Advances in Atomic and Molecular Physics Volume 15

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, NEW YORK

VOLUME 15

@ 1979

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COPYRIGHT @ 1979, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF 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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79 80 81 82 9 8 7 6 5 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

This Page Intentionally Left Blank

Contents

ILISI OF C O N T R I B U l O R S SIR HARRIE 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

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. Dissociative Attachment

VI. Ionic Reactions at Thermal and Epithermal Energies V . Photodetachment and Photodissociation

VII. Ionic Reactions at High Energies VIII. Negative Ions in Electric Discharge and Breakdown Phenomena

References

Atomic Physics from Atmospheric and Astrophysical Studies

A . L k i l g r r n i o

I . Introduction 11. Dissociative Recombination

111. Ion-Molecule Reactions 1V. Neutral-Particle Reactions V . Accidental Resonance Charge Transfer

VI. Charge Transfer of Multiply Charged Ions VI I . Fine-Structure Transitions

VII I . Radiative Association IX. Microwave Spectroscopy X . Oscillator Strengths and Branching Ratios

XI. Radiative Recombination X11. Spontaneous Radiative Dissociation of Diatomic Molecules

XIII. Relativistic Magnetic Dipole Transitions References

vii

... X l l l

x v

2 7 9

13 18 23 26 28 33

37 38 42 44 46 50 53 55 56 59 62 62 67 69

... V l l l CONTENTS

Collisions of Highly Excited Atoms

R . F . Stehhings

I. lntroduction

References 11. Thermal Collisions with Heavy Particles

Theoretical Aspects of Positron Collisions in Gases

J . W . Hirmherston

1. Introduction 11. Positron-Hydrogen Elastic Scattering

111. Positron-Helium Elastic Scattering IV. Annihilation in Positron-Atom Scattering

V1. Positronium Formation in Positron- Atom Collisions V. Angular Correlations in Positron Annihilation

VII. Concluding Remarks References

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

Reactive Scattering: Recent Advances in Theory and Experiment

Richard B. Bernstein

1. Introduction 11. Potential-Energy Surfaces

111. Classical Trajectory Methods and Results IV. Transition-State Theory: New Developments V. Collisional Ionization: Nonadiabatic Reactions

77 77 99

101 102 105 1 I8 124 126 130 I3 1

135 138 142 146 159 164

167 168 171 173 175

CONTENTS ix

VI. Accurate Quanta1 Scattering Calculations VII. Information-Theoretic Approach to Reactive Scattering

V I I l . Molecular-Beam Chemistry IX. Crossed-Beam Chemiluminescence

XI. Influence of Different Forrns of Energy upon Reactivity X. State-to-State Cross Sections

XII. Translational Thresholds References

Ion- Atom Charge Transfer Collisions at Low Energies

. I . B . Hri s t r d

I . Introduction 11. Symmetrical Resonance Processes

Ill. Nonresonant Atomic Chatge Transfer Processes IV . ‘Total Cross Sections of Pseudocrossing Atomic Charge

Transfer Processes V. Curve-Crossing Spectroscopy

VI . Charge Transfer Processes of Excited Ions References

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

References V. Recombination in an Ambient Neutral Gas

The Theory of Fast Heavy Particle Collisions

B . H. Brotisclcti

I. Introduction 11. Excitation of Atoms by Ions

111. Electron Capture from Atoms by Fast Ions IV. Ionization and Charge Exchange into the Continuum

References

179 i80 181 183 187 189 193 198

205 206 21 1

214 22 1 229 23 I

235 235 238 245 250 259

263 266 214 286 288

X CONTENTS

Atomic Collision Processes in Controlled Thermonuclear Fusion Research

H . B . Cilhody

I . Introduction 293 11. Classification of Relevant Heavy-Particle Collision Processes 295

111. Experimental Studies 300 References 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

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

329 329 335 362 377

38 1 382 391 394 398 40 I 415 419

Coherence and Correlation in Atomic Collisions

H . Kleinpoppon

I . Introduction 423

Impact Ionization 425

455

11. Angular Correlation and Spin Experiments as Tools for Studying

111. Particle-Photon Angular Correlations 437 1V. Electron-Ion Angular Correlations from Autoionizing States

CONTENTS xi

V. Summary and Conclusions 460 Appendix: Coherent Excitation of Degenerate States with

Different Angular Momenta 462 References 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


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 Univer- sity, 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, North- ern Ireland, and Science Research Council, Daresbury Laboratory, Daresbury Warington, England (471 )

A. DALGARNO, Harvard-Smithsonian Center for Astrophysics, Cam- bridge, 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, Uni- versity of London, Egham, Surrey, England (381)

J. W. HUMBERSTON, Department of Physics and Astronomy, Univer- sity 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,

H. S. W. MASSEY, Department of Physics, University College London,

R. F. STEBBINGS, Department of Space Physics and Astronomy, Rice

Bielefeld, West Germany (423)

London WClE 6BT, England (1)

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-more- parent.

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 prevent- ing 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 geo- physicist 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 i t the wrong way the

x v

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~zncvs in 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 the mutually beneficial amalgamation of the Physics and Astronomy Depart- ments; 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 Re- search.

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 “Elec- tronic 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 fast- growing 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 ~ ~ J c , . R. Soc. Londori. S e r . A 130, 655. Arnot, F. L. (1933). “Collision Process in Gases,” p. 59. Methuen, London. Bullard, E. C. , and Massey, H. S. W. (1931). Proc. R. S i r . Lond(~/ i . Ser. A 130, 579. Burhop, E. H. S. (1968). Adv. At . Mol. P h y s . 4, I . Massey, H. S. W. (1938). “Negative Ions,” 1st ed. Cambridge Univ. Press, London and

Massey, H. S. W. (1969). “Electronic and Ionic Impact Phenomena,” Vol. 2. Oxford Univ.

Massey. H. S. W. (1971). “Electronic and Ionic Impact Phenomena,” Vol. 2. Oxford

Massey, H . S. W. (1976). ”Negative Ions,’‘ 3rd ed. Cambridge Univ. Press, London and

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,”

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.

New York.

Press (Clarendon), London and New York.

Univ. Press (Clarendon), London and New York.

New York.

Vol. 4. Oxford Univ. Press (Clarendon), London and New York.

Atl\*nticc.v in ATOMIC AND MOLECULAR PHYSICS

VOLUME I5


ADVANCES IN ATOMIC AND 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 Ions . . . . . . . . . . . . . . . . . . . . A. Determination of Atomic Electron Affinities . . . . . . . . . B. Calculations for Li- . . . . . . . . C. B, C, N, 0, and FAtorns . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. AI ,S i ,P ,S , and CI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. New Methods for Calculating Electron Affinities . . . . . . . . . . . . . . . . .

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

B. States Metastable toward Aut 111. Electron Affinities and Structures s . . . . . . . .

. . . . . . . . . . B. Polyatomic Ions

A. Measurements of Total Dissociation Cross Sections . . . . . . . . . . . . . .

C. Attachment to H N 0 3 . . .

E. Effect of Vibrational Ex sociative Detachment-

F. SF6 ......................................................... V. Photodetachment and Photodissociation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A. Measurement of Photodissociation Cross Sections. . . . . . . . B. Experimental Results-Photodetachment from NO- . . . . . . . . . . . . .

D. Angular Distributions of ......................

H, and . . . . . . . . . . . . . . . . . . . . . . .

c . c0,-. . . . . . . . . . . . . . . . . . . . VI. lonic Reactions at Th

A. Observations of Infrared Luminescence in Ionic Reactions . . . . . . . .

C. Energy Dependence of 0- Reaction Rates. . . . . . . . . . . . . . . . . . . . . . . , D. Negative-Ion Reactions with Atmospheric Trace Constituents . . . . .

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

B. The Reactions 0: + CO, COT + 0, . . . . . . . . . .

VII. Ionic Reactions at High Energies . . . . . . . . . . . . . . . . . . . . . . .

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 ATOMIC ELECTRON AFFINITIES

During the last few years the accurate measurement of the electron affinities of atoms has become possible through the availability of high- resolution 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

NEGATIVE IONS 3

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 J l a , +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 though Gaussian functions either of the form, P"Wrz or e--p(r-rl)z are also 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 Lr-

As 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

4 H . S. W . Massey

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. Neverthe- less 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; uA ranged 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

NEGATIVE IONS 5

TABLE I

CORRELATION ENERGY IN ATOMIC UNITS, WITH SIGN REVERSED, FOR ATOMS OF

THE FIRST R o w OF T H E PERIODIC TABLE ( E X C L U D I N G Li)"

K shell KL intershell L shell

SDb SD TQb SDTQb SD TQ SDTQ

B- B C- C N - N 0- 0 F- F

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

a 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.

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

ELECTRON A F F I N I T I E S ( I N eV) OF B . C, N , 0, A N D F

B C N 0 F ~~~~~ ~ ~ ~

Calculated Sasaki and Yoshimine (1974b) 0.15 1.11 -0.52 1.13 3.12 Weiss (1971). Marchetti er a / . (1972) - - - 1.47 3.47 Moser and Nesbet (1971) 0.223 1.29 -0.12 1.43 3.37 Staemmler and Jungen (1975) 0.243 1.322 -0.299 1.505 3.621

Observed Hotop and Lineberger (1975) 0.28 1.25 -0.07" 1.465 3.448

~~~

a See Section IV.

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 Yo- shimine (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 Yoshi- mine 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, A N D 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 quad- ratic extrapolation from isoelectronic sequences.

TABLE 111

ELECTRON AFFINITIES (eV) FOR SECOND Row ATOMS ~ ~~

Al Si P S CI

Calculated

Observed Moser and Nesbet (1975) 0.49 1.52 0.74 2.18 3.79

Hotop and Lineberger (1975) - I .39 0.743 2.0772 3.715

NEGATIVE IONS 7

E. NEW METHODS FOR 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 al . , 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 r r l . , 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. RESONANCE STATES

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. STATES METASTABLE TOWARD 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 Senas- kenko (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 long- lived *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 9

HeZ+ + H, -+ He(2'S) + H+ + H+ (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 ~ ) ~ P + H: (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 , l lu, , 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 Fz very 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 I 1

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?r21 of the molecule and f is between 0.05 and 0.10.

Turning now to excited states of diatomic ions, strong experimental evidence of a state of H; with a lifetime of at least sec has been obtained 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 D2 in the duoplas- motron 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. Obser- vations 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 ~ ~ 8 , from 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.

12 H . S. W. Massey

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 hexa- fluorides 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/Nz flames 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 (Dun- ning, 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 TOTAL DISSOCIATION 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

14 H . S. W. Massey

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 cm2 at zero energy, declines rapidly to 8 X cm2 as 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 AND 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 14N15N as the source gas and found peaks of equal height at mass numbers 14 and 15. The appearance energy for the ions from N2 was found to be close to 16 eV as compared with 8-9 eV for the 0- ions resulting from an N2-02 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)

and from NO via (9)

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

NEGATIVE IONS 15

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 trapped- electron 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 disap- peared 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

N2 + 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; (12)

is very fast, with a rate constant of 5 2 3 x 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. EFFECT OF VIBRATIONAL EXCITATION O N ASSOCIATIVE DETACHMENT-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

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 high- energy 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 Nikoto- poulou 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

cm2 for H, and 8 x cm2 for D,).

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

(13) SF, + e - SF; + F.

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 34SF6 isotope 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 high- temperature 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 2 laser 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

NEGATIVE IONS 19

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 lo5 ions per second. If I and I , are the numbers of ions counted with the laser beam respectively on and off, the photodestruction cross section QPd is 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 Qpd is known. Thus if measurements for 0- are distinguished by the suffix s,

where P denotes the laser power output and ions.

P v ps us --

u the speed of the drifting

In general QPd includes 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 suf- fered 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.


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 O2 with a trace of N2 contained 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 NO, from observations at the lower photon energy, 1.4 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.

by 0, and 5.9 x and 9.7 x

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 mass- analyzing 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


Thus with E,(C03) = 2.69 eV, E,(O) = 1.462 eV, D(C02-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(C02-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 predisso- ciating 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 os- cillation 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 OF I N F R A R E D LUMINESCENCE IN IONIC

REACTIONS

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 REACTIONS 0; + 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 O2 and a small quantity of 0, into the helium buffer gas at a point past the ion source. 0-and 0; were produced by attachment of slow electrons and these were rapidly converted to 0; by charge transfer reactions with 03. As the 0; produced 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 03. 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(02-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 variable- temperature 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 03, respectively, T AS at 600 K = - 0.66 kcal mole-', so (17) gives

D(C02-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-

NEGATIVE 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. ENERGY DEPENDENCE OF 0- REACTION RATES

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 0- with 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 TRACE CONSTITUENTS

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 ter-

minated. (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 H20 from 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 dif- ferently. 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 = 8 ~ a ~ ( a z / / P z ) f d (22) 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 " << 1. For elastic collisions the cross sections Qw are also given by (22), with Jd replaced by f w , which equals 0.125 -+ 0.003 and 0.262 5 0.005 for H and He, respectively. More complicated expressions are found for em, the total cross sections for inelastic collisions in which the H- remains unaffected. Com- parison with observed total cross sections (Rose et a f . , 1958: Berkner et al., 1964; T. D. Hayward and J . Tesmer, 1977, private communication to G. H. Gillespie; Smythe and Toevs, 1965; Dimov and Dudnikov, 1966), at impact energies greater than 200 keV shows good agreement with all observations for He and with the first three experiments referred to for H (derived from H,). The remaining two give considerably lower cross sections.

At somewhat lower but still quite high energies the semiclassical approximation of Bates and Walker (1967) is usually regarded as providing the best description. This approximation has been extended to detachment from He-(4P) by Snyder (1973). Heinemeier er al. (1976) have recently carried out measurements by the single-growth method of the single- and double-electron loss cross sections for H- and He- passing through H,, He, N,, Ne, and Ar, the impact energies being 50-500 keV for H- and 100-3000 keV for He-. Their results agree with those calculated by the Bates- Walker method to better than 40%, the agreement being especially good for He- in Hz, N P , and Ar. The observed and predicted variation with ion energy agrees very well, but for H- the observed cross sections are all about 30% lower than predicted.

Total cross sections for detachment in which the resulting H atom is left in an excited state have been measured recently by Harnois et al. (1975, 1977) and by Risley et a f . (1978).

NEGATIVE IONS 27

Esaulov et al. (1978) have measured energy loss spectra for the H atoms projected at different scattering angles in detachment collisions of H- with He, Ne, and Ar in the energy range 0.08-2.0 keV. From these observations differential cross sections were obtained as a function of angle of projection for collisions in which the H atom is produced in the ground state and in excited states. Differential cross sections for elastic scattering of H- were also obtained.

Peart et al. (1976a) have determined cross sections for detachment from H- atoms by proton impact, in the c.m. energy range 1.49 to 35.2 keV. Using an inclined beam apparatus they measured the total cross section for production of H atoms in H+-H- collisions. These result either from detachment or mutual neutralization (H+ + H- + H + H). Subtracting cross sections observed for the latter process (Moseley et af., 1975b; Peart er ul., 1976b) the required detachment cross sections are obtained. Some typical values of the detachment and mutual neutralization cross sections are given in Table IV.

B . ION PAIR PRODUCTION I N DISSOCIATION OF H 2 A N D H:

The production of H+-H- ion pairs through dissociation by impact with rare gas atoms of H2 and H i of 6 keV energy has been observed by coincidence counting methods by Brouillard et at. (H,. 1975) and by Oliver rr af. (HZ, 1976). Measurements of cross sections for ion pair formation and for total negative-ion production (from H:) have been made as well as of the energy and angular distribution of the ffagment ions. Total pair production cross sections of order to 10-19 cm2 are observed for H, and considerably smaller for H:.

C. COLLISIONS INVOLVING OTHER NEGATIVE IONS

Fayeton er ul. (1978) have measured differential cross sections for collisions between CI- ions (of energy between 80 and 2000 eV) and rare gas

TABLE 1V

CROSS SECTIONS FOR DETACHMENT (Qd) A N D M U T U A L NEUTRALIZATION (em) I N

H+-H- COLLISIONS"

Kinetic energies in c .m. system (kV) 1.49 2.51 4.95 6.31 9.22 15.6 22.46 31.5 35.2

Q d ( cm2) 0.45 2.06 3.59 4.09 4.37 3.70 3.36 2.87 2.73 Q," cmZ) 8.23 5.50 2.79 2.19 1.46 0.88 0.62 0.44 0.39

Observed by Peart et al. (1976a.b).

28 H . S . W. Mussey

atoms leading to detachment and other inelastic processes. Relative differential cross sections for detachment from C1- and 0- with energies between 20 and 200 eV have been measured by Doverspike et ul. (1977), while Herbst et al. (1977) have studied detachment from OH- with energy below 100 eV, in both cases in collision with rare gas atoms.

In recent years two specially interesting charge transfer experiments have been carried out. Mathis and Snow ( 1974) used a crossed-beam apparatus to measure cross sections for charge transfer collisions between 0- ions with energies between 10 and lo4 eV and both ground-state and metastable excited ('Ag) 0, molecules. Serenkov e l a/. (1975) observed the formation of C - ions in both the ground (4S) state and in the excited (2D) state through collisions of C+ ions in Na, Mg, NP, and 02. The fraction of 2D ions produced was determined by detaching electrons from these weakly bound ions in an electric field (Oparin et al., 1970).

VIII. Negative Ions in Electric Discharge and Breakdown Phenomena

It has long been believed that the presence of negative ions in an electric discharge often leads to instabilities but, until the last few years, there has been little progress in developing an adequate theory of such effects. This has been partly due to concentration on discharges in which electron loss is dominated by ambipolar diffusion rather than to the simpler cases in which it occurs primarily in the gas phase. However, the development of the COz gas laser has drawn urgent attention to the problem, which in this case is one of the simple type. Haas (1973) developed the theory of instabilities for this case while Nighan and Wiegand (1974) have extended the work in a very interesting and thorough paper in which they concentrate on the physical interpretation of the phenomena as well as reporting the results of confirmatory experiments.

A. EQUILIBRIUM CONDITIONS I N DISCHARGES CONTAINING NEGATIVE IONS

In order that electron loss by transport processes should be unimportant the minimum characteristic dimensions of the discharge must be of order several centimeters and the operating pressure greater than 10 torr. The plasma is supposed to be weakly ionized so that electron-electron and electron-ion collisions are unimportant. Because the discharge behavior depends on collision processes in the gas phase a great number of possibilities arise. For this reason, to avoid unnecessary complications

NEGATIVE IONS 29

Nighan and Wiegand considered specifically discharges in a mixture of helium and nitrogen containing a small fraction of CO, and occasionally 0, and/or CO. Under these conditions the main gases determine the electron energy distribution for each value, F / p , of the ratio of applied electric field F to gas pressurep and, moreover, these distributions, which are far from Maxwellian, are known. The high rate of energy loss by electrons in nitrogen due to vibrational relaxation ensures that the energy loss collision frequency is high.

In the absence of CO,, O,, and CO the discharge will operate at an electron temperature T , , determined by F / p , such that the rate of production of electrons by ionization balances the rate of loss by recombination, i.e.,

kinnp = an,"

where k i , a are the ionization and recombination coefficients, respectively, n and n , the respective concentrations of neutral molecules and of electrons. ki in general is a rapidly increasing function of T , , while a varies quite slowly. Equation (23) will be satisfied when T , corresponds to an energy of 1 eV or less.

With CO, present at a concentration n(CO,), 0- ions are produced by dissociative attachment,

CO, + e-- , CO + 0- (24)

at a rate k,n(CO,)n,, where k , is the attachment coefficient. 0- ions will, in turn, be destroyed by the inverse reaction

0- + C 0 - t CO, + e (25)

which will be important if the CO concentration n ( C 0 ) is high enough. Alternatively, the 0- ions may combine with CO, to produce COT in three-body collisions

co, + co, + 0- - co; + co, (26)

The importance of this reaction is that COT is stable toward detachment in collisions with the neutral constituents.

If n ( C 0 ) is kept small by working with a flowing gas so that the CO produced by (24) does not build up, then even with quite a small fractional concentration of CO,. at any time almost all of the electrons will be attached in negative ions. Equation (23) will then be replaced by

kin(CO2)ne = a n : + k,n(CO,)n, (27)

where k , is now the ionization coefficient for CO,, which is so much larger than for N, and He that it will alone be of importance under most operating conditions. The discharge will now operate at such an electron tempera-

30 H . s. w. Mnssry

ture that k i = k , because under most circumstances an , << k,n(CO,). This temperature will, in general, be higher than for the pure He-N2 mixture.

The discharge may be operated under conditions intermediate between the two extreme cases we have discussed by arranging for a suitable fraction of detaching gas such as CO to be present. Thus in a discharge in an He-N,-C0,-CO mixture the equilibrium electron temperature will depend strongly on the ratiof = n(CO)/n(CO,) + n(C0).

For a typical C0,-N,-He mixture in proportional concentration 0.05 : 0.35: 0.60, for example, at a pressure 20 torr and with n , = lo9 c m 3 , f must be greater than 0.1 in order that detachment through (25) should predominate over the cluster-forming reaction (26). Forf > 0.1 the equilibrium value of T , remains practically constant at 0.8 eV but for smaller values it rises gradually to the value 1.15 eV atf = 0 characteristic of an attachment-dominated discharge. If the cluster reaction did not occur T , would remain constant forf > 3 x

If the CO, is replaced by 0, in the same relative concentration the cluster reaction is replaced by

0- + 0 2 + 0, + 0; + o* (28)

which has a much lower rate coefficient so that detachment dominates for values off > f being now n(CO)/n(CO) + n(0,). T, at equilibrium is therefore constant and close to 0.8 eV forf > lo+. At smaller values it rises gradually to a value close to 1.5 eV atf = 0. This is higher than for CO, because the rate of attachment to 0, under the assumed conditions is faster .

These conclusions are in good agreement with observed data taken under conditions in which the concentration of CO does not build up through reaction (24).

While it is clear that by varying the CO fractionfa range of equilibrium values for T , and for the electron-negative-ion ratio may be attained in principle. we shall see that not all these modes of operation are stable.

B. STABILITY OF THE DISCHARGE

To discuss the stability of a discharge it is necessary to consider the orders of magnitude of the different relaxation times concerned. That for the space charge is of order 10-lo-lO-g sec, which is much shorter than those for changes in the electron and ion concentrations (nki)-l, ( n k , ) - l ,

etc., which are of the order 10-8-10-4 sec. On the other hand, the concentrations of the neutral species are effectively frozen over all of these time intervals. Finally for electron energy relaxation the time 7e is of order v i l ,

NEGATIVE IONS 3 1

where v E is the energy loss collision frequency. Since for the mixture containing a high proportion of N,, vE is of order lo8- 1 O ' O sec-', T~ is also short compared to the relaxation times for the electron and ion concentrations.

Haas (1973) then showed that, at times still short comparable with the latter, a relation may be obtained between the change AT, in T, and that An, in n, for a disturbance propagating in the plasma in a direction making an angle C#I with the direction of the steady-state curent flow. This is given by

AT, 2 C O S ~ C # I An, I, fik n,

- -- - -

where the caret denotes logarithmic differentiation with respect to T,, i.e.,

a In uE tE = ___ a In T , (30)

PE is given by 1 + vE vM cos 24 where vM is the momentum loss collision frequencies. For almost all practical circumstances 6; > 0 , and so it follows from (29) that an increase in T , is associated with a decrease of n, and vice versa.

The stability or otherwise of the discharge then depends on whether the later changes in the charged particle concentrations due to an increase in T , tend to increase or decrease n , . If they increase n , , they will damp down the decrease, which has already arisen due to the increase of T, , and the discharge will be stable. On the other hand, if they decrease n , the decrease of n , already due to increase of T , will be enlarged and unstable conditions ensue.

For a discharge in which attachment is unimportant, increase of T, will always increase n , through the increase of k i and the discharge will be stable. Again, in the other extreme when attachment is the dominant loss mechanism, increase of T , can only increase n , if k , > k^, , which is never valid under practical conditions. Again the discharge is stable but, at certain intermediate modes of operation, instability arises.

DetaiIed analysis (see Smith and Thomson, 1978) shows that, if cluster formation is ignored, this will occur if

-2 COSZf$ (I - %) n k i l i - (,,a + hnkd + hn,q + - I nk, ) > 0 (31) fi; A

Here ai is the ion-ion recombination coefficient and X the ratio n - / n , of the negative-ion concentration to that of the electrons.

Since all the quantities in the fast term of (31) are >O and GE > 0 a net-

32 H . S. W. Mussey

essary condition for instability is that kaia/klii > 1. For the CO,-N,-HE mixture discussed above, this condition is satisfied for T , > 1.06 eV and for the corresponding mixture with 0, replacing CO,, for T , > 1.22 eV. However this is not a sufjicient condition. In addition the quantities in the last term of (3 I ) must be small enough. Numerical evaluation shows that in the C0,-N,-He mixture considered above, with the CO fraction f adjusted to give different values of A, there exists an unstable region in the A, n , plane that, for n , > lo9, is approximately triangular with a base on the A axis between 0.2 and 5 and vertex at A = 0.6, n, = 1O'O cmP3. This region can only be reached withf > 4 x and a 2 X lo-*. A somewhat 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

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 photo- multipliers 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 offmin for the mixtures of CO, and of 0, with N,-He already referred to above. For the 0, mixturefmln was close to

6 3 x 10-4.

NEGATIVE IONS 3 3

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 collision- dominated discharges. We have confined the discussion above to self- sustaining 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.

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ADVANCES IN ATOMIC A N D MOLECULAR PHYSICS. VOL. 15 / I

ATOMIC PHYSICS FROM ATMOSPHERIC AND ASTROPHYSICAL STUDIES

A . DALGARNO Hurvard-Smithsonian Center for Astrophysic.! Cumbridge. Massachuserts

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Dissociative Recornbinatio . . . . . . . . . . . . . . . . . . 38

111. Ion-Molecule Reactions . . . . . . . . . . . . . . . . . . . . . . . . . 42 IV. Neutral-Particle Reacti V. Accidental Resonance

VI. Charge Transfer of Multiply Charged Ions .......................... 50 VII. Fine-Structure Transiti

VIII. Radiative Association . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 IX. Microwave Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 X. Oscillator Strengths and Branching Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . 59

XI. Radiative Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 XII. Spontaneous Radiative Dissociation of Diatomic Molecule

XIII. Relativistic Magnetic Dipole Transitions.. . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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 H e + t e + e + H e + 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

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-

cm3 sec-I.

ATMOSPHERIC A N D ASTROPHYSICAL S T U D I E S 39

400.

360.

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 popu- lating excited vibrational levels, is also consistent with the coefficient measured by Walls and Dunn (D. G . Torr et al., 1976a).

- E 3201 x

2 280

k w l c

240/

+ x .

+ X.

+ X .

+ Y + K

t X.

m + m + u

U + .X

o(

+ . X .X + .X

U

1 I I I , 1 1 1 1 1 I I I I 1 , , , I I I 1 1 1 1 1 1 l 1 I , 1 1 1 ,

(02 lo3 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). (Oppen- heimer 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. Ton et al., 1976b; Op- penheimer 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%. Vi- brationally excited 0: ions are probably quenched efficiently in the atmosphere by an atom interchange mechanism (Bates, 1955a)

O$(V) + 0 4 O$(IJ' < IJ) + 0 and it is unlikely that the atmospheric and laboratory ions have a common vibrational distribution. The agreement shown in Fig. 2 suggests that dissociative recombination of 0: is not sensitive to the initial vibrational level.

The situation regarding N$ is more uncertain. Oppenheimer et al. (1976, 1977b) obtained satisfactory agreement with Atmospheric Explorer N t data by using the laboratory values of Mehr and Biondi (1969) for the rate coefficient for dissociative recombination of Nz ions, values that are 30% smaller than the beam measurements of Mu1 and McGowan (1979). A more extensive study of the Atmospheric Explorer data led Orsini et ul. ( 1977a) to argue that the atmospheric rate coefficient varied between 1.4 x and 3.8 x cm3 sec-I in the conditions experienced by the


01 I I I I I I I 2 4 6 8 10 12 14

R E C O M B I N A T I O N C O E F F I C I E N T ( I O - ~ ~ ~ ~ sec-’)

FIG. 2. The rate coefficient for the dissociative recombination coefficient of 0: ions as a function of electron temperature derived from atmospheric data (D. G . Torr ef f t / . , J. Geophy.~. Rrs. 81, 5578, 1976, copyrighted by the American Geophysical Union.) The circles are the atmospheric values. The solid curve (a) is obtained from Walls and Dunn (1974) and the solid curve (b) from Mu1 and McGowan (1979).

AE-C satellite. Orsini et al. attributed the variation to changes in the vibrational population of N l ions associated with changes in the electron density. For vibrational levels u 2 1 , they derived rate coefficients in excess of 2 x

The conclusions drawn by Orsini et al. have been challenged by Biondi (1978) on theoretical grounds. They are sensitive to the detailed nature of the ion chemistry, and the close agreement between laboratory measurements of Mehr and Biondi (1969) and of Mu1 and McGowan (1979) indicates that another interpretation of the atmospheric data may be more plausible (cf. Torr and Torr, 1978; Torr and Orsini, 1978).

The process of dissociative recombination, identified by Bates and Massey from a consideration of the ionization balance, is also a significant contributor to the airglow of the planets, and airglow observations are a

cm3 sec-*.

42 A . DN Ig art 1 o

valuable source of information about the dissociation paths followed in the recombination process (cf. Hays et al . , 1978). The red and green lines of atomic oxygen are produced in the terrestrial atmosphere by 0; recombination, and the 5 199 8, line of atomic nitrogen is produced by NO+ and N2+ recombinations. On Mars and Venus, recombination of CO: ions produces the Cameron bands of CO.

The product atoms may possess considerable kinetic energy. Viking data on the atmospheric composition of Mars (Nier and McElroy, 1977) have demonstrated a low fractional abundance of nitrogen and an enhanced 15N/14N isotope ratio. Detailed analyses (McElroy et a/ . , 1977) have provided strong support for the existence of a substantial initial reservoir of nitrogen on Mars, which has diminished in time as nitrogen has escaped from the planet. One source of escaping nitrogen is dissociative recombination of N: (Brinkman, 1971; McElroy, 1972).

The process is also an essential element in the gas phase chemical schemes that have been developed for the formation of molecules in interstellar clouds (cf. Watson, 1976; Dalgarno and Black, 1976; Huntress, 1977).

111. Ion- Molecule Reactions

Bates and Massey (1947) obtained the ionospheric molecular ion for removal by dissociative recombination by postulating charge transfer to some unspecified molecule, which must be either O2 or NO producing 0: or NO+. The theory was extended by Bates (1955b), who pointed out that ion-atom interchange processes may be rapid so that

O+ + N , + N O + + N

determines the loss rate of O+ ions. Further progress toward a quantitative understanding came with the ad-

vent of rocketborne mass spectrometers with which the ionic composition was measured. Bates and Nicolet (1960) used the ion composition data of Johnson et a/ . (1958) in conjunction with an effective recombination coefficient derived from remote radio sounding data to show that O+ reacted with either O2 or N2 in only a small fraction of collisions.

Estimates then became available of the rate coefficients for dissociative recombination (Kasner et al., 1961; Doering and Mahan, 1962) and the major uncertainty in the interpretation of the data concerned the ion- molecule reactions.

Norton et al. (1963) incorporated the laboratory data in a quantitative

A T M O S P H E R I C A N D A S T R O P H Y S I C A L STUDIES 43

analysis of the rocket data, from which they derived a rate coefficient for the reaction of O+ with N2 and more importantly introduced the reaction

N i + O - + N O + + N

Bates and Nicolet (1960) had argued that the reaction was negligibly slow because it could not proceed by a simple ion-atom interchange but required an electronic transition. The atmospheric data were unequivocal in demonstrating that the reaction is rapid, as direct laboratory measurements (Ferguson et d . , 1965) soon established.

Laboratory studies of ion-molecule reactions, particularly the flowing afterglow experiments (Ferguson, 19731, determined the further development of a quantitative understanding of the ionosphere and detailed models were constructed in which the remaining uncertainties concerned the complications introduced by vibrationally excited species and by metastable species.

Production rates of metastable species in the initial photoionization of the atmospheric gases were calculated by Dalgarno et al. (1963) and the ionospheric importance of the reaction

O+(aD) + N 2 -+ 0 + N:

was noted by Dalgarno and McElroy (1965, 1966). Laboratory investigations of thermal reactions of metastable species are rarely practicable but the quality, extent, and simultaneity of the Atmospheric Explorer data have made it possible to identify the important atmospheric reactions involving the metastable O+(2D> 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 a l . , 1977) and it has also been detected by ground- based 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.

44 A . Dulgarno

TABLE I

RATE COEFFICIENTS OF REACTIONS OF

O+(*D) WITH Nz A N D 0 (cm'sec-')

Of(2D) + N,-+ 0 + N J 1 x 10-8 Oppenheimer ef a/., (1976) 5 x 10-10 Torr and Orsini (1977) 1 x 10-10 Torr and Orsini (1978)

O+(*D) + 0 -+ O W ) + 0 1 x 10-10 2 x 10-11

<<3 x 10-1' Torr and Orsini (1978)

Oppenheimer ef a/. (1976) Orsini et a / . (1977b)

IV . Neutral- Particle Reactions

Beginning with the work of Chapman (1931) on the emission of the green line of atomic oxygen in the night airglow, upper atmosphere studies have been a source of information about neutral-particle reactions. Of special importance are reactions of metastable O'D and N2D atoms, which have significant effects not only in the upper atmospheric regions but also in the stratosphere and troposphere (cf. Nicolet, 1975, 1976; Logan, et al., 1978).

The O'D atoms emit the red lines of atomic oxygen. There are many possible modes of excitation of the O'D atoms (Chamberlain, 1961; Bates, 1954; Noxon, 1968) including dissociative recombination of 0: (Bates, 1946), but not of NO+ (Dalgarno and Walker, 1964), photodissociation (Barbier, 1959), and electron impact (Dalgarno and Walker, 1%4). The intensities of the red line, measured by ground- and rocket-based instruments, are much smaller than the calculated O'D excitation rates and it was clear that the OID atoms undergo severe quenching. Analyses of the altitude profiles of the emission demonstrated that deactivation occurred mainly in collision with nitrogen molecules. Thus atmospheric data demonstrated that reactions of simple systems such as

OPD) + N ~ + OPP) + N~

involving a change in spin multiplicity could nevertheless be rapid. The most recent of many studies is that of Hays et al. (1978) based upon

the visible airglow observations on Atmospheric Explorer AE-C. Figure 3 illustrates the volume emission rate of 6300 A radiation derived for a particular orbit and compares it with a theoretical calculation in which the efficiency of production of O( 'D) atoms from dissociative recombination is 1.33 and the rate coefficient for deactivation by N2 is 3 x lo-" cm3 sec-I.

ATMOSPHERIC A N D ASTROPHYSICAL STUDIES 45

- 1 ' 1 ' I I '

.O .. .

0

.O

..

c ' J J I J I I I I I I I I

14 16 18 20 22 24 26 213

ANGLE ALONG TRACK (DEG) FIG. 3. The volume emission rate of 6300 A photons measured along the track of orbit

1536 of the AE-C satellite ( 0 ) the theoretical predictions based on a deactivation rate coefficient of 3 x 10-I' cm3 sec-' for collisions of O('D) with N, 10) IP. B. Hayset d., Re\*.,Geo- phys. Spuce Ph.vs. 16, 225, 1978, copyrighted by the American Geophysical Union.)

The derived value for deactivation by N2 is consistent with the laboratory determinations of Streit et al. (1976).

The emission from N(*D) at 5199 A has been analyzed similarly. That dissociative recombination of N: might be a source of N(2D) atoms was noted by Bates (1954). There are in fact many other sources including dissociative recombination of NO', the ion-molecule reactions

N: + 0 + NO+ + N(2D)

Nf + Oz+ NPD) + 0: and photodissociation and electron impact dissociation

N2 + hu 4 N('S) + N('D)

N, + e -+ N('S) + N(*D) + e

46 A. Dalgarno

(cf. Oran et al., 1975; Ogawa and Shimazaki, 1975; Rusch et al., 1975; M. R. Torr et al., 1976; Strobe1 et al., 1976; Kondo and Ogawa, 1977; Fred- erick and Rusch, 1977).

Destruction occurs by radiation of 5199 8, photons, by electron impact deactivation, and most important by reaction with O2 to form NO,

NPD) + O2 + NO + 0

(Norton and Barth, 1970; Nicolet, 1970). Analyses of Atmospheric Explorer data have shown that N(2D) is also

deactivated in collisions with atomic oxygen. The most recent analysis (Frederick and Rusch, 1977) leads to a rate coefficient of 4 x cm3 sec-l, which is a factor of about four below the laboratory value of Daven- port et al. (1976). The process plays an important role in the determination of the diurnal variation of NO (Stewart and Cravens, 1978).

The discrepancy in the value of the deactivation rate coefficient suggests a source of N2D has been omitted from the chemistry. It appears that the associative ionization of N(2D) with atomic oxygen (Zipf, 1978) followed by dissociative recombination of NO+ is too slow to provide the missing source (Black and Dalgarno, 1978).

V. Accidental Resonance Charge Transfer

The asymmetric charge transfer process O+ + H + 0 + H+

is close to resonance and Dungey (1955) drew attention to its possible importance as a source of H+ ions in the topside ionosphere. If the reaction is rapid at atmospheric temperatures, the charge transfer reaction is a larger source of protons than photoionization of atomic hydrogen. The reaction is the source of the polar wind flux of H+ ions (cf. Banks and Holzer, 1968, 1969).

Symmetric resonance charge transfer processes are rapid at thermal energies and a direct application of the adiabatic hypothesis of Massey (1949) suggests that asymmetric resonance charge transfer processes are also (Gurnee and Magee, 1957). There are, however, fundamental differences between symmetric and asymmetric resonance processes (Bates and Lynn, 1959) and for simple atomic systems most ion-atom charge exchange reactions will proceed slowly at thermal energies, despite the accelerating effect of the long-range polarization force.

Atmospheric data and astrophysical data have demonstrated that the 0+-H reaction is an exception. An analysis of the altitude distributions of


O+ and H+ in the terrestrial ionosphere by Hanson el 01. (1963) showed that the ions are closely coupled by charge transfer and a rate coefficient of 4 x cm3 sec-' was derived. The ambient temperature was about 1000 K.

The value is consistent with an extrapolation (Rapp, 1963) of beam measurements at high energies (Stebbings et al., 1960, 1964; Fite et al., 1962; Stebbings and Rutherford 1968), an extrapolation that assumed that the process varied with energy as does a resonance charge transfer process.

The importance of the reaction in astrophysical contexts was emphasized by Chamberlain (1956) who pointed out that in nebulae at high temperatures the equilibrium densities n(H+) of H+ and n(O+) of O+ are related to the densities of neutral hydrogen n(H) and oxygen n ( 0 ) by the simple formula

n(O+) 8 n(H+) n(O) 9 n(H)

The charge transfer process has been incorporated into discussions of the ionization distribution in a variety of astrophysical plasmas (cf. Wil- liams, 1973; Zeilik, 1977; Pequignot e? d., 1978; Raymond, 1978).

The charge transfer reaction is not exactly resonant and at low temperatures the effects of the different energy defects associated with the 3Pz, 3P1, and 3P0 fine-structure states of atomic oxygen must be considered. The individual reactions are

-- - --

O('P2) + H+ + O+('S) + H(?3) -0.020 eV

O(3P,) + H+ --* O+('S) + H(?3) +0.00007 eV

O(3P0) + H+ + O+('S) + H(%) +0.008 eV

In the upper atmosphere the atoms are distributed among the fine- structure levels according to thermal equilibrium characterized by the kinetic temperature of the neutral atmosphere, the equilibrium being established by neutral-particle collision processes that excite and deactivate the fine-structure levels with high efficiency. In low-density astrophysical plasmas, collisions are infrequent and the populations of the 3P, and 3P0 levels cannot be maintained against destruction by the emission of magnetic dipole radiation, so that almost all the oxygen atoms are in the ground 3Pz level.

Theoretical models of the process in which the differing fine-structure level populations are taken into account have been presented by Field and Steigman (1971) and by Brown (1972). Field and Steigman used the Langevin orbiting approximation (cf. Gioumousis and Stevenson, 1958)

48 A . Dalgurno

and Brown used a more elaborate approach based upon a semiclassical model of resonance charge transfer (Bates and Mapleton, 1%6) modified to treat asymmetric processes. The adequacy of these theoretical models, both of which employ arbitrary assumptions about the efficiencies of reaction at close collisions, has not been established. Although there are differences in detail between the predictions of the models, they both suggest that the rate coefficient for the exothermic reaction paths exceeds 10-lo cm3 sec-l for temperatures above 20 K.

The theoretical models are consistent with the laboratory measurement of Fehsenfeld and Ferguson (1972) at about 300 K, which yielded a rate coefficient of 3.8 x 10-Io cm3 sec-' for the reaction of H+ with 0. Rate coefficients at lower temperatures have been derived from theoretical analyses of the abundance of the hydroxyl radical in interstellar clouds.

The abundance of OH has been measured in the direction toward the three stars 5 Ophiuchi, 5 Persei, and o Persei (Crutcher and Watson, 1976; Snow, 1976, 1977; Chaffee and Lutz, 1977). In interstellar clouds, H+ ions are produced by cosmic rays. They may be removed by radiative recombination or by charge transfer to form O+ ions. In the presence of H2 molecules, any Of ions are rapidly transformed into H30+ ions by a sequence of reactions, followed by dissociative recombination to yield OH or H,O (Black and Dalgarno, 1973b; Watson, 1973; Herbst and Klemperer, 1973). In diffuse clouds, H 2 0 is photodissociated to produce OH, so that every H+-0 charge transfer leads to the production of an OH molecule.

The chemical scheme is illustrated in Fig. 4. The abundance of OH is then related directly to the rate coefficient for the H + - 0 charge transfer process and to the cosmic ray ionizing flux 5 that produces the H+ ions.

The molecule HD is also produced from H+ ions by the sequence H+ + D -+ H + D+

D' + H , + HD + H+

(Dalgarno ef al., 1973a) so that 5 can be determined from measurements of the abundances of HD provided some assumption is made about the deuterium- hydrogen abundance ratio (Black and Dalgarno, 3973b: O'DonneH and Watson, 1974; Barsuhn and Walmsley, 1977).

The most probable value of [D]/[H] in the local interstellar medium is 1.8 x low9 (York and Rogerson, 1976), though there is some evidence that variations may occur (Vidal-Madjar et al., 1977; Dupree et al., 1977). It has been suggested (Vidal-Madjar et al., 1978) that the variations may indicate the presence of a nearby interstellar cloud in front of the Scorpius-Ophiuchus association approaching the solar system. The consequences of an encounter of the planet Earth with an interstellar cloud have been discussed most recently by McKay and Thomas (1978).


FIG.

" "2

" H2

4. The chemical scheme for the formation of OH is diffuse interstellar clouds.

Adopting 1.8 x lop5, Black and Dalgarno (1977) were able to reproduce the OH and HD abundances measured toward 5 Ophiuchi by postulating [ = 1.6 x lo-" sec-I and a rate coefficient for

H+ + o(~P, ) + H + O+PS)

of 5 x 10-lo exp(-232/T) cm3 sec-I. Their estimate should be more accurate than the value of 2 x exp( - 232/T) cm3 sec-' obtained by Crutcher and Watson (1976) using similar arguments but a less detailed model of the cloud structure.

With the same form for the rate coefficient, the OH abundance toward 5 Persei can be reproduced by an ionizing flux 5 of 2.2 x lo-" sec-' and a [D]/[H] ratio between 1.8 x and 9.0 x lo+, the uncertainty arising

50 A . Dalgarno

from observational errors in the determination of the HD abundance (Black et al., 1978).

The OH abundance toward o Persei is larger than that toward 6 Persei. The temperatures at the cores of the two clouds are both about 45 K and it appears that the measured OH abundance must be due to an enhanced cosmic ray flux. Unless OH is produced by the endothermic reaction

0 + H, + OH + H in some shocked region (Aannestad, 1973; Elitzur and de Jong, 1978), an ionizing flux 5 = 2.5 x sec-' is required.

The star o Persei may be located in the neighborhood of a supernova ex- plosion (Sancisi, 1974). If so, the measured abundance of OH provides the first observational evidence to support the general view that supernovas are the source of galactic cosmic rays. Within the observational uncertainties the derived [D]/[H] ratio is normal (Hartquist et al., 1978). Thus the measured abundances of OH and HD tend to argue that supernovas are not a source of deuterium.

The accidentally resonant charge transfer process of O+ and H has proved to be of major importance in a wide range of atmospheric and astrophysical phenomena. Atmospheric, astrophysical, and laboratory data all demonstrate that the exothermic reaction is rapid even at temperatures as low as 20 K, but no quantitative theoretical explanation of the charge transfer mechanism has been advanced. Presumably the high efficiency arises because of the close coupling over a wide range of accessible internuclear distances of the low-lying 3C. and 311 states of the quasi-molecule OH+ formed during the collision. A detailed quanta1 study would help to remove the arbitrariness contained in the phenomenological analyses.

VI. Charge Transfer of Multiply Charged Ions

The emission strengths of lines arising from transitions in neutral and singly ionized atomic systems such as CI, NI, NII, NeII, 01,011, SI, and SII in gaseous nebulae are considerably stronger than the line strengths obtained in theoretical models. The measured intensities of the 011 doublet at Ah3727 and 3729 8, in particular are much greater than the intensities calculated for quasars (Davidson, 1972), planetary nebulae (Miller, 1974), Seyfert galaxies (Shields and Oke, 1975), and diffuse HI1 regions (Mathis, 1976). Detailed models of the planetary nebula NGC 7027 that in- voke shadowing and condensations still fail to reproduce the measurements (Pequignot et d., 1978; Shields, 1978).

It has been recognized since the work of Bates and Moiseiwitsch (1954)


that charge transfer reactions of multiply charged ions would in many cases proceed rapidly at thermal velocities. Bates and Moiseiwitsch (1954) employed a description of the energy levels of the initial and final valid at large nuclear separations and applied the Landau-Zener approximation to calculate the cross sections for charge transfer of SiIII, MgIII, and Be111 in collisions with atomic hydrogen. Similar calculations for other ionic species were carried out by Dalgarno (1954) and by Boyd and Moiseiwitsch (1957).

The Landau-Zerner approximation is satisfactory only under conditions that are rarely met in practice (Bates, 1960; Mordinov and Firsov, 1961; Nikitin, 1961, 1962; Ovchinnikova, 1964; Yagisawa, 1976) but it is qualitatively useful in identifying reactions that are likely to be among those which proceed rapidly. It suggests that reaction is probable in the neighborhood of avoided crossings of the interaction potential curves associated with the initial and the possible final states provided the avoided crossings occur at some favorable intermediate value of the internuclear distance and not at large or small values.

Charge transfer processes involving multiply charged ions in astrophysical plasmas were, however, overlooked until Steigman (1975) drew attention to their possible importance.

The two-phase model of the interstellar medium (Field et al., 1969), which achieved considerable success in explaining many of the characteristics of the interstellar gas, is based upon the postulate of an ionizing source uniform throughout the galaxy with a frequency of about sec-'. The ionizing source was attributed either to low-energy cosmic rays or to soft X rays, neither of which can be detected in the solar neighborhood. Low-energy cosmic rays are excluded by the solar wind and soft X rays are excluded by absorption.

Because of the Auger effect large differences occur in the production rates of highly ionized systems corresponding to cosmic rays and X rays (Weisheit and Dalgarno, 1972) and the observation of ions in high stages of ionization is a potential diagnostic probe of the character and intensity of the ionization sources (Weisheit and Dalgarno, 1972; Weisheit, 1973; Steigman et al., 1974). However, Steigman (1975) then pointed out that charge transfer processes could reduce substantially the equilibrium abundance of the multiply charged ions and he presented a compilation of potentially significant astrophysical processes. The compilation is based upon a consideration of the location of the avoided crossings as is a list produced by Dalgarno and Butler (1978) that takes into account charge transfer into excited states.

It appears that for systems stripped of three or more electrons numerous channels exist with favorable avoided crossings so that reaction

52 A . Dalgarno

almost surely occurs rapidly. For less highly stripped systems, the rate coefficients vary widely, depending upon the systems involved.

More elaborate calculations of the interaction potentials of several multiply charged ions with atomic hydrogen have appeared recently and Landau-Zener or two-state approximations have been used to derive rate coefficients for CIII (McCarroll and Valiron, 1979, CIV (Blint et al., 1976), SiIII (McCarroll and Valiron, 1976), and NIV (Christiansen et al., 1977). The reactions of CIV, NIV, and SiIII are fast, and the reaction of CIII is slow.

Following the analysis by Steigman (1975), Pequignot et al. (1978) suggested that charge transfer with atomic hydrogen and helium could resolve most of the discrepancies between theoretical models and observations of planetary nebulas and they attempted to reproduce the observations by treating the rate coefficients for charge transfer with atomic hydrogen as disposable parameters.

Table I1 lists the recommended values. Calculations support a high rate coefficient for NIV (Christiansen et al., 1977) and for CIV (Blint et al., 1976) and the potential energy curves of Butler et al. (1977) are consistent with a rapid rate for NIII. Elementary considerations suggest that at first sight reactions with 0111 and NeIII proceed slowly (Dalgarno and Butler, 1 978).

Pequignot et al. (1978) also derived a rate cpefficient for the charge transfer process

N + + H A N + H+ The value is 2 x 10-l2 cm3 sec-', two orders of magnitude slower than that calculated by Steigman et al. (1971) using a method similar to that adopted by Field and Steigman (1971) in discussing 0+-H. The value is in

TABLE I1

EMPIRICAL COEFFICIENTS FOR CHARGE TRANSFER REACTIONS WITH ATOMIC

HYDROGEN

Ion Rate coefficient

0111 NIII NelII SlV CIV NIV

~~

1.6 2 0.2 2.0 * 1.0 0.3 t 0.1 1.5 t 0.3 2.0 c 0.6

-3.0

In units of 10-o cm3sec-'. The values were derived from a model of NGC 7027 by Pe'quignot et a/ . (1978).


agreement with a semiclassical calculation by Melius (1973), who attri- butes the charge transfer to a spin-orbit interaction between 211 and 4Z states of NH+ that cross.

In a second paper Pequignot et CJI. (1979) have explored further the consequences of charge transfer processes in planetary nebulas and they have stressed the importance of the charge transfer of OIV.

In view of the derivation of several rate coefficients that are in harmony with theory, a more detailed examination of 0111 charge transfer is needed. Preliminary analysis by Butler et ul. (1979) suggests indeed that an interaction involving quartet states of OH2+ may be favorable.

The doubly charged ion 0111 has been detected in the terrestrial ionosphere (Hoffman, 1967, 1969, 1970; Hoffman et ul. (1969; Breig e f al., 1977). It is produced by photoionization of 0 and of O+ (Walker, 1970; Breig et al . , 1977) and removed at high altitudes mainly by charge transfer

0 2 + + 0 4 0’ + o+ A rate coefficient of lo-” cm3 sec-’ has been derived from the upper atmosphere data by Breig et (11. (1977).*

VII. Fine-Structure Transitions

The importance of the fine-structure transitions O(3P0,,) 4 OPPd + h~

in cooling the neutral component of the upper atmosphere was pointed out by Bates (1951) and the emission at 63 pm has been detected by Feldman and McNutt (1969) and by Offermann and Grossmann (1978).

Its importance in cooling the ionospheric electron gas was recognized later in an attempt to resolve a persistent discrepancy between the measured and calculated ionospheric electron temperatures. Dalgarno and Degges (1968) used cross sections calculated by Breig and Lin (1966) to compute cooling rates for the electron gas due to the collision processes

e + O(3P,) 4 e + (3P,,0)

and demonstrated that it dominated the cooling efficiency of the electron gas in the F region of the ionosphere.

A comparison of the heating and cooling curves for selected orbits of the AE-C satellite is reproduced in Fig. 5 (Hoegy and Bracy, 1978). The cooling curve is based upon improved calculations of the cross sections for fine-structure excitation (Le Dourneuf and Nesbet, 1976; Tambe and Henry, 1976; Hoegy, 1976).

* See, however, Victor, G . A., and Constantinides, E. R. (1979). Geophys. Res. Lett. 6, 519.

54 A . Dalgarno

T, K ) T, ( K I

FIG. 5. The electron temperatures T, measured on two orbits of the AE-C satellite are shown as functions of altitude and latitude; ---, obtained by equating the local heating and cooling rates; ..., obtained by including heat conduction. The full lines are the measurements. The inclusion of heat conduction improves the overall agreement but does not reproduce the wavelike structure (W. R. Hoegy and L. H. Brace, Geophys. Res. L e f t . 5, 269, 1978, copyrighted by the American Geophysical Union).

Generally good agreement is obtained at altitudes below 200 km and provided the effects of thermal conduction are included also at higher altitudes. The agreement is sufficient at least to argue that the major heating and cooling mechanisms have been identified.

In astrophysical contexts, the possible importance of oxygen fine- structure cooling in interstellar clouds was recognized by Burgess et d. (1960) but the cross sections were uncertain. Dalgarno and Rudge (1964) pointed out that excitation of the fine-structure levels of C+ by hydrogen atom impact excitation

C+(*P,,,) + H + C+(2P,,2) + H

would dominate the cooling of diffuse interstellar clouds and gave an estimate of the cross section. Several theoretical investigations were carried out (Smith, 1966; Weisheit and Lane, 1971; Wofsy et al., 1971; Launay and Roueff, 1977b) and the associated cooling rate is reasonably well established. In many clouds excitation by H2 is more significant. Theoretical studies were initiated by Chu and Dalgarno (1975) and Toshima (1975). More accurate calculations have now been carried out by Flowers and Launay (1977).

Calculations of cross sections of fine-structure excitation by hydrogen atom impact of neutral oxygen (Smith, 1966, 1968; Wofsy et al., 1971; Launay and Roueff, 1977a) and of neutral carbon (Yau and Dalgarno, 1976; Launay and Roueff, 1977b) have also been performed and for these


cases empirical determinations have been made from observations (Snow, 1977) of absorption by oxygen and carbon in excited fine-structure levels in the direction toward the star 4 Persei.

Black et al. (1978) have constructed a model of the interstellar cloud lying toward 5 Persei. According to it, the excited J = 1 level of atomic oxygen exists mostly in a warm 120 K envelope, where it is populated by hydrogen atom impact excitation. From the observations of Snow (1977) a rate coefficient of 1 x lo-" cm3 sec-' has been derived, which is in harmony with the calculated rate coefficient at 120 K of 8 x cm3 sec-l (Launay and Roueff, 1977a). The neutral carbon is mostly located in a cold 45 K inner core where the excited fine-structure levels are populated by collisions with H,. The ratio of J = 0 to J = 1 populations corresponds to a rate coefficient of 5 x lo-" cm3 sec-' (Black el al., 1978). No calculation of cross sections for collisions of neutral carbon with H, has been reported. The rate coefficient calculated for collision with H at 45 K is 2.8 x

There is, however, some question about the interpretation of the astrophysical data. The data imply a rate coefficient at 45 K for the J = 1 to J = 2 excitation of carbon by H, impacts that is about twice that for the J = 0 to J = 1 excitation, which seems improbable.

Observations of the emission produced by transitions between the fine-structure levels could be instructive. The 63 Fm line of atomic oxygen arising from the J = 2 to J = 1 transition has been detected recently in the Orion and Omega nebulae (Melnick et a / . , 1979).

cm3 sec-l (Launay and Roueff, 1977b).

VIII. Radiative Association

In a diffuse gas, the formation of molecules can occur only by a reaction sequence initiated by a radiative process. In a discussion of the chemistry of interstellar CH+ and CH molecules, Bates and Spitzer (1951) introduced the radiative association process

C + + H + CH+ + hu

but calculations of its rate coefficient (cf. Giusti-Suzor et al., 1976; Ab- grall et al., 1976) show that it is too slow to explain the observed abundances of CH and CH+. No calculations have been reported for the similar radiative association process

C+ + H,+ CH: + hu

but it should be more rapid (Black and Dalgarno, 1973a; Herbst et d., 1977).

56 A . Dalgarno

511- 051

FIG. 6. The chemical scheme for the formation of CH in diffuse interstellar clouds.

Detailed model calculations of the interstellar clouds toward 5 Ophiuchi (Black and Dalgarno, 1977) and 5 Persei (Black el al., 1978) employing the chemical scheme illustrated in Fig. 6 are consistent with observations of CH abundances, provided a rate coefficient of about cm3 sec-l is adopted for the C+-H2 reaction. Laboratory confirmation is needed. The reaction is a critical step in ion-molecule schemes for the formation of more complex interstellar molecules.

It should be noted that the chemical reaction scheme of Fig. 6 does not reproduce the measured abundances of CH+. For CH+, it appears that special assumptions must be made about the physical environments in which the molecular ions are located (Bates and Spitzer, 1951; Stecher and Williams, 1972, 1974; Dalgarno, 1976; Elitzur and Watson, 1978).

IX. Microwave Spectroscopy

Astronomical observations of emission lines in the visible, ultraviolet, and soft X-ray regions of the spectrum have long been an abundant source of spectroscopic data on atomic and ionic species. Of more recent origin is the development of spectral line radio astronomy, which has led to the discovery in interstellar clouds and circumstellar shells of a wide variety of molecules by measurements in the microwave and millimeter regions.

For reactive radical and ionic molecules, the astronomical measurements can often be performed with greater precision than can laboratory


experiments, and indeed for several astronomical species no laboratory experiment using microwave spectroscopy has been attempted. Thus radio-astronomical observations have been made of the radical species CH (Rydbeck et al., 1974, 1976; Turner and Zuckerman, 1974; Robinson et al., 1974; Zuckerman and Turner, 1975; Hjalmarson et al., 1977; Lang and Willson, 1978) and CN (Jefferts et al., 1973; Turner and Gannon, 1975), which have provided highly accurate data on the A-doublet splitting of CH and on the hyperfine structure of CN.

As is common in observations of astronomical objects, many unidentified lines exist. The line at 89.2 Ghz, due to X-ogen, is now definitely attributed to the molecular ion HCO+, as Klemperer (1970) originally proposed. The identification, which is strongly supported by elaborate quanta1 calculations of the potential energy surface (Bruna, 1975; Kraemer and Diercksen, 1976), was made certain by the microwave laboratory experiment of Woods et al. (1975) and by the detection of the isotope HI3CO+ (Snyder et af., 1976).

An unidentified triplet of lines was discovered at 93.2 GHz by Turner (1974), who argued that the triplet structure arose from nuclear quadrupole hyperfine structure and suggested that the molecule contained a nitrogen atom in the "outboard" position. Green et ul. (1974) then proposed that the molecule was the ionic species NzH+ and supported their pro- posal by a self-consistent field calculation of the rotation constant and hyperfine structure that was in harmony with the observational data. The identification was confirmed by resolving the hyperfine structure due to the inner nitrogen nucleus (Thaddeus and Turner, 1975) and by a laboratory microwave study of a glow discharge in a mixture of hydrogen and nitrogen (Saykally et al., 1976). A comparison of the theoretical spectrum with that observed in the Orion source OMC-2 is shown in Fig. 7.

A quartet of lines near 87.3 GHz was detected by Tucker et al. (1974), who were able to identify it positively as due to the ethynyl radical CzH. Gas phase ethynyl has not been studied in the laboratory, but experiments with electron spin resonance, infrared and optical spectroscopy (Cochran et al., 1964; Graham et af., 1974) of ethynyl trapped in inert gas matrices at liquid helium temperatures have been performed. The astronomical data have yielded values of the rotational, spin-doubling, and hyperfine constants of CzH.

Astronomical observations have also provided accurate data on conjugated carbon chains. The cyanopolyynes HC5N, HC7N, and HC,N have been detected and accurate transition frequencies have been determined (Avery et al., 1976; Kroto et al., 1978; Broten et al., 1978). A tentative identification of the cyanoethynyl radical C3N in the envelope of IR + 10216 has been made (Guelin and Thaddeus, 1977), based largely on theo-

58 A . Dulgurno

1 1 1 I 1 1 1 1 I

N,H' in OMC-2

I 1 1 1 1 -L- I -.I -4 -2 0 2 4

V-V, (MHz) FIG. 7. The theoretical and observed spectra of N2H+ in the Orion molecular source

OMC-2 (P. Thaddeus and B. E. Turner, Astrophys. J . Lerr. 201, L25, 1975, copyrighted by the American Geophysical Union).

retical predictions of its microwave spectrum (Wilson and Green, 1977), and the butadiynyl radical C4H has been observed also in IR -t 10216 (Guelin et al., 1978). Electron spin resonance has been used to investigate C4H trapped in frozen inert gas matrices (Dismuke et al., 1975) but no gas phase studies exist. The rotation and spin-doubling contrasts inferred from the astronomical data, assuming that C4H is the origin of the emissions, are generally consistent with Hartree-Fock calculations (Wilson and Green, 1977) and the identification of C4H is apparently unques- tioned.

Interstellar space is a valuable laboratory for fundamental investigations of the properties of highly unstable reactive molecules, not accessible in easily achieved terrestrial laboratory conditions.


X. Oscillator Strengths and Branching Ratios

Astrophysical abundance measurements have often provided reliable estimates of transition oscillator strengths and branching ratios. A recent example is the oscillator strength of the ultraviolet 2Z--X211 transition of OH.

The equivalent widths of the absorption lines of OH at 3078.4 A in the directions of 5 Ophiuchi, 5 Persei, and o Persei have been measured (Crutcher and Watson, 1976; Chaffee and Lutz, 1977) as has the equivalent width of the absorption line at 3081.7 A toward 6 Persei (Chaffee and Lutz, 1977). The lines belong to the AZZ+-XzIl transition and their oscillator strengths are accurately known (cf. Elmergreen and Smith, 1972). For 6 Persei Chaffee and Lutz (1977) derived an OH column abundance of 5.25 x 1013 cm-*.

The OH radical has also been detected toward 5 Persei (Snow, 1977) by absorption in the 1222.1 A line of the 2C--XzIl transition. The measured equivalent width and the column density of 5.25 x 1013 cm-2 imply an oscillator strength of 5.1 x which is a factor of about three larger than that suggested by Hartree-Fock calculations (Ray and Kelly, 1975).

The 1221.1 8, is actually an unresolved doublet. Chaffee and Lutz (1977) have worked out the separate contributions from each line and have constructed a table of individual line oscillator strengths. Their results are reproduced in Table 111.

Recent atmospheric measurements of the extreme ultraviolet spectrum of auroras (Christensen, 1976; Park et al., 1977; Christensen et al., 1977) have proved to be a useful source of data on branching ratios in atomic oxygen, at least to the extent of showing that some of the values listed by Wiese et al. (1966) must be seriously in error. Thus emissions at 989 and

TABLE 111

ABSORPTION OSCILLATOR STRENGTHS OF T H E

'8--X2n TRANSITION IN OH"

Line Astronomicalf value x lo3

60 A . Dalgnmo

at 7990 8, have been detected (Christensen et al., 1977). The lines origi- nate in the 2p33s' 3D0 level of atomic oxygen, the 989 8, emission corresponding to a transition to the ground level 2p4 3P and the 7990 8, emission to a transition into the 2p33s' 3P level. The branching ratio derived from the oscillator strengths tabulated by Wiese et al. (1966) is 0.89 : 0.11.

The 989 8, data show a directional anisotropy, which can be achieved under optically thick conditions only if the branching into 7990 8, emission is inefficient, and Christensen et al. (1977) suggest that the branching ratio may be too large by about three orders of magnitude.

More refined calculations of oscillator strengths (Pradham and Saraph, 1977) do indeed lead to the considerably reduced ratio of 1 : 4 x low4, a value consistent with laboratory experiments of Christensen and Cun- ningham (1978).

The atmospheric data contained also a strong component of 1027 8, radiation arising from the 2p33d3D0-2p4 3P transition of atomic oxygen. The atmosphere is optically thick to 1027 8, radiation and branching to the 2p33d3Do-2p33p3P emission line at 11287 8, is enhanced. The 11287 8, line was not observed but indirect arguments suggest that the ratio of the emission probabilities of 11287 and 1027 8, must be smaller than the value of 0.38 computed from the tables of Wiese et al. ( 1966). The quanta1 calculations of Pradhan and Saraph (1977) lead to a slightly smaller branching ratio of 0.28, whereas laboratory experiments (Christensen and Cun- ningham, 1978) give a value of 0.10.

The branching ratios are important in several astrophysical contexts and may directly influence the derived oxygen abundances. A brief survey has been given by Christensen (1979).

An intense component of the Martian dayglow is produced by emission from the A%: and BVI, states of COI: (Barth et al., 1971, 1972; Stewart et al., 1972). The possible sources are fluorescent scattering by C o t ions, photoionization of COz by solar ultraviolet radiation, and photoelectron impact ionization of COz (cf. Dalgarno et al., 1970a; Stewart, 1972). Pho- toionization is surely a major mechanism and the ratio of the intensities of the A and B band systems measured by Mars can be reproduced by an effective relative photoionization efficiency of 1.4 for the population of the A and B states (Fox and Dalgarno, 1979). The value obtained by combining the solar spectrum with the laboratory cross sections of Samson and Gardner (1973) is 0.7. There has been some difficulty in reconciling the measurements of Samson and Gardner (1973) with other laboratory studies (Lee and Judge, 1972; Bahr et al., 1972; Gustaffson et al . , 1978) on the production of the A and B states of COi through photoabsorption by CO,. The atmospheric analyses of the Martian data suggest that the


01 I 1 I I I

I 1 15 20 25 30

ENERGY eV FIG. 8. The measured photoelectron energy distribution at altitudes between 206 and

181 km ( J . S. Lee ef ol., Geophys. Reg. Lerf . 5, 581, 1978, copyrighted by the American Geophysical Union).

branching ratio has not been derived correctly from the laboratory measurements.

As a final example of the determination of branching ratios from atmospheric data, consider the interpretation of the photoelectron velocity distribution in the terrestrial ionosphere, measured on the AE-E satellite (Lee ef nl., 1978). A sample of the data is reproduced in Fig. 8.

Detailed theoretical calculations (Victor rt al., 1976) show that the main peaks between 21 and 28 eV are produced mostly by the photoionization by the solar line at 304 A of atomic oxygen leading to Of ions in the 4S, 2D, and 2P states and of molecular nitrogen leading to N,+ ions in the XzZi and A2n, states. From the magnitudes of the peaks relative cross sections can be obtained. Table IV lists the cross-section ratios for the absorption of 304 A radiation resulting in the production of the 4S, 2D, and 2P states of the atomic oxygen ion (Constantinides ef nl., 1979).

The atmospheric analysis agrees better with the laboratory data of Dehmer and Dehmer (1977) than with the theoretical calculations of Henry (1967) or of Starace et d. (1974).

TABLE 1V

BRANCHING R A I I O S FOR r H E P H O T O l O N l Z A r l O N OF ATOMIC O X Y G E N A 1 304 A

2D/4S ZP/%

1.48 0.94 Henry (1967) I .49 0.99 Starace c f ril. (1974)

1.67 ? 0.17 0.92 rt: 0.10 Dehmer and Dehmer (1977) 1.73 t- 0.12 1.04 2 0.08 Constantinides e f ul (1979)

62 A. Dalgarno

XI. Radiative Recombination

Radiative recombination is the dominant recombination mechanism in high-temperature low-density plasmas. The hydrogen recombination spectrum is a prominent feature of astrophysical nebulae and the hydrogen atoms in energy levels of high principal quantum numbers that emit at radio frequencies from HI1 regions are populated by radiative recombination.

Radiative recombination was considered by Bates and Massey ( 1946, 1947) as a mechanism for the removal of O+ ions in the ionosphere but dis- missed by them as too slow, partly on the basis of quanta1 calculations of the process by Bates et al. (1939). It does have some importance nevertheless as a source of ultraviolet emissions in the atmosphere and the observations of the emissions have led to an empirical determination of the rate coefficient.

Ultraviolet emissions were detected in the noctural tropical zone in two bands north and south of the geomagnetic equator, the spectrum consisting of the 1304 and 1356 8, lines of atomic oxygen (Hicks and Chubb, 1970; Barth and Schaffer, 1970; Meier and Opal, 1973; Anderson et af . , 1976). The emissions have recently been observed at midlatitudes (Brune et al., 1978).

The possible sources are radiative recombination Of + e + 0' f hv

and mutual neutralization Of + 0- + 0' + 0

(Hanson, 1970; Knudsen, 1970; Tinsley and Bittencourt, 1975). A quantitative interpretation of the midlatitude data leads to a recombination coefficient for the production of 1356 8, radiation of (7.8 2 2.3) x cm3 sec-I and of 1304 A radiation of (4.9 2 2.2) x cm3 sec-' at a temperature of about 700 K (Brune et al., 1978) in good agreement with theoretical calculations (Julienne et al., 1974).

The results are consistent with a rate coefficient of 1.5 x 10+'cm3 sec-I for mutual neutralization calculated by a modifled Landau-Zener approximation (Olson et al., 1971).

XII. Spontaneous Radiative Dissociation of Diatomic Molecules

A substantial part of the mass of the interstellar medium exists in the form of molecular hydrogen. The molecular hydrogen is located mostly in


cold interstellar clouds where it is produced by association on the surfaces of grains and destroyed by the absorption of photons (Solomon and Wickramasinghe, 1969; Hollenbach ef af., 1971).

Absorption from the ground X'Z; electronic state to the repulsive b3Z$ state dissociating into two hydrogen atoms in their ground states has a threshold of 2765 8, but because of the change in spin multiplicity it is an inefficient process. The cross section at 1500 8, has been estimated to be of the order of cm2 (Gould and Salpeter, 1963) but could be as large as cmz.

Dissociation may also occur at wavelengths shorter than 2765 A by absorption directly into the vibrational continuum of the ground electronic state. The selection rules permit only electric quadrupole photons and the overlap of the nuclear wavefunctions of the initial discrete vibrational level and the final continuum is very small. The cross section is unlikely to exceed cm2 and neither quadrupole nor the spin-forbidden absorption is significant in the interstellar gas.

Direct photodissociation occurs effectively only by absorption into excited electronic states, which dissociate into an excited hydrogen atom and a ground-state hydrogen atom. The threshold for direct photodissociation is 845 A. In the interstellar gas, photons with wavelengths shorter than 912 A are absorbed by atomic hydrogen, producing HI1 regions in which the hydrogen is ionized. Thus the hydrogen atoms shield the hydrogen molecules from direct photodissociation except when an interstellar cloud happens to encounter an HI1 region. Such an encounter happens but rarely, perhaps every lo7 or lo8 years. However, it was pointed out by P. M. Solomon (cf. Field et a/ . , 1966) that the dissociation of molecular hydrogen can be accomplished by photons longer in wavelength than 912 8, by absorption into the discrete vibrational levels of excited electronic states followed by spontaneous emission into the vibrational continuum of the ground electronic state and that despite the small overlap of the initial and final nuclear vibrational wave functions the process might be important in the interstellar gas. Its threshold lies at 1107 8, where atomic hydrogen is transparent. Approximate estimates of the efficiency of the process (Stecher and Williams, 1967; Nishumura and Takayanagi, 1%9) soon established that discrete absorption followed by spontaneous radiative dissociation is the dominant destruction mechanism of interstellar molecular hydrogen.

The process is illustrated in Fig. 9. Fluorescence spectra generated by discrete absorption followed by emission into the discrete vibrational levels of the lower electronic states are familiar in experimental spectroscopic studies (cf. Herzberg, 1950). Clearly the discrete emission lines are accompanied by underlying continua but no such continuum emission had

64 A . Dalgarno

I L

CONTINUUM E'

* 2

- -

L I I _ 0 0.4 I . 2 2 .o 2.8 3.6

INTERNUCLEAR DISTANCE (i) FIG. 9. A schematic illustration of the process of fluorescent photodissociation of H2 pro-

ducing hydrogen atoms with energy E ' .

been identified in laboratory spectra. The continuum emission is usually weak and because it lacks regular features is not easily recognized. A general discussion has been given by Mulliken (1971).

Stimulated partly by the need for greater precision in astrophysical applications, Dalgarno and Stephens (1970) carried through an accurate calculation of the dissociation efficiencies of individual vibrational levels u' and showed that for H2 in the excited B'Z: state, levels with v ' 3 12 decay preferentially by emission into the vibrational continuum. Earlier, G. Herzberg (cf. Dalgarno et al., 1970b) had observed in flash discharges in H2 and in D, a continuous emission spectrum in the region between 1200 and 1600 8, underlying the Lyman bands. A continuous emission was subsequently observed in fast-electron impact studies of H2 by Schmoranzer (1975).


Detailed calculations of the intensity distribution of the continuum emission resulting from the absorption of a uniform radiation field are compared in Fig. 10 with microphotometer traces of the spectra from which the discrete lines have been eliminated (Dalgarno et al., 1970b). The positions of the intensity maxima are reproduced to within SA and the relative intensities of the different maxima are in better than qualitative agreement. A similar situation is found for D,, for which the continuum emission has a distinctly different appearance, also reproduced by the theoretical calculations. The calculations ignore rotational effects. They are discussed by Stephens and Dalgarno (1972). who presented a detailed development of the theory of spontaneous emission into a continuum.

Although there can be no doubt about the identification of the continuous emission observed by Herzberg, the quantitative comparison of

1 I I I I I I I I I I I I 1400 1450 1500 1550 1600 1650 I700

A 0

b

1400 1450 I500 1550 1600 1650 I700 0

A

FIG. 10. Microphotometer tracings of the continuous spectrum emitted in fluorescence of molecular hydrogen. Curves (a) are measured tracings at different exposures and curve (b) is calculated (Dalgarno et a / . . 1970b).

66 A . Dcrlgurno

theory and experiment is limited by the assumptions about the efficiencies with which the vibrational levels of the excited state are populated. Re- cently Schmoranzer and Zietz (1978) have observed the continuous emission from selectively excited vibrational levels of the B'X: state. A comparison of the theoretical and experimental intensity distributions for the u' = 9 level is presented in Fig. 11. The calculations refer to the level with a rotational quantum number J' of zero; higher J' values tend to broaden and shift the maxima toward longer wavelengths and to raise the minimum intensities (Stephens and Dalgarno, 1972).

The experimental and theoretical distributions agree closely as does the ratio of the intensities of the continuous and discrete emissions. For

1 300

I I I I I I I

0 I I ,

5 200 H

0 4 I i

I I I I 1400 I500 1600 I700

EMISSION WAVELENGTH ( b l FIG. 1 1 . Emission spectrum arising from the v' = 9 level of the B'C: state of H, (Sch-

moranzer and Zietz, 1978).

A T M O S P H E R I C A N D A S T R O P H Y S I C A L STUDIES 67

u' = 9 Schmoranzer and Zietz (1978) measured a value of 0.34 and theory gives a value of 0.36.

The continuum emissions were the first clear example of the diffraction bands predicted by Condon (1928). They may be useful in the construction of vacuum-ultraviolet tunable lasers (Schmoranzer and Zietz, 1978). A similar process in the iodine molecule has also been suggested as the basis of an ultraviolet tunable laser (Tellinghuisen, 1974).

In interstellar clouds the H2 dissociation process is a heating mechanism (Milgrom et al., 1973; Stephens and Dalgarno, 1973). In a tuned laser it would be a source of monoenergetic hydrogen atoms.

XIII. Relativistic Magnetic Dipole Transitions

In a general discussion of the decay paths of metastable states of hydrogen and helium, Breit and Teller (1940) noted that the selection rules permitted a single photon magnetic dipole transition from the metastable P S I state to the ground l1S0 state of helium but concluded on the basis of an estimate of the singlet admixture of the triplet state that the magnetic dipole transition probability was negligibly small. They suggested that two-photon decay

He(2?3,) -+ He(l 'S,) + hu, + hu2

would proceed more rapidly. Proceeding by analogy with the two-photon decay of the 2'S0 metas-

table state, Mathis (1957) estimated a decay rate of 2.2 x sec-I. The analogy is misleading: in the case of a transition between states of zero angular momentum J = 0 the two-photon transition probability A(ul , u2) reaches its maximum when u1 = u2 (Drake and Dalgarno, 1968). For a transition from J = 1 to J = 0, the transition probability vanishes when u1 = u2 and severe cancellation occurs in the evaluation of the two-photon matrix elements. The decay rate of the z3S, state of helium is reduced to 4 x lo+' sec-' (Drake and Dalgarno, 1968).

The two-photon decay rate increases rapidly with nuclear charge (Drake et a / . , 1969; Bely and Faucher, 1969) and remains larger than the single-photon decay mechanism of Breit and Teller (1940).

Our understanding was transformed by the identification by Gabriel and Jordan (1969a,b) of emission lines in the solar spectrum that could be attributed only to the 23S1-11S0 transition of the heliumlike ions CV, OVII, NeIX, and MgXI. The Z3Sl- llSo emission lines of heavier helium- like ions ranging up to FeXXV have also been detected in the solar spectrum (Walker and Rugge 1970; Neupert and Swartz, 1970).

68 A. Dalgarno

The identification of the lines established that a single-photon emission mechanism was operating and rigorous selection rules permitted only magnetic dipole transitions (Gabriel and Jordan, 1969a,b).

The theoretical explanation of the radiative decay mechanism was advanced by Griem (1969) though it was contained in its essential aspects in the work of Breit and Teller (1940). In discussing the 22S1,2 state of atomic hydrogen, which decays preferentially by two-photon emission, Breit and Teller noted that a single-photon magnetic dipole transition is also possible. If nonrelativistic wavefunctions are used, the calculated transition matrix element vanishes because of the orthogonality, but with the inclusion of relativistic corrections a nonzero matrix element is obtained. For consistency it is necessary also to retain the leading two terms in the expansion of the spherical wave component describing the photon (Zhu- kovskii et al., 1970).

For the 22S,,2 state of atomic hydrogen the two-photon decay is relatively rapid though for nuclear charges exceeding Z = 45 single-photon decay becomes the dominant mode. For the 23S, states of the helium isoelectronic sequence, the two-photon decay is slow and the relativistic magnetic dipole emission is the dominant decay process for all Z .

The identification of Gabriel and Jordan (1969a,b) of magnetic dipole emission lines was soon followed by the development of relativistic theories in which terms up to order a2 were consistently retained (Drake, 1971; Feinberg and Sucher, 1971).

sec-I (Drake, 1971) in harmony with the subsequent measurement (Moos and Woodworth, 1973, 1975) of 1 . 1 5 0.3 x sec-I. The new value for the radiative lifetime of the triplet metastable helium atom resolved a discrepancy of long standing between observations of the 23P-23S emission at 10,830 8, in planetary nebulae and theoretical models of the abundance of the z3S atoms (cf. Drake and Robbins, 1972).

Measurements of the lifetimes of the 23S1 states of several highly ionized members of the helium isoelectronic sequence have been carried out using beam-foil spectroscopy (Marrus and Schmieder, 1972; Gould ef al., 1973, 1974; Cocke et al., 1973; Bednar et al., 1975; Gould and Marrus, 1976). The experimental results and the theoretical calculations (Drake, 1971; Feinberg and Sucher, 1971; Johnson and Lin, 1974) are listed in Table V. Discrepancies with earlier measurements on ArXVII and ClXVI appear to have originated in the nonexponential character of the 23S decay curves at short time intervals.

The extension of radiation theory to include relativistic magnetic dipole transitions has been accomplished and it produces radiative lifetimes in agreement with measurement over many decades. The development was

For helium the predicted decay rate is 1.27 x

ATMOSPHERIC AND ASTROPHYSICAL STUDIES 69

TABLE V

THE 23s,-l’s, RADIATIVE LIFElIME (nSeC) OF THE HELIUM ISOELECTRONIC SEQUENCE

Ion Z Theory Experiment

S CI A Ti V Fe Kr

16 17 18 22 23 26 36

697 38 1 210 27.4 17.4 5.0 0.18

706 ? 86 350 2 24 202 ? 20

75.8 5 1.3 16.9 ? 0.7 4.8 ? 0.6

0.20 f 0.06

stimulated by astrophysical observations. Relativistic magnetic dipole transitions may well be important in other atomic systems of atmospheric and astrophysical relevance, especially those species with unusually long-lived metastable states.

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I . 11.

ADVANCES IN ATOMIC

COLLISIONS

AND MOLECULAR PHYSICS. VOL I5

OF HIGHLY EXCITED ATOMS

R . F . STEBBINGS Depuriment of Spuce Physics and Aslronomv Rice University Houston. Texus

lntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 77

A . Collisional Mixing.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 B. Collisional Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Thermal Collisions with Heavy Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I. Introduction

Within the last few years Rydberg atoms have been the subject of numerous experimental investigations. A major stimulus for this activity has been the increasing availability and variety of tunable lasers, which can provide the resolution necessary to excite discrete individual Rydberg states.

The spectroscopy of Rydberg atoms has experienced great advances but will not be discussed here; we shall be concerned only with collisions. This topic was the subject of a recent review (Stebbings, 1977) and this chapter will emphasize only the most recent developments, although where necessary for reasons of clarity the earlier work will be described. This chapter is in no sense intended to provide an exhaustive overview of the field but rather seeks to highlight a few measurements that are representative of a given class or give indication of opening new avenues for exploration.

11. Thermal Collisions with Heavy Particles

By far the greatest concentration of experimental effort has been in the area of inelastic thermal collisions between Rydberg atoms and other atoms and molecules. Such collisions may proceed along a variety of

77 Copyrighi 0 1979 by Academic Press, Inc

All rights of reproduction in any form renerved. ISBN 0-12-003815-3

78 R . F . Strhbings

pathways including those open to metastable atoms, such as Penning ionization, and associative and dissociative ionization. In addition, however, new pathways are accessible as a consequence of the very high density of Rydberg states and because the ionization potentials of Rydberg atoms are typically much smaller than thermal energies. It is on these latter collisions that this chapter primarily focuses.

In considering them it is useful to view a Rydberg atom as comprising an electron moving in a very distant orbit about its ionic core. Its spatial and momentum distributions are then determined by its quantum state. In collision with a thermal neutral target the effects of the core can often be neglected and the collision can be viewed as occurring between the “essentially free” Rydberg electron and the target molecule. Broadly speaking, in such collisions the excited electron may either be transferred to another orbital leading to “collisional mixing,”

(1) A(n/) + BC + A(n’/‘) + BC

or it may acquire energy sufficient for “collisional ionization,”

A(n/) + BC + A+ + BC + e (2)

These processes may be accompanied by changes in the internal state of BC, while in (2) negative-ion formation may result. Collisional mixing and ionization may occur concurrently and their collective effect causes “collisional depopulation” of the initial Rydberg state. It frequently happens, however, that one of these processes is dominant and, for the purposes of discussion, we shall consider them separately.

A . COLLISIONAL MIXING

1 . Atomic Targets

Foremost in studies of collisional mixing have been groups at the Uni- versity of Reading (Deech et al . , 1977), Saclay (Gounand et al . , 1977), SRI International (Gallagher et al . , 1977a), and Rice University (Smith et al . , 1978). The experiments of the first three groups are conceptually very similar. In each case alkali atoms contained within a cell were stepwise photoexcited to Rydberg states by tunable lasers, and time-resolved fluorescence was used to monitor the subsequent Rydberg population.

In the absence of collisions this fluorescence decays with the radiative lifetime T,, , characteristic of the laser-excited Rydberg state. However, collisions with the parent gas, or with an added perturbing gas, cause the time dependence of this fluorescence to change. In this work at SRI where collisions between highly excited d states of sodium and helium, neon,

COLLISIONS OF HIGHLY EXCITED ATOMS 79

and argon were investigated, three different pressure regimes of the target gas were identified. For the 8d state, for example, the fluorescent decay was unaffected for rare gas pressures up to about 1 mtorr. In the 1- 10 mtorr range, a two-component decay was observed in which the prompt component decayed with a lifetime shorter than T~ while the subsequent component decayed with a lifetime longer than T ~ . At pressures above about 10 mtorr and up to 1 torr the fluorescent decay reverted to a single exponential with a pressure-independent decay time 7 that is considerably longer than T ~ . Within the accuracy of the measurements the total number of photons emitted was observed to be independent of the gas pressure, and thus there was no indication of radiationless quenching.

These observations are interpreted by Gallagher et a / . ( 1977a) using the model illustrated in Fig. 1. The two states in this model are the laser- excited d state and a “reservoir” state comprising all states of the same n with I 3 3 . For collisions with helium this scheme may thus be represented by

Thus loss from the d state occurs by radiation with lifetime T~ and by collisions that transfer atoms to the reservoir state with a rate constant lo. The reservoir, which is assumed to be statistically populated among all possible I states, in turn decays both radiatively with a lifetime 7, given by

(where T ~ , ~ is the radiative lifetime of an individual n,/ state) and by collisions that repopulate the d state with a rate constant d , .

The observations are then explained as follows. At sufficiently high pressures of Ar, He, and Ne (0.1- 1 torr) the collisional mixing time is short compared to the radiative lifetime of the d state. The sodium d atoms therefore mix very rapidly with the nearly degenerate higher I

‘ 0 r = f ,g ,h a d

FIG. I . The two-state model showing the radiative lifetimes and T , and the collisional mixing rates /o and d , .

80 R. F. Stebbings

states of the same n and the resulting mixture of 1 states decays radiatively with the average radiative lifetime of all the states in the mixture (i.e., all states with 1 3 2). A single exponential decay is thus observed with a lifetime that is found to be independent of the rare gas pressure over two orders of magnitude. Furthermore, when corrected for the missing s and p states this lifetime is found to increase as -n4.5, the predicted behavior of the average lifetime of all the states of the same n for hydrogen (Bethe and Salpeter, 1957).

At intermediate rare gas pressures, when the two-component decay is observed, the fast component is interpreted as being due to the decay of the d state both by collisions and by radiation and the slow component is due to the decay of the mixture of 1 2 2 states by radiation. In this intermediate pressure regime, a two exponential fit matches the data well within the statistical fluctuations of the measurement.

Of particular relevance to this chapter is the initial fast decay since the effective lifetime Teff here is given by

where the second term on the right-hand side represents the loss of d state atoms due to collisions with rare gas atoms. Thus 1 / ~ ~ ~ ~ = do, where n is the number density of the target gas and lo the associated mixing rate constant. From measurements of Teff as a function of rare gas pressure, Gal- lagher et al. (1977a) were able to determine lo. The most obvious feature of their results, which are shown in Fig. 2, is that the 1 mixing cross sections increase rapidly as the principal quantum number increases from 5 to 10 but decrease as n increases from 10 to 15. The n dependence of the cross section can be interpreted in the following way. The 1 mixing is produced by the strong short-range interactions of the valence electron and the rare gas atom. In the lower n states the probability of finding the valence electron at any point in its orbit is sufficiently high that the passage of the rare gas atom through any part of the orbit will induce a transition from the d state to a higher 1 state. Consequently, the cross section for the process reflects the geometric size of the excited Na atom's orbit and increases as n4. As n increases, the volume of the excited Na atom increases as ns and the probability of finding the valence electron at any point decreases accordingly. For n = 10- 15 the electron density is sufficiently reduced that passage of the rare gas atom through the valence electron's orbit does not automatically induce a transition. The decreased electron density is partially offset by the increased volume with the result that the cross sections decrease slowly as n increases from 10 to 15. It is in this


I _

t In15L1 4

12 I I 20" I 4 I I2 I --..--;4 Prtncipol qiiontum number, n

Fic;. 2. Collisional deactivation of Na(n2D) by He, Ne, and Ar: 0, experimental values of Gallagheref a / . (1977a); 0, calculated Na(n2D) + Na(nZF) cross sections Olson (1977).

range of n that the / mixing cross section is most sensitive to the collision partner.

These measurements show clearly that the / mixing cross sections increase from Ne to He to Ar. The scattering lengths of Ne, He, and Ar (in atomic units) are given by O'Malley (1963) as 0 . 2 4 ~ ~ . l .19u0, and - 1 . 7 0 ~ ~ . The fact that the 1 mixing cross sections increase in the same manner as the scattering lengths suggests that I mixing may be viewed as a low-energy electron-scattering process. The view is encouraged by the recent work of Olson (19771, who has calculated / mixing cross sections for He, Ne, and Ar based on their scattering lengths and polarizabilities. His calculated cross sections are also shown in Fig. 2 and are seen to be in excellent agreement with the experimental values.

In a rather similar experiment, Gounand et al. (1977) studied the collisional depopulation of high-lying nP states (12 s n s 22) of rubidium induced by thermal collisions with ground-state rubidium or rare gas atoms. They argued that although various collision mechanisms can lead to the depopulation of a given highly excited level the dominant process is that of / mixing in accordance with the model of Gallagher et a/. (1977a). They find that for rubidium atoms these cross sections are one or two orders of magnitude smaller than they are for sodium d states. They note, however, that whereas the nd states of sodium are nearly degenerate with the ( n , I > 2) states (the energy gap AE is less than 1 cm-l for n = 14), the np states of rubidium are, in contrast, well separated from all other states (at least 43 cm-I for tz = 14). Intuitively, and on the basis of the classical pic-

82 R . F. Stebbings

ture of kinetic energy transfer between the rare gas atoms and the valence electron of the alkali atom, one would expect a smaller hE to lead to a larger cross section. Thus, they argue, it is not unexpected that the I changing cross sections for the np states of rubidium are lower than those for the nd states of sodium.

This apparent strong dependence of the I mixing cross sections upon the energy gap between the initially excited level and adjacent levels prompted Gallagher et al. (1978) to extend their studies to cases where the energy gap was further reduced by a factor of 10. To this end they undertook collisional depopulation studies with Na(nf) states, since for n = 14 the f-g energy gap A& is -0.1 cm-', a factor of ten less than hEdt.

Their experimental approach (for n = 13) may be understood with reference to Fig. 3. Two pulsed lasers were used to stepwise excite ground-state Na atoms to the 13d5,2 state. A resonant microwave field was used to drive the d5,2-f5,2 and d512-f,12 transitions. The microwaves were plane polarized so that only Am = 0 transitions were induced and the resulting mixture was thus one-third d and two-thirds f. The time dependence of the fluorescence from the f state was then determined as a function of rare gas pressure. The results were qualitatively the same as these obtained earlier for the d states, in that when an appropriate amount of rare gas was added, a two exponential decay in the time-resolved fluorescence was observed. The initial fast component was pressure dependent and reflected the depopulation of the d-f mixture. The second slow component reflected the average radiative lifetime of all the I 5 2 states. Fol-

5890 i/

3s 112'

FIG. 3. The straight arrows indicate the laser pumping steps and the wavy arrows indicate the observed infrared fluorescence. The double-ended arrows indicate the microwave mixing of the d,,, withf,,, andf,,, states. The collisions are indicated by the horizontal arrow.


lowing Gallagher et al. ( 1978) we define the following I mixing cross sections:

U d : d -+ 1 > 2 uf: f + d , l > 3

(Td-f: d,f + 1 > 3,d’,f’

where d’ and f ’ are those parts of the d and f states not coupled by the microwave field. From the pressure dependence of the initial fast decay the (Td-f cross sections may be derived. On the assumption (Gallagher et al., 1978) that there are no A1 selection rules for the collisions, (Td-f may be expressed as

The factors 3 and % reflect the proportions of d and f in the d-f mixture while the factors in parentheses account for the fact that certain of the final states for the (Td and uf processes are not present in U d - f because the microwave field makes most d + f and f += d processes unobservable.

Then, from measurements of u d and U d - f , values of ur may be obtained. Rather than use the u d values of Fig. 2. Gallagher et al. (1978) drew a smooth curve for u d vs. n and used the smoothed values of u d to determine ge from Eq. (6). The values of uf so obtained are plotted in Fig. 4, together with values of u d . The most striking aspect of these results is that, contrary to expectation, uf is smaller than u d for Ar and He. This result is not yet understood.

N

0

V

.- c

m 2000 m m

* 1000

13 14 15

400

300

2ooL’ I00 0

13 14 15

Principal quantum number, n

FIG. 4. Values of u,, (A) obtained by smoothing the data of Gallagher er a/ . (1977a) together with values of U, (0).

84 R . F. Stebbings

In their analysis of the time dependence of the fluorescence Gallagher et al. (1978) assumed that the mixture of the 1 states was essentially statistical after the first collision, i.e., that there was no A1 selection rule for the collisions. This assumption leads to a model in which there are only two exponentials in the time-resolved fluorescence and which was thus consistent with their experimental observations. However, their data were such that the possibility of more than two components in the decay could not be definitively ruled out. A multiexponential decay would imply that I mixing occurred by a stepwise process and therefore that Af selection rules were operative. In order to clarify this question Gallagher et al. (1978) analyzed the distribution of final states using the technique of field ionization. In this experiment a beam of Na was excited to the 17d state in the presence of a low-pressure (- torr) of Ar, which induces I mixing collisions. At this pressure about 10% of the atoms in the 17d state are removed by collisions so that the results of only a single collision may be observed. At 4 psec after each laser pulse a pulsed electric field was applied to ionize the excited atoms and to collect the ions so formed. The amplitude of these ionizing pulses was swept and the ion signal determined as a function of the amplitude of the pulse. Since each I , rn state has a well-defined threshold field E at which it is field ionized, the dependence of the ion signal on the amplitude of the ionizing pulse E gives information on the final-state distribution after one collision.

Typical data are shown in Fig. 5 . The locations of the lowest (Iml = 0) thresholds of the n = 17, I 2 2 , states are indicated at the top of Fig. 5 . As noted by Gallagher et al. (1977b) the Irn( = I thresholds are 3% higher, the Irnl = 2 thresholds are 19% higher, and states of Irnl 2 3 require yet higher ionizing fields than those of the Irnl = 2 states of the same 1. Thus in Fig. 5 significant contributions from only Irnl = 0 and 1 states are observed. While states of higher Irnl are undoubtedly populated, their ionization thresholds are masked by the much larger signal from ionization of the 17d state. With no added Ar, the lower trace of Fig. 5 was observed. The 17d threshold at E = 4.36 kV/cm is quite evident. At fields <4.36 kV/cm there is a constant, nonzero signal that is due, not to field ionization, but to laser photoionization of the 17d state. The upper trace was taken with 4 x torr of Ar in the system and the 17d threshold at 4.36 kV/cm is still evident. From E = 3.50 kV/cm to E = 4.36 kV/cm there is roughly linear increase in the ion signal, which is due to the higher-1 states of the n = 17 manifold, which are populated by collisions. Below E = 3.50 kV/cm there is a constant photoionization signal. To interpret the signal one must recall that the ion signal is an integrated signal in the sense that the signal at field Eo represents the photoions plus the ions from all the states that ionize at fields E s E,,. Thus, the roughly linear increase in

COLLISIONS OF HIGHLY EXCITED ATOMS

ln e c

g C c

E! V 3

- 0 C

85

1 ' 1 ' 1 ~ 1 ' 1 ' 1 ' 1 ~ 1 a

a * .

***** . ***** ************* .

****** . '*********** .......mm..m....~

I I I I

Electr ic field , k V / c m FIG. 5 . Ion current vs. the amplitude of the field ionizing pulse when the Na 17d state is

excited in a vacuum (m) and under conditions of a single collision with Ar (0). At the top of the figure are the ionization thresholds for the n = 17. I 3 2 states.

signal from E = 3.50 kV/cm to E = 4.36 kV/cm represents a more or less flat population distribution over the higher4 states for m = 0 and 1.

The observation of no A1 selection rules supports the original interpretation of the SRI data and, more fundamentally, shows that the interaction responsible for the I mixing is short range.

2. Moleculur Turgrts

Collisions between Rydberg atoms and ammonia molecules have been reported by Smith et al. (1978). The use of NH, as target adds a new di- mension to this class of experiments since in addition to I mixing, which is observed for atomic targets, an additional depopulation mechanism occurs as a consequence of the dipole moment of NH,. This effect was discussed by Flannery (1973) and Matsuzawa (1971, 1974), who predicted that in collisions between Rydberg atoms and polar molecules, rotational deexcitation of the molecules could provide the energy needed to further excite or ionize the Rydberg atoms. To investigate this prediction Smith e f d. (1978) made an experimental study of collisions between xenon Ryd- berg atoms Xe(nf) with 26 < n G 40 and ammonia molecules.

The rotational term energies for NH, are given by

Ej = B J ( J + 1 ) + ( A - B ) K 2 (7) where J and K are rotational quantum numbers and A and B are constants. In dipole-allowed deexcitations ( J + J - 1, K + K ) the energies released, pE,, are approximately

86 R. F. Stebbings

hEj = E j - Ej-1 = 2BJ (8)

If these energies are transferred to Xe(nf) Rydberg atoms, with term values T,,, the resulting states will have term values given approximately by

(9)

Clearly if 2BJ exceeds IT,,[, TA will be positive and ionization will occur, while for 2BJ less than IT,,/ hrther excitation will result.

For the particular case of Xe(27f) atoms the possible reactions are

TA = T,, + 2BJ, J = 1, 2, 3, . . .

Xe(27f) + NHAJ) -+ Xe+ + NH,(J - I ) + e-, J > 7 (1 0 4

+ Xe(n'1') + NH,(J - I ) , J c 7 (lob) These processes are illustrated in Fig. 6. The arrows have lengths 2BJ (with J = 1,2, 3, . . .) and widths that are proportional to the room temperature populations of the upper rotational levels involved.

Collisions of the type (lob) lead to the production of seven discrete Rydberg states (or groups of states), which may in principle be separately

ION~ZING EXCITING TRANSITIONS TRANSITIONS

FIG. 6. Abbreviated term diagram for xenon showing the effect of Xe(27f)-NH, collisions. The lengths of the arrows show the energies released in the rotational deexcitations indicated. The widths of the arrows are proportional to the room temperature populations of the upper rotational levels involved.


detected and identified by the technique of field ionization since each of them will have its own characteristic critical field.

The apparatus used in these studies is shown in Fig. 7. A beam of xenon atoms was formed by effusion from a multichannel array. A fraction of these atoms was excited to the Xe 3P,,z metastable levels by electron impact. Charged particles formed by electron impact were removed from the beam by a strong transverse electric field. The metastable beam then passed through the experimental chamber, which contained the target gas at a pressure of a few microtorr, and impacted upon a surface detector, which was used to monitor the metastable flux by secondary electron ejection. Near the center of the experimental chamber the metastable beam was crossed by a dye laser with an output pulse width of - 4 nsec and a pulse repetition rate of -lO/sec. A small number (typically -5/laser pulse) of the Xe 3P0 atoms were thereby excited to a selected high Rydberg state (nf). Approximately 7 psec after each laser pulse, the electric field in the excitation region was increased from 0 to 1100 V/cm in -2 psec. As the field strength grew, the different groups of Rydberg states present were successively ionized, and the resulting free electrons were swept into an electron multiplier. The electrons arising from the ionization of the different Rydberg states were thus resolved in time, and the arrival time of each electron was then related to the field strength at which it was liber- ated.

A time-to-amplitude converter (TAC) was started at the beginning of the ionizing voltage ramp and was stopped by the first electron pulse subsequently registered by the detector. The TAC output was fed into a standard multichannel pulse analyzer (MCA). For sufficiently low count rates ((1 per laser shot) the MCA output displays the probability of a field ionization event per unit time during the 2 psec ramp.

,--, Laser beam

Fused quartz windo

teraction region

arged porticle deflector

Xenon beam source

FIG. 7. Schematic of the Rice University apparatus.

88 R . F. Strbbings

Typical MCA outputs are shown in Figs. 8 and 9. For zero target gas pressure a single peak (Fig. 8a) was observed corresponding to field ionization of the unmodified laser-excited Xe(27f) atoms. When - torr NH3 was introduced, spectra of the type shown in Figs. 8b and 9 were obtained. The most striking feature of these data is the series of well-defined peaks, at low field strengths, which result from the ionization of different groups of Rydberg atoms and thus explicitly demonstrate the occurrence of discrete energy transfer.

It is expected both theoretically (Herrick, 1976) and experimentally (Schiavone et af., 1977) that the term values T; of the product Rydberg states are related to their critical field values E; by

(1 1) EL = cR-' (TC)'

EL = cR-' (T , + 2BJ)'

where R is the Rydberg constant. Combining Eqs. (9) and (11) yields

(12) where J is an integer in the range 0 s J

Using Eq. (12) and the assumption that adjacent low-field peaks in Fig. 8b result from ionization of Rydberg states whose term values differ by

7.

ffl

c 3

t .-

-e 0

a, c C 0 c V

a, Q

ffl

c 3 0 0

-

I

t

NO N H 3

Xe(27 f l

h

- 500 100

Field strength, V/cm FIG. 8. Field ionization signal as a function of ionizing field strength (a) without NH3, (b)

tom NH3. Ionization field strengths obtained from Eq. (12) (c = 10.1 x 108 V/cm) for the products of reaction (lob) are indicated.

0 500 Kx FIELD STRENGTH I V k m )

FIG. 9. Field ionization of the products of reaction (lob) for different initial values o f )7.

The peak positions obtained from Eq. (12) are again noted. The position denoted 0 corresponds to collisions involving no change in J .

90 R. F . Strhhirigs

2B, one can identify the peaks by determining those J assignments that, together with a fixed value for c , yield values for El, most consistent with the low-field peak positions in Figs. 8b and 9. The best fit of Eq. (12) with the data is obtained for a c value of 10.1 x lo8 V/cm and the corresponding E; values are shown in Figs. 8b and 9. Adopting the transition identifications shown, one can immediately determine the principal quantum numbers of the product states. For example, the peak occurring at the lowest field in Fig. 8b is due to ionization of Xe(n -75) atoms formed by the reaction

Xe(27f) + NH3(J = 7) - X e ( n -75) + NH,(J = 6) (13)

Several features of the data require explanation, namely, (i) the absence of peaks associated with J + J - 1 transitions when J < 3; (ii) the absence in Fig. 8b of the narrow 27f peak and its replacement by the broader and displaced peak labeled P o ; (iii) the fact that the 27f peak in Fig. 8a and the Po peaks in Figs. 8b and 9 do not consistently coincide with critical-field values given by Eq. (12) when c = 10.1 x lo8 V/cm; (iv) the appearance for n 2 31 of an additional set of peaks labeled P I located on the high-field side of Po. We shall consider each of these effects in turn.

Collisions involving low-lying rotational levels of NH, are expected to be infrequent because the room temperature population of these levels falls abruptly for J < 3. Furthermore, such collisions tend to populate Rydberg states with 27 d n d 40 (Fig. 6). The density of states in this range of n is rather low and near-resonant transitions of the type (lob) are thus less likely than when the initial J values, and thus the final n values, are higher. As a consequence of these two effects, the peaks associated with J = 2 + J = 1 and J = 1 + J = 0 transitions are small enough to be obscured by the larger peaks labeled Po and P I .

The broadening of Po in Fig. 8a is attributed to I mixing collisions of the type discussed earlier:

Xe(271) + N H 3 ( J ) + Xe(271’) + NH,(J) (14)

Each of the Xe(271’) product Rydberg states has its own critical field, and the broad Po peaks correspond to a mixture of such product states.

The locations of the Xe(27f) peak in Fig. 8a and the Po peaks in Figs. 8b and 9 are not consistent with the value for c = 10.1 x lo8 V/cm obtained for the low-field peaks. This apparent paradox may be explained with the aid of the peaks labeled P , that emerge for n > 3 1. In a subsidiary experiment it was observed that, at a fixed value on n , the area of the P , peak grows relative to that of Po as the rate of change of ionizing field strength is increased. This fact points strongly to the conclusion that both Po and P I arise from Rydberg atoms of the same n and that the two peaks simply

COLLISIONS OF H I G H L Y EXCITED ATOMS 91

represent the outcome of two different routes to ionization in a time- dependent electric field. Littman r t d. (1976) have observed that the field ionization characteristics of a particular level are often determined by the interactions of that level with more easily ionized levels. These interactions occur in the vicinity of avoided crossings between levels of the same (m,l belonging to manifolds of different principal quantum numbers. As noted by Gallagher t>t ul. (1977b) the details of the path taken to ionization depend on whether these avoided crossings are traversed diabatically or adiabatically. Po and P , are provisionally interpreted therefore as resulting from ionization of products of reaction (14) in processes occurring along predominantly adiabatic and diabatic paths, respectively. As shown in Fig. 9, the height of PI grows relative to Po as n increases. From this one may infer that the low-field peaks, which are associated with high n values, are characterized by diabatic ionization. It follows that the critical fields for the P I peaks should also be given by Eq. (12) with c = 10.1 x lo8 V/cm. Inspection of Fig. 9 indeed shows that as n increases the P, peaks become more fully developed and their locations increasingly tend toward the series origins (i.e., EL for J = 0) of the low-field peaks. Thus, with the exception of the Po peaks and the Xe(27f) peak in Fig. 7a, the positions of all peaks are well represented by Eq. (12) with a c value of 10.1 x lo*. The Po peaks and the Xe(27f) peak, on the other hand, result from adiabatic ionization and are characterized by a value of c = 4.8 x lo8 V/cm.

B . COLLISIONAL IONIZATION

In the process of collisional ionization at thermal energies, the energy required to liberate the Rydberg electron many be provided either by the translational energy of the reactants

A ( d ) + BC --+ A+ + BC + e (15)

by the energy of attachment in the formation of negative ion A(n/ ) + BC - A+ + BC- (16)

or by transfer of rotational energy when the targets are polar molecules

A(n/ ) + BC(J)- A+ + BC(J - I ) + e (17)

Experiments in a number of laboratories have provided data on each of these mechanisms. Until quite recently all such measurements used electron impact excitation to produce the Rydberg atoms, which therefore had a broad and indeterminate range of quantum numbers; many of the fine details of the collision process were, in consequence, obscured. Nev-

92 N. F. Stebbings

ertheless, such measurements were of a pioneering nature and provided considerable insight into the nature of Rydberg collisions. Chupka (1974) made a significant step forward when he photoexcited rare gas atoms to single Rydberg levels and undertook a series of relative collision measurements.

Absolute collision rates for atoms excited to single Rydberg levels were first reported by West et al. (1976) and subsequently extended by Foltzet a/ . (1977) and Hildebrandt er af. (1977). The apparatus used in these studies is shown in Fig. 7 and was described earlier. The ion signals resulting from collisions between the laser-excited Rydberg atoms and the target gas were observed with a particle multiplier. The interpretation of these signals may be understood as follows. Following each laser pulse the number N ( r ) of Xe(nf) Rydberg atoms present in the interaction region is given by

N(r) = N(O) exp(-tl.r,d (18)

where N ( 0 ) is the number of Rydberg atoms excited by the laser and Teff is given by

where kd is the rate constant for collisional depopulation of the (nf) state, n the number density of the target gas, and T the radiative lifetime. The dominant processes that contribute to collisional depopulation of the initial state are collisional ionization and mixing and therefore

where k , and I , are the rate constants for ionization and mixing, respectively. Independent investigations demonstrated that the mixing rates I,, for electron-attaching targets can be as large as lo-' cm3 sec-', and that the rates for ionization from the resulting states are of comparable magnitude. However, despite these large rate constants, the experimental conditions may be chosen such that only a small fraction (typically 2-3%) of the laser-excited Rydberg atoms suffer a mixing collision during the short time interval ( - 5 psec) following the laser pulses, which is required for the collisional ionization measurements. Furthermore, the few atoms that do experience a change in quantum state do not significantly contribute to the ion signal since their rate of ionization is comparable to that of the initial (nf) state. Thus, at early times the observed collisional ionization is appropriate to an essentially pure (nf) state.


The ion production rate resulting from collisions of Rydberg atoms Xe(nf) with the target gas is thus given at short times by

dS/dt = N ( t ) n k , (21)

and thus has the same time dependence as N ( r ) .

pulse is The number of ions produced in the time interval t following the laser

S ( r ) = N(t)nk , dt = N(O)n[l - e x p ( - r / ~ , ~ ~ ) ] ~ , ~ ~ k ~ (22) l The number density n of the target gas is obtained by means of an ion

gauge calibrated against an MKS baratron gauge. S(r) is determined as follows: Subsequent to the laser pulse the Rydberg atoms are allowed to collide with the target gas under essentially field-free (60 .1 V/cm) conditions for a time 1 , typically -5 psec, during which the Rydberg atoms and any ions formed move a negligibly small distance (-- 1 mm). A small electric field is then applied across the interaction region accelerating these ions to the particle multiplier where they are detected. This field is kept sufficiently small (S40 V/cm) so that no field ionization of the remaining Rydberg atoms occurs. The ion transit time to the multiplier is determined in a subsidiary experiment, and the counting system is gated on for a short interval about this transit time.

N ( 0 ) is not directly measurable. Instead, N ( t ’ ) , the number of high Ryd- berg atoms present after a short time t ‘ (typically I psec) following the laser pulse, is determined by the application at time t ’ of an electric field, in excess of their critical ionization field. The remaining Rydberg atoms are thereby ionized, essentially instantaneously, and the resulting Xe+ ions [equal in number to N ( r ’ ) ] are accelerated to the multiplier. The counting system is once again gated on for a short interval about the ion transit time.

A minor complication arises in the determination of N ( t ’ ) because the target gas is present during this measurement. Consequently, a small number of Xe+ ions [<0.05 N ( r ‘ ) ] that are formed by collisional ionization during the interval r ’ are also detected when the ionizing field is applied. It is, however, straightforward to make a correction for this small effect.

From Eqs. (18) and (22), S(?) may be expressed as

S O ) = exp(r’/Teff)n[l - exp( -r/7,ff)l~,ffko (23)

It is evident that following any given laser pulse it is not possible to measure both N(t ’ ) and S(r) . These quantities are thus determined alter- nately over a sufficiently large number of laser shots that good statistics are obtained. Because the determination of both N ( t ’ ) and S ( r ) involves

94 R . F . Stebbirigs

the detection of Xe+ ions it is not necessary that the absolute detection efficiency for these ions be known. It is sufficient to ensure that the impact energy of the Xe+ ions upon the multiplier first dynode is the same for the determination of N ( t ‘ ) and S(t ) , so that the multiplier detection efficiency is the same for both measurements. Using this approach, rate constants have been determined for Xe+ production in collisions between X3(nf) atoms and various molecular targets.

Theoretical studies of collisional ionization have been made by Matsu- zawa (1974) and Flannery (1973). Matsuzawa (1974) predicted on the basis of the “essentially free electron” model that, for electron attaching targets, the rate constants for transfer of bound electrons from high Ryd- berg atoms should equal the rate constants for the attachment to the same target of free electrons having the same velocity.

Representative data for Xe(nf)-CH,I collisions are given in Fig. 10 together with the measurement of Lee (1963) for the attachment to CHJ of free thermal electrons. The measured rate constants using Rydberg atoms are located on the energy axis by ascribing to the excited electrons their time-averaged kinetic energy, which is equal to their binding energy.

PRINCIPAL QUANTUM NUMBER, n 40 35 30 25

I I I I

Xe I n f > + CH,I

m J ’ I 1 I I 1 I ,

10 50 I ELECTRON ENERGY (meV)

0

FIG. 10. Rate constants for the attachment of electrons to CHJ as a function of mean electron energy. 0, Rydberg electrons (Hildebrandt er a/. , 1977); ., free electrons (Lee, 1963).

COLLISIONS OF H I G H L Y EXCITED ATOMS 95

A reasonable extrapolation of the Rydberg data is seen to be consistent with the free-electron result.

In order to determine the identities of the charged reaction products an alternative experimental procedure was used in which the high Rydberg atoms collide with the target gas in the presence of a weak dc electric field (-2 V/cm) rather than in zero field as described above. Depending on the polarity of the field, either the positive or negative collision products were accelerated toward the multiplier immediately upon their formation. The output of the multiplier was then fed into a synchronized multichannel analyzer. The shape of the ion signal as a function of time will then mirror the temporal behavior of the high Rydberg population. In cases when mixing is negligible only the laser-excited Rydberg atoms are present and the ion signal has the same time dependence as N ( r ) . Furthermore since the flight time of an ion from its point of formation to the multiplier depends on its mass, the time delay following a laser pulse before the onset of the ion signal provides a measure of its mass.

For Xe(nf) = CHJ collisions the negative-ion arrival time spectrum exhibits a single peak, whose onset time is consistent with that calculated for I- (126.9 amu), although the resolution is not sufficient that CH,I- (141.9 amu) can be definitely discounted. In that I- results from the attachment of a free thermal electron to CHJ, this result lends further support to the theoretical model of collisional ionization discussed above. Similar investigations have been also carried out for other electron- attaching molecular targets. In all cases the results are in satisfactory accord with those for the attachment of free electrons to the same molecules.

Collisional ionization with polar targets has been investigated by Chupka and associates (Chupka, 1974; Matsuzawa and Chupka, 1977). As discussed earlier for these targets the energy required to ionize the Ryd- berg atoms may come, according to the model of Matsuzawa (1974), from the transfer of rotational energy. Furthermore, this model (Matsuzawa, 1974; Latimer, 1977) predicts steplike structure in the n dependence of the ionization cross section. Chupka (1974) has investigated this effect in collisions between krypton Rydberg atoms and HF and HCl, and representative data are given in Fig. 11.

Using mass spectrometry he observed SF; ions produced by photons of fixed wavelengths in mixtures of SF, and Kr. The SF; ions result from

Kr** + SF, + Kr+ + SF, (24)

When a small amount of a dipolar gas such as HCI is admitted, collisional ionization also occurs through

(25) Kr** + HCI(J) -+ Kr+ + e + HCI(J - 1)

96 R . F. Strhhings

v/o i C I

’! ., . , I , 1 1 1 1 885.5 886.0 886.5 887.0

PHOTON WAVELENGTH (a) FIG. 1 1 . SF, ions produced in Kr**-SF, mixtures, with and without HCI, as a function

of the photon wavelength.

The resulting electrons are quickly accelerated by an applied electric field and are not captured by the SF,. The influence of the HCI on the SF; production is shown on the lower curve in Fig. 1 1 , where steplike structure is indicated. Also shown and labeled by the rotational quantum number J are the thresholds for electron detachment by the polar molecule making a transition from the rotational state J to the final state J - I . The theory predicts clear steplike structure; however, as discussed by Matsuzawa and Chupka (1977) a number of effects may be operative that tend to obscure these steps.

Collisional ionization in Xe(nf)-NH, encounters has been studied by the group at Rice University. They found that the time dependence of the Xe+ ion signal did not exhibit the behavior required by Eq. (18). Instead they observed that significant ion production occurred after each laser pulse, at times long compared to the natural lifetime of the laser-excited Rydberg states. Representative data for laser-excited Xe(3 If) atoms are given in Fig. 12. Clearly the ionization occurring at late times results from long-lived Rydberg atoms created by collisional mixing and the interpretation of these data requires a reaction scheme that takes account of such effects. The simplest model capable of reproducing the observed time


Lifetime \ \ \

I I I I

0 10 20 30 40 50 TIME ( i 0 -6sec i

FIG. 12. Time dependence of the ion signal resulting from Xe(31)-NH3 collisions: 0, exp.: -, computer fit using the two-reservoir model.

dependence of the ion signal is as follows: I , , I ,

X e ( 3 l f i + N H , + Xe(1) + NH, - Xe(21 + NH,

Xe’ + N H , + e Xe’ + N H 3 + --- e X e + + NH, + e

(26)

Thus collisions between Xe(3 If) and NH, depopulate the initial Ryd- berg level by ionization with rate constant k , and by mixing with rate constant /,. This mixing leads to an “intermediate” reservoir state Xe(l) , which in turn decays by ionization and mixing into a second reservoir state Xe(2), which is in turn depopulated by collisional ionization. Radia- tive decay from Xe(1) and Xe(2) is unimportant, relative to loss by collisions. While this model should not be taken too literally, it appears to incorporate the essential features of the overall reaction process and certain rates derived with its use are meaningful. In particular, collisional depopulation of the initial Rydberg level occurs with a rate constant that is the sum of ko and /,. Furthermore, ionization at late times will be determined by k , and k , .

The full line in Fig. 12 is the computer fit obtained using this two- reservoir model. The rate constants so derived are’ (in units of cm3 sec-’)

I, = 7 x 10-7, k , = 1.4 x 10-7 I , + k , = 3.5 x k , = 1.7 x

98

lo-"

R . F. Stebbings

L

:

X e ( 3 1 ) t NH,

P = 2 x IO-' torr

2 - ,yo* YI +y2 -

m z 0

= I -

0 10 20 30 40 50 TIME ( p s e c )

FIG. 13. Time dependence of the ion signal resulting from Xe(3 1) + NH3 collisions. Total signal is given by Yo + Y , + Y , . Individual contributions from the laser-excited Xe(3lf) atoms and reservoirs 1 and 2 are given by Y o , Y,, and Y2, respectively.

10 20 30 40 50 100 PRINCIPAL QUANTUM NUMBER, n

FIG. 14. Cross sections for ionization in Xe(nf)-NH, collisions as a function of n. 0, Theory (Latimer, 1977); 0, exp.(field free collision); A, exp. weak-field (3.7 V/cm) collisions.


The model also allows separate determination of the ion production rates from the initial Rydberg atoms Yo and from the two reservoirs ( Y , and Y2), respectively. Such data in Fig. 13 clearly show that the overwhelming contribution to the initial ( t s 5 psec) ion production comes from the laser-excited Rydberg atoms. As a consequence it is possible to derive values of k,, under field-free collisions, using the approach outlined earlier that led to Eq. (22).

At low values of n the ko values appropriate to the field-free and to the weak-field experiments are in close agreement, while at n -40 the field-free values exceed the weak-field values by about a factor of 2. The experimental values are shown in Fig. 14 together with the values calculated by Latimer (1977). Satisfactory agreement is seen although the step structure predicted theoretically is not discernible in the experimental results.

REFER EN CES

Bethe, H. A., and Salpeter, E . A. (1957). "Quantum Mechanics ofOne- and Two-Electron Atoms," p. 266. Academic Press, New York.

Chupka, W. A. (1974). Bull. Am. Phys. Soc. [2] 19, 70. Deech, J . S., Luypaert, R., Pendrill, L. R., and Series, G. W. (1977).J. Phys. B [1]10, L137. Flannery, M. R. (1973). Ann. Phys. (N. Y . ) 79, 480. Foltz, G . W., Latimer, C. J., Hildebrandt, G. F. , Kellert, F. G., Smith, K. A,, West, W. P.,

Gallagher, T. F., Edelstein, S. A., and Hill, R. M. (1977a). Phys. Re\,. A [3] 15, 1945. Gallagher, T. F., Humphrey, L. M., Cooke, W. E., Hill, R. M., and Edelstein, S. A.

Gallagher, T. F., Cooke, W. E., and Edelstein, S. A. (1978). Phys. Rev. A [3] 17, 904. Gounand, F., Fournier, P. R., and Berlande, J. (1977). P h w . Re\,. A [3] 15, 2212. Herrick, D. R. (1976). J. Chem. Phys. 65, 3529. Hildebrandt, G. F., Kellert, F. G., Dunning. F. B., Smith, K. A., and Stebbings, R. F.

(1977). J. Chem. Phys. 68, 1349. Latimer, C. J. (1977). J. Phys. B [I ] 10, 1889. Lee, T. E. (1%3). J . Phys. Chem. 67, 360. Littman, M. G., Zimmerman, M. L., and Kleppner, D. (1976). Phvs. Rev . Lett. 37, 486. Matsuzawa, M. (1971). J. Chem. Phys. 55, 2685. Matsuzawa, M. (1974). J. Electron. Spectrosc. Relaf . Phenom. 4, 1 . Matsuzawa, M., and Chupka, W. A. (1977). Chem. Phys. Lett. 50, 373. Olson, R. E. (1977). Phys. Rev . A [3] 15, 631. O'Malley, T. F. (1963). Phys. Rev. A [2] 130, 1020. Schiavone. J . A., Donahue, D. E., Herrick. D. R.. and Freund, R. S. (1977). Phys. Re\*. A

Smith, K. A,, Kellert, F. G., Rundel, R. D., Dunning, F. B., and Stebbings, R. F. (1978).

Stebbings, R. F. (1977). Proc. Int . Cor1.f: Phys. Electron. At. Colli.Fions, loth, 1977. p. 549,

West, W. P., Foltz, G. W., Dunning, F. B., Latimer, C. J. , and Stebbings, R. F. (1976).

Dunning, F. B., and Stebbings, R. F. (1977). J. Chem'. Phys. 67, 1352.

(1977b). Phys. Rev. A [3] 16, 1098.

[3] 16, 48.

Phys. Rev. Let t . 40, 1362.

North-Holland, Amsterdam.

Phys. Rev. L e f t . 36, 854.


ADVANCES IN ATOMIC A N D MOLECULAR PHYSICS. VOL IS 1 1

~ POSITRON COLLISIONS 1 IN GASES

J . W . HUMBERSTON Department of Physics and Astrononry University College Unii,ersitp of' London London. England

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 11. Positron-Hydrogen Elastic Scattering . . . . . . . . . . . . . . . . .

111. Positron-Helium Elastic Scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 IV. Annihilation in Positron- Atom Scattering . . . . . . . . . . . . . . 118 V . Angular Correlations in Positron Annihilat

VI. Positronium Formation in Positron-Atom A. Positronium Formation in Positron-Hydrogen Collisions . . . . . . . . . . 126 B. Positronium Formation in Positron-Helium Collisions . . . . . . . . . . . . . 129

VII. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I31

I. Introduction

The past few years have seen great advances in the study of positron scattering by atoms and molecules. Theoretical investigations of such systems have been made for many years, one of the eariiest being that of Massey and Mohr (1954), but the recent availability of accurate experimental data against which to compare theoretical predictions has stimulated further theoretical interest in the subject. An account of some of the more important techniques that have been used to obtain these experimental results is given by Griffith elsewhere in this volume. In this chapter we shall discuss recent progress in the scattering and annihilation of low-energy positrons by simple atoms, principally hydrogen and helium. Brief mention will be made of some earlier work in the field, but for a more complete historical review of the subject the reader is referred to the articles by Fraser (1968), Brandsden (1969), Drachman (1972a), Massey (1971), and Massey et al. (1974).

Superficially, positron-atom scattering might seem to be very similar to electron-atom scattering, the only difference being the sign of the charge of the incident particle. But this change of sign of the charge has

101

Copyright 0 1979 by Academic Press, Inc All rights of reproduction in any form reserved

ISBN 0-12-003815-3

102 J . W . Humberston

several very important consequences. The positron is distinguishable from the electrons in the target atom and therefore exchange effects with the incident particle are absent. In their place, however, are electron- positron correlations that take the form of states of positronium, the bound system of an electron and a positron. These correlations are generally much more important than those in electron-atom scattering, and detailed descriptions of them are required if accurate results are to be obtained for the various scattering parameters. The potential energy of the positron in the static approximation is positive, whereas the polarization potential is negative, so that two major components in the positron-atom interaction tend to cancel each other. As a result, the low-energy scattering cross sections are much smaller than the corresponding electron- atom cross sections, and the actual values are quite strongly dependent on the accuracy of the polarization potential.

A further feature of positron-atom collisions that has no counterpart at all in electron-atom collisions is electron-positron annihilation into two or three y rays. The annihilation rate, which is proportional to the probability density of the positron in the immediate vicinity of the electron with which it is about to annihilate (see Section IV) is readily calculated from the trial function used in the scattering calculations, but the accuracy of the result is usually much less than that of the scattering cross section or phase shift.

These various features make the calculation of the low-energy scattering and annihilation parameters of the positron-atom system a very sensitive test of approximation methods.

11. Positron- Hydrogen Elastic Scattering

Although it is still not amenable to direct experimental investigation, positron- hydrogen scattering continues to be of considerable theoretical interest. It is the simplest positron-atom system and is therefore one upon which various approximation methods and techniques can fairly readily be tested. The first successful attempt to obtain accurate results for elastic scattering was that of Schwartz (1961), who used the Kohn (1948) variational method with trial functions containing up to 35 linear variational parameters to calculate the scattering length and s-wave phase shifts. A similar method was used by Armstead (1968) to calculate positron-hydrogen p-wave phase shifts. The Kohn method is not a bounded variational method (except under certain conditions at zero energy), and it may produce very inaccurate results (Schwartz, 1961). How- ever, it is simple to use and, by avoiding those regions in the space of the

THEORETICAL ASPECTS OF POSITRON COLLISIONS I N GASES I03

nonlinear parameters in the trial function where erratic behavior is found, very accurate results can be obtained. Indeed, the s-wave phase shifts of Schwartz remained the most accurate values available until Bhatia et al. (1971) performed their much more elaborate lower-bound calculations with trial functions containing up to 84 linear variational parameters. Both sets of results are given in Table I , and it can be seen how accurate the results of Schwartz are.

Bhatia et al. (1974a) have calculated very accurate lower bounds on the p-wave phase shifts, and their values and those of Armstead (1968) are also given in Table I. The accuracy of these results obtained with the Kohn variational method is again very satisfactory.

In most calculations of positron- hydrogen scattering phase shifts no use has been made of the approximate wavefunction, which is produced as a by-product of the main calculation, other than for calculating the positron annihilation rate (see Section IV) and the "cusp" value (Lebeda and Schrader, 1969). A quite detailed investigation of various aspects of the wavefunction for zero-energy s-wave scattering has, however, been carried out by Humberston and Wallace (1972). The wavefunction was obtained from a calculation of the scattering length using the Kohn variational method with a trial function containing 72 linear variational parameters. One of the more interesting features of the wavefunction was the almost complete absence of monopole distortion of the hydrogen atom when the positron was further than the electron from the nucleus. This result provides the explanation for what Drachman (1963, in his adiabatic

TABLE I

POSITRON-HYDROGEN PHASE SHIFTS

s-Wave phase shifts p-Wave phase shifts

Schwartz Bhatia er a/ . (1%1) (1971)

Armstead (1968)

Bhatia et a / . ( 1974a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

-2.10" 0.151 0.188 0.168 0.120 0.062 0.007

-0.054

-2.103" 0.1483 0.1877 0.1677 0.1201 0.0624 0.0039

-0.0512

0.009 0.033 0.065 0.102 0.132 0.156 0.178

0.009 0.0325 0.0659 0.1010 0.1303 0.1541 0.1792

* The zero energy entries are values of the scattering length in units of a,,. The phase shifts are in radians.

104 J . W. Humberston

treatment of positron-hydrogen scattering, had found empirically: that almost complete suppression of the monopole distortion in his wavefunction was required in order to reproduce the s-wave phase shifts of Schwartz.

There have been a number of other quite recent calculations of positron-hydrogen phase shifts, but the main interest in such work has usually been in investigating inelastic processes, particularly positronium formation, and the elastic-scattering calculations have only been carried out to provide a check on the suitability of the method being used and the accuracy of the computer programs. We shall defer discussion of such calculations to Section VI.

The accurate calculations of the s- and p-wave phase shifts by Bhatia et al. (1971, 1974a) seem to have established quite definitely that no elastic-scattering resonances exist in positron-hydrogen scattering below the threshold for positronium formation ( E = 6.8 eV). It is possible, however, that there are resonances at energies just below the threshold for excitation of the hydrogen atom to the n = 2 state ( E = 10.2 eV), similar to those known to exist in electron - hydrogen scattering (Gailitis and Dam- burg, 1963), and a number of searches for them have been carried out recently. The resonances occur, in principle, because the projected Hamil- tonian of the system QHQ, where Q is the closed-channel projection operator in the Feshbach formalism (Feshbach, 1962), has an infinite set of eigenvalues below the n = 2 threshold (Mittleman, 1966). Their positions are not, however, identical with the eigenvalues of QHQ, but are shifted because of the coupling between the open and closed channels. They may even be shifted above the n = 2 threshold and disappear. In positron- hydrogen scattering a further complication is the existence of the open positronium channel at energies above 6.8 eV in addition to the positron channel.

The first calculation that seemed to establish the existence of these "Feshbach" resonances in positron- hydrogen scattering was that of Seiler et al. (1971). The algebraic close-coupling method was used, but without the open positronium channel, and three s-wave and two p-wave resonances were found. Chan and Fraser (1973) also searched for s-wave resonances in their elaborate variational calculations of the cross section for positronium formation but failed to find any, and concluded that either the resonances were very narrow or they did not exist. They suggested that the inclusion of the positronium channel in their wavefunction might be responsible for the disappearance of the resonances found by Seiler et al. The correctness of this suggestion was confirmed by Drachman (1975) in a calculation in which the positronium channel was added to the wavefunction used by Seiler et al. An even more thorough search than that of

THEORETICAL ASPECTS OF POSITRON COLLISIONS I N GASES 105

Chan and Fraser was conducted by Shimamura (1975). He used the stabilization method (Hazi and Taylor, 1970) with trial functions containing up to 50 variational parameters, but found no resonances. Resonances were found again, however, by Wakid (1973, who used the close-coupling approximation with an expansion of the wavefunction that included the ground state and the 2s and 2p excited states of positronium. The position of the lowest s-wave resonance was 10.09 eV with a width of 0.42 eV. It is rather surprising that the correlation functions of Chan and Fraser and Shimamura are not sufficiently flexible to reproduce the structure of Wakid's wavefunctions in the resonant states, and one might have been inclined to suspect that these resonances were spurious but for the fact that a very narrow s-wave resonance with an energy of 10.10 eV has also been found by Doolen et al. (1978). This calculation employed the coordinate rotation method with very elaborate trial functions containing as many as 680 correlation functions. At least one such resonance does, therefore, seem to exist, but clearly further investigations are required.

111. Positron- Helium Elastic Scattering

The elastic scattering of positrons by helium atoms is of particular interest because it is the simplest system for which experimental results are available. Although much more complicated' than positron-hydrogen scattering, it is nevertheless still sufficiently simple that accurate values can be obtained for scattering parameters with an acceptable computational effort.

The experimental situation in positron- helium scattering has been transformed in recent years with the total cross section measurements of Canter et rr l . (1972, 1973), Jaduszliwer and Paul (1973), Brenton rt al . (1976), Burciaga e t c r l . (1977), Stein c t r r l . (1978), and Wilson (1978). and detailed comparisons between experimental and theoretical values can now be made. Such comparisons provide an important test of the accuracy of both theoretical and experimental results.

The positron-helium system also provides a further important testing ground for theoretical methods that are to be used for more complicated systems. Only if a method were found to give reasonably accurate results for positron-hydrogen and positron-helium scattering would one usually have any confidence in applying it to more complicated systems. I t is therefore very important to have accurate values for the positron-helium scattering parameters.

The most accurate calculations of positron-helium elastic-scattering parameters are probably those of Humberston (1973) and Campeanu and

106 J. W . Humherston

Humberston (1975, 1977a). The method used by these authors was similar to that used so successfully by Schwartz (1961) and Armstead (1968) for positron-hydrogen scattering, namely, the Kohn variational method with trial functions containing large numbers of linear variational parameters. For each partial wave the trial function was written as

(1) where +He is the helium atom wavefunction and the nomenclature is as defined in Fig. 1. The form of the trial function $: for s-wave scattering was taken to be

*i = +H&Z r3)$l(rl, rz rd

$6 = Yo,o(~l){ jo(krl) - tan qAno(krl)(I - e-ArL) N

+ C Ci ~ X P [ - ( W ~ + pr2 + prdI(1 + Pz3)rVk$r?r%r%} ( 2 )

where k is the wave number of the incident positron and P23 the space exchange operator for the two electrons. For the calculation of the scattering length the trial function was the zero-energy form of Eq. (2) with additional polarization terms, the most important of which is a dipole term with an inverse square dependence on the positron coordinate (Houston and Drachman, 1971; Humberston, 1973). Without this term the convergence of the scattering length with respect to increasing the number of correlation terms in Eq. (2) is slow. In the summation over i in Eq. ( 2 ) all terms were included such that

i = l

ki + f i + mi + n, + pi + qi s o

where k i , I,, m i , n i , pi, q i , and o are nonnegative integers. value of o provided a convenient means of systematically

(3)

Increasing the improving the

- FIG. 1 . The coordinates of the positron-helium system.

THEORE-rICAL ASPECTS OF POSITRON COLLISIONS I N GASES 107

wavefunction. The numbers of distinct correlation terms according to Eq. (3) for w = 1. 2. 3, and 4 are 4, 14, 36, and 84. respectively.

The wavefunction, Eq. ( I ) , was inserted into the Kohn expression

where q' is a trial value of the phase shift and q" the variational value. Varying the linear parameters, the C,s ( i = 1 , . . . , N ) and tan q', in Eq. (4) produces a set of linear simultaneous equations that are readily solved to yield the optimum values of the linear parameters and hence the stationary value of qv and the wavefunction.

A problem in scattering by any target system other than a hydrogenic atom is that the target wavefunction is not known exactly, and some approximate function must be used instead. The usual procedure is to ac- cept that the approximate target function is no longer an eigenfunction of the target Hamiltonian and to use the expectation value of the target Ha- miltonian with the approximate target function as the energy of the target system. This procedure can, however, produce very inaccurate results (Peterkop and Rabik, 1971; Houston, 1973; Wardle, 1973; Hoe1 d., 1975; Page, 1975, 1976), even when the method of calculation gives a bound on the scattering length or phase shift with the exact target function. Various attempts have been made by some of the authors listed hnre to devise means of overcoming this problem. The most satisfactory of these, and the one used in the calculations being described, is what has been termed by Drachman (1972b) "the method of models." The method involves replacing the exact Hamiltonian of the target system by a model Hamil- tonian, for which the approximate target function is an exact eigenfunction, and the exact target energy by the eigenvalue of the model Hamil- tonian. No change is made in the interaction between the incident particle and the particles in the target system, and so the method can only be used consistently when the incident particle is distinguishable from the particle in the target. It cannot be used for electron-atom scattering. Provided the total wavefunction of the system is written as the product of the approximate target function and some other function, as in Eq. ( l ) , it is not even necessary, so far as the Kohn variational method is concerned, to know the form of the model potential or the value of the model energy.

It is important to appreciate that any approximate phase shift obtained with the method of models is an approximation to the exact phase shift for the particular model target being used, and not to the exact phase shift for the exact target. This is particularly important when bounds on scattering parameters are being calculated. Nevertheless, i t is reasonable to expect that as the accuracy of the model target wavefunction is increased, the re-

I08 J . W . Hiitnhersfoti

sulting phase shifts will converge in some sense to the exact values. This is the basis of the techniques adopted by Campeanu and Humberston.

Three helium model functions were used in the calculations of the s-wave phase shifts and two in the calculations of the p- and d-wave phase shifts, the genera1 form of each function being

( 5 ) +He(t.Z, r 3 , rz3) = e - Y ( r * + r d ) 2 Di(r2 + r3)Ll(rz - r3Pr& i

where the summation includes all terms such that Li + Mi + Ni S wHe

with M i and Ni even, and Li, M i , Ni, and wHe are nonnegative integers. The three models correspond to setting wHe = 0 , 2 , and 4 and are referred to as H1, H5, and H14, respectively, where the number in each name spe- cifies the number of linear parameters in each function according to the above prescription. Model H1 is the same as that referred to previously by Humberston as model DB. The values of the linear parameters were determined by the Rayleigh-Ritz variational method and, at the same time, the value of the nonlinear parameter y was adjusted so that the dipole polarizability of each model had the approximate value 1.38~; (Dalgarno and Kingston, 1960; Thomas and Humberston, 1972). This condition was imposed because the magnitude of the polarization potential has an important influence on the values of the low-energy scattering parameters. Other properties of the three helium models are given in Table 11.

The convergence of the scattering length and the s-wave phase shifts with respect to w [see Eq. (311 is such that the results for w = 4, the highest value used in these calculations, are not fully converged. I t was therefore assumed that the convergence with w is given by

7, = 7, + c/w" (6)

where 7) must be determined separately for each energy and helium model. The extrapolated results vrn for the three helium models are given in Table I11 and plotted in Fig. 2. They are believed to be accurate to within 1 in the least significant digit.

TABLE I1

PROPERTIES OF T H E HELIUM MODELS

HI H5 HI4 Exact

Energy (Ryd) -5.680 -5.789 -5.800 -5.8074 Polarizabili ty 1.376 1.372 1.387 1.3834 (4) I . 1730 1.2117 1.1968 1.1935

THEORETICAL ASPECTS OF POSITRON COLLISIONS I N GASES I09

0.08 7 0.06

0

- - 0.OL

2

0 -0.08

U - F

- 0 12

- 0 I6

0 0 2 O L 06 0 8 10

k I ao-' I FIG. 2. The s-wave phase shifts for positron-helium scattering: ---, model HI; -,

model HS; - -, model H14.

TABLE 111

POSITRON-HELIUM SCATTERING LENGTHS A N D S-WAVE PHASE SHIFTS FOR THREE HELIUM MODELS*

k@') HI H5 H14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .o

-0.524 0.038 0.052 0.045 0.028 0.002

-0.030 -0.066 -0.097 -0.133 -0.166

-0.472 0.032 0.041 0.030 0.007

-0.022 -0.056 -0.093 -0.127 -0.162 -0.193

-0.48 0.033 0.042 0.03 1 0.009

-0.020 -0.053 -0.090 -0.124 -0.159 -0.190

The k = 0 entries are the scattering lengths in units of a, . Phase shifts are in radians. The phase shifts published by Campeanu and Humberston (1977a) are for w = 4, and are therefore slightly different from the present extrapolated values.

I10 J . W . Humberston

The results for helium models H5 and H14 are very similar, and, as model H14 is a very good approximation to the exact helium wavefunction, it is very unlikely that significantly different results would be obtained with an even more accurate target wavefunction.

The phase shifts increase monotonically with the number of variational parameters in the trial function, and the convergence is quite rapid and smooth. It is therefore likely that the results for o = 4 are lower bounds on the exact results for each model. The model H1 results are actually very slightly more positive than the rigorous lower bounds calculated by Ho and Fraser (1976) for the same model, but this may be because the correlation functions used by Ho and Fraser are of a slightly less general form than those used by Humberston.

It was found, when increasing the number of correlation terms in the trial function, that the addition of terms to I+ involving r Z 3 , the electron-electron coordinate, had a negligible effect on the phase shifts for models H1 and H5, and such terms were therefore dropped from the calculations for model H14 and for the higher partial waves. It should not, however, be thought that electron-electron correlations are unimportant, for, even with no term involving rZ3 in I+, the total wavefunction, Eq. ( l ) , for models H.5 and HI4 still has a dependence on rZ3 through the presence of this variable in the target wavefunction. An indication of the importance of electron-electron correlations is provided by the difference between the phase shifts for model H1, and those for model H5. Much of the difference seems to be due to these correlations in the target function. The justification for this statement is based on the fact that the results for model H1 are similar to the very accurate results obtained by Ho and Fraser (1976) with a different helium function, which also has the correct dipole polarizability but no electron-electron correlations. This function, referred to as HF2, is an analytic approximation to the Hartree-Fock function. The results for models HI, H5, and HF2 are plotted in Fig. 3, together with the results of some other calculations. It can be seen that the differences between the results for the two uncorrelated target functions HI and HF2 are much less than the differences between the results for models H I and H5.

Calculations of the s-wave phase shifts for helium model H1 have been made by Drachman (1966, 1968), Houston and Drachman (1971), and Ho and Fraser (1976) using a variety of computational methods. All the results, with the exception of Drachman’s (1966), are in quite good agreement with the values of Humberston. Almost identical results to these have also been obtained for a numerical Hartree-Fock helium function by McEachran el al. (1977), using a very similar form of the polarized orbital method to that used by Drachman (1966). In the remaining calcula-

THEORETICAL ASPECTS OF POSITRON COLLISIONS IN GASES I 1 1

006

0 04

0 02

0

- 0 02 - ‘13 I -

- 0 0 4 F

-006

-008

- 0 I0

- 0 I 2

- 0 14

I 1 I I 1 1

0 01 0 2 03 0 4 05 0 6

k I ao-’)

7

FIG. 3. The s-wave phase shifts for positron-helium scattering: A, Humberston (1973) H5: B , Humberston (1973) HI; C , Ho and Fraser (1976) HF2; D, Ho and Fraser (1976) HY I ; E, Amusia r / NI. (1976): F, Aulenkamp e / nl. (1974): G , Drachman (1966). The results of Ho and Fraser for model HY2 and those of McEachran P I NI. (1977) are very close to the HI results of Humberston and are therefore not plotted.

tions referred to in Fig. 3, either the helium wavefunction does not have the correct dipole polarizability (for example, the HY 1 function of Ho and Fraser) or the computational method being used is not expected to yield such accurate results as those obtained from elaborate variational calculations.

The p-wave phase shifts have been calculated in a similar manner to


that described above for the s-wave phase shifts, but with trial functions of the form

$ = Ylo@J[j1(krl) - tan v:n,(kr,)(l - e ~ * ‘ , ) ~ ]

As in Eq. (2) the summation over i a n d j includes all terms with ki + d i + mi + ni + pi s w1 and kl + l j + mj + nj + pl s w2. At low positron energies the second symmetry terms, those involving (1 + P23)Y,0(iZ), are unimportant but become increasingly important as the positron energy is raised above a few electron volts.

Many correlation terms of both symmetries are then required to obtain accurate results. Fortunately, however, for positron energies below 9 eV one can extrapolate from relatively small values of wz to infinity with reasonable confidence by exploiting the similarities between the convergence patterns for positron -hydrogen and positron-helium p-wave phase shifts (Campeanu, 1977). At higher energies, and particularly as the positronium formation threshold is approached, the coupling between correlation terms of the two symmetries becomes much stronger and the extrapolation procedure is less reliable. The extrapolated results for models H1 and H5 are plotted in Fig. 4. They are somewhat higher at the higher positron energies than the values published by Campeanu and Humberston ( 1973, because these authors neglected all second symmetry correlation terms in their calculations.

For sufficiently low positron energies, all phase shifts with I > 0 are given by the formula (O’Malley et al., 1962)

nPk2 = (21 - 1)(21 + 1)(21 + 3 )

where the remainder term is of order k3 for I = I and of order k4 for I > 1. Both sets of p-wave phase shifts are consistent with this formula with P = 1.38~3, as k + 0, but as the positron energy is increased the higher-order corrections to Eq. (8) rapidly become important. Even so, the results do not become sensitive to the detailed form of the helium wavefunction, provided it has the correct dipole polarizability, until the positron energy exceeds a few electron volts.

Also plotted in Fig. 4 are the results of a number of other calculations of positron-helium p-wave phase shifts. The methods used to obtain these results are essentially the same as those used to obtain the corresponding

0.07 cl

0 02 0.1, 0.6 0 8 1 0

k21 a L 2 )

FIG. 4. The p-wave phase shifts for positron-helium scattering: -, Campeanu and Hum- berston H5; ---, Campeanu and Humberston H1; ---, McEachran et a/. (1977); -'-, Drachman (1%6); -.-, Aulenkampet a/ . (1974). The results of Amusiaef a / . (1976) are very close to the H5 results of Campeanu and Humberston and are therefore not plotted.

s-wave results plotted in Fig. 3. The results of Aulenkamp et 01. (1974) are clearly very different from all others, and at very low energies are inconsistent with Eq. (8) with P = 1.38~3, . Possible explanations of the discrepancy are that the polarizability of the target atom is quite different from 1.38~3, (the value of the polarizability is not specified by these authors), or that the wavefunction is so poor that the results are nowhere near their fully converged values. This would be rather surprising, however, with trial functions containing 95 correlation functions.

Results of several calculations of the d-wave phase shifts are plotted in Fig. 5. They are all in reasonably good agreement with the values given by Eq. (8) over quite a wide range of positron energies, although the results of Aulenkamp et al. (1974) have a rather different behavior from that of all other results at energies greater than 7 eV. The method used by Cam- peanu (1977) is very similar to that used by Campeanu and Humberston (1975) in the p-wave calculations referred to earlier, but only correlation terms of the first symmetry were included in the trial function. Conse-

114 J . W . Hirmberston

0.OL

0.03

- U 2 - 0.02 N F

0.01

0 0.2 0.4 0.6 0.8 1.0

k 2 I a;* I FIG. 5 . The d-wave phase shifts for positron-helium scattering: -, Campeanu (1977)

H5;---, Campeanu H1;--, Drachman (1966);-..-, Amusiaefttl. (1976); ..., Aulenkamper a/ . (1974); ---, Eq. (8).

quently, these results are slightly low at the higher positron energies, and the most accurate phase shifts are probably those of Drachman (1966).

Phase shifts for even higher partial waves have been calculated by Amusia et a/ . (1976) and McEachran et a/. (1977), and the results, particularly those of McEachran et al., agree well with the values given by Eq. (8) over the entire energy range below the positronium formation threshold.

The total cross section is given in terms of the partial wave phase shifts by

and the values calculated from the s- and p-wave phase shifts for helium models HI and H5 and Drachman’s (1966) d-wave phase shifts are plotted in Fig. 6 together with the results of a number of experimental measurements. Also plotted are the values calculated from the H14 s-wave, the H5 p-wave, and Drachman’s (1966) d-wave phase shifts. Partial waves with I > 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

0 3

1 4 + I

T

0 0 2 0 4 0 6 0 8 1 0

Positron wavenumber 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 PI 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

(10)

As an example of this agreement, the total cross section at zero positron energy, the value of A . in Eq. ( lo) , is 0.883~~120, whereas the value calculated from the scattering length is 0 . 9 2 ~ ~ 2 0 for helium model H 14 and 0.89 for model HS.

Above 6 eV the results of Canter et al . (1973), Burciaga rt NI. (19771, and Stein et L J I . (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 ul . and Burciaga e f

(+ = A . + A,k + A,k2 In k + A3k2 + A,k4


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 significantly larger than the values 0 . 0 4 ~ ~ ~ and 0 . 0 5 5 ~ ~ : measured by Burciaga 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

C 4 9OeV * .-

2 I * .-

2.18 eV 2 - Q)

t UI .-

- e b

0.54 eV

0 136eV

0 30 60 9 0 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 ~ 1 ~ for 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 = n r k n (14) where ro (= e2/mc2) 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 = micNZef f (15) where Zerr is the effective number of electrons per atom. The value of Zeff will in general vary with the positron velocity. The annihilation cross sec-


tion (T, is defined as A 1

Nu L] (Ta = - = - 7rY:cZ,[f(u) (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+l is P(r l , r z , . . . . rN+l), then

where P i s normalized to unit positron density as Y, -+ =. The calculation of Zeff is 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 Zeff is 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 Zeff is 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 Zeff at 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


0 1 0 2 03 0 4 05 0 6 07

k (a,,-')

FIG. 8. ZeH for 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.

good as that of the corresponding phase shifts, the value of 3.86 for Zeff for 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 Zeff at energies other than thermal energy, but the experimentally measured quantity is Zeff(t), the value of Zeff averaged 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 ~ ( 0 , t)zefXv) du z e d t ) = (19) lom 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

4 2

II .O

3 8

3 6 Zeff

3 2

0 0 2 0 4 0 6 0 8 1 0

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 Hum- berston 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 ac- quires 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)

L J N U , ( L . ) ~ T ) 2 ( u , 1 )

M au

M

+ ac 3 d L ~ v u , ( U )

- 2 e ' F

a [( e2F

as - ( u , t ) = - at

3m2L~'NCr~j( U )

(20)


where ( T ~ ( u ) is 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

0 3 - N

m 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. __


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 Zeff for 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 rf NI., 1972; Grover, 1978), but the lack of

I I I I I I I 0 MO moo 1500 axil 2500 3000

time Insec amaqatsl

FIG. 1 1 . The time dependence of Zetf for 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 H5: calculated from the data for model HI.


TIK) 0 5 00 lo00 1500 2 m

1 I

- - - - - -i 10 20 30 LO

E I v cm-' arnogat-'1

FIG. 12. The dependence of the equilibrium value of Zeff on 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. Theo- retical results: -, variation of Zen with 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 COLLISIONS I N GASES I25

+m +m w) = j-m r(ps.pu.pz = 1nc.o) dp, dpu ( 2 2 )

where T ( p ) is the momentum distribution function of the annihilating electron-positron system. For positron-helium scattering this has the form

u p ) = I dr3 I j dr, exp(-ip rl)*(rlrr2 = r l , r 3 ) 1' ( 2 3 )

- 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 paraposi- tronium, 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.


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 i at 12 eV and then decreased more slowly, so that at 24 eV the value was 2 ~ 4 . These authors also used a form of the distorted-wave 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

0

- C

” ; 0.OOL

ln v)

u e

0.003 0 .- ” 2 p 0.002

.- E, E 0.001

4- - In a“

0 5 0 6 0 7

kZ (a;‘ )

FIG. 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 LL = 0.5.


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 Hyl- leraas 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 QHQ (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 FORMATION CROSS SECTIONS' IN POSITRON-HYDROGEN SCATTERING

k(a;') OD(/ = O)* up(/ = I ) C OD(/ = 0) + OD(/ = 1)

0.71 0.0015 0.0147 0.0162 0.75 0.0029 0.357 0.360 0.80 0.003 1 0.505 0.508 0.85 0.0032 0.584 0.587 0.866 0.0033 0.667 0.670

The cross sections are in units of sa', . * Chan and Fraser (1973). Chan and McEachran (1976).

THEORETICAL ASPECTS OF POSlTRON COLLISIONS IN 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


cross section. According to these calculations the positronium formation cross section is dominated by the d-wave contribution. Fels and Mittle- man (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 polarized- orbital 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 Zeff for 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 at- tracted 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 f 3 1

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 elastic- scattering 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 (Cam- peanu, 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 Zeff at 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.

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17, 1600.


ADVANCES IN ATOMIC A N D MOLECULAR PHYSICS, VOL. 15 I i EXPERIMENTAL ASPECTS OF POSITRON COLLISIONS IN GASES

T. C . GRIFFITH

Department of Physics and Astronomy Unii*ersiiy to l fege Uniwrsily of London London, England

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

142 11. Scattering Techniques with Positron Beams

111. Accuracy of the Cross Section Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Total Cross Section Measurements . . . .

B. Neon and Other Inert Gases at Low C. Inert Gases-Intermediate Energies D. Inert Gases-Positronium Formation Cross Section . . . . . . . . . . . . . . . . 153 E. Cross Sections for Molecular Gases . . . . . . . . . . . . . . . . . . . . . . . . . . F. Inelastic Collisions of Positrons in Helium above the Ionization

Threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A. Positron-Helium Scattering at Low . . . . . . . . . . . . . . . . . . . . . . . 146

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

136 T . C . Grijjith

10 2' 50 70 90 110 130 150 170 190 210

Channel Number - 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 I37

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 Ja- duszliwer and Paul (1973) and Brenton ct a / . (1976) have used static monochromators for the same purpose. The yield of positrons depends criti- cally 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 low- energy 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-

138 T. C . Grijjirh

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 Hum- berston (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 al . (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 elastic- scattering 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 modifi- cations 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

EXPERIMENTAL ASPECTS OF POSITRON COLLISIONS IN GASES 139

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

Proton Beam 4 I5 MeV

Solenoid lenath I = I meter) -

--Gas

Movable Aperture

-Aperture Coil

Retarding Element

Channelt ron Detector

Scak L-U-LI 0 5cm

FIG. 2. Schematic diagram of the system used by Kauppila pi a / . (1976).

I40 T . C . Grijjith

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

E X P E R I M E N T A L ASPECTS OF POSITRON COLLISIONS I N GASES I4 I

length 8 cm, has a total length of 130 cm and is evacuated with three S O 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 cen- trally 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 PI (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-

2" pump scattering cell

llghl

2" pump below i.' 1 2*5bump

L" pump below pressure measuring port

not to scale

Flc. 3 . Schematic diagram of the time-of-flight system of Twomey P I r r l . (1977) with localization of the scattering region.

142 T . C. Gr$$th

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 e t a / . (1972), have used slow positron beams for the same purpose. Variants and modifica- tions 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 uT directly, but it is generally sufficient to deduce uT from 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 I43

where (Nv)j and (N,I j 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. Subtrac- tion 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-

so00

4000

3000

2000,

- 0 C

m L u IOOC L

E 80C

60C

5 40C 3

200

C

80C

60C

40C

2 oc C

-2oc

-4oc

la I !P

P Q

0

X

2 2 8 * 2 0 eV

I I I I I I I I I 10 20 30 40 50 60 70 80 90'

Channel number

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).

144 T. C . Griffith

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. Fluctua- tions 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 com- pensates 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

EXPERIMENTAL ASPECTS OF POSITRON COLLISIONS I N GASES 145

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 Can- ter 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. In- stead 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". Cor- rections 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.

146 T . C . Grij j th

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 Bur- ciaga 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-

EXPERIMENTAL ASPECTS OF POSITRON COLLISIONS I N GASES 147

0 8

0 7 -

-0

06 - 5 05 c u w ln

ul ln 0

0 4

0 3 - J

I- 4

2 02.-

HELIUM

I

i

4y f

0 2 4 6 8 10 I2 I4 I6 I8 20 22 24 26 28 30

POSITRON ENERGY lev)

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 c t a / . (1978): A, Burciagaer a / . (1977); +. Wilson (1978). The theoretical calculation of Cam- peanu and Humberston (1977) is represented by the solid curve.

terjng and spiraling. discussed in Section 111 suggest that the data of Can- ter 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 OTHER INERT GASES AT 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

148 T. C . Griffith

+Pi i i + 1, NEON

f f I 5

_--------- rv

m t! Net

I21 5 eVI

-

-

I I I 0 2 4 6 8 10 12 I4 I6 I8 20 2 2 2 4

I I I I

‘I Net

I21 5 eVI

L

POSITRON ENERGY lev1

FIG. 6. Positron-neon total cross sections at low energies. Experimental points: A, Ja- duszliwer 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 Gil- lespie 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


0 ' I I I 1 1 I I I 1 I 0 2 I 6 8 10 12 IL 16 18 20 22

POSITRON ENERGY l e v )

FIG. 7 . Positron-argon total cross sections at low energies. Experimental points: 0. Ja- duszliwer 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

150 T . C . Grv$th

30 0

25 0

- *a la !2 I

20 0 2 0 + V UJ u)

u) u) 0 E 0

U I5 0

_I

+ 2

10 0

5 0

3 0

I

XENON and KRYPTON A I

i I

1

P ' G A ' 9

0 i I 6

I I I I I I I I I I I I

0 2 L 6 8 I0 I2 1L 16 20 LO 60 80 100

POSITRON ENERGY lev)

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-INTERMEDIATE ENERGIES

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-

I52 T. C . Grjffith

0 3 0 1 '

O Z 0 I 0 1

0 1 I Y L I I I I I I I I

0 50 100 200 300 LOO 5w 600 700 Bw 900


FIG. 9. Positron-helium total cross sections at intermediate energies. Experimental points: +, Brenton ef a/. (1977h 0. Jaduszliwerei a / . (1975); 0, Coleman ef d . (1976); 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.

EXPERIMENTAL ASPECTS OF POSITRON COLLISIONS IN GASES I53

3 0

- *a m c .

b

z

-

p 2 0 - V w m

m m -

U

_I

2 1 0 - 2

-

- 0

\'k 6 ' ' \\'\

e' - Neon

+ t 0

0 D O

I I I 1 I I 1 1

3 0 200 300 LOO 500 600 700 800 900

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 CROSS SECTION

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 ti l . (1976) have calculated the positronium formation cross section ups as 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,s can 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 eV to the 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

l o t

0

I 1 I I I I I I I 1 I

0 50 700 200 300 LOO 500 600 700 800 900


FIG. 1 1 . Total cross sections for e+-Ar scattering at intermediate energies. Experi- mental points: 0, Griffith et ( I / . (1979); f, Brenton et a/. (1978); 0, Tsai et a/ . (1976); 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 values are quoted as 0 . 1 5 ~ ~ 2 , at 16.7 eV by Griffith and Heyland (1978) and 0 . 2 3 ~ ~ 2 , by Stein et al . (1978). In argon the value at 11.5 eV is about 2 T U 2 .

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 ups can be set as 0 . 1 6 ~ ~ 2 , at 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 ups is assumed to increase linearly from zero at threshold. At the same energy the values of ups deduced from the beam and lifetime measurements are in good agreement.


E. CROSS SECTIONS FOR MOLECULAR GASES

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

- 8 0 -

i . .

O L '\..,..' I 1 I , , I I I I I I 1 1 I I

0 2 L 6 8 I0 12 1.4 16 18 20 LO 60 80 I00 250 LOO


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. (1977) for N,; 0. Griffith and Heyland (1978) for H,; 0. Kauppila rr a / . (1977) foi H,; 0, D. 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 N I . (1974) for H,. The thresholds for positronium formation, for molecular dissociation. and for ionization are indicated by the arrows.

156 T . C . Grxtith

0 -

CARBON DIOXIDE

I

I

I I

2 - I I I

I I I 1 I I I I I I

Q I P

Vt€€ if

I 1 I I I I I I I I I I I I I I I I

P I 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 COLLISIONS 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 rt 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 HELIUM ABOVE THE 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 kine- matics 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 dis- similar 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

158 T . C . Grijjjth

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 AE. This behavior is consistent with the so- called 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 to ' note that the classical positron-atomic hydrogen ionization calculation of Percival and Valen- tine (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

ENERGY DISPLACEMENT OF THE INELASTIC PEAKS

E," (eV) 50 75 100 150 200 250 300 400 500 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.


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

160 T . C . Cr$3th

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

(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

Y ( t ) = A exp(- A f t ) + B exp( - A,t)

Af = wpZefr9 A, = OX, + 4wpiZeff ( 5 )

where p is the gas density in amagats, the vacuum decay rate of ortho-positronium and o p = .rrr',cn has already been defined by Humber- ston (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 LIFETIME PARAMETERS FOR THE I N E R T GASES

Density Shoulder range width

Gas (amagats) Zeir ,Z,, (nsec amagat) F

He 2-60 3.94 2 0.02 0.125 t 0.002 1700 t 50 0.23 N e 7-40 5.99 -t 0.08 0.235 2 0.008 1700 t 200 0.26 Ar 0-280 27- 18" 0.314 t 0.003 362 2 5 0.33 Kr 0- I17 66.8-39.5" 0.445 t 0.005 296 -t 6 0.16-0.28" Xe 1-6 320 ? 10 1.03 2 0.08 200 t 20 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 Grif- fith and Heyland (1978).

I62 T . C. GrifJith

7 I I I I L

h '\

P' f-\r -

B 25-

'L' \

N

=-4 20-

L 1 7 I I

L

0.46- ,--I----*-,

-%-

,Y' 0.30 - /

!L

NITROGEN 1 = 297K /i 0-20 - ?/

/ I I I I

0 50 100 150 200 250

shoulder region is well reproduced by the calculated phase shifts. The calculated value of 3.88 for Zeff at 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

I

I62 T . C. GrifJith

shoulder region is well reproduced by the calculated phase shifts. The calculated value of 3.88 for Zeff at 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

NITROGEN 1 = 297K //

0.20 Ld/A I I I I I

0 50 100 150 200 250

DENSITY p (AMAGATS)

FIG. 13. Len. &w. and h'for Nz at ZY7 K and densities up to 240 amagats. Upper curve shows that lZ,n is constant at 0.26. Lower curve shows that F has a maximum value at 140 amagats (see Griffith and Heyland, 1978, for details).


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. LIFETIME PARAMETERS FOR MOLECULAR GASES

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. VACUUM LIFETIME OF ORTHO-POSITRONIUM

An account of lifetime studies would not be complete without a brief reference to the important measurements and calculations of that have 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 VALUES OF THE LIFETIME PARAMETERS FOR MOLECULAR GASES

Density range

Gas (amagats) ZeK ~zeff F

D2 12-39 13.6 t 0.1 0.186 t 0.004 - H2 19- I70 13.6-12.6 0.186 -0.45 N2 0-234 30.6- 18.6 0.260 t 0.005 0.19-0.36 co 0- 172 38.5- 24 0.285 ? 0.01 0.28-0.55 co2 0-48 so- 120 0.48 t 0.02 0.50-0.60

7-215 26- 19 80- 14 0.32-0.56 CCIZF2 0.3-5 750- 1500 0.57 2 0.02 0.4s 0 2

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 a l . (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 illustra- tions, the Photographic Section for photographs, and Mrs. Pat Crowther for typing the manuscript.

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ADVANCES IN ATOMIC AND MOLECULAR PHYSICS. VOL. IS

REACTIVE SCATTERING: RECENT ADVANCES IN THEORY AND EXPERIMENT

RICHARD B . BERNSTEIN Chemistry Deparrment Columbia University N e n York, New York

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 11. Potential-Energy Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

111. Classical Trajectory Methods and Results . . . . . . . . . . . . . . . . . . . . . . . . . . 171 IV. Transition-State Theory: New Developments . . . . . . . . . . . . . . . . . . . . . . . 173 V. Collisional Ionization: Nonadiabatic

VI. Accurate Quantal Scattering Calculations .................... VII. Information-Theoretic Approach to

VIII. Molecular-Beam Chemistry . . . . . . . IX. Crossed-Beam Chemihminescence X. State-to-State Cross Sections . . . . . .

XI. Influence of Different Forms of Ene

B. Vibrational Enhancement. . . . . .

D. Translational Excitation of Reac

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

A. Electronically Excited Reagents.. . . . . . . . . . 189

C. Rotationally Excited Reagents . . . . . . . . . . . . 190

XII. Translational Thresholds . . . . . . . . . . . . .

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 Bern- stein (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 Dis- cussion 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, less- sophisticated 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 + H 2 + H F + 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

/ / / / I \ \ \‘lo 1.00 1.25 1.50 1.75 2.00 20

IF-,, ( A ) FIG. 1 . Preferred semiempirical potential energy surface for FH2. collinear configura-

tion: 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 re- lease &, = 0.24, compared to the experimental values (Polanyi and Wood- all, 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

: 8.0 taPO

O0 2.0 4.0

RHg-I'

FIG. 2. Empirical potential surface for Hg + I2 system (Mayer er al., 1977b). Left portion, C,, collinear configuration; right, CZv insertion approach, separated by oblique line. Note different coordinates for the two configurations. Solid contours based on observations, dashed contours estimated.


I HgI L

( - I .45eV)

FIG. 3. Minimum energy path for the Hg + I, reaction (Mayer C I al., 1977a). Note der Waals minimum, then barrier preceding deep IHgl potential well responsible complex-mode dynamicat behavior.

van for

of a potential surface are sometimes useful. From a crossed-molecular- beam study of the endoergic reaction

H g + I , + H g l + 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 H2 and 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]. Clas- sical, 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 Csiz- madia e f al. (1970). Hopping locations were chosen to be the avoided in- tersections 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, REACTIONS

0 2 4 6 0 2 4 6 0 2 4 6 8 - E + , ( e V )

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).


I 5.5 1

0 0 2 4 6

Eir ( e v )

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).

D+ + HD H + + D t + H D + + D I 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 quantal- scattering (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,


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 + H2 exchange 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,

0

a 0

0.5

0.0 0

STATE THEORY

UNIFIED STATISTICAL THEORY

POINTS: EXACT CALC.

1.0 2.0 E ( e V )

3

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 S- matrix 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 Landau- Zener-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 + Br low E

K + Br2+ K+ + Br, I 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 + B r 2

- E + ~ (KJ 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).


__________----------

0 102 103 1’04 - 100 10’

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 cf ~ 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 HI2 were 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).

178 Richard B . Bernsrein

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 a l . (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 EAxy is the vertical electron affinity of the halogen molecule. Fig- ure 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.

. * - N a + X Y

A - K + X Y

- cs + XY

I I I 1 I 1 I

4 5 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.


4 ai a a

0

; 0.5

1

z I-

cn v)

0

V z t-

a

0

VI. Accurate Quantal Scattering Calculations

- D+H~--DH+H (EXP’T’L) ----

-

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 rt al. (1972).

The detailed quantal results are of great value not only because they are essentially predictive of experiment, but because they serve as bench- marks for testing various approximation schemes, such as the semiclassi- cally 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

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 + H2 exchange reaction by Kuppermann and Schatz (197.5) and Schatz and Kupperrnann (1976).


(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 simplifica- tions 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 Hz product of the ,H + H2 exchange 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),


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 up- to-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., Par- rish and Herschbach, 1973; see also Kingrt al., 1974; Dixon and Hersch- bach, 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. Co- reagents 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


reactions of considerable significance has been reported by Bauer et a/ . (1978), namely, the D + HX exchange reactions

+ D + H +F: HI 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,,*)

h u 5tNO 8, I 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 ADVANCES

3 - w - n

0

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.

184 Richard B. Bernstein

uCL of -160 A, for this reaction, and -90 h2 for 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 ucL of 40 and 20 A,, respectively, for the reactions (with Br, and ClJ of Hg(63Po) reagent atoms. This was confirmed experimentally by Hayashi et al. (1978), who measured a value of cCL = 3 Az for the reaction 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., Red- path 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 I I I I I NO + 0, - No; + 0,

0.06 o*Mpy 0.02 0 2 3 4 5

1 . - I I I 0 2 4 6 0 1 0 -

E , ~ ( KCAL MOL-')

FIG. 14. Cherniluminescence cross section function for the reaction NO + 03+ NO,* + 0, (Redpath er ul . , 1978).


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 + N 2 0 +

PbO* + N, (Wicke er al . . 1978).

I86 Ritlzard 3. 3rrnstvin

Manos and Parson (1978). Figure 16 shows the CL translational energy dependence for the three concurrent reactions:

(CZrIl,2)

La + O2 + 0 + La0 (B22) 1 (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.

a Lo + 02 - LOO + 0

10

Lao 1 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).


FIG. 17. Chemiluminescence cross section for the dioxetane reaction of O,(’AJ with methyl vinyl ether (Alban pr d., 1978).

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 to be - 5 xz. The ground-state reaction is exoergic by 7.2 eV, and light emission from Pb atoms excited up to this limit was observed. A related reaction,

N 2 0 + B + BO* + N,

has been studied by Tang et 01. (1976) and Brzychcy et al. (1979).

surfaces in dealing with the dynamics of chemiluminescent reactions. Clearly one must take cognizance of the many excited-state potential

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,

188 Richard B . Bernstrin

i.e., k,-,f(T), but also the more fundamental quantities, state-to-state cross sections, i.e., as a function of collision energy. Here i refers to a 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 laser- induced 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 I 1 I

649 650 651 652 653 654 Xnrn

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).


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

I + I + F + F 0

-\ 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).

190 Richard B. Bernstein

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 (Va- lentini 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 (Evlb z= 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 Wol- frum (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 EXCITED REAGENTS

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

] + N a + H + E:Fl HF(J = 0-14) HCI(J = 0-19)

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.


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 via a long-lived complex (Stolte et d., 1976). The measurements were of the so-called reactive branching fraction, i.e., (T~/(T,-, where mR is the 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 1 I I 3 4 5 6

E t r l h c o l m o l - ’ )

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 “thermal” reagents (&,, 2.3 kcal rnol-I); the abscissa is the average collision energy


I I I !

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 Bern- stein, 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

U R E,'(E - 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 I I 1 I I I 1 I


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 Bern- stein, 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

U R E,'(E - Eo)", 0 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 I 1 I I I 1 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).


I

0 5 10 15 Etr (KCAL M O C ' I

0 -

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

H C I + K - + K C I + 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 + 1 2 + HgI + I

194 Rich urd B . Be rns rein

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 + Cst

{CsXet Xe + CsCl+ C1- +

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; Bern-

t 0 -I

>

I

w U CI,

I I I I I I

Hg+12- HgI+I - L r”

- E+,(eV 1

FIG. 23. In-plane yield of HgI product from the Hg + I, reaction vs. average collision energy. Solid curve: calculated, best fit of convoluted line-of-centers cross section function, with indicated translational threshold energy (/it,, = 1.14 f 0.03 eV). From Wilcomb rt a / . ( I 976).


- 2 0.75 ._ c

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).

-

K+CH3Br-KBr +CH3 ''@I----- * /

I 0.8 I

''mr R b + CH3Br - RbBr + CH,

3

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).


stein, 1973). This introduces nonnegligible uncertainties in the experimentally derived Eth and 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 trihal- ogen 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 HI to produce IIF, CIIF, and HIF. respectively; thresholds indicated. From Valentini er a / . (1976).


I + I + F + F 0

(- 36)

\ \

I *x

\ (-66) \-3--F

(-73!,-;?’ I I F + F I 2 + F 2

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 IIF

F, + C I I + F + C l l F 11 /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 0 W 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.


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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 . Leu. 19, 1. Light. J . C. (1979). In "Atom-Molecule Collision Theory: A Guide for the Experimentalist"

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. At. Cnllisions, Brh, 1973. Invited Lectures. Los, J . , and Kleyn, A. W. (1979). In "Alkali Halide Vapors" (P. Davidovits and D. L. Mc-

Love. R. L., Herrmann, J. M.. Bickes, R. W., Jr., and Bernstein, R. B. (1977). J . Am.

McClure, G . W., and Peek, J. M. (1972). "Dissociation in Heavy Particle Collisions." Wiley

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.

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.

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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.

Micha, D. A. (1975). A h . Chem. Phys. 30,l. Miller, W. H., ed. (1976a). "Dynamics of Molecular Collisions," Parts A and B. Plenum.

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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.”

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. Opt . 10, 1848. Ottinger, C., and Zare, R. N. (1970). Chem. Phys. Lef 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.

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A D V A N C E S IN ATOMIC A N D MOI.FCUL.AR PHYSICS. V O L I 5 l l ~ ION-ATOM CHARGE

! LOW ENERGIES TRANSFER COLLISIONS AT

J . B . HASTED Birhbech College Univcnitj of' L o t h n London, Eriglond

I . Introduction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 11. Symmetrical Resonance Proc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

111. Nonresonant Atomic Charge Transfer Processes . . . . 211 IV. Total Cross Sections of Pseudocrossing Atomic Charge Transfer

Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 I4 V . Curve-Crossing Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 V1. Charge Transfer Processes of Excited Ions . . . . . . . . . . . . . . . . . . . . . . . 229

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

I. Introduction

At impact energies not in excess of one atomic unit of velocity. ion- atom 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. Distinc- tions 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.

2 0 5

CoDrrlehl ti) I979 hk Academic Re\ \ 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:

(1)

Using the JWKB approximation the phase can be expressed in the form P(u, b) = sin2(r), - vg)

(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 oscil- lates between zero and unity for impact parameter values less than a value bo, outside of which i t falls off exponentially. Without great loss of accuracy, P(b, v ) can be replaced by its mean value of 0.5 for b =s b,, 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 u = 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 ng and nu is different from that between C, and Xu. When the calculations are generalized and averaged over the appro-

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

6

b

4 lo6 lo7

v (cmkec) - FIG. 2. Symmetrical resonance charge transfer cross section function U ' ' ~ ( I J ) for 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) experiments; A, Dillon PI ul. (1955) experiments; 0, Flaks and Solovev (1958) experiments; H, Kushnir P I n l . (1959) experiments; ..., Langevin orbiting limit.


phase shift equations over nonrectilinear trajectories (Sinha and Bardsley, 1976) confirm the prediction that there is an intermediate region of energy in which both the ( A - B in v ) ~ behavior and the Langevin limit 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 in- stances of oscillating behavior, and the Li+Li collision illustrated in Fig. 3

I I I I I

b

120

100

i 8 12 16 20

v-'(a.u.) - FIG. 3 . Li+Li charge transfer cross section as a function of inverse velocity; U, calcula-

tions of Sinha and Bardsley (1976); .. calculations of McMillan (1971); A, experimental data of Perel et c r l . (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

N \

- b

3.4

3.2

3.0

2.8

2.6

E (keV) - 2 4 7 10 1

I I I I 1 2.5 3 4 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 difficulties of accurate low-pressure (- Torr) determination as well as 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 Krf2P1,2 ion 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

212 J . B . Hasted

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

H12(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:

(6)

This leads to an expression for the transition probability P at impact parameter b:

IHdRII = !ilVi(Rc) - VARAI = !dAV(Rc)I

+V(RC)l P = sech2 ( 2AY(R,) ) sin2 (!- n 1- --m H12 dr) (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

c1 . . . 11 - - -hEm-2+

H A a ? + C . . . 2 ~ 4 R= 22 - (9)

ION-ATOM CHARGE TRANSFER COLLISIONS AT LOW ENERGIES 2 I3

b (a,) - FIG. 5. Transition probability P as a function of impact parameter calculated (Olson,

1972) for the following situation:

H I 2 = e-m, V(R,) = 0.0 a.u. . R, = 15.350,

,4 = 0.30;'

-. Calculated from equation; ---, calculated with the sinz term approximated by 1.

where C1,2 are van der Waals constants and A a is the difference between the polarizabilities a l , a2 of 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 -2ala2/R7.

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 TOTAL CROSS SEcrioNs cmZ) AT M A X I M U M

Calc. Exp. Reference

NaLi+ S Na'Li 5.5 6.4 Daley and Perel (1969) RbK+ S Rb+K 7.5 9.5 Perel and Yahiku (1967) Li+K + Li(2p) 3.1 5.4 Daley and Perel (1969)

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 (1 1 )

X = (Ei/13.6)1/2 (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 E ~ ~ / E ~ ' ~ ) graph giving 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-

ION-ATOM 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 avoided- crossing 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’-

where p is the reduced mass of the system, the separation at the avoided crossing.

The total cross section is

(T = 47rR:I(r))

where

(13)

(14) R=R,

V’ denotes aV/aR, and R , is

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 1 4iv+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.

216 .I. B . Hasted

An asympotically exact determination of Hlz 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 H12 upon R , using experimental data and computations. The most recent empirical parametrization, due to Olson er a / . (1971) is

(19) HT2 = 0.662 exp( - 0.858R:)

R: = (a - y)Rx (20)

H12 = 2Hlz/aYRX (21)

where

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 AEm 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.

,5p'FIx \ '... .

, .. \ '.. 7d, m 5 f l S ! \ ,... 5f 1x

i 1 IX

I It \ '.. \ '.. 5! m5dFI):

I J J .I.... j I j I %ASS I 1 2 3 4 5 6 7 8 9 10 11

i , '. ...

AV ( 1 o 3 C ~ ' ) - FIG. 7. Variation of statistically weighted thermal energy cross section with energy de-

fect (AC, wavenurnbers) for metal-ion laser charge transfer processes (Turner-Smith et al. , 1973). .-., He+Cd, ---. Ne+Mg, calculated using Landau-Zener approximation with empirical H , 2 .

ION-ATOM CHARGE TRANSFER COLLISIONS AT LOW ENERGIFS 2 I7

10"

10'

1 104 - 0

* N

?

1-

Y

loe

lo-'"

, 1 " ' ~ ' 1 '

\

0 4 8 12 16 20 24 28

Rf (au.) - FIG. 8 . Empirical variation (Olson er l i / . , 1971) with H: of 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

( 2 2 )

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 caicula- tion is given in Fig. 9. Only two channels contribute to the process, and H I 2 calculations were made using Morse wavefunctions. It is seen that the measured cross section function, for which the error is given rather con- servatively as *7%, exceeds the calculated values by a factor of almost two. Other tests of Landau-Zener calculations show a similar overesti-

B3+ + He - B2+ + He+ '

218 J . B . Hcisted

(EXP) v i ~ ~ ' c r n / s e c ) - 0 4 06 1 2 4 6 8 10 20 40 60

I I I 8 1 1 1 I , I 1 1 1 l 1 1 , ' ( I

30

25

I t 2o

3 15 N O 0

I

.! 10 b

5

2 3 4 5 0

(BJS) log E (eV) - 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 Wiese- mann, 1978) of ArZ+Ne and Ar2+Ar there 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(2 are the probabilities Pl,2 of the system being on potential energy curves 1,2. For the ground-state Be2+H collision both types of calculation were made, and are illustrated in Fig. 9.

ION-ATOM CHARGE TRANSFER COLLISIONS A T LOW ENERGIES 2 I9

C" He 2 keV

r.=4 35 i

I I I P o 1 I I

C" He 2 keV

r.=4 35 i

I I I P o 1 I I 1 3.0 H,,(01) law 0

0 1 2 3 4

b (a.u.) - FIG. 10. Comparison of measured (-) and calculated (.. .)oscillating transition probabili-

ties P ( h ) at 2 keV impact energy for CZ+He ground-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 Landau- Zener 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 im- pacting 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-

220 J . B . Hrrstrd

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 10 *

E(keV) - 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).

I O N - A T O M 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 move- ment 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

2 2 2

I , , , , I , I l l 1 , I , , ,I>

J . B . Hasted

1 I , , I I , , 1 , 1 1 , I j

2600

1300

0 128

I $ c

0

?! c C 3

8 +a, I

200 eV

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

- I l l I I I I I I I I I I I I I

20

10

19

8

4

- 1IIII11lIIIIIII11111 I

18

,-.

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).

224

A 0

J . B. Hustrd

0 5 10 15

AE (eV) - Fic. 14. Energy loss spectrum at E = 500 eV, 0 = 0.75" for single-electron capture in

H+Xe(Kubacheta/ . , 1976) . (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 move- ment 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 united- atom 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 united- atom 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,


I \ \ i i

01

0

H CS' -0-35

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 spec-

TABLE I 1

PUBLISHED CURVE-CROSSING SPECTRA

H+He, Ne, Ar, Kr, Xe H+Xe He+Ne He+He He+Ne. Ar, Kr , Xe Ne+He

He2+Ar

C2+He C2+Ne 1 I

OZ+He A P H e Ar2+Ne Ar2+Ar

Abignoli el a/ . (1972) Kubach et 11/. (1976) Barat ef a / . (1976) Rrenot e t ( I / . (1975)

Baudon P I a/ . (1970)

Chen et t i / . (1974): Chen (1972)

Makhdis et c r / . (1976)

Sat0 and Moore ( 1978)

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 small- est 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+Ar spectra (Fig. 13) (Chen, 1972) show additional features of interest, among which is the possible contribution of the continuum process

(26) Hez+ + Ar + He+ + Ar+ + e

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+Kr spectrum.

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 energies at large separations, so that the crossings appear as in the ~ ( b ) diagram in Fig. 17, at 10ao and 6u0.

FIG. 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

Q) 73

2 - 200

0

-200

-400

0 2 4 6 8 1 0

b (a,) - FIG. 17. ~ ( h ) function for production of Xe+ *PI,, from H+Xe. Numbering refers to the

paths labeled in Fig. 16.

228 f. B . Hasted

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 reduced classical deflection functions ~ ( b ) . For 7 s 417 eV deg, four impact 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+He angular 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/ . . process A; (1976). 0, Process A; A, process B (Fig. 14); convoluted calculations:

, process B . _ _ _


10

5

2

t 10 I

v) - E 5

e : 2 3

10

5

2

1

02'He

- \ o'2P

Ir2D 0 1000 2000

z (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 Sec- tions Ill and IV. It would also be desirable for checks to be made using the translational spectroscopy technique of Section V.

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p. 720.


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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Radiative e -O+ Recombination a Complex Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 A . Ion-Ion Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 B. Electron-Ion Recombination . . . . . . . . . . . . . . . . . . . . . . . . 241 Recombination in an Ambient Ele Recombination in an Ambient Neutral Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 A . Electron-Ion Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 B. Ion-Ion Recombination . . . . . . . . . . . . . . . . . . . . 255 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 iono-

2 3 5 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 after- ward 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, 5.8 x cm3 sec-' at 1000 K, they found close agreement between the 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 Cou- lomb 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 (7.5)-.' to 5.3 x cm3 sec-lbut reduces a(1356) and 47774) by factors 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 ex- isted 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).

and 5.2 x

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. Impor- tant 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 H20 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

112 300 112 a, = 9.8 x 10-9 R, 6) (y) 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-


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. Subse- quent 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 aI for 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 I O N - I O N RECOMBINATION

Reduced Rate coefficient Deduced mass at 300 K Rc or RT

Reactant ions ( a m ) cm3sec-l) Ref. ('4)

H30+.( HZO), + NO,*HNO, 46.1 5.7 h 8.8 NO+.(NO,), + NO,*(HNO,), 82.0 3.5 h 7.2 H@+.(H,O)3 + NOT 33.6" 5.5 h 7.3 NH:.(NH3), + CI- 20.9 7 .9 C 8.2 NH:*(NH,), + NO, 24.5 4 .9 C 5.5 NH:*(NH& + NO, 27.6 6.7 C 8.0 H,O+'(H&)3 + CI- 23.6 4.8 C 5.3

" The measurement was made at 182 K and a T-", power law assumed. Smith and Church (1976). Smith ct ul. (1978).

240 D. R. Bates

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 Cou- lomb 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 recom-

TABLE I 1

DIPOLE MOMENTS"

Dipole moment Species (debye units)

1.9 1.3 0.3 2.2

a Hodgman (1951).

ASPECTS OF RECOMBINATION 24 1

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 - neutrals (10)

H,O+-(H,O), + e - 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

Recombination coefficient Electron temperature ( cm3sec-') (K)

NH:.(NH,),s family 0 1 L

3 4

0 1 2 3 4 5

H,o+-(H,O)~ family

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

540 300-8000 300-6000 300-6000

about as above about as above

Leu er a / . (1973), Huang er al. (1976, 1978)

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 Ei above 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


Provided

ELIkT >> 1 (16)

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 lop6 cm3 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

CH+(J = 0) + e 4 CH+(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, /kT , << 1 (20)

Hence (18) must be replaced by

244 D. R . Bates

so that the rate coefficient for given C (or for given excitation cross section) is very much reduced, and moreover a c312 variation is followed.

In their later investigation Huang et al. (1978) postulated that rotational- vibrational excitation is rapid and is the initial rate-limiting step. Modes in which the effective bonds binding the cluster molecules to the ion are bent must be easily excited because just those couples that bring about rotational excitation are involved. However, granted that the excitation cross section may be large enough and, with greater hesitation, granted also that its fall off with energy is as desired, there remains a difficulty: Although the strong inequality (16) may be satisfied at the lower temperatures at which measurements were made, it can scarcely be satisfied at the higher temperatures. The difficulty is grave.

Consequently it is worth turning from indirect to direct dissociative recombination even though a strong case for the indirect process has been skillfully expounded by Bottcher (1978). Developing earlier work by Smirnov (1977) he achieved considerable success in the low temperature region. He took p = 1 and in accord with (17) obtained a c1’2 variation. A check on the observed variation is needed.

For definiteness consider the hydronium family of ions. The main information on the family comes from the equilibrium measurements of Ke- barle et al. (1967) on

H,O+*(HzO),-1 + HZO + M * H,0+.(HzO)8 + M (22)

These show that as s is increased in steps of unity from 1 to 5 the energy D(s) with which an H 2 0 molecule is bound to the ion is successively about 1.47, 0.95,0.72, 0.66, and 0.56 eV. The corresponding values of the mean binding energy B(s) per H20 molecule in the clustered ion are 1.47, 1.21, 1.05,0.95, and 0.87 eV. If the binding were due to the interaction between the permanent dipole of the H20 molecule and the Coulomb field of the ion the separations, R(s) would be 3.7, 4.1, 4.4, 4.6, and 4.8 a.u. A Ryd- berg level would not be expected to be degenerate or nearly so, unless the quantum numbers

n , l b R(s) (in a.u.)

The direct dissociative recombination process envisaged is

(24)

in which rlss denotes a radiationless transition from the continuum to the (nlm,) state around the ion and diss denotes dissociation into the products indicated because of mutual repulsion (which stabilizes the recornbina- tion). Regarded as a function of the separation R between an H 2 0 mole-

(a) { H,O+*(H,O), + e A {H30(nlm,)}.(H,0),

a {H,O(nlm,)}-(H,O),-, + H,O (b)


cule and the remainder of the clustered ion on the left of (24), the potential energy curve of H,O+(H,O), is attractive. At large R it lies above that on the right of (24b), but provided the principal quantum number n is low enough this last may conceivably be weakly repulsive partly because of the direct interaction between the (nlm,) orbital and the H20 dipole and partly because of the effect of the change of R on the orientations that the (s - 1 ) other H 2 0 molecules of the cluster take up in the presence of the orbital.

There are two reasons why the recombination coefficient may be large. First, the repulsive force U ’ [ R ( s ) ] on the H 2 0 molecule concerned is likely to be weak enough to make crU’[R(s)], where a is the amplitude of the initial vibrational motion, smaller than is usual for a diatomic ion and therefore (cf. Bardsley, 1%8; Bardsley and Biondi, 1970) to make acorre- spondingly larger. Second, more than a few (nlm,) orbitals may have binding energies rather less than D ( s ) , so that they could contribute to the recombination. The standard T;”, fall off would tend to be flattened by the existence of contribution orbitals in a wide energy band.

The introduction of processes like (24) appears worth considering but is, of course, highly conjectural.

IV. Recombination in an Ambient Electron Gas

Collisional-radiative (CR) recombination is the name give to the passage of an electron from the free to the ground state by a random walk, each step of which entails a collisional or radiative process, for example, three body recombination,

(25) H + + e + e + H(p) + e

and its inverse collisional ionization,

H(p) + e + Ht + e + e

collisional deexcitation and excitation,

H(p) + e + H ( 4 ) + e

radiative recombination,

(26)

and radiative discrete transitions,

H(p) + H(q) + h u (29)

the inverse of which may normally be ignored unless 4 refers to the ground state, when in the important case of a plasma optically thick

246 D. R . Bates

toward the Lyman lines the inverse can be taken into account by letting the radiative transition probabilities of these lines vanish.

A statistical method of treating the complex of interacting collisional and radiative processes was developed by Bates et al. (1962a,b) and has subsequently been used in all quantitative investigations. Its physical basis is simple.

In thermodynamic equilibrium the ratio nE( p ) / n ( e ) of the number density n , ( p ) of atoms in level p to the number density n(e) of free electrons or of ions is given by

It is hence evident that for a wide range of plasma parameters

nE(p)/n(e) << 1, p z 1 or 2 (32)

p(p) = n(p)/n(e) << 1, p # 1 or 2 (33)

and that for an even wider range’

where n( p ) is the number density while recombination is proceeding. Consequently a quasi-equilibrium number density of excited atoms is established effectively instantaneously-and indeed detailed examination of the problem shows that this applies even to atoms in the first excited level except in cool dense plasmas and in any event causes no real complication (Bates et al., 1962b). The rates at which the excited atonis are produced or destroyed in the quasi-equilibrium are very much greater than the rates at which their number densities change. The rates of production and destruction of atoms in each level p > 1 may therefore be equated. This leads to a set of linear equations for p(p) , which may be restricted in number to s - 1 by assuming that levels s + 1 and above are populated as in thermodyriamic equilibrium and the ground level is empty. Having found p(p) the recombination coefficient aCR may readily be computed.

The binary collisional rate coefficients K ( p , q ) are the main source of uncertinty especially at the lower temperatures. Bates el al. (1962a,b) adopted coefficients obtained from a minor modification of the classical impulse approximation given by Gryzinsky (1959). Johnson and Hinnov (1973) sought to improve the position by utilizing the self-consistent semiempirical electron hydrogen atom cross sections that Johnson (1972)

Actually condition (33) need only be satisfied in the defile of Byron e f a / . (1%2).


had compiled from the available experimental and theoretical information. At the lower temperatures their values of acR are considerably less than those of Bates et al. (1962b). The most reliable calculations on collisional radiative recombination to date should be those carried out by Stevefelt er al. (1975). They are based on rate coefficients K ( p , 4) that were obtained for high p and y by Mansbach and Keck (1969) from classical trajectories using the Monte Carlo method. They are especially well suited to the problem because of the major role played by rate coefficients between levels having binding energies several times the thermal energy and being therefore in the defile of Byron el al. (1962). The derived aCR are approximately midway between the other two sets reported in this paragraph. Stevefelt et al. (1975) found that their results for a plasma optically thick toward the Lyman lines for 250 K s T , S 4000 K and for loe ~ m - ~ s n(e) d 1013 cm-3 could be reproduced to within 10% by the simple formula

aCR = [ l S S x 10-10T,-0.s3 + 6.0 x 10-sT,-2.18n(e)0.37 + 3.8 x 10-sT;4.5n(e)] cm3 sec-l (34)

The first term of this represents purely radiative recombination, the third term represents purely collisional recombination, while the middle term represents the complex nonadditive effect of the collisional and radiative processes occurring. For reference purposes (34) was evaluated at selected T , and n(e) and the results presented in Table IV. Stevefelt et al. (1975) compared the theoretical aCR of themselves and of the earlier investigations with selected experimental data (obtained mainly in helium plasmas). They found discrepancies of up to a factor of about two in the case of their own results, presumably the most reliable. The improvement since 1962 was rather disappointing doubtless partly because the mea-

TABLE IV

COLLISIONAL-RADIATIVE RECOMBINATION COEFFICIENT aCR AS GIVEN BY (34)"

aCR (cm3sec-')

250 I .4-'0 8.0-1° 6.6-@ 6 P 6.2-l 500 2.3-11 7.0-11 3.7-10 2.9-e 2.8-8

1000 5.8-12 I .2-11 3.4-11 1.7-10 1.3-0 2000 2 . 1 - 1 9 3.3-12 6.3-12 1.7-11 7.9-11 4000 1.0-12 1.3-12 1 3.4-12 8.6-12

"The indices give the power of 10 by which entries in the columns for aCR must be multiplied.

248 D. R. Bates

surement of aCH is difficult. Indeed there is a difference of opinion on the interpretation of some of the more recent measurements (cf. Bates, 1979a). These were conducted in helium afterglows, which as the tortur- ous history of the subject reveals, are frustratingly complicated despite the simplicity of the parent gas.

In an influential paper on the decay of a helium afterglow at a temperature of 300 K and pressure p in the range 15 sz p d 60 torr where He: is the dominant ionic species, Johnson and Gerardo (1972a) pointed to the presence of an intense source S of ionization due to collisions between metastable atoms and molecules. They attempted to allow for S by heating the electrons for a brief interval by a low-level current pulse. Shortly after the cessation of the pulse the electron temperature T, and also S dropped to their original values, this last being verified by monitoring the number density of helium atoms in the second quantum levels (Johnson and Gerardo, 1972b). However, the electron density, enhanced owing to the reduction in the recombination coefficient a when T , was raised, recovered more slowly. Consistent with measurements they made on the dependence of the helium band intensity on the electron density, Johnson and Gerardo assumed that the recombination coefficient may be expressed in the form

a = a0 + kenn(He) (35)

and from their results deduced that

(yo = 1.1 x cm3 sec-', ken = 1.2, X cm6 sec-' (36)

Although aCR of (34) refers to H+ ions it should also describe the rate of collisional Y'adiative recombination to He+ ions and He: ions unless the latter rate were enhanced (cf. Collins, 1965) by the loss of electronic energy being facilitated by suitably placed repulsive potential energy curves. The conflict between (34) and (35) and (36) might therefore seem a matter of serious concern in that the n(e) range covered by the measurements extended to 5 x 10" ~ m - ~ . However, Johnson and Gerardo (1976) have reported that their work does not exclude the presence in (35) of a term

k,,n(e), with k,, < 2 x cms sec-' (37)

According to (34) the value of k,, at 300 K is 2.7 x there is in fact little cause for unease.

tive recombination

cm6 sec-I so that

It would be natural to attribute the constant term a. in (35) to dissocia-

(38) He: + e + He + He'


especially as the monitoring carried out by Johnson and Gerardo (1972b) led to the important result that at least half the He,+ ions that recombine yield He atoms in the second quantum levels. However, there are strong reasons (Mulliken, 1964; Cohen, 1976a) for believing that the rate of (38) is very slow unless the ions are vibrationally excited; and vibrational relaxation proceeds too rapidly (cf. Bates, 1979b) for this possibility to be invoked. The difficulty casts doubt on the reality of such a large constant term the magnitude of which was obtained by an extrapolation to zero ambient gas density rendered somewhat uncertain by the absence of the k,,n(e) term.

Deloche et ci / . (1976) also studied recombination in a high-pressure, 5 4 p G 100 torr, helium afterglow. Their approach was quite different. They measured atomic and molecular ion currents to the walls, atomic and molecular metastable number densities, electron number densities, elastic electron collision frequencies, and electron radiation temperatures as functions of the time for a wide range of conditions and compared the data (40,000 points) with the solutions of a system of five coupled partial differential equations, which they judged to allow for all the processes occurring. In this way they were able to obtain the various rate coefficients including that for e-HeJ recombination, which they found to be

The last term is in satisfactory agreement with the last term in formula (34) for aCH. which is the major term in the [n(e), T,] range covered. Con- sistent with the results of Johnson and Gerardo (1972b), Deloche et (11. (1976) found that 70% of the recombined molecular ions provide metastable (itoms.

Convincing experiments have been carried out by Boulmer et ul. ( 1977) on recombination in a low-pressure ( 3 torr) helium afterglow, where He+ ions predominated and T, could be controlled between 300 and 600 K by microwave heating. The afterglow could be subjected to a 10.6 pm laser beam as in an earlier study by Drawin et al. (1974). When acting, the beam photoionizes at a known rate helium atoms in levels with principal quantum number 1 1 or greater. On solving the quasi-equilibrium equations for p(p) it is found that lines originating from levels with p < l l are quenched equally by the perturbation. The extent of the calculated quenching naturally depends on the collisional rate coefficients K( p, 9) adopted. From their measurements on quenching Boulmer et al. (1977) verified that those adopted by Stevefelt et a/ . in computing formula (34)

250 D . R . Bates

for aCR cannot be much in error. The rate coefficients being the only significant source of uncertainty it follows that formula (34) is reliable Boulmer et al. also determined the recombination coefficient from the absolute photon intensity of all lines ending on the second quantum levels, thus avoid the difficulty (Johnson and Gerardo, 1972a) the presence of an ionization source causes. Their value aeXp differs from aCR of (34) by an amount

a1 a e x p - ~ C R (40)

= 3.5 x 10-5c1.9 cm3 sec-I (41) apparently independent of n(e), which supports the conclusion already reached that (34) describes collisional radiative recombination quite accurately. Boulmer et al. (1977) attributed aI to

He+ + e + He + He + He (42) Understandably it is only under afterglow conditions making a1 small compared with aexp that the quenching by the laser beam is as predicted from simple theory.

It is at first surprising that collisional-radiative recombination to He: ions in high-pressure afterglows should give the observed (Johnson and Gerardo, 1972b, Deloche et a l . , 1976) large fraction of He atoms in the second quantum levels. According to Bates (1979b) the explanation is probably that a collision between an excited (n > 2) helium molecule and a normal helium atom leads to a long-lived complex

He + He; + He; (43) and that this complex then dissociates along one of the exothermic channels

He; + He(llS) + He(1'S) + He(21S,21P,23S,or23P) (44)

yielding two normal atoms and an atom in one of the second quantum levels (which are the only excited levels accessible energetically). Con- sistent with this it has been observed (cf. Deloche er al., 1976) that the proportion of recombining molecular ions producing molecular metas- tables decreases with increasing gas pressure.

V. Recombination in an Ambient Neutral Gas

A. ELECTRON-ION RECOMBINATION

Several theoretical treatments have been given of the recombination of electrons and ions X+ in an ambient neutral gas,


X+ + e + A + X + A (45) Massey and Burhop (1952) were the first to estimate the rate of the

process. Their estimate was based on the well-known theory of Thomson (1924). In developing this Thomson adopted equal masses for the three colliding particles. Massey and Burhop took into account that a free electron, because of its very small relative mass, can experience only a slight change of kinetic energy in an elastic collision with an atom. The modified Thomson formula for the recombination coefficient is

(46)

where M is the reduced ion-atom mass and u the momentum transfer cross section for electron-atom collisions. Even if the problem were purely classical, some uncertainty would be introduced by the approximations made. Pitaevskii (1962) sought to reduce this. He noted that a characteristic feature of the elastic collisions is that whereas the energy change of the electron is minute, the momentum change may be large and that therefore the time needed for the energy distribution to reach equilibrium greatly exceeds the other relaxation times. In consequence the recombination process may be regarded classically as diffusion in energy space in the direction of negative energies and may be described by the Fokker-Planck equation. On solving the equation Pitaevskii found the recombination coefficient to be

128(3d)1’2 e6m1I2 n ( A b 81 M(kT)5/2

(YT =

The main defect of the model is that the bound energy levels of the electron in the field of the ion are treated as forming a continuum, which would tend to make ap an overestimate. Another defect (unimportant except in a low gas density and high-temperature region not usually of interest) is that radiative transitions are ignored.

To allow for the bound levels being discrete it is necessary to know the binary rate coefficients for collision-induced transitions between the levels. In obtaining approximations for these coefficients Bates and Khare (1%5) attributed the appropriate velocity distribution to the electron in the initial level p and, considering it as free to make elastic collisions with the atoms, supposed that a transition to level y takes place if its kinetic energy relative to the core is changed between A and A + d A , where

252 D. R . Bates

They found transitions between neighboring levels to be dominant. The binary rate coefficients K ( p , q ) obtained were shown to give a recombination coefficient that falls toward ap of Pitaevskii in the classical limit. Rather than proceeding as for collisional -radiative recombination (Sec- tion IV) Bates and Khare were content to calculate the recombination coefficient from a simple approximate formula based on the concept (Byron et al . , 1962) of a defile. Crude allowance was made for radiative transitions. Less uncertainty arises from the formula than from the binary rate coefficients.

Gousset et a f . (1977) have measured the rate coefficient for Cs+ + e + He + Cs + He (50)

cms sec-' x n(He). Laboratory

(51)

at 625 K, finding it to be about 4 x determinations of the rates of

He+ + e + He --* He + He

(Boulmer et al., 1977) and of He: + e + He 4 He + He + He (52)

(Johnson and Gerardo, 1972a; Deloche et a f . , 1976) have already (Section IV) been mentioned. Table V compares the experimental results with the theoretical results obtained from the formulations of Pitaevskii (1962) and

TABLE V

ELECTRON -ION RECOMBINATION I N AMBIENT NEUTRAL GAS

Tertiary recombination coefficient

Temperature Measured Calculated Ion (K) (cmasec-') Ref. (cm%ec-') Ref.

Gas: helium cs+ 625 4 x 10-20 a 3.8 x lo-"

3.5 x 10-29

He+ 300 1 . 1 x d 2.4 x 10-27

1.2 x 10-27 He: 300 1.3 x e 1 . 8 x 10-27

5 x 10-27 f 7 x 10-28

Pb+ 1680 4.8 x 10-27 g 2.1 x 10-27 to 5 x 10-27

Gas: nitrogen, hydrogen, water vapor mixture

2270 1.9 x 10-27 9 x 10-28 to 2.1 x 10-27

h h

Gousset er al. (1977). Pitaevskii (1962). Bates and Khare (1965). Boulmer er al. (1977). Hayhurst and Sugden (1965).

Johnson and Gerardo (1972a). Deloche er al. (1976). Bates er a / . (1971b).


of Bates and Khare (1963, the elastic cross section for helium being taken to be 2Oai (cf. Moiseiwitsch, 1962). Recalling that given the elastic scattering model the first of these formulations necessarily overestimates the effectiveness of the collisions, attention will be focused on the second. For completeness the table also compares the rate coefficient for

Pb+ + e + (N, , H,. H,O mixture) - Pb + (N, , H,. H,O mixture) (53)

inferred by Hayhurst and Sugden (1965) from their studies of recombination in flames at atmospheric pressure with the lower and upper limits to the rate coefficient calculated by Bates et al. (1971b) using an extension of the method of Bates and Khare, which takes rotational and vibrational transitions into account. For a system as far from hydrogenic as Pb the upper limit was expected to provide a better approximation than the lower.

It may be seen that the recommended theoretical recombination coefficient is in good agreement with experiment for Cs+ and Pb+ ions. As has been noted (Section IV) a complication arises with He: ions. Consider- ation of it will be deferred.

The remaining case is anomalous. Although it concerns the simplest atomic ion, He+, the success attained with Cs+ and Pb+ ions is not repeated. Instead the calculated rate coefficient is much less than the measured rate coefficient.

A vital feature of the case is the strong attraction between an He atom and an He+ ion in the lui luu %C: configuration, which gives a well depth of 2.469 eV (Liu, 1971), wher-2s the well depth of HeCs+, for example, is only arourl 0.014 eV (Mason and Schamp, 1958). Excited He, molecules with a lug2 luu 5; core also have attractive diabatic potentials, while those with a lug lui ?5: core have repulsive diabatic potentials. An obvious possibility is that transition between Rydberg levels of an atom are facilitated by the crossings between the two sets. Multicrossings occur (Cohen, 1976b). An approach that begins on an attractive potential may switch to a lower repulsive potential that crosses the He: potential at an energetically accessible position and end with associative ionization

(54) adding to the complication of the problem. An assured conclusion must await the results of very extensive computations. The results of quite extensive computations that Cohen ( 1 976a) carried out in another connection are useful only for a preliminary exploration. On the two-state approximation the Landau-Zener probability for staying on the same diatomic curve in a single transit through a crossing is

He(n/. I3L) + He + He:(u) + e. n 3 3

P = exp(- w) ( 5 5 )

254 D. R. Bates

where following an obvious notation

in which

Cohen (1976a) has computed the parameters appearing on the right of (56), or other parameters directly related to them, for 80 of the lower (nf s 6,nf s 4) crossings. As far as can be judged by extrapolating the values of w, the rates of deactivation of atoms in high (ni P 10) Rydberg levels are markedly enhanced by the crossings as required to explain the discrepancy in Table V. In contrast to the position with the original model, transitions characterized by a large change in the principal quantum number n are important.

Turning now to the recombination of He: ions in ambient helium the general sequence of (43) followed by

( 5 8 ) He; -+ He + He,(n’)

which includes (44) as a special case, increases the rate provided the helium molecules taking part in the first stage are in a Rydberg level n that is within or above the defile of Byron el al . (1962) and provided

n’ < n (59)

Geometrical and kinetic considerations suggest that the rate coefficient of (43) may be around 3 x cm3 sec-I, so that even at quite low gas pressure He,(n) may experience a number of collisions during its lifetime (cf. Bates, 1979b). The complex may decay along the inverse elastic channel or along one of the inelastic channels. Channels that lead to a large change in n are naturally especially effective in assisting recombination. The termolecular recombination coefficient cannot be calculated until the various branching ratios are known. All that can yet be said is that the sequence proposed provides a possible explanation of the high measured rate (Table V).

Warman et al . (1979) have made further comparisons between theory and experiment. The disagreement they mention in the case of ambient carbon dioxide is being investigated (Bates, 1979~). Allowance may have to be made for the complex nature of the ions.


B. ION--ION RECOMBINATION

Ion-ion recombination in an ambient neutral gas is the kind of recombination studied earliest (cf. Bates, 1975a) but much remains to be done.

The high-density region was treated theoretically by Langevin ( 1903). Regarding oppositely charged ions as drifting together under the influence of their mutual attraction he showed the recombination coefficient to be2

(YL = 4 ~ r ( K + + K - ) [ (60)

where K+ and K - are the mobilities at the temperature and density concerned and [ is the probability that the ions adhere when they collide, which probability is in fact unity when (60) may be used. Langevin neglected diffusion of the ions. The neglect has long been known to be quite unjustified. Curiously, on taking only this diffusion (treated as Brownian motion) into account, Jaffe (1940) recovered formula (60) except that there was some uncertainty, which he covered by including a factor 4, which may differ slightly from unity. It has now been proved that 5 is exactly unity and the remarkable success of the erroneous simple mobility model of ionic recombination is understood (Bates, 1975b): success is attained because the rate-limiting process is mobility-controlled drift along the fixed electric field, and not diffusion down a predetermined concentration gradient (rather the concentration gradient adjusts itself to give the required rate).

Considerable effort has been devoted to the low-density region. The theoretical work of Thomson (1924), which was mentioned in Section V,A, specifically concerns ions and neutral gas particles of equal mass. It is based on an approximate mathematical artifice and cannot be refined and generalized. The artifice is to introduce a trapping radius

R T = 2e2/3kT (61)

and to suppose, rather arbitarily, that recombination occurs if either ion experiences one or more collisions when within a distance R T of the other. The probability W of this happening may be written

w = w+ + w- - w+w- (62)

where w+ and w- are the corresponding probabilities for positive ions and for negative ions. The general form of the dependence on the ambient gas number density N is obvious on Thomson's model. Remembering that only the first collision within the trapping sphere contributes it is apparent

This is sometimes called the Langevin-Harper formula (with aL replaced by ctH) in rec- ognition of research on the topic by Harper (1932. 1935).

256 D . R. Bates

that W is directly proportional to N in the low-density limit and that the W - N curve is convex upwards with W tending to unity as N tends to infinity. The variation of the recombination coefficient aT on N is naturally similar. In view of its simplicity Thomson’s intuitive model is quite successful. However, a properly based general theory is needed.

In the low-density limit a quasi-equilibrium statistical method similar to that introduced by Bates et al. (1962a) for collisional-radiative recombination but modified to allow for the bound levels forming in effect a continuum is applicable. This method has been used to determine how the recombination coefficient there depends on the ion-neutral interactions and on the relative masses of the three species involved (Bates and Moffett, 1966; Bates and Flannery, 1968). It has been verified that little error is caused by assuming that even when recombination is proceeding the centers of mass of the ion pairs have thermal motion at the temperature of the ambient gas (Bates et al., 1971a) and that the microcanonical angular momentum distribution is preserved (Bates and Mendas, 1975).

The extension of the quasi-equilibrium statistical method beyond the region near the origin where the recombination coefficient a is directly proportional to the ambient gas number density N is simple in principle but difficult in practice. A guide as to how to proceed is provided by Thomson’s assumption that only the first collision within his trapping sphere contributes to the recombination. Clearly the ratio of the number density when recombination is proceeding to the number density in thermodynamic equilibrium is not just a function p ( E ) of the energy E but is a function p(E,R) both of E and of the distance R between the ions. Treating ion pairs having inward radial motion separately from ion pairs having outward radial motion, Bates and Mendas (1978a) derived a set of coupled integro-differential equations for p(E, R) that must be solved numerically. An N-power series expansion for a may be obtained but the convergence is too slow for it to be useful.

In order to study the intermediate density region, Bates and Mendas (1 978b) carried-out so-called computer-simulated experiments. Only the general principle of the design of these will be outlined here. Let y(R,) be the rate coefficient describing collisions in which oppositely charged ions first approach within distance R, of each other and let P(rec,m) be the probability that ions having so approached recombine instead of separating to infinity. The recombination coefficient is obviously given by

a = y(R,)P(rec,m) (63)

Of the factors on the left, y(R,) may be found from Fink’s difhsion equation provided

RoIA12 >> 1 (64)


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 partial- parting 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 ( 69)

Introducing the rate coefficient defined by

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

(72) 0: + 0, + o*+ [O,] + o2

and

0: + 0, + o*+ [O,] + o* (73)

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 low- and high-gas-density regions. As the density is raised isothermally, the contribution ( Y ~ to the total recombination coefficient from the binary 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


TABLE VI

RATE COEFFICIENTS CKM FOR BINARY RECOMBINATION"

Temperature ( K ) Rate coefficient (cm3secc1)

Assumed value for isolated ions

Computed value in molecular oxygen at 1.09 atm (0: + 0; recombination) 300 4-7 2-7 I -7 5 -8 2.5-8

300 6.6-7 4.1-7 2.5-' 1.4-7 0.8-7 600 7.7-7 4.6-7 2.7-7 1 . 5 - 7 0.8-7

lo00 6.7-7 3.7-7 .2.1-7 1.2-7 0.7-7 1500 4.0-7 2.1-7 I . 2 - 7 0.6-' 0.4-' 2000 3.0-7 I .5-7 0.8-7 0.4-7 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 aM and 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

(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]021H2 flames 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.

K+ + Cl-+ K + C1(2P3,2,,,2) + 0.72 or 0.61 e V

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

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ADVANCES IN ATOMIC A N D MOLECULAR PHYSICS. VOL. IS 1 1 I

I I THE THEORY OF

FAST HEAVY PARTICLE COLLISIONS

B . H . BRANSDEN University of Durham Durham, 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 Gil- body (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 ZAe is 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 re-

263 Copyright 0 1979 by Academic Ress. Inc

All rights of reproduction in any form reserved ISBN 0-12-003815-3

264 B . H . Bransden

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 time- dependent Schrodinger equation (in atomic units2),

subject to suitable boundary conditions as t + ? m.

( Z B ) Z +----t ---*

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, = ZB a.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 indepen- dently.

In the high-velocity region, it is usually sufficient to put W = 0 and take a rectilinear trajectory for the nuclear motion

( 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

R(t) = b + vt, v . b = 0

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-

266 B . H . Brnnsden

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

(7)

the Schrodinger equation (4) can be reexpressed in a coupled-channel representation as

(8)

target ion (B + e-), and expanding NrB, t ) as

WrB, t ) = 2 an@, f)+n(re)e-i~c n

i in(b , t ) = C Vnm(R)e*(C-'n'U,(b 9 I ) 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 p v sin 8/2)

Qf, = 27r lom l ~ r ( h , m ) ) ~ h db

(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

( 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 x m 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 neg le~ ted ,~ except of course in discussion of reactions that can occur only through exchange, such as

(12) H(ls) + He(1'S) 4 H(n/) + He(n3L)

A. THE BORN A 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).

268

lo’

B . H . Bransden

- - \ - \

\

‘ ’ ’ 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 “close- coupling” 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 i He, 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. HIGHER ORDER METHODS

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 ar , 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 ignoring the energy differences (a , ~ am) in all channels other than those treated explicitly. He obtained

(15)

where uf is the amplitude calculated from the coupled-channel calculation, ufG the Glauber amplitude, and UFO” a correction term to avoid double counting:

ut(b,w) = N F + a:‘ - N F O ~


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 Sul- livan (1967) expanded both the interaction and the wavefunction q(rB, 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 + He system include pseudostates to represent the continuum.

T H E THEORY OF FAST HEAVY PARTICLE COLLISIONS 27 1

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 U-

TABLE I

CROSS SECTIONS FOR EXCITATION OF HYDROGEN ATOMS B Y PROTON IMPACTn

E(keV) Cross sections in mi:

1 s - 2 s

B c 4 CIO 60 0.161 0.329 0.301

105 0.0975 0.174 0.158 200 0.0530 0.080 0.074

Is- 3s

B CIO F 60 0.032 0.089 0.041

105 0.019 0.043 0.024 200 0.011 0.018 0.012

Is - 2p

S F B C4 C10 S F 0.29 0.182 1.36 1.23 1.16 0.88 0.95 0.16 0.108 1.02 1.03 1.00 0.86 0.85 0.075 0.057 0.70 0.72 0.72 0.67 0.65

Is- 3p Is ---f 3d

B CIO F B C10 F 0.239 0.237 0.168 0.019 0.083 0.040 0.178 0.182 0.147 0.103 0.043 0.023 0.120 0.124 0.109 0.0071 0.016 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 Moi- seiwitch (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 Park e l a/. (1975, 1977); ---. Born approximation: -, U-matrix calculations of Fitchard 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


calculations agree reasonably well with the accurate results. The four- state 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 (660 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 equa- tions8

(19) iH + N; = Ha + Kc, iN+H + I! = Ka + Rc

A derivation can be found in Bransden (1972).


with

H , = F, H - I - Fj , Hnj = (G,, IH - i t 1 Gj) (20) ( I A )

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

( 2 2 ) lim a,(b, t ) = a,,, lim c j ( b , t ) = 0 t+-m t+- m

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. INTERMEDIATE ENERGIES

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,


about A and B and the center of charge C (Anderson et al . , 1974; Antal et a l . , 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 pseudostate orbitals, which were designated % and p, were chosen to maximize the overlap with the He+ united-atom intermediate states. An- other 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

(24) 1

Snl(r) = - ~ 2(/ + 1 ) Snl(r )

1 d2 / ( I + 1) [ - z p + - 2 ; " - - - r

where the parameter aAl is 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


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 1 I-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 CAPTURE FROM THE K SHELL INTO THE K SHELL, PER TARGET ELECTRON I N THE TWO-STATE APPROXIMATION^

Cross-sections in 10-20cm2

Process Energy (MeV) u l u e q t h U e x p

N7+ + Ne 14 0.79 368 355b 19 0.92 342 350

OR+ + Ne 24 0.96 368 435 30 1.08 269 330 35 1.17 195 300

FQ+ + Ne 20 0.8 I 516 440 25 0.90 440 430 30 1 .oo 360 410

19.0 0.52 2.52 2.34 22.6 0.57 5.40 5.83

N7+ + Ar 14.7 0.42 5 . 1 3.2 26.3 0.56 12.6 13.2

Fet + Ar 20 0.42 25.2 9.7d 30 0.52 32.4 29.0 46 0.64 52.8 47.7 76 0.82 39.6 44.3

C6+ + Ar 12.6 0.42 1.30 1.79'

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 ob- taine 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 , << ZB). Reading et al . (1978) have suggested that in these circumstances the projectile-electron interaction can be treated as a perturbation and the complete wavefunction can be accurately approximated by

T H E THEORY O F FAST HEAVY PARTICLE COLLISIONS 279

1 2 3 4 5 6

PROTON E N E R G Y I M t V l

FIG. 4. Electron capture cross sections from the K shell of neon atoms by fast protons. The solid curve represents the calculations of Lin el a / . (1978). 0 are from Rpdbro et a / . (1978), and A are from Cocke el ul. (1977).

a single-center expansion in atomic states and pseudostates centered on the target nucleus B. Thus, the probability of charge exchange is

where q,(rB, t ) is determined by making the single-center experiment (7) and solving the coupled Eq. (8), either directly or in the equivalent integral form. However, it might be expected that the distortion of the final-state function Gf(rB1 t ) would be severe, since after the collision the atom (A + e-) is perturbed by the larger of the charges ZB, and this effect is not included in the formulation of Reading rt a/ . If this is the case, it is possible to allow for distortion in the final state by replacing Gf by a wavefunction Qf, also determined by a single-center expansion, this time about A. Numerically, the determination of two separate single-center expansions is more economical than solving the complete two-center set of equations, although some care needs to be taken to orthogonalize the functions Qf and qi.

Reading e t a / . apply their method to a number of inner-shell capture processes as energies of a few MeV per nucleon, for ZB/ZA in the range 4 to 18. The computed cross sections are from 0.3 to 0.5 times the

280 B. H . Bransden

Brinkman-Kramers results, and scale like ZS,. In general, the agreement with experiment is good.

3 . The Classical Model

An entirely different approach, which appears to provide reasonably accurate (within 20%) total charge exchange cross sections at intermediate energies, is the purely classical model of Abrines and Percival (1966). The model works well for the proton-hydrogen system in a limited energy region, the predicted cross section falling below the data when u < u,, and when u > 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 CHARGE TRANSFER

Following the early worklo of Drisco (1955), it has been shown by a number of workers (Brodskii et al . , 1970; Dettmann, 1971) that the high- energy 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 ZA from a target with charge ZB has the form, in the high-velocity limit,

WB2 = ~BK[0.295 f 5nV/211(zA + ZB)] a.u. (27)

(+BK = n218(zAzB)5/5u12 a.u. (28)

where uBK is the Brinkman-Kramers first-order cross section

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).

THE 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

(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

cB2 = + U , U + L I ~ U ~ + . . . + u ~ ~ - , u ~ ~ - ~ ) (30)

Since the Brinkman-Kramers cross section behaves like u-12-21, the behavior of cBz is always of the form u-l ' , for sufficiently high velocities.

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) =

= cBK[0.319 + .5ru/211(ZA + Z B ) ]

(31)

where +(k, r, 1 ) is a wavefunction for Coulomb scattering in the potential Z A / R , and +, is the Fourier transform of the initial bound-state function. The results obtained with this approximation were disappointing (Co- leman, 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 im-

- i J dq [/:mexp(-iqfr) ( X f ( f A ) IM - +(q - v, rA,t) ) dt 1 1 +(q) exp(iq b)

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

C f ( h , r n ) =

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


I l l 30 50 70 100 200 300 500 700 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 dr N ( u A ) N ( u B ) ( G g ( r A 9 t ) l F 1 ( i u B , I ; v r B + ivrB)IO I_m_ IFi (rB, f ) I F I ( i V A , l ; v r A + i U r A ) ) (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

284 B. H . Bransden

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 2 8

PROJECTILE CHARGE 2, lau)

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.


da - = / i p v dR h db exp[i6,(b) + i8f(h)]Jo(2,vh sin 8/2)cf(h,z)12 (35)

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 0 06 0.08

e 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 the internuclear potential: ---. excluding the internuclear potential; A, experimental data of Bratton el u / . (1977).

286 B. H. Bransden

IV. Ionization and Charge Exchange into the Continuum

A. TOTAL CROSS 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 << ZB, Born’s approximation may be accurate at lower incident velocities because the perturbation (ZA/rA) is small. A useful discussion of this situation, which occurs in the treatment of the ionization of the inner shells of heavy atoms, has been given by Briggs and Taulbjerg (1978). The impact parameter Born approximation usually employs a straight-line trajectory for the heavy-particle motion, but if (ZAZ,) is large the integration can be carried out without difficulty along a Coulomb trajectory (Kocbach, 1976; Aashamar and Kocbach, 1976; Pauli and Trautmann, 1978).

The calculation of ionization cross sections when the first Born approximation is not accurate is a matter of some difficulty. The binary encounter model, in which the interaction between the incident nucleus A and the electron to be ionized is treated as a two-body classical collision, does not seem well founded, but is quite effective in predicting cross sections, when ZA < ZB (Garcia et a/., 1973). The purely classical model of Abrines and Percival (1966), which we noted was rather successful in a limited velocity region for the calculation of charge exchange cross sections, is equally successful for ionization and has been employed extensively by Olson and Salop (1977) to calculate total cross sections for the ionization of atomic hydrogen by a variety of fully or partially stripped positive ions. At intermediate energies ( A - I ) , where coupled-channel calculations can be performed, an indirect calculation of the total ionization cross section is possible, provided a sufficient number of pseudostates have been included in the basis set to represent the continuum states. In this case, the ionization cross section is simply identified with the difference of the overall total cross section (calculated from the flux lost to the entrance channel) and the integrated cross sections for elastic scattering and discrete excitations. With a large basis set, this technique appears to produce accurate cross sections as shown, for example, by Fit- chard et al. (1977) in the case of a single-center expansion for p + H. In


contrast, in the work of Shakeshaft (1976), also for p + H, based on a two-center expansion, the calculated ionization cross section was about a factor of one-half smaller than the data of Park et al. (1977), although the cross section peaked at the correct energy of 55 keV. As Shakeshaft suggests, this discrepancy must be due to the inadequate size of the basis set, as the method is certainly sound in principle.

B. DIFFERENTIAL CROSS SECTIONS

When the charges on the incident and target nuclei are comparable in size, the bulk of the ionized electrons are ejected with small velocities, as predicted by the Born approximation. However, there is a significant enhancement of the cross section, above the Born approximation predictions, for ejected electrons that have a velocity uE close to that of the incident proton u , and in this region the cross section is cusplike, behaving approximately like ( l / ( v - vEl) (Crooks and Rudd, 1970; Harrison and Lucas, 1970). Since the incident ion is not appreciably deflected during the collision, the electrons concerned are those that emerge close to the forward direction. This effect is very clearly seen in the experiments of Manson et al. (1975) on an ionization of helium by protons of energies between 300 keV and 5 MeV, and in Fig. 8 we show the ratio of their experimental data to the Born cross section for electrons with the same speed as the incident proton for a number of angles of ejection as a function of incident energy. One way of looking at this enhancement is as an

I I I 1 1 I I I I

I I I I 1 I 1 , 0 1 2 3 L 5

PROTON ENERGY (MeV)

FIG. 8. The ratio R of the experimental to the theoretical in Born differential cross sections for the ionization of helium by proton impact, for the case in which the ejected electrons have the same speed as the incident proton.

288 €3. H . Bransden

electron capture process, in which the electron is captured into a continuum rather than a discrete state of the incident ion. This explanation was put forward by Macek (1970; see also Macek and Rudd, 1972), who coined the term charge exchange into the continuum. The fact that the discrepancies in the forward direction could be attributed to a final-state interaction between ejected electron and the scattered proton was suggested earlier by Oldham (1967). Both Macek (1970) and Salin (1969) were able to show that the forward enhancement could be explained by introducing a final-state interaction into the Born approximation. Band ( 1974) has given the generalization of the Bates distortion approximation to capture into the continuum, and Dettmann et al. (1974) have made use of the second Born approximation. The theories have not been completely successful in reproducing the data quantitatively. For the case of highly stripped carbon and oxygen ions incident on argon, the ZA dependence of the differential cross section near the forward direction is found to

1978), but the evidence from this system and from experiments in solids (Meckbach et al . , 1977) may be clouded by the complexity of the target.

Clearly, in a proper theoretical treatment one should not attempt to describe the ionization cross section as a Born background with a charge exchange process superimposed; rather, a good model should contain both features automatically. A promising new approach has been developed by BelkiC (19781, who has made use of the multiple-scattering expansion of Dodd and Greider (1966) to derive an ionization amplitude that allows for the distortion of the wavefunction by the nucleus A in both the initial and final states. The final expression for the probability amplitude is a generalization of the continuum distorted-wave approximation (see Section 111) and is similar in form to Eq. (34). It can be derived within an impact parameter formulation along the lines originally used by Cheshire (1964).

BelkiC has applied his model to ionization of atomic hydrogen by protons and has made a detailed comparison with data of Park et al. (1977) and also with Kuyatt and Jorgensen (1963) and Rudd el al. (1966), which relate to molecular hydrogen. Above 50 keV, the agreement is satisfactory provided the distortion terms in the incident channel and in the final channel are both retained.

be Z2.220.2 A , in contrast to the theoretical prediction of Z3, (Vane et al . ,

REFERENCES

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ADVANCES IN ATOMIC A N D MOLECULAR PHYSICS, VOL. I S

~~

ATOMIC COLLISION PROCESSES IN CONTROLLED THERMONUCLEAR FUSION RESEARCH

H . B . GILBODY

Department of Pure and Applied Physics The Queen 's University o f Belfast Belfast, Northern Ireland

I. In . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 11. CI eavy-Particle Collision Processes

A. Plasma Heating, Energy, and Particle Loss. . . . . . . . . . . . . B. Fast-Beam Injectors for Plasma Heating or Fueling . . . . . . . . . . . . . . . . 297 C. Plasma Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

111. Experimental Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 A . Collisions between Multiply Charged Ions and Hydrogen . . . . . . . . . . . 300 B. Collisions between Positive Ions . . . . . C. Electron Capture Neutralization of Fast Ions D. Formation of Fast Negative Hydrogen Ions a

Neutrals . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

I. Introduction

Over the past 25 years we have seen steady progress toward the practical realization of controlled thermonuclear energy through fusion of the isotopes of hydrogen in a high-temperature plasma. To date, by far the greatest effort has been applied to schemes in which the plasma is heated during magnetic confinement. However, in recent years attention has also been directed to heating and inertial confinement by high-power lasers, or by fast beams of electrons or heavy ions. Schemes for a number of reactor systems based on several magnetic confinement configurations and on inertial confinement by lasers have been described in a recent review by Ribe (1975).

291 Copyright 0 197') hy Aczrdemic h e r , . Inc.

All right\ of reproduction in any form I-exrved. ISBN 0- I ?-I1018 I ( - 3

294 H . B . Gilbody

Of several possible fusion reactions under consideration the process D + T + (‘He + 3.52 MeV) + (n + 14.06 MeV)

is attractive since this attains reaction rates sufficient for ignition at temperatures greater than only 4-5 keV. The 3.52 MeV a particles formed in this process remain in the fully ionized plasma giving up their energy in collisions with the constituents. The fast neutrons pass out of the plasma and, in the scheme currently envisaged, are then trapped in a lithium blanket where their kinetic energy is converted into recoverable heat. At the same time, tritium breeding occurs through the reaction

BLi + n -+ ‘He + T + 4.80 MeV

The general role of atomic collision processes in the design and operation of fusion devices has been reviewed recently by Barnett (1976), Lorentz er al. (1976), and Harrison (1977). Steady improvements in the characteristics of magnetically confined plasmas, particularly with regard to toroidal plasmas of the Tokamak type, have produced a new awareness of the importance of such processes in the mechanisms of plasma heating, energy and particle loss, plasma fueling, and the diagnostics of high- temperature plasmas. With the upgrading of present fusion devices in the approach toward a practical reactor, a better understanding of the relevant atomic collision processes is now of particular concern.

Many types of collision processes are involved and, apart from the partially or fully ionized primary constituents of the plasma, the role of impurity species is of special interest. Data are required on collisions between heavy particles involving charge changing, ionization, or excitation, electron collisions involving excitation or ionization, electron -ion recombination processes, bremsstrahlung production, and photoionization. Ef- fects resulting from the interaction of the plasma constituents with metal electrodes or the walls of the confining vessel are also of considerable importance.

In this short review, the scope is limited to a discussion of some heavy-particle collision processes of importance mainly to Tokamak systems. The relevant processes are identified and the results of recent experimental studies that are beginning to provide a better understanding of specific processes are considered in relation to current theoretical descriptions. Most of the measurements have been carried out with ions or atoms of hydrogen rather than deuterium. For this reason and since isotope effects do not arise for the processes considered, reference is usually made to hydrogen rather than to deuterium in the ensuing discussion.

ATOMIC COLLISION PROCESSES IN FUSION RESEARCH 295

11. Classification of Relevant Heavy-Particle Collision Processes

A. PLASMA HEATING, ENERGY, A N D PARTICLE LOSS

In the operation of a Tokamak device the induced toroidal current provides plasma confinement and equilibrium and at the same time supplies energy through ohmic heating to the electrons and hence to the ions by coulomb collisions. This mode of ion heating becomes ineffective at energies beyond 2-3 keV, which are not high enough to initiate effective further heating by the a particles from the fusion reaction. A form of supplementary heating is therefore required.

The most promising scheme for supplementary heating is by the injection of an intense fast neutral beam of an appropriate isotope of hydrogen (e.g., deuterium) through the magnetic confining field and into the plasma where it undergoes electron loss by both charge transfer

(1) H+ + H ---f H + H+

and ionization

H + + H + H + + H + + e ( 2 )

in collisions with the plasma protons. The resulting fast protons are trapped in the confining field and give up their energy in collisions with the plasma constituents. Cross sections for (1) and (2) are fairly well established (Fig. 1) and indicate that, while resonant charge transfer is the dominant trapping process at the lower impact energies, above about 50 keV the ionization process is the most effective.

In practical devices, plasma heating is complicated by the presence of small fractions of partially and fully ionized impurities of carbon, nitrogen, and oxygen and partially ionized metal atoms of high atomic number 2 such as iron, molybdenum, titanium, niobium, vanadium, and tungsten arising from the interactions at the chamber walls. While multiply charged impurities can produce an enhancement of ohmic heating through an increase in the resistivity of the plasma, their presence leads to serious power loss, which increases with charge state through radiation from electron-ion recombination, line radiation, and free-free bremsstrahlung arising from the acceleration of electrons during close encounters with the ions. Radiative recombination is of great importance for highly ionized ions of high Z since the power loss for a charge state q is proportional to q4. In present Tokamak devices, spectroscopic observations (Isler et al., 1977: Schwob et al. , 1977) indicate the presence of ions of tungsten,

296 H . B . Gilhody

ENERGY ( k e V )

FIG. I . Cross sections u, for charge transfer and U, for ionization in H+-H collisions. The curves shown are approximate values based on the experimental data of Fite et al. (1958), McClure (1966), Gilbody and Ireland (1%3), and Wittkower el a/. (1966).

molybdenum, and iron with charge states 4 as high as 34, 32, and 23, respectively. Meade (1974) has estimated that power losses from a 0.2% tungsten impurity level with 4 - 70 (which is feasible at reactor temperatures) would be sufficient to prevent ignition at any temperature.

Impurity ions in the plasma may severely limit the efficiency of neutral beam heating through charge transfer

(3 ) XP+ + H + x(a - I)+ + H+

and ionization

XO+ + H + X O + + H+ + e (4)

High cross sections for these processes would result in shallow energy deposition profiles for the incoming fast hydrogen atoms, particularly if the impurities are present near the plasma boundary. The fast protons formed in this region may be magnetically deflected to strike the chamber walls thereby producing further high-Z atoms by sputtering, which in turn provide enhanced neutral-beam trapping at the plasma boundary rather than the core (Hogan and Howe, 1975).

In the charge transfer reaction (3) if, at the plasma temperatures con-


cerned, electron capture takes place into high-lying short-lived excited states of the impurity ion, the subsequent radiative decay of these states can provide significant loss of power. The effect of capture into metastable states that may be long-lived even in the presence of high magnetic confinement fields (e.g., the helium-like metastable states of C4+, Ns+, and 06+) must also be taken into account. In addition, a high cross section for the ionization process (4) or the process

Xu+ + H -+ X'p+l)+ + H(H;c) + e ( 5 )

[where H(Z c) indicates all final bound and continuum states of HI leads to cold electrons that can be detrimental to the density and temperature distribution of the plasma (Lorentz ('1 L I I . , 1976).

A possible serious source of particle loss and plasma cooling can occur through ion-ion collision processes of the type

(6)

where X may be helium or an impurity species of the type already mentioned. Fast hydrogen atoms formed in this way escape from the magnetic confinement while the charge state of X is increased, leading to enhanced power loss by radiation in the manner already discussed.

Finally, in addition to processes relevant to heating by fast neutral beams, a complete understanding of the energy and particle loss mechanisms associated with a particle heating requires quantitative data on charge transfer processes such as

He2+ + H H e + + H+ (7) H e + + H E H e + H + (8)

The extent to which collisions with multiply ionized impurity ions limit the effectiveness of a particle heating is also of considerable interest.

H+ + Xu+ -+ H + X(u+1)+

B . FAST-BEAM INJECTORS FOR PLASMA HEATING OR FUELING

Many atomic collision processes are relevant to the design and efficient operation of fast neutral-beam injectors for the supplementary heating of Tokamak plasmas referred to in Section I1,A. While beam energies in the range 25- 100 keV are appropriate to present devices, rather higher energies appear to be necessary for practical reactors and powers at the MW level are contemplated.

An intense pulsed beam of ions of hydrogen (or deuterium), derived from a suitable source, may be accelerated to the required energy and then passed through a gas or metallic vapor target, where partial conver-

298 H . B . Gilbody

sion to fast neutral atoms or molecules takes place by electron capture collisions. Electron capture cross sections for H+, H:, or Hi ions in a wide range of targets are therefore of considerable interest. The extent to which dissociative processes involving molecular ions of hydrogen and deuterium affect the neutral yield and the scattering of the beam is also important.

For optimum conversion of ions to fast neutral atoms, target thicknesses must be increased to levels such that single collisions no longer apply and a knowledge of the cross sections for electron capture and loss for beam components of all charged states is necessary for a complete understanding of the growth and equilibration of the neutral component. The possible strong influence of fast electronically excited atomic species and of both electronically and vibrationally excited molecular species is also important in this context. At the very high beam power levels concerned, the target gas or vapor will be ionized to a significant extent by either pure ionization or transfer ionization processes of the type

(9)

(10)

The interaction of the fast primary components of the ion beam with the electrons and ions formed by these processes may be of considerable importance in thick targets.

The very high primary ion beam currents are such that space charge considerations preclude magnetic analysis to remove impurity species from the beam before neutralization. The extent to which typical impurity species (e.g., C+, 0+, N+, and various metallic ions) undergo electron capture in the neutralizer cell is therefore important. Although the impurity content in the emergent atom beam may be quite small, when these atoms enter the plasma and become multiply ionized, significant power loss may occur for the reasons given in Section II,A.

The relative fractions of fast atomic and molecular neutral species and the excited-state populations of the beams formed in different targets are required so that the degree of penetration and trapping of the injected beam within the plasma can be optimized.

Charge transfer neutralization of ion beams becomes less efficient at the higher energies likely to be required for beam injectors of prototype reactors. For this reason, consideration is being given to the production of fast H or D beams by the electron detachment process

H- + X + H + X(Z;c) + e (1 1)

which occurs when an intense beam of negative ions is passed through a suitable gas or vapor. It may be possible to produce negative-ion beams of

H+ + X + H + + Xn+ + ne

+ H + Xn+ + (n - 1)e

ATOMIC COLLISION PROCESSES I N FUSION RESEARCH 299

the required intensity by charge changing collisions during the passage of H+, H:, and H: through alkali metal vapors. A detailed knowledge of the processes involved is required in order to optimize the H- yield.

One method being considered for the supply of fuel to the plasma in a fusion reactor is by the injection of fast neutral beams. While beams of the hydrogen isotopes are of primary interest, the injection of 3He beams is also a possibility. Such injectors may operate in a different energy range from those used for plasma heating but the general design considerations in terms of the basic collision processes are similar.

Finally, it is worth noting that, while the design of intense neutral-beam injectors for the heating of magnetically confined plasmas poses severe problems, difficulties are also inherent in recent schemes for DT pellet heating and inertial confinement by fast heavy-ion beams. Heavy ions such as Xe+ at high current levels would be accelerated to very high bombardment energies. However, ion-ion collisions within the beam of the type

x+ + x+ + x + x2+ (12)

occurring either within an accelerator or storage ring would result in the loss of both ions. High cross sections for such processes at collision energies -0.5 keV amu-' (see Kim, 1976) would severely impair the operation of such a scheme.

C. PLASMA DIAGNOSTICS

The successful design and development of controlled thermonuclear fusion devices clearly must rely heavily on techniques for plasma diagnostics. The use of fast-atom or -ion beams as probes can provide useful data on spatial ion densities and temperatures.

Kislyakov and Petrov (1971) have used 4-14 keV beams of hydrogen atoms to probe plasmas in Tokamak devices. A study of the attenuation of such beams and a knowledge of the cross sections for ionization of the beam atoms by charge transfer and ionization in collisions with plasma protons (processes 1 and 2) and ionization in collisions with the velocity-averaged plasma electrons can provide a useful measure of the path-averaged proton density in the plasma.

Electron capture by plasma protons in collisions with fast injected H or D beams also results in the formation of excited H atoms. Studies of the Doppler-shifted radiation emitted in the subsequent decay of these excited atoms can be used to assess the temperature of the plasma. A knowledge of the cross sections for electron capture into the states concerned (e.g., the n = 2 or 3 states) also allows assessment of the proton density.

300 H . B . Gilbody

Impurity species in the plasma can in principle also be investigated by means of this technique.

Energetic heavy-ion beams (e.g., TI+, K+, Ba+) capable of penetrating the magnetic confinement fields may also be used to obtain data on plasma proton densities if the appropriate ion-ion charge transfer and ionization cross sections are known.

111. Experimental Studies

A. COLLISIONS BETWEEN MULTIPLY CHARGED IONS AND HYDROGEN

1 . Experimental Approach

As noted in Section I1 there is now considerable interest in collisions involving charge changing, excitation, and ionization for a wide range of partially and completely ionized species in collisions with atomic and molecular hydrogen. The production of suitable ion beams to carry out such measurements presents some difficulties. Few types of conventional ion source produce adequate yields of multiply charged ions, especially those of high atomic number 2. However, beams of intensity sufficient for many measurements can be prepared by electron-stripping collisions during the passage of an intense, fast singly charged ion beam of the required energy through a gas or thin foil target. In such a scheme, particular care must be taken to avoid contamination of the beam by impurity ions. While momentum analysis after acceleration is usually sufficient to remove atomic impurity ions emitted directly from the ion source, a second stage of energy analysis is desirable in order to reject break-up products of molecular ions that might otherwise be present in the beam. Careful account must also be taken of the possible metastable population of the beam, which may have a pronounced effect on the measurements (see review by Gilbody, 1978).

The production of completely stripped ions by charge-changing collisions at energies of interest to fusion is limited to species of very low Z. Primary ion energies - MeV seem to be required to produce useful yields of completely stripped C, N , or 0 ions. It is also difficult to produce high-Z ions in the very high charge states of interest in Tokamak devices. Ion sources employing pulsed high-power lasers interacting with gases and solids may prove attractive for future work.

Atomic hydrogen targets suitable for cross section measurements can be provided in several different configurations each of which offers cer-

ATOMIC COLLISION PROCESSES IN FUSION RESEARCH 30 I

tain advantages. Since detailed descriptions may be found elsewhere (see, for example, Massey and Gilbody, 1974), the main features and recent developments only need be summarized here.

( a ) The modulated crossed-beurn technique. This technique, which was pioneered by Fite et al. (1958) for studies of H+-H collisions, has since been widely used by several different investigators. The primary ion beam is arranged to intersect at right angles a thermal energy target beam of highly dissociated hydrogen from a tungsten tube furnace or gas discharge source. By recording the yields of the ions, electrons, or photons formed in the beam intersection region, it is possible to obtain cross sections for collisions involving charge changing, ionization, and excitation. The main difficulty in such measurements arises from the fact that hydrogen beam densities (- lo9 atoms ~ m - ~ ) are usually comparable with residual gas densities in the crossed-beam intersection region. However, by mechanically chopping the hydrogen beam at a fixed frequency, the signal from the process of interest may be distinguished from that arising from the interaction of the ion beam with the residual gas by its specific frequency and phase. Since accurate determination of effective target thickness is a difficult problem, the method is best suited to the determination of ratios of cross sections for targets of H and H,, where the cross section in H, is known from other measurements. The necessary allowance for undissociated H, molecules, which typically comprise at least 10% of the hydrogen beam, can be made on the basis of an electron impact mass spectrometric analysis.

The modulated crossed-beam method is well suited to studies of both change transfer and ionization by multiply charged ions in collisions with H from an analysis of the slow protons and electrons formed in the beam intersection region. However, to date, apart from some measurements with He2+ ions, experiments have so far been carried out only with singly charged ions. Collisions involving electron capture into excited states of the primary ion or direct excitation of the hydrogen target can also be studied by the use of selective detectors to record the spontaneous decay of the excited collision products. However, photon yields are very low and, for excited multiply charged ions, the relevant emissions usually occur in the vacuum ultraviolet region, where the design of selective detectors is difficult. Measurements of this type have so far been confined to studies of Lyman a emission from excited H atoms using a method first employed by Stebbings et al. (1965).

( h ) The furnace target technique. An atomic hydrogen target of thickness several orders of magnitude greater than that for an atomic

302 H . B . Gilbody

beam can be obtained by using a tungsten tube furnace to contain highly dissociated hydrogen. This technique first used by Lockwood et al. (1964) has since been adapted by a number of investigators for studies of electron capture by both singly and multiply charged ions in H over a wide energy range. In these experiments, the primary beam passes along the axis of a cylindrical tungsten tube a few centimeters long, which can be electri- cally heated to temperatures -2500 K. Hydrogen gas is introduced at the midpoint of the tube and confined mainly to the hot central region by metal buttons within the tube. Apertures in these buttons must be adequate to define the beam and accommodate fast scattered collision products. Electrostatic analysis of the emergent beam allows the charge state populations and hence particular charge-changing cross sections to be determined.

Allowance for the fraction of H, molecules remaining undissociated in the furnace may be made by the use of a fast probe of H+ or He2+ ions to study the formation of fast H- or He, respectively, by two-electron capture in single collisions. The accurate determination of the total density of hydrogen nuclei present is quite difficult. McClure (1966), in studies of H+-H charge transfer, obtained the density from measurements of the angular distribution of fast scattered H and H- collision products. In the analysis it could be assumed that H2 molecules are twice as effective as H atoms in producing scattering that can be described accurately in terms of a simple Coulomb potential. In subsequent experiments involving various primary ions, some investigators have obtained target densities by normalization to the H+-H and Hf-H2 data of McClure. Bayfield (1969) has also shown that calibration may be carried out from measurements when argon is substituted for hydrogen in the furnace. Measurements are then normalized to known H+-Ar and H+-H2 charge transfer cross sections.

The furnace target method is well suited to measurements of electron capture or loss by fast multiply charged ions. However, determination of cross sections for collisional excitation from spectroscopic observations is severely limited by the fact that many of the excited states of interest are so short-lived that the fast ions undergo radiative decay within the furnace. Those formed in metastable states (e.g., the 2s states of H or He+) may be conveniently detected beyond the furnace. Useful information on some excitation and ionization processes may also be obtained from the energy loss spectrum obtained when the beam emerging from the furnace is subjected to high-resolution energy analysis. In this way Park et al. (1975, 1977) have obtained cross sections for excitation for the n = 2 state of H and for ionization of H by proton impact in the range 15-200 keV.

( c ) Merged-beam technique. This technique, first applied successfully to ion-neutral collisions by Trujillo et al. (19661, is designed to pro-


vide cross sections from studies of collisions between particles in two beams moving along the same path. In this way, collisions extending down to very low center of mass energies may be studied using beams of comparatively high laboratory energy provided these are nearly equal. Since both beams interact with the residual gas, signal to background ratios may be very unfavorable. Considerable care must also be taken in defining the path length from which the collision products arise.

Belyaev et a / . (1967) first applied the method to collisions involving H. A beam of fast H atoms formed by charge transfer neutralization of protons was merged with a proton beam of slightly lower energy to obtain cross sections for charge transfer in H+-H collisions. Of major concern in experiments of this type is the effect on the measurements of small fractions of long-lived highly excited atoms present in the H beam. In recent merged-beam studies of H+-H and He2+-H collisions, Koch and Bay- field ( 1975) and Burniaux et ai. (1977) have employed electric field ionization to specify high-n-state populations of the H beams so that the cross sections pertaining to the excited species could then be determined.

2. Charge Transfer Data

An indication of the importance of electron capture by multiply charged ions in collisions with atomic hydrogen (process 3) is provided by a number of recent theoretical studies concerned mainly with completely stripped ions. At intermediate and high velocities V > 1 a.u. (2.2 x lo8 cm sec-') application of the first Born approximation is inappropriate to the strong interaction between multiply charged ions and H. Olson and Salop (1977) have applied a classical Monte Car10 method to this problem in order to obtain capture cross sections (see Fig. 2) for fully stripped ions with initial charges q in the range 1-36. The method entails solving the 12 coupled Hamilton equations of motion for a three-body system with a random distribution of initial conditions. The electron distribution about the hydrogen ion is described classically by a method used by Abrines and Percival ( 1955) and Coulomb forces represent the interactions between all bodies. Olson and Salop have also estimated cross sections for some partially stripped ions with q a 3 by the use of effective charges qefr that allow for electron screening.

At low velocities V d 1 a.u., it is well known that electron capture by multiply charged ions may occur through one or more pseudo-crossings of the adiabatic potential energy curves of the molecular system. Such pseudocrossings arise in exothermic reactions through the presence of a Cou- lomb repulsion between the collision products and occur at internuclear separations R, = (q - I)/AE, where AE is the energy defect for the process. For endothermic reactions and for the special case of accidental

3 04 H . B . Gilbody

EH I k e W 37.5 50 75 100

I I I I I 2 3 L 5 6 I

VELOCITY (lo* cm sec-') FIG. 2 . Electron capture cross sections calculated by Olson and Salop (1977) for the

process Xa+ + H -+ X'q-')+ + H+ for completely stripped ions with various charge states q.

resonance when AE = 0, no outer crossings can occur and the potential energy curves gradually diverge with decreasing R .

Theoretical studies of electron capture based on a molecular description have been applied to a number of completely stripped ions in collisions with atomic hydrogen (Salop and Olson, 1976; Vaaben and Briggs, 1977; Hare1 and Salin, 1977). These calculations indicate that the total one-electron capture cross section is determined primarily by curve crossings leading to selective population of one or a limited number of excited states in the outer shells of the multiply charged product ion. For example, electron capture in C6+-H collisions may be understood by considering the outer molecular orbitals of the (CH)6+ system in Fig. 3. The initial Cs+-H(ls) system at infinite separation correlates to the 6hc state.


-0.5 c 6 ha I

- 3 .

I I I 1 .0 10 0 100.0

INTERNUCLEAR SEPARATION (a.u.1

FIG. 3 . Molecular orbitals of the (CHIsf system (Vaaben and Briggs, 1977)

A pseudocrossing occurs at an internuclear separation R = 16 a.u. between the 6hm and 5gu states, which at low velocities ( V 6 0.5 a.u.) exhibits diabatic behavior. Thus the 5gu curve can be regarded as joined to the initial states through the crossing. I t can be shown that transitions occur primarily by radial coupling through the 5gm-4fm avoided crossing at R = 8 a.u., leading to selective population of the tz = 4 state of the C5+ product ion.

Approximate cross sections for transitions of this type may be obtained in terms of the two-state Landau-Zener approximation. However, mul- tistate calculations in which allowance for rotational coupling of the m orbitals to orbitals of higher angular momentum due to rotation of the internuclear axis provide a better estimate. At velocities V = 0.5 a.u., cross sections - 5 x cm2 are predicted for C6+-H collisions. In the same way, it can be shown that, for electron capture by 08+ ions in H , while 07+ formed in the n = 8 state corresponds to LIE = 0, curve crossing considerations lead to the expectation that 07+ ions will be formed predominantly in the n = 5 state with a cross section -4 x cm2 for V =

0.5 a.u. Experimental evidence of electron capture into excited states of 07+

has been provided by Isler (1977) who injected 10-30 keV H atoms into a Tokamak device. A sharp enhancement of the Balmer (Y ( n = 3 + ti = 2 ) emission line of 0 VIII was observed. Although this observation apparently contradicts the theoretical prediction that charge transfer proceeds mainly through the n = 5 level, the limitations of the spectroscopic


technique do not rule out the possibility that the observation results from cascading.

Olson and Salop (1976) have calculated cross sections for one-electron capture by ions with initial charge states 4 s q s 54 at velocities s 108cm sec-' using an absorbing-sphere model based on the Landau-Zener method. The model requires a high density of curve crossings around an easily defined critical internuclear separation R,. Cross sections only weakly dependent on velocity and increasing roughly linearly with q for high values of q are predicted. Measurements by Crandall et al. (1977) of one-electron capture cross sections for carbon, nitrogen, and oxygen ions in H, with q = 3-6 in the velocity range 0.3- 1.2 x 108cm sec-'are found to be in good general accord with this model. Further support is provided by the extensive experimental data presented by Miiller and Salzborn (1977) on electron capture by multiply charged neon, argon, krypton, and xenon ions in a wide range of gas targets.

Winter et al. (1977a,b) have carried out studies of electron capture into excited states during the passage of 25-800 keV Ne*+ ions (with q = I , 2, 3 ,4 ) through He, H,, and Ar by spectroscopic observations of the spontaneous radiative decay of the excited products of single collisions. As q is increased, electron capture into excited states of the projectile is found to contribute increasingly to the total one-electron capture cross section. The observations show that electron capture into states involving moderate exothermic energy defects is favored. These correspond to curve crossings at intermediate internuclear separations between 3 and 10 a.u.

Experimental studies of electron capture in atomic hydrogen by a number of partially stripped ions of direct relevance to fusion are now available but measurements with completely stripped ions have so far been limited to H+, He2+, and Li3+ ions. Capture cross sections in H cannot easily be inferred from measurements in H,. The multiplicity of reaction channels in the latter case makes it difficult to determine the extent to which H2 can be regarded as equivalent to two free-hydrogen atoms.

The experimental and theoretical data now available for charge transfer in He2+-H collisions are quite extensive. This case is of special interest since resonant capture into the n = 2 state of He+ is believed to provide the main contribution to the total capture cross section over a wide energy range. In Fig. 4, measured total electron capture cross sections vZ1 and cross sections uz,* for capture into the 2s metastable state are compared with a number of theoretical predictions. Of the measurements obtained using the furnace target technique, the cross sections uZ1 measured in the range 1.5-343 keV for 3He2+ ions in this laboratory (Shah and Gilbody, 1974, 1976, 1978; Nutt et al., 1978a) are in good accord with experimental


I I 1 I

10 100 'He2* ENERGY I keV 1

FIG. 4. Cross sections uzl* for electron capture into the 2s state and uZ1 for capture into all bound states of He+ in HeZ+-H collisions. 0: uZlr Shah and Gilbody (1978); a: uZ1, Nutt et a / . (1978a); 0: n2], Bayfield and Khayrallah (1975); A: uZ1, Fite er a / . (1962); 0: uZ1, Shah and Gilbody (1978): B: uzl, Bayfield and Khayrallah (1975); -. theoretical cross sections for capture into (Is + 2s + 2p) states; ---, theoretical cross sections for capture into 2s state: ..., theoretical cross sections for capture into Is state. Curves R: Rapp (1974), eight-state impact parameter approximation using atomic states. Curves M: Malaviya ( 1%9), five-state impact parameter approximation using atomic states. Curves PS: Piacen- tini and Salin (1977). impact parameter approximation using molecular states. Curves WL: Winter and Lane (1978). impact parameter approximation using molecular states.

data (not shown) obtained by Olson et a/ . (1977) in the range 20-550 keV. However, the results of Bayfield and Khayrallah (1975) obtained for 4He2+ ions in the range 7- 144 keV exhibit differences in magnitude and energy dependence that probably reflect differences in the normalization procedure. It is important to note that, unlike the furnace target data, values of vz1 measured by Fite et 01. (1962) using a modulated crossed-beam technique remain large at low impact energies and probably reflect large errors

308 H . B . Gilbody

Velocity ( x \o8crn/sec-')

1.0 1.5 2.0 2.5 3.0 L.0 5.0 6.0 7.0 1 I 1 1 1 1 , I I ,

I I I 1 1 1 1 1 I 1 1 I I l l 1 I

100 1000 ' L i p + ENERGY ( k e V )

FIG. 5 . Cross sections uI0. uz1, and ua2 for one-electron capture by Li+. Liz+, and Li3+, and cross sections ul, and uZls for one-electron loss by Li+ and Liz+ ions (a) in H (above) (b) in H, . 0, 0, Shah ef ill. (1978: A, Olson and Salop (1977), us, theory; --- Shirai er a/ . (1977), uZ3 theory.

(see Nutt et al . , 1978a) arising from undissociated H, molecules in the beam. At the low impact energies where the discrepancy arises, crZ1 for H2 is larger and falls much more slowly than cr21 for H, but at the higher impact energies, the cross sections are comparable in magnitude and exhibit a similar energy dependence.


1 Ol

- N

E 2 lo1(

z

I- V w v,

v, m 0 a V

0

to1'

1

8 Velocity ( x 1 O cm/sec-')

1 .o 1.5 2.0 2.5 3.0 4.0 6.0 I I I I I l l I I I 1

Fie. 5b

Although there are significant differences between the various theoretical estimates of both (+21 and uZ1* in Fig. 4. the experimental values of (+21

are in rough general accord with the sum of the theoretical cross sections for capture into the Is, 2s, and 2p states. I t is evident that capture into the Is state only becomes significant at high impact energies.

Burniaux et ul. (1977) have used a merged beam technique to obtain cross sections for the process

310 H . B . Gilbody

He2+ + H* + He+ + H+ (13)

in the center of mass energy range 0.25-478 eV. For excited hydrogen atoms with principal quantum numbers n ranging from 8 to 24, cross sections between about lo-', and 10-lo cm2 were obtained.

Figure 5a shows cross sections ulo, u,], and ~ 3 1 for the capture processes

L i e + H + Li(a-l]+ + H + (14)

measured in this laboratory (Shah et a/., 1978) for q = 1,2 , and 3, respectively, in the range 65-1500 keV. Although the energy defects for the ground-state formation channel have the markedly different values AE =

-8.2, +62, and + 108 eV for q = 1, 2, and 3 , respectively, the measured cross sections are large, increase with q , and have a similar energy dependence. For 4 = 2, it is possible that near-resonant capture into the n = 2 states of Li+ with hE = 1 eV predominates, while for q = 3 capture into the n = 3 states of Liz+ would provide accidental resonance. It is also interesting to note that at high velocities, the cross sections scale approximately as q3 (see Fig. 6). The corresponding capture cross sections for lithium ions in H, (Fig. 5b), while generally larger, are found to exhibit a remarkably similar energy dependence to those in H.

Capture cross sections (+32 for Li3+-H collisions obtained by Olson and Salop (1977) (Fig. 5a) from classical trajectory calculations in a Monte Carlo approach are in rough accord with the experimental values. In Fig. 7 the Li3+-H capture cross sections and cross sections for HeZ+-H collisions (Shah and Gilbody, 1978) are shown scaled down by the factor l / q 2 , together with experimental data (McClure, 1966; Wittkower et al., 1966) for Hf-H collisions. It is evident that q2 scaling of the H+-H cross sections is appropriate for these two cases at velocities above about 2.8 x lo8 cm sec-' although this is unexpected on the basis of the classical calculations of Olson and Salop (1977). The extent to which this simple q2 scaling applies to completely stripped ions of higher Z would be of considerable interest.

Phaneuf et al. (1978) have measured cross sections for one-electron capture in H and H, by fast partially stripped N and 0 ions with charge states q ranging from 1 to 5 and for C ions with q between 1 and 4 in the velocity range 0.3-5.2 x lo8 cm sec-'. Capture cross sections in H at high velocities are consistently somewhat smaller than those in H, and are found to increase with initial charge q and scale approximately as q2. Their results for N ions in H and HZ are shown in Fig. 8. In the case of H, cross sections calculated on the basis of the Olson and Salop (1977) classical method with allowance for electron screening are also found to be in reasonable accord with the measured cross sections.

ATOMIC COLLISION PROCESSES IN FUSION RESEARCH 31 1

8 Velocity (10 cm sec-'1

2 3 L 5 6 1 0

EOUIVALENT PROTON ENERGY IkeVl

FIG. 6. Cross sections a / q 3 for electron capture by L i e ions in H for q = I , 2 , 3 (Shah el

a / . . 1978).

Cross sections for one-electron capture by Si ions (with q between 2 and 9) in H and H, measured by Kim et al. (1978) in the velocity range 3.1-6.3 x lo8 cm sec-' show the same general behavior, with cross sections decreasing monotonically with increasing velocities. In this case cross sections scale roughly as q", where n ranges from 2 to 3 as the velocity increases.

Gardner et al. (1977) have measured cross sections for capture by 1.5- 16.3 MeV Fe ions in H for initial charge states q ranging from 4 to 13.


velocity ( x l ~ ~ c r n / s e c - ’ )

1.0 2.0 3.0 4.0 6.0

I I I I I I I \ \

10 100 EQUIVALENT PROTON ENERGY ( k e V )

FIG. 7 . Cross sections u/q2 for electron capture by H+, He2+, and Li3+ ions in H , where y denotes initial charge state (Shah er al . , 1978).

They observe an approximate dependence on charge state that varies from q1.5 for V = 1 a.u. to q3 for V > 2.5 a.u. The q dependence exhibits a discontinuity at q = 8, corresponding to an argon-like closed-shell configuration. Berkner et al. (1978) have carried out similar measurements at 14.7, 15.5, and 61.6 MeV for Fe ions in H2 with q between 11 and 22. In this case, measured cross sections, when divided by two, are shown to be


NC' ENERGY IkeVl 20 100 200 LOO 1000 1600

I I I 1 I I I I I l l

t

1.0 1 I I I I I

0 1 2 3 4 5

Ve loc i t y ( t o8c rn sec- ' ) FIG. 8 . Cross sections for electron capture by NP+ in H and H, for q = 1-5 (Phaneuf rr

a / . , 1978). -, N*+ + Hi ---. No+ + H,.

in good agreement with cross sections calculated for H on the basis of the Olson and Salop (1977) theory. Cross sections are also found to scale approximately according to 93. For electron capture by FeZ5+ ions in H, the calculations indicate cross sections rising from about 4.5 x cm2 at 17 MeV to about 1.2 x 10-14 cm2 at 2.8 MeV.

Experimental data on the reaction

Cz+ + H + C+ + H+ (15) are now available (Fig. 9) down to impact energies of 0.5 keV, where the cross section, although decreasing, is still greater than cm2. In this case the three exothermic channels leading to either C+( 1 ~ ~ 2 s ~ 2 p ) ~ P ground-state ions, C+( 1 ~ ~ 2 s 2 p ~ ) ~ P or C+( 1 ~ ~ 2 s 3 p ~ ) ~ D excited ions with energy defects of 10.8, 5.44, and 1.49 eV involve pseudocrossing of potential energy curves at internuclear separations of about 3, 5 , and 18 a.u., respectively. McCarroll and Valiron (1975) have estimated that none of these channels would lead to significant charge transfer at low impact energies. However, Butler et d. (1977) have shown that in the absence of favorable crossings, the charge transfer process

3 14 H . B . Gilhody

V w v) 1-

r I I I 1 1 1 1 1

C 2 * in H and H2 d d5 -

I + + { i dQ@

[ , , , , I 10 100

N

5 p. 2 * t- - $ V w v)

u) u) 0

V 1" a

C '+ ENERGY ( k e V )

, , , , , , , , , , , , , , , , , I , , ; 0.5 1.0

10- -

FIG. 9. Cross sections uzl(H) and uzl(Hz) for electron capture by Cz+ in H and H,. 0: uzl(Hz), Nutt et al. (1978b); 0: upl(H,), Phaneufet al. (1978); A: uzl(H), Nutt er al. (1978b): A: uZI(H), Phaneuf e t al. (1978).

C '+ ENERGY ( k e V ) FIG. 9. Cross sections uzl(H) and uzl(Hz) for electron capture by Cz+ in H and H,. 0:

uzl(Hz), Nutt et al. (1978b); 0: upl(H,), Phaneufet al. (1978); A: uzl(H), Nutt er al. (1978b): A: uZI(H), Phaneuf e t al. (1978).

C2+ + H + C+ + H+ + hu (16)

occurring with the emission of a photon may be relatively efficient, The reaction

Tiz+ + H + Ti+ + H+ (17)

for which hE = -0.03 eV for ground-state C+ ion formation provides an interesting example of accidental resonance. No exothermic channels involving curve crossing are possible in this case, and the measured cross section (Fig. 10) is not large at low impact energies. The corresponding one-electron cross sections in H, also shown fall less rapidly with decreasing impact energy.

It is interesting to note in Fig. 10 that the two-electron capture cross section for Ti2+ ions in H,, which necessarily involves dissociation of the molecule, becomes comparable with the one-electron capture cross section at the lowest impact energies shown.

Measurements by Crandall et al. (1977) for C, N, and 0 ions in H, with


1 0-l6

- N

E U - Z 0 + W W ln

cn ln 0 LY 0

T i 2 ' in H and H 2

1 I I I I -_

2 5 10 1 017

T i 2 + ENERGY ( k e V )

Fic. 10. Cross sections u,,(H) and uel(Hz) for one-electron capture and ue0(H2) for two-electron capture by Tie+ in H and H2 (Nutt et a/ . . 1978b).

9 = 3-6 for velocities between 0.3 and 1.2 x lo8 cm sec-' show that two-electron capture cross sections are generally about an order of magnitude smaller than the corresponding cross sections for one-electron capture. However, in the case of 08+ in H, the two-electron capture cross section (+64 is found to be rising rapidly with decreasing impact energy. In similar measurements with C4+ in He, Crandall et al. (1976) have found that the two-electron capture cross section begins to exceed that for one-electron capture for velocities below about 6.5 x 10' cm sec-'. In this case, at low velocities the particles have energies sufficient only for a transition through the avoided crossing at 3.5 a.u. between the I Z 4 initial potential energy curve and the lZ3 curve associated with two-electron capture. At higher velocities the particles are increasingly able to transit a second avoided crossing at 2.7 a.u. to the '2, molecular state associated with the one-electron capture process.

3 . Ionization Data

Theoretical estimates of cross sections for the ionization of H atoms (process 4) in collisions with fully stripped ions for charge states 9

316 H . B . Gilbody

between 1 and 36 obtained by Olson and Salop (1977) using the classical trajectory Monte Carlo approach are shown in Fig. 1 1 . The cross sections increase with q but do not scale as q2 at high velocities as might be expected on the basis of the binary encounter approximation (see, for example, Gryzinski, 1965). Comparison with the corresponding cross sections for charge transfer (process 3) in Fig. 2 indicates that while, at the lower impact energies, hydrogen atoms lose electrons mainly by charge transfer, ionization is the dominant process at high energies. In all cases, the predicted total cross section for electron removal as a result of the two processes is only weakly dependent on energy over a wide range.

Berkner et al. (1978) have extended these calculations to collisions of partially ionized FeQ+ ions with H. They show that as q is increased from 10 to 25, the energy at which the calculated ionization cross section begins to exceed that for charge transfer increases from about 110 to 160 keV amu-I.

Experimental data on the ionization of H and H2 by protons are now

E H ( k e V ) 37.5 50 75 100 150 200

I I I I I I I

to-''

N- 1615 E U - b

1 0-l6

36 26 18 1 6

10

7: 5

1

3

2

1

t 10171 I 1 I I

2 3 L 5 6

Velocity (10*crn sec")

4

1 7

FIG. 11. Ionization cross sections calculated by Olson and Salop (1977) for the process Xe + H .+ Xq+ + Hf + e for completely stripped ions with various charge states q.

ATOMIC COLLISION PROCESSES I N FUSION RESEARCH 317

available over a wide energy range. However, measurements with multiply charged ions have so far been extremely limited. In fact the only measurements of direct relevance to fusion are those due to Berkner et NI. (1978), who have used a condenser plate technique to determine cross sections for ionization of H, by FeQ+ ions for q = 11-33 at 1 . 1 MeV amu-I, for q = 9 at 277 keV amu-', and for q = 12 and 14 at 262 keV atnu-'. Measured cross sections, when divided by two, are found to be in good agreement with cross sections calculated for FeQ+-H collisions. At 1 .1 MeV amu-I, where the cross section has decreased beyond its peak value, cross sections are found to scale according to q", where n = 1.43 ? 0.03, rather than 2 predicted by the binary encounter approximation. For q = 22 a cross section of 9.5 x cm2 in H, was determined.

Electron loss from multiply charged ions in collisions with H and H, (process 5) is of much less significance than the ionization process (4) particularly for the higher charge states. In Fig. 5 cross sections v12 and vZ3 for electron loss by fast Li+ and Liz+ ions in H and H2 are shown. Theoret- ical estimates of vZ3 by Shirai et a / . (1977) based on the Born approximation are in poor accord with experiment in the velocity range concerned. Experimental data are very limited for other projectiles. How- ever, cross sections for one-electron loss measured by Dmitriev et (11. (1962, 1975) for Oq+ ( q = 3-7) and Nq+ (q = 0-6) ions in He at energies ranging from about 0.3 to 2.5 MeV amu-' and 0.035 to 0.75 MeV a m - ' , respectively, are in reasonable general accord with cross sections calculated by Rule (1977) using the Born approximation. At 0.3 MeV amu-I, the measured loss cross sections for Oq+ in He range from about 4 x lo-" to cm2 as q is increased from 3 to 6. Dmitriev et al. have also obtained experimental evidence that cross sections for electron loss from fast metastable helium-like ions are considerably greater than those for ground-state ions.

For very highly charged ions, an estimate of the relative importance of processes (4) and (5) is provided by theory. For electron loss by fdst hydrogen-like FeZ5+ ions in H, Rule (1977) has calculated a peak cross section of about 2 x cm2, which may be compared with a peak value of 5 x

cm2 calculated by Olson and Salop (1977) for the cross section for process (4).

B. COLLISIONS BETWEEN POSITIVE IONS

1 . Experimental Approach

Although charge transfer and ionization in collisions between positive ions at keV energies are directly relevant to aspects of fusion research,

318 H . B . Gilhody

experimental studies to date have been very limited. In early measurements, Bogdanov et al. (1965) obtained an approximate estimate of cross sections for electron capture in H+-Li+ collisions from studies of the passage of protons through a lithium arc. More reliable methods based on the use of fast colliding beams, which have been used sucessfully in studies of some electron-ion collisions, low-energy ion-neutral and positive ion- negative ion charge transfer (see review by Dolder, 1969), have so far been applied to only a few cases.

Sindaet al. (1967) first used a fast intersecting-beam technique to obtain data of limited accuracy for 50-250 keV H,+-N,+ collisions, while Brouil- lard and Delfosse (1968) have described a merged-beam method for studies of charge transfer between He2+ and He+ ions at low effective interaction energies. More recently, in this laboratory, we have carried out a series of measurements (Mitchell et al . , 1977; Angel et al., 1978a,b) of charge transfer and ionization in H+-He+ collisions. Total cross sections dHe2+) for He2+ production and separate cross sections cc and u, for the process

(18) H+ + He+ + H(Z) + He*+

[where H(Z) indicates all final bound states of HI and

H+ + He+ + H+ + He2+ + e (19) have been determined within the center of mass energy range 40-400 keV. In our measurements, the two beams were arranged to intersect at right angles. However, P e a t et al. (1977a,b) have studied H+-He+ and H+-Mg+ collisions in experiments where the intersecting beams were inclined at 8.5", an arrangement more suited to studies at low interaction (center of mass) energies using beams of comparatively high laboratory energy. Their studies have been confined to measurements of He2+ and Mg2+ production cross sections in respective center of mass energy ranges 3-29 and 1-45 keV.

In experiments of this type, the main difficulties arise from the low densities of the primary beams and the comparatively small signal count rates. Efficient separation of the collision products formed in the intersection region from the primary beam component, which may be - 1OI2 times more intense, requires very carefully designed electrostatic or magnetic analyzers. In addition, serious signal-to-background problems arise from the interaction of both beams with the residual gas in the collision chamber. Beam modulation in a carefully programmed sequence of pulses synchronized with periodic gating of the detectors of the collision products is usually necessary in order to extract the required signals. Careful account must also be taken of the possible strong influence of any long- lived excited ions present in the primary beams.


In principle, coincidence counting techniques may be used to distinguish between the contributions from charge transfer and ionization. Thus in our recent studies of H+-He+ collisions (Angel et al., 1978b) separate measurements of uc for reaction (18) were obtained by counting the fast H atoms in delayed coincidence with He2+ ions from reactions (18) and ( 19).

2 . Results

In Figs. 12 and 13, cross sections o-(He2+), u,, and uI measured in this laboratory (Mitchell el a l . , 1977; Angel et al., 1978a,b) for He2+ production, charge transfer, and ionization in H+-He+ collisions are seen to be in good accord with the low-energy values of dHe2+) measured by Peart et al. ( 1977a). While charge transfer provides the dominant contribution to dHe2+) at the lower impact energies, the ionization cross section ui = dHe2+) at center of mass energies above about 200 keV.

In spite of its basic simplicity, theoretical studies of the H+-He+ system have been very limited. Olson (cited by Angel et af., 1978b) has used a classical trajectory Monte Carlo method (Olson and Salop, 1977) to obtain charge transfer cross sections u, (included in Fig. 12) that are in reasonable accord with our experimental values. Cross sections for the inverse charge transfer process

(20) HeZ+ + H ( l s ) -+ He+(ls) + H+

z 0 L

u W m

I f

\\ 5 \\\ 9,

\

\

I00 150 200 250 300 350 LOO

-18 10

50 E C M I k e V )

FIG. 12. Charge transfer in H+-He+ collisions. A, Cross sections uc for charge transfer measured by Angel el a / . (1978b) (90% confidence limits): A, cross sections uc for charge transfer calculated by Olson (99% confidence limits); ---, estimate of lower limit to uc based on theoretical data of Rapp (1974); 0, cross sections u(He2+) for He2+ formation from the combined processes of charge transfer and ionization measured by Angel et a/. (1978a) (90% confidence limits); 0, cross sections for He2+ formation from the combined processes of charge transfer and ionization measured by Peart el a l . (1977a) (90% confidence limits).

320 H . 3. Gilhody

r-0.

50 100 150 200 250 300 354 ECM (keV1

FIG. 13 . Ionization in H+-He+ collisions. U, Cross sections u, = dHe*+) - crc from results of Angel ef al. (1978a,b) (see Fig. 12); -, theoretical cross section uI obtained by scaling Born approximation data of Bates and Griffing (1953) for H+-H collisions: ---, theoretical cross section uI obtained by scaling classical estimates of Banks et a / . (1976) for H+-H collisions.

calculated by Rapp (1974) using an impact parameter method are also plotted at the same center of mass energy. These cross sections, which might be expected to provide lower-limit estimates for cc (which involves all bound states of H), lie considerably below measured values of mc at the higher impact energies.

A theoretical estimate for ionization cross sections m, in H+-He+ collisions may be obtained by scaling the ionization cross sections calculated by Bates and Griffing (1953) for H+-H collisions using the Born approximation. Calculations by Bates and Boyd (1962) indicate that, for interactions of a bare nucleus with a hydrogenic ion, the effects of Coulomb repulsion in ionization are negligible except in a region on the low-energy side of the cross section maximum. At energies where the effects of Cou- lomb repulsion are small, they suggest that cross sections for a process

(21)

characterized by projectile and target atomic numbers Z , and Zt may be obtained from the Born cross section for H+-H collisions by multiplying the cross section scale by ZZ,/Zf and the projectile energy scale by M&, where M , is the projectile mass.

Cross sections mi for Hf-He+ collisions obtained in this way are shown in Fig. 13, together with classical estimates obtained by scaling H+-H ionization cross sections calculated by Banks et al. (1976). At the lower

A%+ + B”rL1+ + A%+ + BZt+ + e

ATOMIC COLLISION PROCESSES IN FUSION RESEARCH 32 I

interaction energies, these classical estimates are in better accord with the experimental values of mi than those based on the Born approximation. At the highest interaction energies for which experimental data are available, both these theoretical estimates are significantly larger than our values of u(He2+) = mi. However, it is important to note that the fractional error in the scaled theoretical values for H+-He+ collisions at a center of mass energy of 400 keV would be the same as that pertaining to the calculations for H+-H ionization at a center of mass energy of only 62.5 keV. Unfor- tunately, experimental data for H+-H ionization (Gilbody and Ireland, 1963; Park et a/ . , 1977) involve comparatively large experimental uncertainties and provide only a rough check of theory at 62.5 keV and, indeed, at higher interaction energies.

Figure 14 shows cross sections measured by Peart el al. (1977b) for Mg2+ formation from the combined processes of charge transfer and ionization in H+-Mg+ collisions. They suggest that, since the charge transfer process for ground-state species involves an energy defect of only 1.4 eV, this might be expected to provide the dominant contribution to Mg2+ production over the range of interaction energies considered. Good agreement is obtained between the measured cross sections and cross sections for the Mg2+-H inverse charge transfer process calculated by Bates et a/ . (1%4) using a semiclassical two-state approximation.

I I 1 1 I I I - L - L - W

1 10 100 1 0 "

I N T E R A C T I O N E N E R G Y IkcVI

FIG. 14. 0, Cross sections for Mgz+ formation in Mg+-H+ collisions measured by Peart pr al. (1977b): -, theoretical estimate by Bates el a / . (1964) of cross section for the charge transfer process MgZ+ + H - Mg+ + H + .

H . B . Gilbody 322

c. ELECTRON CAPTURE NEUTRALIZATION OF FAST IONS

Cross sections for electron capture neutralization of fast ions of hydrogen or deuterium in passage through gas and vapor targets have been widely investigated. Alkali metal vapor targets are of particular interest with regard to fusion devices since they provide large cross sections, which increase with the mass of the metal concerned.

For proton impact, in the case of cesium, the reaction

H+ + Cs + H(n = 2) + Cs+ - 0.49 eV (22)

as with other alkali metal vapors, involves only a small energy defect. Capture into the n = 2 states dominates the total electron capture cross section at low impact energies. Theoretical descriptions of the process that make use of semiempirical wavefunctions have been given by Rapp and Francis (1962), Olson (1972), and Olson et a/. (1971). Experimental studies of capture into the 2s metastable state of H have been carried out by recording the Lyman a emission emitted when the H(2s) atoms are quenched in a transverse electric field. In the case of H+-Cs collisions, the measurements of Tuan and Gautherin (1974) indicate that at 0.5 keV about 43% of the atoms formed are in the 2s metastable state.

Electric-field ionization studies of the H atoms formed by electron capture in alkali vapor targets (Oparin e t al., 1967) show that the yield of highly excited hydrogen atoms is comparatively large. The population of levels n > 6 is found to be proportional to n-3, in accordance with theoretical predictions. Such long-lived species when injected at velocity V into a fusion device with a magnetic confinement field B experience an effective electric field E = V x B, which may lead to ionization.

Schlachter (1977) has summarized the available experimental data on total cross sections ml0 for electron capture by protons or deuterons in alkali metal vapors. In these experiments, the target vapor may be either in the form of a beam or contained in an oven or in a recirculatory heat pipe of the type described by Bacal and Reichelt (1974). Large discrepancies in the values of vl0 measured by different observers reflect the difficulties of accurately determining the target gas thickness and, in some cases, inadequate allowance for the angular distribution of the scattered collision products. Nevertheless, the same general features are apparent for all the alkali metals. The cross section vl,, attains its peak value for proton energies below 4 keV and then falls monotonically with increasing impact energy. Typical results for electron capture by 20-100 keV protons in potassium vapor obtained in this laboratory (O’Hare et al., 1975) are shown in Fig. 15.


I

20 40 60 80 100 PROTON ENERGY IkeVI

FIG. 15. Charge transfer and ionization in H+-K collisions measured by O'Hare ef a/ . (1975). u,,,, Cross section for one-electron capture: uK+, u R 2 * , cross sections for K+ and K2+ production: 0,. cross section for electron production.

Meyer et al . (1977) have studied electron capture by H+, H i , and H i ions in cesium vapor within the energy range 30- 160 keV. For the molecular ions, cross sections cr = crl0 + crD were determined, where crD is the cross section for break-up of the molecular ion into one or more atomic ions in processes of the type

H i + Cs 4 (Hi )* + Cs -+ H+ + H + Cs (23)

(24)

At velocities above about 1.5 X lo8 cm sec-1 the measured cross sections decrease more slowly with increasing velocity as the mass of the hydrogen ion increases. However, at lower velocities within the range 0.5- 1.5 x lo8 cm sec-' the cross sections for all three hydrogen ions were found to have approximately the same values ranging from about 1.2 x

to about I .5 x l O - I 5 cmz, Cross sections of approximately the same magnitude were also found for N+, N:, Ar+, and Xe+ incident ions within

4 H+ + H' + Cs + e

324 H . B . Gilbody

-100 s - g 80 - c V

2 60 LL

the same velocity range, a result that provides a useful indication of the extent to which impurity ions in the hydrogen beam would undergo neutralization.

Of more practical interest than the cross sections ul0 is the dependence of the fractional yield of fast neutral atoms on the target thickness p, where p = (effective length x number density). As p i s increased beyond the region where single-collision conditions apply, the fraction Fo of neutrals in the emergent beam gradually increases and attains a maximum value usually when the target is “thick” and the beam composition has attained charge equilibrium in accordance with the various competing electron capture and loss processes. Figure 16 shows the variation in the neutral equilibrium fractions FOm observed by Schlachter et al. (1969) for 0.5-20 keV protons incident on a thick Cs target. While a maximum value Fo, = 0.97 is observed at 5 .5 keV, this has dropped to 0.71 at 20 keV and will continue to decrease at higher impact energies. Measurements in gas targets in the range 5-500 keV (Gilbody et af., 1971; Gilbody and Corr, 1974) show that collisional destruction cross sections for fast metastable atoms are large and indicate that the equilibrium metastable populations in alkali vapor targets should be very small.

In neutralizers employing very intense beams, secondary collision processes involving ions and electrons formed as collision products from both charge transfer and ionization need to be considered. Included in the data for H+-K collisions shown in Fig. 15 are cross sections re, vK+, and q p + measured in this laboratory (O’Hare et af., 1975) (using a modulated crossed-beam technique) for production of electrons, K+, and K2+ ions, respectively, from ionization (processes 9 and 10) and charge transfer.

-

-

-


When considered in relation to wlo these cross sections show that ionization is much more important than charge transfer over the energy range considered.

D. FORMATION OF NEGATIVE HYDROGEN IONS A N D CONVERSION TO NEUTRALS

At proton energies above about 100 keV, where cross sections ul0 for electron capture neutralization are rapidly decreasing with increasing energy, larger fractional yields of neutral hydrogen or deuterium can be obtained by electron detachment when the H- or D- ions are passed through an appropriate target. Although the primary negative-ion beam may be obtained directly from an ion source, as noted in Section II,B, conversion of fast positive ions to negative ions by electron capture collisions is also a possibility. At present alkali metal vapor targets appear to be the most promising.

Experimental data on the conversion of protons or deuterons to negative ions in alkali vapor targets have been summarized by Schlachter (1977). Cross sections c+lifOr two-electron capture by protons are roughly two orders of magnitude less than wl0. Data for incident molecular- hydrogen ions are very limited, but Cisneros et al. (1976) have shown that the cross section for D- formation in Cs by D: ions is less than half that for D+ impact at 1 keV amu-I but approximately the same in the range 1.8-2.5 keV amu-I.

are small, reasonable yields of H- ions can be formed at high target thicknesses through the two-step capture process H+ +- H +- H-. The most efficient conversion takes place at very low impact energies, and in Fig. 16 it will be seen that for Hf in Cs, Fim = 22% at 0.8 keV, falling to 0.4% at 20 keV. However, these particular results are subject to large uncertainties since other investigators (for reasons not clear) have obtained fractions that are up to a factor of two lower. Apart from Rb. which provides values of Fim not greatly different from those in Fig. 16 (Girnius et al., 1977a), lower fractions are obtained with the remaining alkali metals.

Experimental and theoretical studies of electron detachment by fast H- ions in passage through gas targets (see, for example, Gillespie, 1977) show large cross sections, which at high velocities decrease more slowly with increasing velocity than typical one-electron capture cross sections. The measurements by Girnius et ul. (1977b) show that Cs vapor provides a very large detachment cross section mi0 ranging from about 3 x

cm2 at 200 keV. Corresponding values of the two-electron detachment cross section ri1 are more than an order

Although cross sections

cmz at 30 keV to 1.4 x

326 H . B . Gilbody

of magnitude smaller. As the Cs target thickness is increased beyond single-collision conditions, the fraction of fast H atoms formed is found to increase to a peak value F,,,, before decreasing to the equilibrium value FOm for a thick target. Girnius et ul. show that FOm decreases from about 54% at 30 keV to 3% at 200 keV. However, FOmax exhibits a quite different energy dependence, decreasing from about 69% at 30 keV to a constant 58% within the range 100-200 keV. Thus by this means, a comparatively high H- + H conversion efficiency can be maintained at the higher impact energies.

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1312.

ADVANCES IN ATOMIC A N D MOLECULAR PHYSICS. VOL 15

INNER-S HEL L IONIZA TION

E. H. S . BURHOP

CERN Gmerw . Switzerlnnd

I . Introduction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 11. Inner Shell Ionization by Electrons . . . . . 329

A. Total Cross Section.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 B. Differential Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 C. Inner-Shell Alignment of Atoms Following Electron Impact

Ionization ..................................................... 332 111. Inner-Shell Ionization by Atomic Ions

A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C. Inner-Shell Ionization by Heavier Atomic Ions .................... IV. Radiations Following Inner-Shell Ionization .

A. Survey of Structure in the Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Molecular Orbital Radiation . . . . . . . . . C. Miscellany.. .................... References .........................

B. Inner-Shell Ionization by Heavy Pa . . . . . . . . 336 345

362

I. Introduction

The subject of inner-shell ionization is so vast and expanding so rapidly that the topics discussed in this review have had to be limited rather arbi- trarily. The subjects excluded include solid-state phenomena related to inner-shell processes, almost all beam foil analysis work, inner-shell ionization of real molecules, use of Auger spectra for surface studies, and with one or two exceptions all work on inner-shell ionization produced by any mechanism except charged-particle impact.

II. Inner-Shell Ionization by Electrons A. TOTAL CROSS SECTION

Figure la, which compares plane wave Born approximation (PWBA) calculations for Ni K ionization (Burhop, 1940) with experiment

329 Copyright 0 1979 by Academic Ress. Inc.

All rights of reproduction in any form reserved ISBN 0- I2-OO3815-3

330 E . H . S. Burhop

(Pockman et af., 1947; Smick and Kirkpatrick, 1945, shows that at higher energies the calculated cross sections fall below the measured values.

Figure lb shows a similar comparison for Ag K ionization. The discrep ancies between the different experimental measurements are greater than the estimated experimental errors (- 10%) so that it is difficult to draw conclusions about the comparison with theory. There is no such difficulty for the Au K ionization shown in Fig. lc. The experimental cross sections are considerably greater than those calculated using nonrelativistic PWBA.

Comparison is shown in Fig. 1 with the relativistic calculations of Moi- seiwitsch and colleagues (Arthurs and Moiseiwitsch, 1958; Davidovic and

-

(b) Ag - - =-+-<-p&-a--- - -

-

- -

I I I I I I I I I 1 1 14 - -

/ -. - -- - - - - - ---- - ---- --

_-------- - (c) Au -

- -

I I I I I I I I I I I

2 4 6 8 10 12 14 16 18 x) 22

E /E,

FIG. 1. Variation of K ionization cross section with E / E K . (a) Ni: 0, Pockman et a / . (1947). and Smick and Kirkpatrick (1945); A, Green (1962). (b) Ag: 0, Websteret a/. (1933) and Clark (1935); +, Dangerfield and Spicer (1975); x , Daviset a/ . (1972). (c) Au: *, Motz and Placious (1964); *, Rester and Dance (1%6); X , Davis et a / . (1972). For each of the series of measurements a typical error bar is shown on one point. The theoretical curves are given by --, nonrelativistic (Burhop, 1940); ---, relativistic (Davidovic and Moiseiwitsch, 1975); -.-, relativistic (Perlman, 1960); ---, relativistic (Kolbenstvedt, 1967).

INNER-SHELL IONIZATION 33 1

Moiseiwitsch, 1975) in which the incident and scattered electrons were represented by plane-wave Dirac functions and the atomic electron approximated by Darwin wavefunctions in both initial and final continuum states. A retarded Coulomb potential was used for the electron-electron interaction. Comparison is also made with the approximate relativistic calculation of Kolbenstvedt (1967). He classified collisions as “close” or “distant” according to whether the impact parameter is smaller or greater than the K shell radius of the target atom. For distant collisions the incident electron field was represented by an equivalent spectrum of virtual photons as in the Williams- Weiszacker method, the known cross section for photoelectron emission being used to calculate the electron ejection probability. Close collisions were treated in terms of Moller scattering of the incident electron by the bound electron. Ionization was assumed to occur if the energy transfer exceeded the ionization energy of the atom. For Ag the two theories predict similar cross sections over a wide energy range and agree with the measurements of Dangerfield and Spicer (1975). They differ considerably for Ni and Au, although at higher energies they converge toward the experimental results. The results of complete PWBA Dirac relativistic calculations (Perlman, 1960) are also shown in Fig. lc.

The nonrelativistic PWBA exhibits a simple scaling behavior

E ~ ( + K ( U ) = f ( U ) ( 1 )

where U = E/EK; E, EK being the incident electron energy and K shell ionization energies, respectively, and f( U ) a universal function.

The Kolbenstvedt theory suggests the universal relation

E2 (9 - 47rr; - = f (-) 1: E K 0

at very high energies, with 0 = 1 - (16/13)(1 - O ) , 8 being a screening constant and ro the classical electron radius. Alternatively, Helstroom et al. ( 1977) have suggested the semiempirical scaling relation

(3)

with A, B , C, D constants. The experimental errors are so large in the high-energy region, however, that there does not seem much to choose between the expressions (Ishii et a l . , 1977; Genz et u l . , 1976; Middleman et d . , 1970).

If, in Kolberstvedt’s theory the Moller scattering of electrons in close collisions is replaced by Bhabha’s formula for e+-e- scattering an observed reduction of about 2% in K shell ionization by positrons compared with electrons in the energy range 10-20 MeV can be accounted for (Schiebel et a1 . , 1976).

UK = EE2 In(E2/EK)[k? + AEK + C E a + DEE’


Much more realistic calculations of inner-shell ionization cross sections using Hartree- Fock wavefunctions, distorted-wave Born approximation, and allowing for electron exchange and polarization of electron atomic orbitals by the incident electron seem feasible today but have not been carried out. Calculations similar to those for K electrons are available for some L and even M shell ionization by electrons.

It turns out, however, that the total cross section does not provide a very searching test of any theory. Even the classical BEA calculations of Gryzinski (1965) give a passably good account of total inner-shell ionization cross sections-related no doubt to the fact that nonrelativistic classical, Born approximation and exact quantum-mechanical theories of e-e scattering all lead to the same Rutherford scattering formula.

B. DIFFERENTIAL CROSS SECTION

Differential cross section measurements for scattering and emission of the electrons into defined angles and one into a defined energy range would provide a more searching test of theories of inner-shell ionization by electrons. Such very difficult measurements have been made by Ehr- hardt et al . (1971) for the ionization of He and could no doubt be extended to K shell ionization of the rare gases by electrons. Double-differential cross section measurements for K shell ionization of Cu and Ag by Quarles and Faulk (1973) are compared in Fig. 2 with semirelativistic calculations of Das (1972). The agreement is reasonable at small scattering angles but considerable discrepancies are evident round 45". Since most of the cross section is contributed by small scattering angles, it is not surprising that PWBA calculations give good agreement with total cross section measurements.

A single differential cross section has been measured by Camilloni et al . (1972), using a geometry in which the two electrons, scattered and ejected after ionizing C atoms in a thin self-supporting film, were observed in coincidence between two counters set to observe electrons coplanar with the direction of the incident electron and symmetrical about its direction. Figure 3 shows the correlation between the two electrons as a function of the angle 6 either makes with the direction of the incident electron. From the shape of the correlation curve the approximate momentum distribution of K electrons in C can be derived. The results are compared with PWBA predictions in Fig. 3.

C. INNER-SHELL ALIGNMENT OF ATOMS FOLLOWING ELECTRON IMPACT IONIZATION

The inner-shell ionization cross section depends on Irnjl, the magnitude of the projected angular momentum of the ejected electron in the beam

I d d 30 50 70 90 110 130

4 z = 47

I 0'

E (keV1

FIG. 2. Measurements of Quarles and Faulk (1973) of the differential cross section for K shell ionization of Cu and Ag by 140 keV incident electrons as a function of the energy of one of the electrons emitted at a scattering angle of 15" (a) and 45" (b). The curve shows the theoretical calculations of Das (1972). [Adapted from Quarles and Faulk (1973).]

3 34 E. H. S. Rrtrhop

\ \

‘\4 I I I I I I I L L I

36‘ L OM LL’ 40’ 52’

e FIG. 3 . K shell angular correlation measurements of Camilloni et al. (1972) showing coin-

cidences between scattered and ejected electrons from thin C foil (thickness 250 A) . The incident electrons had kinetic energy 9 keV and the direction of scattered and ejected electrons were symmetrical about the incident direction and made an angle 0 with it . The curve shows the theoretical distribution using Roothaan wavefunctions. [After Camilloni et al. (1972). 1

direction. Inner-shell ionization of a shell withj > 4 will leave the ion aligned and X rays emitted following such ionization should be polarized while both the X rays and Auger electrons should have a nonisotropic distribution in angle. For L3 shell ionization this angular distribution can be written.

z(e) = zo(i + A ~ P ~ cos e)/4n ( 4 4 with

o(j,rnj) being the cross section for leaving the shell in the state ( j , m j ) . Measurements of the angular distribution of L3Mz3M23(’SO) Auger electrons from Ar ionized in the L shell by electron impact made by Cleff and Mehlhorn (1971, 1974) and extended by Sandner and Schmitt (1978) using an arrangement in which an electron beam crossed an Ar atomic beam, showed a maximum anisotropy of - 5% for an electron energy of 750 eV. The results shown in Fig. 4 are compared with theoretical PWBA calculations using screened hydrogenic wavefunctions (McFarlane, 1972) and Hartree-Fock functions (Berezhko and Kabachnik, 1977). There is reasonable agreement with the former calculations in the electron energy region below 10 keV but attempts to interpret the measurements in the

INNER-SHELL IONIZATION 3 3 5

0.25 1 5 10 50 T I keV1

FIG. 4. Measurements of the anisotropy coefficient A for the LIMIJM1,I(lSO) Auger electrons as a function of the energy (T) of the electrons producing the initial L3 shell ionization. The dashed curve was calculated by McFarlane (1972) and the full curves (R , quasi- relativistic; N.R., nonrelativistic) by Sandner and Schmitt ( l978), both using screened hydrogenic wavefunctions. The differences seem to be due to the screening parameter used. The chain curve was calculated by Berezhko and Kabachnik (1977) using Hartree-Slater wavefunctions. Experimental data: ., Cleff and Melhorn (1974): 0, Sandner and Schmitt (1978). [After Sandner and Schmitt (197Q.l

higher-energy region were less successful, even when relativistic wavefunctions were used. More recently it has been shown by Sandner et d. (1978) that both theoretical and experimental A2 coefficients when plotted as reduced values, A2(T/EB)/A2(m), nearly coincide in the region of Born’s approximation for all systems studied so far. Here T is the electron kinetic energy and E B the ionization energy of the inner shell concerned.

111. Inner-Shell Ionization by Atomic Ions

A. INTRODUCTION

Many fascinating physical problems have emerged from the vast mass of data accumulated in recent years on inner-shell ionization by heavy particles. It is well known that under conditions in which PWBA is applicable the cross section should depend only on the projectile velocity, not its mass. Of great interest, however, is the adiabatic region where the projectile velocity is much less than that of inner-shell electrons of the target

336 E . H . S . Burhop

atoms, while the projectile kinetic energy may be many times the ionization energy of that shell. This is a region that does not exist in inner-shell ionization by electrons so that it is not surprising that new phenomena are revealed. In the following discussion we distinguish between “heavy particles,” in which the ionization is produced by Coulomb effects predominantly, and “atomic ions,” in which the effects of the electron cloud are predominant. Of course there is no sharp distinction because even protons can capture electrons from target atoms and so leave them ionized. Roughly one can say that “atomic ion” effects are of most importance for projectiles that carry a substantial number of electrons with them and are moving with a velocity less than that of the inner-shell electrons of the target atoms. At higher velocities the effects are largely the same Cou- lomb effects that dominate inner-shell ionization by heavy particles at all velocities.

B. INNER-SHELL IONIZATION BY HEAVY PARTICLES

I. Direct Ionization

Figure 5 compares measurements of K shell ionization cross sections of Al by protons in the energy range up to 2 MeV with PWBA calculations, a region in which the proton velocity is less than that of the A1 K shell electrons. The observed cross sections are smaller than the PWBA cross sections by an order of magnitude or more over a large part of the range. Bang and Hansteen (1959) in a classic and long overlooked paper used a semiclassical impact parameter approach to calculate inner-shell cross sections in this velocity region. Using simple and general uncertainty principle arguments, Bohr (1948) had shown that it was possible to treat the motion of a projectile past an atom classically provided 2Z,Zze2 /hv >> I , Z , e , Z,e being the charges of the projectile and the nucleus of the target atom, respectively, and u the projectile velocity. This condition is certainly satisfied in the energy region considered. Bang and Hansteen used first-order time-dependent perturbation theory to calculate the transition probability for the removal of an electron from the K shell (ionization energy E K ) to the continuum (kinetic energy El) under the perturbation

V ( r , t ) = Z,e2/ lr - R(t)l

due to the passage of the projectile past the atom with impact parameter b. Here r and R are the position vectors, respectively, of the K electron and projectile. Then the differential cross section, summed over all impact parameters for transfer of energy A E = Ef + EK to the atom, is

INNER-SHELL IONIZATION

,005 - -

( 0 ) ,004 -

::I:\ ,001 - - -

337

12

1

::I 04 la

0.2

( b )

f ,003

x .002 + 4 ,001

0 : - d

a

t c n

ax

d

1

OB T I

0.5 ,h 0.4

0.2

0

b (0.u.) - FIG. 5 . Total cross section for A1 K series X-ray production by protons of energy up to

2 MeV. X , Brandt et a / . (1966); 0, Needham and Sartwell (1970); A, Khan er u / . (1965). Theoretical curves: ( I ) Born's approximation (Merzbacher and Lewis, 1958); (2) Bang and Hansteen ( 1959) semiclassical approximation (SCA) with allowance for Coulomb distortion of trajectory of protons; (3 ) calculated by Brandt and Laubert (1969) allowing for change in K shell binding energy due to perturbation by incident projectile. [After Massey i>t a / . (1974).]

338 E. H . S. Burhop

They showed that if the projectile were fast enough so that the trajectory could be assumed to be a straight line, the differential cross section for ejection of the electron into an s state of the continuum, (dcr/dEf)pw was identical with that given by PWBA theory.

If the effect of the Coulomb field of the target atom on the trajectory is taken into account, the amended Bang-Hansteen relation (Amundsen, 1977a) follows:

where qo = AE/hv is the minimum transfer from the projectile and d =

ZlZze2/2E is half the "head-on" closest distance of approach. Brandt and his colleagues (Basbas et al., 1973a) have given nonrelativ-

istic modified PWBA calculations of K shell ionization applicable to ionization by heavy particles of any charge and any target atom. Starting with a PWBA calculation they introduced the relation (6) (without the factor .rrq,d/sinh .rrqod) and integrated over all energies of the ejected electron. They used a perturbation calculation to allow for static polarization effects, which was equivalent to increasing the K shell binding energy by a factor E (Basbas et al., 1973b, 1978). Figure 6 compares the experimental data with the universal curve of Basbas er al. In this figure the quantity

is plotted against r ) / ( E B ) ' , where uK is the K ionization cross section, ZK the effective nuclear charge seen by the K electrons, a. the Bohr radius, B = EK/EKRy the ratio of the experimental K shell energy to that for a screened hydrogenic atom of nuclear charge ZKe,

is the ratio of the velocity of the projectile to that of a K electron, uo being the Bohr velocity. Alternatively, r) = 40E1/PK if El is expressed in MeV/amu:

Elo(x) = e-"/(9 + X)

Figure 6 indicates that the universal curve of Basbas et al. represents the observed cross sections for K shell ionization of a variety of atoms by H+, D+, 3He+, 4He+, s.7Li+ over a range of four decades in cross section.

INNER-SHELL IONIZATION 339

P X

N

+w2 FIG. 6. Universal plot of K ionization of a variety of nuclei by light ions taken from the

original paper of Basbas et a / . (1973a) which should be consulted for references to the measurements. The nomenclature is described in the text.

As u/uK approaches 1 (qK/eZ between 0.05 and l), however, the theoretical curve falls below the experimental measurements. The data in Fig. 6 were obtained from measurements of K shell X rays. For light elements the fluorescent yield is small and not well known. In principle, it is better to measure the total Auger electron emission as has been done by Stolter- foht el a/. (1973a, 1975). Measurements using this method by McKnight and Raines (1976) showed that the theory of Brandt and his colleagues does indeed substantially underestimate the experimental K shell ionization cross section in this region. Ashley et a / . (1972) had suggested the need for a Z3-dependent dynamical polarization correction in this region such as would be introduced by a second-order Born approximation calculation. The situation is somewhat unclear however because while second-Born approximation calculations do indeed introduce a Z3- dependent interference term between the first- and second-order amplitudes, calculations of Ford et al . (1977) indicate that the interference term, as well as the use of Hartree-Fock wavefunctions, substantially reduces (by 10-25%) the theoretical cross sections in this velocity region.

3 40 E. H . S . Burhop

The universal curve of Basbas et al . does not allow for relativistic effects. Caruso and Cesati (1977) allowed for relativistic effects in the inner-shell wavefunctions by replacing the screening parameter 8 by

(a is the fine structure constant) with a marked improvement in the fit to experiment. Relativistic inner-shell wavefunctions had in fact been used in a much earlier calculation of inner-shell ionization by protons by Jamnik and Zupancic (1957). Amundsen (1977b) modified the semiclassical approximation (SCA) calculations of Bang and Hansteen using rela-

FIG. 7 . Comparison between measured K shell ionization cross sections and SCA calculations of Amundsen (1977a.b). Relativistic, full curve; nonrelativistic, dashed curve. Exper- imental points: M, Deconninck and Longree (1977); 0, Hardt and Watson (1973); A, Lark (1962) (after Deconninck and Longree).


tivistic wavefunctions. His calculations are compared with the measurements of Deconninck and Longree (1977) of K shell ionization of heavy elements by (Y particles of energy up to 25 MeV per amu in Fig. 7.

K shell ionization cross sections have been measured for 176 MeV protons by Jarvis et al . (1972) and at 4.88 GeV by Anholtet al. (1976). In Fig. 8 the results of these measurements are compared with the calculations of Davidovic et al. (1978), who used a method similar to that of Arthurs and Moiseiwitsch (1958) for electrons. Davidovic et al . found the ratio of K ionization cross sections for protons to that of electrons of the same velocity ( p = 0.979) increased from 1.00 for 2 = 25 to 1.20 for 2 = 95.

Extensive studies of L subshell ionization by heavy particles are just beginning (Li et al . , 1976; Chen, 1977; Morgan et al., 1977).

2. Inner-Shell Ionization by Electron Capture

Protons, He2+, or other nearly stripped atoms can produce inner-shell ionization of target atoms by electron capture. Guffey et a l . (1977) passed fully stripped nuclei in the energy range 0.25-2.3 MeV/amu through He gas and observed the emission of Lyman series radiation due to electron capture into excited hydrogenic states. MacDonald et a / . (1974) measured cross sections for electron capture by protons of energy up to 12 MeV

40 60 80 100

Atomic number 2

FIG. 8. Relativistic K-shell ionization cross sections for 4.88 GeV protons on various targets. -. Davidovic et a / . (1978); .... Davidovic et irl. neglecting spin change of atomic electron: x. Anholt et nl. (1976). partially relativistic calculations. The experimental points shown were obtained by Anholt et a/. (1976) (after Davidovic et d.).

342 E. H . S . Burhop

from the K and L shells of argon (Fig. 9). The argon X rays were observed in coincidence with an incident particle but no outgoing charged particle.

Lapicki and Losonsky (1977) have calculated cross sections for electron capture in this case. The usual Oppenheimer (1928) and Brinkman and Kramers (1930) (OBK) approximation using PWBA and neglecting internuclear interaction gives much too high a cross section for this process. Lapicki and Losonski obtained a drastically reduced cross section for low-energy ions by allowing for the effects of Coulomb deflection and static polarization in the same way as Brandt and his colleagues in the case of direct ionization (Section III,B,l). For high velocities in the inner-shell orbits they adapted the second-Born approximation calculation of Drisko (1955), developed for electron capture of protons in hydrogen. Figure 9 compares their calculated cross section with the measurements of MacDonald et a l . and also with the OBK approximation. They obtain much better agreement with experiment.

3 . Direct Excitation from Inner t o Outer Shells

F+ ions in charge states 7+ , 8 + , 9+ and energy up to 60 MeV were observed to emit F K series radiation when passing through Hz and He gas (Hopkins et a l . , 1976). The measured cross sections for emission of the radiation are shown in Fig. 10. The energy dependence of the cross section is markedly different for the fully stripped F9+ compared with F7+ and

I 5 10

E , (MeV)

FIG. 9. Comparison of electron capture cross sections by protons of energy E from K and L shells of Ar, calculated by Lapicki and Losonsky (1977) (-) with calculations using the OBK approximation (Nikolaev, 1966) (---)and the experimental results of MacDonald er al . (1974); 0, K shell; 0, L shell. [After Lapicki and Losonsky (1977).]


I I I I I

t e \,9+

343

L \

\ \ 0

10 20 30 40 50 60 70

E (MeV)

FIG. 10. Cross section for emission of K series radiation from nearly stripped ions of F passing through H, and He gas (taken from figure given by Hopkins et a l . (1976). The shape of variation with projectile energy E , is consistent with electron capture for F8+, and with direct inner- to outer-shell excitation for F'+ and Fa+.

Fa+. To emit a K X ray, F9+ must first capture an electron into a shell with n > 1 and the variation with energy in this case is consistent with an electron capture process (compare Fig. 9). The very different energy dependence for F" and Fa+ is consistent with that expected from PWBA in this energy region for direct excitation from inner to outer shells.

4 . Inner-Shell Alignment of Atoms Following Ionizution by Hcwvy Purticlrs

Scholer and Bell (1978) have measured the angular distribution of Cu L and Ge L X radiation excited by the impact of protons of energy 100 keV. The results, shown in Fig. 1 1 , indicate an asymmetry of the form

where 7~ - 8 is the angle between the proton beam and the direction of X-ray emission. As expected, no asymmetry is seen for C K radiation. The coefficient P corresponds to the weighted mean of polarization of the La, (L, + M5), La, (L, + M4), and LI (L, + M,) radiations. The mea-


m- 554 0 5 552

5 550 In - 0 0

548

-' ' ' ' ; ' ' 1

I 79 c

0.2 0.4 0.6 0.8 1.0 cos2 8

FIG. 1 1 . Corrected X-ray angular distribution with respect to (backward) direction of proton beam for (a) C(K), (b) Cu(K), (c) Ge(L) radiation emitted from thin foils. The error bars (AC) are statistical (after Scholer and Bell, 1978).

surements gave RCu) = 3.7 5 0.4%, RGe) = 3.2 5 0.4%, compared with theoretically estimated values Pm(Cu) = 6.8%, Fth(Ge) = 6.2%.

Although the K X-ray diagram lines are not expected to be polarized, K a satellite spectra involving initial KL, KM vacancies would be expected to show evidence of inner-shell alignment. Jamison and Richard (1977) found that the K a satellite spectra of A1 target X rays produced by bombardment by 1.9 MeV H+ and He+ ions showed strong polarization. This was measured directly using a 4 m curved crystal spectrometer as a polarimeter, the intensities I , , , Zl being measured with the Rowland circle normal to and coplanar with the beam axis, respectively. Then

z,, - z,. 1 + C O S ~ 28 P = Z,, + I , sin2 28

where 8 is the Bragg angle (Warren, 1969). Jamison and Richard found polarizations of the A1 Ka' satellite line of 52 2 5% for He+ and 28 5 5% for H+ projectiles. They showed that the measured results are consistent


with values of the ratio u,,/u, = 1.8 for H+ and 3.2 for He+ impact, u,,,u, being, the partial ionization cross sections for L = 1 states with lMLl = 0, 1 respectively.

C . INNER-SHELL IONIZATION B Y HEAVIER ATOMIC IONS

1 . The Obsertvd Phenomena

When the projectile ions are atomic systems instead of simply positively charged structureless particles a remarkable change is observed in the size of the inner-shell ionization cross section at low velocities. It may be many orders of magnitudes larger. Several other new phenomena are observed. The whole field becomes richer in physics content.

In such collisions inner-shell ionization of either (or both) the target or projectile atoms occurs. If targets of different Z 2 are bombarded by projectiles of given Z1 and of the same energy the intensity of K or L series radiation from the projectile changes periodically with Z 2 , the amplitude of the fluctuations being two or more orders of magnitude. A similar behavior, periodic in Z1, is observed if the same target is excited by projectiles of different 2,.

The characteristic X rays emitted exhibit small changes in frequency, and under high resolution this is seen to be associated with a large increase in the number and relative intensity of satellite lines. A similar phe- nomenon is observed for the emitted Auger electron spectra.

2 . The Total Ionization Cross Section

Figure 12 shows the Ar L shell emission cross section observed by Saris and Onderdelinden (1970) for Ar+, Ne+, and H+ ions of energy up to 120 keV incident on Ar. Even at an incident energy of 10 keV the K shell ionization cross section is close to cm2. In Fig. 13, given by Brandt (1973), cross sections for collisions of ionized atoms with Z1 = Z 2 are seen to cluster roughly around the "universal" modified PWBA curve (Fig. 6) at higher energies. At projectile energies around one-tenth of the energy corresponding to the maximum of the curve they break away from it. At the lowest energy the cross sections may be ten orders of magnitude larger than that for a structureless projectile.

Figure 14 illustrates the fluctuations in cross section with Z 2 . It shows results of Kubo et a / . (1973) for Br K a production cross sections from Br+ ( Z = 35) ions of various energies incident on a large variety of targets. Figure 15, given by Meyerhof and Anholt, shows similar curves for Br+ and I+ ions incident on various targets as well as the cross sections for K shell ionization of the targets themselves.

Ion energy, E, (keV)

FIG. 12. Measurements of Sans and Onderdelinden (1970) of cross sections for emission of Ar series radiation by Ar+, Ne+, and H+ ions of energies up to 120 keV.

FIG. 13. Departure of K shell cross sections from universal curve at low velocities when projectile is atomic ion with Z, comparable to Z2 (after Brandt, 1973). In this figure the parameter 5 which replaces the binding energy parameter c of Fig. 6, takes account of the polarization correction at velocities near the maximum of the cross section curve.

346

4 4

b

1 30 40 50 60 70 00 90

FIG. 14. Cross section UB for emission of Br K a radiation by Br ions incident on various targets of atomic number Z 2 . Energy of projectiles: 0, 110 MeV; +, 85 MeV; A, 60 MeV; 0, 45 MeV (after Kubo et a / . , 1973).

z2

FIG. 15. Variation with target atomic number Z2 of K-vacancy cross sections produced in both targets (0) and projectiles (A) when (a) 43 MeV Br ions and (b) 47 I ions are incident on various targets (taken from Meyerhof et a/ . 1977).

341

348 E. H. S . Burhop

3 . Energy Shift and Satellite Structure of X Rays and Auger Electrons Excited by Atomic Ions

The X rays emitted in these processes exhibit energy shifts, dependent on the energy of the projectile and amounting in some cases to several hundred eV. Figure 16 shows typical shifts obtained by Betz et al. (1972) for various target X rays induced by Br+ ions as a function of projectile energy. When examined under high resolution the apparent energy shifts are seen to arise from a large increase in number and intensity of satellite lines compared with other modes of inner-shell ionization. Figure 17 (Knudson et al., 1972) shows A1 K a spectra excited, respectively, by 5.0 MeV zoNe+, 3.0 MeV 4He+, and 6-10 keV e-. The satellites on the high-energy side of the K a doublet excited by Ne+ are considerably stronger than the parent doublet.

A similar effect is observed in Auger spectra but in this case the satellites are on the low-energy side of the parent lines. Figure 18 (Stolterfoht et a f . , 1977) shows the Ne K Auger spectrum excited by various projectiles. The spectrum of Fig. 18d excited by protons of energy 4.2 MeV is similar to that produced by electrons. (This is not the case for low-energy protons. For example, 300 keV protons produce satellite lines of higher intensity.)

Bromine

300 Targets:

_I' - 6 .

d

50 I I I 10 20 30 50 100

Pro jec t i le Energy [MeV] FIG. 16. Energy shift of various X-ray lines excited by Br ions of energies up to 100 MeV

(Betz ef al . , 1972). ( I am indebted to Dr. H . D. Betz for supplying the updated version of the figure used.)


6 -

5 0 MeV 20Ne -

2 -

I 04

gp -

a - -

t

t 2 2 3.0 MeV 4 H e -

X

w cl 103 -

-

9 6 -

1 - 4 z

z E 2

- 6-10 KeV e-

102 - - - -

-

lo" 14bO 1500 I& 15;O 1430 1440 1450

349

It is well known that X-ray and Auger electron satellite lines arise from atoms multiply ionized in inner shells. Compared with inner-shell ionization by structureless particles, inner-shell ionization produced by atomic ions must produce a far greater proportion of multiple inner-shell ionization. Figure 17 indicates that inner-shell configurations containing up to five L vacancies in addition to the K vacancy are frequently produced.

4. Multiple Ionization und Fluorescence Yield

The degree of inner-shell ionization has a marked effect on both radiative and Auger transition rates and thence on the mean fluorescence yield W, so that the latter must depend on the mode of inner-shell ionization. For ionization by charged atomic particles, W can be expected to depend on the energy and charge states of the projectiles. Stolterfoht (1976) has


2 0 0 - M e V Xe3"+Ne FWHM' 2 . l e V

L S - M ~ V CI'*+ 1 5eV -

n

C 3

0 a

- - -

(ldl I L . 2 - M e V H *

1 2e\ I 650 700 750 800 850 9

Electron Energy CeVl

3

FIG. 18. Ne K Auger spectra excited by (a) 200 MeV Xe31+ ions (Stolterfoht et a / . , 19771, (b) 45 MeV CllZf ions (Schneider et a / . , 1976). (c) 30 MeV OSt ions (Burch et a / . , 1975), and (d) 4.2 MeV Hf (Stolterfoht er a / . , 1973b).

discussed attempts to obtain Li for Ne K shell ionization as a function of the mean number of L vacancies accompanying the K vacancy. Mea- surements have been made of WK and the K Auger spectrum of Ne ionized by projectiles of different types and energy. From a knowledge of the intensity of the main diagram line K-Lz3Lz3(ID) relative to the total Auger electron intensity and of the mean energy EA of the Auger electrons, and assuming the probabilities 4n of the production of n L vacancies follow a binomial distribution,

where p is an adjustable parameter, relations between n and and thence between and n were derived. This latter relation is shown in Fig. 19 and compared with theoretical calculations of Chen et al. (1975). These calculations, which assume a statistical population of the multiplet


I I 1 I I I

I 1 I I I I

1 2 3 L 5 6 i i .Meon Number of L Vacancies

F I G . 19. Variation of mean K shell fluorescence yield for Ne with different mean numbers fi of L vacancies due to different modes of excitation ( 0 . 4 . 2 MeV H+; 0.0.1 MeV H+; A, 0.2, 0.4, 0.6 MeV Ne+; 0, 30 MeV 05+; 0, 50 MeV CIe, 9 = 5, 6 , 7; W, 50 MeV C I e , 9 = 12, 13, IS). The experimental points are compared with the calculations of Chen et d. (1975) [after Stolterfoht (1977)l.

states LS in the configuration n, agree with the values of Tj;; derived from experiment, but the assumption of statistical population is open to question.

Since estimates of total inner-shell ionization cross sections require a knowledge of W, the variation of this quantity with experimental conditions introduces considerable uncertainties. To obtain accurate inner- shell ionization cross sections both X-ray and Auger electron intensities should be measured simultaneously.

5 . Difjkulties in Accurate Measurement of Inner-Shell Ionization Cross Sections by Atomic Ions

From the preceding discussion these can be summarized as:

(1) Difficulty of separating direct inner-shell ionization from electron

(2) Difficulty of identifying ionization of target atoms by ionized target

(3) Uncertainty of the charge and excitation state of the projectile ions.

capture by projectile ions from inner shells of target atoms.

atoms recoiling after a previous collision.

352 E. H . S . Birrhop

Irrespective of the initial charge state of the projectiles, as a result of ionizing and electron exchange collisions with target atoms an equilibrium is established. For example, in a typical experiment in which 14N+ ions passed through Al, in equilibrium it was estimated the 14N+ ions had charge states 3+, 4+, and 5+ and that 98% of the projectiles had 1 or 2 K shell vacancies (McDaniel et a l . , 1977).

(4) Uncertainty about the correct value of W to use.

Multiple-collision effects mean that using heavy ions and thick targets the observed number of inner-shell vacancies per incident particle per surface density of target may be many times the number that would be produced in thin targets by projectiles of the initial charge state.

Methods of extracting meaningful cross sections from measurements using thick targets are discussed by Laubert and Losonsky (1976).

6. Interpretation of Phenomena Associated with Inner-Shell Ionization by Atomic Ions

An important step toward the understanding of these phenomena was made by Fano and Lichten (1965) in terms of pseudomolecule formation during the collision and the concept of electron promotion. We illustrate the process by reference to a collision between two Ne atoms. Figure 20 shows a correlation diagram for the Ne+-Ne system calculated by Larkins (1972). Two of the four electrons initially in the 1s levels of the two Ne atoms go via the Is crg molecular orbital (MO) to fill the 1s level of the “united atom” (Ca). The Pauli principle prevents any more electrons going into this level. Instead they follow the 2pa; MO and are “promoted” to populate the 2p level of the united atom. This level is also reached via the 2p7r, orbital from the 2p levels of the separated Ne atoms. The united atom state is never reached but if the projectile Ne+ has a kinetic energy above -50 keV the value of R for a head-on collision is less than the Ne K shell radius (5 x cm) and the energy separation between the 2pu, and 2~77, MOs becomes small. Assuming the Ne+ ionization is in the 2p shell the vacancy will be shared between the 2p7rU and 3du, MOs and a transition 2pcru-2p.rr, can take place. As the atoms separate an electron that came into the collision from the ls level on the 2pvU MO will leave on the 2pr, MO, so that when separated the initial 2p vacancy will have been transferred to the 1s level, leading to the emission of K X rays or Auger electrons. The cross section for the process will depend on the 2pcru-2p7r, transition probably discussed later.

In systems of more than 10 electrons the difficulty of this model lies in estimating the proportion of projectiles that either enter the collision with a vacancy in the 2p shell or in which a vacancy is produced in the

INN E R-S H E L L I 0 N I Z A T I ON 3 5 3

United atom Separated atoms

- 0 5

-1 0 3 P

x

3

Q,

"I Atomic t , , , , , , ]Levels

I S 0 5 10 1 5

Ne Internuclear seperotion ( A U

Ca Levels < r >

FIG. 20. Correlation diagram for the symmetric Net-Ne system calculated by Larkins (1972). In the adiabatic approximation molecular orbitals such as Z s u , , 3dcr,, with the same parity avoid crossing as shown. Near the avoided crossing the diabatic curves are represented by dashed lines. Transitions between one orbital and another are much more probable in regions such as these. where the two orbitals approach each other in energy.

2pr, MO while R is still larger than the separation at which the transition can occur. In solid targets, as already seen (Section 111,C,5) inner-shell vacancies are readily produced and quenched as ions pass through matter. Macek and Briggs (1973) have estimated values of the vacancy number in some cases.

The first calculations of cross sections for inner-shell ionization using the Fano-Lichten model and a perturbed stationary state (PSS) impact parameter method were made for Ne+-Ne collisions by Briggs and Macek (1972) and Briggs (1972). They followed the perturbed stationary state (PSS) method used in an analogous calculation by Bates and Williams (1964) of electron excitation in slow H+-H collisions. The kinetic energy of relative motion of the two atoms provides the perturbation under which the transition 2pu, + 2pr, occurs. The Schrodinger equation for the nuclear motion reduces to a set of coupled equations containing dynamical coupling terms that in the semiclassical limit reduce to U R ( i 1 P R I j ) and fl( i lLzlj) , u R , P R , and R being respectively the relative velocity, relative momentum, and rotational angular momentum of the nuclear motion. The radial coupling term vR( i lPR) j ) induces transitions between states of the same rotational symmetry. For example, in Fig. 20 the MOs ~ s u , , 3dug are seen to approach each other at R = 0.5 a.u. and transition between


them can occur under the influence of this term. The rotational coupling maxtrix element O(ilL,lj) connecting states of different angular symmetry determines the 2pau + 2pr, transition probability and thence the cross section for the creation of a 1s vacancy by this process.

Figure 21 compares the results of these calculations with the measured cross sections of Stolterfoht et al . (1975). There is good agreement at low impact velocities with the theoretical curve calculated using a screened Coulomb internuclear potential. At higher velocities, however, the theoretical curve falls far below the experimental values. This is attributed to a velocity dependence of the number of vacancies the projectile brings into the collision.

The situation is more complicated for collisions in which Z , # Z z . We refer to them as the lighter (L) and heavier (H) atoms, irrespective of which is the projectile and which the target. In the first place the correlation diagram is more complex. Figure 22 shows such a diagram calculated by Larkins (1972) for the AI+-Ar system. In such systems the K shell of the heavier partner (Ar) would not be excited by the Fano-Lichten process because the lowest lscr orbital does not have a close approach to any of the other MOs. It is seen from Fig. 17, however, that the sharing of K vacancies with the heavier partner is observed quite generally although the sharing ratio diminishes rapidly as ZH - ZL increases. Meyerhof (1973) has applied a model of radial coupling due to Demkov (1964) to interpret the observed K shell vacancy sharing between the collision partners in cases where ZH - ZL is not too large. He supposes the sharing to arise from a l s a -+ 2pu transition (Fig. 22) on the outgoing part of the

0 0.5 1.0 1.5 2.0

Impact ve loc i ty (mu.)

FIG. 21. Comparison of measurements of K-shell ionization cross section in Ne+-Ne collisions (Stolterfoht ef a/., 1975) with the calculations of Briggs and Macek (1972). Figure taken from Briggs (1976).


4 s

-1 3 d

3 P

- 3 s 3 a -10 - > (3 U W z W

2 P 9 2 s

$ -100

z 0 U F

1 W

\ .

.-.-a 1 /* /--=

- 3 p A r 3 s Al'

- >(To 3 p A r )

3 s Ar -

- 2 p A I '

- 2 s A r +

2 p Ar 2 sAr

- -

- l s A l f

- I s A r

Atomic Levels

Go+ Levels F I G . 22. Correlation diagram for the asymmetric A]+-Ar system calculated by Larkins

I N T E RNUCL EAR SEPARATION ( A.U.)

(1972). In this case the united atomic system is Ga+.

collision, after the 2pu vacancy has been formed by rotational coupling 2p77 + 2pa 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

( 10a) e-zx P,, = - - M'

- 1 - - w

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 2 x

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 Stol- terfoht (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 /ZL is 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


43-MeV Er

5.

f l

Y

0

L7-MeV I

A)

0.

22 40 60 00

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 Taulb- jerg, 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 or 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 ZL and the L shell of ZH. 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


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

(12b) P S R = exp[2rA~/au] - exp[2nA~ cos2 3O/av]'

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 Demkov- Meyerhof 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!

exp[2rA~ cos2(b 8) /au ] - 1

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 sec-

I

c

3 60 11,

\ \

FIG. 25. Ratio P S R of 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 ( f m )

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/wK where n (= 6) is a statistical factor related to the 2pr, MO vacancy fraction and wK is the fluorescence yield. Compari- son 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.

70-3-J 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) .


IV. Radiations Following Inner-Shell Ionization

A. SURVEY OF STRUCTURE I N 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 re- named "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 hyper- satellites 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 203Hg that 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).


- 2150 2100 2050 2000 1950

ENERGY (cV) FIG. 29. Structure observed on the low-energy side of the K a doublet of sulfur. The ra-

diative Auger effect is shown in (b). The peaks E , , E2 are 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

3 64 E. H . S . Burhop

7

Mn

1061

a )

WAVELENGTH (n’) r 1 I I I 1 I I 1

-200’ 2 6 5 2 6 4 2 6 3 2 d 2 261 2 d o Ib9 ’ FIG. 30. K spectrum of Mn showing structure in the K& hypersatellite. (b) Detail of the

WAVELENGTH ( d l

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


-I W z Z U I V U W a

Z 3

8

J W Z z a

U w a

I- z 3

8 I - I I I J

6.5 7 7.5

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


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 Kaa 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 suc- cessive 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 Demkov- Meyerhoff 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 Kaa

1 o6

E ._

ZI y lo4

I o3

X

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 ab- sorbers.


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 (REC) by 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.

100 200 300 400 500 600

E,, ev

FIG. 33. Auger electron spectrum from 50 keV Ar+-Ar (0) and CI+-Ar ( X ) collisions observed by Afrosimov er a / . (1976) due to transition L2,3LZ.3-MMM.


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 ZUA the 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~t sec, 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,, Cp are 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).


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, Cz are very prominent. In Fig. 35 only C1 appears: CB is completely absent. The difference between the two cases is

3 70 E. H . S. Burhop

2 6

5 5

z W

8 a 4 W

3 3 z

J

1

I 40

0 ' '

10 20 30

ENERGY (KeV)

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 C1 continuum 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 37 1

lo2 10' lo4 lo5 R ( f r n )

6oi I E ( k e V )

FIG. 36. Correlation diagram for the Nb+-Nb system. Owing to the dipping of the Isu MO, the transition giving rise to C1 is more energetic than that giving rise to CI . [After Kaun rr a / . (1976).]

iments of Saris and Hoogkamer (1977) on MO X radiation produced by 200 keV N + collisions in gaseous Nz and NH3. It is seen in Fig. 37 that the MO radiation of the N-N system is seen in Nz, not in NH3. A two- collision 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 Rays

The MO X rays are not emitted isotropically. This was first observed by Greenberg et a l . (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 I

1000 2000 3000

X-RAY ENERGY [eVl

FIG. 37. 15 MeV N + ions bombarding gases targets of N2 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 , + ZH, 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 + ZL between 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


20

8 0 1 0 + C I F e - t 4 1 } ~ ' ~ ~ -NI

0 K r - Z r 200 MeV

40[; 20 ,-..,:"i" 0

45 , 4 40 20 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,

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 positron emission the U ions would need to stay together for - sec.

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. Re- cently Howat et a l . (1978) have introduced two new refinements, orbital relaxation and intercontinuum interactions between the different 2S final 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 a l . 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


-loo0 -

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 negative- energy 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

RELATIVE TRANSITION RATES FOR K-LL TRANSITIONS (LS COUPLING) I N Mg"

Final LL Without With With both state CI or ICC CI CI and ICC Experiment

(2s)-2 1s 10.03 1.34 5.41 6.32 ? 0.67

(2s)-L(2p)-' 3P 8.82 8.80 8.20 8.40 f 0.88 (2s)-'(2p)-' 'P 21.94 22.03 18.80 17.44 ? 1.58

(2p)-2 'S 4.37 6.20 7.82 8.97 -+ 1.37 (2p)-l ID 54.83 55.63 59.77 57.61 f 3.52

a From Howat (1978).


(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 lQ7Au was 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 , , Ka2 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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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

B. Application of the Bethe Approximation . . . . . . . . . . . . . . . 403

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 com-

381 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 wefl- documented 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 BY ELECTRON

(.. + AJl + 2 AJk) 1 S Nj dS k

IMPACT 383

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 gJ is 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 excitation of other states, we can write

where we have written J j to describe the cascade cross section

J j = Qi‘ A d A i

It is clear without detailed calculation that gj will approach unity for van- ishingly 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

i

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 Sec- tion V) where collisional transfer from atoms in these states could inter- fere 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, c* x, + Xk

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 n1F3 and 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

FIG. 2. Partial energy level diagram of helium experiment of Shaw and Webster (1976).

(23s

showing the transitions relevant to the

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

(gk /gJ) 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-

EXCITATION OF ATOMS B Y ELECTRON IMPACT 387

TABLE I

CROSS SFCTIONS FOR E X C I I A T I O N TRANSFER A N D FOR ASSOCIATIVE I O N I Z A T I O N I N H ~ L I U M ~

Initial levelb Initial level' Final Final level 4'P 4'D 4'F 43P 43D 43F level 3'P 3'D 33P 33D

41P - 80 0 0 0 0 3'P - 10.6 0 0 4'D 170 - 71 0 0 38 3'D 32 - 0 0 4'F 0 97 - 0 51 0 33P 0 0 - 0.62 43P 0 0 0 - 150 0 33D 0 <0.02 0.067 - 43D 0 0 110 81 - 100 H e l d 3.1 20 1.6 4.5

He: 94 S6 40 31 19 13 43F 0 160 0 0 140 -

~ ~ ~ ~ _ _ _ _ _ ~

In units of m2. Shaw and Webster (1976). Wellenstein and Robertson (1972a). Wellenstein and Robertson (1972b).

I 1 1 I

0 1 2

Pressure / rn Torr

FIG. 3. The effect of pressure on the apparent cross section for the excitation of the 43D state of helium for several values of electron energy.

388

lo - *

N

E 0 \

10.;

10.;

D. W . 0. Heddle

Electron Energy/eV (4’0)

10 100 I 1 - 1 1 I I I I

FIG. 4. Absolute excitation functions of the 43S and 43D states of helium. + , van Raan ef al. (1974); 0, Showalter and Kay (1975).

larly considering the range (three decades) of values involved. There is a theoretical prediction (Ochkur, 1964) that both of these cross sections should vary as E-3 at high energy, and lines of this slope are shown. The behavior at low energies has been taken from Weaver and Hughes (1%7), Heddle and Keesing (1967) (43S), and Heddle and Lucas (1963) (43D), in each case normalized to the higher-energy values.


C. INSTRUMENTAL POLARIZATION

Any monochromator will be sensitive to the degree of polarization of the light that enters the slit. In prism monochromators this is the result of the different reflection losses for light polarized parallel and perpendicular to the plane of incidence (the angle of incidence on the prism face is quite close to the Brewster angle) and normally varies quite smoothly with wavelength. The behavior is quite different in a grating monochromator as shown in Fig. 5 : abrupt and substantial changes occur. These are known as Wood’s anomalies; their position in wavelength is governed by the condition that some (unobserved) order is diffracted in the plane of the grating, but their magnitude depends on details of the groove profile.

We define k,, and k, as the sensitivities of the detection system to light polarized parallel and perpendicular to the entrance slit and consider light from the collision chamber to have dipole components I,, and ZL, which are the fluxes per unit sold angle observed perpendicular to the electron beam with electric vectors, respectively, parallel and perpendicular to the electron beam. The radiance transmitted by a polarizing filter oriented to transmit light having its electric vector at an angle 8 to the beam direction is

I , , COSZe + I , sin%

and this transmitted light is now polarized in the direction 8, so that the signal recorded by the detecting system will be

S(90) = (I,, cos28 + 1,. sin28)(k,, cos2e + k , sin%)

FIG. 5. The response of an S-20 photocathode to light from a quartz-iodine lamp transmitted by a grating monochromator.

3 90 D. W. 0. Heddle

Clout and Heddle (1969) have shown that this can be written

( 1 ) I 1 - P(sin2B - cos28)

1 - P / 3 S(90) = - 8 7 ~

(kl, cos28 + k, sin2B)

where P, the degree of polarization of the light, is defined by

and I is the total emitted flux. Equation (1) shows that S(90) will be independent of P if sin% -

cos28 = -5, i.e., if e = 54'44'. A polarizing filter placed at this angle im- poses a polarization sensitivity ratio of 3 regardless of the values of k,, and k, for any subsequent optics.

Observations have frequently been made at an angle of 54'44' to the electron beam in the belief that this takes the correct average of the emitted light and the signal is independent of P. This is not true, in general, because the light is still polarized at this angle and any polarization sensitivity will lead to error. Clout and Heddle (1969) have shown that this difficulty can be overcome by the use of a polarizing filter placed with its axis at 45" to the beam direction or by orienting the plane containing the electron beam and the optic axis at 45' to the slit, as depicted in Fig. 6. This latter method, which works because monochromators have their polarization axes parallel and perpendicular to the slit direction, has been applied by de Jongh and van Eck (1972), Donaldson et al. (1972), van

Electron Beam

FIG. 6. An orientation of the electron beam relative to the monochromator, which elim- inates polarization effects.

EXCITATION OF ATOMS BY ELECTRON IMPACT 39 1

Raan (1973), and Westerveld et al. (1979) to measurements below 100 nm where only inefficient reflection polarizers can be used.

While it is important to eliminate the effects of instrumental polarization the measurement of the degree of polarization of impact radiation is also important. Clout et al. (1971) describe the use of a half-wave plate to rotate the planes of polarization and show that a wave plate of any retar- dation (except, of course, A/4, 3A/4, etc.) can be used as a polarization modulator and that the effect of imperfect polarizing filters can be allowed for.

The angular distribution of impact radiation is directly related to the polarization by

provided the detection system has no polarization sensitivity, and this provides a method well suited to studies in the vacuum ultraviolet. Care is needed to allow for the changes in the volume of the excited gas that occur at different angles of observation. Mumma ef af. (1974) have applied this technique to the unresolved resonance lines of helium, making measurements at angles between 40 and 140". They measured the angular distribution of the 130 nm line of 0 formed by dissociative excitation of O2 as a test of the geometry of their system. Their results are in reasonable agreement with those of Heddle and Lucas (1%3) for the 3lP-2lS line at 501.6 nm.

111. Behavior near Threshold

A. RESONANCES

At energies close to those of excited states of an atom an additional electron can be bound to the atom by either a polarization or a centrifugal potential. This attachment is generally very short-lived and the system au- toionizes in less than 1 psec. These temporary negative ion states or resonances can have a very marked effect on the excitation cross section and especially on the polarization of the light. Heddle (1976) has reviewed this topic and there has been some more recent work on undispersed resonance lines by Brunt et al. (1977a,b). The details and sometimes even the existence, of resonance structure can only be studied using velocity- selected electrons, and there are many factors that limit the energy resolution that can be attained (Read, 1975).

Figure 7 shows the excitation functions of the 3ID and 33D states of helium (Heddle et al., 1977). The excitation thresholds differ by less than

392 D . W . 0. Heddle

. . , . . .

i ..,... ..i’.,:..: ..., ..: - :.. +.,:.. .,_.. , , ,. L.. . .*’ .

. . . . .. .” . i ’. . .

1 mV and yet the onsets of excitation are so different that they could well differ by as much as 100 mV.

The study of electron scattering at these resonance energies would require the “photon labeling” technique of Pochat e f al . (1973), which is discussed briefly in Section VI1,C.

B. POLARIZATION

Percival and Seaton (1958) have shown that the polarization of impact radiation can be expressed in terms of the cross sections Qm for the excitation of states with components m of orbital angular momentum about the beam direction. They point out that, provided the atomic potential is of finite range, excitation by an electron having only the threshold energy can excite only the state with m = 0 and the polarization is then determined purely by geometric factors. Their argument is based on the conservation of angular momentum.

They give extensive tables that enable the polarization to be calculated from values of Qm. Flower and Seaton (1%7) have extended the analysis in a way that allows for the nonzero nuclear spin of the alkali atoms. We shall show in Section VI,B several examples where the behavior of the polarization does not tend smoothly toward this threshold value. Figure 8


I 1 I I I I 1 I

0 -&w///mz2m I 1 I I I I I 1 23.6 24.0 24.2 2 4 4

Electron EnergyleV ,

FIG. 8. Near-threshold polarization of the 492 nm line of helium (4ID-2lP). 0, McFar- land (1967); -, Heddle et a/ . (1974).

shows the polarization within 1 eV of threshold of the 4lD-2IP line of helium at 492 nrn. The threshold polarization is expected to be 0.6 for this transition. The results appear to extend below the excitation threshold, but this is merely the consequence of identifying the energy of the peak of the distribution of energies in the electron beam as the energy. This is per- fectly reasonable for energies well above the threshold where the excitation cross sections change but little over the range of energies in the beam. However, even if the energy of the peak lies below the threshold there will be some electrons with sufficient energy to produce excitation. The two sets of data compared in Fig. 8 were obtained under very different conditions. McFarland (1967) used a crossed electron-atom beam observed at 90" with a photomultiplier tube preceded by a narrow-band interference filter, the system being essentially free from any polarization sensitivity; Heddle et a] . (1974) used a monochromator that had considerable polarization sensitivity, but they eliminated it using the method described by Clout and Heddle (1969).

The polarization values shown in Fig. 8 have in no way been normalized. The difference is entirely a consequence of the very different energy homogeneities of the electron beams.

Because the prediction of threshold polarization rests on the fact that at

394 D . W . 0. Heddle

threshold the electron that has excited the atom has no kinetic energy left and has consequently zero angular momentum, in an experiment that ob- serves light emitted by atoms excited by electrons of any energy, but which are scattered in the forward direction and therefore have no angular momentum about the electron beam, the polarization of this light should have the “threshold” value. This has been demonstrated by King et al. (1972) for the 501.6 nm line of helium. Their experiment also demonstrates that the tolerance on “forward scattering” is quite strict as measurements made at incident energies above 100 eV showed appreciable depolarization. The technique can probably be only expected to give reliable results if the scattering is not too strongly peaked in the forward direction. The constraint on the range of angles that can be accepted is more severe than in experiments such as those of Eminyan et al. (1974) in which angular correlations are measured.

IV. Measurements by Different Techniques

The normal technique used for the study of electron impact excitation is the observation of the light emitted by the excited atoms in their subsequent decay, but other methods are possible and may indeed be preferred if the radiation lies at wavelengths beyond the range of photoemissive cathodes [though Jobe and St. John (1967a,b, 1972) have used a lead sul- fide detector at longer wavelengths]. Figure 9 shows excitation functions of the 63P states of mercury obtained by four quite different techniques. The 63P1 excitation function of Shpenik et al. (1976) was obtained by the conventional method, using a trochoidal electron monochromator (Sta- matovic and Schulz, 1970) to produce a beam of electrons homogeneous in energy to rather better than 0.1 eV. The effect of resonances is very marked. The 63P0 excitation function of Krause et al. (1977) was obtained by observation of the line at 265.6 nm emitted in the transition 63P,-61S0. This is normally considered to be forbidden, and for the even isotopes of mercury this is indeed the case. However, the odd isotopes have nonzero nuclear spin and the selection rule on J is superseded by a condition that F must change by 0 or ? 1. Garstang (1962) has calculated a radiative lifetime of 5.56 sec for this electric dipole transition for the terrestrial isotope ratio. The measurements were made in a crossed-beam system, the light being detected some 3 cm downstream from the excitation region. A mercury vapor cell was used to absorb light of 253.7 nm from atoms in the 63P1 state. The results of Bogdanova and Marusin (1971) were obtained by the sensitized fluorescence of sodium. Their excitation chamber contained both sodium and mercury vapor. They observed that certain of the


0 2

I 8 10 0 4

Electron EnergyloV

, 1.0

0.8

0.6

0.1

0.2

0 >

!

F I G . 9. Excitation functions of the 63P levels of mercury. 63P0: ---, Korotkov (1970); -, Bogdanova and Marusin (1971); 0, Krause P I ( I / . (1977). 63P,: -, Shpeniket c r / . (1976). 63P2: ---, Korotkov and Prilezhaeva (1969); -, Borst (1969); * , Krause et a / . (1977).

sodium lines, notably the 6S-3P, 7S-3P, and 9S-3P, were enhanced to an extent proportional to the ratio of the mercury and sodium concentrations. The 9s-3P line is enhanced as a result of collisions with mercury atoms in the 63P, state and they showed that this enhancement gave an excitation function for this state that agreed very well with direct measurements made with similar energy resolution. The other two lines were enhanced in a very similar fashion and they ascribed this to transfer of excitation from mercury atoms in the 63P, state. Figure 10 shows the part of the energy level diagrams of sodium and mercury relevant to these processes. Some of the enhanced population of the 6 s and 7 s states will be due to cascade from the 8P and 9 s states, which are enhanced by collision

396 D . W . 0. Heddle

Na "g

4.5L 6s ~

FIG. 10. Partial energy level diagram of sodium and mercury.

with 63P1 atoms. Some 12% of the 8P excitation (Anderson and Zilitis, 1%4) will reach each of the lower levels, and so the results of Bogdonova and Marusin (1971) almost certainly represent an overestimate, especially at the higher energies, though there is good agreement with Krause et al. (1977) in the presence of a feature on the rising edge of the function and in th,: form of the feature around 8 eV. The measurement of Korotkov (1970) was made by detecting the electrons superelastically scattered in the reaction

Hg(63Po) + e + Hg(6'So) + e + 4.67 eV

and applying the Klein-Rosseland relation to determine the cross section for the reverse, excitation process. He found two well-separated maxima though the details of the lower maximum must be in some doubt because of the indirect method of measurement. Shpenik et al. (1976) have observed the electrons scattered in the forward direction after exciting the 63P states and they find a very sharp peak at 4.70 eV for those which have excited the 63P0 state. Their measurements for this state only extend to 4.9 eV.

Korotkov and Prilezhaeva (1969) used this same technique to determine the excitation function of the 63P2 state and there are again signs that the analysis has overenhanced the values at low energies. The results of Krause et al . (1977) were obtained by sensitized fluorescence of the 253.7 nm line following the reaction

Hg(6SP2) + Nz -., Hg(63P,) + N$

A beam of nitrogen was crossed with the beam of mercury atoms and this gave a much stronger signal than could be obtained from the direct decay of the 63P, atoms at 227 nm. As in the case of the 63P0 state, radiative decay is possible for the odd isotopes; Garstang (1962) has calculated a ra-


diative lifetime of 6.67 sec for this transition. The measurements of Borst (1%9) use yet another technique. A pseudomonoenergetic electron beam produced by the RPD technique causes excitation on the axis of a cylindrical collision chamber. Coaxial grids prevent the passage of charged particles, but metastable atoms and photons can reach an outer tungsten cylinder and, if they have sufficient energy, eject electrons. 63P0 atoms and 253.7 nm photons have insufficient energy and the ejected electrons give a measure of the 63P2 atoms for electron energies up to 8.5 eV. At higher energies other sources become important, notably atoms in the 5d%s26p3D metastable state at 8.785 eV (Krause et al., 1975). Krause et al. (1977) have also measured the 63P2 excitation function directly.

It is perhaps appropriate in the light of the data collected in Fig. 9 to reconsider the classification of the resonances that appear in electron- mercury scattering below 6 eV. Albert et al. (1977) have made a major advance in the classification by measuring the differential cross sections for elastic scattering and the spin polarization of the scattered electrons. They have shown that both the prominent resonances that are observed in the 63P, excitation function have total angular momentum of 5/2, in contrast to the suggestion of Heddle (1975) that the lower-energy resonance would have J = 3/2. The elastic scattering measurements show that one resonance with J = 3/2 does not appear. However, the excitation functions of the 63P0 and 63P, states show (the first very prominently) a resonance at 5.2 eV that has no counterpart in elastic scattering. Albert el d. (1977) suggest that this is the “missing” J = 3/2 resonance. We show in Table I1 the energies of the low-lying resonances, the channels in which they appear, and term descriptions of the Hg states involved.

The resonances between 4 and 6 eV are all of even parity (configuration 6s6p2) and so the electron scattered after exciting a 63P state can carry

TABLE I 1

ENERGIES OF RESONANCES IN ELECT RON-MERCURY SCATTERING

EnergyleV Elastic 63P, 63P, 63P2 Term

0.63 J ZP ‘P,n ‘P3i2

4.55 J 4.71 J 4.94 J 4- ’P,,, 5 . 2 J J h 2D3i2

5.51 J J J- 2D,,2

(I -

a Resonances of lower energy cannot appear Very weak.

398 D. W. 0. Heddle

away at most 3/2 units of angular momentum. In the case of the 63P0 state the resonances having J = 5/2 are therefore excluded on the grounds of conservation of angular momentum and the identification of the resonance at 5.2 eV as having J = 3/2 is directly confirmed. The absence of this resonance in elastic scattering and its extreme weakness in the 63P, excitation function require a more detailed and quantitative description.

V. Time-Resolved Measurements

The light observed following impact excitation is not only the result of the direct excitation process, but has a component due to excitation of higher states and subsequent cascade. If the cascade transitions (or other transitions from the same states) can be observed then it is possible in principle to make a proper allowance. Frequently this is not possible and the cascade contribution must be assessed in a different fashion. The radiative lifetimes of a state of interest and of states from which cascade populations arise are sometimes very different, and time-resolved measurements can help to discriminate between the various excitation processes.

Figure 11 shows data of Anderson er al. (1969) for the intensity of the 668 nm line of helium (3lD-2lP) following excitation by a pulse of 100 eV electrons with a cutoff time of some 7 nsec. The basic data show an approach to an exponential decay with a time constant exceeding 1 psec.

100

>. c

.A .- C 0

c c

- 10 0 >

0 0

D?

._ c -

1 0 2 4 6

-7 Tirne/ lO sec

FIG. 11. Decay of the 668 nrn line of helium (3lD-2lP) following pulse excitation.

EXCITATION OF ATOMS B Y ELECTRON IMPACT 3 99

The origin of this is not explained but is probably the consequence of cascade from states of large n and 1. Subtraction of this asymptote from the data gives the values represented by open circles, which approach an exponential decay with a time constant of 71 nsec. This agrees very well with the lifetime of the 4F states calculated by the Coulomb approximation. Subtraction of this asymptote gives values that lie on a line showing a time constant of 16 nsec, in good agreement with the lifetime of the 3lD state itself. Without the use of the sink for resonance radiation mentioned in Section II,A, this shortest lifetime could not be resolved.

The intercepts of the lines on the ordinate enable the contribution of cascade from the 4lF state to be found. Because the 4F states are themselves populated very readily by excitation transfer and cascade from other states (notably n'P; see Section II,B) Anderson er al. (1969) made similar observations over a range of pressures and deduced values of the excitation cross sections of the 4lF states (and 43F, from similar observations at 587.6 nm), which are shown in Fig. 12. This figure also shows some results of van Raan and van Eck (1974) obtained from their measured 43D cross sections. They adopted a particular model to describe the mixing of the 4lF and 43F states and their results might be expected to be between those for the two separate states.

In contrast to the results shown in Fig. 4 there is no clear E-3 dependence for the 43F or E-1 dependence for the 4lF. In view of the small size of the cross sections, the contributions from higher states and the indirect measurement methods, it is almost surprising that the discrepancies are not greater.

The energy levels of hydrogenic ions are practically degenerate and the excitation of the various 1 states of a given n can only be separated by methods that exploit their different lifetimes. The excitation function of the 468.6 nm line of He+ (n = 4-3) has been extensively studied (for a review, see Heddle and Keesing, 1968), but the excitation of the individual I states had not been separately measured. Sutton and Kay (1974) measured the decay curve following excitation by a 1.2 nsec pulse with a turn- off time <0.1 nsec at a repetition rate of 1 MHz. The lifetime of the 42P state is only 0.78 nsec and so its contribution to the decay is important only for times comparable with the exciting pulse and the proportion of 42P excitation can consequently be determined only with low precision. The results of Sutton and Kay (1974) for 200 eV are given in Table I11 together with theoretical predictions of Lee and Lin (1965). There is no agreement.

Mahan er ul. (1976) have studied the excitation of the Balmer cy line of hydrogen at energies between threshold and 500 eV. They determined the

400 D. W . 0. Heddle

0

Electron Energy/eV

FIG. 12. Apparent excitation cross sections of the 4'F (0) and 43F (0) states of helium (Anderson el al . , 1969) and of the mixed 4F state (+) (van Raan and van Eck, 1974).

relative contributions of the 32S, 32P, and 32D states by modulating the electron beam at frequencies between 0.1 and 30 MHz and measuring the modulation depth of the light signal in phase and in quadrature with the modulation. The proportion of the three components and of cascade depend on the energy, but have been determined within some 20%. Abso- lute calibration was made in terms of a Born approximation calculation at 500 eV. In Section VI1,A we shall return to this point, but just note here that the so-calibrated values are likely to be too large.

TABLE I l l

RELATIVE EXCITATION CROSS SECTIONS FOR THE 41 STATES OF He+

Experiment Theory

42s I 1

0.36 + 10

l 6 -14 42P

42 D +0.4 1.1

-0.6 0.084

42F 0.74 5 0.2 0.001

EXCITATION OF ATOMS BY ELECTRON IMPACT 40 1

VI. The Determination of Cross Sections in Absolute Terms

A. ABSOLUTE MEASUREMENTS

I . The Relation between Line and Level Cross Sections

Observation of light corresponding to a given transition will yield directly only the cross section for excitation of that spectrum line. The upper state may be populated by other processes (see Section 11) of which only cascade cannot, at least in principle, be avoided and it may be depopulated via other paths. The conversion of the measured line cross section to one for the excitation of a particular state or level requires knowledge of the cascade contribution and of the branching ratio. The latter are frequently only known from calculation (though often of quite good precision), but the former may be deduced either from theory or experiment. The line cross section is often more valuable if one is seeking to model (for example) conditions in a stellar atmosphere, and we think that it is really the duty of the theoretician to calculate line cross sections and not just to expect the experimenter to make corrections that are only necessary for comparison with theory.

2 . “Static Gas” Measurements

In order to deduce the excitation cross section for a spectrum line it is necessary to determine the atomic number density, the electron beam current, the length of the interaction region, and the total photon flux. None of these are trivial matters if high accuracy is to be obtained. The measurement of electron current is possible with excellent precision and the problem is to ensure that the electrons forming this current have all passed through the observed region and have done so only once and with the correct energy. The length of the interaction region is usually determined by the width of the monochromator slit and the magnification of a lens system, but will need an energy-dependent correction if a magnetic field is used to confine the electron beam (Taylor et al . , 1974). One of the major problems in the radiometric calibration is the difference in radiance of any standard lamp and the light from the collision chamber. Showalter and Kay (1975) have used an attenuator box (Stair et a l . , 1965) with a standard lamp and claim that the possible systematic error in their light intensity calibration is 8.3%. Taylor and Dunn (1973) used an intermediate standard consisting of a 150 W quartz-iodine lamp, a monochromator, a light-pipe, and a small integrating sphere, which they calibrated against a standard lamp that had been itself calibrated against a copper melting-

402 D. W. 0. Heddle

point (1357.8 K) black body. They claim an accuracy of some 5% at the 98% confidence level.

The measurements of gas density and pressure are not quite the same thing because the temperature in the collision chamber may be significantly higher than that at the pressure gauge because of the proximity of the hot cathode. The capacitance manometer calibrated directly against a point-contact oil manometer (Ruthberg, 1972) is probably the simplest standard of good accuracy, but Van Zyl et al. (1976) have demonstrated that it is possible to produce gas pressures in the 10-3-10-4 Torr range, with an uncertainty of 1% at approximately three standard deviations, by using a dynamic expansion method in which the pressure in a reservoir at about 1 Torr is measured with an oil manometer and the pressure in the collision chamber deduced from the conductances of a porous plug at the input and two apertures (to pass the electron beam) at the output. Van Zyl et al. (1976) also describe the use of light from the collision chamber as a means of extending the calibration to lower pressures. Unfortunately they do not specify the transition or the energy. Jobe and St. John (1972) used a McLeod gauge to measure pressure in the range from 4 to 70 mTorr and, finding that the apparent excitation cross section of the 5IS-2lP line of helium (444 nm) at 32 eV did not change over this range, used this to extend their pressure scale down to 15 pTorr.

3 . Crossed-Beam Measurements

It is exceedingly difficult to make a truly absolute measurement if the target is a beam of neutral atoms. The problems of density determination are immense and such measurements are probably best calibrated against some theoretical model for one transition. There are, of course, exceptions: Williams (1976) determined, in essence, the atomic concentration by a phase-shift analysis of a resonance in elastic scattering. In the case of electron excitation of beams of positive ions the ion concentration follows, in the same way as the electron concentration, from a measurement of the ion current. For electron and ion beams that cross at right angles, the excitation cross section can be expressed in terms of observable quantities by

where S(90) is the signal measured at right angles to the electron beam, Ii and I , the ion and electron currents, vl and v, the ion and electron velocities, V a volume defined by the overlap of the ion and electron beams, and A(P) a factor that takes account of the anisotropy of the radiation and can be expressed as


A(P) = ( I - P / 3 )

provided that observation is limited to a fairly small solid angle, or as -

A(P) = (1 - P / 3 ) / ( 1 - P COS'O) -

where cos20 is the average value over the collection solid angle. S(90) must be in absolute terms and, for the highest accuracy, any variation in the collection efficiency of the radiometric system over the volume V must be allowed for. Taylor and Dunn (1973) discuss this in greater detail. If polarization measurements are to be made it must be remembered that the direction of the impact is not that of the electron beam, but the resultant of u, and - u,. In the case of excitation of the resonance line of C3+ by Taylor et al. (1977) the ion beam has a kinetic energy of 29 keV and at the excitation threshold of 8 eV the velocities of the electrons and ions are almost equal, so that the resultant velocity makes an angle of 45" with either beam.

B. APPLICATION OF THE BETHE APPROXIMATION

It has been known for many years (Bethe, 1930; Inokuti, 1971; Inokuti et af., 1978) that for excitation of an optically allowed transition by fast electrons the cross section can be represented asymptotically by

Q - - - 1 n ( F ) 4ra; f 4cjE ' - EIR Ej/R

Where E is the kinetic energy of the electron, Ej the excitation energy, R the Rydberg constant, uo the radius of the first Bohr orbit, andfj and ci parameters that describe the magnitude of the cross section and the angular distribution of the scattered electrons. f, is known as the optical oscillator strength and is dimensionless. The occurrence of energies only in the form EIR means that the units of Q j are those chosen for a t . Fano (1954) pointed out that Eq. (2) could be cast in the form

Q@ = A In(4c@E/R) (3)

and suggested that a plot of QiE against In E for experimental data should tend at high energies to a straight line with a slope proportional to h/Ej and an intercept on the abscissa at E = Eo = R/4c,. If we substitute numerical values and change to common logarithms we can write Eq. (3) as

(4) where the energies are now measured in electron volts and we have omitted the suffixj on Q to simplify some later notation. The two parame-

QE = 1.50 x 10-17(fj/Ej)(log E - log E,) m2eV

404 D. W. 0. Heddle

ters of Eq. (4) can be calculated, but the accuracy of the calculated oscillator strength is generally greater than that of the cj, or Eo, parameter.

A major application of this equation is in the absolute calibration of an optical excitation experiment or, if radiometry of very high precision can be done, as a test of theory. There are certain practical problems in the application of this method. The light emitted in the decay is polarized to a degree that depends on the electron energy and it is not always possible to measure the polarization. If it is to be measured and even i f it is nor, any polarization sensitivity in the detecting system must be eliminated. Not all of the excitation observed is the direct result of electron excitation to the statej but may be a consequence of excitation to higher states followed by radiative decay to the statej. There is also the possibility that some excitation may be the result of processes that could be avoided in principle, such as collisions of slow electrons that have been produced in ionizing collisions or at the electrode surfaces. These last processes become of greater importance as the electron energy is increased because the direct excitation cross sections become small. There is consequently a risk of a progressive overestimation of the excitation cross section, and hence QE, at high energies, which can lead to the matching of experimental data to a line of too large a slope. Values of the intercept E,, will also tend to be too high.

Figure 13 is a vector diagram showing the directions of the incident electron ( k o ) , the scattered electron (k , ) , the momentum transfer (K), and the direction of observation, which is in a plane perpendicular to ko and at an angle 4 to the scattering plane. Averaging over this angle gives the polarization P as

Po(3 COS2x - 1) P =

2 - PO(1 - cos2x) where Po is the polarization that would be observed perpendicular to the momentum transfer vector K, and the average value of cos2x is that weighted by the generalized oscillator strength. For excitation of neutral atoms, Po is equal to the threshold polarization and can be calculated as discussed in Section III,B. In the case of excitation of a positive ion the threshold polarization cannot be calculated so easily, but Po is still a valuable parameter because it is simply related to the polarization approached at infinite impact energy, P , = -Po / (2 - Po), where the influence of the Coulomb potential can be ignored.

McFarlane (1974) has developed a Bethe theory for the polarization of impact radiation and his results appear to offer a means of determining the parameter cj from measurements at moderate energies and under conditions such that the effect of secondary excitation processes is minimal.


/- 3 \ ' 'J

FIG. 13. Vector diagram showing the directions of the incident electron k O , the scattered electron k , , the momentum transfer K, and the direction of observation.

He has shown that the weighted mean value of cos2x can be expressed as

cos2x = [In(4cjE/R)]-'

and that the expression for the polarization becomes

P0[3 - In(4cjE/R)] ( 2 - Po) In(4cjE/R) + Po

P =

Equation ( 5 ) has two consequences of great practical importance. The most direct result is that, for any value of P o , the polarization P is zero if 1n(4cjE/R) = 3 , that is to say, for an electron energy of e3Eo.

As the effect of secondary excitation processes is almost without exception to reduce the degree of polarization they can have no effect where P = 0. The exception is cascade population from certain states, but as the polarization of different spectrum lines of an element is observed to reach small values at similar energies this is probably not a serious effect either.

Equation (5 ) can also be used to modify Eq. (2) so that it applies to the

406 D . W. 0. Heddle

apparent cross section deduced from observations made at right angles to the electron beam on the assumption that the light is emitted isotropically . This is a very common situation experimentally: ideally the polarization should be measured, but it is frequently the case that the signal is too weak to permit this. Following Heddle and Keesing (1968) we define cross sections Q,, and QL leading to the production of light polarized with electric vector, respectively, parallel and perpendicular to the electron beam and a cross section QgO = Q,, + QI as that leading to light emitted at right angles to the electron beam. This light is polarized and we define the degree of polarization by

QII - QI QII + QI P =

We can then show that

Q = I - P / 3

For excitation to a 'PI state Q,, and Ql are the cross sections for exciting the levels with MJ = 0 and f 1, respectively. Substitution of Eq. ( 2 ) into Eqs. (6) gives

Equations ( 5 ) and (7) are represented in Fig. 14 for the case of Po = 1, which is appropriate to a 'So to 'PI excitation. The ratio QI/QII is also shown. Note that the lines for QE and Qg& intersect at ln(E/Eo) = 3, the value for which P = 0: this is true for any value of P o . For Po # 1 the slopes of the Qg&, Q,@, and QLE lines change as can be shown from Eqs. (7). They are, of course, only high-energy asymptotes. The graph of QL/Q,, is nowhere a straight line unless Po = 1.

The polarization changes in a nonlinear manner for Po # 1. The value at ln(E/Eo) = 1 is always P o , but the high-energy asymptote becomes


FIG. 14. The behavior of QE, Q&. Q,,E. Q I E . QI/QII, and P a s functions of E / E o for the case Po = I .

- P o / ( 2 - Po) = P , and is always smaller in magnitude than P o . The behavior of P deserves further comment. It is plainly wrong at ln(E/Eo) < 1 for it takes values > P o , which is a limiting value reached only at threshold and the threshold energy is rarely as much as 2Eo and can be less than Eo. It might therefore be expected that the true polarization would lie below the Bethe curve and would go through zero at an energy less than e3Eo. Table IV shows values of the energy E, at which the polarization is observed to be zero and of calculated values of E o . If the above supposition were well founded we would expect the ratio E,/Eo to be significantly less than e3, but this is plainly not the case.

The one atom for which an exact calculation off, and c, is possible is hydrogen. Figure 15 shows the polarization of the Lyman (Y line. The curve is based on a value of 0.408 for cZp (Inokuti, 1971) and Po = 0.42 (Percival and Seaton, 1958). The experimental results of Ott et al. (1970) are in excellent agreement from 50 to 110 eV (the divergence at lower energies is not unexpected), but show some structure at higher energies. Figure 15 also shows the experimental Qw, data of Long el al. (1968). Equation (7a) shows that the slope of the asymptotic line is (1 - P 0 / 2 ) / ( 1 - P0/3 ) times the slope of the QE line (in this case the factor is 0.92) and the intercept is given by In(E/E,) = - P 0 / ( 2 - Po) = P , = - 0.266. The value of 0.408

408 D . W. 0. Heddle

TABLE IV

Li Na K Rb c s Mg Ca Sr Ba Ca+ Bat TI

0.69a 1.080 0.87" 0.82" 0.79" 3.10b 1 .90b I .87b 1.556 1.88' I .67" 5.5"

20.6d 25.3' 17.4' 17.9' 18.6' 59.48 37.0" 35.7' 3 1.3' 39.2k 40' 85"

29.9 23.4 20.8 21.8 23.5 19.2 19.5 19.1 20.2 20.8 24 15.5

a Vainshtein ef a/. (1965). Kim and Bagus (1973). Burgess and Sheorey (1974). Leep and Gallagher (1974). Enemark and Gallagher (1972).

IChen and Gallagher (1978). Leep and Gallagher (1976). Ehlers and Gallagher (1973). Chen er 0 1 . (1976). ' Chen and Gallagher (1976).

Taylor and Dunn (1973). ' Crandall ef a/. (1974). " Chen and Gallagher (1977).

for czp implies Eo = 8.34 eV, so the intercept is at E = 6.39 eV. The line in Fig. 15 has this intercept and has been drawn through the point at highest energy. The slope of this line is probably less than that of the true asymptote because the data have clearly not reached an energy where they approach any line through E = 6.39 eV in a satisfactory manner. Nevertheless it is instructive to examine this line and use it to calibrate the measurements, which were made only on a relative scale. On this scale the slope of the line is 6.59, which we should write as 6.59F, where F is a normalizing factor. For fi, = 0.416, E,, = 10.2 eV, and Po = 0.42, Eqs. (4) and (7a) show that the true slope is

0.416 0.79 10.2 0.92 1.50 x 10-17 x - x - or 5.62 x m2eV

The calibration factor is then given by F = (5.62/6.59) x 10-19 = 8.53 x m2


10

8

6

Q E

4

2

0 10 100 1000

Elec t ron E n e r g y eV I FIG. 15. Excitation and polarization of the Lyman a line of hydrogen. The polarization

data of Ott et a / . (1970) are shown by 0 and the excitation date of Long el a / . (1968) by * .

Long et al. (1968) found a calibration factor of 8.68 x m2. The difference of some 2% is accounted for by the cascade contribution for which we have made no allowance. However, the true asymptotic line probably has a greater slope, which would lead to a smaller calibration factor. All in all it would appear that the Long et al. (1%8) calibration, which was made at a single energy, must represent an upper limit.

Ehlers and Gallagher (1973) have measured the excitation and polarization of the calcium resonance line at 423 nm. Their results are shown in Fig. 16. The line representing the Bethe approximation to the polarization has been drawn with Po = 1 and E , = 37 eV. The line representing-QE is given by QE = (-403 + 1023 log E).rrai and the intercept is at 2.48 eV. The experimental data include cascade and so these energies would not be expected to be in the ratio of e3 : 1. Ehlers and Gallagher (1973) discuss the effect of cascade on the QE line and show that if the cascading states are excited solely by electron impact and their cross sections therefore vary as B / E at high energy, the effect is to add a constant amount B to QE at all

410 D . W . 0. Heddle

1.0

0.5

c 0 .- c

.- z o

a" L

O -

- 0.5

2 I I I I I I I

'.. . \

10 100 1000 E l e c i r o n E n e r g y / e V

.. ' I 1 I I

10 100 1000 E l e c i r o n E n e r g y / e V

FIG. 16. Excitation and polarization of the 423 nm resonance line of calcium.

energies, but not to change the slope. If there is a contribution from higher P states via S or D states, then this has the type of energy dependence given by Eq. (2) and the effect is primarily to increase the slope though the intercept may also be affected because the cj will not all be the same. For this line they estimate that such cascade increases the slope by only 4%. At 1400 eV they estimate from measurements of Karstensen (1%8) that B / E cascade contributes some 7% or that B = 1.7 x lo-'* m2 eV. Sub- traction of this amount from the plotted line changes the intercept to 3.7 eV. Whereas the Ep value of 37 eV is in excellent agreement with the calculated E,, of Burgess and Sheory (1974) this latter value suggests that the data do not extend to energies high enough to reach the asymptotic region.

Crandall el af. have measured the excitation and polarization of several lines of Ba+. Their results for the 62P312-62S112 line at 455 nm are shown in Fig. 17 together with the calculated values for Q E of Burgess and Sheory (1974) and a line to which these calculated values are asymptotic. The

EXCITATION OF ATOMS B Y ELECTRON IMPACT

FIG. 17. Excitation and polarization of the 455 nrn line of Ba+. The experimental data of Crandall et al. (1974) are shown by Owith the polarization below 9eV as a full line. Values of the excitation cross section calculated by Burgess and Sheorey (1974) are shown by 0.

experimental data include cascade and are absolute. The polarization data are well described by Eq. (3, with E, = 40 eV and Po = 0.57 for energies > 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-


Electron EnergY/eV

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


I I I , 1 I I I

10 100 1000 E l e c t r o n 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 QsO for 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+.


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 QE. 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

E l e c t r o n E n e r g y / e V

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.


93% of the excitation to the 32P state 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 increase in QE by some 0.2 x m2 eV.

VII. Miscellaneous Measurements

A. THE BALMER LINES OF 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 col-

lisions 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 (Vainsh- tein, 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

N

E P N

2 \

Z 0 I- 0 W m m cn 0 U 0

- 50

4c

30

2 0

10

0

H P d H

8 00

6 00

4 00

2 0 0

5 c

4 c

3 c

2 c

I0

1 1 I 1 I 0 100 200 300 400 500

H €

E L E C T R O N E N E R G Y ( e V )

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.).


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 Eq. (2)] and a term 1 - (Er/E)1’2, which vanishes at the excitation threshold.

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. PHOTON LABELING

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

418 D . W . 0. Heddle

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 - 1OOeV 0

N

* 10

4

0 2WeV + +

0 * +

0

* + o ' t o

. o + . +

0; 10 20

Scattering Angle/'

FIG. 22. Differential cross sections for the excitation of the SIS state of helium.


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 . Phys . B 10,

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. Rev . 181, 198. Bell, K. L., Kennedy, D. J.. and Kingston. A. E . (1969). J . Phys . B 2 , 26. Bethe, H . (1930). Ann. Phys. (Leipzig) [5] 5, 325. Bogdanova, 1. P. , and Marusin, V . D. (1971). Opt. Spectrosc. ( U S S R ) 31, 184. Borst, W. L. (1969). Phys . R e v . 181, 257. Brunt, J . N. H. , King, G. C., and Read, F. H. (1977a). J . Elecfron. Spectrosc. Relal.

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ADVANCES I N ATOMIC A N D MOLECULAR PHYSICS. VOL. I5 l l 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 . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

A. (e, 2e) Experiments.. ........................................... 425

437

Appendix: Coherent Excitation of Degenerate States with Different Angular Momenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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-OO38 15.3

424 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

426 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; Pe- terkop, 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 character- izes 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


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 al . , 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 (Pe- terkop, 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,”

COHERENCE A N D CORRELATION

,,o

I N ATOMIC COLLISIONS 429

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 f rom (e, 2e) Experiments

Although this review is mainly concerned with applications in atomic collisions it would be inappropriate to exclude the most impressive infor-

430 H . Kleinpoppen

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, = -0b for the scattering angles for coplanar electron- electron 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 Mc- Carthy (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 electrons (k , , kb), and the residual ion in an eigenstate If). The amplitude Mf for the (e, 2e) reaction is then written as

Mf(k0 7 ka 7 kb) = (ka kbl (fl Tlg) 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 state vector If), which contains neither electron, commutes with T, the (e, 2e) amplitude becomes

COHERENCE A N D CORRELATION IN ATOMIC COLLISIONS 43 I

Mf = ( k a , kblTl(flg)ko) (3 1 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 target Hartree-Fock functions as basis sets for 18) and I f) , we obtain

[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).

432 H . Kleinpoppen

0 1 9 ( a u )

i 2

FIG. 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 EFFECTS IN ELECTRON IMPACT IONIZATION

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 al. , 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' - Qint ( 8)

By applying partially polarized electrons (polarization P,) and partially polarized atoms (polarization PA, 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,, PA can 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

DYE LASER 0

wtc T W

KLU C W P L R

KCELERATOR Iw k V

K)N DETECTOR

0;:; / H ) T WIRE DETECTOR

FIG. 6. Scheme of the experiment for the measuremen >f relative cross sections of interference in th

f - 0 rn

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 AND 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 was typically about A with a measured polarization of 20%. The relative 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/Qo for sodium and potassium

0 3

0 1

0

1 2 3 4 5 6 7 8 9 1 0 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/Q from 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 NE RGY (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 angle- dependent 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).


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 im- pinging 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 IMPACT EXCITATION

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

438 H . Kleinpoppen

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 If') 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 state vector before the collision after the collision

polelllldl Ul 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, momentum po spin So, and and spin

component MF,, component m,

electron

We restrict ourselves to ' S + 'P excitation with So = 0, S , = 0. We then have


Spin conservation requires M,, + m, = M s , + m, or m, = m,. Therefore Imo) cancels out and we have

or

P o d = l W 1 ~ l ~ ~ l P 1 ~

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:

I = pll + p22 = 1 (normalized to unity)

(10) P I = p,1 - p2z = 1(0°) - I(90”) P2 = plz - pzl = I(45”) - I( 135”) P3 = i(pzl - p12) = I(RHC) - I(LMC),

pl l , . . . 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

( 1 1 ) [PI = (IPi12 + IP212 + IP312)1’2

440 H . Kleinpoppen

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 .--, 2lSO of helium could be verified by measuring the Stokes parameters of the helium A = 5016 8, line radiation (Standage and Klein- poppen, 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 AND CORRELATION I N ATOMIC COLLISIONS 441

SCATTERING ANGLE IDEG) SCATTERING ANGLE IDEG) 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 difference between the two amplitudes aML=o and aM,+, , and with the relation D = uo + 2u1 between 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 state vector with the amplitude representation CMLaMJLML); e.g., the 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

( C O S ~ 8, + 1) dW 3 ( 1 - A) - [A sin2 8, + - day 8lr 2 - _ -

1 - A 2

+ [A(A - sin e, cos 2(4, - &)] --

cos 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 + l1S0 process 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

442 H . Kleinpoppen

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 a l . , 1974; Kleinpoppen, 1975) that the expectation values for the orbital angular momenta of the excited 'P, state,

are calculable from A and x with respect to i = x, y, z directions, e.g.,

(L , ) = ( 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 I 1 1 1 I 1 1 1 .

0 10 20 30 4 0 0 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 1 , , , , I ,

i =" 1.5

I ' ' ' ' 1

0 10 20 30 10 50 e( DEG)

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 1 L O 160 1

ELECTRON SCATTERING A N G L E I

FIG. 13. A = u ~ / u data 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).

444 H . Kleinpoppen

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); Scott and McDowell (1976).

(L:) = A, (L:) = 1 , (Lg) = 1 - A (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

AZC+"' = + ( L i - L:) = + ( A - I ) = (PI - 1)/4 Of?:-.' = +(I!,,) = - P3/2

A C O l = (18) I+ +(L,L, + L,L,) = [X(1 - A)]"' cos x = -Pz/2

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


i

oo 40 80 120 I60 200

-

,,

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 l lS, + 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

0 .L

0 . 3

P P

5 n

P P

0-1 ' I 1 0 20 L O 60 80 100 120 140 ELECTRON SCATTERING ANGLE 101 (degrees)

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).

446 H . Kleinpoppen

IF

" w

N

\ - 0

a

0, I I \ \ * \

I \

-0.41

i

-as+ FIG. 17. Alignment parameters A::: ,2 (in units of fi2/2) calculated from experimental A

2*P1electron impact 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-


H ( 1 s - 2 p )

I I 1 I I I 0 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, inco- herently populated magnetic substates are assumed in the theory of impact radiation of atoms (Percival and Seaton, 1958; Baranger and Ger- juoy, 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 + JlddI2 representing the charge density of co-


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 the Mg+ ions with the rare gas atoms the Mg I1 (3s * 3p) excitation/deexcitation occurs. The coherency (see Section III,A,l,a) of the photons from this transition is modified by the depolarizing influence of the fine- structure 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)112 5 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-

450 H. Kleinpoppen

He Ne A r

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 111.1 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).

452 H . Kleinpoppen

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') (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 depo- larizes 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 Na23 by 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 polarization (PI and the degree of coherence JpJ [Eqs. ( 1 1 ) and (12)] are restricted within the above limits, which depend on h(*) and G"). These considerations were first used to test the coherence properties of the He(3lP,) excitation by electrons. In that case (G") = G@)) one finds IPI =

IpI = 1 (see Fig. lo), independent of the scattering angle of the electrons. Since the observed values for (p( and (PI of 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 P1 and P2 data; most accurate values for x were obtained from A and P3 data: less accurate values for x re- sulted from P, and P2 and by using P3 only 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 quasi- molecular 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:

(21)

p 3 - - -6h(l)G(l) [A( 1 - sin x/(4 - h(2)G(2))

x = l 4 ' / { l ~ Z l 2 + lanI2}


( b ) no

ECH: 93 sV

' 1 15 10 aa 06 b I A I

1 ' 1 ' I 10 08 06 01 0 2 b& x . ' ' '

h 1 1 . 1 '

I 08- .

06 - 06

02 - 5 1o TlkeV desl

/ , I I I I 8

10 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.

1 x = - 1: dt(V, - v,) ( 2 2 )

with Vn and V , as the potential energy in the two states A211 and B2C. Fig- ure 22 shows results for x 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

454 H . Kleinpoppen

- X

(b)

2M

100

0

- x 08

06

01

0 2

10 [deql io

-I K- Ne

1 E,, 13LOeV

15 10 08 06 b l h I 1 ' 1

10 20 T lkeV degl

FIG. 22. x 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 ml = 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 ~ ~ ) ~ A ~ 8 , and ( l s ~ , ) ~ 4 d 4 2 , H e t electronic energy curves, these authors could derive expressions for the amplitudes a. and al with fixed 90" phase relation. This 90" phase difference results from the inability of


He* + He 3.05 kev 3889 a

z

I I

1.5 2 0 ( 0 ) SCATTERING ANGLE (deg)

t

0- 1.0 15 2.0

SCATTERING ANGLE (deq)

H i t He 3.05 k p 3889 A

I I I I I

1 15 20 SCATTERING ANGLE (dog)

FIG. 23. (a) Amplitude probabilities (relative units Pb and Pi in numbers of particle- photon coincidences per lo0 scattered 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)’D and (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-

45 6 H . Kleinpoppen

2 0

I I 1 3 30 35 40

I I r 10 0 @=O

(b 1

i0 35,OL 40

electron energy I eV )

FIG. 24. Energy spectra of the electrons ejected from the unresolved autoionizing states He(2pZ)'D and He(2s2p)lP in He+/He collisions at 2000 eV, 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


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:

+2 +1

qi = d m $D.m + P m + P m (23) m=-2 m=-1

The final state is represented by the product of a wavefunction for

J l r = +,oneUFr (24) The transition amplitude for autoionization is then given by the matrix

element M = (qr IV, with V , as transition operator for autoionization that is a monopol operator: taking also the relation for the amplitudes urn = (- l ) m ~ ~ - m gives for M = M I , + MP

He+( 1s) and a plane wave for the ejected electron:

with ldn,l, p m and Ipml, x m as moduli and phases of the complex excitation amplitudes. IMDI2 and IMPl2 represent 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.

45 8 H . Kleinpoppen

- *

00 30' 60" 90' 120" 150' 180" Op 330" 500" 270' 240' 210"

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

MODULI A N D PHASES OF AMPLITUDES FOR THE AUTOIONIZING STATES 'P AND ID OF He"

State, m = O m = l m = 2

'P lpml 5.4 7.6 x m h 0.6 1

ldml 23 25 39 'D

pmlm 0 1.7 0.9

With reference to the incoming He+ ion beam (energy 2 keV) as axis of quantization (Kessel er a/. 1978); Mor- genstern and Niehaus, 1979). (Scattering angle of He+ ions e = e.)


\ 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 (198, 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 (Her- mann 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 AND 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). Theo- retical studies of simultaneous electron-photon excitation of atoms opened up 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; Jit- schin et al. (1979) Lutz et al., 1979). Coincidence studies between ions and Auger electrons or characteristic X-ray photons in ion atom collisions re- sulted 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 momentum (Blum and Kleinpoppen, 1977; Blumet d., 1978). We consider as 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


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. (Al ) :

with ( L M , L‘ - M ‘ J K - Q) as a standard Clebsch-Gordan coefficient (Blum and Kleinpoppen, 1977). From the properties of the Clebsch- Gordan coefficients follow the limitations K s L + L ’ , - K s Q s K. Note the relations

(A4a) ( T(L’L) i Q ) = ( T( L ’ ) K Q ) * 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 (A2- , Al+, Ao), which contribute to the transition from the coherently excited hydrogenic n = 3 levels to the n = 2 level in the field- free case:

(T(2, 2),+,) = (8/7Y2 Re(f2d2*2) - (3/7)1/2(1fz112)

(~(22)2+~2) = (8/7)1’2(lj;2p) - (2/7)1/2[lh112) - (2/7)1/2(lfi012)

(T(22)A) = - (12/7)1/2 Re(.f,,fg2) - (2/7)’” Re(.h0f2*1)

(T(22)L) = i(4/5)”’ Im(f,l.f$z) + i(5/6)1’2 Im(.fiofA)

(T(22),+,) = (l/5)112d3d),

(T(20),:) = ( h I f l T O ) 7

(T(20);z) = ( j i z f & ) ( T(20GO) = (.f,o.flTo)

(T(oo)&) = U(35’),

(T(ll)$z) = (r(3p)A2+, ( T ( I 1 ) l o ) = (2/3)-”2~(3P)Ao

(T(l1);i) = a(3p)0,-

(T(11);1) = -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. Impor- tant 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.

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1967 p. 648.

I 1 ADVANCES IN ATOMIC A N D MOI.ECULAR PHYSICS. VOL. 15

1 COLLISIONS

1 1 THEORY OF LOW ENERGY

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

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 low- energy 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

472 P . G. Burke

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 Sec- tion 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 OF 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 RADIAL EQUATIONS

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 .I2 and 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 an- tisymmetrization 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 rA and 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 k j are given by

kf = 2(E - El) (6 )

where El is the energy of the target in thejth rotational state, Vj:Aat is 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-


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 formu- lated 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.

476 P. G. Burke

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 a2 are related to the polarizabilities along and perpendicular to the molecular axis by

(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

a 0 = +(all + 2aJL a 2 = #(a,, - all


where C ( r ) is a cut-off factor

C ( r ) = 1 - exp[- ( ~ I Y ~ ) ~ ] 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 CROSS SECTION

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 : < O r - t m

where the second pair of indicesj’l’ distinguishes the linearly independent solutions. The total cross section for a transition from statej t o j ’ is

where the T-matrix in this equation is obtained from the K-matrix defined by Eq. (13) by the matrix equation

T J = 2iKJ/(I - i K J ) (15)

In order to obtain the angular distribution cross section for a rotational transition it is convenient to introduce, following Fano and Dill (1972), the quantum number I , , which corresponds to the angular momentum transferred from the orbital angular momentum of the scattered electron to the molecular rotation. The resultant cross section has been determined in terms of the transformed T-matrix

by Chandra (1975). He shows that

478 P . G . Burke

where

[(21 + 1)(21' + 1)(21" + 1)(21" ' + l)]",

0 0 0 0 0 0 "> (1 I" ')(" l f ' '

and

are 3 j and 6j coefficients, respectively (Brink and Satchler, 1971). Finally, the momentum transfer cross section is given by

f A l >

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 I I I I I

16' I eV

10

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 ~ r l i e

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(cos B)/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 Nz 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. In 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). How- ever, 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 ELECTRON-MOLECULE COLLISIONS 48 I

A. DERIVATION OF T H E R A D I A L EQUATIONS

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

(20) q m = d 2 W l . . . . , N;R) , . - 'Gl"(r )Yl"( i ) 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 i t 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 - 'YT( i ) then yields the coupled equations

where the channels are now all degenerate with the wave number defined

k2 = 2 ( E - E,) (23)

The direct potential VF, can be expressed in terms of the V h ( r ) defined

by

where Eo is the target energy.

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 i t 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

(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

Tm = 2iKm/(I - iKm)

B. RELATIONSHIP TO T H E LABORATORY FRAME REPRESENTATION

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


= (- l ) l + l - J = (- I ) l ’ + j ‘ - J

Substituting Eq. (28) into Eq. (16) then yields

Tflixit

(30)

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 de- formed 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

.fj*,j(a; R ) = 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. SOME RECENT CALCULATIONS

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 en- forcing orthogonality between the scattering orbitals and all molecular or-

484 P . G. Burke

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-HF scattering and by Chandra (1977) for e-CO scattering.

I I I a

I I I 1 I 2 3 4 5 6 7 8 ,

ENERGY [ e V ]

FIG. 4. Comparison of theory and experiment for the total e-N, cross section. 0, Mor- rison and Collins (1978); BB, Buckley and Burke (1977); BC, Burke and Chandra, (1972); G. measurements of Golden (1966). (From Morrison and Collins, 1978.)


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 Buck- ingham 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

486 P . G. Biirlie

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 I J 81 represented 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 Au 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 -

corre ia i ians a r e aisregaroeui

L 1 I I

‘0 rl r2 r

dis tance 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 photo- electrons 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 single- center expansion near the nuclear singularities. For example, in the work of Morrison (’1 ul. (1977) on e-CO, scattering, terms up 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. L2 Methods

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. INTRODUCTORY BACKGROUND

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 Mor- rison 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


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 0.05

h

0.04 - 3

0.03

.- + 0.02 0 C 3 6- 0.01 0, > B o - .!? -0.01 U rg CtI

1.0 2.0 3.0 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

3 O W u b 30

functions giving

d7T -

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).

c 2 0 -

I n' ! O -

c r

B. T-MATRIX METHOD

0 0

20 -

-

-

I 1 I 1 i 1 I

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 approxi-

Momentum

FIG. 7. Comparison of L2 and close coupling p r and d r phase shifts for e-N, scattering. 0, La results of McCurdy et nl. (1976); -, close coupling results of Burke and 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 = 2V, 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 plane- wave 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-MATRIX METHOD

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


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 el- lipsoid 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

+ 4j(l, . . . , N + l ;R)c i j (45 ) I

The molecular orbitals contained in the target channel wavefunctions (Dj and the “bound” functions dj are 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 ci jk and cij are determined by diagonalizing the operator H + L b ,

( q j , ( H + Lb)qj) = E$ij (46)

494 P . G. Burke

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 (48 1

VIE = ( H - E + Lb)-lLbqE

which has a formal solution

(49) Finally, we expand the inverse operator in this equation in terms of the eigenfunctions defined by Eq. (46) and evaluate qE on 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 COLLISIONS 495

' 0 2 4 6 8 10 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 Herzen- berg (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

496 P . G. Burke

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. An- other, 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 Her- zenberg (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

(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 Sul- livan (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 specific j + 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 xU(R) and xUt (R) in Eq. (51) by the solution of the equation

h f u t , j u ( f l ) = 1 xut(R)fjr,j(n; R)xu(R) d R

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-


tional quantum number to the corresponding vibrational cross section with A j = 1 -+ 3 . The theoretical calculations were carried out using x , ~ ( R ) and in the insert figure by replacing xUj(R) by 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.)

498 P . G . Birrkr

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 Her- zenberg (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

(53 1

where the vibrational levels are closely coupled but the adiabatic-nuclei

qrn = &i W l , . . . , N ; R)r-'Hl",(v)X,(R)Yl"(f) ill

E-(R) VS. (R-Ro)

E-(R) 'k,'(R) *EN,(R)-EN,(R~)

O 4.4 4.2 0.2 0.4 0.6 R-Ro(ao)

FIG. 1 1 . Width and position of the Ina resonance 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.)


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 Sec- tion 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 az as pointed out by Morrison and Hay (1977), Temkin ( I978), and Kendrick ( 1978).

C. OTHER APPROACHES

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

5 00 P. G . Burke

0.40 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 un- derway 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 )

( 5 5 )

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

W ( R ) = EAR) - f i U R )

9 = Ql( l , . . . , N + 1 , R ) + $I+(l, . . . , N + I ; R ) [ + ( R ) + C(1. . . . , N + l ; R ) [ - ( R ) + A+[ (56)

where QL is 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

(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)

in terms of the R-matrix states defined by Eq. (45). The (,,(R) then satisfy the equation

( T + W ( R ) - E ) [ , ( R ) = {,(R)xo(R)

If2

502 P . G. Burke

ELECTRON ENERGY (eV)

FIG. 13. Differential cross section at 90” for u = 1 + 2 e-N, excitation. (a) -, Wong (1978); 0. Dube and Herzenberg (1979); (b) -, Wong (1978); x, Chandra and Temkin (1976a). (From Dube and Herzenberg, 1979.)

( E t q ‘ , (T + EdR) -k 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

(60) S,, = (pIS(E - H, ;R)Ju)

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.


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. How- ever, 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

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more, Maryland.

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,

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.

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

302, 321,327

461,464

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,

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

503

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 Baliunas, S., 48, 71 Band, Y. B., 288,289, 447,466 Bandel, H. W., 478,505 Bang, J., 286,289, 336, 337,377 Banks, D., 320,326 Banks, P. M., 46,69 Baragiola, R. A , , 8.33 Baranger, E., 437, 488,465 Barat, M., 27, 34, 221, 224, 225, 228, 231,

Barbe, R., 17,35 Barbier, D., 44,69 Bardsley, J. N., 16,33, 38, 40,69, 209, 210,

Barnes, K. S., 320,326 Barnes, W. S., 18.35 Barnett. C. F., 294, 325,326 Barth, C. A., 46,60,62,69, 73, 75, 236, 260 Barsuhn, J., 48,69 Basbas, G., 338, 339,377 Bassel, R. H., 277,289 Bastide. R. P., 26,35 Bastrom, C. O., 61, 73 Basu, D., 130, 133, 153, 166 Batalli-Cosmovici, C., 194, 198 Bates, D. R., 26.33, 37, 38, 40, 42, 43, 44,

45,46,48,49,50,51,53,55,56,62,69, 70, 174,198, 215,218,228,231, 235,236, 246, 247,248,249,250,251, 252,253,254,255, 256,257,258,259,260, 274, 277,286,289, 320, 321,326, 353, 361,377

232

211,231, 233, 235, 242, 245,259

Baudon, J., 225,231 Bauer, E., 178,202, 216, 217, 218,233, 322,

Bauer, S. H., 168, 198 Bauer, W., 182,.183, 198 Baum, G., 460,468 Baun, W. L., 349,378 Bauschlicher, C. W., 171, 198 Baye, D., 272,289 Bayfield, J. E., 283,289, 302, 303, 307, 3 11,

Beaty, E. C., 21, 25,33, 36, 426,464 Bederson, B., 437,460,465, 469 Bednar, J. A., 68, 70 Begum, S., 269.289 Behnke, H. H., 461,465 Behrens, R., 181, 199, 202 Belkic', Dz., 266,282,283,284,285,288,289

327

326, 327

Bell, F., 343, 344,380 Bell, K. L.. 267,268,270,286,289, 418,419 Bely, O., 67, 70 Belyaev, V. A., 303,326 Bender, C. F., 10, 14,35, 53, 70, 169, 171,

198, 489,506 Bennett, R. A., 11, 18, 21, 22, 33, 35, 36,

238, 239, 240,260 Bennett, S. L., 10,33 Benoit, C., 221, 224, 225, 228,232 Benoit-Cattin, P., 457,465 Benson, S. W., 238, 239, 240,260, 261 Ben-Shaul, A., 180, 181, 198, 201 Bentz. E., 331,380 Berezhko, E. G., 334, 335,377, 462,465 Berkner, K. H., 26,33, 280, 289, 312, 316,

Berko, S., 137,164 Berkowitz, J., 194, 199 Berlande, J., 78, 81, 99, 252,260 Bernstein, R. B., 168, 169, 170, 171, 175,

180, 181, 184, 185, 188, 190, 191, 192, 193, 194, 195, I%, 198, 199,200, 201,202, 203

317,326

Berrington, K. A., 269,289 Berry, H. G., 448, 461,465, 469 Berry, M. J., 169, 180, 198 Bethe, H. A., 80,99, 403,419 Betz, H. D., 348, 367, 373, 375,377, 380 Bhatia, A. K., 103, 104, 119, 120, 131 Bickes, R. W., Jr., 181, 194, 201, 203 Bierbaum, V. M., 23.33 Billing, G. D., 172, 199 Billingsley, F., 39, 70 Biondi, M. A., 38, 39, 40,41, 42,69, 70, 72,

73. 235,241, 242, 243, 244, 245,259, 261 Birkinshaw, K., 214,219,225,228,229,231,

232 Birtwistle, D. T., 498, 500,504 Bisgaard, P., 461,465 Bittencourt, J . A,, 62, 75, 236, 237,262 Bjorkholm, P. J., 324,328 Black, G., 46,71 Black, J. H., 42, 46, 48, 49, 50, 55, 56, 70,

Blackwell, B. A., 169, 190, 198 Blake, A. J., 60,69 Blatt, S. L., 341,379 Blint, R. J., 52, 70 Bloch, C., 494,504 Bloemen, E. , 306,328

71, 72

AUTHOR INDEX 509

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,

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,

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,

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

373,380

482,504

74, 75

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,

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

151, 152, 153, 154, 164

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,

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,

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,

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

202

504

149, 164, 165

510 AUTHOR INDEX

Byron, F. W., Jr., 138, 151, 152, 153, 154,

Byron, S. , 246, 247, 252,254,260 164, 166, 269,289

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,

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

498,504

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,

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, 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

464

Chu, S.-I., 54, 70, 243,260

Collins, C. B., 248,260

AUTHOR INDEX 5 1 1

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., 7,33 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,

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,

Dance, W. E., 330,380 Dangerfield, G. R., 330, 331,378 Darewych, J . W., 131, 132, 138, 155, 157,

Das, J. N., 332, 333,378 Dassen, H. W., 446,468 Datz, S., 183,200, 394, 395, 396, 397,420

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,

Dehl, P., 461,465 Dehmer, J . L., 61, 71 Dehmer, P. M., 61, 71 de Jong, T., 50, 71 de 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

504

164, 165

46 I , 465

306.328, 441, 412,420

512 AUTHOR INDEX

Diana, L. M., 105, 115, 132, 142, 146, 147,

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,

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,

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

149,164, 165

203

327, 401,421

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 Dub6 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,

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

403, 408, 410, 419, 421

E

Eastman, D. E., 60, 72 Edelstein, S. A., 78, 79, 80, 81, 82, 83, 84,

Edwards, H. D., 208,232 Ehlers, V. J., 408, 409, 417,419 Ehrhardt, H., 47,75, 157, 158,165, 301,328,

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

91.99

332,378, 426, 427, 428. 429,466, 467

AUTHOR INDEX 5 13

Everhart, E., 221, 222,232, 233, 302,327 Evers, C. W. A.. 176, 177,198. 199

F

Fainberg, Yu. A., 317,326 Faisal, F. H. M., 461,466, 475,483,504,506 Faist, M. B., 175, 199 Fajen, F. E., 417,420 Falk, R. A., 26,34 Fano, U., 352,378, 403,419, 437, 438, 444,

447,448,460,462,463,466, 472,477,482. 485, 487, 489,504

Farago, P. S., 437, 443,465, 469 Farazdel, A., 123, 132 Faris, J . L., 391,420 Farrar, J . M., 168, 181, 199 Fastie, W. G., 59, 62, 70, 74, 236, 260 Fastrup, B., 339, 354,380 Fatceva, L. N . , 317.326 Faucher, P., 67, 70 Faulk, J . D., 332, 333,380 Fayeton, J. , 27,34, 225,231 Fedorenko, N. V. , 28.35, 209, 226,232 Fehsenfeld, F. C., 15, 22, 23, 25,34, 35, 43,

Feinberg, G., 68, 71 Feldman, P. D.. 46,53,59,62,70, 71, 74, 75.

Fels, M. F., 127, 128, 129, 130, 132, 153,

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 Finkenthal, M., 295,328 Fiquet-Fayard, F., 1 I , 35, 36 Firsov, 0. B., 51, 73, 206,232 Fischer, D. W., 349,378 Fischer, 0.. 419 Fisk, J. B.. 480, 504 Fitchard, E., 270, 271, 272, 273. 278, 286,

290, 291, 339,378, 448, 462, 464,465 Fite, W. L., 47, 71, 75, 179, 199, 230, 232,

296, 301, 307,326. 407.409.420 Flaks, I . P., 208, 209, 220, 226,232 Flannery, M. R., 85, 94, 99, 235, 2S6, 258.

260, 267, 268, 272,289. 290, 441, 443,466 Fliflet, A. W.. 491,504

48, 71

236,260

165, 488,505

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,

290, 291. 339,378 Ford, J . C., 283,290 Ford, R. L., 331,379 Fornari, L., 455,467 Fortner, R. J. , 286,290, 358,380 Fournier, P. R., 78, 81, 99 Fox, J . L., 60, 71 Francis, W. E.. 206, 207,208, 210, 212,233,

Franco. V., 272,290 Frank, W., 366, 368, 370, 371,379 Franklin, J. L., 10, 12.33, 35 Fraser, P. A., 101, 104, 107, 110, 1 1 1 , 119,

121, 123, 124, 127, 128, 129,132, 133, 153, 166

322,328

Frederick, J . E., 46, 71 Freed, K. F., 180, 199 Freedman, A., 181, 199, 202 Freund, R. S., 88.99 Friichtenicht, J. F., 185, 187, 203 Frilley, M., 364,377 Froese-Fischer, C., 432,466 Fryar, J. , 443,469 Fung, K . H., 180, 199 Fuss, I . , 428,470 Fussell, W. B., 401,421 Futrell, J. H., 23,33, 208,233 Fyfe, W. I. , 14,33

G

Gabler, H., 350,380 Gabriel, A. H., 67, 68, 71. 385,419 Gabrielse, G., 447,466 Gailitis, M., 104, 132, 428,470 Gallagher, A. C., 8, 34, 36, 399, 408, 409,

415, 417,419, 420 Gallaher, D. F., 273, 276,289, 290, 436,466 Gannon, R. H., 57, 75 Garcia, J . D., 286.290 Gardner, F. F., 57, 74 Gardner, J . L., 60,69, 74 Gardner, L. D., 311,327 Gardner, R. K., 279,289

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,

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,

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

465, 472, 480,504, 505

430,465

Gillen, K. T., 181,200 Gil1espie.E. S., 130, 131,132, 138, 148, 149,

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,

155, 157, 165

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,

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

462,469

AUTHOR INDEX 5 1 5

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,

Hardin, D. R., 181,203 Hardt, T. L., 340,378 Harel, C., 304,327 Harnois, M., 26.34 Harper, W. R., 255,261 Hams, 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

165, 479,505

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,

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,

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

222, 225, 228, 229,231. 232

75, 256,260

Heenan, P. H., 272,289 Hegerberg, R., 210, 211,232 Heidemann, H. G. M., 385, 391, 421, 461,

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.

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,

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,

466

478, 479, 479,505

202

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,

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,

Huber, B. A, , 20.34, 218, 225.232

466, 467, 470

26 I

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,

Hughes, V. W., 432, 436,464 Huguenin, G., 57, 72 Humberston, J. W., 103, 105, 106, 108, 109,

I 1 1 , 112, 113, 115, 116, 119, 120, 122,132, 133, 138, 146, 147, 148, 164, 165

400,419, 421

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,

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

412,420

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 517

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,

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,

Johnson, S. G., 183,200, 394,395,396,397,

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,

Jungen, C., 487,504, 505 Jungen, M., 5 , 6.36 Junken, B. R., 210, 21 1,231

26 I

232

420

429,466, 467

K

Kabachnik, N. M., 334, 335,377, 429, 462, 464, 465

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,

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

260, 267, 268, 270, 286,289, 418,419

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,

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,

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,

Kushnir, R., 207, 208,232 Kutner, M. L., 57, 75 Kuyatt, C. E., 288,290 Kwei, G. H., 173,200

396, 397,420

232, 233

200, 202, 203

L

LaBahn, R. W., 127, 130,133,147, 148, 149,

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,

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,

Launay, J. M., 54, 55, 71, 73 Laurent, C., 48, 75 Law, J., 277,290 Lawley, K. P., 168, 199, 200

166

479, 480, 485, 487, 488, 489,505

379, 380

AUTHOR INDEX 519

Lawson, J., 123, 130, 133, 147, 148, 149,

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,

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,

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,

166, 461,468

505, 506

192, 198, 199, 200. 201

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,

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

200, 201

M

Ma, M., 491,504 MacAdam, K. B., 394, 418,419. 441, 442,

McCann, K. J. , 267, 268,290, 441, 443,466 McCarroll, R. W., 52.73, 266,274,284,289,

McCarthy, I. E., 426, 430, 431, 432, 465,

McClure, G. W., 168,201, 296,302,310,327 McConkey, J. W., 390,419, 443, 444, 446.

443, 444,445, 446, 448,466

290, 313,327

467. 470

467, 468,469

520 AUTHOR INDEX

McCullough, R. W., 306, 307,308,314, 315,

McCurdy, C. W., 10,35, 489,490,491,505,

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

322, 323, 327,327

506

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,

McGregor, I., 461,464, 467 McGuire, J. H., 272,273,277,284,287,288,

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,

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,

McWalters, K. D., 43, 73 McWhirter, R. W. P., 246, 247, 256,260 Madison, D. H., 287, 290, 442, 443, 445,

Magee, J. L., 46,72

165, 288,291

291, 436,466

328, 339,379

149,164, 165

460,468

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,

Mazeau, J., 11, 15.34, 35 Meade, D. M., 2%, 327 Meadows, E. B., 42, 72 Meckbach, W., 288,291

201, 203

AUTHOR INDEX 5 2 1

Mecklenbrauck, W., 450,468 Megunov, A. I., 429,464 Mehlhorn, W., 334, 335, 362, 375, 376,378,

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,

Meyerhof, W. E., 347, 354, 356, 357, 369,

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,

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

379,380, 461,469

314, 323,327, 328

379

505

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,

Morgan, J. F., 341,379 Morgan, L. A., 446, 447,467, 468 Morgan, T. J., 283,291 Morgenstern, R., 455, 457, 458, 459, 467,

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,

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,

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

I33

468

499,505

467. 469

N

Nagamiya, S., 341,377 Nagel, D. J., 348, 349,379 Nagy, A. F., 43, 73 Nagy, S. W., 211,232

5 2 2 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,

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,

Nienhuis, G., 460,468 Nier, A. O., 39,40,41,42,43,44,53,70, 73,

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

180, 199, 202, 461,468, 502,505

466, 467, 468

74, 75

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,

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

321,327

AUTHOR INDEX 5 2 3

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.

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

321,327

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,

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,

202, 203

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,

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,

291, 307, 319, 320, 322,328

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,

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,

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

505, 506

202

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,

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,

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,

Sartwell, B. D., 337,379 Sasaki, F., 4, 5, 6,35

420, 421

469

380

AUTHOR INDEX 525

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,

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,

Schlecht, W., 436,467 Schmeltekopf, A. L., 15, 25.34. 43, 45, 71,

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,

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,

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

3 79

325,326, 328

75

461.469

506

Schrader, D. M., 103, 133 Schraeder, R. J.. 341,379 Schreiber, J. L., 168, 169, 171, 181, 190,

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,

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

199, 202

496,499,506

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,

Siegel, M. W., 11.33.36, 221,222,223,225,

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

232, 233

231

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,

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,

326

420, 437. 448, 449,467, 468

Smith, W. D., 7,36 Smith, W. H., 59, 71 Smythe, R., 26,36 Snow, T. P., 48, 55 , 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,

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,

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

26 I

94, 99, 301, 307,326, 328

AUTHOR INDEX 527

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,

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,

Stokes, E. D., 283,290 Stoller, C., 363,364,365,366. 371, 372, 373,

Stoke, S., 191,202, 203 Stolterfoht, N., 278,287,290,291, 339, 348,

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,

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

126, 132

373,380

380

349, 350, 351, 354, 357,377, 380

372, 373,380

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

Taulbjerg, K., 267, 286, 289, 291, 356, 357,

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

377, 379

Tench, A. L. V., 475,504 Teplova, Ya. A,, 317,326 Tesmer, J. , 26.34 Teubner, P. J. 0.. 429, 430, 432, 443, 444,

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

445,464,467,470

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,

Torr, M. R., 39,40,41,43,44,46,53,70, 74, 74, 75

75

5 2 8 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,

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

200, 202, 203

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,

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 ,33 Vorburger, T. V., 25,36 Vroom, D. A., 230,233 Vujovic, M., 230, 231,233

391, 399, 400,421

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,

Valentine, N. , 158, 166 Valentini, J. J., 181, 190, 196, 197,203

400,419,421

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,

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,

Walton, D. R., 57, 73 Wannier, G. H., 426,470 Wardle, C. E . , 107,133

44, 45, 46, 53, 70, 71, 72, 73, 74, 75

I65

AUTHOR INDEX 529

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.

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.

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,

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

74, 76

341,379

201, 203

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,

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,

380

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 ,34

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.

Zavilopulo, A. N., 394, 395, 396,421 Zehnle, L., 450, 451, 452,470 Zeilik, M., 47, 76

202, 203

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

53 I

A

Activation energy, zero, treatment of, 174 Adiabatic approximation, energy-modified,

for electron-molecule collisions 502- 503

Adiabatic maximum rule and Massey cn- tenon, 213-214

Adiabatic-nuclei approximation, vibrational excitation and, 4%-497

Alignment tensor, 446-448 Alkali atom, electron impact ionization,

Alkali-halogen atom collisions, 175- 178 Alkali-hydrogen halide system, potential

energy surfaces and, 168-170, 171- 173

Alkali-proton collisions, electron capture neutralization and, 322-325

Aluminum, K shell ionization, 336-337 Aluminum ion-argon collisions, 354-356 Ammonium ion cluster

433 -436

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 -

Annihilation of positrons 335

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 vibra-

Atmosphere, negative-ion reactions with

Atomic structure, from (e,2e) experiments,

tional excitation on H, and D,. 16-17

trace constituents in, 25

429-431

Attenuation method, in charge transfer collisions, 230-231

Auger electron spectra effects of inner-shell ionization on, 348-

theory of, 374-376 Autoionization, 455-459

350, 362-367

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

for heavy-particle excitation, 267-268,

for heavy particle ionization, 286 inner shell ionization by electron impact,

second-order, high-energy charge transfer

and, 441-443

271 -274

329-331

and, 280-281 Boron, electron affinity of, 5 Branching ratio, importance in astrophysics,

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

59 -6 I

C

Carbon electron affinity of, 5 fine structure levels of ion, excitation of

K shell ionization, 332, 334, 343-344 in interstellar clouds, 54

532 SUBJECT INDEX

Carbon ion-hydrogen collisions, 54, 304-

Carbonate ion, photodestruction of, 21-23 CDW approximation, see Continuum dis-

Cesium

305, 313-314

torted-wave approximation

in electron-ion recombination, 252 ion-atom charge transfer collisions, 207-

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 colli-

sions, 303-315 nonresonant, 21 1-214 psuedocrossing, total cross sections,

symmetrical resonance 206-21 I

208

2 14 -220

Chemiluminescence, study by crossed-beam,

Cheshire-Sullivan model, for heavy-parti-

Chlorine ion, collisions with rare gas atoms,

Chlorine ion-nickel collisions, 361 Chlorine ion-titanium collisions, 361 CIS approximation, see Continuum interme-

Close-coupling approximation, for heavy-

Closure approximation, for heavy-particle

Coherence effects, 461, 462 Coherent excitation of degenerate states

with different angular momenta, 462- 464

Collisions, see also Alkali-halogen atom collisions; Charge transfer collisions; Fast heavy particle collisions; Hydro- gen-hydrogen exchange reaction

coherence and correlation in, 423-464

183- 187

cle excitation, 270-271

27-28

diate-state approximation

particle excitation, 267-268, 271-272

excitation, 268

electron-molecule, at low energies, 471 -

future studies, 460-462 503

Collisional detachment, 26-28, 325-326 Collisional ionization, atom-atom

of highly excited atoms, 91 -99 of normal atoms, 175-178

atomic targets, 78-85 molecular targets, 85-91 of Rydberg atoms, 78-91

Collisional mixing

Collisional-radiative recombination defined, 245, see Electron-ion recombination in ambient electron gas

Continuum distorted-wave approximation,

Continuum intermediate-state approxima-

Copper

283-285

tion, 282-285

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 fu-

sion reactor research, 301 CR recombination, see Collisional-radiative

recombination Curve-crossing spectroscopy, 22 1-228

D

Demkov-Meyerhof model, for inner-shell

Demkov radial coupling model, 354-356 Demkov two-state approximation, 2 12-

2 14 Deuterium-hydrogen abundance ratio, in

local interstellar medium, 48 Deuterium-hydrogen halide exchange reac-

tions, crossed-beam study, 181- 183 Diatomic negative ions, 9-1 1 Diatomic molecules, spontaneous radiative

ionization, 357

dissociation of, 62-67

SUBJECT INDEX 533

Dioxetane reaction, chemiluminescence cross section, 186-187

Dissociative attachment, 13- 18 angular distribution of product ions, 15-

of HN03, I5 total cross section, measurement of,

16

13-14 Dissociative recombination, 38-42. 45 Distorted-wave approximation, electron-

photon angular correlations and, 442- 443

Double-electron radiative transitions, 362- 366

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 colli-

negative ion formation and, 325-326 by pure rotational excitation, 243 radiative, 264, 367-368

sions, 303-315

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,

Electron-electron angular correlations,

Electron-hydrogen molecule collisions,

Electron impact, excitation of atoms by, see

322-325

from impact ionization, 425-429

478 -480

Excitation, electron impact

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

collisional-radiative: recombination de-

formation of excited atoms in high pres-

Electron-ion recombination in ambient

electron gas, 245-250

fined, 245

sure helium afterglow, 250

neutral gas, 250-254 cesium ions in helium, 252 helium-atomic and molecular ions in

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, 473-

Lz methods, 488-495 at low energies, theory of, 47 1-503 molecular-frame representation, 480-485 radial equations, derivation of, 473-477,

R-matrix method, 492-495 7'-matrix method, 490-492

helium, 252-254

480, 482-483

481-482

Electron-nitrogen molecule scattering,

Electron-oxygen ion recombination, and nightglow, 62, 235-237

Electron-photon angular correlations, amplitudes and state parameters from, 440-448

483 -485

Electron promotion, 352 Energy loss spectra, for ion-atom charge

Energy-modified adiabatic approximation,

Excitation

transfer collisions, 221 -224

502-503

of atoms by ions, 266-274 of hydrogen atoms by protons, 271-

Excitation of atoms, by electron impact, 274

381 -419

534 SUBJECT INDEX

absolute cross section measurements,

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,

secondary effects, 382-391 time-resolved measurements, 398-400

Excitation, of atoms by fast ions, theory,

401-415

382-384

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

Fluorine, electron affinity of, 5 Fluorine-hydrogen system, potential sur-

face studies, 168-170 Franck -Condon overlap approximation and

quanta1 reactive scattering, 179-180 Furnace target technique, 301-302 Fusion reactor

and, 349-351

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,

Heavy-particle collisions 336-345

effect on Auger spectra, 362-367 effect on X-ray spectra, 368-373 electron capture from atoms by fast ions,

excitation of atoms by ions, 266-274 impact parameter studies of, 358-360

Heavy particle-photon angular correla-

Helium

274-285

tions, 449-455

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 10-

two-photon decay of metastable, 67 211

Helium afterglow, 248-250 Helium ion-hydrogen collisions, 306-309,

Helium-like ions 312

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,

collisional detachment from, 26-27 electron affinity of, 2 fluorescent photodissociation of, 63-64

charge transfer from multiply charged

electron impact excitation, 446-448 electron impact ionization, 436-437

52

Hydrogen atom

ions, 303-315

SUBJECT INDEX 5 3 5

ionization, 3 15-3 17 proton impact excitation, 271 -274 two-photon decay of metastable, 68

Hydrogen-deuterium exchange reaction,

Hydrogen-hydrogen exchange reaction,

quantal scattering calculations, 179- 180

fast beam studies, 325-326

172-173

173- 174

Hydrogen ion, negative, 2

Hydrogen ion-helium ion collision, 3 17- 321

Hydrogen ion-magnesium ion collisions, 32 1

Hydrogen molecules, in interstellar medium, 62 -67

Hydronium ion in electron-ion recombination, 241, 244-

in ion-ion recombination, 239-241 245

Hypersatellite X-ray, 362

I

ICC term, see Intercontinuum coupling term Impulse approximation, 281 -283 Information-theoretic method, for reactive

Infrared luminescence, in ionic reactions, 23 Inner-shell ionization, 329-376

alignment of atoms and, 332-335, 343-

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, 362-

366 Intercontinuum coupling term and Auger

electron spectra, 374 Internuclear vector, 264-266 Interstellar cloud

CH formation in, 55-56

scattering, 180- 181

345

cooling in, 54-55 OH formation in, 48-50

hterstellar medium, molecular hydrogen

Ion, atomic, inner-shell ionization by, 335-

Ion, negative, see Negative Ion Ion- hydrogen collisions

in. 62-67

361

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

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,

Ionization in fast heavy particle collisions, 286-288, see a1.w Collisional ionization. atom-atom; Inner-shell ionization

gas, 255-259

258-259

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,

Iron, K shell ionization, 342-343, 364-366 Iron ion-hydrogen collisions, 311-313 IT method, see Information-theoretic

and H:, 27

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 ioni-

universal curve for, 338-339, 345-346 zation

L

Landau-Herring method, 216 Landau-Zener-Stueckelberg approxima-

tion, 175-176 .andau-Zener two-state approximation in

ion-atom collisions, 217-220, 253- 254, 305

.asers, uses in studies of photodetachment 2, 8, 20, 22 reactive scattering, 188 recombination, 249

electron affinity of, 3-4 ion-atom charge transfer collisions, 209-

ithium

2 10 Lithium ion-hydrogen collisions, 308-312 Lz methods in electron -molecule collision

LZS approximation, see Landau-Zener- theory, 488-495

Stueckelberg approximation

M

Magnetic dipole transitions, relativistic, 67- 69

Manganese, doubly ionized K shell, 362- 364

Many-body approximation, electron-photon angular correlations and, 442

Mercury, electron Impact excitation, 394- 398

Mercury-halogen system, crossed-beam chemiluminescence and, 183- 184

Mercury-iodine system, potential surface

Merged-beam technique, in fusion reactor

Molecular-beam chemistry, 181 -183 Multichannel eikonal approximation, elec-

tron-photon angular correlations and, 441

study, 170-171

research, 302-303

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

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 autodetach-

Negative ion, clustering, effect on reaction

Negative ion, molecular, electron affinities

Negative ion reactions at high energies,

Negative ion reactions at thermal and epi-

with atomspheric trace constituents, 25 flowing afterglow studies, 23-25 infrared emission, 23 0- reaction rates, 25

Negative ion, atomic

ment, 8-9

rates, 25

and structures of, 9- 13

26-28

thermal energies, 23-25

Neon ion-neon collisions, inner-shell ioni-

Neon-krypton collisions, inner-shell ioniza-

Nickel, K shell ionization, 329-331 Nightglow, 44-46, 62, 235-237 Nikitin formalism, 357-358 Nitrate ion, in ion-ion recombination, 239-

Nitrogen, atomic

zation and, 352-354, 360-361

tion, 358

24 1

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,

Nonadiabatic collision, approximation 184-185

methods for, 179-180

0

OBK approximation, see Oppenheimer- Brinkman-Kramers approximation

SUBJECT INDEX 537

Oppenheimer- Brinkman- Kramers approximation, 342

Orthopositrium. 126

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

5577 and 6300 A lines of, in airglow,

Oxygen ion, in electron-ion recombination,

Oxygen ion-hydrogen collision, 46-49.305,

vacuum lifetime, 137, 163-164

astrophysics, 53-55

44-45

235-237

3 10

P

Parapositrium. 126 Partial-parting concept in computer simu-

lated experiments on ionic recombination, 257-259

Particle- photon angular correlations, 437 - 455

PCI, see Postcollision interaction Perturbed stationary state impact parameter

method, 353, 356 Photodetachment, 18 -23 Photodissociation of negative ions, 18-23 Photon labeling in study of optical excitation

Plasma diagnostics, 299-300 Plasma heating

function, 417-4 18

energy and particle loss and, 295-297 fast-beam injectors for, 297-299

Polarization, instrumental, 389-391 Polarization of impact radiation

angular correlations and, 449-452 Bethe approximation and, 403-415 near-threshold, 392 -394

Polyatomic negative ions, 12-13 Positron annihilation

angular correlations of y-rays, 124- 126 in atom scattering, 118- 124

Positron-argon scattering, at low energy, 148- 149

Positron-atom scattering, theoretical, 101 - 131

annihilation in, 118- 124 overview, 101 - 102

Positron beam, generation of, 138- 142 Positron emission, in heavy atom collisions,

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 - 1 13 s-wave phase shifts, 108- 1 I I system coordinates, 106

Positron-hydrogen scattering resonances in, 104- 105 theoretical, 102-105, 130- 131

definition, 126 formation in inert gas, 153-154 formation in positron-helium collisions,

formation in positron-hydrogen colli-

lifetime, 126 types, 126 vacuum lifetime of ortho-positronium,

Positron-krypton scattering, at low energy,

Positron lifetime parameters

373-374

Positronium

129-130

sions, 126- 129

137, 163-164

149-151

for inert gas, 159-163 for molecular gas. 163 methods of study, 159

Positron-neon scattering, at low energy,

Positron scattering, experimental, 135-159 cross section data, accuracy, 142- 145 cross section measurement, 146- 159 intermediate energy, inert gases, 151 -153 in molecular gas, 155 - 157 positrium formation cross section in inert

Positron-xenon scattering, at low energy,

Postcollision interaction and Auger effect,

147- I49

gases, 153-154

149-151

374-376

538 SUBJECT INDEX

Potential energy surfaces ab inirio computations, 168-171 classical tragectories on, 171 -173

Proton, inner-shell ionization by, 336-338,

Proton-rare gas collisions, 224-228 Pseudostate expansions, for heavy-particle

Pseudostates, 270 PSS impact parameter method, see Per-

turbed stationary state impact parameter method

358-360

excitation, 270, 271

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 ap-

Rydberg atom proximation

collisional mixing of, 78-91 thermal collisions with heavy particles,

77 -99

S R

Radiation, following inner-shell ionization,

Radiation, impact, Bethe approximation for

Radiation, molecular orbital, 368-374

362-376

polarization of, 403-415

anisotropy of, 371-373 characteristics of, 369-371

237, 382-385 Radiation, resonance, imprisonment of,

Radiative electron rearrangement, 362 Reaction rate, transition-state theory and,

173- 175 Reactive molecular scattering, 167- 197, see

also Potential energy surfaces; Transi- tion state theory

accurate quanta1 calculations, 179- 180 collisional ionization: nonadiabatic reac-

crossed beam chemiluminescence, 183-

with electronically excited reagents, 189-

information-theoretic approach to, 180-

with rotationally excited reagents, 190-

state-to-state cross sections, 187-188 translational excitation of reaction and,

translational thresholds, 193- 197 use of laser, 188 vibrational enhancement, 190

tions, 175-178

I87

190

181

192

192- 193

REC, see Electron capture, radiative

SCA, see Semiclassical approximation Scattering, see Positron scattering; Reactive

molecular scattering Second-order potential model, for heavy-

particle 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,

Static gas measurements, of excitation cross

Stueckelberg model, 180 Sulfur, double-electron radiative transitions

Surface hopping, 176-178 Symmetrical resonance charge transfer pro-

56-58

sections, 401-402

in, 362-363

cess, 206-21 1

T

Time-of-flight positron beam system, 140-

Titanium ion-hydrogen collision, 314-315 T-matrix method, 490-492 Tokamak plasma device, 294, 295, 297 Trajectory methods, classical, in reactive

Trajectory surface-hopping method, 176-

Transitions, fine-structure, in aeronomy and

142

scattering, 171-173

178

astrophysics, 53-55

SUBJECT INDEX 539

Transition-state theory. 173- 175 adiabatic-nuclei approximation and, 496-

hybrid theory and, 497-499 unified statistical theory, 174-175 497 zero activation energy, treatment of, 174

Two-photon decay, 67-69

W

U

Uranium, inner-shell ionization, 359

Wentzel theory, Auger effect and, 374

X

X-radiation, quasi-molecular formation, V 368 -373

anistropy of, 371 -373 Vibrational excitation by electron impact, X-ray spectra, heavy-particle collisions and,

495 -503 368-373


Contents of Previous Volumes

Volume 1

Molecular Orbital Theory of the Spin Properties of Conjugated Molecules, G. G. H d 1 r r i rd A . T. Arms

Electron Affinities of Atoms and Mol- ecules, 6. L . Moisc+wifsc,lr

Atomic Rearrangement Collisions, B. H . Brcrrisderi

The Production of Rotational and Vi- brational 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 Molec- ular Beams, J . B. Aiiderson, R . P . Atidrcs, r r t i d J . B . Fvriir

AUTHOR INDEX-SUBJECT I N D E X

Volume 2

The Calculation of van der Waals Inter- actions, A . Dtrlgcirrio r r r i d W . D . Drr i i i . \or i

Thermal Diffusion in Gases. E . A . Mirsorr. R. J . Mrrrn, ( / / id F1.tr17c.i.~ J . Sriiiih

Spectroscopy in the Vacuum Ultra- violet, W . R. s. Gurtorr

The Measurement of the Photoioniza- tion Cross Sections of the Atomic Gases, J o t w J A . R. Srrrn.wri

The Theory of Electron-Atom Colli- sions, R. P ~ l ~ r A o i p r i n d V . VPILJI.P

Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, F. J . cir Hrrr.

Mass Spectrometry of Free Radicals,

AUTHOR INDEX-SUBJECT INDEX S . N . Foricr

Volume 3

The Quanta1 Calculation of Photoioni- zation Cross Sections, A . L . Stricwt

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 Crystal- Surface van der Waals Scattering, F. Clirrrioc,h Bptier.

Reactive Collisions between Gas and Surface Atoms, Henry W i s ~ ( / t i t1

Borirrird J . Wood


Volume 4

H . S. W. Massey-A Sixtieth Birthday Tribute, E. H . 5'. Birrhop

Electronic Eigenenergies of the Hy- drogen 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~hirrghtrin rind E. Gtrl

Positrons and Positronium in Gases, P. A . Frciwr.

S41

542 CONTENTS OF PREVIOUS VOLUMES

Il1t2P.s B. R . Jiitltl

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. M o h r

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 .

The Direct Study of Ionization in L)(II,qNrIIO

Space, R . L . F . Boyd A U T H O R INDEX-SUBJECT INDEX

Volume 5

Flowing Afterglow Measurements of Ion-Neutral Reactions, E . E . Fc~gir- . s o i l , F. C. Fdi.wiifi4d. Lind A. L. S d i i n r l t r h o p f

Experiments with Merging Beams, Roy H . Nevritrher

Radiofrequency Spectroscopy of Stored Ions 11: Spectroscopy, H . G . Delimclt

The Spectra of Molecular Solids,

The Meaning of Collision Broadening of Spectral Lines: The Classical- Oscillator Analog, A . Brti-Rriiwn

,The Calculation of Atomic Transition Probabilities, R. J . S . Crossley

Tables of One- and Two-Particle Coeffi- cients of Fractional Parentage for Configurations . s ? ~ ’ ~ p ~ , C. D. H . Chisholnz. A . Daigrirno. niicl F . R .

0. SCII I I Ppp

Relativistic Z Dependent Corrections to Aomic Energy Levels, Holly Thoniis Doyle

AUTHOR INDEX-SUBJECT INDEX

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 irJm uti

The Rotational Excitation of Molecules by Slow Electrons, Kaziio Ttihtrvrr-

m g i and YuXikuz~~ I f i kar iw

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 l R 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

AUTHOR INDEX-SUBJECT I N D ~ X

Volume 7

Physics of the Hydrogen Master, C‘. Aiiiloit/. .I. P . S c / i o r i ~ i t i t i t i , ( i t i d

P . Grirvt Molecular Wave Functions: Calcula-

tion and Use in Atomic and Molecular Processes, J . C. B r o w e

Localized Molecular Orbitals, Hurl)/ Weiizstrin, Ridhen Ptrrrncz, titit1

Miiiiricr Cohcn General Theory of Spin-Coupled Wave

Functions for Atoms and Molecules, J . Gerratr

Diabatic States of Molecules-Quasi- stationary Electronic States, Tliorntr~ F . O’MLiilry

Selection Rules within Atomic Shells,

CONTENTS OF PREVIOUS VOLUMES 543

Green‘s Function Technique in Atomic and Molecular Physics, Gy. f s c i t i d . H . S . Tuylor. rr 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. Gr~~i i , f ie ld

A u T HO R 1 N D EX - S u BJ ECT 1 N I) EX

Volume 8

Interstellar Molecules: Their Forma- tion 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 i C. Y . Chrii ciiitl AirpistiiiP C . Chrn

Photoionization with Molecular Beams, R. B. Coirtis, H~rl.vti~(ril Htirrisuii, triitl

K . I. J ” ~ h 0 P I I

The Auger Effect, E . H . S. Birrhop ti i i t l

W. N . A.strritl


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 Colli- sions, D. Aticfrii,X

Molecular Beam Electric Resonance Spectroscopy. J tws C. Zorii toid

l l r o i ~ i ( i . s C. Eiiglish

Atomic and Molecular Processes in the Martian Atmosphere, Mic,/itrrl B . McEIro?.


Volume 10

Relativistic Effects in the Many-Elec- tron Atom, Lloyd Arn i s t ro i~~y . J r . rititl Srrgr Ft2iicwillcj

The First Born Approximation, K . L . Rtill cind A . E. Kitigstoii

Photoelectron Spectroscopy, W. C. Priw

Dye Lasers in Atomic Spectroscopy,

Recent Progress in the Classification of the Spectra of Highly Ionized Atoms, u. c. F c r nYet t

A Review of Jovian Ionospheric Chem- istry, Weslay T . Hiiiitrcxs , .Is.

SUBJECT INDEX

W . LfItigP. J . LiiII i t fF, tJtlll A . St;,iid[,l

Volume 11

The Theory of Collisions Between Charged Particles and Highly Ex- cited 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 Nu- clei, Jolirrtiiie.~ M . H ~ r t i ~ t c ~ ~ r r

Stark Broadening, Htiiis R . Grietii

Chemiluminescence in Gases, M . F .

Au I HOR INDEX-SUBJLCT INDEX Goldr cind U. A . Tlirr~.th

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 . SdI in Time-of-Flight Scattering Spectros-

Ion Chemistry in the D Region, Gtwrge

AUTHOR INDEX-SUBJECT INDEX

copy, Wilhc4tn Rciith

C. R t i d

Volume 13

Atomic and Molecular Polarizabilities -A Review of Recent Advances, Tliomris M. Miller rind Benjamin Bedcrson

Study of Collisions by Laser Spectros- copy, Paiil R . Biwncin

Collision Experiments with Laser Excited Atoms in Crossed Beams, I . V . Hrrtcl rind W . Stol l

Scattering Studies of Rotational and Vibrational Excitation of Molecules, Mtrnfird Fnirbi.1 tirid J . Peter Torn flies

Low-Energy Electron Scattering by Complex Atoms: Theory and Calcu-

Microwave Transitions of Interstellar Atoms and Molecules, W . B . Soinrrvillr


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~igold rind 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.s M. 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 Astro- physics, A . K . Dicpriv

lations, R . K . Neshrt AUTHOR INDEX-SUBJECT INDEX

Documents

Advances in Atomic and Molecular Physics Volume 15