Excitation Completely coherent impact excitation should be possible under the assumption that both the projectile particles and the target atoms are in "pure" quantum-mechanical states initially. Due to the linearity of the Schrodinger equation the final state resulting from the collisional excitation is again a pure state. In representing the projectile particles by the state vector )'fI and the target atoms by the state vector [ A ) , the initial state I$,") = 1P)IA) is "transferred" to the final state by the following scheme: state vector before the collision
state vector after the collision
polelllldl U l l ) dcling during the collivondl excitdlion
According to such a scheme the knowledge represented by the state vector I$out) represents maximum information on the excitation process (Fano, 1957). a . Construction of an example: completely coherent excitation by electron impact. We discuss the electron impact excitation e(Eo) A + A* + e(Eo - Ethr)according to the above scheme with pure states; we require the transfer of an initial-state vector to the final-state vector as a result of the collisional excitation:
+
with L o = 0, spin So,and component MF,,
electron
momentum po and spin component m ,
We restrict ourselves to ' S have
+
'P excitation with So = 0, S ,
=
0. We then
COHERENCE A N D CORRELATION IN ATOMIC COLLISIONS
Spin conservation requires M,, Imo) cancels out and we have
439
+ m, = M s , + m ,or m, = m,.Therefore
or P o d =
lW1~l~~lP1~
whereby the final-state vector is represented by pure states. The excitation process under consideration is described by a linear superposition of the magnetic substates ILML) (coherent sublevel excitation) with a given momentum ( p , ) for the electron having excited the atom and being scattered in a fixed direction. The important point in connection with this example is that the spin dependence falls out in the excitation under consideration so that we can carry out the excitation process with unpolarized electrons. In order to test these assumptions we have to include the decay process of the excited state: llPl) + IISo)lhv). In other words, if the decay process “conserves the coherence” both the atom in the lower state and the photon should be in pure states. A critical test of these assumptions can be based upon a study of the state of the photon from the lP + ‘ S transition. Being in a pure state the photon radiation is expected to be completely coherent. The coherence properties of the photon radiation can be measured by means of the classical Stokes parameters. In an electron-photon coincidence experiment, the Stokes parameters are related to polarized intensity components with reference to the scattering plane as follows:
+ p22 =
1 (normalized to unity) 1(0°) - I(90”) I(45”)- I( 135”) P3 = i(pzl - p12)= I(RHC) - I(LMC),
I = pll P I = p,1
p2z = P2 = plz - pzl = -
(10)
. . . are the components of the photon density matrix p = (gig;;), I(cu), I(RHC), I(LHC) are the linearly or circularly polarized components of observed photons (angle LY with reference to the beam direction of the incoming electrons; see Fig. I). Criteria for coherence of the photon radiation (Born and Wolf, 1965) are the vector polarization
pll,
[PI = (IPi12 +
IP212
+
IP312)1’2
(11)
H . Kleinpoppen
440
and the coherence correlation factor
with lpl as degree of coherence and p as effective phase (e.g., the phase between two orthogonal light vectors). Complete coherence is valid for /PI = lpl = 1. Such a complete coherence of the excitation/deexcitation 1 9 , .--, 31P1 .--, 2lSOof helium could be verified by measuring the Stokes parameters of the helium A = 5016 8, line radiation (Standage and Kleinpoppen, 1976). Figures 9 and 10 show the experimental scheme and the results for the Stokes parameters lpl and IPI for 80 eV incident electron energy. This confirmation for complete coherence in the lS -+ ‘P excitation of helium permits us to analyze this process in terms of excitation amplitudes and their phases and also in terms of state parameters.
6 . Amplitudes and state parameters from electron -photon angular correlations. We only report here on recent results of such investigations; with regard to both technological aspects of coincidence techniques as well as to the theoretical analysis of angular correlation experiments we refer to recent reviews (Macek, 1976; Kleinpoppen et a l . , 1976). The information to be obtained can be briefly summarized as follows: The bracket
ANALYZER
*
(b)
0
LENS
SCATTERING PLANE
FIG. 9. Schematic diagram for studying the Stokes parameters in an electron-photon coincidence experiment (Standage and Kleinpoppen, 1976). The photons, detected normal to the scattering plane, are analyzed by means of (a) a linear polarizer and (b) by a A/4 wave plate to provide circular polarized light analysis.
COHERENCE A N D CORRELATION I N ATOMIC COLLISIONS
SCATTERING ANGLE IDEG)
SCATTERING ANGLE IDEG)
441
SCATTERING ANGLE IDEGI
FIG.10. (a-c) Experimental data for the vector polarization components P I ,P I . and P2, respectively, of the He 3'P-Z1S (5016 A) coincident photons at 80 eV incident electron energy vs. electron-scattering angle (Standage and Kleinpoppen, 1976). Solid line, first Born approximation: dashed line, multichannel eikonal approximation (Flannery and McCann, 1979,(d) vector polarization lPl;(e) degree of coherence 11.~1.
can be normalized to the total differential cross section with uML=l= - aML1-l(because of reflection invariance) and aML=o as excitation amplitudes for exciting the magnetic sublevels ML = 0, ? 1 , with X as the phase and aM,+,, and with the redifference between the two amplitudes aML=o lation D = uo+ 2u1between the partial cross sections for exciting the magnetic sublevels and the total differential cross section. Angular correlations between the electrons and photons can be calculated with the knowledge of the electric dipole matrix element (lSolr[(lPl))where the ket is the e.g., the state vector with the amplitude representation CMLaMJLML); probability density for the photon emission after electron scattering in a particular direction (upon which aML=O= uo and uM,=*] = ul depend) is given by dW -day _ - - 8lr3 [A --1
sin2 8,
- A sin 2
+ [A(A
-
( 1 - A) +( C O S ~8, + 2
e, cos 2(4, cos
1)
&)]
x sin 28, cos(4,
- &)
(14)
with A = m0/w and the polar and azimuthal angles for the electrons (e) and photons ( y ) . In the experiments of electron-photon angular correlations from the 1 'So + 21P1 + l1S0process of helium (Eminyan et a / ., 1973, 1974)A and x parameters were extracted from angular correlations between inelastically scattered electrons and the A = 584 A photons from the
H . Kleinpoppen
442
2lP + 1% transition for the first time. It then was shown by Standage and Kleinpoppen (1976) that the phase x between the excitation amplitude was identical to the phase difference p between the two orthogonal light vectors of the 5016 8, radiation observed from the 3lP + 2lS transition perpendicular to the scattering plane (the identity x = p proved the assumption of complete coherence of the excitation process). Examples for the measurement of A and x are demonstrated in Figs. 11 - 14. Absolute calibration of u0 and ul for the l1S0+ 31P1 excitation have been carried out by Chutjian (1976). Additional information on target parameters can be obtained from the angular correlation data. A complete analysis of orbital angular momentum transfer to the atom can be obtained from the knowledge of A and Xdata. For example, it can be shown (Eminyan et al., 1974; Kleinpoppen, 1975) that the expectation values for the orbital angular momenta of the excited 'P, state,
are calculable from A and
(L,)
=
x with respect to i = x, y, z directions, e.g.,
( L , ) = 0,
(L,)
=
-[A(l -
sin
x
(16)
(in units of h). It also follows (in units of h2)that
E- BDeV
/
/
4
0.2
-
0 0
I
1
1
10
20
30
1
4 0 0
I
1
1
1
10
20
30
40
.
SCATTERING ANGLE (DEG)
FIG.1 1 . Experimental data of A = uo/ufor the He, l'S, + 2*P1+ 1'S0excitation/deexcitation process [Eminyan er a / ., 1974) at 60 and 80 eV compared to theoretical predictions: -first Born approximation; ---, distorted-wave approximation (Madison and Shelton, 1973); -.-. many-body approximation of Csanak et a / . (1973).
2.01
1.5
1
1
,
,
'
'
,
,
I
,
i
="
I
0
10
20
'
30
e( DEG)
'
1
10
50
FIG. 12. Experimental data of 1x1 for He I'S + 2'P 4 1% excitation/deexcitation at 80 eV compared to theory: -.-, distorted-wave approximation (Madison and Shelton, 1973); eikonal distorted-wave approximation with Glauber distorting potentials (Joachain and Vanderpoorten, 1974) at 81.6 eV: -, 10-state eikonal (Flannery and McCann, 1975).
---.
0
0
20
LO
60
80
100 120 1LO SCATTERING A N G L E
- -
ELECTRON
160
1 I
FIG. 13. A = u ~ / udata for 1's 2'P 1's excitation/deexcitation of helium at 81.2 eV. 17,Ugbabe er a / . (1977); 0, Eminyan er a / . (1974); A, Tan et a / . (1977); 0,Sutcliffe er a / . (1978); 0, Hollywood er a / . (1978);---, first Born approximation; -, Madison er a/. (taken from Sutcliffe er a / . . 1978); -, Thomas er a / . (1974); -.-, Scott and McDowell (1976).
H . Kleinpoppen
444
P.
T
ELECTRON SCATTERING ANGLE (degrees)
FIG. 14. Modulus of phase difference x between excitation amplitudes a,, and a, for 1 3 , --* 2IP1 + I S o excitation/deexcitation of helium and electron impact energy 81.2 eV. 0, Hollywood e r a / . (1978); 0, Eminyan ef a / . (1974); 0 , Ugbabe er a / . (1977); McDowell (1976).
(L:) = A,
(L:) = 1 ,
(Lg) = 1 - A
Scott and
(17)
with (L2) = L(L + 1) = 2 for L = 1 . Fano and Macek (1973) connected the anisotropic population of the magnetic substates with alignment tensors (ACo1)and an orientation vector (OCo1) of the excited atom, which can be calculated from A and x or from the Stokes parameters P1,Pz, and P3 (Standage and Kleinpoppen, 1976): AF' = 4 (3L: - L') = ( 1 - 3A)/2 = -(1 + 3P1)/4 A C O l = +(L,L, + L,L,) = [X(1 - A)]"' cos x = -Pz/2 I+ AZC+"'= + ( L i - L:) = + ( A - I ) = (PI - 1)/4 Of?:-.'= +(I!,,) = - P3/2
(18)
Figures 15- 17 present such alignment and orientation parameters for helium 'PI excitation. It is interesting to note the large variations of these quantities depending on scattering angle and impact energy. While most of the experimental work has been confined to the electron impact excitation to helium to date, some data on atomic hydrogen, neon, argon, and even molecular hydrogen have become available recently. We only briefly discuss these results and refer to more extended reviews (Blum and Kleinpoppen, 1979; Kleinpoppen and Williams, 1979). Electron-photon angular correlations and Stokes parameters have been measured for the two resonance lines of argon at 1067 and 1048 A by McConkey and Malcolm (1979). Different A and x parameters were reported for these two lines. Total coherence for the excitation of the
COHERENCE A N D CORRELATION IN ATOMIC COLLISIONS
445
oo 40
80
120
200
I60
Electron energy (eV)
Scattering angle (degrees)
FIG. IS. Fano-Macek orientation vector as a function of electron energy and different scattering angles (left) and of scattering angles and fixed excitation energy (80 eV) for the helium llS, + 2'P, transition. Left-hand diagram includes only experimental data (Eminyan ef a / ., 1974). whereas the right-hand diagram also displays theoretical predictions based upon distorted-wave theory for 78 eV (curve from Madison and Shelton. 1973).
1067 8, was demonstrated at an electron scattering angle of 5". Arriola et af. (1975) studied the A and x parameters of the first two resonance lines of neon, which they were unable to resolve. In connection with this experiment the question still remains for what scattering angles and energies the A and x parameters have to be separately treated in the singlet and triplet excitation in neon. A parameters were also measured for the excitation of the Lyman a
0.5
P
P
0 .L
5
n 0.3
P P
1 - 00
-
i
,, '
I
1
20 LO 60 80 100 120 140 ELECTRON SCATTERING ANGLE 101 ( d e g r e e s )
FIG. 16. Fano-Macek orientation vector vs. electron scattering angle at 81.2 eV impact energy for llS, + 2IP, excitation of helium. 0 , Sutcliffe el a/. (1978); 0, Eminyan el a / . (1974); 0, Ugbabe ef a / . (1977).
H . Kleinpoppen
446
N
IF
-0\
0,I
I I
\
\
* \ \
"w
a
i
-0.41
-as+
FIG. 17. Alignment parameters A::: ,2 (in units of fi2/2) calculated from experimental A 2*P1electronimpact excitation of helium and xparameters (Eminyaner a / . , 1974) for I'S, at 80 eV (Kleinpoppen and McCregor, 1976).
radiation of atomic hydrogen. Figure 18 displays results of such measurements in comparison with theoretical predictions. Further recent results have been reported by Hood et ul. (1979). First experimental investigations of electron-photon angular correlations from excitations of the Werner band of molecular hydrogen were recently reported by McConkey and Malcolm (1979) and Malcolm ef al. (1978). Lyman and Werner band polarizations were measured for the H2 radiation in the range from threshold to 300 eV. Stokes parameters for the Werner band emission were related to relevant multipole moments as introduced by Blum and Jakubowicz (1978). Extensive theoretical investigations have been carried out with regard to theoretical predictions of electron-photon angular correlations from impact excitation of atomic hydrogen. Two-dimensional contour maps for electron-photon angular correlations between electrons and Lyman a photons have been calculated by Morgan and McDowell(l975). The exci-
COHERENCE A N D CORRELATION IN ATOMIC COLLISIONS
447
H (1s -2p)
0
I
I
1
I
I
I
30
60
90
120
150
180
8 e DEGREES FIG. 18. A parameter for electron impact excitation of Lyman a radiation of atomic hy-
drogen vs. electron scattering angle at 100 eV. O.Experimental data of Dixon ef a / . (1978); 0, J. F. Williams (private communication, 1978). Full curve, Born approximation;dashed curve, distorted-wave polarized orbital approximation (McDowell er a / . , 1975: Morgan and McDowell, 1975).
tation of atomic hydrogen is of special interest in connection with angular correlations from coherently excited states with different angular momentum. Because of the near degeneracy of hydrogenic states with the same principal quantum number n but different orbital angular momentum, these states will be coherently excited. The excitation of atomic states can be described in terms of the relevant scattering amplitudes. Fano and Macek (1973) developed a theory characterizing atomic states in terms of expectation values of angular momentum operators (orientation vector and alignment tensor). This method, however, gives no complete characterization of coherently excited states with different angular momentum. In order to describe the interference terms between these coherent states, additional multipoles must be introduced. The necessary generalization of Fano and Macek’s method has been given recently by Gabrielse and Band (1977) and Blum and Kleinpoppen (1977), and for the general background and theory we refer to these papers. Gabrielse and Band (1977) parametrized the density matrix of coherently excited states in terms of the various electric and magnetic multipoles and their time derivatives. They particularly showed that the “coherent” multipoles, characterizing the interference terms, provide new information about the scattering process, and they stressed the importance of a determination of these “interference parameters.” Blum
448
H. Kleinpoppen
and Kleinpoppen developed an equivalent method in terms of “state multipoles” [following earlier work by Fano (1953) in nuclear physics]. If atomic states with sharp angular momentum have been excited, these multipoles can be related to certain expectation values of angular momentum operators (e.g., to the spherical components of the Fano-Macek “orientation vector” and the “alignment tensor”). For atomic ensembles with different angular momenta the coherently excited states are not completely described by these orientation and alignment parameters. A more general approach is then required in terms of state multipoles, which allows us to define multipole parameters (Blum et a1 ., 1978) characterizing interference between coherently excited states of different angular momentum (see Appendix). 2. Incoherent Impact Excitation Incoherent excitation will occur in collisional excitation, where at least either the projectile or the target particle are in a state represented by an incoherent mixture of eigenstates. The collisional excitation process can then be described by density matrices or the scheme
which represents the “transfer” of the colliding partners before (density matrix pin)and after the collision (density matrix pout).For example, incoherently populated magnetic substates are assumed in the theory of impact radiation of atoms (Percival and Seaton, 1958; Baranger and Gerjuoy, 1958); no information on phases of amplitudes can then be obtained (Eminyan et al., 1974). A collisional system loses its axis of symmetry (e.g., direction of incident beam) if another axis besides this axis is introduced [e.g., in the coincidence experiments as discussed above the momentum of the outgoing particle represents a second direction as axis; or in beam-foil experiments with tilted foils (Berryet al., 1974), where the normal to the foil is a second axis]. Loss of reflection symmetry with regard to a plane perpendicular to the axis of the ingoing beam can be achieved by an electric field, parallel or antiparallel to the beam. As in the above examples this loss of symmetry has the result that coherent excitation will occur instead of incoherent excitation in zero electric field. As reported in detail by Mahan and Smith (1977) and Krotkov (1975) an electric field superposed parallel to the beam of electrons, which excite Balmer a radiation of atomic hydrogen, results in a coherent excitation of states of opposite parity. In this case the electronic charge cloud of the hydrogen atom is displaced along the electric field. A calculation of the term lJleven + JlddI2representing the charge density of co-
COHERENCE A N D CORRELATION IN ATOMIC COLLISIONS
449
herently excited states of even- and odd-parity hydrogen states yields cross terms that change sign across the plane transverse to the electron at the nucleus. This results in an intensity asymmetry of the Balmer a radiation (Mahan and Smith, 1977), which is seen as an intensity difference between the electric field being parallel or antiparallel to the ingoing electron beam, in other words, while no interference effects occur in excitation of hydrogen with axial symmetry and zero electric field coherent excitation occurs with nonzero electric field. 3. Heavy-Particle -Photon Angular Correlcrtions
Particle -photon angular correlations have recently been reported in connection with ions and atoms as projectiles. Quasi-molecular mechanisms involved in such excitation processes cause additional complications in the interpretation of angular correlations from particle-photon excitation processes. On the other hand studies of coherence properties can provide additional information on the mechanism of heavy-particle impact excitation. Polarized-photon-ion angular correlations from Mg+-He, -Ne, and -Ar collisions were extensively studied by N . Andersen et al. (1979). The scheme of their experiment corresponded to that of Fig. 9: Coincident photons emitted from the projectile ions were detected perpendicular to the scattering plane defined by the ingoing and outgoing beam of Mg+ ions; a complete analysis of the photon polarization state based upon the measurement of the Stokes parameters was carried out. In collisions of 3p) excitation/deexthe Mg+ ions with the rare gas atoms the Mg I1 (3s citation occurs. The coherency (see Section III,A,l,a) of the photons from this transition is modified by the depolarizing influence of the finestructure coupling of the excited 2P state, which develops after the collisional excitation is completed. Andersen et a l . (1979) showed that the polarization P = (PI+ P; + Pi)1125 1 for the photons of the 2P + ?3 transition is modified by the fine-structure interaction to
*
P’
=
(APT
+ + P i + Pi)1’2
(19)
which is expected to be unity for completely coherent excitation. The experiments carried out by Andersen et al. (1979) at 15 keV projectile energy are summarized in Fig. 19. The modified polarization vector P‘ is large (0.7 s P’ s 1 ) for Mg+-He and Mg+-Ar, whereas for the Mg+-Ne system P’ decreases steadily above T = 15 keV deg. The deviation of P’ from unity has been estimated to be attributed to cascades and contributions from the Mg I1 (3p + 3d) transition. The interpretation of the results of Fig. 19 has been suggested by Andersen et al. (1979) as follows. At en-
H . Kleinpoppen
450 He
Ne
Ar
0
LT
a 0'
10
20
30
0
10
20
30
0
10
20
30
T(keV deg)
FIG. 19. Mg I1 (3s + 3P) emission probability and modified vector polarization P' [see Eq. (19)] for 15 keV Mg+-He, Ne, Ar collisions as a function of reduced scattering angle T = EO (Andersen et al., 1979).
ergies above 10 keV, the Mg I1 (3s + 3p) excitation predominantly occurs due to direct electrostatic interaction between the Mg+ 3s ion and the rare-gas atom (mechanism I). The target electrons of the rare-gas atoms are expected to play a rather passive role in this direct excitation mechanism. The data in Fig. 19 indicate that the Mg I1 (3s + 3p) excitation in collisions with the rare-gas atoms is highly coherent for the direct mechanism (small T values). The alternative excitation mechanism occurs at energy below 10 keV where the excitation of the Mg+ 3s valence electron occurs at molecular-curve crossings and where the electron shells of projectile and target interpenetrate each other. In this mechanism (II), the rare-gas electrons will play an active role during the formation and the break-up of the quasi-molecule. If this quasi-molecular excitation mechanism dominates (large T values) the Mg I1 (3s + 3p) excitation is still highly coherent for the asymmetric systems (He, Ar targets) whereas the excitation is incoherent for the quasi-symmetric Mg+-Ne system. Similar experiments for photon-scattered atom coincidences including a vector polarization analysis of photons have been carried out by Zehnle et al. (1978, 1979) for potassium excitation in collisions between K atoms and rare-gas atoms. The fast beam of potassium atoms was produced by surface ionization of K+ ions on a hot Re ribbon (Mecklenbrauck, 1976). The fast scattered K atoms are detected by surface ionization on a tungsten ribbon kept at about 700 K. The reflection of the fast atoms as ions is a process fast enough to conserve the time correlation between the coincident photons and the atoms. The photons of the transitions K(42P3/2,1+ /2
COHERENCE A N D CORRELATION I N ATOMIC COLLISIONS
45 1
4'%,,,) are detected perpendicular to the scattering plane. The Stokes parameter analysis again corresponds to the scheme as described in Fig. 9. Figure 20 shows the results for the vector polarization If'(,the degree of coherence 111.1, and the effective phase p for the excitation of the above transition in collisions between K atoms and He, Ne, and Ar atoms. Based upon both the assumption of coherent magnetic ml sublevel excitation and the depolarizing influence of fine and hyperfine interaction in the . excited 42P state of potassium, /PI varies between 0.16 and 0.62 and 1111 from 0 to 0.62. It follows from the data in Fig. 20 that coherent excitation of the K 42P state occurs in the K-He and K-Ne excitation; with regard to the K-Ar system Zehnle et a l . (1979) suggested that both single excitation of the potassium atom and simultaneous excitation of both the potassium and argon atom occur with comparable probability. Under the assumption that at least for the K-He and K-Ne the excitation is coherent for the ml sublevels of the K(42P)states, X = m o / u and x parameter (see Section III,A, 1,a) can be derived from the following relations between the Stokes parameters and the A and x parameters:
IPI
K-Ar
E ,,=501eV
FIG.20. Modulus of vector polarization (PI, degree of coherence (PI. and effective phase difference p vs. the reduced deflection angle 7 and the impact parameter h for K(4*P) excitation in K-Ar collisions (Zehnle ef a / . , 1979).
H . Kleinpoppen
452
PI
3/~'~'G'~'(l - 2A)/(4 - h'2'G'2') Pz = 6h'2'G'21A(l - A)]"z cos x/(4 - h'2'G'2') p3 -- -6h(l)G(l)[A( 1 sin x/(4 - h(2)G(2)) =
(20)
where h(*)and G") (i = 1,2) are defined by Eqs. (8) and (40) in the paper of Fano and Macek (1973). The values of G"' and GC2'were calculated by Zehnle et al. (1979) by taking into consideration the hyperfine splitting and the lifetime of the K(42P) states. The factors G(" and G(2'are due to the fact that after the excitation the spin orbit and hyperfine interactions couple I, s, and I before the radiation decay occurs. [This process depolarizes the line radiation and accordingly reduces the degree of the coherence in the photon radiation: this resembles the depolarization of the resonance lines of Lis, Li', and Na23by the influence of the hyperfine interaction as studied by Hafner et al. (1965) and Flower and Seaton (1967).] As pointed out by Zehnle et al. (1979) the importance of the Stokes parameters is that they can be used as tests for coherent excitation. The vector [Eqs. ( 1 1 ) and (12)] are polarization (PI and the degree of coherence JpJ restricted within the above limits, which depend on h(*)and G"). These considerations were first used to test the coherence properties of the = G@))one finds I PI = He(3lP,) excitation by electrons. In that case (G") IpI = 1 (see Fig. lo), independent of the scattering angle of the electrons. Since the observed values for ( p (and (PIof the excitation of K(42P)from the K-He and K-Ne collisions are falling within the calculated intervals for coherent excitation, Zehnle et al. (1979) determined A and x parameters (Fig. 21). A was calculated from the P1and P2 data; most accurate values for x were obtained from A and P3 data: less accurate values for x resulted from P,and P2 and by using P3only for the sign of x (crosses in Fig. 21). While the A and x data shown in Fig. 21 directly refer to the atomic 2P state of potassium, further information can be extracted from a quasimolecular picture of the collisional excitation process at low energies. As discussed by Zehnle et al. (1978, 1979) nonadiabatic transitions can occur from the quasi-molecular ground state X2C [emerging from K(42S)He,-Ne] to the excited states A211 and B22 [emerging from K(42P)He, -Ne]. Under the assumption that orbital angular momenta remain strongly coupled to the internuclear axis of the quasi-molecule the molecular states IC) and Ill)correlate with the atomic substates Ilml) of potassium for t + 30: In) correlates with 110) and IC) with ( l / d ) { l l l ) - 11 I)}. From the analysis of the Stokes parameters quantities similar to the A and x parameters can then be extracted for the description of the excitation quasi-molecular state:
x
= l4'/{l~Zl2
+ lanI2}
(21)
453
COHERENCE A N D CORRELATION IN ATOMIC COLLISIONS
(b)
no
15
10
1
1
h 08-
aa '
06
1
' 1
b IAI 10
I
'
ECH: 93 sV
08
x.' '
.
06
01
'
1
02 .
1
b& '
I 06 -
06
02 5 /
10
,
I
I
I
I
1o
TlkeV
desl
8
20
rlkeV degi
FIG.21. A and x vs. 7 (energy times scattering angle) and impact parameter h for K(42P) excitation in collisions between potassium atoms and (a) N e and ( b ) He atoms (Zehnle e / ( I / . 1979). For data and points see text.
.
x = - 1 1: dt(V,
-
v,)
(22)
with Vn and V , as the potential energy in the two states A211 and B2C. Figure 22 shows results for and X of the collisional systems K-Ne and K-He. It has been suggested by Zehnle et uf. (1979) that the strong impact parameter dependence of X in the K-He system is due to the fact that transitions between the quasi-molecular ground state X2C and the quasi-molecular excited states A211 and B2C occur over a relatively extended range of internuclear distance although there are no crossings between the states involved. Transitions based on potential curve crossings at a well-localized range of internuclear distances are expected to have a weak dependence of X on the impact parameter. This would be the case for the K-Ne system though further evidence is thought to be necessary to establish such a mechanism. Very interesting results on the excitation mechanism for 3.0 keV He+-He charge transfer collisions leading to He(33P) have recently been reported by Eriksen ef a / . (1976) and Jaecks ef al. (1979). The scheme of their experiment can again be related to that of Fig. 9: The polarization
x
454
H . Kleinpoppen
X
io
10
[deql
(b)
-I
K- Ne E ,,
2M
1
13LOeV
100
0
-
x
15
10
I
1
08 '
06
blh
1
08
06 01 02
10
20 T lkeV degl
x
FIG.22. and X parameters for the collisional excitation system (a) K-He and (b)K-Ne (Zehnle et nf., 1979).
dependence of the coincidence rate between the 3889 8, photons [He(33P + 23S) transition] and scattered neutral He atoms is studied whereby the photons are observed perpendicular to the plane defined by the incoming He+ ions and the scattered neutral helium atom in the excited state. Without explaining further details of the experiment it can be shown that a complete polarization analysis of the coincident photon radiation results in a determination of the excitation amplitudes a,, and u1 for the magnetic sublevel excitation of the 33P state with m l= k 1, including information on the phase difference between the amplitudes. Figure 23 shows the examples of results. The most surprising feature of these data is the nearly constant phase difference of ? 90" between the two excitation amplitudes for all angles and energies measured. An interpretation of this result was given by Eriksen et al. (1976). Using the electron promotion model and assuming a Landau-Zener crossing between the l s u , ( 2 p ~ ~ ) ~ Aand ~ 8 ,( l s ~ , ) ~ 4 d 4 2 , H eelectronic t energy curves, these authors could derive expressions for the amplitudes a. and a l with fixed 90" phase relation. This 90" phase difference results from the inability of
COHERENCE A N D CORRELATION IN ATOMIC COLLISIONS
455
He* + He 3.05 kev
3889
a
z I
I
1.5 20 SCATTERING ANGLE (deg)
(0)
H i t He 3.05 k p 3889 A
t
-
0
1.0 15 2.0 SCATTERING ANGLE (deq)
I 1
I
I
I
I
15 20 SCATTERING ANGLE (dog)
FIG.23. (a) Amplitude probabilities (relative units Pb and P i in numbers of particlephoton coincidences per lo0scattered particles for exciting m, = 0 and nil = t 1 substates in He+ + He + He(33P) + He+ charge exchange. ( b ) Phase difference between the two amplitudes.
the molecular electron cloud to follow the rapidly rotating internuclear axis; the excited-state molecular wavehnction is “frozen in space” (Jaecks ef af., 1979) when it is populated at a Landau-Zener crossing. Therefore in this situation the phase difference between the two amplitudes a. and a , should be rather insensitive to the exact scattering potential. However, theoretical predictions of uo and a, should be sensitive to the molecular potentials of the He$ system, which have not been calculated up to now.
IV. Electron- Ion Angular Correlations from Autoionizing States Excitation amplitudes and phases of the autoionizing states (2p2)’Dand (2s2p)’P of He were recently determined from measurements of electron-ion angular correlations (Kessel rf af., 1978; Morgenstern and Niehaus, 1979). The experimental scheme was as follows. He+ ions were scattered by helium atoms. Electrons emitted from the above autoionization states were energy analyzed and detected in coincidence with the scattered ions. Figure 24 shows noncoincident and coincident energy-
H . Kleinpoppen
45 6
2 0
r 10 0
I
30
I
1
40
35
3
I I
(b 1 @=O
i0
40
35,OL electron energy
I eV )
FIG.24. Energy spectra of the electrons ejected from the unresolved autoionizing states He(2pZ)'Dand He(2s2p)lP in He+/He collisions at 2000 e V , angle of electron ejection 0 = 135". (a) Noncoincident spectrum; (b-d) energy spectra measured in coincidence with ions scattered into 0 = 6" and with azimuthal angles for the ejected electrons at 4 = 0, 90, and 180", respectively.
analyzed spectra of the ejected electrons resulting from 2 keV He+/He collisional excitation of the two autoionizing states (angle of ejection 8 = 135"). The top figure is a noncoincident spectrum, which shows two peaks, the one due to the electron emission of the doubly excited helium atom as target (at about energy 35 eV) and the other one due to the electron emission from the doubly excited helium atom as projectile (at about 31 eV). The two peaks are separated from each other by Doppler effect: the nominal energies for the autoionizing states He2+('D)and He2+('P)are
COHERENCE A N D CORRELATION IN ATOMIC COLLISIONS
457
35.30 and 35.54 eV, respectively. Although the spectrometer resolution was sufficient the two autoionizing states are not resolved due to the postcollision interaction (Morgenstern et a l . , 1977). In the coincidence measurement (Fig. 24b-d) the projectiles are scattered through 8 = 6" and energy analyzed (Gerber et al., 1973). In coincidence with these ions were the electrons from the target atom at 8 = 135" and azimuthal angles 4 = 0, 90, and 180" ($ = 0 lies in the scattering plane defined by the ingoing and scattered ion). Note the slight shift in energy of these three coincidence spectra due to Doppler effect. A quantitative description of these spectra is based upon the assumption that the two autoionizing states are indistinguishable and their wavefunction is a coherent superposition of the 'D and 'P magnetic substates with excitation amplitudes d, and p t n , respectively:
qi
+2
+1
dm
= m=-2
$D.m
+
Pm
+Pm
(23)
m=-1
The final state is represented by the product of a wavefunction for He+(1s) and a plane wave for the ejected electron:
Jlr
= +,oneUFr
(24)
The transition amplitude for autoionization is then given by the matrix with V , as transition operator for autoionization element M = (qrIV, that is a monopol operator: taking also the relation for the amplitudes urn = (- l ) m ~ ~ -gives m for M = M I , + MP
with ldn,l,p m and Ipml, xm as moduli and phases of the complex excitation amplitudes. IMDI2and IMPl2represent the combined angular electron-ion angular correlation, for which Fig. 25 gives an example. A fitting procedure of the coincidence data to the angular correlation function provides all the amplitudes and phases for the 'D and 'P autoionizing states. Table I gives an example. As can be seen the excitation probability of the 'D state is -97% whereas only -3% for the 'P state. Bordenave-Montesquieu et al. (1975) obtained similar results from the angular distribution of the ejected electrons with respect to the beam axis in He+/He collisions at 15 keV (noncoincidence type of experiment); however, they neglected the contributions from the 'P state.
H . Kleinpoppen
45 8
00
30' 330"
* 60" 500"
90' 270'
120" 240'
150' 210"
180"
Op
FIG. 25. Azimuthal angular distribution of coincident electrons (Kessel er a l . , 1978) of the He+/He autoionizing process with He+ ions scattered at 6". ..., Azimuthal angles from 0 to 180"; x x x , from 180 to 360"; -, theoretical angular correlation curve calculated from the parameters of Table I.
Morgenstern and Niehaus (1979) calculated the complete electron angular distribution using the 'D amplitudes of Table I and neglecting the 'P amplitudes. Figure 26 gives their result, which reflects the density of the excited electron cloud ("image" of the He atom in the autoionizing 'D state). The distribution shows rotational symmetry around an axis in the scattering plane; the axis (73" with primary beam axis) nearly coincides TABLE I MODULIA N D PHASESOF AMPLITUDES FOR THE AUTOIONIZING STATES'P A N D ID OF He" State,
m=O
m = l
lpml
5.4
7.6
x m h
0.6
1
m=2
'P
'D ldml pmlm
23 0
25 1.7
39 0.9
With reference to the incoming He+ ion beam (energy 2 keV) as axis of quantization (Kessel er a/. 1978); Morgenstern and Niehaus, 1979). (Scattering angle of He+ ions
e
=
e.)
COHERENCE A N D CORRELATION IN ATOMIC COLLISIONS
459
\ FIG.26. ”Image” of the intensity distribution of the ejected electrons from autoionizing state ID of He. The scattering direction of the He+ ion as well as the polar angle of 135” and the azimuthal angles r$ = 0, 90, 180” for the ejected electrons are indicated. The measured azimuthal angular distribution of the electrons of Fig. 25 can be qualitatively derived from this figure (Morgenstern and Niehaus, 1979).
with the momentum transfer axis (75.6’). Taking the symmetry axis of the electron distribution as axis of quantization a remarkable “transfer” of population into the m = 0 sublevel takes place at the expense of the m = k 1 and m = 2 sublevels. This suggests the following interpretation of the excitation process of the ID state (Morgenstern and Niehaus, 1979): Near the distance of closed approach the electron cloud is “blown up” by the electron promotion via the 2pu, orbital. This has also the consequence that the electron cloud does not follow the rotation of the internuclear axis; it stays frozen in space (this is similar to the He+/He excitation of He(33P) previously discussed; again a phase difference of 909 between the amplitudes for m = 0 and m = I occurs). In a rotating frame this corresponds to a 2 p u - 2 ~rotational ~ coupling. Another interesting information can be calculated from the amplitudes, namely, the transfer of orbital angular momentum (or orientation) to the electron cloud of the autoionization states; taking the amplitudes of Table I the orbital angular momentum perpendicular to the scattering plane is ( L , ) = - 0.95h.
*
460
H . Kleinpoppen
V. Summary and Conclusions It has only very recently become possible to extract data from atomic collision processes, which provide us with a description in terms of scattering amplitudes and their phases, of target parameters such as orientation, alignment, and state multipoles, and of coherence parameters (e.g., degree of coherence of the excitation). Of course, some of the underlying physical concepts, such as quantum-mechanical interference phenomena in scattering processes were already obvious in the 1920s and 1930s (e.g., the Bullard-Massey experiment on electron angular distribution and the Ramsauer-Townsend effect resulting from interference effects of partial waves). While coherent excitation of atoms has been the focus of attention in angular correlation experiments (and in this review) there are many more applications in atomic collisions (which could not be reported here), where coherence and/or correlation effects have been investigated. We only briefly mentioned the theoretical analysis and the dynamics of excitation in collision experiments and refer to Fano (1979) with regard to more recent studies of the geometrical characterization and identification of target states excited by collision. Detailed and extensive theoretical studies of the effect of optical potentials in distorted-wave calculations of angular correlation parameters A and x have been reported by Madison ( 1 9 8 , 1979). It was found that the optical potential produced a relatively small change in the calculated results and made agreement for x parameters worse. The possibility of a complete determination of scattering amplitudes for S + D transitions has been theoretically analyzed by Nienhuis (1978, 1979). New studies of spin polarization effects, caused by interference in electron-atom and ion-atom scattering and also in photoionization processes, require higher sophistication in experimental technology (“third generation” type experiments as discussed by Bederson and Jaduszliwer, 1979; Hanne, 1978, 1979; Heinzmann et al., 1978, 1979; Kessler, 1979; Raith et al., 1979; Reichert, 1979; Schussler, 1979). Coherence effects in heavy ion-atom collisions occur in total and differential cross sections of excitation, and also in charge capture and ionization processes (Aquilanti e f al., 1978, 1979; Bottcher, 1978, 1979; Hasselkamp er al., 1979; Sellin, 1979;Tolk, 1979). Recent scattering experiments with laser-excited atoms reveal through coherence effects and correlation experiments information on scattering potentials (Duren, 1979)and alignment and orientation (Hermann and Hertel, 1979; Register et d.,1978, 1979). Correlation and spin effects decisively influence multiple escape of electrons near the threshold of ionization (Klar, 1979). It remains to be seen how the experimental investigations on threshold laws of ionization with polarized electrons and
COHERENCE A N D CORRELATION I N ATOMIC COLLISIONS
46 I
polarized target atoms (Section I1,B) will progress to test theoretical predictions accounting for spin effects. Interference effects, many-electron correlations, and energy and angular momentum exchanges between electrons in a Coulomb field are the main topics related to postcollision interactions in atomic collisions (Amusia et ( I / ., 1979"; Heidemann, 1979; Nesbet, 1978, 1979; Read and Comer. 1979: Matreev and Parilis, 1978, 1979; Roy, 1978, 1979). New results on coherence effects and orientation and alignment in collisional processes of ions with surfaces and a crystal lattice have been summarized by Schectman et a / . (1978), Datz (1979), and Tolk (1979). Theoretical studies of simultaneous electron-photon excitation of atoms opened u p a completely new area of impact excitation processes (Faisal, 1979). In small-angle elastic electron-atom scattering, the effect of angular coherence can be studied in a crossed-beam experiment (Rubin, 1979). While electron-, ion-, and atom-atom collision experiments have reached a high degree of sophistication in technology and analysis (as reported here) it is exciting now to follow up the recent developments in positron-atom scattering (see reviews by Humberston, Chapter 4, and Griffith, Chapter 5 , in this volume): the Ramsauer-Townsend effect has been observed in positron-rare-gas scattering according to the predictions by Massey et al. (1966) and Thompson (1966) as resulting from interference between the induced polarization attraction and the mean repulsive atomic field. Recent progress in studies of coincidences, alignment, and polarization for inner-shell excitation processes show a similar development as with excitations of outer-shell transitions. In impressive experiments, Nakel and collaborators (Nakel, 1979) studied electron-photon coincidences of the atomic-field Bremsstrahlung by applying a Compton polarimeter for linear polarization measurements of the coincident Bremsstrahlung photons (Behnke and Nakel, 1978). Bremsstrahlung angular distributions from electron impact on free atoms have been reported (Aydinol et al., 1978, 1979). Strong interference effects in K shell vacancy sharing were found in collisions of sulfur charge 15+ on Ar (Schuch et a/., 1978, 1979). Angular distribution studies of Auger processes by electron and ion impact gave information on alignments of inner-shell substates in the target or projectile, respectively (Sandner et al., 1978, 1979; Bisgaard et al., 1978; Bruch et a / ., 1979). Polarization measurements of characteristic L - M X-ray transitions have been carried as a function of electron or ion impact energy with free rare-gas atoms (Aydinol et al., 1978, 1979; Jitschin et al. (1979) Lutz et al., 1979). Coincidence studies between ions and Auger electrons or characteristic X-ray photons in ion atom collisions resulted in detailed studies on the impact parameter dependence of ioniza-
462
H . Kleinpoppen
tion probabilities in inner shells (Lutz et a / . , 1979). Such information is particularly important for the understanding of the role of rotational coupling mechanisms in ion impact excitation of inner shells. Angular correlations between Auger electrons or characteristic photon radiation and scattered ions or electrons are of topical interest for detailed studies of inner shell excitation (Berezhko et al., 1978; Lutz et al., 1979). Coherent processes play an important role in quasi-molecular collisions and even in the production of positrons in heavy-ion collisions (Reinhardt et a / . , 1979).
Appendix: Coherent Excitation of Degenerate States with Different Angular Momenta In the excitation of atomic states with energy differences smaller than the energy spread A E of the initial beam of projectiles, the atomic states will be coherently excited (as, for example, with hydrogenic levels whose energy splittings are small compared to AE). As pointed out in Section III,A, 1,a if atomic ensembles with different angular momenta have been excited coherently the excited states are not completely described by the orientation vector and the alignment tensor (Fano and Macek, 1973). A more general approach is then required, which can be based on the introduction of state multipoles (Fano, 1953). This approach particularly allows us to define multipole parameters that characterize interference terms in the excitation of atomic states with different orbital angular mo1978). We consider as mentum (Blum and Kleinpoppen, 1977; Blumet d., an example coherent excitation of hydrogenic states with same principal quantum number n , but different L and M by electron impact. We denote the spin-averaged scattering amplitudes by
where S is the total spin of projectile and atomic target. We normalize the scattering amplitudes to the partial differential cross sections for exciting a level with quantum numbers n , L , M averaged over all spins:
The quantities of Eq. (Al) are elements (L'M'lpJLM)of the density matrix p describing the atomic ensemble. Quantities with L # L ' , M' # M in Eq. (Al) are in interference terms due to coherent excitation of these
463
COHERENCE A N D CORRELATION I N ATOMIC COLLISIONS
states. It can be shown that there is, for each pair of L , L ’ , a relationship between “state multipoles” (Fano, 1953) and the quantities of Eq. ( A l ) :
with ( L M , L‘ - M ‘ J K- Q ) as a standard Clebsch-Gordan coefficient (Blum and Kleinpoppen, 1977). From the properties of the ClebschGordan coefficients follow the limitations K s L + L ’ , - K s Q s K. Note the relations ( T(L’L)i Q ) = ( T(L ’
)KQ
)*
(A4a)
where the asterisk stands for complex conjugation, and ( T ( L ’ L ) i Q )= ( - I)L’+L+K+Q(
T(L‘L)i-Q)
(A4b)
which follows from reflection invariance in the scattering plane, and (T(L’L)kQ)=
l)’(T(LL’)k)
(-
(A4c)
which follows from (A4a) and (A4b). The state multipoles of Eq. (A3) can be related to the Fano-Macek “orientation vector” (0, and “alignment tensors” (Ao,Al-, A2+) of atomic states with sharp angular momentum L. For example, the quantities (T(LL):Q)and (T(LL)&)are proportional to the spherical components of the orientation vector ( L Q )and of the alignment tensor, respectively. As an example we list here the state multipoles and their connections to the scattering amplitude ( f L M ) , the orientation vector (O1-), and the alignment tensor ( A 2 - ,Al+,Ao), which contribute to the transition from the coherently excited hydrogenic n = 3 levels to the n = 2 level in the fieldfree case: (T(2, 2),+,) = (8/7Y2 Re(f2d2*2)- (3/7)1/2(1fz112) (T(22)A)
= - (12/7)1/2Re(.f,,fg2) -
(2/7)’” Re(.h0f2*1)
(~(22)2+~2) = (8/7)1’2(lj;2p)- (2/7)1/2[lh112)- (2/7)1/2(lfi012) (T(22)L)
=
i(4/5)”’ Im(f,l.f$z) + i(5/6)1’2Im(.fiofA)
(T(22),+,) = (l/5)112d3d), (T(20),:)
=
(hIflTO)7
(T(oo)&)= U(35’), (T(ll)$z) = (r(3p)A2+, ( T (I 1 ) l o )
=
(T(20);z)
=
(jizf&)
( T(20GO) = (.f,o.flTo)
(T(l1);i)
=
(T(11);1)
(2/3)-”2~(3P)Ao
a(3p)0,=
-g(3p)A1+
464
H . Kleinpoppen
If levels with n = 3 have been excited and the light emitted in the n = 3 + n = 2 decay is observed in coincidence with the scattered electrons, the intensity and polarization of the radiation depend on all the terms (T(L'L);,) with L = 0, 1, 2 and K s 2 and on the multipoles describing s-d-interference, that is, (T(0,2)i4) with K = 2. Numerical results for the state multipoles of n = 3 have recently been calculated by Blum et al. (1978). Furthermore, angular distribution and photon polarization (Stokes parameters) can directly be expressed in terms of these state multipoles both for the direct n = 3 += n = 2 decay or for cascade transitions n = 3 -+ n = 1 (i.e., observing Lyman a radiation in coincidence with electrons having excited n = 3 states). ACKNOWLEDGMENTS The author gratefully acknowledges the hospitality of the University of Bielefeld. Important parts of the paper are the outgrowth of the International Workshop on "Coherence and Correlation in Atomic Collisions," held at University College London, 18- 19 September, 1978. Numerous discussions with colleagues during the meeting have been of great value in preparing this chapter.
REFERENCES Alguard, M. J., Hughes, V. W., Lubell, M. S., and Wainright, P. F. (1977). Phys. Rev. Lett. 39, 334. Amusia, M. Ya., Kuchiev, M. Yu., and Sheinerman, S. A. (1979a).In "Coherence and Correlation in Atomic Collisions" (H. Kleinpoppen and J. F. Williams, eds.). Plenum, New York, in press. Amusia, M. Ya., Sheftel, S. I., Cherepkov, N. A,, and Schulz, M. (1979b). To be published. Andersen, N., Andersen, T., Cocke, C. L., and Pedersen, E. H. (1979).I n "Coherence and Correlation in Atomic Collisions" (H. Kleinpoppen and J. F. Williams, eds.). Plenum, New York, in press. Aquilanti, V., Casavecchia, P., and Grossi, G. (1978). Abstr. Comtribured Pup. oflnrernat. Workshop. "Coherence and Correlation in Atomic Collisions," p. 28, UCL London. Aquilanti, V.. Casavecchia, P., and Grossi, G. (1979). I n "Coherence and Correlation in Atomic Collisions" (H. Kleinpoppen and J. F. Williams, eds.). Plenum, New York, in press. Arriola. H., Teubner, P. J. O., Ugbabe. A., and Weigold, E. (1975). J . Phys. B 8, 1275. Aydinol, M., McGregor, I., Hippler, R.. and Kleinpoppen, H. (1978). Abstr. Contributed Pup. of Internat. Workshop. "Coherence and Correlation in Atomic Collisions," p. 6, UCL London. Aydinol. M., McGregor, I., Hippler, R., and Kleinpoppen, H. (1979). In "Coherence and Correlation in Atomic Collisions" (H. Kleinpoppen and J. F. Williams, eds.). Plenum, New York, in press. Balashov, V. V., Lipovertsky, S. S . , and Senashenko. (1972). Phys. Rev. Lett. 35, 209. Balashov, V. V., Grum-Grzhimailo, A. N., Kabachnik, N. M., Megunov, A. I., and Strak-
COHERENCE A N D CORRELATION I N ATOMIC COLLISIONS
465
hova, S. I. (1979). In “Coherence and Correlation in Atomic Collisions” (H. Kleinpoppen and J . F. Williams, eds.). Plenum, New York. Baranger, E., and Gejuoy, E. (1958). Proc. Phys. SOC. London 72, 326. Beaty. E. C.. Hesselbacher, K. H.. Honz, S. D., and Moore, J. H. (1978a). Proc. ICPEAC I n f . Conf. Phys. Electron. A f . Collisions. lOrh, 1977 p. 374. Beaty, E. C., Hesselbacher. K. H., Honz, S. D., and Moore, J. H. (1978b). Phys. Rev. A 17, 1592. Bederson. B. 11%9a). Comments A f . Mol. Phys. 1, 41. Bederson. B. (1%9b). Comments A t . Mol. Phys. 2, 65. Bederson, B., and Jaduszliwer, B.. (1979). In “Coherence and Correlation in Atomic Collisions“ (H.Kleinpoppen and J . F. Williams, eds.) Plenum, New York, in press. Behnke, H. H., and Nakel, W. (1978). Phys. Rev. A 17, 1679. Berezhko. E. G., Kabachnik, N. M., and Sizor, V . V. (1978). J. Phys. B 11, 1819. Berry, H. G., Curtis, L. J., Ellis. D. G., and Schectman, R. M. (1974). Phvs. Rev. Leti. 32, 751. Bisgaard, P., Dehl, P., RBdbro, M.. and Bruch, R. (1978). Absfr. Consrributed Pap. of Infernat. Workshop. “Coherence and Correlation in Atomic Collisions,” p. 33, UCL London. Blum, K., and Jakubowicz, H. (1978). J. Phys. B 11, 909. Blum, K., and Kleinpoppen, H. (1974). Phys. Rev. A 9, 1902. Blum, K.. and Kleinpoppen, H. (1975). I n f . J . Quunfum Chem., S y m p . 9,415. Blum. K.. and Kleinpoppen, H. (1976). I n f . J. Quunfum Chem., S y m p . 10, 231. Blum, K., and Kleinpoppen, H. (1977). J. Phys. B 10, 3283. Blum. K., and Kleinpoppen, H. (1979). Phys. Rep. in press. Blum, K., Fitchard, E. E., and Kleinpoppen. H. (1978). Z. Phys. A287, 137. Bordenave-Montesquieu, A . , Benoit-Cattin. P., Gleizes. A.. and Merchez, H. (1975). J . Phvs. B 8, L350. Born. M., and Wolf. E. (1965). In “Principles of Optics,” 3rd ed. Bottcher. C. (1978). Ahsfr. Contribufed Pop. of Infernut. Workshop. “Coherence and Correlation in Atomic Collisions,” p. 26, UCL London. Bottcher. C. (1979). In “Coherence and Correlation in Atomic Collisions“ (H. Kleinpoppen and J. F. Williams, eds.). Plenum, New York, in press. Brady, E. L . , and Deutsch, M. (1947). Phy.s. Reba. 72, 870. Bransden, B. H., Smith, J. J., and Winters, K. H. (1978). J. Phys. B 11, 3095. Bruch, R., Bisgaard, P., Dahl, P., and R#dbro, M. (1979). In “Coherence and Correlation in Atomic Collisions” (H. Kleinpoppen and J. F. Williams, eds.). Plenum, New York, in press. Campbell. D. M.. Brash, H. M.,and Farago, P. S. (1971). Phys. Left. 36A, 449, and (1972) Proc. Roy. Soc. Edinburgh 70A, 165. Camilloni, R.. Giardini-Guidoni, A , , Tiribelli, R., and Stefani, G. (1972). Phys. Re\’. Left. 29, 618. Camilloni, R.. Giardini-Guidoni, A , , McCarthy, I. E . , and Stefani, G. (1978). Phys. ReLl. A 17, 1634. Chutjian, A. (1976). J . Phys. B 9, 1749. Coplan, M. A., and Brooks, E. D. III (1977j.Ahsrr. o f P a p e r s X f h Int. ICPEACCor&erenc.c>, p. 378. Csanak. G., Taylor, H. S.. and Tripethy, D. N . (1973).1.Phys. B 6, 2040. Cvejanovic, S . , and Read, F. H. (1974). J. Phys. B 7, 1841. Datz (1979). In “Coherence and Correlation in Atomic Collisions” (H. Kleinpoppen and J. F. Williams, eds.). Plenum, New York. in press.
466
H . Kleinpoppen
Diiren, R. (1979). In “Coherence and Correlation in Atomic Collisions” (H. Kleinpoppen and J. F. Williams, eds.). Plenum, New York, in press. Dixon, A. J . , Hood, S. T. and Weigold, E (1978).Phys. Rev. Lett. 40, 1262. Ehrhardt, H., Schulz, M., Tekaat. T., and Willmann, K. (l%9). Phys. Rev. Lett. 22, 89. Ehrhardt, H., Hesselbacher, K. H., Jung. K., and Willmann, K. (1972).J.Phys. B 5,2107. Ehrhardt, H., Jung, K.. and Schubert, E. (1979).In “Coherence and Correlation in Atomic Collisions” (H. Kleinpoppen and J . F. Williams, eds.). Plenum, New York, in press. Eminyan, M., MacAdam, K. B., Slevin, S., and Kleinpoppen, H. (1973).Phys. Rev. Lett. 31, 576. Eminyan, M., MacAdam, K. B., Slevin, S . , and Kleinpoppen, H. (1974).J. Phys. B 7 , 1519. Eriksen, F. J . , Jaecks, D. H., de Rijk, W., and Macek, J . (1976).Phys. Rev. A14, 119. Faisal, F. H. M. (1979).In “Coherence and Correlation in Atomic Collisions” (H. Kleinpoppen and J. F. Williams, eds.). Plenum, New York, in press. Fano, U. (1953).f h y s . Rev. 90, 577. Fano, U.(1957).Rev. Mod. Phys. 29,74. Fano, U. (1979).I n “Coherence and Correlation in Atomic Collisions” (H. Kleinpoppen and J. F. Williams, eds.). Plenum, New York, in press. Fano, U., and Macek, J . (1973).Rev. Mod. Phys. 45, 5 5 3 . Flannery, M. R., and McCann, K. J . (1975).J . Phys. B 8, 176. Flower, D. R., and Seaton, M. J . (1967). Proc. Phys. Soc. London 91, 59. Froese-Fischer, C. (1972).A t . Data 4, 301. Gabrielse, G., and Band, Y. B. (1977).Phys. Rev. Lett. 39,697. Gallaher, D. F. (1974).J . Phys. B 7 , 362. Geltman, S . , and Hidalgo, M. B. (1974).J. Phys. B 7 , 831. Geltman, S . , Rudge, M. R. M., and Seaton, M. J. (1963).Proc. f h y s . Soc. London 81, 315. Gerber, G., Niehaus, A.. and Thielmann, U. (1973).J . Phys. B 10, 1039. Goldin, J . E., and McGuire, J . H. (1974).Phys. Rev. Lett. 32, 1218. Hafner, H., Kleinpoppen, H., and Kruger, H. (1%5). fhys. Lett. 18, 210. Hamilton, D.R. (1940).Phys. Rev. 58, 122. Hanne, G. F. (1978).Abstr. Contributed P a p . offnternat. Workshop. “Coherence and Correlation in Atomic Collisions,” p. 21, UCL London. Hanne, G. F. (1979).In “Coherence and Correlation in Atomic Collisions” (H. Kleinpoppen and J. F. Williams, eds.). Plenum, New York, in press. Hanne, G. F., and Kessler, J. (1974).Phys. Rev. Lett. 33, 341. Hanne, G . F.,and Kessler, J. (1976).In “Electron and Photon Interactions with Atoms” (H. Kleinpoppen and M. R. C. McDowell, eds.). p. 445. Plenum, New York. Hasselkamp, D., Scharmann. A., and Schartner, K. H. (1979).In “Coherence and Correlation in Atomic Collisions” (H.Kleinpoppen and J . F. Williams, eds.). Plenum, New York, in press. Heidemann, H. G. M. (1979). In “Coherence and Correlation in Atomic Collisions (H. Kleinpoppen and J. F. Williams, eds.). Plenum, New York, in press. Heinzmann, U., Schonhense, G., and Wolcke, A. (1978).Abstr. Contributed Pap. of Internut. Workshop. ”Coherence and Correlation in Atomic Collisions,” p. 21, UCL London. Heinzmann, U., Schonhense, G., and Wolcke, A. (1979).In “Coherence and Correlation in Atomic Collisiorrs” (H. Kleinpoppen and J . F. Williams, eds.) Plenum, New York, in press. Hermann, H. W.. and Hertel, I. V. (1979).I n “Coherence and Correlation in Atomic Collisions” (H.Kleinpoppen and J. F. Williams, eds.). Plenum, New York, in press.
COHERENCE A N D CORRELATION I N ATOMIC COLLISIONS
467
Hils, D.. and Kleinpoppen, H. (1977). Sutrlli/e Meeting v f X t h ICPEAC Conference. Paris, 1977 (private communication ). Hils. D., and Kleinpoppen, H. (1978). J . P h j ~ B. 11, L283. Hils. D.. McCusker. V.. Kleinpoppen, H., and Smith, S. J . (1972). Phys. Rev. Lett. 29, 398. Hils, D.. Rubin, K., and Kleinpoppen, H. (1979). In “Coherence and Correlation in Atomic Collisions” (H. Kleinpoppen and J . F. Williams, eds.). Plenum, New York, in press. Hollywood, M. T., Crowe. A , , and Williams, J . F. (1979). J . Phys. B 12, 819. Hood, S. T., McCarthy. I. E., Teubner, P. J. O., and Weigold, E. (1974). Phys. Rev. A 9, 260.
Hood, S. T.. Hamnett, A., and Brion, C. E. ( 1977). J . Electron Spectrvse. R e l i t . P h m o m . 11, 205.
Hood, S. T.. Weigold, E., and Dixon, A . J . (1979). J . Phys. B. 12, 631. Jacob, G., and Maris, A. J . (1973). Rev. Mod. Phys. 45, 6. Jaecks, D. H., Eriksen, F., and Fomari, L., (1979).In ”Coherence and Correlation in Atomic Collisions” (H. Kleinpoppen and F. J. Williams, eds.). Plenum, New York, in press. Jitschin. W., Kleinpoppen. H., Hippler, R., and Lutz. H. 0.. (1979). J. Phys. B . to be published. Joachain. C. J., and Vanderpoorten, R. (1974). J . Phys. B 7, L528. Jung, K., Schubert, E., and Ehrhardt, H. (1978).Ahstr. Pup., (2). I n / . ICPEACConf Phys. Electron. At. Collisions. 1977 No. 2, p. 670. Kessel, Q. C., Morgenstern. R., Miiller, B., Niehaus, A., and Thielmann, U. (1978). Phys. Rev. Lett. 40, 645. Kessler. J. (1969). Rev. Mod. Phys. 41, 3. Kessler, J . . (1979). In “Coherence and Correlation in Atomic Collisions” (H. Kleinpoppen and J . F. Williams. eds.). Plenum, New York, in press. Kieffer, L. J., and Dunn, H. G. (1966). Rev. Mod. Phys. 38, 1. Klar, H. (1979). In “Coherence and Correlation in Atomic Collisions” (H. Kleinpoppen and J . F. Williams, eds.). Plenum, New York, in press. Klar, H., and Schlecht, W. (1976). J . Phys. B . 9, 1699. Kleinpoppen, H. (1967). JILA (unpublished report) (see also Bederson. 1969). Kleinpoppen, H. (1975). Invited Lect. At. Phys., Proc. I n t . Conf:. 4th. 1974 449. Kleinpoppen, H. (1971). Phys. R e v . A 3, 2015. Kleinpoppen, H., and McGregor, I. (1976). Poi.. I n / . Summer Sch. Phys. Ionized Cases Duhrovnik, (B. NavinSek, ed.) p. 71. Kleinpoppen, H., and Williams, J. F., eds. (1979). I n “Coherence and Correlation in Atomic Collisions. Plenum, New York, in press. Kleinpoppen, H., Blum, K., and Standage, M. C. (1976). Phys. Electron. At. Collisions Invited Lect. I n t . Conf., 9th. 1975 (J. S . Risley and R. Geballe, eds.), p. 641. Krotkov, R. (1975). Phys. Rev. A 12, 1793. Lutz, H. 0.. Luz, N., Sackmann, S., Jitschin, W., and Hippler, R. (1979). I n “Coherence and Correlation in Atomic Collisions” (H.Kleinpoppen and J. F. Williams, eds.). Plenum, New York, in press. McCarthy, I. E. (1979). In “Coherence and Correlation in Atomic Collisions” (H. Kleinpoppen and J. F. Williams, eds.). Plenum, New York, in press. McCarthy, I. E., and Weigold, E. (1976). Phys. Rep. 27c, 277. McConkey, J. W., and Malcolm, I. C. (1979).1n “Coherence and Correlation in Atomic Collisions” (H. Kleinpoppen and J . F. Williams, eds.). Plenum, New York, in press. McDowell, M. R. C., Morgan, L. A., and Myerscough, V. P. (1975). J. Phys. B 8, 1838. Macek, J. (1976). Invited Lect. I n / . ICPEAC Conf:, 9rh, 1975(J. S . Risley and R. Geballe, eds.). p. 627.
468
H . Kleiripoppen
Macek, J., and Jaecks, D. H. (1971). Phys. Rev. A 4, 2288. Madison, D. H. (1978). Abstr. Constribitird P a p . of Internat. Workshop. “Coherence and Correlation in Atomic Collisions,” UCL London. Madison, D. H. (1979). I n “Coherence and Correlation in Atomic Collisions” (H. Kleinpoppen and J. F. Williams, eds.). Plenum, New York, in press. Madison, D. H., and Shelton, W. N. (1973). Phys. Rev. A 7, 449. Mahan, A. H., and Smith, S. J . (1977). Phys. Rev. A 16, 1789. Malcolm, I. C . , Dassen, H. W., and McConkey, J. W. (1978). To be published. Massey, H. S. W. (1976). “Negative Ions,” 3rd ed. Cambridge Univ. Press, London and New York. Massey, H. S. W., and Mohr, C. B. 0. (1931). Proc. Roy. Soc. Lond. Ser. A 132, 605. Massey, H. S. W., and Mohr, C. B. 0. (1933). Proc. Roy. Soc. Lond. Ser A 140, 613. Massey, H . S. W., Lawson, J., and Thompson, D. G. (1966). I n “Quantum Theory of Atoms, Molecules, and the Solid State” (P.-0. Lowdin, ed.), p. 203. Academic Press, New York. Massey, H. S . W., and Burhop, E. H. S., (1969). “Electronic and Ionic Impact Phenomena,” 2nd ed., Vols. I and 2. Oxford Univ. Press, London and New York. Massey, H. S. W., Burhop, E. H. S., and Gilbody, H. B. (1971). ”Electronic and Ionic Impact Phenomena,” 2nd ed., Vol. 3. Oxford Univ. Press, London and New York. Massey, H. S. W., Burhop, E. H. S., and Gilbody, H. B. (1974). “Electronic and Ionic Impact Phenomena,” 2nd ed., Vol. 4 . Oxford Univ. Press, London and New York. Matreev, V. I . , and Parilis, (1979). I n “Coherence and Correlation in Atomic Collisions” (H. Kleinpoppen and J. F. Williams, eds.). Plenum, New York, in press. Mecklenbrauck, W. (1976). Ph.D. Thesis, Freiburg University. Morgan, L. A., and McDowell, M. R. C. (1975). J . Phys. E 8, 1073. Morgenstern, R., and Niehaus, A. (1979). I n “Coherence and Correlation in Atomic Collisions” (H. Kleinpoppen and J. F. Williams. eds.). Plenum, New York, in press. Morgenstern, R., Neihaus, A., and Thielmann, U. (1977). J . Phys. B 10, 1039. Mors, P. M., Herscovitz, V . E., and Jacob, G. (1977). Rev. Eras. Fis. 1, 283. Nakel, W. (1979). I n “Coherence and Correlation in Atomic Collisions“ (H. Kleinpoppen and J. F. Williams, eds.). Plenum, New York, in press. Nienhuis, G. (1978). Absrr. Constrihuied Paper of Internat. Workshop. “Coherence and Correlation in Atomic Collisions,” p. 18. UCL London. Nienhuis, G. (1979).In “Coherence and Correlation in Atomic Collisions’’ (H. Kleinpoppen and J . F. Williams, eds.). Plenum, New York, in press. Nesbet, R. K . (1978). Abstr. Contributed Pup. of‘lnterntrtionol Workshop. “Coherence and Correlation in Atomic Collisions,” p. 20. UCL, London. Nesbet, R. K . (1979). I n “Coherence and Correlation in Atomic Collisions’’ (H. Kleinpoppen and J . F. Williams, eds.). Plenum, New York, in press. Ochkur, V. I. (1%4). Z h . Eksp. Teor. Fiz. 47, 1746. Peach, G. (1965). Proc. Phys. Soc. London 85, 709. Peach, G . (1966). Proc. Phys. Soc. London 87, 381. Percival, I. C., and Seaton, M. J. (1958). Philos. Trans. R . Soc. London. Ser. A 251, 113. Peterkop, R . (l%la). Z h . Eksp. Teor. Fiz. 41, 1938. Peterkop, R. (1961b). Proc. Phys. Soc. London 77, 1220. Peterkop, R. (1962). Sov. Phys.-JETP (Engl. Transl.) 14, 1377. Peterkop, R. (1976). J . Phys. E 4, L283. Peterkop, R. (1971). J . Phys. E 9, L283. Raith, W., Baum, G., Caldwell, D., and Kisker, E. (1979). In “Coherence and Correlation in
COHERENCE A N D CORRELATION I N A T O M I C COLLISIONS
469
Atomic Collisions“ (H. Kleinpoppen and J. F. Williams, eds.). Plenum, New York, in press. Rau, A. R. P. (1976).J . Phys. B 9, L283. Read, F. H. , and Comer, J. (1979). I n “Coherence and Correlation in Atomic Collisions” (H. Kleinpoppen and J . F. Williams, eds.). Plenum, New York, in press. Register. D., Jensen, S . , and Trajmar, S . (1978). Abstr. Contributed Pup. of Irrternurionol Wurk.shop. “Coherence and Correlation in Atomic Collisions” p. 1. UCL, London. Register. D., Jensen, S., and Trajmar, S. (1979). I n ”Coherence and Correlation in Atomic Collisions” (H. Kleinpoppen and J . F. Williams, eds.). Plenum, New York, in press. Reichert, R. (1979). In ‘-Coherence and Correlation in Atomic Collisions‘‘ (H. Kleinpoppen and J . F. Williams. eds.). Plenum, New York. in press. Reinhardt, J., Muller, B., and Greiner, W. (1979).I n “Coherence and Correlation in Atomic Collisions” (H. Kleinpoppen and J. F. Williams. eds.). Plenum, New York, in press. Roy. D. (1978). Absrr. Conrribured puper uf Inrerntrr. Workshop “Coherence urld Corrrltrtion in Alontic Collisions,” p. 15. UCL London. Roy. D. (1979). I n “Coherence and Correlation in Atomic Collisions” ( H . Kleinpoppen and J. F. Williams, eds.). Plenum, New York. in press. Rubin, K. (1979). In “Coherence and Correlation in Atomic Collisions” (H. Kleinpoppen and J . F. Williams, eds.). Plenum. New York. in press. Rubin, K., Bederson, B., Goldstein. M., and Collins, R. E . (1969). Phys. Rev. 182, 201. Rudge, M. R. M. (1968). Rev. Mod. Phys. 40,564. Rudge, M. R. M. (1978). J . Phys. B 11, L149. Rudge, M. R. M.. and Schwartz, S. B . (1966). Pruc. Phys. Soc. London 88, 563. Rudge. M. R. M., and Seaton, M. J. (1965). Proc. R . Soc. London, Ser. A 283, 262. Sandner. W., Weber, M., and Mehlhorn, W. (1978). Abstr. Contributed Pup. c!f Internutiorictl Workshop. “Coherence and Correlation in Atomic Collisions,“ p. 30. UCL, London. Sandner, W.. Weber, M., and Mehlhorn. W. (1979). In “Coherence and Correlation in Atomic Collisions” (H. Kleinpoppen and J . F. Williams, eds.). Plenum, New York, in press. Schectman, R. M., Curtis, L. J., and Berry, H. G . (1979). In “Coherence and Correlation in Atomic Collisions“ (H. Kleinpoppen and J. F. Williams, eds.). Plenum, New York, in press. Schuch. R . , Nolte. G . . Lichtenberg, W., and Schmidt-Backing. H. (1978). Abstr. Contrihrtted Pup. o f Inrrrnutionul Wurk.~hop.“Coherence and Correlation in Atomic Collisions,” p. 36. UCL, London. Schuch. R., Nolte, G.. Lichtenberg, W., and Schmidt-Backing, H. (1979). I n “Coherence and Correlation in Atomic Collisions” (H. Kleinpoppen and J . F. Williams. eds.). Plenum. New York, in prress. Schiissler, H. A . (1979). /ti “Coherence and Correlation in Atomic Collisions’’ (H. Kleinpoppen and J . F. Williams, eds.). Plenum. New York, in press. Scott, T., and McDowell, M. R. C. (1976). J . Plry.\. B 9, 511. Sellin, I . A . (1979). I n “Coherence and Correlation in Atomic Collisions” (H. Kleinpoppen and J . F. Williams, eds.). Plenum, New York, in press. Standage, M . C.. and Kleinpoppen, H. (1976). Phys. Rev. Lerr. 36, 577. Sutcliffe, V . C.. Haddad, G . N., Steph. N . C . , and Golden, D. E . (1978).Phys. Rev. A 17, 100.
Tan, K. H., and Bnon, C. E. (1978). J . Electron Spec~trosc.Relur. Phenom. 13, 1 1 . Tan. K. H., Fryar, J . , Farago. P. S . , and McConkey. J. W. (1977). J . Phys. B 10, 1073.
470
H . Kleinpoppen
Thomas, L. D., Csanak, Gy., Taylor, H. A., and Yarlagadda, B. S. (1974). J. Phys. B 7,71. Thompson, D. G. (1966). Proc. R . SOC.London, Ser. A 294, 160. Tolk, N . (1979). In “Coherence and Correlation in Atomic Collisions’’ (H. Kleinpoppen and J. F. Williams, eds.). Plenum, New York, in press. Ugbabe, A., Teubner, P. J., Weigold, E. and Arriola, H. (1977). J . Phys. B 10, 71. van der Wiel, M. J., and Brion, C. E. (1974). J. Electron Spectrosc. Relat. Phenom. 1,439. Veldre, V. Ya., Damburg, R. Ya., and Peterkop. R. K. (1966). I n “Atomic Collisions” (V. Ya. Veldre, R. Ya. Damburg, and R. K. Peterkop, eds.) Butterworth, London. Vinkalns, I., and Gailitis, M. (I%@.Proc. Int. Conf. Phys. Electron. Ar. Collisions, Sth, 1967 p. 648. Wannier, G . H. (1953). Phys. Rev. 90, 817. Weigold, E., and McCarthy, I. E. (1978). Adv. Ar. Mol. Phys. 14, 127. Weigold, E., Hood, S. T., and Teubner, P. J. (1973). Phys. Rev. Lert. 30, 475. Weigold, E., Ugbabe, A., and Teubner, P. I. 0. (1975). Phys. Rev. Lett. 35, 209. Wetzol, W. W . (1933). Phys. Rev. 44, 25. Zehnle, L., Clemens, E., Martin, P. J., Schauble, W., and Kempter, V. (1978). J . Phys. B 11, 2865. Zehnle, L., Clemens, E., Martin, P. J., Schauble, W., and Kempter, V. (1979). In “Coherence and Correlation in Atomic Collisions” (H. Kleinpoppen and J. F. Williams, eds.). Plenum, New York.
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ADVANCES IN ATOMIC A N D MOI.ECULAR PHYSICS. VOL. 15
THEORY OF LOW ENERGY
1 COLLISIONS P . G . BURKE Queen's University of Belfast Belfast, Northern Ireland, and Science Research Council, Dareshury Laboratory Dareshury Warington, England
11. Laboratory Frame Representation .
A. Derivation of the R B . Relationship to the
.......... IV. Frame Transformation V. L z Methods . . .
...................................... VI. Vibrational Excitation . . A. Adiabatic-Nuclei Ap
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C. Other Approaches.. . . . . VII. Conclusions ...................................................... References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47 1 473 413 417 419 480 48 I 482 483 485 488 488 490 492 495 4% 491 499 503 504
I. Introduction The excitation of molecular rotation and vibration by electron impact plays an important role in the energy loss mechanisms of low-energy electrons in molecular gases. These processes are thus of importance in the transport of energy in the ionospheres of the earth and of other planets and in electrical discharges in the laboratory. In addition, a knowledge of such processes is required to understand pumping mechanisms in gas lasers such as the C02-N2 laser. In the last few years considerable progress has been made in the ab initio theory and calculations of lowenergy electron molecule collisions. This is partly due to new theoretical 47 1 Copyright 0 1979 by Academic Ress. Inc. All rights of reproduction in any form reserved. ISBN 0- 12-0038 15-3
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methods that have been introduced and partly to the ability to make use of sophisticated bound-state computer program packages. In this chapter we shall concentrate on developments since the early review by Craggs and Massey (1959) and the more recent reviews by Takayanagi (1967), Massey (1969, 1976), Takayanagi and Itikawa (1970), and by Golden et al. (1971). Some of these developments have been outlined in recent comments by Temkin (1976a,b). The first systematic ab inirio calculations of electron-molecule collision cross sections were made by Sir Harrie Massey and co-workers in the early 1930s. Thus calculations using the Born approximation for e-H, scattering were made by Massey (1930) and by Massey and Mohr (1931, 1932) and using an independent atom-scattering model for e-Np scattering by Bullard and Massey (1930) and Massey and Bullard (1933). These were compared with the pioneering angular distribution measurements for H, , N2,and CH4 made by Bullard and Massey (1931). From this work it became clear that at low energies a partial-wave expansion must be used, except, in the case of polar molecules, where Massey (1932) showed that because of the dominance of the long-range potential the Born approximation for the total cross section could be used at all energies. We start in Section I1 from a single-center partial-wave analysis of electron scattering by a rigid rotating molecule given by Arthurs and Dalgarno (1%0). This uses a laboratory fixed frame of reference. We then show how exchange and polarization effects can be systematically incorporated into the theory. We also review the basic total and angular distribution cross section formulas. Many successful calculations of elastic scattering and rotational excitation cross sections have been carried out using this approach, but difficulties are experienced both due to the slow convergence of the single-center expansion and to the inclusion of exchange. In Section 111 we therefore consider an alternative representation in which a molecular fixed frame of reference is used. This frame is used in molecular bound-state calculations and exchange can be included in a more natural way. In addition, by removing one degree of freedom, that of rotation of the molecule, the number of coupled equations that must be solved is reduced. The validity of this approximation is discussed and the relationship with the laboratory frame representation is given. We then discuss in Section IV the frame transformation theory introduced by Chang and Fano (1972) in which they show that different representations can be used in different regions of space. In particular, they show that the molecular frame is appropriate at all energies in an internal region. In Section V this fact is made use of to discuss recent applications of Lz methods or discrete basis set methods to describe the electron-molecule wavefunction in the internal region. We show that these approaches are very
THEORY O F LOW ENERGY ELECTRON-MOLECULE
COLLISIONS
473
similar to molecular bound-state methods and that molecular bound-state codes can be used with little modification. This is a very imRortant advantage when it is realized how much work is required to develop new computer program packages describing the collision of an electron with a complex molecule. Finally, in this section we show that the relationship of L2 methods with frame transformation theory can be established using an R-matrix approach similar to that first used in nuclear reaction theory by Wigner and Eisenbud. So far our discussion has been restricted to the collision of an electron with a molecule that is constrained to the equilibrium nuclear separation. In Section VI we relax this condition and consider the possibility of vibrational excitation occurring in the collision. We show that in order for efficient transfer of energy to occur between the incident electron and the molecular vibrational motion the electron has to stay in the vicinity of the molecule for an appreciable period. This happens most efficiently in the neighborhood of a resonance. We then show that vibrational excitation can be described in either an adiabatic nuclei representation or in a representation in which the nuclear vibration is taken explicitly into account. Which is appropriate depends on the lifetime of the resonance. We then conclude this section by reviewing briefly some other approaches for vibrational excitation. Finally in Section VII we draw some general conclusions.
11. Laboratory Frame Representation A . DERIVATION OF
THE
R A D I A LEQUATIONS
We commence our discussion by considering the scattering of an electron by an N-electron rigid rotating closed shell diatomic molecule constrained to its ground electronic state, which is assumed to have l2 symmetry. The wavefunction describing the collision can be written following the approach adopted by Arthurs and Dalgarno (1960) as
Here is the wavefunction describing the target molecule, the radial functions F $ ( r ) describe the radial motion of the scattered electron, the unit vectors R and i describe the orientation of the molecular axis and the angular location of the scattered electron as in Fig. I , and the angular function 9@ (R,?)is an eigenstate of the total angular momentum opera-
474
P . G. Burke
tor .I2and its Z component .Iz,which are conserved in the collision. It is defined by
where Y;ll and Y$i are spherical harmonics describing the angular motion of the target and the scattered electron, respectively. Finally, &is the antisymmetrization operator necessary to ensure that the total wavefunction is antisymmetric in the space and spin coordinates of all N + 1 electrons. In order to obtain the radial equations satisfied by the functions Fil(r) we substitute Eq. (1) into the Schrodinger equation, which for the present problem can be written in atomic units as
Here p is the reduced mass of the incident electron, which can accurately be set equal to unity, HT is the Harniltonian for the target molecule, H R is the Hamiltonian for the molecular rotation, and the potential interaction V between the scattered electron and the target is given by
Here ZA and ZB are the charges of the two nuclei located at rAand rB, respectively. In principle there are also cross terms involving gradient operators but these are small and are usually neglected (e.g., Choi and Poe, 1977). We now project the Schrodinger equation (3) onto the functions @ ( l , . . . , N ; R)r-l%#J (R,?), yielding the following infinite set of coupled integrodifferential equations.
The wave numbers kj are given by
kf = 2(E - E l )
(6)
where El is the energy of the target in thejth rotational state, Vj:Aatis the local direct potential, and W:I,,l, is the nonlocal exchange potential. The evaluation of the direct potential is straightforward provided that some suitable representation, such as an analytic Hartree-Fock approxi-
THEORY OF LOW ENERGY ELECTRON-MOLECULE
COLLISIONS
475
mation for the target, is available. In most applications a single-center expansion of the static potential
is adopted, where the integral in Eq. (7) is carried out over the space and spin coordinates of all N target electrons and the angle 8 is defined in Fig. I . A method of calculating the VA(r)has been given by Faisal (1970) and Faisal and Tench (1971). The resultant expression for the direct potential is then
where the.fA(j/,j’/’; J ) have been tabulated by Percival and Seaton (1957). Exchange was first included in an lib initio treatment of low-energy electron molecule collisions by Massey and Ridley (1956). They formulated the e-H, collision problem in the molecular frame (see Section 111) using oblate spheroidal coordinates and solved the resultant integrodifferential equations using the Kohn variational method. The inclusion of exchange in the laboratory frame formulation of the problem considered in this section is more difficult due to the need to antisymmetrize the scattered electron, defined in the laboratory frame, with the target electrons attached to a rotating molecule. Thus while some attempts have been made to include exchange exactly within the present formulation of the Problem, most notably by Ardill and Davison (1968) and by Henry and Lane (1969) for e-H, scattering, most workers solving Eq. ( 5 ) have approximated the nonlocal exchange potential by an equivalent parametrized local potential. This approach is justified if the long-range polarization potential, caused by the distortion of the molecular charge distribu-
B FIG. 1. Coordinate system for an electron scattered from a linear molecule.
P. G. Burke
476
tion by the scattered electron is also parametrized, as is often the case. We shall therefore defer further discussion of the ab initio treatment of exchange until we have introduced the molecular frame in Section 111. The final interaction that must be considered is the polarization potential. This is the hardest interaction to include in a completely ab initio treatment of the problem and yet it plays a fundamental role in low-energy collisions. Its importance can best be understood by considering the asymptotic form of the electron molecule potential. Since the exchange potential vanishes exponentially at large distances then the total potential can be written as
where the first two terms arise from the static potential defined by Eq. (7) and the last two terms from the polarization potential. The quantities p and Q are the dipole and quadrupole moments, respectively, while a. and a2are related to the polarizabilities along and perpendicular to the molecular axis by a 0
= +(all +
2aJL
a2
= #(a,, - all
(10)
Although the polarization potential has a simple asymptotic form, it is very difficult to obtain an accurate representation for it at short distances, where it develops a complex nonlocal energy-dependent character. One approach, adopted by Lane and Henry (1968) for e-H, scattering, was similar to the well-known polarized orbital method of Temkin (1957) used for electron-atom collisions. In this theory a variational trial function is used to represent the distortion of the molecule in the adiabatic field of the electron. More recently the polarizability has been represented by including a pseudostate in the expansion of the wave function. This approach was introduced by Damburg and Karule (1967) in the case of e-H scattering and has been widely used in electron-atom scattering (e.g., Le Dourneuf et al., 1977). In the case of molecules this would involve including two pseudostates in Eq. ( 1 ) to represent aI,and aL. It has only been carried through for e-H, scattering using the R-matrix method, which will be discussed later. As we have already mentioned, both exchange and polarization are often parametrized in the laboratory frame formulation of interest in this section. One common approach is to write the polarization potential as
THEORY OF LOW E N E R G Y ELECTRON-MOLECULE
COLLISIONS
477
where C ( r ) is a cut-off factor C(r)= 1
-
exp[-
(~IY~)~]
and rc is an adjustable parameter. This potential is included in addition to the static and exchange potential and the parameter used to give the best agreement with low-energy scattering data. In this way the short-range part of Eq. ( 1 I ) represents in some average way the short-range part of the polarization.
B. THE CROSSSECTION We conclude the theoretical development in this section by giving the expressions for the cross section. Following Arthurs and Dalgarno (1960) we look for solutions of Eq. ( 5 ) that are regular at the origin and satisfy the asymptotic boundary conditions.
-
0,
k: c. ILLUSTRATIVE
RESULTS FOR e-H,
COLLISIONS
Many calculations have been carried out in the laboratory frame of reference using a single-center expansion of the static and polarization potential with some parametrization of exchange. In this section we consider by way of illustration the results obtained for e-H, scattering, where the most detailed and accurate work has been carried out. We show in Fig. 2 total cross section results obtained by Henry and Lane (1969), who included t h e j = 0 and 2 states in Eq. ( 5 ) for each J. The importance of both exchange and polarization effects is clear from this figure and both have to be included to obtain agreement with the experiments of Golden et a l . (1966) and Ramsauer and Kollath (1930). The broad peak in the total cross section at 3 -4 eV is due to a "2: resonance, which is important in dissociative attachment. For rotational transitions in low-energy electron-molecule collisions the long-range potentials defined by Eq. (9) play a much more important role. This is particularly true for scattering of electrons by polar molecules where p is nonzero and, as mentioned in the introduction, Massey (1932) pointed out that the Born approximation can give reliable results at all energies. However, in the case of homonuclear diatomic molecules such as H, and N,, where p is zero, there is a subtle balance between long- and short-range effects and detailed coupled channel calculations are necessary to obtain accurate results. This is illustrated in Fig. 3, which
THEORY OF LOW ENERGY ELECTRON -MOLECULE
COLLISIONS
479
I-
,
,
I
16'
I
I
I
I
I I
10
eV
FIG. 3. Comparison of theory and experiment for the j = 0 + 2 e-H, cross section. HL, Henry and Lane (1969); H, Hara (1969); LG, Lane and Geltman (1967): DM, Dalgarno and Moffett (1963): GS. Gerjuoy and Stein (1955); 0, measurements of Crompton et [ I / . (1969). (From Takayanagi and Itikawa, 1970.)
480
P. C. B~rlie
for exchange and polarization. An earlier calculation by Lane and Geltman (1967) using an empirically determined potential lies below experiment. Finally, the calculations of Gerjuoy and Stein (19551,who included the point quadrupole potential - QP2(cos B)/r3 in the Born approximation, and those of Dalgarno and Moffatt (1963), who extended these calculations by including the polarization term - azPz(cosB)/r4, while satisfactory at low energies fail to reproduce the 22: resonance peak at higher energies.
111. Molecular-Frame Representation While the theory developed in the previous section can and has been used for heavier molecules such as N z and CO, a difficulty arises because of the larger number of terms that must be retained in the single-center expansion of the potential and wavefunction in order to adequately represent the nuclear singularities. In addition, as we have already noted, the inclusion of exchange and polarization is both arduous and raises fundamental questions of principle in the laboratory frame. These difficulties can to a large extent be overcome by an approach similar to that used in molecular bound-state calculations. I n this case the Born -0ppenheimer separation of electronic and nuclear motion is adopted and the electronic motion is determined in the fixed field of the nuclei. The effect of the collision on the molecular rotation is then treated in a second step of the calculation. Such an approach is often called the fixed-nuclei or adiabatic approximation. From the physical point of view such an approach is expected to be valid if the time of collision is much shorter than the time of nuclear rotation. It is thus expected to be valid when the scattered electron energy is not close to threshold or when the cross section is dominated by the potential interaction close to the nucleus. However, a straightforward application of the method breaks down when the energy is close to a narrow resonance or when the long-range tail of the potential is important. Such an approach is not new. It was first used to describe the scattering of electrons by homonuclear diatomic molecules using prolate spheroidal coordinates by Stier (1932) and by Fisk (1936). It was also used by among others Massey and Ridley (1956) and Temkin and Vasavada (1967). However, in the last few years it has gained acceptance as the basis of a method that is capable of yielding the most accurate low-energy results for complex molecules.
THEORY OF LOW ENERGY E L E C T R O N - M O L E C U L E C O L L I S I O N S
48 I
A. DERIVATIO OF N T H E R A D I A LEQUATIONS
In analogy with Eq. (1) we limit our discussion to the collision of an electron with a closed-shell diatomic molecule in a 'Z state. Following Burke and Sinfailam (1970) we then write the collision wavefunction as
q m= d
2 Wl.
. . . , N;R),.-'Gl"(r)Yl"(i)
(20)
I
where the Z axis is now defined to lie along the internuclear axis. Although we have again used a single-center expansion for the target electron wavefunction, in order to relate it directly with Eq. ( I ) , this assumption is not essential and will be relaxed later. The Schrodinger equation describing this collision is now
where the notation is the same as in Eq. (3) and we note that the Hamiltonian for the molecular rotation is omitted since the molecular orientation is fixed. Projecting Eq. (21) onto the functions @ ( l , . . . , N ; R ) r - ' Y T ( i ) then yields the coupled equations
where the channels are now all degenerate with the wave number defined by k2
=
2(E
-
E,)
(23)
where Eo is the target energy. The direct potential VF, can be expressed in terms of the V h ( r )defined by Eq. (7). We find that
(24)
The determination of the nonlocal exchange potential W$ now proceeds in a straightforward way once the target state W l , . . . , N ; R ) has been defined. Burke and Sinfailam (1970) have given an explicit formula for it when the target state is represented by a single-center closed-shell configuration. and the extension to open-shell molecules would not present a serious problem. The polarization potential can be represented in this ap-
482
P . G . Burke
proach by the inclusion of polarized pseudostates multiplied by further arbitrary radial functions in Eq. (20). However, this would give rise to additional integrodifferential equations for these functions coupled to Eq. (22), which would make the whole problem more difficult to solve. In practice, therefore, a model potential similar to that defined by Eqs. (1 1) and (12) has usually been included. The cross section is determined from the asymptotic form of the solution of the integrodifferential equations (22). Asymptotically we have
where, as in Eq. (131, the second index I’ on the solution vector labels the linearly independent solution of Eq. (22). Introducing the T-matrix in the molecular fixed frame by the relation
Tm = 2iKm/(I - iKm)
(26)
then the total cross section for scattering from the molecule, after averaging over all incident electron beam directions relative to the molecular axis, is
B. RELATIONSHIP TO
THE
LABORATORY FRAMEREPRESENTATION
We can relate these equations to the results obtained in the previous section in the laboratory frame of reference. This relationship has been considered by Bottcher (1969), Burke and Sinfailam (1970), Chang and Fano (1972), Chandra and Gianturco (1974), and Chandra (1975). If the moment of inertia tends to infinity then the rotational energy levels Ej in Eq. (6) become degenerate and the channel energies kjz are then equal. In this case Eqs. (5) and (22) and the corresponding potentials and T-matrices are related by a unitary transformation. We find that
where the elements of the transformation operator are given by
and the parity quantum number
THEORY OF LOW ENERGY ELECTRON-MOLECULE COLLISIONS
= (- l)l+l-J = (-
I)l’+j‘-J
483 (30)
Substituting Eq. (28) into Eq. (16) then yields
Tflixit
It follows that in this limit the solutions of Eqs. (5) and (22) are equivalent, although from the numerical point of view Eq. (22) is clearly much simpler to solve since the additional quantum number j is not involved. This means that the number of coupled equations required to obtain convergence for each value of m in the case of Eq. (22) is less than the number of coupled equations required for each J in the case of Eq. ( 5 ) . This approach is closely related to the adiabatic-nuclei approximation introduced by Chase (1956). who was concerned with the excitation of deformed nuclei. If we define .f’(@, n:R ) as the scattering amplitude describing the scattering of an electron into a laboratory angle n by a h e d molecule whose orientation is defined by the Euler angles @, then the transition amplitude from an initial rotational state I,!J~(@) to a final rotational state $jl(@) is
R)
.fj*,j(a; =
1
$$(Plf(@,a;R)$j(P) dP
(32)
The condition for this result to be valid is that the collision time is very much shorter than the rotation time, which is the same condition used in the derivation of Eqs. (22) and (28). The angular integral in Eq. (32) can be determined analytically (Chang and Temkin, 1969; Temkin and Faisal, 1971 ) and the formal equivalence between these approaches deduced. C. SOMERECENTCALCULATIONS Many calculations have been carried out in recent years based on Eq. (22). Calculations of e-N, scattering in the static exchange approximation by Burke and Sinfailam (1970) were improved by Burke and Chandra (1972) by including a polarization potential and also by including more terms in the single-center expansion of the potential and the wavefunction. However in order to do this and still leave the equations numerically tractable they replaced the exchange potential by a procedure of enforcing orthogonality between the scattering orbitals and all molecular or-
P . G. Burke
484
bitals of the same symmetry. This had the effect of replacing the integrodifferential equations (22) by a set of inhomogeneous differential equations. More recently Buckley and Burke (1977), with the help of better computing facilities, returned to the full integrodifferential equation (22) and obtained convergence with a polarization term included. The approach where exchange is represented by an orthogonality condition has also been used by Gianturco and Thompson (1976, 1977) for e-CH,, e-HCl, and e - H F scattering and by Chandra (1977) for e-CO scattering.
a
I I I
, I
2
3
4
ENERGY
I
I
I
5
6
7
[eV]
1 8
FIG.4. Comparison of theory and experiment for the total e-N, cross section. 0 , Morrison and Collins (1978); BB, Buckley and Burke (1977); BC, Burke and Chandra, (1972); G. measurements of Golden (1966). (From Morrison and Collins, 1978.)
THEORY OF LOW ENERGY ELECTRON-MOLECULE
COLLISIONS
485
An alternative approach of replacing the nonlocal exchange potential in Eq. (22) by a local free-electron gas potential has been adopted by some authors. Work using this approach has been carried out by Morrison et ul. (1977) for e-CO, scattering, by Morrison and Collins (1978) for e-H, and e-N, scattering, and by Collins and Norcross (1977, 1978) for electron collisions with highly polar molecules. Various theoretical results for the e-N, total cross section are compared with measurements of Golden (1966) in Fig. 4. We see that there is very close agreement between the work of Burke and Chandra and that of Morrison and Collins even though they used completely different procedures for including the exchange. We also see that all the theories are too large near the ,rig shape resonance at 2.4 eV owing to the neglect of the vibrational motion of the nuclei. This problem is considered by Chandra and Temkin (1976a,b) and discussed later in this chapter. Finally we consider further the work on the scattering of electrons by methane. The first calculation for this system was carried out by Buckingham et ul. (19411, who used a spherical potential obtained from a self-consistent field calculation. They found a strong peak at about 2 eV due to a low-energy p-wave resonance. Recently Gianturco and Thompson ( 1976) have reconsidered this problem and included exchange by the orthogonalization procedure of Burke and Chandra and a model polarization potential with a form similar to Eq. (1 1). They found a broad peak at 6-8 eV in the T2 symmetry state, which had a strong d-wave component, and a low-energy Ramsauer-Townsend minimum in the A l symmetry state. Their results are in excellent agreement with the measurements by Brode (1923, Bruche (1930). and Ramsauer and Kollath (1930).
IV. Frame Transformation Theory We now return to a more detailed discussion of the region of validity of the molecular-frame approach. When the incident electron energy is large compared with the energy spacing between the rotational levels of importance in the collision, then the unitary transformation defined by Eq. (28) is accurate. However as the energy decreases toward the thresholds of interest this transformation breaks down. It was pointed out by Chang and Fano (1972) that the energy at which this breakdown is significant depends on the radial distance Y of the electron from the molecule. When the electron is close to the molecule, within its strong attractive field, its velocity is high even at low incident electron velocities and the time spent in this region is short compared with the rotation time. Thus in this internal region the molecular frame can always be used. However, at large
P . G. Biirlie
486
distances the laboratory frame must be used at low incident velocities since it is important to distinguish which rotational state the molecule is left in. This is illustrated in Fig. 5 , where A denotes the region where the molecular frame can always be used, and where the angular momentum component IJ 81represented by the quantum number A is conserved, and B denotes the region where the dependence of the wavefunction o n j is essential at low energies. An extension of this argument given by Chang and Fano to include vibrational motion, where the characteristic time is shorter than the rotational time, implies that region A must be subdivided into two regions Aa and A b , where only in the inner region A u is the motion “adiabatic” in vibration. This is discussed further in Section VI. At this point we note that there is another division of configuration space that can be made. If we consider the coupled integrodifferential equations ( 5 ) and (22) then we have already remarked in connection with Eq. (9) that while the direct potential goes over asymptotically to a summation over inverse powers of the radius, the nonlocal exchange potential vanishes exponentially. It is therefore possible to define a distance, called r,, in Fig. 5 , beyond which the exchange potential is effectively zero, leaving of course a residual long-range direct potential. Simple considerations then show that for molecules in their ground states r,, is contained within region A . This discussion then suggests the following method of solution. For r 6 ro we solve Eq. (22) retaining the nonlocal exchange potential on the RHS. This enables the wavefunction GE,(r) and its derivative dGEl/dr, or what is equivalent, the R-matrix introduced by (Wigner and Eisenbud, 1947) Region B
-
c o r r e i a i i a n s a r e aisregaroeui L
1
I
I
‘0
rl
r2
r
d i s t a n c e of e l e c t r o n from molecule FIG. 5. Partition of configuration space into different regions relevant to electron molecule collisions according to frame transformation theory. (From Chang and Fano, 1972.)
THEORY OF L O W ENERGY ELECTRON-MOLECULE
COLLISIONS
487
to be calculated on the boundary. In this equation we have written the complete set of linearly independent solutions as a matrix G” and we have introduced an arbitrary diagonal constant matrix b. We then transform the R-matrix on this boundary to the laboratory frame using the transformation given by Eqs. (28) and (29). Finally, the coupled Eqs. ( 5 ) are integrated out to the asymptotic region from r = r,, using boundary conditions defined by the transformed R-matrix. In this last step the direct potential is given by its asymptotic form Eq. (9), and the exchange potential, which is negligibly small, is omitted. This approach has recently been used by Chandra (1977) in a study of low-energy scattering of electrons by CO. Using a static potential obtained from a LCAO-SCF calculation by McLean and Yoshimine (1967), polarization represented by Eqs. (1 I ) and (12), and exchange represented by the orthogonalization method previously described, good agreement was obtained with the low-energy momentum transfer cross sections of Hake and Phelps (1967). Earlier calculations in the laboratory frame by Crawford and Dalgarno (1971)had used a semiempirical potential to fit the experimental data at thermal energies but this gave unreliable results at higher energies. It is important to note here that the frame transformation theory has also been used by Fano (1970b, 1975) and collaborators to extend the multichannel quantum defect theory of Seaton (1966a,b) to molecular ions. The internal region is characterized by eigenchannels a parametrized by a transformation matrix Ui, and eigen-quantum-defects p,, . These eigenchannel parameters then serve as a boundary condition on the surface of the internal region, which vary smoothly with the total energy. This approach has been used to analyze the angular distribution of photoelectrons from H, by Dill (19721, to study the photoabsorption spectrum of the Rydberg levels of molecular hydrogen by Atabek ef a/. (1974) and by Jungen and Atabek (1977). and to analyze the dissociative recombination of e + NO+ by Lee (1977). Returning to a discussion of the tih inirio calculation of the electron molecule wavefunction in the internal region, we note that the outstanding computational problem remaining is the slow convergence of the singlecenter expansion near the nuclear singularities. For example, in the work of Morrison (’1 ul. (1977) on e-CO, scattering, terms u p to A = 40 had to be retained in Eq. (7) to give convergence. In the next section we discuss recent developments that remove this difficulty.
488
P . G . Burke
V. L2Methods We have already remarked in the previous section that the adoption of a molecular-frame representation is analogous to that used in molecular bound-state calculations. Further, we have seen in our discussions of frame transformation theory that this representation is appropriate at all energies in an internal region. This argument therefore suggests that we look for an approach to the collision problem more closely related to bound-state methods. Further, it can be hoped that in adopting such an approach it will be possible to make use of standard bound-state packages, so reducing the very large amount of effort required to develop new electron-molecule collision programs. A . INTRODUCTORYBACKGROUND
The basic idea behind this approach was discussed by Temkin (1966) and studied in detail in a one-dimensional model problem by Hazi and Taylor (1970) and Hazi and Fels (1971). An early calculation of this type was carried out for e-N, scattering by Krauss and Mies (1970). We expand the wavefunction describing an electron molecule collision as
where the 4, are a basis set of square integrable ( N + 1) electron configurations (Lz integrable functions). We then determine the coefficients ail by diagonalizing the molecular-frame Hamiltonian defined in Eq. (2 1 )
Since only a finite numljer of terms can be retained in the expansion, the oscillating form of the exact wavefunction cannot be represented at large distances since the wavefunction described by Eq. (36) must vanish asymptotically. However, and this is the basis of the method, Eq. (34) can give an accurate description of the collision wavefunction when the scattered electron is close to the molecule and the total energy is at one of the eigenvalues El defined by Eq. (36). We illustrate this by considering the Lz calculation carried out by Morrison and Lane (1975) for e-H, scattering. They used a two-center STO basis for the three-electron system, and in this way avoided the problem of the slow convergence of the single-center expansion. We show in Fig. 6
THEORY OF LOW ENERGY ELECTRON-MOLECULE COLLISIONS
489
the sa component of their wavefunction compared with the equivalent J = 0 close-coupling wavefunction of Henry and Lane (1969) obtained by solving Eq. ( 5 ) . Clearly in the region of configuration space illustrated in this figure the agreement between the two wavefunctions is excellent. In order to extract the phase shift and the cross section from the L2 wavefunction it is necessary to assume that enough terms have been retained in Eq. (34) so that the wavefunction is accurate out to a region r 2 r,,, where exchange effects are negligible. Further, in the work of Morrison and Lane and also McCurdy et al. (1976), who also considered e-N, scattering, and Rescigno et a / . (1976), who considered e-F, scattering, a low I-spoiling approximation is used. In this it is assumed that while many angular momenta I may be coupled strongly at short range, the centrifugal barrier prevents the partial waves from being strongly coupled at large r (Fano, 1970a). This should hold particularly well for homonuclear diatomic molecules, where the center of symmetry in any case causes only even-/ or odd-/ values to be coupled. Assuming that the I-spoiling approximation is valid it is then possible to project out the radial wavefunction I?[&) from qt corresponding to a particular partial wave and to fit it asymptotically to the appropriate spherical Bessel
n
3
2
-3
0.05
h
0.04
0.03 .+ 0
0.02
C
3 6-
0.01
0,
>
B o -
?!.
U
-0.01
rg
CtI
1.0
3.0
2.0
4.0
r (a,) FIG. 6. Comparison of Lz and J = 0 close coupling radial wavefunctions for e-H, scattering. L2 wavefunction, Morrison and Lane ( 1975); close coupling wavefunction. Henry and Lane (1969). (From Morrison and Lane, 1975.)
490
P . G . Burke
functions giving
where r 5 r g . If long-range polarization and quadrupole potentials are important in this region, then the spherical Bessel functions would have to be replaced by the appropriate linearly independent solutions of the radial Schrodinger equations including these potentials. We show in Fig. 7 the phase shifts obtained by McCurdy et al. (1976) for e-N, scattering for the pn and d n phase shifts using a three-center Gaussian basis wavefunction compared with results obtained solving Eq. (22) by Burke and Sinfailam (1970). In both calculations, only the static exchange approximation was considered. Agreement is satisfactory, bearing in mind that the 1-spoiling approximation used by McCurdy et al. introduces some error and full convergence in the single-center expansion used by Burke and Sinfailam was not achieved. However, it is important to note that because a multicenter basis was used in these L2 calculations, the convergence problems of the single-center expansion has been avoided. In concluding this introductory section we note that a review of L2 methods with particular emphasis on atomic and molecular photoabsorption has been given by Reinhardt (1979). B. T-MATRIXMETHOD One of the most interesting L2 methods in which the T-matrix is expanded in a discrete basis, has been proposed by Rescigno et ul. (1974a,b). This approach, has been applied in the static exchange approxi3 O W u
d7T
30 -
b
0
0 c
c r
I
20 -
20-
-
n' ! O I
1
I
1
i
1
I
Momentum
FIG. 7. Comparison of L2 and close coupling p r and d r phase shifts for e-N, scatterclose coupling results of Burke and ing. 0,La results of McCurdy et nl. (1976); -, Sinfailam (1970) (momentum in atomic units). (From McCurdy et nl., 1976.)
THEORY OF LOW ENERGY ELECTRON-MOLECULE
COLLISIONS
49 1
mation to obtain elastic cross sections for e-H, scattering (Rescigno et
NI.. 1975) and for e-N, scattering (Fliflet et ul., 1978). Recently, Klonover and Kaldor (1977, 1978, 1979) have included polarization using perturbation theory and obtained rotational and vibrational excitation cross sections for e-H, scattering. This method starts from the Lippmann-Schwinger equation for the T-matrix T
=
U
+
UGZT
(38)
where U = 2 V , V being the nonlocal potential interaction between the electron and the target and G,+the free particle Green's function with outgoing wave boundary conditions. We now expand U in a basis set of square integrable functions + a , giving
Inserting the truncated potential U' into Eq. (38) we obtain a matrix equation for the T-matrix that can be solved, yielding T' = ( 1
-
UU;,+)-lU'
(40)
The matrix G$ is defined by
where E = fq' and k and k ' denote plane-wave states of the form
The scattering amplitude for the elastic process is then given by f(k,
+
kf) = -2~~(krlT'(E)lki)
Ostlund (1975) has shown how the matrix elements involving the planewave states may be evaluated analytically if Gaussian basis €unctions &(r) are used. If rotational excitation cross sections are required, the adiabatic approximation of Chase defined by Eq. (32) may be used except close to threshold giving
492
P . G . Burke
where the integration is carried out over the orientation R of the molecule in the laboratory frame of reference. The importance of this method again stems from the fact that standard molecular bound-state packages can be used in calculating the matrix elements of the potential. Further, an important advantage is that the extraction of the scattering amplitude from the wavefunction is built into the method and does not rely on approximations such as low I-spoiling and fitting procedures such as those based on Eq. (37). C. R-MATRIXMETHOD
A difficulty with the L2 methods discussed so far is that since they are based on the use of the molecular frame representation over all space they cannot easily take advantage of the frame transformation theory to enable rotational and vibrational excitation cross sections to be obtained close to threshold. Further, the effect of long-range potentials on the wavefunction has to be represented in the expansion basis, which is also representing the short-range interactions. This could lead to slow convergence when the long-range potentials are important as is the case for the scattering of electrons by highly polar molecules. An L2 approach that enables the frame transformation theory to be applied in a straightforward way and separates out the treatment of short-range nonlocal potentials from long-range local potentials is the R-matrix method. The R-matrix method was first introduced by Wigner (1946a,b) and Wigner and Eisenbud (1947) in a study of nuclear reactions. Recently the method has been used to describe a broad range of atomic processes including electron -atom and -ion scattering, atomic photoionization cross sections and polarizabilities, van der Waals coefficients, and nonlinear optical coefficients. These applications have been reviewed by Burke and Robb (1975) and at the symposium on the R-matrix method at the X ICPEAC (see Burke, 1978, and following papers). The method was lirst extended to treat electron molecule scattering by Schneider (1975a) and it was then applied to e-H, scattering by Schneider (1975b), to e-F, scattering by Schneider and Hay (1976), and to e-N, scattering by Morrison and Schneider (1977). An independent R-matrix method was introduced by Burke et al. (1977) and results for e-H, and e-N, scattering were reported by Buckley et al. (1979). In this method configuration space is divided into two regions by a boundary surface as illustrated in Fig. 8. In the internal region exchange between the scattered electron and the molecule is important, and the wave function is expanded in terms of a multicenter discrete basis in the molecular frame. In the external region only the local long-range dipole
THEORY OF LOW ENERGY ELECTRON-MOLECULE
COLLISIONS
493
External
Region
Boundary Surface FIG.8. Division of configuration space into two regions in the R-matrix method.
quadrupole and polarization potentials remain and these can be treated by a single-center expansion using only a few partial waves in either the laboratory or the molecular frame as appropriate. The link between these two regions is provided by the R-matrix on the surface defined by Eq. (33). In the work of Schneider and collaborators on diatomic molecules, prolate spheroidal coordinates were used and the surface is then naturally an ellipsoid of revolution. In the work of Burke ef a/. (1977) the surface was a sphere. The advantage of this latter choice is that it is more easily extended to polyatomic molecules where the system of prolate spheroidal coordinates offers no special advantage, and it relates more closely to the frame transformation theory and the partial-wave analysis of the cross section discussed in earlier sections. In the work of Burke ef a / . (1977) the wavefunction describing the electron-molecule system is expanded in the internal region in a basis that essentially combines Eqs. (20) and (34), giving
+
I
4j(l, . . . , N + l ; R ) c i j
(45 )
The molecular orbitals contained in the target channel wavefunctions (Dj and the “bound” functions djare expanded in terms of STOs centered on A and B , which vanish on the boundary while the continuum orbitals +k also involve STOs centered on the center of gravity G , which are nonzero on the boundary. The coefficients cijkand cijare determined by diagonalizing the operator H + L b , ( q j ,
(H
+ Lb)qj)
= E$ij
(46)
P . G. Burke
494
which replaces Eq. (36). The integrals in this equation are carried out over the internal region and Lb is a surface operator introduced by Bloch (1957), which for a sphere of radius ro (see Figs. 5 and 8) is defined by
which ensures that H + Lb is Hermitian for arbitrary b in the given basis. We write the Schrodinger equation in the form
(H
-
E
+ Lb)*E
= LbqE
(481
which has a formal solution
VIE
=
(H
-
E
+ Lb)-lLbqE
(49)
Finally, we expand the inverse operator in this equation in terms of the eigenfunctions defined by Eq. (46) and evaluate qEon the boundary,
Comparing this equation with Eq. (33) we see that after projecting out the channels the quantity in square brackets reduces to the R-matrix. Since the only nonzero STOs on the boundary are centered on G this projection is particularly straightforward. Once the R-matrix on the boundary surface is determined, the calculation in the external region proceeds as in our discussion of frame transformation theory. The most time-consuming part of this calculation is setting up and diagonalizing the matrix H + Lb in Eq. (46). This can be done by a small modification of standard three-center molecular structure codes, which involves subtracting the tail from those integrals that involve STOs on G that extend beyond the boundary. The results obtained for e-H, and e-N, scattering in the static exchange approximation 6 y Schneider (1975b), Morrison and Schneider (1977), and Buckley er af. (1979) are in essential agreement. Schneider (1978) has also reported calculations on e-H, scattering, including the polarizability by means of a pseudostate in expansion (45). His total cross section results are compared with the T-matrix results of Klonover and Kaldor (1978) and with the experiments of Linder and Schmidt (1971) in Fig. 9. These are the first completely ab initio calculations that have been carried out in which all important effects have been included. In concluding this section we remark that L2 methods are now capable of yielding accurate elastic scattering and rotational excitation cross sections for the low-energy scattering of electrons by diatomic molecules.
THEORY OF LOW ENERGY ELECTRON -MOLECULE
' 0
2
4
6
8
10
COLLISIONS
495
12
E (eV)
FIG.9. Comparison of theory and experiment for the total e-H, cross section. -, Schneider (1978); ---, Klonover and Kaldor (1978); A. measurements of Linder and Schmidt (1971). (From Schneider, 1978.)
We can expect efforts to extend this work to more general molecules, based on general configuration interaction bound-state codes that are currently available. In addition, electronic excitation of molecules at low energies can be expected to be included within the same framework by the inclusion of the appropriate electronically excited states in the expansion of the wavefunction.
VI. Vibrational Excitation The mechanisms responsible for strong vibrational excitation of. molecules by low-energy electrons have been reviewed recently by Herzenberg (1978). As shown by Massey (1935) the Born approximation predicts that the cross section should be of the order of lo-'* cm2 if there are no long-range dipole potentials involved. However, observations for e-N, scattering by Schulz (1962, 1964) and others, which have been reviewed by Schulz (19731, showed that the cross section was in fact of order
P . G. Burke
496
cmz near 2 eV. The explanation of this result is that in this energy range there is as we have already seen a resonance. The incident electron gets captured into this resonance state and stays in the vicinity of the molecule for a time commensurate with the vibration time. It thus has an enhanced probability of transferring energy to the vibrational motion. Another, perhaps not completely disconnected reason for strong vibrational excitation occurs in the case of polar molecules. Rohr and Linder (1976) and Rohr (1977) have shown that for HF, HCI, and HBr there is very strong vibrational excitation close to threshold. This is probably caused by a *Z+ virtual state whose location is strongly influenced by, if not caused by, the long-range dipole potential as discussed by Dube and Herzenberg (1977). A. ADIABATIC-NUCLEI APPROXIMATION The adiabatic nuclei approximation of Chase (1956) described by Eq. (32) can be readily extended to include vibrational excitation. The scattering amplitude for a transition involving rotational and vibrational excitation is hfut,ju(fl)
=
1
xut(R)fjr,j(n; R)xu(R)d R
(51)
where xU(R)are the vibrational wavefunctions of the nuclei. It follows that the scattering amplitudef,t,j(fl;R ) for fixed internuclear separation R , calculated by one of the procedures described in the previous sections, must be obtained over the range of R , where x J R ) and xu,(R)are nonzero. This approximation has been successfully extended by Temkin and Sullivan (1974) to describe the vibrational excitation measurements in e-H, scattering by Linder and Schmidt (1971) and by Wong and Schulz (1974). We have already seen that the low-energy e-H, scattering cross section is dominated by a resonance, which consists principally of an I = 1 partial wave with small admixtures of higher odd partial waves. If we assume that only one partial wave dominates the scattering, then a straightforward application of Eq. (51) shows that the ratios of transitions between specificj + j’ states are independent of the vibrational quantum numbers u and u ’ . However, observation of this ratio revealed a substantial dependence on u’ for fixed u. To explain this result Temkin and Sullivan replaced x U ( R )and x U t ( R in ) Eq. (51) by the solution of the equation
where V ( R ) is the ‘Z,+ ground state potential of H,. We show in Fig. 10, taken from the work of Temkin and Sullivan, the ratio of the vibrational excitation cross sections without change of rota-
THEORY OF LOW E N E R G Y ELECTRON-MOLECULE
COLLISIONS
497
tional quantum number to the corresponding vibrational cross section with A j = 1 -+ 3 . The theoretical calculations were carried out using x , ~ ( Rand ) in the insert figure by replacing x U j ( Rby ) X,~(R where ) in both cases 1 = 1 and 3 partial-wave phase shifts were included in .fjr,j(a, R). The comparison shows that the j dependence of the vibrational functions is indeed important in explaining the difference in the u‘ = 2 and u ‘ = 1 ratios. We remark in conclusion that Chang (1974)has carried out an independent analysis of these cross sections in terms of molecular-frame phase shifts.
B. HYBRID THEORY The criterion for the validity of Eq. (5 1) is that the collision time is short compared with the vibration time. This is indeed true in the case of e-H, scattering considered above, since the width of the resonance is several eV, giving a lifetime shorter than 10-15 sec. However, that this is not the case near the ,IIg resonance in e-N, scattering can be seen from Fig.
FIG. 10. Comparison of theory and experiment for the ratios of the vibrational excitation cross section without change of rotational quantum number to the corresponding vibrational cross section with A j = 1 -+ 3. The inset curves are given for the same ratios but where the rotational dependence of the vibrational wavefunctions is suppressed. -, calculations of Temkin and Sullivan (1974); ( u ’ = 1). Linder and Schmidt (1971); 0 ( u ’ = I), A (0’ = 2) 0 ( u ’ = 3) Wong and Schulz (1974). (From Temkin and Sullivan, 1974.)
P . G . Birrkr
498
11. This shows the potential energy curve of the ' X i ground state of N2 and the real and imaginary parts of the 211g resonance calculated by Chandra and Temkin (1976a)using a single-center expansion in the molecular frame of reference. Also shown in the figure are calculations using the boomerang model (described in the next section) by Birtwistle and Herzenberg (1971)and the L2 method by Krauss and Mies (1970). At the equilibrium nuclear separation of the molecule R o , the width of the resonance is -0.4 eV, which is somewhat larger than the vibrational spacing AEv = 0.29 eV of the molecule. However, for R > R o t the resonance width rapidly becomes smaller and violates the condition for the adiabatic nuclei approximation to be valid. Chandra and Temkin (1976a,b) have thus introduced a hybrid theory to describe the e-N, collision. In their work Eq. (20) is replaced by the expansion
qrn= &iW l ,
...
,N ; R)r-'Hl",(v)X,(R)Yl"(f)
(531
ill
where the vibrational levels are closely coupled but the adiabatic-nuclei
E-(R) E-(R) 'k,'(R)
O
4.4
4.2
0.2
VS.
(R-Ro)
*EN,(R)-EN,(R~)
0.4
0.6
R-Ro(ao)
FIG. 1 1 . Width and position of the Inaresonance in e-N, scattering as a function of the internuclear separation R . -, Chandra and Temkin (1976a); ---, Birtwistle and Herzenberg (1971); x , Krauss and Mies (1970). The lowest curve is E,,(R) - EN,(Ro). (From Chandra and Temkin, 1976a.)
T H E O R Y OF LOW E N E R G Y ELECTRON-MOLECULE
COLLISIONS
499
approximation (or molecular-frame approximation) is used for the rotational motion. Substituting this expansion for Vr,, into the Schrodinger equation for the electron-molecule system, which now includes the kinetic energy operator for the vibrational motion, we obtain the following coupled integrodifferential equation for the Hg(r.):
The direct potential is obtained by evaluating the fixed nuclei potential defined by Eq. (24) as a function of R and projecting onto the vibrational states x , ( R ) .In the work of Chandra and Temkin the exchange interaction was approximated by an orthogonalization procedure as discussed in Section II1,C and the polarization potential was represented by the parametrized form of Eq. ( 1 l ) , where a. and a, are both functions of R. Although expansion (53) was only used by Chandra and Temkin for the resonant211, state and the adiabatic theory based on Eq. (51) was used for the other states, the calculation was formidable. For each vibrational level included in expansion (53), convergence in the single-center expansion variable I has to be achieved. In practice, up to 10 vibrational states were included and three I values were retained, although previous work at a fixed internuclear distance indicates that this last expansion is unlikely to be converged. We show in Fig. 12 the differential vibrational excitation cross sections calculated by this method compared with the experimental result of Ehrhardt and Willmann (1967) at 20". The theory shows structure in the cross section in agreement with experiment although the details are different owing to the convergence problems discussed above. In addition, the theory still involves a cutoff parameter in the polarization potential and there is some question concerning the dependence on R assumed for a,, and azas pointed out by Morrison and Hay (1977), Temkin ( I978), and Kendrick ( 1978). C. OTHERAPPROACHES Although the development of the hybrid method is a substantial step forward, and yields good results for e-N, vibrational excitation, it is extremely costly in terms of computational effort to implement in the single-center form considered by Chandra and Temkin. In this section we therefore mention some other approaches that have been suggested recently and that may be easier to apply to complex molecules. We have already noted in our discussion of Fig. 5 that in the context of the frame transformation theory of Chang and Fano there is an inner
P. G . Burke
5 00
0.40
INCIDENT ELECTRON ENERGY (cVI
FIG. 12. Differential vibrational excitation cross sections at 20" for e-N, scattering calculated by Chandra and Temkin (1976a). Inset experimental curves, Ehrhardt and Willmann (1967). (From Chandra and Temkin, 1976a.)
region Aa in which the motion is expected to be adiabatic in vibration. If this region includes the region where exchange is important, this suggests that an L2 method, such as the R-matrix method, be used to determine the R-matrix on the boundary as a function of the internuclear coordinate R. The R-matrix is then transformed as in Eq. (51) to the vibrational representation and the integration in the external region completed as in the work of Chandra and Temkin using this transformed R-matrix to start the integration. This approach would clearly avoid the difficulties associated with the inclusion of exchange and the multiplicity of coupled channels occurring in the single-center, hybrid theory and attempts are now underway to evaluate its usefulness in the e-N, case. An alternative approach called the boomerang model has been introduced by Birtwistle and Herzenberg (1971) and extended recently by Dub6 and Herzenberg (1979) to give absolute cross sections for e-N, vibrational excitation. The essence of this approach is to note that the lifetime of the ,&resonance state is such that the nuclei just have time for a single vibrational cycle before the extra electron departs. Herzenberg and
THEORY OF LOW ENERGY ELECTRON-MOLECULE
COLLISIONS
50 I
collaborators parametrize the real and imaginary part of the resonance potential energy (see Fig. l l ) W ( R ) = EAR) - f i U R )
(55)
However, they could just as well have taken them from some accurate fixed-nuclei calculation, leading to an uh inirio theory. The total wavefunction is then written in the form
9 = Ql(l, . .
.
,N
+ C(1. . . .
+ ,N
1 , R ) + $I+(l, . . . , N + l ; R ) [ - ( R ) + A+[
+
I;R)[+(R) (56)
where Q Lis the initial plane-wave state, $I2 are the fixed nuclei wavefunctions with m = +- I satisfying outgoing wave boundary conditions corresponding to the energy W ( R )[using the prescription of Siegert (1939)], and [,(R) describes the nuclear motion and satisfies the inhomogeneous equation ( T + W ( R ) - E ) [ , ( R ) = {,(R)xo(R) (57) where [,(R) is the electron entry amplitude. Finally, A$Ii represents the nonresonant direct scattering. Cross sections can be rapidly calculated once [ J R ) have been determined by taking the matrix element of the interaction potential with the final-state wavefunction. We show in Fig. 13 the u = 1 +. 2 cross section at 90" calculated by Dube and Herzenberg (1979) compared with the experiments of Wong (1978). Also shown in this figure are the results of the hybrid compared with the same experiment. Clearly the Dube and Herzenberg theory gives excellent agreement with experiment in the resonant region. although there is some discrepancy in the nonresonant contribution. In comparing the boomerang model with the frame transformation theory we note that it takes into account the relaxation of the nuclear wavefunction during the collision. It was suggested by Herzenberg (1968) that this relaxation is responsible for the oscillations in the cross section as a function of energy. By contrast, a straightforward application of the frame transformation theory ignores this relaxation in the internal region. Although in principle the boomerang model could be extended to describe the situation where more than one resonance and/or electronic target state takes part in the collision, this would be difficult. In sucha situation it may be preferable to formulate the problem in terms of an expansion 9= 9J1, . . . , N + l;R)&,(R) (58) If2
in terms of the R-matrix states defined by Eq. (45). The (,,(R) then satisfy the equation
502
P . G . Burke
ELECTRON ENERGY (eV) FIG. 13. Differential cross section at 90” for u (1978); 0 . Dube and Herzenberg (1979); (b) -, (1976a). (From Dube and Herzenberg, 1979.)
(Etq‘,
( T + EdR)
-k
=
1 + 2 e-N, excitation. (a)-, Wong Wong (1978); x, Chandra and Temkin
LB)5iq)
= Eiq8qq’
(59)
where L B is a Bloch operator for the nuclear motion within an internal region defined by the coordinate R. This formulation of the problem, which can also be used to describe dissociative attachment, has been considered by Schneider et al. (1979). It combines the frame transformation approach with one that makes explicit allowance for nuclear relaxation in the internal regi0n.t Finally, we mention an interesting new “energy-modified adiabatic approximation” suggested by Nesbet (1979). In this method the S-matrix connecting target states denoted by p and u is defined by S,
= (pIS(E - H , ; R ) J u )
(60)
t Note added in proof: Results presented at the XI ICPEAC by Schneider Le Doumenf and Vo Ky Lan show that this approach accurately reproduces the structure in the vibrational excitation of N, near 2 eV.
THEORY OF LOW ENERGY ELECTRON-MOLECULE
COLLISIONS
503
where S ( E - H,; R ) is the S-matrix calculated at the fixed internuclear separation R at an energy defined by the operator E - H,, where H,, is the target Hamiltonian including the nuclear kinetic energy operator. This enables adiabatic S-matrix elements to be used, and if suitably written in terms of a symmetric K-matrix preserves unitarity and gives the correct threshold behavior. An application to e-N2 scattering gives good agreement with experiment. In concluding this section we remark that a detailed examination of the approaches considered still needs to be carried out for a variety of molecules. However, the present indications are that a basic understanding of how to calculate accurate vibrational excitation cross sections is well advanced.
VI. Conclusions In this review we have traced the development of ab initio calculations of low-energy electron molecule collisions from the work of Massey in the early 1930s. We have concentrated our attention on the rapid developments in this field that have occurred in the last few years. We have seen that methods and computer programs are now being developed that will allow accurate calculations for elastic scattering and rotational excitation to be carried out for arbitrary diatomic molecules within the near future. Further, the problem of vibrational excitation has seen a number of advances in the last two or three years and it can reasonably be hoped that this problem too will rapidly yield to ab initio calculations shortly. We have not considered the problem of electronic excitation and have said little about polyatomic molecules. The former problem at least for diatomic molecules can be expected to yield rapidly to these new techniques once the rotational vibrational problem is fully understood. However, in the case of complex molecules there is an urgent need to develop more approximate methods, perhaps exploring their validity in the diatomic system. We can expect increasing attention to be given to this area in the next few years. ACKNOWLEDGMENTS
I would like to take this opportunity of thanking my colleagues at Queen’s University, Belfast, and elsewhere with whom I have worked and discussed the problem of low-energy electron-molecule collisions over the last ten years. I would also like to acknowledge my great indebtedness to Sir Harrie Massey, in whose department at University College, London, I carried out my first research in atomic and nuclear scattering theory.
P. G . Burke REFERENCES Ardill, R. W. B., and Davison, W. D. (1968). Pror. R . Soc. London, Ser. A 304, 465. Arthurs, A. M., and Dalgarno, A. (1960). Proc. R . Sue. Landon, Ser. A 256, 540. Atabek, 0.. Dill, D., and Jungen, C. (1974). Phys. R e v . Lett. 33, 123. Birtwistle, D. T., and Herzenberg, A. (1971). J . Phys. B 4, 53. Bloch, C . (1957). Nitcl. Phys. 4, 503. Bottcher, C. (1969). Chem. Phys. Leu. 4, 320. Brink, D. M., and Satchler, G. R. (1971). "Angular Momentum." Oxford Univ. Press (Clarendon), London and New York. Brode, R. B. (1925). Phys. R e v . 25,636. Bruche, E. (1930). Ann. Phys. (Leipzig) [ S ] 4, 387. Buckingham, R. A., Massey, H. S. W., and Tibbs, S. R. (1941). Proc. R . Soc. Lundon, Srr. A 178, 119. Bucklev, B. D., and Burke, P. G. (1977). J . Phys. B 10, 725. Buckley, B. D., Burke, P. G., and Vo Ky Lan (1979). Comprct. Phys. C C J ~ ~ 17, ~ ? 175. IU~. Bullard, E. C . , and Massey, H . S. W. (1930). Pro(.. Cambridge Philos. Sor. 26, 556. Bullard, E . C . , and Massey, H. S. W. (1931). Proc. R . Soc. London, Ser. A 133, 637. Burke, P. G. (1978) "Electronic and Atomic Collisions'' (G. Watel, ed.), p. 201. NorthHolland, Amsterdam. Burke, P. G., and Chandra, N . (1972). J . Phys. B 5 , 1696. Burke, P.G., and Robb, W. D. (1975). Adv. A t . Mol. Phys. 11, 143. Burke, P. G.. and Sinfailam, A. L. (1970). J . Phys. B 3, 641. Burke, P. G., Mackey, I., and Shimamura, I. (1977). J . Phys. B 10, 2497. Chandra, N. (1975). J . Phys. B 8, 1338. Chandra, N . (1977). Phys. R e v . A 16, 80. Chandra, N., and Gianturco, F. A. (1974). Chem. Phys. Lett. 24, 326. Chandra, N., and Temkin, A. (1976a). Pl7ys. R e v . A 13, 188. Chandra, N . , and Temkin, A . (1976b). Phys. Rpv. A 14, 507. Chang, E . S. (1974). Phys. R e v . Lett. 33, 1644. Chang, E. S . , and Fano, U . (1972). Phys. R e v . A 6, 173. Chang, E. S . , and Temkin, A. (1969). Phys. R e v . Lett. 23, 399. Chase, D. M. (1956). Phys. R e v . 104, 838. Choi, B. H . , and Poe, R. T . (1977). Phvs. R e v . A 16, 1821. Collins, L. A., and Norcross, D. W. (1977). Phys. R e v . Lett. 38, 1208. Collins, L. A , , and Norcross, D. W. (1978). Phys. Rev. A 18, 467. Craggs, M. D., and Massey, H. S. W. (1959). In "Handbuch der Physik." (S. Flugge, ed.), Vol. 37, Part 1, p. 314. Springer-Verlag, Berlin and New York. Crawford, 0. H., and Dalgarno, A. (1971). J. Phys. B 4, 494. Crornpton, R . W., Gibson, D. K., and McIntosh, A. I . (1969). Aust. J . Phys. 22, 715. Dalgarno, A . , and Moffett, R. J. (1963). Proc. Nor/. Acad. Sci. India, S e e / . A 33, 51 I . Damburg, R., and Karule, E. (1967). Proc. Phys. Soc. Loridon 90, 637. Dill, D. (1972). Phys. R e v . A 6, 160. Dube. L . , and Herzenberg, A . (1977). Phys. R e v . Lett. 38, 820. D u b t , L., and Herzenberg, A. (1979). Phys. R e v . to be published. Ehrhardt, H. and Willmann, K. (1967) Z. Phys. 204, 462. Faisal, F. H. M. (1970). J . Phys. B 3, 636. Faisal, F. H. M., and Tench. A . L. V. (1971). Conrpirt. Phys. Commun. 2, 261. Fano, U. (1970a). Comments A t . M o l . Phys. 1, 140. Fano, U . (1970b).Phys. R e v . A 2 , 353.
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COLLISIONS
505
Fano. U . (1975). J . Opt. Soc. A m . 65, 979. Fano. U.. and Dill, D. (1972).Phys. Rev. A 6, 185. Fisk. J . B. (1936). Phys. R e ) , . 49, 167. Fliflet, A. W.. Levin. D. A . , Ma, M.. and McKoy. V . (1978). Phys. R ~ vA. 17, 160. Gerjuoy, E.. and Stein. S. (1955). Phys. R w . 97, 1671. Gianturco. F. A.. and Thompson, D. G. (1976).J . Phps. B 9, L383. Gianturco. F. A . . and Thompson, D. G. (1977). J . Phps. B 10, L21. Golden, D . E. (1966). Pliys. Re\,. Lett. 17, 847. Golden, D. E . , Bandel, H. W.. and Salerno, J . A . (1966). Phys. Rev. 146, 40. Golden, D. E . . Lane, N . F.. Temkin, A . , and Gerjuoy, E. (1971). Rev. Mod. Phvs. 43, 642. Hake, R. D., and Phelps, A . V. (1967). Phys. Reit. 158, 70. Hara, S. (1969).J . Phys. Soc. Jpn. 27, 1592. Hazi. A. U . , and Fels, M. F. (1971). Chum. Phys. Lett. 8, 582. Hazi. A. U . , and Taylor, H. S. (1970). Phys. Rev. A 1, 1109. Henry, R. J. W.. and Lane, N . F. (1969). Phys. Rev. 183, 221. Herzenberg, A. (1968). J . Phps. B 1, 548. Herzenberg, A. (1978). 1n "Electronic and Atomic Collisions" ( G . Watel, ed.). p. I . North-Holland, Amsterdam. Jungen. C . . and Atabek, 0. (1977).J . Chum. Phys. 66, 5584. Kendrick, J. (1978). J . Phys. B 11, L601. Klonover, A.. and Kaldor, U. (1977). Chem. Phvs. Lett. 51, 321. Klonover, A , , and Kaldor, U. (1978). J . Phys. B 11, 1623. Klonover. A . , and Kaldor, U . (1979).J . P k y s . B 12, 323. Krauss, M.. and Mies, F. H . (1970). Phys. Rev. A 1, 1592. Lane, N . F., and Geltman. S. (1967). Phys. Rev. 160, 53. Lane, N . F., and Henry. R. J. W. (1968). Phys. Re,,. 173, 183. L e Dourneuf. M., Vo Ky Lan, and Burke, P. G. (1977). Comnients A t . Mol. Phps. 7, 1. Lee, C. M. (1977). Phys. Rev. A 16, 109. Linder, F., and Schmidt, H. (1971). Z . Naturforsch., Teil A 26, 1603. McCurdy, C. W . , Rescigno, T. N., and McKoy, V. (1976). J . Phys. B 9, 691. McLean, A. D., and Yoshimine, M. (1967). J . Chem. Phvs. 46. 3862. Massey, H. S. W. (1930). Pro(.. R . Soc. LofldOi7, Sur. A 129, 616. Massey. H . S. W. (1932). Proc. Carnbridge Philos. SOC. 28, 99. Massey, H. S. W. (1935). Trans. Faraday Soc. 31, 556. Massey. H. S. W. (1969). "Electronic and Ionic Impact Phenomena," 2nd e d . , Vol. 11. 0 x ford Univ. Press, London and New York. Massey, H. S. W. (19761. "Negative Ions." 3rd ed. Cambridge Univ. Press, London and New York. Massey. H. S. W., and Bullard, E. C. (1933). Proc. Cambridge Philos. Soc. 29, 511. Massey, H. S. W . , and Mohr, C . B. 0. (1931). Proc. R . Soc. London, Ser. A 132, 605. Massey, H. S . W . , and Mohr, C . B. 0. (1932). Proc. R . Soc. London. Ser. A 135, 258. Massey, H . S. W., and Ridley. R. 0. (1956). Proc. Phys. Soc. London. Sect. A 69, 659. Morrison, M. A., and Collins, L. A. (1978). Phys. R e ~ sA. 17, 918. Morrison. M. A.. and Hay, P. J . (1977).J . Phys. 8 17, L647. Morrison, M. A . , and Lane, N . F . (1975). Pliys. Rev. A 12,236/. Morrison. M. A . , and Schneider, B. I . (1977). Phys. Rev. A 16, 1003. Morrison, M. A., Lane, N . F., and Collins. L. A. (1977). Phys. Re\,. A 15, 2186. Nesbet, R. K. (1979). Phys. Rev. A19, 551. Osllund. N . S. (1975). Cliem. Phys. Lett. 34, 419. Percival. I . C.. and Seaton, M. J. (1957). Pi-oc. Canihridge Philos. Sol.. 53, 654.
506
P. G. Burke
Ramsauer, C., and Kollath, R. (1930). Ann. Phys. (Leipzig) [S] 4, 91. Reinhardt, W. P. (1979). Comput. Phys. Commun. 17, I . Rescigno, T. N., McCurdy, C. W., and McKoy, V. (1974a). Chem. Phys. Lett. 27. 401. Rescigno, T. N., McCurdy, C. W., and McKoy, V. (1974b). Phys. Rev. A 10, 2240. Rescigno, T . N., McCurdy, C. W., and McKoy, V. (1975). Phys. Rev. A 11, 825. Rescigno, T. N., Bender, C. F., McCiirdy, C. W., and McKoy, V. (1976). J . Phys. B 9,2141. Rohr, K. (1977). J . Phys. B 10, L399. Rohr, K., and Linder, F. (1976).J. Phys. B 9, 2521. Schneider, B. I. (1975a). C h e m . Phys. Lett. 31, 237. Schneider, B. I. (1975b). Phys. Rev. A 11, 1957. Schneider, B. I. (1976). Phys. Rev. A 14, 1923. Schneider, B. 1. (1978). In “Electronic and Atomic Collisions” ( G . Watel, ed.), p. 257. North-Holland, Amsterdam. Schneider, B. I., and Hay, P. J. (1976). Phys. Rev. A 13, 2049. Schneider, B. I., Le Dourneuf, M., and Burke, P. G. (1979). J . Phys B 12, L365. Schulz, G. J. (1962). Phys. Rev. 125, 229. Schulz, G . J. (1964). P h y s . Rev. 135, A988. Schulz, G. J . (1973). Rev. Mod. Phys. 45, 423. Seaton, M. J. (1966a). Proc. Phys. Soc. London 88, 801. Seaton, M. J. (1966b). Proc. Phys. SOC.London 88, 815. Siegert, A. J. F. (1939). Phys. Rev. 56, 750. Stier, H. C. (1932). Z. Phys. 76,439. Takayanagi, K. (1967) Prog. Theor. Phys. Suppl. 40, 216. Takayanagi, K., and Itikawa, Y. (1970). Adv. Ar. Mol. Phys. 6, 105. Temkin, A. (1957). Phys. Rev. 107, 1004. Temkin, A. (1966). In “Autoionization” (A. Temkin, ed.), p. 5 5 . Mono Book Corp., Baltimore, Maryland. Temkin, A. (1976a). Comments At. Moi. Phys. 5 , 129. Temkin, A. (1976b). Comments Ar. Mol. Phys. 6, 27. Temkin, A. (1978). Phys. Rev. A 17, 1232; see also Phys. Rev. A 18, 783. Temkin, A , , and Faisal, F. H. M. (1971). Phys. Rev. A 3, 520. Temkin, A., and Sullivan, E. C. (1974). Phys. Rev. Left. 33, 1057. Temkin, A., and Vasavada. K. V. (1967). Phys. Rev. 160, 109. Wigner, E. P. (1946a). Phys. Rev. 70, IS. Wigner, E. P. (1946b). Phys. Rev. 70, 606. Wigner, E. P., and Eisenbud, L. (1947). Phys. Rev. 72, 29. Wong. S. F. (1978). Phys. Rev. (submitted for publication). Wong. S. F., and Schulz, G. J. (1974). Phys. Rev. Lett. 32, 1089.
I N
I I AUTHOR INDEX Numbers in italics refer to the pages on which the complete references are listed.
A Aagaard, B., 339, 354,380 Aannestad, P. A., 50,69 Aasharnar, C., 286,288 Aberg, T., 362, 363, 374,377, 378 Aberth, W., 11, 12,33 Abgrall, H., 5 5 6 9 Abignoli, M.,225,231 Abrines, R., 280, 286,288, 303,326 Adam, M. Y., 375, 376,380 Adams, A., 394,420 Adams, J. T., 172,203 Adrian, F. J., 57, 70 Afrosimov, V. V . , 226,232, 366,377 Alben, K . T., 186, 187, I%, 198 Albert, K., 397,419 Albritton, D. L., 22, 23, 25,34, 35 Alday, J. E., 272, 273, 276, 287, 288, 291, 302, 321,327 Aldrovandi, S . M. V., 47, 48, 52, 53, 74 Algvard, M.J., 432, 436,464 Alhassid, Y.,180, 198 Allen, M.,16,33 Allison, S. K., 230,231, 362,378 Alton, G. D., 288,291. 352,379 Alvarez, I., 325,326 Amundsen, P. A., 338, 340, 359,377 Amusia, M. Ya., 111, 113, 114, 131. 429. 461,464 Anbar, M.,11, 12,33 Andersen, N., 449, 450,464 Andersen, T., 449, 450,464 Anderson, C. J., 323, 325,327 Anderson, D. E., 62,69, 237,262 Anderson, D. G. M.,276,288, 289 Anderson, D. N., 236,259 Anderson, E. M.,3%. 419 Anderson, G. P., 60,69 Anderson, L. W., 323, 324, 325,327, 328 Anderson, R. C., 62, 70, 236,260 Anderson, R. J., 384, 398, 399, 400,4/9
Anderson, T. G., 57, 74 Andrew, E. P., 283,288 Angel, G. C., 318, 319, 320,326, 327 Angreji, P. D., 236, 237, 262 Anholt, R., 341,347,356, 357, 369,377,379 Ankudinov, V. A., 283,288 Anlauf, K. G., 185,200 Antal, M.J., 276,288, 289 Aoiz, F. J., 192,203 Aquilanti, V., 460,464 Ardill, R. W. B., 475,503 Armbruster, P., 369,378 Armstead, R. L., 102, 103, 106,131 Armstrong, D. A., 254 Arnoldi, D., 190, 198 Amiota, H.,443, 444,445,464, 470 Arshade, M.,244,261 Arthurs, A. M.,330, 341,377, 472,473, 477, 503 Asaad, W. N., 374,377 Ashley, J. C.. 339,377 Atabek, 0.. 487,504, 505 Aten, J. A., 176, 178, I98 Audouze, J., 48, 75 Auerbach, A., 186, 187, 1%, 198 Auerbach, D. J., 177, 198 Aulenkarnp, H., 111, 113, 114, 131 Avery, L. W., 57,69, 70, 73 Avnllier, S., 17,35 Aydinol, M.,461,464
B Bacal, M.,16,35, 322,326 Baede, A. P. M., 168, 175, 177, 198 Baer, M.,172, 198, 199 Bagus, P. S., 408,420 Bahr, J. L., 60,69 Bailey, C. L., 288,291 Baille, P., 131, 132, 138, 155, 157,164, 165 Balashov, V. V., 429,464
507
508
AUTHOR INDEX
Balint-Kurti, G. G., 168,198 Bell, F., 343, 344,380 Baliunas, S., 48, 71 Bell, K. L.. 267,268,270,286,289, 418,419 Band, Y.B., 288,289, 447,466 Bely, O., 67, 70 Bandel, H. W., 478,505 Belyaev, V. A., 303,326 Bang, J., 286,289, 336, 337,377 Bender, C. F., 10, 14,35, 53, 70, 169, 171, Banks, D., 320,326 198, 489,506 Banks, P. M.,46,69 Bennett, R. A., 11, 18, 21, 22, 33, 35, 36, Baragiola, R. A , , 8.33 238, 239, 240,260 Baranger, E., 437, 488,465 Bennett, S. L., 10,33 Barat, M.,27, 34, 221, 224, 225, 228, 231, Benoit, C., 221, 224, 225, 228,232 232 Benoit-Cattin, P., 457,465 Barbe, R., 17,35 Benson, S. W.,238, 239, 240,260, 261 Barbier, D., 44,69 Ben-Shaul, A., 180, 181, 198, 201 Bardsley, J. N., 16,33, 38, 40,69, 209, 210, Bentz. E., 331,380 211,231, 233, 235, 242, 245,259 Berezhko, E. G., 334, 335,377, 462,465 Barnes, K. S., 320,326 Berkner, K. H., 26,33, 280, 289, 312, 316, Barnes, W. S., 18.35 317,326 Barnett. C. F., 294, 325,326 Berko, S., 137,164 Barth, C. A., 46,60,62,69, 73, 75, 236, 260 Berkowitz, J., 194, 199 Barsuhn, J., 48,69 Berlande, J., 78, 81, 99, 252,260 Basbas, G., 338, 339,377 Bernstein, R. B., 168, 169, 170, 171, 175, Bassel, R. H., 277,289 180, 181, 184, 185, 188, 190, 191, 192, 193, Bastide. R. P., 26,35 194, 195, I%, 198, 199,200, 201,202, 203 Bastrom, C. O., 61, 73 Berrington, K. A., 269,289 Basu, D., 130, 133, 153, 166 Berry, H. G., 448, 461,465, 469 Batalli-Cosmovici, C., 194, 198 Berry, M.J., 169, 180, 198 Bates, D. R., 26.33, 37, 38, 40, 42, 43, 44, Bethe, H. A., 80,99, 403,419 45,46,48,49,50,51,53,55,56,62,69, 70, Betz, H. D., 348, 367, 373, 375,377, 380 174,198, 215,218,228,231, 235,236, 246, Bhatia, A. K., 103, 104, 119, 120, 131 247,248,249,250,251, 252,253,254,255, Bickes, R. W., Jr., 181, 194, 201, 203 256,257,258,259,260, 274, 277,286,289, Bierbaum, V. M.,23.33 320, 321,326, 353, 361,377 Billing, G. D., 172, 199 Baudon, J., 225,231 Billingsley, F., 39, 70 Bauer, E., 178,202, 216, 217, 218,233, 322, Biondi, M. A., 38, 39, 40,41, 42,69, 70, 72, 327 73. 235,241, 242, 243, 244, 245,259, 261 Bauer, S. H., 168, 198 Birkinshaw, K., 214,219,225,228,229,231, Bauer, W., 182,.183, 198 232 Baum, G., 460,468 Birtwistle, D. T., 498, 500,504 Baun, W. L., 349,378 Bisgaard, P., 461,465 Bauschlicher, C. W.,171, 198 Bittencourt, J. A,, 62, 75, 236, 237,262 Baye, D., 272,289 Bjorkholm, P.J., 324,328 Bayfield, J. E., 283,289, 302, 303, 307, 3 11, Black, G., 46,71 326, 327 Black, J. H., 42, 46, 48, 49, 50, 55, 56, 70, Beaty, E. C., 21, 25,33, 36, 426,464 71, 72 Bederson, B., 437,460,465, 469 Blackwell, B. A., 169, 190, 198 Bednar, J. A., 68, 70 Blake, A. J., 60,69 Begum, S.,269.289 Blatt, S. L., 341,379 Behnke, H. H., 461,465 Blint, R. J., 52, 70 Behrens, R., 181, 199, 202 Bloch, C., 494,504 Belkic', Dz., 266,282,283,284,285,288,289 Bloemen, E . , 306,328
AUTHOR INDEX
Blum, K.,437, 438, 441,444, 446, 447, 448, 462, 463, 464,465, 467 Bobashev, S. V., 283,288 Bogdanov. G. F., 318,326 Bogdanova, I. P., 394, 3%. 419 Bohr, N., 336,377 Bonani, G., 363, 364, 365, 366, 371, 372, 373,380 Bonham, R. A., 146,166 Bonnet, R. M., 48, 75 Borchert, C. L., 376,377 Bordenave-Montesquieu, A., 457,465 Boring, J. W.,221, 222, 223, 225,231 Born, M., 439,465 Borst, W. L., 395, 397,419 Bortz, P. I., 246, 247, 252, 254,260 Bottcher. C., 244, 260, 267, 289, 460, 465, 482,504 Boulmer, J., 247, 249, 250,252, 260, 261 Boving, E. G., 339, 354, 357,377, 380 Bowers, V. A., 57, 70 Bowman, H., 341,377 Bowman, I. M., 172, 179, 198 Boyd, A . H., 320,326 Boyd, T. J. M., 51,70. 215,231 Brace, L. H., 39, 40, 41, 43, 44, 53, 54, 72, 74, 75 Brackmann, R. T., 2%. 301,326 Brady. E. L., 424,465 Brady, K., 306, 307, 308,327 Branscomb, L. M., 21,36 Bransden, B. H., 101. 115, 117, 118, 127, 128,131, 138, 151,164, 264,265,269, 271, 272, 274, 277, 280,289, 291, 427,428,465 Brandt. D., 348,380 Brandt, W.,337, 338, 339, 345, 346,377 Brash, H. M., 437,465 Bratton, T. R.,279, 285,289 Breig, E. L., 53, 70 Breit, G., 67, 68, 70 Brenot, J. C., 225,231 Brenton. A. G., 105, 115,132, 137, 144, 145, 151, 152, 153, 154, 164 Breton, C., 295,328 Breuckmann, B., 377 Brezhnev, B. G., 303,326 Briand, J. P., 362, 364, 376,377, 378 Briggs. J . S., 207, 208. 210, 211, 232, 264, 265,274,281,286,289, 304, 305,328, 353. 354, 356, 360,377, 379, 380
509
Brink, D. M., 478,504 Brinkman, H. C., 342,377 Brinkman, R. T., 42, 70 Brinton, H. C., 39, 40, 41, 43, 44, 74, 75 Brion, C. E., 426,467, 469, 470 Briscoe, C. V., 125, 126,132 Broadhurst, J. H., 359,378 Brode, R. B., 485,504 Brodskii, A. M., 280,289 Brongersma, H., 288,291 Brooks, E. D. 111,465 Brooks, P. R., 168, 187, 188, 193, 198, 199, 202 Broten, N . W.,57,69, 70, 73 Brouillard, F., 27,33. 35, 303, 309, 318,326 Brown, M., 278,291 Brown, R. L., 47, 70 Brown, S. C., 38, 70 Browne, J. C., 225,231, 315,326 Browning, R., 8.34, 318, 319, 324,327 Bruch, R., 461,465 Bruche, E., 485,504 Bruna, P. J., 57, 70 Brune, W.H., 62, 70, 236,260 Brunt, J. N. H., 391,419 Bruston, P., 48, 75 Brzychcy, A., 187, 198 Buchma, I., 207,232 Buckingham, R. A., 62, 75, 236, 260, 485, 504 Buckley, B. D., 484, 492, 494,504 Budzynski, T. L., 193, 199 Bullard, E. C., 472,504, 505 Bunge, C. F., 3,36 Burch, D., 339, 350, 354,377, 380 Burciaga, J. R.,105, 115,132, 142, 146, 147, 149, 164, 165 Burdett, N. A., 259,260 Burgess, A., 54, 70, 408, 410, 411,419 Burhop, E. H. S., 101, 133, 251. 26/, 329, 330, 337, 374,377, 378.379, 423,425,468 Burke, P. G., 7,36, 476, 481, 482, 483, 484, 490, 492, 493, 494, 502,504, 505, 506 Burkhalter, P. G., 348, 349,379 Burniaux, M., 303, 309,326 Burnside, R. G., 46, 75 Burrow, P. D., 15.36 Butler, S. E., 51, 52, 53, 70, 71, 313, 326 Byerly, R., 25,33 Bykhovskii, V. K., 212, 220,231
510
AUTHOR INDEX
Byron, F. W., Jr., 138, 151, 152, 153, 154, 164, 166, 269,289 Byron, S . , 246, 247, 252,254,260
C Cable, P. G., 217, 218,233 Cadez, I., 16,34 Cage, M. E., 359,378 Caldwell. D., 460,468 Callaway, J., 104,133 Camilloni, R.,332, 334,378, 426, 430,465 Campbell, D. M.,437,465 Campeanu, R. I., 105, 106, 109, 112, 113, 114, 122, 123, 131,132, 138, 146, 147, 148, 164 Campos, D., 266,289 Canter, K. F., 105, 115,132, 137, 142, 144, 145, 146, 147, 148, 149, 150, 151, 153, 164 Caplinger, E., 207,232 Carignan, G., 43,72 Carlson, R. W., 43,70 Carnutte, B., 68,70 Carrington, T., 184,202 Carruthers, G. R.,237,260 Caruso, E., 340,378 Carver, J. H., 60,69 Casavecchia, P., 460,464 Caswell, W. E., 164 Catterall. J. A., 362,378 Cederbaum, L. S., 7, 9,33 Celotta, R. J., 11,33, 36 Center, R. F., 14,33 Certain, P. R., 55, 72 Cesati, A., 340,378 Chaco, T., 278,291 ChaRee, F. H., 48, 59, 70 Chamberlain, G. E., 402,421 Chamberlain, J. W., 44,47, 70 Champion, R. L., 28,34 Chan, Y. F., 104, 119, 127, 128, 129,132 Chandler, C. D., 415, 416,420 Chandra, N., 477, 482, 483, 484, 485, 487, 498,504 Chandrasekhar, S., 145, 164 Chang, E. S., 472, 482, 483, 485, 497,504 Chantry, P. J., 13, 17,33, 194, 198 Chapman, F. M., 172,200 Chapman, S., 44,70 Chappelle, J., 249,260
Chase, D. M., 483, 4%, 504 Chen, C. L., 13, 14, 17,33 Chen, J. R., 341,378 Chen, M. H.,350, 351,378 Chen, S. T., 408, 417,419 Chen, Y. H., 221, 222, 223, 225, 226,231 Cherepkov, N. A., 111, 113, 114,131, 429, 464 Cheret, M.,249, 250, 252,260 Chernysheva, L. V., 111, 113, 114, I31 Cheshire, 1. M., 270, 273, 276, 282, 288,289 Chetiovi, A., 362,377 Cheung, J. T., 181, 199 Chevallier, P., 362, 364, 376,377, 378 Child, M. S., 167, 199. 220,231 Chiu, K. C. R., 288,291 Chkuaseli, D., 207,231 Choe, S.-S., 283,290 Choi, B. H., 474,504 Choi, S.-I., 125, 126, 132 Chong, A. T. J., 216, 220,232 Christensen, A. B., 59,60,70, 236, 237,262 Christian, C., 397, 419 Christiansen, R. B., 52, 70 Chu, S.-I., 54, 70, 243,260 Chu, T. C., 331,379 Chubb, T. A,, 62, 72, 236,261 Chupka, W. A., 92,95, 96,99, 194,199 Church, M.J., 239,261 Chutjian, A., 442,465 Cisneros, C., 325,326 Claeys, W., 27,33, 35 Clark, D. L., 341, 359,378, 379 Clary, D. C., 180,199 Cleff, B., 334. 335,378 Clemens, E . , 450, 451, 452,470 Clemens, L., 172,200 Clout, P. N., 390, 391, 393,419 Cobic, B., 230, 231,233 Cocke, C. L., 68,70, 279,285,289, 341,342, 379, 449, 450,464 Cochran, E. I., 57, 70 Coggiola, M. J., 181, 190, 196, 197,203 Cohen, J. S., 249, 253, 254,260 Coleman, 9. P., 264,267,269, 271,272,281, 283,289, 290. 291 Coleman,P. G., 105, 115, 117, 118, 120, 121, 123, 124, 129, 131,132, 136, 137, 140, 141, 142, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 159, 160,164, 165 Collins, C. B., 248,260
AUTHOR INDEX
Collins, L. A . , 485, 487,504, 505 Collins, R. E., 437,469 Comer, J., 25,33, 461,469 Compton. A. H., 362,378 Compton, R. N., 12,33 Condon, E. U., 67,70 Connor, J. N. L., 170, 199 Connor, R. J., 26,35 Constantinides, E. R., 40, 61, 70, 74 Cooke. W. E., 82, 83, 84, 91, 99 Cooks, R. G., 221.23l Cooper, C. D., 12,33 Coplan, M. A., 465 Cornille, M.,7 , 3 3 Corr, J. L., 324,327 Cosby, P. C., 18, 20, 21, 22, 23.33, 34, 35 Costello, D. G., 137, 142, 147, 165 Cowan, R. D., 295,327 Cox, D. M.,407, 409, 415,420 Craggs, M. D., 472,504 Crandall, D. H., 306,314,315,326, 381,403, 408, 410,419, 421 Craseman. B., 350, 351,378 Cravens, T. E., 46, 75 Crawford, 0. H., 11,33, 487,504 Crompton. R. W., 479,504 Crooks, J. B., 287,290 Cross, R. J., Jr., 186, 187, I%, 198 Crothers, D. S. F., 220,231 Crowe, A., 467 Cruse, H. W., 188, 199 Crutcher, R. M.,48, 49, 59, 71 Csanak, G., 442, 443,465, 470 Csizmadia, I. G., 171, 172, 199 Cue, N., 278,290, 342, 343, 362, 363,378 Cunningham, A. J., 60,70 Curnutte, B., 68, 70, 279,289 Curtis, L. J., 448, 461,465, 469 Cvejanovic', S . , 426,465
D Dagdigian, P. J., 181, 184, 188,199, 203 Dahl, P., 461,465 Daley, H. L . , 207, 209, 210, 213, 214,231, 233 Dalgarno, A., 39, 40, 42, 43, 44,46, 48, 49, 50,51,52,53,54,55,56,60,61,64,65,66,
67, 70, 71, 72, 74, 75, 76, 108, 132. 215,
511
232. 236, 243,260, 313,326, 472,473,477, 480, 487,503, 504 Dalvaille, J. P., 367,380 Damburg, R. Ya., 104, 132, 427, 470, 476, 504 Dance, W. E., 330,380 Dangerfield, G. R., 330, 331,378 Darewych, J. W., 131, 132, 138, 155, 157, 164, 165 Das, J. N., 332, 333,378 Dassen, H. W., 446,468 Datz, S., 183,200, 394, 395, 396, 397,420 46 I , 465 Davenport, J. E., 46,71 Davidovic, D. M.,330, 331, 341,378 Davidovits, P., 187, 198, 202 Davidson, J. A., 22, 23,34. 45, 75 Davidson, K., 50, 71 Davis, C. K., 371,378 Davis, D. V., 330,378 Davis, J.. 62, 72, 237,261 Davison, W. D., 475,503 Deconninck. G., 340, 341,378 Deech, J. S., 78. 99 Degaonkar, S. S., 47,72 Degges, T. C., 53, 60,71 DeHaven, J., 187, 198 de Heer, F. J., 26, 35, 152, 153, 154, 165, 306.328, 441, 412,420 Dehl, P., 461,465 Dehmer, J. L., 61, 71 Dehmer, P. M.,61, 71 de Jong, T., 50, 71 d e Jongh, J. P., 390,419 Delfosse, J. M.,27,33, 318,326 Deloche, R., 249, 250, 252,260 Delpech, J. F., 247, 249, 250, 252,260, 261 Delvaille, J. R., 348,377 Delvigne, G. A., 175, 199 de Michelis, C., 295,328 Demkov, U. H., 212,232 Demkov, Y.N., 354,378 Denison, J. S., 42, 74 de Rijk, W., 454,466 Dettmann, K., 264, 280, 288,289, 290 Deutsch, M.,424,465 Devos, F., 249, 250, 252,260 DeVries, A. E., 176, 198. 199, 203 Dewangan, D. P., 138, 151, 152, 153, 158, 165, 269, 272,289, 290 Dhuicq, D., 27,34, 225,231
512
AUTHOR INDEX
Diana, L. M.,105, 115, 132, 142, 146, 147, 149,164, 165 Dickinson, A. S., 243,260 Diercksen, G. H. F., 57, 73 Diestler. D. J.. 168, 199 DiGiuseppe, T. G., 187,202 Dill, D., 477, 487,504 Dillon, J. A., 208,232 Dimoplon, G., 194,195,202 Dimov, G . I., 26,34 Ding, A. M.,190, 199 Dinwiddie, D., 276,291 Dirks, J. F., 127, 128, 132 Dirscherl, R., 194, 199 Dismuke, K. I., 57, 58, 71. 72 Dispert, H., 193,199 Dixon, A. J., 428, 446,447,466, 467, 470 Dixon, T. A,, 57, 74, 76. 168, 181, 199, 200, 203 Dmitriev, I. S ., 317,326 Doannou-Yannou, J. G., 341,377 Dodd, L. D., 282, 288,290 Doering, J. P., 42, 61, 70, 71, 73 Dolder, K. T., 27, 35, 318, 319, 321, 326, 327, 401,421 Doll, J. D., 179, 199 Domcke, W., 7, 9,33 Donahue, D. E., 88,99 Donaldson, F. G., 390,419 Doolen, G. D., 105, 132 Doorn, S., 358,380 Dotan, I., 22, 23,34 Doucet, H. I., 16.35 Doughty, B. M.,283,290 Douglas, D. J., 190, 199 Doverspike. L. D., 28.34 Drachman, R. J., 101, 103, 104, 106, 107, 110, 111, 113, 114, 118, 119, 120, 125,131, 132, 133 Drake, G. W. F., 8.34, 67, 68;71 Drawin, H. W., 249,260 Dressler, K.. 63, 71 Drisco, R. M., 280, 281,290, 342,378 D u b 6 L., 496, 500, 501,504 Dubrevil, B., 249,260 Dudnekov, V. G., 26,34 Duren, R., 460,466 Duff, J. W., 172,200 Duggan. J. L., 352,379 Duke, L., 281,289
Dukel’skii, V. M., 11.35 Dulkiewicz, V.,278,290 Duman, E. L., 211,232 Dungey, J. W., 46, 71 Dunn, G. H., 38, 39, 40, 75, 382, 401, 402, 403, 408, 410, 419, 421 Dunn, H. G., 425,467 Dunn, K. F., 8,34, 318, 319, 320,326, 327 Dunning, F. B., 78, 85, 92, 94, 99 Dunning, T. H., 13.34 Dupree, A., 48, 71 Dutton, J., 105, 115,132, 137, 144, 145, 151, 152, 153, 154, 164, 165 Duveneck, F. B., 330,380
E Eastman, D. E., 60,72 Edelstein, S. A., 78, 79, 80, 81, 82, 83, 84, 91.99 Edwards, H.D., 208,232 Ehlers, V. J., 408, 409, 417,419 Ehrhardt, H., 47,75, 157, 158,165, 301,328, 332,378, 426, 427, 428. 429,466, 467 Eidson, W. W., 341, 342,379 Eisele, F. L., 211,232 Eisenbud, L., 486,492,506 Eiserike, H., 103, 104, 120, 131 Elford, M.T., 210, 211,232 Elitzur, M.,50, 56,71 Elkowitz, A., 179, 199 Ellder, J., 57, 72, 74 Ellis, D. G., 448,465 Ellison, G. B., 23,33 Ellsworth, L. D., 278,291, 341,378 Elmergreen, B. G., 59, 71 Elston, S. B., 288,291 Emard, F., 249,260 Eminyan, M., 394, 418, 419, 441, 442, 443, 444,445,446,448,466 Enemark, E. A., 408, 417,419 Engelberg, P. C., 10,34 Engelke, F., 181, 188, 189, 199 Epstein, I. R., 123, I32 Erastov, E. M., 303,326 Eriksen, F. J., 454, 455,466, 467 Esaulov, V., 27,34 Essen, M.,172, 199 Eu, B. C., 192,199
AUTHOR INDEX
5 13
Flower, D. R.,54, 71, 392,419, 466 Fluendy, M. A. D., 168, 199 Folkmann, F., 369,378. 380 Foltz, G. W., 92.99 Ford, A. L., 270, 271, 272, 273, 278, 286, F 290, 291. 339,378 Fainberg, Yu. A., 317,326 Ford, J. C., 283,290 Faisal, F. H. M.,461,466, 475,483,504,506 Ford, R. L., 331,379 Faist, M. B., 175, 199 Fornari, L., 455,467 Fajen, F. E., 417,420 Fortner, R. J., 286,290, 358,380 Falk, R. A., 26,34 Fournier, P. R.,78, 81, 99 Fano, U., 352,378, 403,419, 437, 438, 444, Fox, J. L., 60, 71 447,448,460,462,463,466, 472,477,482. Francis, W. E.. 206, 207,208, 210, 212,233, 485, 487, 489,504 322,328 Farago, P. S., 437, 443,465, 469 Franco. V.,272,290 Farazdel, A., 123, 132 Frank, W., 366, 368, 370, 371,379 Faris, J. L., 391,420 Franklin, J. L., 10, 12.33, 35 Farrar, J. M., 168, 181, 199 Fraser, P. A., 101, 104, 107, 110, 1 1 1 , 119, Fastie, W. G., 59, 62, 70, 74, 236, 260 121, 123, 124, 127, 128, 129,132, 133, 153, Fastrup, B., 339, 354,380 166 Frederick, J. E., 46, 71 Fatceva, L. N., 317.326 Faucher, P., 67, 70 Freed, K. F., 180, 199 Faulk, J . D., 332, 333,380 Freedman, A., 181, 199, 202 Fayeton, J., 27,34, 225,231 Freund, R. S., 88.99 Fedorenko, N. V., 28.35, 209, 226,232 Friichtenicht, J. F., 185, 187, 203 Fehsenfeld, F. C., 15, 22, 23, 25,34, 35, 43, Frilley, M., 364,377 Froese-Fischer, C., 432,466 48, 71 Feinberg, G., 68, 71 Fryar, J., 443,469 Feldman, P. D.. 46,53,59,62,70, 71, 74, 75. Fung, K . H., 180, 199 Fuss, I., 428,470 236,260 Fels, M. F., 127, 128, 129, 130, 132, 153, Fussell, W. B., 401,421 Futrell, J. H., 23,33, 208,233 165, 488,505 Fyfe, W. I., 14,33 Ferguson, E. E.. 25,34, 35, 43, 48, 71. 72 Feshbach, H., 104, 132 Field, G. B., 47, 51, 53, 54, 63, 70, 71 Filippenko, P. G., 220,232 G Finkenthal, M., 295,328 Gabler, H., 350,380 Fiquet-Fayard, F., 1 I , 35, 36 Firsov, 0. B., 51, 73, 206,232 Gabriel, A . H., 67, 68, 71. 385,419 Gabrielse, G., 447,466 Fischer, D. W., 349,378 Fischer, 0.. 419 Gailitis, M., 104, 132, 428,470 Fisk, J. B.. 480, 504 Gallagher, A. C., 8, 34, 36, 399, 408, 409, Fitchard, E., 270, 271, 272, 273. 278, 286, 415, 417,419, 420 290, 291, 339,378, 448, 462, 464,465 Gallaher, D. F., 273, 276,289, 290, 436,466 Fite, W. L., 47, 71, 75, 179, 199, 230, 232, Gannon, R. H., 57, 75 Garcia, J . D., 286.290 296, 301, 307,326. 407.409.420 Gardner, F. F., 57, 74 Flaks, I . P., 208, 209, 220, 226,232 Flannery, M. R., 85, 94, 99, 235, 2S6, 258. Gardner, J. L., 60,69, 74 260, 267, 268, 272,289. 290, 441, 443,466 Gardner, L. D., 311,327 Gardner, R. K., 279,289 Fliflet, A. W.. 491,504 Everhart, E., 221, 222,232, 233, 302,327 Evers, C. W. A.. 176, 177,198. 199
5 14
AUTHOR INDEX
Garrett, W. R., l1,33 Garstang, R. H., 394, 396,420 Gash, A. S., 153, 166 Gautherin, G., 322,328 Gauyacq, J. P., 27,34 Gavrila, M.,365,378 Gayet, R., 282, 283,289, 290 Geballe, R., 26,34 Geddes, J., 179, 199, 283,291 Gee, D. M., 318, 321,327 Geis, M. W., 193, 199 Geldon, F. M.,52, 75 Geltrnan, S., 429, 436,466, 480,505 Gentry, W. R., 207,232 Genz, H., 331,378 George, J. M., 272, 273, 276, 287, 288,291, 302, 321,327 George, T. F., 179, 180, 199, 200, 201 Gerardo, J. B., 248, 249, 250, 252,261 Gerber, G., 457,466 Gejuoy, E., 117, 132, 277, 289, 437, 448, 465, 472, 480,504, 505 Gersh, M. E., 193,200 Ghosh, A. S., 130, 133, 272,290 Ghosh, S . N . , 208,232 Gianturco, F. A., 482, 484, 485,504, 505 Giardini-Guidoni, A., 332, 334, 378, 426, 430,465 Gibson, D. K., 479,504 Gidley, D. W., 137, 159, 163, 164, 165 Gilbody, H. B., 8, 34, 101, 133, 235. 261, 263,283,290,291, 2%, 300,301,306,307, 308,310,311,312,314, 315, 318,319,320, 321,322,323,324,326,327,328, 337,379, 423,468 Gillen, K. T., 181,200 Gil1espie.E. S., 130, 131,132, 138, 148, 149, 155, 157, 165 Gillespie, G. H., 26,34, 325,327 Gilmore, B. J., 8,34 Giournousis, G., 47, 72 Gippner, P., 366,378 Girnius, R. J., 325,327 Giusti-Suzor, A,, 55,69, 72 Glauber, R. J., 269,290 Gleizes, A., 457,465 Glennon, B. M.,59, 60,76, 237,262 Goddard, W. A., 212,233 Goffe, T. V.,308, 310, 31 1, 312,328 Golden, D. E., 443,445,469, 472, 478, 485, 505
Golden, J. E., 277, 284,291. 436,466 Golden, P. D., 43, 71 Goldsmith, D. W.,51, 71 Goldstein, M.,437,469 Gordeev, Y. S., 366,377 Gordon, R. J., 181, 184,200 Gorman, M. R., 237,262 Gorry, P. A., 181,201 Goscinski, O., 374,378 Gould, H., 68, 72 Gould, R. J., 63, 72 Gould, R. K., 12.34 Gouldamashvili, A., 207,231 Gousset, G., 252,260 Gouveia, H., 236, 237,262 Govers, T. R., 303, 309,326 Grabiner, F. R., 193,202 Graham, W. G., 280,289, 312, 316, 317,326 Graham, W. R. M., 57, 58, 71, 72 Gray, J. C., 172,200 Green, G. W., 330,378 Green, J. M., 216,233 Green, L. C., 415, 416,420 Green, S., 57, 58, 72, 76 Green, T. A , , 274,291 Greenberg, J. S., 366, 371,378 Greenland, P. T., 220,232 Greenlees, G. W., 341, 359,378, 379 Gregory, D., 381,403,421 Greider, K. R., 282, 288,290 Greiner, W., 372, 373, 374, 375, 377, 379, 462,469 Gresteau, F., 11, 15, 34, 35 Grey, R., 27,35, 318, 319,327 Grice, R., 168, 178, 181,200, 20/, 203 Griem, H. R., 68.72 Griffin, P. M., 288,291 Griffing, G., 286,289, 320,326 Griffing, K. M.,10,34 Griffith, T. C., 105, 115, 117, 118, 120, 121, 123, 124, 129, 131,132, 136, 137, 138, 140, 141, 142, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 159, 160, 161, 162, 163, 164. 165, 166 Groce, D. E., 137, 142, 147,165 Grossi, G., 460,464 Grossmann, K. U., 53, 74 Grover, P. S.,123, 132 Grum-Grzhimailo, A. N., 429,464 Gryzinski, M., 246,260, 316,327, 332,378 Guberman, S. L., 52,70, 313,326
AUTHOR INDEX
Gudat, W., 60,72 Guelin, M.,57, 58, 72 Guffey, J . A., 341,378 Guidini, J . , 318,328 Gull, G . E., 55, 73 Gurnee, E. F., 46, 72 Gustaffson, T., 60, 72 Gutman, D., 194, 199
H Haas, R. A , , 28, 31,34, 499.505 Haberman, J. A., 185,200 Habing, H . J., 51, 71 Haddad, G. N., 443, 445,469 Haeberli, W., 324,328 Hafner, H., 466 Hagmann. S., 369,378 Hagstrom, S. A,, 3, 4,36 Hahn, Y., 126, 127, 128, 132 Hake, R. D., 487,505 Halavee, U., 180,200 Hall, J. L., 11.33 Hall, J. M.,1 I , 36 Hall, R. I., 11, 14, 15, 16,34, 35 Halpen, A , , 277,290 Hamilton, D. R., 424,466 Hamnett, A,, 426,467 Hanne, G. F., 437,460,466 Hanrahan, R. J., 14,35 Hansen. J. E., 365,378 Hansen. P. G.,376,377 Hansen, W. W., 330,380 Hanson, W. B., 39,40,43,44,47,53,62,70, 71. 72, 74, 75, 236,260 Hansteen, J. M.,286, 289, 336, 337, 359, 377, 378 Hague, M.A , , 391,419 Hara, S., 123, 131, 132, 133, 138, 155, 157, 165, 479,505 Hardin, D. R., 181,203 Hardt, T. L., 340,378 Harel, C., 304,327 Harnois, M.,26.34 Harper, W. R., 255,261 H a m s , F. M.,105, 115, 132, 137, 144, 145, 151, 152, 153, 154, 164, 165 Harrison, K. G., 287, 288,290, 339,380 Hamson, M. F. A,, 294,327
515
Hartquist, T. W., 50, 5 5 , 56, 70, 72 Harwit, M.,5 5 , 73 Harworth, K., 330,380 Hasselkamp, D., 460,466 Hasted, J . B., 207, 213, 214, 216. 219, 220, 222, 225, 228, 229,231. 232 Hay, P. J., 10.35, 492, 494, 499,505, 506 Hayashi, S., 184,200 Hayes, E. F., 168, 172, 187, 198, 200 Hayes, M. A., 414,420 Hayhurst, A. N., 252, 253, 259,260, 261 Hays, P. B., 40,42.43,44,45,46.72, 73, 74, 75, 256,260 Hayward, T. D., 26,34 Hazi, A. U., 105,133, 488,505 Heddle, D. W. O., 381, 383, 384, 385, 388, 390,391,393,397,399,406,412,418,419, 420 Heenan, P. H., 272,289 Hegerberg, R., 210, 211,232 Heidemann, H. G. M.,385, 391, 421, 461, 466 Heiligenthal, G., 373, 375,377 Heindo&€, T., 397,419 Heinemeier, J., 26.34 Heinzmann, U., 460,466 Heiss, P., 111, 113, 114, 131 Helbig, H. F., 302,327 Helm, H., 211,232 Helmy, E. M.,12, 21.34 Helstroom, R., 331,378 Hender, M.A,, 390,419 Henry, R. C., 62.70, 236,260 Henry, R. J. W., 53, 61, 72, 75, 475, 476. 478, 479, 479,505 Hepburn, J. W., 193,200 Herbst, E., 10, 12, 20, 28,34, 35, 48, 5 5 , 72 Herm, R. R., 168, 181, 185, 199, 200, 201, 202 Hermann, H. W., 460,466 Herrick, D. R., 88, 99 Herring, C., 216,232 Herring, D. F., 137, 142, 147, 165 Herrmann, J. M.,181,201 Herschbach, D. R., 168, 178, 181, 182, 183, 199, 200, 201, 202 Herscovitz, V. E., 431,468 Hertel, I. V., 460,466 Herzberg, G., 63, 64,65, 71, 72 Herzenberg, A., 495, 4%. 498, 500, 501, 504, 505
516
AUTHOR INDEX
Hesselbacher, K. H., 157, 158, 165, 331, 378, 426, 427,428 Heyland,G. R., 105, 115, 117,118,120, 121, 123, 124, 129, 131,132, 136, 137, 138, 140, 141, 142, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 159, 160, 161, 162, 163, 164, 165, 166 Hibbert, A., 7.33 Hicks, G. T., 62, 72, 236,261 Hidalgo, M. B., 429,466 Hildebrandt, G. F., 92, 94, 99 Hill, R. M., 78, 79, 80, 81, 83, 84, 91, 99 Hils, D., 432, 434, 435, 436, 437,467 Hinnov, E., 246,261 Hippler, R., 461, 462,464, 467 Hiraoka, H., 14,34 Hishinuma, N . , 211,232 Hjalmarson, A., 57, 72, 74 Ho, Y. K., 107, 110, 111, I33 Hodgkinson, D. P., 207, 208, 210, 211,232 Hodgman, C. D., 240,261 Hoegy, W. R., 53, 54, 72 Hoffman, J. H., 39,40,46,53,70, 72, 74, 75 Hoffmann, D. H. H., 331,378 Hofstadter, R., 331,379 Hogan, J., 296,327 Hollenbach, D. J., 63, 72 Hollis, J. M.,57, 75 Hollywood, M.T., 443, 444,467 Holrnes, J. C., 42, 53, 72 Holstein, T., 210, 211,231 Holt, A. R., 268, 272,290 Holzer, T. E., 46.69 Homann, K. H., 170,200 Hong, S. P., 12, 21,34 Honz, S. D., 426,464 Hood, S . T., 426, 428, 430, 432. 446, 447, 466, 467, 470 Hoogkamer, T. P., 358, 371, 372,380 Hopkins, F., 278,290, 342, 343,378 Hopper, D. G., 13,34 Hord, C. W., 60,69, 75 Hotop, H., 2, 5 , 6, l8,34 Houston, S. K., 106, 107, 110,133 Houver, J.-C., 225,231 Howard, C. J., 15, 25,34, 45, 75 Howat, G., 374, 375,378 Howe, H. C., 296,327 Huang, C.-M., 39,40,72, 241,242,243,244, 26 I Huber, B. A , , 20.34, 218, 225.232
Hubers, M. M., 177, 178,200 Huebner, R. H., 15.36 Huestis, D. L., 238, 239, 240,260, 261 Huetz, A., 11.36 Hughes, B. M.,230,232 Hughes, R. H., 283,290, 384, 388,398,399, 400,419, 421 Hughes, V. W., 432, 436,464 Huguenin, G., 57, 72 Humberston, J. W., 103, 105, 106, 108, 109, I 11, 112, 113, 115, 116, 119, 120, 122,132, 133, 138, 146, 147, 148, 164, 165 Hummer, D. G., 47, 75 Humphrey, L. M.,84, 91, 99 Humphries, R. R., 221, 222, 223, 225,231 Huntress, W. T., 42, 72 Hurley, R. E., 11,34 Hurtz, A., 15, 35 Hult, P. K., 115, 117, 131, 138, 151, 164 Hvelplund, P., 26,34
I Iguchi, K., 308, 317,328 Il’in, R. N., 28,35, 322,327 Ingram, M.F., 236,261 Inokuti, M., 26,34, 138, 151, 165, 403, 407, 412,420 Iqbal, S. M.,222,232 Ireland, J. V., 296, 321,327 Irvine, W. M., 57.74 Ishii, K., 331,379 Isler, R. C., 295, 305,327 Issa, M., 272,289 Itakawa, Y.,403,420. 472,506
J Jackson, W. M., 391,420 Jacob, G., 431,467, 468 Jaduszliwer, B., 105, 115, 118,133, 137, 142, 146, 147, 148, 149, 152,165, 460,465 Jaecks, D. H., 437, 454, 455,466, 467, 468 Jaffe, G., 255,261 Jakubetz, W., 170,199 Jakubowicz, H., 446,465 Jamison, K . A., 278,291, 344,379 Jarnnik, D., 340,379 Janev, R. K . , 215, 219,232, 235, 238,26/
AUTHOR INDEX
Jansen, R. H. J., 152. 153, 154, 165 Jarvis, 0. N., 341,379 Jefferts, K. B., 57, 72 Jensen, D. E., 12,34 Jensen, S., 460,469 Jesion, G., 105, 115,133, 137, 138, 139, 144, 145, 146. 147, 148, 149, 150, 151, 153, 154, 166 Jitschin, W., 461, 462,467 Joachain, C. J . , 138, 151, 152, 153, 154,164, 166, 272,290. 443,467 Jobe, J. D., 394, 402,420 Jognaux, A., 303, 309,326 Johnsen, 0. M.,359,378 Johnsen, R., 39, 40, 72, 241, 242, 243, 244, 26 I Johnson, A. W., 248, 249, 250, 252,261 Johnson, B. R., 175, 180,199, 201 Johnson, B. W., 350,380 Johnson, C. Y.,42, 53, 72 Johnson, H. C., 218, 228,231 Johnson, L. C., 246,261 Johnson, R. E., 208,211,221,222,223,225, 232 Johnson, S. G., 183,200, 394,395,396,397, 420 Johnson, W. R., 68, 72 Johnston, H. C., 321,326 Jones, G., 121, 124, 133 Jones, K. W., 348, 367,377, 380 Jones, R. A., 144, 145, 151, 152, 164. 165 Jonson, A., 364,377 Jonson, B.,376,377 Jordan, C., 67, 68, 71, 72 Jorgensen, T., Jr., 288,290 Joseph, J., 236,262 Judge, D. L., 60, 73 Julienne, P. S., 46, 62, 72, 74, 237,261 Jundi, Z., 127, 128, 131 Jundt, F. C., 345, 347,379 Jung, K., 157, 158, 165, 332,378, 427, 428, 429,466, 467 Jungen, C., 487,504, 505 Jungen, M.,5 , 6.36 Junken, B. R., 210, 21 1,231
K Kabachnik, N. M.,334, 335,377, 429, 462, 464, 465
517
Kalata, K., 348, 367,377, 380 Kaldor, U., 491, 494,505 Kamiya, M.,331,379 Kaneko, Y., 208,233 Kaplan, H., 181,200 Kaplan, S. N., 26.33 Kari, R. E., 171, 172, 199 Karkhov, A. N., 318,326 Karny, Z., 188,200 Karstensen, F., 410,420 Karule, E., 476.504 Kasdan, A., 10, 12.34, 35 Kasner, W. H., 42, 72 Kasper, J. V., 188,202 Kauffman, R. L., 278,291 Kaufmann, K., 190,198 Kaun, K. H., 366, 368, 370, 371,378, 379 Kauppila, W. E., 105, 115, 133. 137, 138. 139, 144, 145, 146, 147, 148, 149, 150, 151, 153, 154, 155, 156,166, 401,407,409,420, 42 I Kavanagh, T. M.,286,290 Kay, R. B., 384,386,388,399,401,420,421 Kayzer, D. C., 40, 75 Kebarle, P., 244,261 Keck, J . , 247,261 Keesing, R. G. W., 381, 388, 391, 393, 399, 406, 418,420 Keever, W. C., 142, 165 Kellert, F. G., 78, 85, 92, 94, 99 Kelly, H. P., 59, 74 Kelly, K. K., 60,69 Kempter, V., 450,451, 452,470 Kendrick, J . , 499,505 Kennedy, B. C., 43, 72 Kennedy, D. J., 61, 75, 418,419 Kennedy, J. V., 414,420 Kennerly, R. E., 146,166 Kenney, J., 10,35 Kerkdijk, C. B., 26.35 Kessel, Q . C., 455, 458,467 Kessler, J . , 433, 437, 460,466, 467 Khan, J. M.,337.379 Khare, S. P., 251, 252, 253,260 Khayrallah, G. A., 307,326 Kherrman, L., 68, 76 Khvostenko, V. I., 11,35 Kieffer, L. J., 15, 16.36, 425,467 Killeen, T. L., 117, 118, 120, 121, 123, 124, 129, 131,132, 136, 140, 142, 145, 154, 155, 156, 159, 160, 165
518
AUTHOR INDEX
Kim, H. J., 31 I , 327 Kim, Y.-K., 26,34, 299,327, 408,412,420 Kimura, M., 211,232 King, D. L., 181,200 King, G. C. M., 391, 394,419, 420 Kingston, A. E., 108, 132, 246, 247, 256, 260, 267, 268, 270, 286,289, 418,419 Kinsey, J. L., 168, 181, 188,200, 201 Kirby, C., 57. 73 Kirby-Docken, K., 40, 61,74, 75 Kirkpatrick, P., 330,380 Kirkpatrick, R. C., 236, 237,261 Kirsch, L. J., 190, 199 Kisker, E., 460,468 Kislyakov, A. I., 299,327 Klapish, M., 295,328 Klar, H., 436, 460,467 Kleinpoppen, H., 394, 418, 419, 424, 432, 433,434,435,436,437,438,440,441,442, 443,444,445,446,447,448,461,464,465, 466, 467, 469 Klernperer, W. B., 48, 57, 72 Kleppner, D., 91.99 Kleyn, A. W., 168, 175, 177, 178,200, 201 Klimek, D., 193,200 Klonover, A., 491, 494,505 Knudsen, W. C., 62, 72. 236,261 Knudson, A. R., 348, 349,379 Kobayashi, N., 208,233 Kocbach, L., 286,288,290, 359,377 Koch, P. M., 303, 311,327 Kocher, D. C., 307, 314,326 Koebach, L., 359,378 Koeppl, G. W., 175,203 Kohn, W., 102, 133 Kolberstvedt, H., 330, 331,379 Koleshikova, M. M., 68, 76 Kollath, R., 478, 485,505 Kollberg, E., 57, 74 Komornicki, A., 180,200 Kondo, Y.,46, 73 Kong, T. Y., 42, 73 Kontrosh, E. E., 394, 395, 396,421 Korotkov, A. L., 395, 396,420 Kouri, D. J., 179,201 Kozlovsky, B. Z., 51, 75 Kraemer, W.P., 57, 73 Kraft, G., 369,378 Kraidy, M., 107, 129, 133, 153, 166 Kramers, H. A., 342,377
Krause, H. F., 179, 183,199, 200, 394, 395, 396, 397,420 Krause, M. O., 364,377 Krauss, M., 488, 498,505 Krenos, J. R., 173, 185, 192,200 Kroll, N. Q., 117,132 Krotkov, R., 448,467 Kroto, H. W., 57,69, 70, 73 Kruger, H., 266,289, 466 Kubach, C., 221, 224, 225, 226, 227, 228, 232, 233 Kubo, H., 345, 347,379 Kucheryaev, Yu. A., 318,326 Kuchiev, M. Yu., 461,464 Kumar, V., 60,69 Kung, T.J . , 283,290 Kuntz, P. J., 168,200 Kuppermann, A., 172, 175, 179, 180, 198, 200, 202, 203 Kushnir, R., 207, 208,232 Kutner, M. L., 57, 75 Kuyatt, C. E., 288,290 Kwei, G. H., 173,200
L LaBahn, R. W., 127, 130,133,147, 148, 149, 166 Lambert, F., 249, 250,252,260 Landau, L. D., 216,232 Landau, M., 1 I , 36 Lane, A. L., 60.69, 75 Lane, N. F., 76, 307,328, 472,475,476,478, 479, 480, 485, 487, 488, 489,505 Lang, K. R., 57, 73 Langevin, P., 255,261 Lapicki, G., 277,290, 342,379 Lark, N . L., 340,379 Larkins, F. P., 352, 353, 354, 355,379 Larsson, S . , 4.35 Latimer, C. J., 92, 95, 98, 99 Latimer, I. D., 419,420 Latypov, Z. A., 209,232 Laubert, R., 337, 338, 339, 352, 368, 377, 379, 380 Launay, J. M., 54, 55, 71, 73 Laurent, C., 48, 75 Law, J., 277,290 Lawley, K. P., 168, 199, 200
519
AUTHOR INDEX
Lawson, J., 123, 130, 133, 147, 148, 149, 166, 461,468 Lebeda, C. F., 103,133 Lebow, L., 144, 145, 152, 153, 154, 166 LeBreton, P. R., 181,201 Le Dourneuf, M., 7,35,36, 53.73, 476,502, 505, 506
Lee, C. M., 40, 73, 264,290, 487,505 Lee. E. T. P., 399,420 Lee, G. F., 124, 133 Lee, J. S., 61, 73 Lee, K. T., 179, 198 Lee, L. C., 60, 73 Lee, T. E., 94,99 Lee, Y. T., 168, 181, 190, 196, 197,199,200, 201, 202, 203, 207,232 Leep, D., 408, 417,419, 420 Leithauser, U., 350,380 Lemley, J. R., 294, 297,327 Lennard, W. N., 366,379 Leone, S. R.,23,33, 190,200 LePage, G. P., 164 Leu, M.T., 241,261 Leung, C. Y.,124, I33 Levin, D. A., 491,504 Levine, J., 1 I , 33, 36 Levine, R. D., 168, 169, 175, 180, 181, 188, 192, 198, 199, 200. 201 Levy, H., 274,290 Lewis, D. M., 144, 145, 151, 152,164 Lewis, H. W., 337,379 Lewis, J. T., 215,231 Li, T. K., 341, 359,378, 379 Lichten, W., 224,232, 352, 366,378 Lichtenberg, W., 461,469 Liesen, D., 366.379 Lifschitz, E. M., 216, 232 Light, J . C., 168, 201 Lin, C.. 68, 72 Lin, C. C., 53, 70, 276,291, 399, 417,420 Lin, C. D., 277, 278, 279.290 Lin, M. C., 184,200 Lin, S.-M., 185,201 Lindblad, P. 0.. 57, 72 Linder, F., 11.35, 494, 496,505, 506 Lindgren, I., 376,377 Lindinger, W., 25.35 Lindroos, P., 57, 72 Lineberger, W. C., 2,5,6, 10, 12, 18,20,34, 35, 36
Lines, K. S . , 117, 132, 145, 146, 152, 153, 154, 163, 164, 165 Ling, J . H., 21, 22, 23,33 Lipovertsky, S. S., 429,464 Lippincott, C. R.,72 Li-Scholz, A., 362,378 Little, A., 342, 343,378 Littman, M. G., 91, 99 Litvak, H. E., 193,201 Liu, B., 10,35, 171,201, 253,261 Liu, K., 186, 193.200, 201 Lium W. S., 192,199 Lo, H. H., 230,232 Lockwood, G., 222,232, 302,327 Lodge, J. G.. 138, 155, 157, 164 Lowdin, P. 0.. 4,35 Logan, J. A., 44,73 Long, R. L., 407, 409, 415,420 Longree, M., 340, 341,378 Lorentz, A , , 294, 297,327 Los, J., 168, 175, 176, 177, 178, 198, 199, 200, 201 Losonsky, W., 277,290, 342, 352,379 Lovas, F. J., 57, 75 Love, R. L., 181,201 Loyd, D. H., 324,328 Lubell, M. S., 432, 436,464 Lucas, C. B., 388, 391, 412,420 Lucas, M. W., 287, 288,290 Lutz, B. L., 48, 59, 70 Lutz, H. O., 360,380, 461, 462,467 Luypaert, R.,78,99 Luz, N., 461, 462,467 Lynn, N., 46,69
M Ma, M., 491,504 MacAdam, K. B., 394, 418,419. 441, 442, 443, 444,445, 446, 448,466 McCann, K. J., 267, 268,290, 441, 443,466 McCarroll, R. W., 52.73, 266,274,284,289, 290, 313,327 McCarthy, I. E., 426, 430, 431, 432, 465, 467. 470 McClure, G. W., 168,201, 296,302,310,327 McConkey, J. W., 390,419, 443, 444,446. 467, 468,469
520
AUTHOR INDEX
McCullough, R.W., 306, 307,308,314, 315, 322, 323, 327,327 McCurdy, C. W., 10,35, 489,490,491,505, 506 McCusker, V., 437,467 McDaniel, E. W., 18,35 McDaniel, F. D., 352,379 MacDonald, J. R., 181, 182, 183, 199, 201, 278,279,285,289,291. 341, 342,366,378, 379 McDonald, R. G . , 190,200 McDoweII, M. R. C., 118, 131, 138, 151, 165, 264, 267,282,290, 414,420, 443,444, 446, 447,467,468.469 McEachran, R. P., 110, 111, 113, 114, 119, 121, 125, 128, 129, 131,132, 133, 138, 148, 162, 166 Macek, J., 288290, 353, 354, 360,377, 379, 437,438,440,444,447,454,462,466,467, 468 McElroy, M. B., 42,43,44,71, 73, 276,288, 289 McFadden, D. L., 187,202 McFarland, R. H., 393,420 McFarlane, S. C., 334, 335,379, 404,420 McGowan, J. W., 40, 41, 73, 137, 142, 147, 165, 288,291 McGregor, I., 461,464, 467 McGuire, J. H., 272,273,277,284,287,288, 291, 436,466 McGuire, P., 179,201 Mcllveen, W. A., 268, 270,289 McIntosh, A. I., 8,34, 479,504 McKay, C. P., 48, 73 Mackey, I., 492,493,504 McKnight, R. H., 280, 291, 310, 313, 314, 328, 339,379 McKoy, B. V., 10,35 McKoy, V., 489, 490,491,504, 505, 506 McLaughlin, R. W.. 237,262 McLean, A. D., 487,505 MacLeod, J. M., 57,69, 70, 73 McMillan, W., 209, 210,232 McNutt, D. P., 53, 71 McNutt, J. D., 105, 115,132, 142, 146, 147, 149,164, 165 McWalters, K. D., 43, 73 McWhirter, R. W. P., 246, 247, 256,260 Madison, D. H., 287, 290, 442, 443, 445, 460,468 Magee, J. L., 46,72
Mahan, A. H., 399, 415,420, 448, 449,468 Mahan, B. H., 42, 71, 207,232 Makhdis, Y.Y., 219, 225, 228, 229,232 Malaviya, V., 252, 253,260, 307,327 Malcolm, 1. C., 444, 446,467, 468 Mallory, M. L., 307, 314,326 Manaev, Yu. A., 226,232 Mandal, P., 130, 133, 153, 166 Manfrass, P., 366, 368, 370, 371,379 Mann, J. B., 414,420 Mann, R.,380 Manos, D. M., 186,201 Mansbach, P., 247,261 Manson, S. T., 61, 75, 287,290 Manur, C., 318,328 Manz, J., 170, 181, 188, 199, 200, 201 Mapleton, R. A., 48,70 Marchetti, M. A., 5,35 Marchi, R. P., 210, 211,232 Margrave, J. L., 10.33 Manno, L. L., 207,232 Mans, A. J., 431,467 Marko, K. A., 137, 159, 163, 165 Marrus, R., 68, 72, 73 Martin, D. W., 18.35 Martin, P. J., 450, 451, 452,470 Marusin, V. D., 394, 396,419 Mascord, D. J., 181,201 Mason, E. A., 253,261 Massey, H. S . W., 2, 3, 14, 18, 21,35, 37, 38, 42, 46, 62. 70, 73, 101, 120, 123, 127, 129, 130,133, 147, 148, 149, 153,166, 167, 175,201, 206,213,232, 235,236,243,251, 260,261, 263,265, 274,290,291, 301,327, 337,379, 423,425,461,468, 472,475,478, 480,485, 495,504, 505 Mathis, J. S., 50, 67, 73 Mathis, R. F., 28.35 Matic, H., 230, 231,233 Matic, M., 231,233 Matreev, V. I., 468 Matsuzawa, M., 85, 94, 95, 96, 99 Matthews, D. L., 350, 351,378 Mattioli, M., 295,328 Mauclaire, G. H., 19, 22,36 Mayer, T. M., 170, 171, 181, 184, 194,200, 201, 203 Mazeau, J., 11, 15.34, 35 Meade, D. M., 2%, 327 Meadows, E. B., 42, 72 Meckbach, W., 288,291
52 1
AUTHOR INDEX
Mecklenbrauck, W., 450,468 Megunov, A. I., 429,464 Mehlhorn, W., 334, 335, 362, 375, 376,378, 379,380, 461,469
Mehr, F. J., 40, 41, 73 Meier, R. R., 62, 63'69, 73, 237,261 Melius, C. F., 53, 73, 212,233 Melnick, G., 55, 73 Mendas, I., 256, 257, 258,260 Menzinger, M., 184, 185, 192,202, 203 Merchez, H., 457,465 Menwether, J. W., 43, 73 Merzbacher, E., 337,379 Meyer, F. W., 280,291, 307, 310, 311, 313, 314, 323,327, 328 Meyerhof, W. E., 347, 354, 356, 357, 369, 379 Micha, D. A., 168,201 Michel, K. W., 194, 198, 199 Michel, W. L., 230,233 Michie, R. W., 54, 70 Middleman, L. M., 331,379 Mies, F. H., 488, 498,505 Milgrom, M., 67, 73 Miller, J. S., 50, 73 Miller, P. D., 352,379 Miller, T. M., 239,261 Miller, W. H., 168, 174, 179, 180, 199, 201 Miller, W. J., 12,34, 35 Mills, A. P., 137, 164 Mims, C. A., 185,201 Misakian, M., 391,420 Mistry, V. D., 330,378 Mitchell, I. V., 366,379 Mitchell, J. B. A., 318, 319,327 Mittleman, M. H., 104, 127, 128, 129, 130, 132, 133, 153, 165 Moffett, R. J., 256,260, 480,504 Moffett, R. W., 43, 71 Mohr, C. B. O., 101, 127,133, 425,468, 472, 505 Mohr, P. J., 68, 72 Moiseiwitsch, B. L., 50, 51, 70, 215, 231, 253,261, 268,272,290, 330,331, 341,377, 378, 381, 418,420 Mokler, P. H., 369,378 Moll, P.G., 386, 388,421 Monchicourt, P., 249, 250, 252,260 Montgomery, J. A., 57, 72 Montgomery, R. E., 130,133, 147, 148, 149, 166
Moore, C. B., 190,200 Moore, C. F., 350,380 Moore, E., 57, 72 Moore, J. H., 225,233, 426,464 Moos, H. W., 68. 73 Moos, N. W., 68, 73 Mordinov, Yu. P., 51, 73 Morgan, D. L.. 110, 111, 113, 114, 121, 125, I33
Morgan, J. F., 341,379 Morgan, L. A., 446, 447,467, 468 Morgan, T. J., 283,291 Morgenstern, R., 455, 457, 458, 459, 467, 468
Morita, S.,331,379 Morokuma, K., 180,200 Morrell, G. O., 172,200 Morrison, H. G., 276,291 Morrison, M. A., 485, 487, 488, 492, 494, 499,505
Mors, P. M., 431,468 Moseley, J. T., 18, 20,21,22. 23, 27,33,34, 35, 62,74, 235.236,238,239,240,260,261 Moser, C. M., 5, 6, 7,33, 35 Moss, J. M., 359,378 Motz, J. W., 330,379 Moussa, A. H. A., 129, 133, 153, 166 Moustafa Moussa, H. R., 411, 412,420 Muckerman, J. T., 168, 169, 170, 171, 174, 184,201 202, 203 ~
Miiller, A,, 306, 327 Muller, B., 372,374, 375,379, 455,458,462, 467. 469
Mukamel, S., 180,202 Mukherjee, D., 238, 239, 240,260 Mul, P. M., 40, 41, 73 Mulholland, K. A., 28.34 Mulliken, R. S., 64,73, 249,261 Muller, A., 331,380 Mumma, M. J., 391,420 Munch, D., 3.36 Muf~oz.J. M., 243,260 Myerscough, V. P., 414,420, 447,467
N Nagamiya, S., 341,377 Nagel, D. J., 348, 349,379 Nagy, A. F., 43, 73 Nagy, S. W., 211,232
522
AUTHOR INDEX
Nakashima, A., 118,133, 152, 165 Nakel, W., 461,465, 468 Natanson, G. L., 258,261 Needham, P. B., 337,379 Neidigh, R. V., 295,327 Nesbet, R. K., 5,6, 7, 14,33,34,35, 53,73, 180, 199, 202, 461,468, 502,505 Neupert, W. M., 67, 73 Neynaber, R. H., 302,328 Nicolet, M., 42, 43, 44,46, 70, 73 Nicolopoulou, E., 16,35 Niehaus, A., 374, 379, 455, 457, 458, 459, 466, 467, 468 Nienhuis, G., 460,468 Nier, A. O., 39,40,41,42,43,44,53,70, 73, 74, 75 Nighan, W. L., 28,35 Nikitin, E. E., 51, 73, 168, 202, 212, 220, 231, 233, 357, 358,379 Nikolaev, V. S., 317,326, 342,379 Nikoleishvili, V., 207,231 Nishimura, S., 63, 73 Nolte, G., 461,469 Norcross, D. W., 414,420, 485,504 Norrington, P. H., 341,378 Norton, R. B., 42, 46, 73, 74 Norton, T. G., 384, 398, 399, 400,419 Noxon, J. F., 44,74 Nutt, W. L., 306, 307, 308, 314, 315,327 Nuttall, J., 105, 132
0
Oberoi, R. S., 104,133 Ochkur, V. I., 388,420, 436,468 Ochs, G., 172,202 Odiorne, T. J., 188,202 O’Donnell, E. J., 48, 74 Opik, U . , 276, 291 Offermann, D., 53,74 Ogawa, T., 46, 73, 74 O’Hare, B. G., 322, 323, 324,327 Oka, T., 57,69, 70, 73 Oke, J. B., 50, 74 Okuno, Y., 208,233 Oldham, W. J. B., 288,291 Oliver, A., 27.35 Ollison, W. M., 186, 187, 196, 198 Olson, R. E., 27.35, 62.74, 81.99, 178,202,
213,216,217,218,225,231, 233, 235, 236, 238,239,240,258,260,261, 280,286,289, 291, 303, 304,306,307,308, 310,312,313, 315, 316, 317, 319, 322,326, 327, 328 O’Malley, T. F., 16,35, 40, 74, 81,99, 112, I33 Omidvor, K., 277, 284,291 Onderdelinden, D., 345, 346,380 O’Neil, S. V., 169, 171, 198 Opal, C. B., 63, 73, 237,261 Oparin, V. A., 28,35, 322, 327 Oppenheimer, J. R., 342,379 Oppenheimer, M., 39, 40, 43, 44, 74, 75 Opykhtin, V.,408,421 Oran, E. S., 46, 62, 72, 74, 75, 237,261 Ormrod, J. H.,230,233 Orsini, N.. 40, 41, 43, 44,74, 75 Orth, P. H. R., 121, 124, 133 Osman, P. E., 119, 133 Osterbrock, D., 74 Osterbrock, E. E., 74 Ostlund, N. S . , 491,505 Ott, W. R., 407, 409,420 Ottinger, C., 185,202 Ovchinnikova, M. Ya., 51.74, 212,219,233 Oxley, C. L., 301,328 Oyamada, M., 331,379
P Pace, S. A., 193,202 Pack, R. T., 179,202 Page, B. A. P., 107, 133 Page, T., 237,260 Palyukh, B. M., 207, 208,232 Panagia, N., 67, 73 Pang, H. F., 192, 193, 195, 196,202. 203 Panov, M. N., 226,232 Parilis, A., 468 Park, H., 59, 74 Park, J . T., 272,273,276,287,288,291, 302, 321,327 Parkin, A , , 391,420 Parks, E. K., 194, 195,202 Parlant, G., 11,35 Parr, T. P., 181, 199, 202 Parrish, D. D., 181,202 Parson, J. M., 181, 186,201, 202 Patterson, T. A., 12, 20,34
523
AUTHOR INDEX
Patterson, T. N . L., 47, 72 Paul, D. A . L., 105, 1 IS, 118, 124,133, 137. 142, 144, 145, 146, 147, 148, 149, 152. 153. 154, 163, 164,165, 166 Pauli, M.,286,291 Peach, G., 435,436,468 Peacher, J. L., 272, 273, 287, 288,291, 302. 321,327 Pearce, J. B., 60,69 Pearson, P. K., 169, 198 Peart, B., 27.35, 318, 319, 321,327 Pechukas, P., 168. 174, 175,202 Pedersen, H., 279,291. 449, 450,464 Peek, J. M.,168,201 Pendrill, L. R., 78, 99 Penzias, A. A., 57,72 Pequignot, D., 47, 48, 52, 53, 74 Percival, 1. C., 158, 166, 280, 286,288, 303, 326, 392,407,420, 437, 448.468, 475,505 Perel, J., 207, 209, 210, 213, 214,231, 233 Peresse, J., 392, 418,420 Perinotto, M.,74 Perlman, H. S., 330, 331,379 Perry, D. S., 190.199 Peterkop, R., 107, 133, 426, 427, 428, 433, 436,468,470 Peterson, J. R., 18,20,21,22,23,27,33,34, 35, 62.74. 235,236,238,239,240,260,261 Peterson, W. K., 61, 70 Petrov, M. P., 299,327 Petters, E. W., 278,291 Petty, R. J., 331,378 Phaneuf, R. A., 280,291, 307,310,311,313. 314,327, 328, 381, 403,421 Phelps, A. V., 383,420, 487,505 Phillips, D., 366,379 Phillips, J., 294, 297,327 Piacentini, R. D., 307,328 Pichon, F., I I, 36 Pitaevskii, L. P., 251, 252,261 Placious, R. C., 330,379 Plummer. E. W., 60, 72 Pochat, A., 392. 418,420 Pockman, L. T., 330,380 Poe, R. T., 474,504 Pol, V., 105, 115, 133. 137, 138, 139, 144, 145, 146, 147, 148, 149, 150, 151, 153, 154, 155, 156, 166 Polanyi. J . C., 168, 169, 170, 171, 172, 181, 190, 193, 198, 199, 200, 202
Pollak, E., 174, 175,202 Pommier, J., 221, 224, 225, 228,231, 232 Pope, W. M.,191,203 Potapov, V. S., 280,289 Potemra, T. A , , 61, 73 Potter, D. L., 337,379 Poulaert, G., 2 7 , B Pradhan, A. K., 60, 74 Prather, M. J., 44, 73 Prengel, A. T., 187, 198 Presnyakov, L. P., 220,233, 408,421 Preston, R. K., 171, 172. 173, 176,198, 200, 202, 203 Prilezhaeva, N. A., 395, 396,420 Proctor, A. E., 191,202, 203 Pruett, J. G., 188, 193,202 Purser, K. H., 345, 347,379 Pyle, R. V., 26,33, 280,289, 312, 316, 317, 326
Q Quarles, C. A , , 330, 332, 333,378, 380
R Rabik, L., 107, 133 Rafelski, J., 374, 375,379 Raff, L. M.,182, 183,202 Raines, R. G., 339,379 Raith, W., 460,468 Ramsauer, C., 151, 166, 478, 485,505 Randall, R., 68, 70 Randell, R., 68, 70 Rapp, D., 206, 207, 208, 210, 212,233, 276, 291, 307, 319, 320, 322,328 Rapp, D. W., 47, 74 Rasmussen, J. 0.. 341,377 Rasulov, D. H., 366,377 Rau, A . R. P., 426,469 Rauscher, E., 341,377 Raun, H. L., 376,377 Ray, J. A,, 325,326 Ray, S., 59, 74 Raymond, J. C., 74 Read, F. H., 391, 394, 419, 420, 426, 461, 465, 469
524
AUTHOR INDEX
Reading, J., 270, 271, 272, 273, 278, 286, 290, 291, 339,378 Reck, G. P., 12,35 Redmon, M. J., 179,202 Redpath, A. E., 184,202 Rees, M. J., 51, 75 Refaey, K. M. A., 12,35 Register, D., 460,469 Reichelt, W., 322,326 Reichert, E., 397,419 Reichert, R., 460,469 Reid, R. G. H., 54, 76 Reinhard, T. J., 373, 375,377 Reinhardt, J., 11, 15.34, 36, 462,469 Reinhardt, W. P., 12,33, 490,505 Rescigno, T. N.,10, 14, 35, 489, 490, 491, 505, 506
Rester, D. H., 330,380 Reynolds, R. M.,324,327 Ribe, F. L., 293,328 Rice, S. A., 181,202 Rich, A., 137, 159, 163, 164, 165 Richard, P., 278,291, 344,379 Richter, A., 331,378 Riddell, G. I., 324,327 Ridley, R. 0.. 475, 480,505 Riley, M. E., 274,291 Risberg, J. S . , 26,34 Risley, J. S., 26,34, 35, 350,377 Ritchie, B., 267,291 Ritchie, R. H., 338, 339,377 Roach, A. C., 168, 169, 171, 172, 199, 200, 202 Robb, M.A., 171, 172, 199 Robb, W. D., 414,420, 492,504 Robbins, R. R., 68, 71 Robertson, W. W., 386, 387,421 Robinson, B. J., 57, 74 Roble, R. G., 42, 44,45, 72 Rodgers, S . R., 284,291 Rodbro, M.,279,291, 461,465 Rogers, W. A., 42, 72 Rogerson, J. B., 48, 76 Rohr, K., 4%, 506 Romick, G. J., 59, 60, 70 Rose, P. H., 26,35 Rosen, N., 212,233 Rosenberg, L., 112,133 Ross, J., 180,202, 203 Rothe, E. W., 12,35
Rother, E. W., 302,328 Roueff, E., 54, 55.69, 72, 73 Roy, D., 461,469 Rozet, J. P., 362, 364,377 Rozuel, D., 392,418,420 Rubin, K., 435, 436, 437, 461.467, 469 Rudd, M. E., 287, 288,290, 291 Rudge, M.R. M.,54,71, 433,436,466, 469 Rugge, H. R., 67, 75 Rule, D. W., 317,328 Rulis, A. M.,176, 181,200, 203 Rundel, R. D., 78, 85,99 Rusch, D. W., 42, 43, 44,45, 46, 71, 72, 74 Rush, P. P., 415, 416,420 Rusin, L. Y.,182, 183, 198 Russek, A., 274,291, 325,326 Ruthberg, S., 402,420, 421 Rutherford, J. A., 47, 75, 230,233 Rydbeck, 0. E. H., 57, 72, 74 Ryding, G., 283,291, 2%, 310,328 Ryman, A. G., 110, 111, 113, 114, 121, 125, 131, 133, 138, 148, 162, 166
S
Sackmann, S . , 360,380, 461, 462,467 St. John, R. M.,394,402,415,416,417,419, 420, 421 Salerno, J. A., 478,505 Salin, A., 266, 282, 284, 285, 288,289, 290, 291, 304, 307,327, 328 Salop, A., 280, 286,291, 303, 304, 306, 308, 310, 313, 316, 317, 319,327, 328 Salpeter, E. E., 63, 67, 72, 73, 80,99 Salvatelli, E. R., 8, 33 Salzborn, E., 306,327, 331,380 Samsom, J. A. R., 60,74 Samuel, M. J., 383, 384,420 Sancisi, R., SO, 74 Sandner, W., 334, 335, 375. 376, 380, 461, 469 Sangvist, A . , 57, 72 Sapirstein, J., 164 Saponova, U. I., 8,35 Saraph, H. E., 60, 74 Sans, F. W., 345, 346, 358, 368, 371, 372, 380 Sartwell, B. D., 337,379 Sasaki, F., 4, 5, 6,35
AUTHOR INDEX
Satchler, G. R.,478,5oQ Sato, Y.,225,233 Sauter, C. A., 288,291 Sayer, B., 252,260 Saykally, R. J., 57, 74, 76 Saylor, T. K., 279,289, 347, 356, 357, 369, 379 Scarborough, J., 244,261 Schaefer, H. F., 168, 169, 171, 198, 202 Schafer, T. P., 181,202 Schafher, S., 62.69, 236,260 Schamp, H. W., 253,261 Scharmann, A., 460,466 Schartner, K. H., 460,466 Schatz, G. C., 175, 179, 180,200, 202 Schauble, W., 450, 451, 452,470 Schectman, R. M.,448,461,465, 469 Schermann, C., 15, 16,34 Schermann, J. P., 17.35 Schiavone, J. A., 88, 99 Schiebel, U., 331, 380 Schiff, H. I., 45, 75 Schlachter, A. S., 312, 316, 317, 322, 324, 325,326, 328 Schlecht, W., 436,467 Schmeltekopf, A. L., 15, 25.34. 43, 45, 71, 75 Schmerling, E. R.,39, 71 Schmidt, H., I1,35, 494, 496,505 Schmidt, J. J., 294, 297,327 Schmidt, V., 375, 376,377, 380 Schmidt-Bleek, F. K., 183,200, 397,420 Schmidt-Backing, H., 361, 366, 379. 3c10, 461.469 Schmieder, R. W., 68, 72, 73 Schmitt, W., 334, 335,380 Schmoranzer, H., 64, 66, 67, 74 Schneider, B. I . , 10,35, 492, 494, 502,505, 506 Schneider, D., 339, 348, 350, 354,377, 380 Schneider, W. E., 401,421 Schneiderman, S. B., 274,291 Schnitzer, R., 11, 12,33 Schnopper, H. W., 348, 367,377,380 Scholer, A., 343, 344,380 Schon, S., 397,419 Schonhammer, K., 7, 9,33 Scholackter, A. S., 280,289 Scholz, W., 362,378 Schonhense, G., 460,466
525
Schrader, D. M.,103, 133 Schraeder, R. J.. 341,379 Schreiber, J. L., 168, 169, 171, 181, 190, 199, 202 Schubert, E., 429,466,467 Schubert, J. G., 55, 72 Schuch, R.,361, 366,379, 380, 461,469 Schule, R.,361, 366,379, 380 Schiissler, H.A., 460,469 Schult, 0. W. B., 376,377 Schulz, G. J., 16, 25.33, 36, 394,421, 495, 496,499,506 Schulz, M.,426, 427, 429,464, 466 Schutten, J., 411, 412,420 Schwartz, C., 102, 103, 106, 128, 133 Schwartz, S. B., 436,469 Schweitzer, N., 295,328 Schwob, J. L., 295,328 Scott, T., 443, 444,469 Seaton, M.J., 207,233, 392, 407,419, 420, 436,437,448,466,468,469, 475,487,505, 506 Searles, S. K., 244,261 Sega, R. S., 341,379 Seiler, G . J., 104, I33 Sellin, I. A., 288,291, 337,377, 460.469 Sena, L. A., 207, 208,232 Senaskenko, V. S., 8,35, 429,464 Sennhauser, E. S., 254,262 Sera, K., 331,379 Serenkov, I. T., 28,35 Series, G. W., 78,99 Sewell, E. C., 318, 319, 320,326 Shafroth, S., 364,377 Shah, M. B., 306, 307, 308, 310, 311, 312, 327, 328, 341,379 Shakeshaft, R.,267,273, 276, 281, 287,291 Shapiro, M.,169, 180,200, 203 Shapiro, S. G., 111, 113, 114, I31 Shaporenko, A. A., 209,232 Sharpton, F. A , , 417,420 Shaw, M. J., 385, 386, 387,420 Sheen, S. H., 194, 195,202 Sheftel, S. I . , 429,464 Sheinerman, S. A., 461,464 Shelten, W. N., 442, 443, 445,468 Sheorey, V. B., 408,410, 411,419 Shepherd, G. G., 43, 72 Shergin, A. P., 366,377 Sheridan, W. F., 208,232
526
AUTHOR INDEX
Shields, G., 50, 74 Shimamura, I., 105, 133, 492, 493,504 Shimazaki, T., 46, 74 Shipman, H. L., 48, 71 Shipsey, E. J., 225,231, 315,326 Shirai, T., 308, 317,328 Shobatake, K., 181,202 Showalter, J. G., 384, 386, 388, 401,420 Shpenik, 0. B., 394, 395, 396,421 Sides, G. D., 14.35 Sidis, V., 221, 224, 225, 226, 227, 228,231, 232, 233 Siegel, M.W., 11.33.36, 221,222,223,225, 231 Siegert, A. J. F., 501,506 Sil, N . C., 130,133, 272,290 Simons, J., 7, 10.34, 35, 36 Simpson, F. R., 8, 26,34 Sims, J. S., 3, 4,36 Sinclair, M. W., 57, 74 Sinda, T., 318,328 Sinfailam, A. L., 481, 482, 483, 490,504 Sinha, S., 209, 210, 211,231, 233 Siska, P. E., 181,202 Sivjee, G . G., 59, 60, 70 Sizor, V. V., 462,465 Slanger, T. G., 46, 71 Slevin, J., 394, 418,419, 441, 442, 443, 444, 445, 446,448,466 Sloan, J. J., 169, 190,198. 199 Smart, J. H., 105, 115, 133, 138, 144, 145, 146, 147, 148, 149, 150, 151. 153, 154, 155, 156,166 Smick, A. E., 330,380 Smirnov, B. M.,206,2 11,232,233, 244,261 Smith, A. C. H., 47, 71, 75, 207, 232, 307, 326 Smith, B. T., 28,34 Smith, D. L., 208,233, 239,261 Smith, F. T., 54,74, 178,202, 207,209,210, 211,216,217,218,232,233, 238,239,240, 260, 261, 322,327 Smith, G. P., 188,202 Smith, J. J., 427, 428,465 Smith, K. A., 31,36, 78, 85, 92, 94, 99 Smith, M. W., 59, 60,76, 237,262 Smith, R. A., 206,232, 265, 274,291 Smith, R. K., 373, 375,377 Smith, S. J., 381, 399, 407, 409, 415, 418, 420, 437. 448, 449,467, 468
Smith, W. D., 7,36 Smith, W. H., 59, 71 Smythe, R., 26,36 Snow, T. P., 48, 5 5 , 59, 75 Snow, W. R., 28,35, 296, 301,326 Snyder, L. E., 57,75 Snyder, R., 26,36 Sohval, A. R., 348, 367,377, 380 Sokolova, A. A., 68, 76 Solomon, P. M.,63, 75 Solomon, W. C., 170,200 Solov’ev, E. S., 28,35, 208,232, 322,327 Somerville, W.B., 63, 71 Soong, S. C., 277, 279,290 Souter, V. V., 394, 395, 396,421 Spence, D., 15,36 Spencer, N. W., 39,71 Spicer, B. M.,330, 331,378 Spitzer, L., 55, 56, 70 Sprevak, D., 256,260, 361,377 Spruch, L., 112, I33 Sridharan, U.C., 187,202 Staab, E., 325,327 Stabler, R. C., 242, 246, 247, 252, 254,260, 26 I Staemmler, V., 5 , 6,36 Stair, R., 401,421 Stamatovic, A., 16.36, 394,421 Standage, M. C., 412, 421, 437, 440, 441, 442, 444,467, 469 Starace, A. F., 61, 75 Stasinska, G., 47, 48, 52, 53, 74 Stauffer,A. D., 110, 111, 113, 114, 121, 125, 131, 133, 138, 148, 162, 166 Steams, J. W., 280,289, 312, 316, 317,326 Stebbings, R. F., 47,71, 75, 77, 78, 85, 92, 94, 99, 301, 307,326, 328 Stecher, T. P., 56, 63, 75 Stefani, G., 332, 334,378, 426, 430,465 Stefansson, T., 210, 211,232 Steigman, G., 47, 51, 52, 53, 71, 75 Stein, H. J., 369,378 Stein, J., 127, 128, 133 Stein, S., 480,504 Stein, T. S., 105, 115, 133, 137, 138, 139, 144, 145, 146, 147, 148, 149, 150, 151, 153, 154, 155, 156, 166 Stelson, P. H., 311,327 Steph, N. C., 443, 445,469 Stephens, T. L., 64,65, 66, 67,71, 75
AUTHOR INDEX Sternlich, R., 127, 128, I33 Stevefelt, J., 247, 249, 250, 252,260, 261 Stevenson, D. P., 47, 72 Stewart, A. I., 46, 60,69, 71. 74, 75, 125, 126, 132 Stewart, I., 218, 228,231, 321,326 Stier, H. C., 480,506 Stigers, C. A,, 283,290 Stockli, M.,363, 364, 365, 366, 371, 372, 373,380 Stokes, E. D., 283,290 Stoller, C., 363,364,365,366. 371, 372, 373, 380 Stoke, S., 191,202, 203 Stolterfoht, N., 278,287,290,291, 339, 348, 349, 350, 351, 354, 357,377, 380 Strakhova, S. I., 429,464, 465 Streit, G. E., 22, 23,34, 45, 75 Strickland, D. J., 237,261 Strobel, D. F., 46, 74, 75 Stuckelberg, E. C. G., 212, 219,233 Sucher, J., 68, 71 Sugden, T. M.. 252, 253,261 Suiter, H.R., 341,379 Sullivan, E. C., 270,289, 496, 506 Sullivan. J., 269, 271, 272,291 Sume, A., 57, 72, 74 Sutcliffe, V. C., 443, 445,469 Suter, M.,288,291, 363, 364, 365, 366, 371, 372, 373,380 Sutton, J. F., 399,421 Suzukawa, H. H., 182, 183,202 Suzuki, K., 43, 70 Sverdlik, D. I . , 175,203 Swaf€ord, G. L., 270, 278,291 Swartz, M.,67, 73 Szento, P. G., 57, 74, 76 Szucs, S., 303, 309,326
T Takahashi, H., 236, 237,262 Takayanagi, K., 63,73, 472,506 Tambe, B. R., 53, 75 Tan, K. H., 426, 443,469 Tang, S. P., 185, 187,203 Tang, S. Y., 12,35 Tashaev, Yu. A., 317,326
527
Taulbjerg, K., 267, 286, 289, 291, 356, 357, 377, 379 Tavernier, M., 362, 364, 376,377, 378 Tawara, H., 331, 368,379, 380 Taylor, A. J., 273, 276, 289 Taylor, H. S., 16, 35, 105, 133, 442, 443, 465, 470, 488,505 Taylor, P. 0.. 382, 401, 403, 408, 410,419, 42 1 Teillet-Billy, D., 1 I , 36 Tekaat, T., 426,427,466 Teller, E., 67, 68, 70 Tellinghuisen, J., 67, 75 Teloy, E., 172,202 Temkin, A., 103, 104, 119, 120, 131. 472, 476,480,483,485,488,496,498,499,504, 505, 506 Tench, A. L. V., 475,504 Teplova, Ya. A ,, 317,326 Tesmer, J., 26.34 Teubner, P. J. 0.. 429, 430, 432, 443, 444, 445,464,467,470 Thaddeus, P., 57, 58, 72, 75 Thielmann, U., 455, 457, 458,466, 467, 468 Thoe, R. S., 288,291 Thomas, B. K., 272,290 Thomas, E. W., 283,290 Thomas, G. E., 48, 73 Thomas, L. D., 443,470 Thomas, M. A., 108, 133 Thompson, D. G., 130, 131, 132, 133, 138, 147, 148, 149, 155, 157,165, 166, 182, 183, 202, 461,468, 470, 484, 485,505 Thomson, J. J., 251, 255,261 Thomson, R. M.,31,36 Thorsen, W. R., 274,290 Tibbs, S. R., 485,504 Tidemand-Peterson, P., 376,377 Tiernan, T. 0.. 13, 14,34, 35, 230,232 Tinsley, B. A., 62, 75, 236, 237,262 Tiribelli, R., 430,465 Toburen, L. H., 287,290 Toennies, J. P., 168, 182, 183, 198, 203 Toevs, J. W., 26.36 Tolk, N., 460,461,470 Tolmadev, V. V., 280,289 Torr, D. G., 39,40,41,43,44,46,53,70,73, 74, 75 Torr, M. R., 39,40,41,43,44,46,53,70, 74, 75
528
AUTHOR INDEX
Toshima, N., 54.75 Touati, A., 362, 364,377 Trainor, D. W., 14,33 Trajmar, S., 460,469 Trautmann, D., 286,291 Trebino, F. P., 39, 74 Trelease, S., 283,289 Tretyakov, Y.P., 366,378 Tripethy, D. N., 442,465 Triribelli, R., 332, 334, 378 Tronc, M., 11, 16.34, 36 Trotter, J., 362,378 Truhlar, D. G., 168, 171, 172, 174,200, 203 Trujillo, S . M., 302,328 Tsai, J. S., 144, 145, 152, 153, 154, 166 Tserruya, I., 361,380 Tuan, V. N., 322,328 Tucker, K. D., 57, 75 Tully, F. P., 181,202 Tully, J. C., 168, 172, 173, 176, 185, 192, 200, 202, 203 Tunnell, L . N., 277, 279,290 Turner, B. E., 57, 58, 75, 76 Turner, B. R., 208,233 Turner, J. E., 403,420 Turner-Smith, A. R., 216,233 Twomey, T. R., 117,132, 138, 140, 141, 142, 144, 145, 146, 151, 152, 153, 154, 157, 158, 163, 164,165
Valiron, P., 52, 73, 313,327 van Brunt, R. J., 8, 15, 16,36 Van der Meulen, A., 176,203 Vanderpoorten, R., 272,290, 443,467 van der Weg, W. F., 368,380 van der Wiel, M. J., 426,470 Vandespoorten, R., 151, 154, 166 Vane, C. R., 288,291 Van Itallie, F. J., 185,200 van Raan, A. F. J., 384, 385, 386, 388, 390, 391, 399, 400,421 van Regemorter, H., 55,72 Van Zandt, T . E., 42, 74 Van Zyl, B., 402,421 Vasavada, K. V., 480,506 Veldre, V. Ya., 427,470 Vernon, R. H., 207. 209,233 Vestal, M. L., 19, 22,36 Vichon, D., 11.34 Victor, G. A., 40, 61, 67, 70, 71, 74, 75 Vidal-Madjar, A,, 48, 75 Vincent, P., 366, 371,378 Vinkalns, I., 428,470 Vo Ky Lan, 7,35,36, 476,492,494,504,505 von Niessen, W., 7, 9 , 3 3 Vorburger, T. V., 25,36 Vroom, D. A., 230,233 Vujovic, M., 230, 231,233
W U
Ugbabe, A., 429, 443, 444, 445,464,470 Ulantsev, A. D., 220,233 Ulich, B. L., 57, 75 Unwin, J. J., 62, 70,236,260 Ureia, A. G., 192, 193,201, 203 Utterback, N. G., 187,203 Uyriainen, J., 362, 363,377
V Vaaben, J., 304, 305,328 Vainshtein, L., 408, 415, 416,421 van Eck. J., 384, 386, 388, 390, 391, 399, 400,419,421 Valentine, N., 158, 166 Valentini, J. J., 181, 190, 196, 197,203
Wadehra, J. M., 16,33 Wagner, H. G., 170,200 Wahl, A. C., 13,34 Wainright, P. F., 432, 436,464 WaKid, S. E. A., 105, 127, 133 Walker, A. B. C., 67, 75 Walker, J. C. G., 26,33, 39, 40, 41, 42, 43, 44, 45, 46, 53, 70, 71, 72, 73, 74, 75 Walker, J. D., 415, 416,421 Wallace, J. B. G., 103, 119, 120, 133 Wallace, S. C., 193,200 Walls, F. L., 38, 39, 40, 75 Walmsley, C. M., 48,69 Walters, H. R. J., 138, 151, 152, 153, 158, I65 Walton, D. R., 57, 73 Wannier, G. H., 426,470 Wardle, C. E . , 107,133
529
AUTHOR INDEX
Warman, J. M., 254,262 Warnatz, J., 170,200 Warren, B. E., 344,380 Watanabe, T., 211,232, 308, 317,328 Watkin, R. D., 393,420 Watson, R. L., 340,378 Watson, W. D., 42,48,49,52,56,59,70, 71. 74, 76 Weaver, F. W., 277, 284,291 Weaver, L. D., 388,421 Webb, C. E., 216,233 Weber, M., 335,380, 461,469 Webster, D. L., 330,380 Webster, M. J., 385. 386, 387,420 Wegner, H. E., 348, 367,377, 380 Weigold, E., 426, 428, 429, 430, 431, 432, 443, 444,445,446, 447,464, 466, 467, 470 Weill, G., 236, 262 Weiner, J., 186, 187, I%, 198 Weisheit, J. C.. 48, 51, 71, 76 Weiss, A. W., 3, 4, 5.36, 414,421 Wellenstein, H. F., 386, 387,421 Weller, C. S., 62.69 Welsh, J. R., 14.34 Welther, W., 57, 58, 71, 72 Welzol, W. W., 425,470 Wentzel, G., 374,380 Werner, M. W., 52, 63, 72, 75 West, W. P., 92, 99 Westerveld, W. B., 391, 413,421 Wexler, S., 168, 194, 195,202, 203 Wherry, C. J., 105, 132 Whitaker, M., 241, 242, 244,261 Whitehead, J. C., 181, 188, 189, 199, 203. 341,379 Whiteoak, J. B., 57, 74 Wichmann, E., 111, 113, 114, 131 Wicke, B. G., 185,203 Wickramasinghe, N. C., 63, 75 Wiegand, W. J., 28,35 Wieman, H., 350,377 Wiese, W. L., 59, 60,76, 237,262 Wiesemann, K., 218, 225,232 Wigner, E. P., 385,421. 486, 492,506 Wilcomb, B. E., 170, 171, 181, 184, 194, 201, 203 Willets, L., 276,290 Williams, D. A . , 56, 63, 75, 353,377 Williams, J. F.. 402,421. 424, 444.467 Williams, R. E., 47, 76
Williamson, K., 157, 158, 165 Willmann, K., 332,378, 426, 427, 428,466 Willson, R. F., 57, 73 Wilson, J. McB., 320,326 Wilson, R. W., 57, 72 Wilson, S., 58, 76 Wilson, W. G., 105, 115, 117, 133, 142, 145, 146, 147, 166 Winter, H., 306,328 Winter, T. G., 268, 270, 276,289, 291, 307, 328 Winters, K. H., 115, 117,131, 138, 151, 154, 164. 166, 427. 428,465 Winters, L. M., 278,291 Wittkower, A . B., 283,291, 296, 310,328 Wolfli, W., 363,364,365,366,371,372, 373, 380 Woerlee, P. H., 358,380 Wofsy, S. C., 44, 54, 73, 76 Wolcke, A., 460,466 Wolf, E., 439,465 Wolf, F. A., 208,233 Wolfgang, R., 173,200 Wolfrum, J., 190, 198 Wong, S . F., 16, 25,33, 36, 496, 501,506 Wong, Y. C., 181, 197,202, 203 Woo, S. B., 12, 21, 25,34, 36 Woodall, K. B., 169, 170,202 Woods, C. W., 278,291 Woods, R. C., 57, 74. 76 Woodworth, J. R., 68, 73 Worley, R. D.. 337,379 Wren, D. J., 185,203 Wu, K. T., 192, 193, 195, 196,202, 203 Wu, R. L. C., 13.34 Wuilleumier, F., 375, 376,380 Wyatt, R. E., 168, 174, 179, 180, 199, 202, 203
Y Yagisawa, H., 51, 76 Yahiku, A. Y.,213,233 Yardley, R. N., 168, 198 Yarlagadda, B. S., 443,470 Yau, A., 54, 76 Yokozeki, A., 192,203 York, D. G., 48, 75, 76 York, G., 8 , 3 4
530
AUTHOR INDEX
Yoshimine, M.,4, 5, 6,35, 487,505 Young, J . M.,53, 72 Young, N. A., 252, 253,260 Young, R. A., 301,328 Yousaf, M.M., 222,232 Yung, Y. L., 42, 73
Z
Zandee, L., 191,203 Zapesochnyi, I. P., 394, 395, 396,421 Zare, R. N., 181, 185, 188, 189, 199, 200. 202, 203 Zavilopulo, A. N., 394, 395, 396,421 Zehnle, L., 450, 451, 452,470 Zeilik, M.,47, 76
Zeiri, Y., 169,203 Zener, C., 212,233 Zetzsch, C., 170,200 Zhukovskii, V. C., 68, 76 Ziemba, F. P., 221,233 Zietz, R., 66, 67, 74 Zilitis, V. A., 396,419 Zimmerman, I. H., 270,291 Zimmerman, M. L., 91, 99 Zinoviev, A . N., 366,377 Zipf, E. C., 40, 46, 76, 237,262 Zittel, P. R., 10,36 Zitzewitz, P. W.,137, 163, 164, 165 Zolla, A., 244,261 Zuckerman, B., 57, 75, 76 Zupancic, C., 340,379 Zvijac, D. J., 180,203 Zwally, H. J., 217, 218,233
1 1 SUBJECT INDEX A
Activation energy, zero, treatment of, 174 Adiabatic approximation, energy-modified, for electron-molecule collisions 502503 Adiabatic maximum rule and Massey cntenon, 213-214 Adiabatic-nuclei approximation, vibrational excitation and, 4%-497 Alignment tensor, 446-448 Alkali atom, electron impact ionization, 433 -436 Alkali-halogen atom collisions, 175- 178 Alkali-hydrogen halide system, potential energy surfaces and, 168-170, 171173 Alkali-proton collisions, electron capture neutralization and, 322-325 Aluminum, K shell ionization, 336-337 Aluminum ion-argon collisions, 354-356 Ammonium ion cluster in electron-ion recombination, 241 in ion-ion recombination, 239-241 Angular correlation for electron-ion collisions, 455-459 electron-photon, 440-448 heavy-particle-photon, 449-455 for particle-photon collisions, 437-455 in positron annihilation, 124-126 use in impact ionization studies, 425-437 Anisotropy coefficient, determination, 333 335 Annihilation of positrons in atom scattering, 118-124 Trays, angular correlations of, 124- 126 Argon L shell ionization, 334-335, 342, 345-346 recoil momentum distribution, 431 -432 Association, radiative, 55-56 Associative detachment, effect of vibrational excitation on H, and D,. 16-17 Atmosphere, negative-ion reactions with trace constituents in, 25 Atomic structure, from (e,2e) experiments, 429-431 53 I
Attenuation method, in charge transfer collisions, 230-231 Auger electron spectra effects of inner-shell ionization on, 348350, 362-367 theory of, 374-376 Autoionization, 455-459
B Balmer lines excitation by electron impact, 399, 415 incoherent effects, 448-449 Bethe approximation, in excitation cross section determination, 403-415 Binary encounter model, 286, 426 Bismuth, inner-shell ionization, 359 Boomerang model, 500-502 Born approximation electron-photon angular correlations and, 441-443 for heavy-particle excitation, 267-268, 271 -274 for heavy particle ionization, 286 inner shell ionization by electron impact, 329-331 second-order, high-energy charge transfer and, 280-281 Boron, electron affinity of, 5 Branching ratio, importance in astrophysics, 59 -6 I Breakdown phenomena, negative ions in electric discharge and, 28-33 Bnnkman-Kramers model, for charge transfer, 277, 280, 281, 284 Bromine, K shell ionization, 345, 347
C
Carbon electron affinity of, 5 fine structure levels of ion, excitation of in interstellar clouds, 54 K shell ionization, 332, 334, 343-344
532
SUBJECT INDEX
Carbon ion-hydrogen collisions, 54, 304305, 313-314 Carbonate ion, photodestruction of, 21-23 CDW approximation, see Continuum distorted-wave approximation Cesium in electron-ion recombination, 252 ion-atom charge transfer collisions, 207208 Charge transfer, see also Electron capture accidental resonance, 46-50 Abrines-Percival classical model, 280 asymmetric, 278-280 curve crossing spectroscopy and, 221 -218 during plasma heating, 2% into continuum, 286-288 involving excited ions, 229-231 high energy, 280-285 low energy, 205-231 of multiply charged ions in astrophysical plasmas, 50-53 in multiply charged ion-hydrogen collisions, 303-315 nonresonant, 21 1-214 psuedocrossing, total cross sections, 2 14 -220 symmetrical resonance 206-21 I Chemiluminescence, study by crossed-beam, 183- 187 Cheshire-Sullivan model, for heavy-particle excitation, 270-271 Chlorine ion, collisions with rare gas atoms, 27-28 Chlorine ion-nickel collisions, 361 Chlorine ion-titanium collisions, 361 CIS approximation, see Continuum intermediate-state approximation Close-coupling approximation, for heavyparticle excitation, 267-268, 271-272 Closure approximation, for heavy-particle excitation, 268 Coherence effects, 461, 462 Coherent excitation of degenerate states with different angular momenta, 462464 Collisions, see also Alkali-halogen atom collisions; Charge transfer collisions; Fast heavy particle collisions; Hydrogen-hydrogen exchange reaction coherence and correlation in, 423-464
electron-molecule, at low energies, 471 503 future studies, 460-462 Collisional detachment, 26-28, 325-326 Collisional ionization, atom-atom of highly excited atoms, 91 -99 of normal atoms, 175-178 Collisional mixing atomic targets, 78-85 molecular targets, 85-91 of Rydberg atoms, 78-91 Collisional-radiative recombination defined, 245, see Electron-ion recombination in ambient electron gas Continuum distorted-wave approximation, 283-285 Continuum intermediate-state approximation, 282-285 Copper doubly ionized K shell, 366 K shell ionization, 332-333 L shell excitation, 343-344 Crossed-beam chemiluminescence, 183- 187 Crossed-beam measurements, of excitation cross sections, 394-397.402-403 Crossed-beam method, potential surface and, 171 Crossed-beam reaction, with electronically excited reagents, 189- 190 Crossed-beam technique, modulated, in fusion reactor research, 301 CR recombination, see Collisional-radiative recombination Curve-crossing spectroscopy, 22 1-228
D Demkov-Meyerhof model, for inner-shell ionization, 357 Demkov radial coupling model, 354-356 Demkov two-state approximation, 2 122 14 Deuterium-hydrogen abundance ratio, in local interstellar medium, 48 Deuterium-hydrogen halide exchange reactions, crossed-beam study, 181- 183 Diatomic negative ions, 9-1 1 Diatomic molecules, spontaneous radiative dissociation of, 62-67
SUBJECT INDEX
Dioxetane reaction, chemiluminescence cross section, 186-187 Dissociative attachment, 13- 18 angular distribution of product ions, 1516
of HN03, I5 total cross section, measurement of, 13-14 Dissociative recombination, 38-42. 45 Distorted-wave approximation, electronphoton angular correlations and, 442443 Double-electron radiative transitions, 362366
E Eikonal distorted-wave approximation, 443 Electric discharge, negative ions in, equilibrium conditions, 28-30 stability of, 30-33 Electron, inner-shell ionization by, 329-335 Electron affinities calculation of, 7 of diatomic molecules, 9-1 1 of first row atoms, 4-6 of polyatomic molecules, 12-13 of second row atoms, 6 Electron capture, see also Charge transfer in multiply charged ion-hydrogen collisions, 303-315 negative ion formation and, 325-326 by pure rotational excitation, 243 radiative, 264, 367-368 Electron capture from atoms by fast ions theory, 274-285 asymmetric charge transfer, 278-280 classical model, 280 high energies, 280-285 intermediate energies, 275 -280 preservation Galilean invariance, 274 two-center expansions, 275-278 Electron capture neutralization of fast ions, 322-325 Electron-electron angular correlations, from impact ionization, 425-429 Electron-hydrogen molecule collisions, 478 -480 Electron impact, excitation of atoms by, see Excitation, electron impact
533
Electron-ion recombination, binary with complex ions, 241 -245 dissociative and nightglow, 44-46 radiative and nightglow, 62, 235-237 Electron-ion recombination in ambient electron gas, 245-250 collisional-radiative: recombination defined, 245 formation of excited atoms in high pressure helium afterglow, 250 Electron-ion recombination in ambient neutral gas, 250-254 cesium ions in helium, 252 helium-atomic and molecular ions in helium, 252-254 lead ions in flame gases, 253 modified Thomson formula, 25 1 Pitaevskii model, 251 semiquantal treatment, 251 -252 Electron-molecule collisions boomerang model, 500-502 cross section, expressions for, 477-478 frame transformation theory, 485-488 laboratory frame representation, 473480, 482-483 Lz methods, 488-495 at low energies, theory of, 47 1-503 molecular-frame representation, 480-485 radial equations, derivation of, 473-477, 481-482 R-matrix method, 492-495 7'-matrix method, 490-492 Electron-nitrogen molecule scattering, 483 -485 Electron-oxygen ion recombination, and nightglow, 62, 235-237 Electron-photon angular correlations, amplitudes and state parameters from, 440-448 Electron promotion, 352 Energy loss spectra, for ion-atom charge transfer collisions, 221 -224 Energy-modified adiabatic approximation, 502-503 Excitation of atoms by ions, 266-274 of hydrogen atoms by protons, 271274 Excitation of atoms, by electron impact, 381 -419
534
SUBJECT INDEX
absolute cross section measurements, 401-415 Bethe approximation and, 403-415 coherent, 437-448 collisional transfer of, 385-388 comprehensive studies, 416-417 incoherent, 448-449 instrumental polarization and, 389-391 measurement techniques, 394-398 quasi-molecular approach, 452-455 resonance radiation, imprisonment of, 382-384 secondary effects, 382-391 time-resolved measurements, 398-400 Excitation, of atoms by fast ions, theory, 266-274 Born approximation for, 267-268 Cheshire-Sullivan model, 270-271 close-coupling approximation, 267-268 Glauber approximation, 269-270 pseudostate expansions, 270 second-order potential model, 269 Excitation, vibrational, by electron impact, 495 -503
F Fano-Lichten model, 352-353 Fano-Macek orientation vector, 444-448 Fast-beam injector, 297-299 Fluorescence yield, inner-shell ionization and, 349-351 Fluorine, electron affinity of, 5 Fluorine-hydrogen system, potential surface studies, 168-170 Franck -Condon overlap approximation and quanta1 reactive scattering, 179-180 Furnace target technique, 301-302 Fusion reactor atomic collision processes in, 293-326 fueling, fast-beam injectors for, 299 ion-hydrogen collisions, 300-317 G Germanium, L shell excitation, 343-344 Glauber approximation, for heavy-particle excitation, 269-270
Gold inner-shell ionization, 376 K shell ionization, 330-331 Ground-state reagents, molecular beam reactions. 183- 187
H Halogen-halogen system, studies with electronically excited reagents, 189- 190 Heavy particle, inner-shell ionization by, 336-345 Heavy-particle collisions effect on Auger spectra, 362-367 effect on X-ray spectra, 368-373 electron capture from atoms by fast ions, 274-285 excitation of atoms by ions, 266-274 impact parameter studies of, 358-360 Heavy particle-photon angular correlations, 449-455 Helium autoionizing states, 455-459 in electron-ion recombination, 252-254 excitatioddeexcitation data, 441 -446 excitation functions, 418-419 excitation transfer, 385-388 ion-atom charge transfer collisions, 2 10211 two-photon decay of metastable, 67 Helium afterglow, 248-250 Helium ion-hydrogen collisions, 306-309, 312 Helium-like ions emission lines in solar spectrum. 67 two-photon decay of metastable, 68-69 Helium-rare gas collisions, 220-222 Hydrogen Balmer lines of, 399, 415, 448-449 coefficients for charge transfer reactions, 52 collisional detachment from, 26-27 electron affinity of, 2 fluorescent photodissociation of, 63-64 Hydrogen atom charge transfer from multiply charged ions, 303-315 electron impact excitation, 446-448 electron impact ionization, 436-437
535
SUBJECT INDEX
ionization, 3 15-3 17 proton impact excitation, 271 -274 two-photon decay of metastable, 68 Hydrogen-deuterium exchange reaction, 172-173 Hydrogen-hydrogen exchange reaction, 173- 174 quantal scattering calculations, 179- 180 Hydrogen ion, negative, 2 fast beam studies, 325-326 Hydrogen ion-helium ion collision, 3 17321 Hydrogen ion-magnesium ion collisions, 32 1 Hydrogen molecules, in interstellar medium, 62 -67 Hydronium ion in electron-ion recombination, 241, 244245 in ion-ion recombination, 239-241 Hypersatellite X-ray, 362
I ICC term, see Intercontinuum coupling term Impulse approximation, 281 -283 Information-theoretic method, for reactive scattering, 180- 181 Infrared luminescence, in ionic reactions, 23 Inner-shell ionization, 329-376 alignment of atoms and, 332-335, 343345 by atomic ions, 335-361 direct ionization, 336-341 by electron capture, 341-342 by electrons, 329-335 Fano-Lichten model, 352-353 by heavier atomic ions, 345-361 by heavy particles, 336-345 measurement difficulties, 35 1-352 radiations following, 362-376 three-electron transitions, 366-367 two-electron radiative transitions, 362366 Intercontinuum coupling term and Auger electron spectra, 374 Internuclear vector, 264-266 Interstellar cloud CH formation in, 55-56
cooling in, 54-55 OH formation in, 48-50 hterstellar medium, molecular hydrogen in. 62-67 Ion, atomic, inner-shell ionization by, 335361 Ion, negative, see Negative Ion Ion- hydrogen collisions furnace target technique, 301-302 in fusion reactor research, 300-317 merged-beam technique, 302 - 303 modulated crossed-beam technique, 301 Ion-ion collisions, 317-321 Ion-ion recombination in ambient neutral gas, 255-259 computer-simulated experiments, 256-259 Langevin (-Harper) formula, 255 partial-parting concept, 257 quasi-equilibrium statistical method, 256 Thomson's theory, 255-256 Ion-ion recombination, binary with complex ions, 238-245 mutual neutralization. 62, 236, 238-239, 258-259 Ionization in fast heavy particle collisions, 286-288, see a1.w Collisional ionization. atom-atom; Inner-shell ionization differential cross sections, 287-288 of impurities in plasma, 296-297 total cross sections, 286-287 Ionization by electron impact angular correlation and, 425-437 interference effects in, 431-437 Ion-molecule reactions, 23-25, 42-45, 174 Ion pair production, in dissociation of H, and H : , 27 Iron, K shell ionization, 342-343, 364-366 Iron ion-hydrogen collisions, 311-313 IT method, see Information-theoretic method
J JWKB approximation for phase shift, 206
K Krypton ion-atom charge transfer collisions, 208-209
536
SUBJECT INDEX
K shell ionization, see also Inner-shell ionization universal curve for, 338-339, 345-346
L Landau-Herring method, 216 Landau-Zener-Stueckelberg approximation, 175-176 .andau-Zener two-state approximation in ion-atom collisions, 217-220, 253254, 305 .asers, uses in studies of photodetachment 2, 8, 20, 22 reactive scattering, 188 recombination, 249 ithium electron affinity of, 3-4 ion-atom charge transfer collisions, 2092 10 Lithium ion-hydrogen collisions, 308-312 Lz methods in electron -molecule collision theory, 488-495 LZS approximation, see Landau-ZenerStueckelberg approximation
M Magnetic dipole transitions, relativistic, 6769 Manganese, doubly ionized K shell, 362364 Many-body approximation, electron-photon angular correlations and, 442 Mercury, electron Impact excitation, 394398 Mercury-halogen system, crossed-beam chemiluminescence and, 183- 184 Mercury-iodine system, potential surface study, 170-171 Merged-beam technique, in fusion reactor research, 302-303 Molecular-beam chemistry, 181 -183 Multichannel eikonal approximation, electron-photon angular correlations and, 441 Multipole parameters, 447-448 Mutual neutralization, 62, 236, 238-239, 258-259
N Negative ions and discharge stability, 30-33 equilibrium in discharges, 28-30 Negative ion, atomic energy dependence of reaction rates, 25 excited states of, 7-9 first row ions, electron affinities of, 4-6 ground states of, 2-7 resonance states, 7-8 second row ions, electron affmities of, 6 states metastable toward autodetachment, 8-9 Negative ion, clustering, effect on reaction rates, 25 Negative ion, molecular, electron affinities and structures of, 9- 13 Negative ion reactions at high energies, 26-28 Negative ion reactions at thermal and epithermal energies, 23-25 with atomspheric trace constituents, 25 flowing afterglow studies, 23-25 infrared emission, 23 0- reaction rates, 25 Neon ion-neon collisions, inner-shell ionization and, 352-354, 360-361 Neon-krypton collisions, inner-shell ionization, 358 Nickel, K shell ionization, 329-331 Nightglow, 44-46, 62, 235-237 Nikitin formalism, 357-358 Nitrate ion, in ion-ion recombination, 23924 1 Nitrogen, atomic deactivation of metastable, 46 electron affinity of, 5 5199 A line of in airglow, 45-46 production of, from N, and NO, 14-15 Nitrogen ion-hydrogen collisions, 310, 313 Nitrogen oxides, molecular beam reactions, 184-185 Nonadiabatic collision, approximation methods for, 179-180
0
OBK approximation, see OppenheimerBrinkman-Kramers approximation
SUBJECT INDEX
Oppenheimer- Brinkman- Kramers approximation, 342 Orthopositrium. 126 vacuum lifetime, 137, 163-164 Orientation vector, 444-448 Oscillator strengths in astrophysics. 59-61 of OH, 59 Oxygen, atomic charge transfer with H+, 46-49 deactivation of metastable, 44-45 electron affinity of, 5 fine structure transition, in aeronomy and astrophysics, 53-55 5577 and 6300 A lines of, in airglow, 44-45 Oxygen ion, in electron-ion recombination, 235-237 Oxygen ion-hydrogen collision, 46-49.305, 3 10
537
Positron-atom scattering, theoretical, 101131 annihilation in, 118- 124 overview, 101 - 102 Positron beam, generation of, 138- 142 Positron emission, in heavy atom collisions, 373-374 Positron-helium scattering at low energy, 146- 147 inelastic collisions above the ionization threshold, 157- 159 theoretical, 105-118, 130-131 d-wave phase shifts, 113- I14 parameter calculation, 105- 107 positronium formation, see Positronium p-wave phase shifts, 11 1 - 113 s-wave phase shifts, 108- 1 I I system coordinates, 106 Positron-hydrogen scattering resonances in, 104- 105 theoretical, 102-105, 130- 131 Positronium P definition, 126 formation in inert gas, 153-154 Parapositrium. 126 formation in positron-helium collisions, Partial-parting concept in computer simu129-130 lated experiments on ionic recombinaformation in positron-hydrogen collition, 257-259 sions, 126- 129 Particle- photon angular correlations, 437 lifetime, 126 455 types, 126 PCI, see Postcollision interaction vacuum lifetime of ortho-positronium, Perturbed stationary state impact parameter 137, 163-164 method, 353, 356 Positron-krypton scattering, at low energy, Photodetachment, 18 -23 149-151 Photodissociation of negative ions, 18-23 Positron lifetime parameters Photon labeling in study of optical excitation for inert gas, 159-163 function, 417-4 18 for molecular gas. 163 Plasma diagnostics, 299-300 methods of study, 159 Plasma heating Positron-neon scattering, at low energy, energy and particle loss and, 295-297 147- I49 fast-beam injectors for, 297-299 Positron scattering, experimental, 135-159 Polarization, instrumental, 389-391 cross section data, accuracy, 142- 145 Polarization of impact radiation cross section measurement, 146- 159 angular correlations and, 449-452 intermediate energy, inert gases, 151 -153 Bethe approximation and, 403-415 in molecular gas, 155 - 157 near-threshold, 392 -394 positrium formation cross section in inert Polyatomic negative ions, 12-13 Positron annihilation gases, 153-154 Positron-xenon scattering, at low energy, angular correlations of y-rays, 124- 126 149-151 in atom scattering, 118- 124 Positron-argon scattering, at low energy, Postcollision interaction and Auger effect, 374-376 148- 149
538
SUBJECT INDEX
Potential energy surfaces ab inirio computations, 168-171 classical tragectories on, 171 -173 Proton, inner-shell ionization by, 336-338, 358-360 Proton-rare gas collisions, 224-228 Pseudostate expansions, for heavy-particle excitation, 270, 271 Pseudostates, 270 PSS impact parameter method, see Perturbed stationary state impact parameter method
Recombination, 235-262, see Electron-ion recombination; Ion -ion recombination Relativistic semiclassical approximation for inner-shell ionizations, 359-360 RER, see Radiative electron rearrangement R-matrix method, 492-495, 500 RSCA, see Relativistic semiclassical approximation Rydberg atom collisional mixing of, 78-91 thermal collisions with heavy particles, 77 -99
S
R Radiation, following inner-shell ionization, 362-376 Radiation, impact, Bethe approximation for polarization of, 403-415 Radiation, molecular orbital, 368-374 anisotropy of, 371-373 characteristics of, 369-371 Radiation, resonance, imprisonment of, 237, 382-385 Radiative electron rearrangement, 362 Reaction rate, transition-state theory and, 173- 175 Reactive molecular scattering, 167- 197, see also Potential energy surfaces; Transition state theory accurate quanta1 calculations, 179- 180 collisional ionization: nonadiabatic reactions, 175-178 crossed beam chemiluminescence, 183I87 with electronically excited reagents, 189190 information-theoretic approach to, 180181 with rotationally excited reagents, 190192 state-to-state cross sections, 187-188 translational excitation of reaction and, 192- 193 translational thresholds, 193- 197 use of laser, 188 vibrational enhancement, 190 REC,see Electron capture, radiative
SCA, see Semiclassical approximation Scattering, see Positron scattering; Reactive molecular scattering Second-order potential model, for heavyparticle excitations, 269-271 -272 Semiclassical approximation, for inner-shell ionization, 340-341, 359-360 Silicon ion-hydrogen collisions, 31 1 Silver, K shell ionization, 330-333 Spectroscopy, microwave in astrophysics, 56-58 Static gas measurements, of excitation cross sections, 401-402 Stueckelberg model, 180 Sulfur, double-electron radiative transitions in, 362-363 Surface hopping, 176-178 Symmetrical resonance charge transfer process, 206-21 1
T Time-of-flight positron beam system, 140142 Titanium ion-hydrogen collision, 314-315 T-matrix method, 490-492 Tokamak plasma device, 294, 295, 297 Trajectory methods, classical, in reactive scattering, 171-173 Trajectory surface-hopping method, 176178 Transitions, fine-structure, in aeronomy and astrophysics, 53-55
539
SUBJECT INDEX
Transition-state theory. 173- 175 unified statistical theory, 174-175 zero activation energy, treatment of, 174 Two-photon decay, 67-69
adiabatic-nuclei approximation and, 496497 hybrid theory and, 497-499
W U Uranium, inner-shell ionization, 359
V Vibrational excitation by electron impact, 495 -503
Wentzel theory, Auger effect and, 374
X X-radiation, quasi-molecular formation, 368 -373 anistropy of, 371 -373 X-ray spectra, heavy-particle collisions and, 368-373
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Contents of Previous Volumes
Volume 1 Molecular Orbital Theory of the Spin Properties of Conjugated Molecules, G. G. H d 1 rrird A . T. A r m s Electron Affinities of Atoms and Molecules, 6. L . Moisc+wifsc,lr Atomic Rearrangement Collisions, B . H . Brcrrisderi
The Production of Rotational and Vibrational Transitions in Encounters between Molecules. K . Trtkrrycrt~tr,yi The Study of Intermolecular Potentials with Molecular Beams at Thermal Energies, H . P[rrrly irncl J . P . Toer I I I ir s High Intensity and High Energy Molecular Beams, J . B. Aiiderson, R . P . Atidrcs, r r t i d J . B . Fvriir AUTHOR INDEX-SUBJECT INDEX
Volume 2 The Calculation of van der Waals Interactions, A . Dtrlgcirrio r r r i d W . D .
Mass Spectrometry of Free Radicals, S . N . Foricr AUTHOR INDEX-SUBJECTI N D E X
Volume 3 The Quanta1 Calculation of Photoionization Cross Sections, A . L . S t r i c w t Radiofrequency Spectroscopy of Stored Ions. I. Storage, H . G . Ddinielf Optical Pumping Methods in Atomic Spectroscopy, B . Birdid Energy Transfer in Organic Molecular Crystals: A Survey of Experiments, H . C. Wolf' Atomic and Molecular Scattering from Solid Surfaces, Rohivt E . S t i c h i r y Quantum Mechanics in Gas CrystalSurface van der Waals Scattering, F . Clirrrioc,h Bptier. Reactive Collisions between Gas and Surface Atoms, Henry W i s ~( / t i t 1 Borirrird J . Wood AUTHOR INDEX-SUBJECT I N D E X
Drr iii.\ori
Thermal Diffusion in Gases. E . A . Mirsorr. R . J . Mrrrn, (//id F1.tr17c.i.~ J. Sriiiih
Spectroscopy in the Vacuum Ultraviolet, W . R. s. Gurtorr The Measurement of the Photoionization Cross Sections of the Atomic Gases, J o t w J A . R . Srrrn.wri The Theory of Electron-Atom Collisions, R. P ~ l ~ r A o ri ipn d V . VPILJI.P Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, F . J . cir Hrrr. S41
Volume 4
H . S. W. Massey-A Sixtieth Birthday Tribute, E. H . 5'. Birrhop Electronic Eigenenergies of the Hydrogen Molecular Ion, D. R. Brites r r r i d R . H . G. Rcitl Applications of Quantum Theory to the Viscosity of Dilute Gases, R. A . Bric~hirrghtrinrind E. Gtrl Positrons and Positronium in Gases,
P. A .
Frciwr.
542
CONTENTS OF PREVIOUS VOLUMES
Classical Theory of Atomic Scattering, A . Biirp>.ss titid 1. C. Puci\wl Born Expansions,A. R . Hoit cind B . L. Moisc~ici,itsc~li
Resonances in Electron Scattering by Atoms and Molecules, P . G. BirrLr Relativistic Inner Shell Ionization, C. B. 0.Mohr Recent Measurements on Charge Transfer, J . 13. Htistrd Measurements of Electron Excitation Functions, D . W . 0 . Hrddle lrnd R . G. W . Keesirig Some New Experimental Methods in Collision Physics, R . F . Stehhings Atomic Collision Processes in Gaseous Nebulae, M . J . Scritoti Collisions in the lonosphere, A . L)(II,qNrIIO
The Direct Study of Ionization in Space, R . L . F . Boyd A U T H O R INDEX-SUBJECT INDEX
Volume 5
Relativistic Z Dependent Corrections to Aomic Energy Levels, Holly Thoniis Doyle AUTHORINDEX-SUBJECTINDEX
Volume 6 Dissociative Recombination, 1. N . Bnrdslry m d M . A . Biotidi Analysis of the Velocity Field in Plasma from the Doppler Broadening of Spectral Emission Lines, A. S. Kti irJmuti The Rotational Excitation of Molecules by Slow Electrons, Kaziio Ttihtrvrrm g i and YuXikuz~~ Ifikariw The Diffusion of Atoms and Molecules, E . A . M u w r i find T. R . Mcirrmi Theory and Application of Sturmain Functions, M a t i i r ~ i lR o t e n h ~ r g Use of Classical Mechanics in the Treatment of Collisions between Massive Systems, D . R . BLite.5 ( i d A . E . Kiiigston AUTHORINDEX-SUBJECTI N D ~ X
Flowing Afterglow Measurements of Ion-Neutral Reactions, E . E . F c ~ g i r Volume 7 . s o i l , F. C. Fdi.wiifi4d. Lind A. L. Sdiinrltrhopf Physics of the Hydrogen Master, Experiments with Merging Beams, C‘. Aiiiloit/. .I. P . S c / i o r i ~ i t i t i t i , ( i t i d Roy H . Nevritrher P . Grirvt Radiofrequency Spectroscopy of Stored Molecular Wave Functions: CalculaIons 11: Spectroscopy, H . G . Delimclt tion and Use in Atomic and Molecular Processes, J . C. B r o w e The Spectra of Molecular Solids, 0. S C I I I I Ppp Localized Molecular Orbitals, Hurl)/ Weiizstrin, Ridhen Ptrrrncz, titit1 The Meaning of Collision Broadening Miiiiricr Cohcn of Spectral Lines: The ClassicalOscillator Analog, A . Brti-Rriiwn General Theory of Spin-Coupled Wave Functions for Atoms and Molecules, ,The Calculation of Atomic Transition J . Gerratr Probabilities, R. J . S . Crossley Tables of One- and Two-Particle Coeffi- Diabatic States of Molecules-Quasistationary Electronic States, Tliorntr~ cients of Fractional Parentage for F . O’MLiilry Configurations . s ? ~ ’ ~ pC.~ , D. H . Chisholnz. A . Daigrirno. niicl F . R . Selection Rules within Atomic Shells, Il1t2P.s B . R . Jiitltl
CONTENTS O F PREVIOUS VOLUMES
Green‘s Function Technique in Atomic and Molecular Physics, Gy. f s c i t i d . H . S . Tuylor. r r i i t l Rohtw Ytrris A Review of Pseudo-Potentials with Emphasis on Their Application to Liquid Metals, Nnthrrii Wi.\rr triirl A . .I. G r ~ ~ i i , f i e l d A u T HO R 1N D E X -S u BJ ECT 1N I) E X Volume 8
Interstellar Molecules: Their Formation and Destruction, L). McNol!\ Monte Carlo Trajectory Calculations of Atomic and Molecular Excitation in Thermal Systems. JciniPs C. K c d Nonrelativistic Off-Shell Two-Body Coulomb Amplitudes, J o s ~ p I iC. Y . Chrii ciiitl AirpistiiiP C . Chrn Photoionization with Molecular Beams, Htirrisuii, triitl R. B. Coirtis, H~rl.vti~(ril K . I. J”~h0PII The Auger Effect, E . H . S.Birrhop t i i i t l W. N . A.strritl AUTHOR INDEX-SUBJECTINDEX Volume 9
Correlation in Excited States of Atoms, A . W. W d s s The Calculation of Electron-Atom Excitation Cross Sections, M . R . H . Riidgr Collision-Induced Transitions Between Rotational Levels, Trrl;rshi (3kri The Differential Cross Section of Low Energy Electron-Atom Collisions, D . Aticfrii,X Molecular Beam Electric Resonance Spectroscopy. J t w s C. Zorii toid l l r o i ~ i ( i . sC.
Eiiglish
Atomic and Molecular Processes in the Martian Atmosphere, Mic,/itrrl B . McEIro?. AUTHORINDEX-SUBJECTI N D E X
543
Volume 10
Relativistic Effects in the Many-Electron Atom, Lloyd A r n i s t r o i ~ ~Jyr.. rititl Srrgr Ft2iicwillcj
The First Born Approximation, K . L . Rtill cind A . E. Kitigstoii Photoelectron Spectroscopy, W. C. Priw
Dye Lasers in Atomic Spectroscopy, W . LfItigP. J . LiiIIitfF, tJtlll A . St;,iid[,l Recent Progress in the Classification of the Spectra of Highly Ionized Atoms, u. Fcr nYet t A Review of Jovian Ionospheric Chemistry, Weslay T . Hiiiitrcxs , .Is. SUBJECT INDEX
c.
Volume 11
The Theory of Collisions Between Charged Particles and Highly Excited Atoms, 1. C. Prrciiwl trritl D. Ric Ii(/rils Electron Impact Excitation of Positive Ions, M . .I. &’fif/Jtl The R-Matrix Theory of Atomic ProceFs. P . C. BiirXo ciiid W . I). R(JI>II Role of Energy in Reactive Molecular Scattering: An Information-Theoretic Appsoach, R . B. Beriistc.iii r r i i d K . n. L t l i t 1 6’ Inner Shell Ionization by Incident Nuclei, Jolirrtiiie.~M . H ~ r t i ~ t c ~ ~ r r Stark Broadening, Htiiis R . Grietii Chemiluminescence in Gases, M . F . Goldr cind U . A . Tlirr~.th Au I H O R INDEX-SUBJLCT INDEX Volume 12
Nonadiabatic Transitions between Ionic and Covalent States, R. K . J ~ i w i . Recent Progress in the Theory of Atomic Isotope Shift, J . Batit,/io c i i i d K.-J. C l i ~ i ~ l / ~ t i / /
544
CONTENTS OF PREVIOUS VOLUMES
Topics on Multiphoton Processes in Atoms, P . Lirmbropoiilos Optical Pumping of Molecules, M. Bi-oycv-, G. Gorietlrrrd. J . C'. Lehr~crni,r i n d J . Vigiic Highly Ionized Ions, hwn A . SdIin Time-of-Flight Scattering Spectroscopy, Wilhc4tn Rciith Ion Chemistry in the D Region, Gtwrge C. R t i d AUTHOR INDEX-SUBJECT INDEX
Volume 13 Atomic and Molecular Polarizabilities - A Review of Recent Advances, Tliomris M . Miller rind Benjamin Bedcrson
Study of Collisions by Laser Spectroscopy, Paiil R . Biwncin Collision Experiments with Laser Excited Atoms in Crossed Beams, I . V . Hrrtcl rind W . Stoll Scattering Studies of Rotational and Vibrational Excitation of Molecules, Mtrnfird Fnirbi.1 tirid J . Peter Tornflies Low-Energy Electron Scattering by Complex Atoms: Theory and Calculations, R . K . Neshrt
Microwave Transitions of Interstellar Atoms and Molecules, W . B . Soinrrvillr AUTHOR INDEX-SUBJECT I N D E X Volume 14
Resonances in Electron Atom and Molecule Scattering, D. E . Goltlrn The Accurate Calculation of Atomic Properties by Numerical Methods, Brimn C. W e h t n , MiLhrrd J . Jrrniieson, und Roniild F . Strwtrrt (e, 2e) Collisions, Erich Wc~igoldrind Icrn E. McCarthy Forbidden Transitions in One- and Two-Electron Atoms, Riclitrrd Murriis irrrd PiJter.I. Mohr Semiclassical Effects in Heavy-Particle Collisions, M . S. Child Atomic Physics Tests of the Basic Concepts in Quantum Mechanics, Frtrt7c~i.sM . PipXin Quasi-Molecular Interference Effects in Ion-Atom Collisions, S . V . Bo has he I! Rydberg Atoms, S. A . Edelstein and T . F . Grilltrghrr UV and X-Ray Spectroscopy in Astrophysics, A . K . Dicpriv AUTHOR INDEX-SUBJECT INDEX