Theory of Slow Atomic Collisions

444 Pages·1984·7.403 MB·English

by  Professor Evgenii E. Nikitin, Stanislav Ya. Umanskii (auth.)

30
Springer Series in Chemical Physics
Edited by 1. Peter Toennies
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Springer Series in Chemical Physics
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E. E. Nikitin S.Ya. Umanskii
Theory of
Slow Atomic Collisions
With 92 Figures
Springer-Verlag
Berlin Heidelberg New York Tokyo 1984
Professor Evgenii E. Nikitin
Stanislav Ya. Umanskii
Institute of Chemical Physics, Academy of Sciences of the USSR, Kosygin Street 4
Moscow V- 334, USSR
Series Editors
Professor Vitalii I. Goldanskii Professor Dr. Fritz Peter Schafer
Institute of Chemical Physics Max-Planck-Institut fiir
Academy of Sciences Biophysikalische Chemie
Kosygin Street 4 D-3400 Gottingen-Nikolausberg
Moscow V-334, USSR Fed. Rep. of Germany
Professor Robert Gomer Professor Dr. J. Peter Toennies
The James Franck Institute Max-Planck-Institut fiir Stromungsforschung
The University of Chicago BottingerstraBe 6-8
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Chicago, IL 60637, USA Fed. Rep. of Germany
Title of the original Russian edition:
Neadiabaticheskiye perekhody pri medlennykh atomnykh stolknoveniyakh
© by "Atomizdat" Publishing House, Moscow 1979
ISBN-13:978-3-642-82047-2 e-ISBN-13:978-3-642-82045-8
DOl: 10.1007/978-3-642-82045-8
Library of Congress Cataloging in Publication Data. Nikitin, E. E. (EvgeniI Evgen'evich), 1933-. Theory
of slow atomic collisions. (Springer series in chemical physics; v. 30). I. Collisions (Nuclear physics)
2. Scattering (Physics) I. Umanskii, Stanislav IAkovlevich. II. Title. III. Series. QC794.6.C6N545 1984
539.7'54 84-1345
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Preface
The theory of atom-molecule collisions is one of the basic fields in chemi
cal physics. Its most challenging part - the dynamics of chemical reactions
- is as yet unresolved, but is developing very quickly. It is here a great help
to have an analysis of those parts of collision theory which are already
complete, a good example being the theory of atomic collisions in process
es specific to chemical physics.
It has long been observed that many notions of this theory can also be
applied successfully to reactive and unreactive molecular collisions. More
over, atomic collisions often represent a touchstone in testing approaches
proposed for the solution of more complicated problems. Research on the
theory of slow atomic collisions carried out at the Moscow Institute of
Chemical Physics has been based on just these ideas. A general viewpoint
concerning the setting up and representation of the theory came out of
these studies, and appeared to be useful in studying complicated systems
as well. It underlies the representation of the theory of slow atomic colli
sions in this book.
Analytical approaches to the theory of slow atomic collisions, includ
ing the calculation of electronic states of diatoms, nonadiabatic coupling
models, calculation of differential and total cross sections, are discussed
here. These approaches are based on the exact and approximate symmetry
properties of an electronic subsystem, and on simplifications provided by
slow quasi-classical nuclear motion. A transparent correlation between
interatomic interactions and the characteristic features of scattering
emerges from these approaches.
The theory presented in this book is self-contained. It can be applied to
the interpretation of various processes occurring in atomic collisions over
a relatively wide energy range, from thermal energies to hundreds of eV .
The authors wish to thank Dr. E.1. Dashevskaya, Dr. G. K. Ivanov,
Dr. M. Ya. Ovchinnikova and Dr. A.1. Reznikov for their valuable ad
vice, as well as colleagues at the Moscow Institute of Chemical Physics
for very fruitful discussions. Thanks are also due to Mrs V. D. Grammat
chikova for her help in preparing the text.
Moscow, January 1984 E. E. Nikitin . S. Ya. Umanskii
Contents
1. Introduction 1
2. General Formulation of Scattering Problem Under Quasi-
Classical Conditions 5
2.1 Scattering Amplitudes and Cross Sections 6
2.1.1 Representations of Amplitudes and Cross Sections. 6
2.1.2 Scattering Amplitudes and Cross Sections Under
Quasi-Classical Conditions 14
2.2 Scattering Equations 22
2.2.1 Atomic Basis 22
2.2.2 Molecular Basis 26
2.3 Collisions of Two Many-Electron Atoms 34
2.3.1 Scattering Matrix and Scattering Equations 34
2.3.2 Collisions of Identical Atoms 41
2.4 Integral Cross Sections for Isotropic Collisions 43
3. Diatomic Electronic States 53
3.1 Quantum Numbers and Wave Functions of a Free Atom. 53
3.2 Quantum Numbers and Wave Functions of Diatoms 59
3.2.1 General Classification of Adiabatic Diatomic
States. 59
3.2.2 Wave Functions of a Diatom at Large Internuclear
Separations 62
3.2.3 Molecular-Orbital Approximation 68
3.3 Adiabatic States, Diabatic States, and Correlation
Diagrams 74
3.3.1 The Noncrossing Rule and Adiabatic Correlation
Diagrams 74
3.3.2 Diabatic States and Diabatic Correlation
Diagrams 78
3.3.3 One-Electron Correlation Diagrams. 85
3.4 Coupling Between Electronic States. Selection Rules . 98
4. Approximate Calculation of the Electronic States of Diatoms . 103
4.1 Atomic Potential and Atomic Orbitals . 103
VIII Contents
4.1.1 Hartree-Fock Screening Function and Atomic
Orbitals. 103
4.1.2 The Pseudopotential Method for Valence
Electrons of Atoms . 109
4.2 Diatomic Interactions at Large Distances and the
Heider-London Approximation. 113
4.2.1 Effective Hamiltonian. 114
4.2.2 Coulomb Interaction 117
4.2.3 Dispersion Interaction 120
4.2.4 Exchange Interaction 125
4.3 Pseudopotential Method for Interatomic Interactions 136
4.3.1 The Model Potential Method 137
4.3.2 MUltiple Scattering Method 141
4.4 Short-Range Atomic Interactions . 148
4.4.1 The Energy of Atomic Interaction at Small
Distances 150
4.4.2 Electronic Potential in a Diatom at Small R 154
4.5 Coupling Between Electronic States . 158
4.5.1 Spin-Orbit Coupling 158
4.5.2 Radial Coupling in the A voided Crossing Region 162
5. Elastic Scattering 167
5.1 Quasi-Classical Scattering Amplitude 167
5.2 Quasi-Classical Scattering Matrix . 170
5.2.1 JWKB Scattering Phase Shifts 171
5.2.2 Violation of Quasi-Classical Conditions in
Localized Regions. Connection Problem 173
5.2.3 Isolated Turning Point 176
5.2.4 Two Close Turning Points . 179
5.3 Classical Scattering . 186
5.4 Integral Cross Sections 191
5.5 Differential Cross Sections . 195
5.5.1 Scattering Through Classical Angles-Repulsive
Potential 195
5.5.2 Scattering Through Classical Angles-Potential
with a Well 196
5.5.3 Scattering Through Small Angles . 199
6. Approximate Calculation of a Multichannel Quasi-Classical
Scattering Matrix . .202
6.1 Common-Trajectory Approach . .202
6.1.1 Common-Trajectory Scattering Equations .204
Contents IX
6.1.2 Eikonal and Impact-Parameter Approximations . 209
6.1.3 Semiclassical Limit of the Quasi-Classical
Approximation. . . . . . . . . . . .. . 214
6.2 Matching Approach ........... .. . 222
6.2.1 Matching Solution of Scattering Equations . 222
6.2.2 Near-Adiabatic Matching . 229
6.2.3 Near-Sudden Matching . . . . . . . .. . 234
6.3 Perturbation Approach . . . . . . . . . . .. . 237
6.3.1 First-Order Perturbation Treatment. The Born
and Adiabatic Distorted-Wave Approximations .. 237
6.3.2 Unitarized Distorted-Wave Approximation . 240
7. Two-State Scattering Problem. . . . . . . . . . .243
7.1 The Two-State Model. Adiabatic and Diabatic
Representations . . . . . . . . . . . . . . . . . . 243
7.2 Construction of the Two-State Quasi-Classical S Matrix
by the Matching Method . . . . . . . . . 248
7.3 Two-State Semiclassical Models . . . . . . . . . . . 254
7.3.1 Derivation of Semiclassical Equations . . . . . . 254
7.3.2 Classification of Semiclassical Two-State Models . 258
7.3.3 Approximate Two-State Transition Probabilities . 262
7.4 Differential Cross Sections and Deflection Functions . 266
8. The Linear Two-State Landau-Zener Model. . . . . .. . 273
8.1 Formulation of the Model . . . . . . . . . . .. . 273
8.2 Nonadiabatic Transitions Far from the Turning Point.
Landau-Zener-Stueckelberg Solution . . . . .. . 276
8.3 Nonadiabatic Transitions Near the Turning Point. . 280
8.3.1 Terms with Slopes of the Same Sign . 280
8.3.2 Terms with Slopes of Different Signs . .. . 284
8.4 Validity of Linear Model and of Analytical Expressions
for Transition Probabilities ..... . . . . . 288
8.5 Cross Sections for the Linear Model . . . . . .. . 292
8.5.1 Integral Cross Sections - Radial Coupling .. . 292
8.5.2 Integral Cross Section - Rotational Coupling . 300
8.5.3 Differential Cross Sections - Threshold Effects . 302
9. Nonlinear Two-State Models of Nonadiabatic Coupling 313
9.1 Exponential Model . . . . . . . . . . . . . 314
9.1.1 Formulation of the Model. . . . . . . 314
9.1.2 Transition Probability and Dynamic Phases . 316
X Contents
9.1.3 Specific Cases of the Exponential Model -
Probabilities and Cross Sections . 322
9.2 Linear-Exponential Model . . . . . . . . . . . 327
9.2.1 Formulation of the Model. . . . . . . . 327
9.2.2 Transition Probabilities and Cross Sections . 329
9.3 Other Nonlinear Models ........... . 332
9.3.1 Hypergeometric Models. . . . . . . .. . 333
9.3.2 Power Models - Large Interatomic Separations. . 334
9.3.3 Power Models - Small Interatomic Separations. . 337
10. Multistate Models of Nonadiabatic Coupling ........ 340
10.1 Transitions Between Degenerate States .. . . . . . . 340
10.1.1 Collisional Depolarization of an Isolated Atomic
State .................... 340
10.1.2 Resonant Excitation Transfer by Dipole-Dipole
Interaction ................. 348
10.1.3 Transitions Between Degenerate Hydrogen States
in Collisions with Ions . . . . . . 351
10.2 Transitions Between Highly Excited States . . . .. . 355
10.3 Generalizations of the Linear Model . . . . . .. . 359
10.3.1 Interaction of a Diabatic Term with a Set of
Parallel Diabatic Terms and a Continuum . . 359
10.3.2 Nonadiabatic Coupling Between Two Quasi-
Stationary States . . . . . . . . . . .. . 364
11. Case Study - Intramultiplet Mixing and Depolarization of
Alkalis in Collisions with Noble Gases . . . . . . . . 366
11.1 Formulation of the M* - X Scattering Problem . 367
11.1.1 Scattering Equations and Couplings . 367
11.1.2 M* - X Interaction .371
11.2 The Scattering Matrix. . . . . . . . . . . 375
11.2.1 Matching Approximation . . . . . . 375
11.2.2 Semiclassical Comparison Equations . 379
11.2.3 Scattering Matrix for 2 Pl/2 Substate . . 384
11.3 Transition Probabilities and Cross Sections for
Isotropic Collisions . . . . . . . . . . . 387
11.3.1 Intramultiplet Mixing . . . . . . . 387
11.3.2 Reorientation in the 2 P1/2 Subs tate . 392
Appendix . . . . . . . . . . . . . . . . . . . 395
A. Quantum Theory of Angular Momentum . 395
A.l Rotation Matrices and Spherical Functions . 395
Contents XI
A.2 Coupling of Angular Momenta, Clebsch-Gordan
Coefficients and 3 n - j Symbols . . . . . 398
A.3 Matrix Elements of the Irreducible Tensor
Operators . 401
References . . .403
Subject Index . 427