Advances in ATOMIC AND MOLECULAR PHYSICS
VOLUME 14
CONTRIBUTORS TO THIS VOLUME S. V. BOBASHEV
M. S. CHILD A. K. DUP...
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Advances in ATOMIC AND MOLECULAR PHYSICS
VOLUME 14
CONTRIBUTORS TO THIS VOLUME S. V. BOBASHEV
M. S. CHILD A. K. DUPREE S. A. EDELSTEIN
T. F. GALLAGHER D. E. GOLDEN MICHAEL J. JAMIESON IAN E. McCARTHY RICHARD MARRUS PETER J. MOHR FRANCIS M. PIPKIN RONALD F. STEWART BRIAN C. WEBSTER ERICH WEIGOLD
ADVANCES IN
ATOMIC AND MOLECULAR PHYSICS Edited by
Sir D. R. Bates DEPARTMENT OF APPLIED MATHEMATICS AND THEORETICAL PHYSICS THE QUEEN’S UNIVERSITY OF BELFAST BELFAST. NORTHERN IRELAND
Benjamin Bederson DEPARTMENT OF PHYSICS NEW YORK UNIVERSITY NEW YORK, NEW YORK
VOLUME 14
@ 1978 ACADEMIC PRESS New York San Francisco London A Subsidiary of Harcourt Brace Jovanovich, Publishers
COPYRIGHT @ 1978, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART O F THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.
ACADEMIC PRESS,INC. 111 Fifth Avenue,
New York, New York 10003
United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London N W 1 7 D X
LIBRARY O F CONGRESS CATALOG CARD NUMBER:65-18423
ISBN 0-12-003814-5 PRINTED IN THE UNITED STATES OF AMERICA
Contents ix
LIST OF CONTRIBUTORS
Resonances in Electron Atom and Molecule Scattering
D. E. Golden 1 5 10 36 78
I. Introduction 11. Theoretical Considerations 111. Experimental Considerations IV. Results References
The Accurate Calculation of Atomic Properties by Numerical Methods
Brian C. Webster, Michael J. Jamieson, and Ronald F. Stewart 88 92 106 109 121 122
I. Introduction 11. Time-Independent Applications
111. The Solution of Coupled Equations IV. Time-Dependent Applications V. Conclusion References
(e, 2e) Collisions
Erich Weigold and Ian E. McCarthy I. Introduction 11. Experimental Methods 111. Basic Theory
IV. Reaction Mechanism at Intermediate to High Energies V. Structure of Atoms and Molecules VI. Conclusions References V
127 130 139 151 164 176 177
CONTENTS
vi
Forbidden Transitions in One- and Two-Electron Atoms Richard Marrus and Peter J. Mohr I. Introduction 11. Preliminary Survey
111. IV. V. VI. VII. VIII.
Magnetic Dipole Decay Magnetic Quadrupole Transitions Two-Photon Decay Intercombination Transitions Nuclear-Spin-Induced Decays Electric-Field-Induced Decays References
182 183 188 194 199 209 21 1 214 220
Semiclassical Effects in Heavy-Particle Collisions
M . S . Child I. 11. 111. IV.
Introduction Elastic Atom-Atom Scattering Inelastic and Reactive Scattering Nonadiabatic Transitions V. Summary References
225 233 246 262 274 275
Atomic Physics Tests of the Basic Concepts in Quantum Mechanics Francis M . Pipkin I. Introduction 11. Conceptual Framework of Quantum Mechanics 111. Experimental Tests
IV. Conclusions References
281
284 293 336 337
Quasi-Molecular Interference Effects in Ion-Atom Collisions S . V . Bobasheu I . Introduction 11. Quasi-Molecular Interference in Inelastic Scattering 111. Total Cross Sections for Inelastic Ion-Atom Collision Processes
341 342 348
CONTENTS
IV. Long-Range Interaction and Polarization of Emitted Light V. Conclusions References
vii 355 36 1 362
Rydberg Atoms
S . A . Edelstein and T. F. Gallagher I. Introduction 11. Spectroscopy and Field Ionization
111. Lifetime and Collision Studies of Rydberg Atoms IV. Directions for Future Research References
365 368 379 389 389
UV And X-Ray Spectroscopy in Astrophysics A . K. Dupree I. Introduction 11. General Considerations 111. The Beryllium Sequence IV. The Boron Sequence V. The Sodium Sequence VI. The Nonequilibrium Solar Plasma VII. Concluding Remarks References
393 396 407 414 42 1 422 426 428
AUTHOR INDEX SUBJECT INDEX CONTENTS OF PREVIOUS VOLUMES
433 45 1 46 1
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List of Contributors Numbers in parenthesis indicate the pages on which the authors’ contributions begin.
S. V. BOBASHEV, A. F. Ioffe Physico-Technical Institute of the Academy of Sciences, Leningrad, USSR (341) M. S. CHILD, Department of Theoretical Chemistry, University of Oxford, Oxford OX1 3TG, England (225)
A. K. DUPREE, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138 (393)
S. A. EDELSTEIN, SRI International, Menlo Park, California 94025 (365) T. F. GALLAGHER, SRI International, Menlo Park, California 94025 (365) D. E. GOLDEN, Department of Physics and Astronomy, University of Oklahoma, Norman, Oklahoma 73019 (1) MICHAEL J. JAMIESON, Department of Computing Science, University of Glasgow, Glasgow G12 8QQ, Scotland (87) IAN E. McCARTHY, Institute for Atomic Studies, School of Physical Sciences, The Flinders University of South Australia, Bedford Park, S.A. 5042, Australia (127) RICHARD MARRUS, Materials and Molecular Research Division, Lawrence Berkeley Laboratory, Berkeley, California 94720 (181) PETER J. MOHR, Materials and Molecular Research Division, Lawrence Berkeley Laboratory, Berkeley, California 94720 (18 1 ) FRANCIS M. PIPKIN, Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts 02138 (281) RONALD F. STEWART*, Center for Astrophysics, Harvard College Observatory, Cambridge, Massachusetts 02138 (87) BRIAN C. WEBSTER, Department of Chemistry, University of Glasgow, Glasgow G12 SQQ, Scotland (87)
* Present address: ICI Corporate Laboratory, P.O. Box No. Cheshire WA7 4QE, England. ix
11, The Heath, Runcorn,
X
LIST OF CONTRIBUTORS
ERICH WEIGOLD, Institute for Atomic Studies, School of Physical Sciences, The Flinders University of South Australia, Bedford Park, S.A. 5042, Australia (127)
Advances in ATOMIC AND MOLECULAR PHYSICS
VOLUME 14
This Page Intentionally Left Blank
ADVANCES I N ATOMIC AND MOLECULAR PHYSICS, VOL.
14
RESONANCES IN ELECTRON ATOM A N D MOLECULE SCATTERING* D . E. GOLDEN Department of Physics and Astronomy University of Oklahoma Norman, Oklahoma
.......................................... .......................................... 111. Experimental Considerations. ........................... I. Introduction.
XI. Theoretical C
1
5
A. The Functions of Monochromators and Energy Analyzers B. Transmission Experiments ................ C. Crossed-BeamExperiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18 24
C. e--H, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. e--N, ..... ..... References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55 67 78
IV.
I. Introduction While resonance effects in atomic and molecular scattering processes have been known to exist for more than 50 years, the study of resonances in electron scattering and photoabsorption has rapidly developed during the past 15 years. Three reasons for this are that toward the end of the 1950s, experimental techniques became sufficiently sensitive to detect the structure of resonances, theoretical understanding became sufficiently detailed to accurately predict resonance positions and shapes, and computer power became sufficientto be able to handle large-scale calculations. This chapter will be restricted to the significance of the techniques used and the observations of effects in electron-atom and molecule scattering for four simple target systems. While these simple targets have been studied carefully by a large number of investigators, it will be seen below that our
* Supported in part by funds from AFOSR and NSF. 1 Copyright @ 1978 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-003814-5
D.E. Golden
2
detailed understanding of resonance effects in these systems is still not complete, which indicates the necessity of additional work. For past reviews of the subject covering a larger number of target systems, the reader is referred to the works of Burke and Smith (1962),Burke (1965, 1968), Smith (1966), Bardsley and Mandl (1968), Taylor (1970), Golden et al. (1971), Schulz (1973a,b, 1976), and Andrick (1973). See also Massey et al. (1969) and Massey (1976). Resonance states of both target and extra-electron systems can be observed in electron scattering. Resonance states of a target (atom/molecule/ion) A can be observed in inelastic electron scattering by e-
+ A + e - + A*
e-
+ (B + C)
(1) where B and C are the decay products of A. In the case of atomic targets, the decay products can be an electron and an ion or a photon and an excited- or ground-state atom. In the case of molecular targets, dissociation products (including stable negative ions) can be formed. Such resonance states can also be observed if the incoming electron is replaced by an incoming photon, atom, or molecule or atomic or molecular ion. In addition, the same resonance states can be observed by studying the inverse reactions. Such resonance states can be thought of as being due to an interaction between discrete and continuum states. Resonance states of an extra-electron system can be observed in inelastic or elastic scattering by e-
+ A +[A-]*
--*
+B
+C
(2) where B and C are now the decay products of [A-I*. As before, resonance states can be observed by studying the inverse reactions. In the experiments with neutral targets denoted by reaction (l), the reaction cross section is determined by the amplitude for the production of the compound state A* and not by the decay width of A*. Therefore, the observation of such a resonance state is not dependent on having high-energy resolution in the incident electron beam. In the experiments denoted by reaction (2), narrow resonances can be observed only if the energy resolution is sufficient. This is the reason that the early observations of resonances in electron scattering were for the most part confined to those involving reaction (1). However, it should be noted that when making a resonance calculation one can always treat the problem as that of an extra-electron resonance. That means the problem of calculating resonances in reaction (I)can be treated by calculating resonances in reaction ( 2 ) for a different target configuration. When dealing with resonances for molecular targets, the situation becomes more complicated. A large number of target states must be considered, and in addition the possible vibrational and rotational structure of the resonance states must also be considered.
ELECTRON ATOM AND MOLECULE SCATTERING
3
Resonances occur at fairly well defined energies and can be thought of as giving rise to a “time delay” in the passage of the incident particle past the target. However, when the time delay becomes comparable to the collision time, the idea of a resonance becomes “fuzzy.” Resonances are classified either as “closed-channel” or “open-channel” resonances. A closed-channel resonance can be thought of as being due to an interaction between the incoming particle and an excited state of the target that is strong enough to support a “temporary bound state.” That is, the incoming electron virtually excites a target configuration, which in turn creates a well of sufficient depth to trap the incoming electron briefly. On the other hand, an open-channel resonance can be thought of as being due to the interaction between the incoming particle and the target such that an intermediate state is formed without changing the configuration of the target. In the usual case, we would expect the potential representing this type of interaction to have a repulsive barrier. Since the shape of the potential is important in producing this type of resonance it is often referred to as a “shape resonance.” Since closed-channel resonances are below the threshold energy of the target state or states to which they are most strongly coupled while open-channel resonances are above, closed-channel resonances are typically much narrower than openchannel resonances. In other words, since there is an additional decay mode close by that is open, an open-channel resonance is usually broader than a closed-channel resonance. A geneology of extra-electron resonances has been discussed by Schulz (1973a) in his comprehensive compilation of experimental results. In the geneology, a target configuration is called a “parent.” He has then distinguished between closed- and open-channel resonances in that closed-channel resonances lie below the parent, while open-channel resonances lie above. The “grandparent” is the particular positive ion configuration such that when an electron is added, a Rydberg state of the neutral target (parent) is formed. When an extra electron is added to the parent an extra-electron resonance is formed. The first possibility of the existence of resonance states is found in the work of Franck and Grotrian (1921). However, the discovery of the Auger effect (Auger, 1925)is probably the earliest observation of a phenomenon involving resonances. Predissociation (see Herzberg, 1950) is another process that depends on resonance effects, but this subject will not be discussed here. Other early observations of target resonance states were made by Compton and Boyce (1928) and by Kruger (1930). They found vacuum UV emission lines in helium that were attributed to transitions between continuum and bound states. Shenstone and Russell (1932) observed autoionizing states of calcium using absorption spectroscopy, and Beutler (1935) observed autoionizing states in argon, krypton, and xenon. While some theoretical work
4
D.E. Golden
on the interaction between discrete and continuum states had been done by Rice (1933), the results of Beutler (1935) were interpreted by Fano (1935) as due to series of autoionizing Rydberg levels converging on the first 2Plj2 excited states of the rare gas ions. The first direct observations of target resonance states in electron scattering were made by Whiddington and Priestley (1934,1935).They observed two states of helium above the first ionization threshold (near 60eV) by studying inelastically scattered electrons. These observations were explained by Massey and Mohr (1935)with direct calculations of the excitation process. The calculations showed that the observations were compatible with the excitation of several doubly excited helium configurations. Other calculations d positions and lifetimes of target resonance states of helium were made by Fender and Vinte (1934), Wu (1934), and Wilson (1935). These target resonance states are doubly excited states that can decay to Hef in the ground state by ejecting an electron (see Smith et al., 1974). However, it has been shown by Hicks et al. (1974) that in collisions between electrons and atoms in which a short-lived autoionizing state is formed, the ejected electron may have its energy displaced to higher values. This is due to the “postcollision interaction,” which can occur between the scattered and ejected electrons. This interaction can be significant for impact energies close to threshold when the scattered electron is moving slowly and can have an additional repulsive interaction with the outer “bound electrons.” The first calculation of an extra-electron resonance in electron scattering was made by Massey and Moisewitsch (1954). They calculated a ’S resonance just above the 13S threshold in the e--He system, which qualitively accounted for the 23S excitation observations of Maier-Leibnitz (1935). The rapid growth of the field in the last 15 years is in large part due to the development of technology, which allowed high-energy resolution electron scattering experiments (10-100 meV) to be performed (Schulz and Fox, 1957; Simpson, 1964). This development allowed the first detailed observations of an extra-electron resonance by Schulz and Fox (1957) just above the Z13S threshold in e--He scattering. Baranger and Gerjuoy (1957, 1958)fitted the observation of Schulz and Fox (1957) with a single-channel Breit-Wigner formula and postulated the existence of another extra-electron resonance in the e--He system below the first excitation threshold of the target. The existence of extra-electron resonances in elastic e--H scattering at energies slightly below the first target excitation threshold was first indicated by the strong-coupling calculations of Smith et al. (1962). However, the closecoupling approximation used by Burke and Schey (1962)for the eC-H system, was able to resolve the anomalous increases in the ‘S and 3Pscattering phase shifts in the calculation of Smith et al. (1962)and thus clearly define a narrow resonance at about 0.6eV below the first excitation threshold of the target
ELECTRON ATOM AND MOLECULE SCATTERING
5
system. The closed-channel extra-electron resonance in elastic scattering of the e--He system was first observed by Schulz (1963) at about 0.5 eV below the first excitation threshold of the target system. Schulz (1964a) was also the first to observe a closed-channel extra-electron resonance in the e--H system at about 0.6 eV below the first excitation threshold of this target.
11. Theoretical Considerations This section begins with a mathematical discussion of the single-channel Schrodinger equation, which leads to the Siegert definition of an isolated resonance. The discussion of the multichannel problem is then outlined and various computational methods based on a partial wave expansion of the wavefunction are discussed. A thorough quantum-mechanical treatment of formal scattering theory has been given by Newton (1966) and the theory relevent to the problem of electron-atom collisions has been given by Geltman (1969),Bransden (1970), Burke (1972), and Nesbet (1977). See also Mott and Massey (1965), and Massey et al. (1969). Using the analytic properties of the S-matrix first introduced by Wheeler (1937), Heisenberg (1943) and M6ller (1945) have shown that if one knows the analytic and unitarity properties of the S-matrix and if completeness can be assumed, knowledge of the S-matrix allows the prediction of all observable quantities. With this in mind, we can use the definition of a resonance given by Siegert (1939).The Siegert definition says that a resonance is a pole in the S-matrix located at an energy such that the Schrodinger equation has a solution with outgoing waves in all channels. These poles occur at complex energies k Z , with the resonance width determined by the imaginary part of k2. This means that the proximity of the pole to the real axis determines how well the corresponding resonance can be observed. A resonance may be sufficiently broad (sufficiently far from the real axis) so that it cannot be distinguishedfrom the nonresonant scattering. Alternatively, a resonance may be sufficiently narrow (sufficiently close to the real axis) so that it cannot be detected with experimentally available energy resolution. To introduce the Siegert definition of an isolated resonance, we begin with the single-channel Schrodinger equation, and formulate the partial wave cross section near an isolated resonance. This problem has already been discussed by many authors including, Hu (1948), Humbelt (1952), Humbelt and Rosenfeld (1961), Ross and Shaw (1961), Dalitz (1963), Martin (1964), and Burke (1968). It is outlined here again because it is easy to do and it will help to put the rest of the discussion in proper context.
D.E. Golden
6
The radial equation for the Ith partial wave in the single-channel case is given by
d2 dr2
l(1
+ 1) + k2 - V ( r ) u,(k,r)= 0
1
r2
(3)
We assume the potential has a finite range, so that r >a
V ( r )= 0,
(4)
In the single-channel case, the S-matrix is a number S,(k), which is defined from the solution of Eq. (3) so that it satisfies the boundary conditions
u , ( k , ~= ) 0,
u,(k,r)
Y
r-
,-Ikr
cc
- ep'rrlSf(k)elkr
(5)
It is easy to demonstrate that S,(k) is unitary by using the relations
S,(k) = s:( - k*)
S,(k) = s; y k ) ,
(6)
in Eq. (3). The phase shift 6,(k) is defined by exp(2i6,(k)] = S,(k)
(7)
For real k, S,(k) is real. Therefore, if S,(k) has a pole at k, it must also have a pole at - k*, and zeros at k* and - k. At a pole of S,(k), the wavefunction given by Eq. (5) has an asymptotic form u,(k,r)
K
elkr
r+n
(8)
which is the Siegert definition of a resonance state. For negative k2, write k = i~ with K real and positive. In this case, Eq. (8) becomes the asymptotic form for a bound state UI K r-
eCKr
(9)
I
For an isolated pole in the S-matrix close to the real k axis, the energy position E , is defined by
k: = E , - +ir
(10)
Using the definition given by Eq. (10) and the unitarity conditions given by Eq. (6), there will be a zero in the S-matrix at E, + )ir. If there are no other singularities in S close by, for E E , ,
-
S,(k) = exp(2i6,)
E - E , - iir E - E , + +ir
where 6, is the slowly varying background phase shift. Thus, E , is the resonance energy and r the resonance width. The resonant contribution to the
7
ELECTRON ATOM AND MOLECULE SCATTERING
total phase Sf in Eq. (11) is given by The partial wave cross section in the neighborhood is given by
where F = (E - E , ) / i r and q = -cot 6,. If the background is taken to be zero in Eq. (13), one obtains the one level Breit-Wigner formula (Breit and Wigner, 1936). In most cases, Eq. (13) provides an adequate description of a resonance. The single-channel Schrodinger equation can be generalized to a Schrodinger equation representing a finite number of channels N with two outgoing particles in each channel. This, of course, does not allow the consideration of processes such as ionization. Nevertheless, in this case, one obtains N coupled radial equations instead of one radial equation :
The potential matrix is symmetric and in general there are N independent solutions to Eq. (14). The resulting S-matrix Sij, is now an N x N matrix such that each of its elements is a function of the momenta k,, . . . ,kN. As in the single-channel case, the S-matrix is defined from the boundary conditions on the wavefunction : Uij(0) = 0
uij(r)r +ccm k;'/'{exp[
-i(kir
-
1
lin)]Sij - enp[i(kir -
(15) l i n ) ] S i j } (16)
If the energy is insufficient to excite all of the channels included in the Schrodinger equation, the solution will have increasing exponentials in the closed channels, which is not acceptable physically. To get around this problem, an open-channel S-matrix is introduced from the boundary conditions: Eq. (15) runs from i = 1 , . . . , N, i = 1 , . . . , N , ; Eq. (16) runs from i = 1, . . . , N,, j = 1, . . . , N , ; uij(r) oc N i jexp( - Icir) r+
m
(17)
( K C = - i ~ i ) i = N , + I,..., N , j = 1, . . . , N, (18) for the N , open channels. The problem in electron atom or molecule scattering is to solve, by some approximation, the Schrodinger equation for which the potential is
8
D.E. Golden
completely known but cannot be solved exactly. A quantitative theory should include such effects as electron exchange, target polarizability, and electron correlation. The close-coupling approximation was introduced by Massey and Mohr (1932) and has been shown by Feshbach (1958,1962,1964) to naturally lead to a description of resonances. This method of calculation has produced a description of resonances in many systems. The close-coupling formalism and the related computational techniques are discussed in detail by Burke (1965, 1968), Burke and Seaton (1971), Smith (1971),and Seaton (1973). The application to electron excitation has been reviewed by Rudge (1973) and the specific application to electron excitation of atomic ions has been reviewed by Seaton (1975). In the close-coupling approximation, the overall wavefunction is expanded in terms of the incoming particle plus the target in terms of eigenstates of the target Hamiltonian, which are assumed known. The infinite series of target states is truncated (to make the calculation possible) and only those target states close to the impact energy are retained. Then, there are M unknown coefficients in the expansion. Thus instead of solving the Schrodinger equations, one must solve an M-channel problem for the radial motion. There will be N channels that are open and for these the coefficients will oscillate as t-+ 00. There will also be M - N closed channels, and for these the coefficients go to zero as r 4 co. The coupled equations have been discussed by the above authors, and will not be discussed here. However, it may sometimes be more convenient not to use the target states as the expansion for the wavefunction (Gailitis, 1963, 1964).For example, a few closed-channel terms from the close-coupling expansion have been shown by Burke (1963) in certain circumstances to be slowly convergent, and for the two-electron case Burke and Taylor (1966) introduced a set of basis functions that represent the electron much better than a few-term close-coupling expansion. When the close-coupling technique was used by Burke et al. (1969b) to solve the e--He scattering problem using the ground state and all n = 2 states as target states, the equations were sufficiently complex so that the exchange terms had to be simplified in order to numerically solve the equations. Because of the numerical problems associated with doing detailed close-coupling calculations with many target states, Burke et al. (1971) introduced the R-matrix method for electron-atom collision problems. The method was originally developed in nuclear physics by Wigner and Eisenbud (1947) and Lane and Thomas (1958). For a complete discussion of the method applied to electron scattering see Burke and Robb (1975). In the R-matrix method, both target states and pseudostates are written as linear sums of Slater orbitals. Then the wavefunction is expanded in orthogonal orbitals, which satisfy logarithmic boundary conditions on a spherical surface. The sphere is taken of sufficient size so that exchange between
ELECTRON ATOM AND MOLECULE SCATTERING
9
scattered and bound electrons need only be considered inside the sphere, and the long-range potentials only need be considered outside the sphere. This procedure allows the Hamiltonian to be diagonalized inside the sphere and the appropriate cross section calculated as an asymptotic problem. The approach allows a great deal of flexibility in the choice of target states, but the main advantage is the great saving in computer time. This means that approximations for numerically solving the equations are not necessary. The matrix variational method is an application to electron scattering of techniques used to study bound states for complex atoms. The method has been reviewed by Harris and Michels (1971), Truhlar (1974), and Nesbet (1973, 1975). For a complete discussion of the method see Nesbet (1977). This method was first used by Schwartz (1961) to do precise calculations for e--H scattering. In this method, a trial wavefunction of proper symmetry is written in terms of a few target eigenstates and a finite number of pseudostates. The trial function can be adjusted to ensure proper dipole polarizabilities and other properties. The wavefunction is then folded into an effective matrix optical potential, which acts in the channel orbital space. The matrix equations are solved by varying the coefficients of the basis in any of the standard variation methods (see Kohn, 1948). The equations to be solved can be described as the variational equivalent of a hierarchy of n-electron continuum Bethe-Goldstone equations (Mittleman, 1966; Nesbet, 1967). The calculations are usually carried out in the first order of the hierarchy, which is equivalent to solving two-electron continuum equations for electron pairs consisting of the electron projectile and each of the target valence electrons. In the general case, all N + 1 electron configurations are included in the basis. The approximation includes static-exchange, dynamic effects of target multipole polarizability, and electron pair correlation of negativeion states, but pair correlation in the target atom has not generally been included. The work of Knowles and McDowell (1973) has extended the work of Pu and Chang (1966) to apply many-body theory to evaluate the optical potential with a Hartree-Fock basis set. A similar calculation using a different many body formulation has been performed by Yarlagadda et al. (1973).Finally, the complete polarized orbital calculation including exchange polarizations has been done for helium by Duxler et al. (1971). In the case of e--H scattering, the wavefunctions of the target states are completely known, so that one may find the phase shifts for the scattering problem to any degree of accuracy required. In the case of more complicated targets, the degree of accuracy to which phase shifts can be calculated has not been specified by calculations. In the future this aspect of theory will hopefully receive more attention. Finally, in the case of molecular targets, additional approximations have thus far been necessary in order to make the calculations tractable.
10
D . E. Golden
111. Experimental Considerations This section contains a review of some of the experimental techniques used to study resonances. It has already been stated that the experimental study of resonances was to a very large extent made possibly by the advances in technology, which allowed the preparation of and the energy analysis of monoenergetic electron beams (energy width 10-100rneV). In order to prepare and analyze such beams, a variety of dispersing elements have been used. The properties of a few of the dispersing elements are briefly described here. However, the major emphasis of this section is on the question of the function and significance of the various techniques. There were many other advances in technology necessary before resonances could be carefully studied. These include the technology of ultrahigh vacuums, low-current measurement, differential pumping, atomic and molecular beam sources and detectors, sensitive pressure measurement, high current-density electron sources, sophisticated electron optics, particle counting, high-speed digital signal processing, and low-noise electronics. However, for lack of space, these developments will not be discussed here. For a discussion of these techniques, the reader is referred to Hasted (1964),McDaniel(1964), Bederson (1968),Massey e f al. (1969), Bederson and Kieffer (1971), Golden et al. (1971), Schulz (1973a), Andrick (1973), Massey (1976),and the original experimental papers. Resonances have been investigated by studying both total, and differential scattering cross sections. The total cross section has usually been studied in electron transmission experiments and this subject is discussed in a separate section below. In differential cross section measurements, one may study either the scattered electrons, the scattered target atoms/molecules/ions, or any of the final products of the interaction. However, most of the work has used a dispersing element to form a more or less monoenergetic beam of electrons that has been scattered from a target gas contained in a cell or a target gas beam. The scattered electrons have been studied as a function of scattering angle by passing them through a second dispersing element in order to analyze their energy distribution. We begin the discussion with a description of some of the more popular dispersing elements and a discussion of crossed electron beam atom or molecular beam experiments that study the scattered electrons are discussed in a separate section. The first high-resolution (100-200 meV) observation of an extra-electron resonance was made by Schulz and Fox (1957).They introduced the retarding potential difference (RPD)technique to obtain the energy resolution in their electron beam. The principle of this techniques is explained with the aid of Fig. la. For further details see, for example, Golden et al. (1972).Consider a parallel beam of electrons of mean energy E with a distribution of energies
-
ELECTRON ATOM AND MOLECULE SCATTERING
YARIPIBLE HElGHT SQUARE WAVE GENERATOR
11
SENSITIVE PHASE DETECTOR
FIG. 1. (a) Schematic arrangement of the electron transmission experiment by Golden et al. (1972) and the electron distribution function obtained by subtraction (clear portion of the plot). (b) The einzel retarding field analyzer of Golden and Zecca (1971).
such as that shown in Fig. la. Suppose that a potential -V is applied to the grid shown such that el V (= E. Electrons with energies less than el VI will be repelled by the grid, while those with energies greater than elV1 will be transmitted. In Fig. l a the distribution of electrons transmitted by the grid is represented by the solid curve. If the voltage on the retarding grid is adjusted to have two different states ( E + E , and E + E 2 ) at different times as shown in Fig. la, the distribution function of transmitted electrons in the two different cases will be composed of all the electrons to the right of the vertical lines at E + E l and E + E , , respectively. If the effect for state E E , is subtracted from that for state E + E,, the net effect is characteristic of the distribution between E + E , and E + E , . However, it should be noted that the noise is characteristic of the distribution containing the high-energy tail. Thus, while the difference signal is composed of a narrow distribution of electron energies, the associated noise must be calculated on the basis of essentially the full distribution. The important assumption in using the RPD technique is that all electrons cross the retarding plane perpendicular to it. The one-dimensional retarding field will not affect the transverse velocity components. If the electron beam is aligned by using an axial magnetic field, there will be transverse velocity components due to the spread in angles of electrons leaving the cathode. However, it is possible, with great care, to electrostatically force essentially all the electrons to cross
+
12
D . E. Golden
the retarding plane perpendicular to it. A very high-energy resolution electron beam (-8meV) has been obtained by Golden and Zecca (1971) in this way. The technique used by Golden and Zecca (1971) to force the electrons to cross the retarding plane perpendicular to it is illustrated in Fig. lb. If one brings a beam of electrons incident from the left to a focus at voltage I/ in the left-hand aperture of Fig. l b while the retarding voltage I/, is applied to the center element, the electrons that pass through the second aperature at voltage I/ must have crossed the retarding plane perpendicular to it if the apertures are equidistant from the retarding grid, according to symmetry arguments. This argument is only valid for apertures of infinitesimal dimensions, and so in the case of Golden and Zecca (1971) a limiting resolution of about 8 meV was achieved. The technique of energy modulation, first introduced by Morrison (1954), can be used to energy differentiate the signal in transmission experiments. This technique is helpful with regard to studying structure in transmission experiments since it is not sensitive to slowly varying effects such as variation of cross section with energy and variation of beam current with energy (due to space charge and focusing effects). Various forms of this technique have been used by Golden (1971), Sanche and Schulz (1972a), and Golden et a/. (1974b), for example, to study resonances in electron transmission experiments. The earliest description of the use of electrostatic deflectors as a method of energy selection has been provided by Hughes and Rojansky (1929) for concentric cylindrical deflectors and Purcell (1 938) for concentric hemispherical deflectors. Electrostatic deflectors now provide the most commonly used method of energy selection. The cylindrical deflector provides firstorder focusing in one plane perpendicular to the beam direction, while the hemispherical deflector provides first-order focusing in both planes. Hence the hemispherical deflector provides a more intense well-collimated beam of electrons. However, only relatively recently was it shown that the hemispherical deflector and its associated lens system can be made to approach its optimum performance so that it could be used in electron scattering experiments (Simpson, 1964; Kuyatt and Simpson, 1967).The hemispherical electrostatic monochromator-energy analyzer of Kuyatt and Simpson (1967) is shown in Fig. 2. This system is usually operated with both the monochromator and energy analyzer at the same energy resolution. The problems associated with the focusing of charged particles have been discussed by Wollnik (1967), and the general problems associated with designing a hemispherical electron energy selecting system have been discussed by Read et a/. (1974). Brunt et a[. (1977) have numerically calculated the potential distribution within conductors of axial rotation symmetry and designed a set of correcting hoops and lens deflector electrodes in order to obtain a
ELECTRON ATOM AND MOLECULE SCATTERING
13
ELEC TAON SPECTROMETER (-
30. l o -t90'
SCAN)
1
'ROTATED
90-
FIG.2. Plan view of the hemispherical monochromator-energy analyzer of Kuyatt and Simpson (1967).
hemispherical deflector energy selector capable of observing electron scattering features of 12 meV full width at half-maximum intensity. This system is shown schematically in Fig. 3 and does not utilize an electron energy analyzer. The cylindrical deflector is simpler to construct and its electrodes are easily made from grids. This allows stray electrons to pass through the grids and be collected on positively biased additional cylindrical electrodes external to the grids. In this way, the space charge field at the electron path is reduced to some extent. The development of lens systems especially for cylindrical deflectors has not received as much attention as has that of lens systems for hemispherical deflectors. In many cases the tube lenses (ideally suited for hemispherical deflectors) have been used in conjuction with cylindrical deflectors. A schematic diagram of the cylindrical monochromatic double cylindrical energy analyzer described by Andrick and Bitsch (1975) is shown in Fig. 4. The electron gun delivers a beam current of about 10- A of width about 50 meV for crossed electron-atom and/or -molecule scattering experiments. Recently a monoenergetic source (energy width 1-2 meV) without a dispersing element has been described by Gallagher and York (1974). The electron beam from this source is produced by near threshold photoionization of a metastably excited barium beam in a field-free region inside the cavity of a CW UV laser. A beam current of 10-12-10-13A has been produced in this fashion, which so far is less than the 10- l 1 A predicted for the process. This looks like a very promising source of monoenergetic electrons for future experiments and so bears mentioning.
-
D.E . Golden
14
2 18.6
12
168 72 GL1 GE
Interact ion
'
5cm
I
FIG. 3 . Approximate scale drawing of the electron monochromator of Read The target beam enters from below perpendicular to the drawing.
et al. (1974).
Electron
FIG.4. The 327' cylindrical monochromator, double 127' cylindrical energy analyzer of Andrick and Bitsch (1975). (Copyright by The Institute of Physics. Reprinted with permission.)
15
ELECTRON ATOM AND MOLECULE SCATTERING
A. THEFUNCTIONS OF MONOCHROMATORS A N D ENERGY ANALYZERS This section contains a discussion of experiments that study either scattered or transmitted electrons since this is the most popular way to make highenergy resolution measurements. In such experiments, one has an electron source and an electron detector each of which has an energy resolution. One can ask what is the function of each of these resolutions. In the usual arrangement. a dispersing element is used. A dispersing element will transmit an energy width AE, which is proportional in first order to the energy E in the dispersing element. The constant of proportionality depends on geometrical quantities. However, because of this proportionality, one is usually forced to make the pass energy relatively low (of the order of 10eV or less) and thus encounters the problems associated with space charge. The space-charge-limited current within a given solid angle is proportional to the 312 power of the energy, and the current issuing from a dispersing element in the usual case will depend on the 512 power of the energy width passed. This 512 dependence is made up of a 312 power dependence due to space change and a 212 power dependence from the width of the slice taken by the monochromator from the cathode distribution. In the case of the hemispherical monochromator described by Kuyatt and Simpson (1967) the output current I , is given by I,
=9 x
10-5 A E : ~ A
(19)
where AE,,, is the full energy width at half-maximum current in electron volts. In the general case, an electron scattering experiment is composed of an electron beam of width A E t , an interaction region, and an electron detector that defines the electron energy detected to within some width A E , . The current through the electron detector I, is thus given by = kAE:l2G AE21AEl,
I.={
=
k AE;l2G,
AE2/AEl < 1 AE21AEI > 3
(20)
where the function G is introduced to represent the attenuation due to scattering, the size of the interaction volume, the gas density, the solid angle subtended by the detector, etc. It has been assumed in Eq. (20) that the current reaching the energy analyzer is sufficiently small so that space charge considerations do not apply within the energy analyzer itself. In much of the past work, which has used both a monochromator and an energy analyzer, it has been assumed that A E , should be equal to A E , . In fact, in many experiments, in order to maximize the output signal, A E , should not be set equal to A E 2 . Rather, the energy width A E , necessary to do a particular
D . E. Golden
16
experiment should be defined by the energy separation of the atomic or molecular states that need to be separated by the energy analyzer in the experiment. Then the energy width AE, necessary in the same experiment may be defined by the energy resolution that is to be used in the experiment. Thus, in certain cases one need not limit the output current of the monochromator by demanding high-energy resolution of it, although one may still perform high-energy resolution scattering measurements. Furthermore, when further discriminants are added such as in an electron-photon coincidence measurement, the above conditions defining AE, may be further relaxed in certain cases. If, in addition, energy discrimination is added to the photon detector, even further relaxation of the restriction on AE, may be possible. Let us consider two states separated by AE, and excited by an electron beam. The electrons arriving at the energy analyzer will be composed of two distributions, which are separated by AE,. If the resolution of the electron beam is AE, and that of the energy analyzer is adjusted to select a slice of width AE, from the distribution, where AE, AEz and t , is the counting time used with AE, = AE,. For an energy resolution of 50meV, in order to make R = 1, t , = 15t,,. At 1meV resolution, to make R = 1, t , N 5.3 x 103t, . These factors are considerable and at extremely high-energy resolution represent the difference between being able and not being able to do an experiment. Looking at it in a slightly different way, the signal achieved with AE, = 0.3 eV and AE, = 0.001 eV is comparable to that achieved with AE, = AE, = 0.05eV. That is, using this method one can do an elastic scattering experiment with 1meV resolution in approximately the same time as the same scattering experiment at 50meV resolution using the usual
,
17
ELECTRON ATOM AND MOLECULE SCATTERING
method. Using the method suggested here at 1 meV energy resolution (that is, AE, = lO-jeV and A E , = 3 x 10-'eV) for an elastic cross section of 10-'6cm2 with a solid angle dQ and a density times path length of 1013/cm2,the signal to noise ratio achieved is
-
-
SIN (104ft1,)1'2 (22) If we demand SIN 10 at each energy and angle used with f 0.8, we need 1/80 sec/point to do an elastic scattering experiment at 1meV energy resolution. So an elastic scattering experiment at extremely high-energy resolution is possible within the present technology. As a further example we consider the separation of the excitation of the n = 2 levels in helium. In this case the resolution of the monochromator is defined by the closest spacing of the n = 2 levels, which is 254 meV. Thus we take AEl Y 0.25eV, and it then takes about 11 times longer to achieve the same signal-to-noise ratio at 50 meV resolution by the usual method than by using the method suggested here to measure these excitation cross sections. For a total excitation cross section cm2 with a solid angle dQ lop4 and a density times path length lOl3/crn2, if we demand a signal to noise ratio of 10 at each energy and angle studied with f = 0.8, this means the difference between about 4.4 sec and 0.4 sec counting timelpoint. At 1meV resolution under the same conditions, this means the difference between 7 x 10" and 18 sec/point. When coincidence measurements are done between electrons and photons, a further reduction in signal will be suffered due to the addition of another solid angle factor. The above method of doing the experiment would in this case help to compensate for such loss and therefore make some experiments possible that were either not possible or marginally possible. Suppose we consider the case we have already considered above, of the excitation of the a = 2 levels of helium. The 2'P level is the only level that can radiate to the ground state. If the photon detector detects only the 2lP 3 1'S transition, and cascade can be eliminated by time-resolved spectroscopy, we could open up the resolution of the monochromator to its full extent while keeping the energy analyzer at the resolution desired. In experiments where the full energy width of the source can be utilized, one must consider the possibility of increasing the electron beam signal further by completely eliminating the monochromator. For example, if we write the space-charge-limited current
-
-
-
-
-
i
-
I = 3.85 x
10-5~3/2(d/1)2
A
(23)
where E is in eV and d/l is the ratio of beam diameter to length in a drift space defining the beam. For high electron energies Eq. (23) can be made to yield significantly more beam current than predicted by Eq. (19).
18
D.E. Golden
B. TRANSMISSION EXPERIMENTS The oldest method used to make quantitative electron-atom and -molecule total scattering cross section measurements is the transmission method of Ramsauer (1921). This method has been refined by Golden and Bandel (1965a) by utilizing differential pumping, high electron energy resolution (20-100 meV), and ultrapure target gases, and used to study resonance effects. The basis of measurement in all transmission experiments is the attenuation of a more or less monoenergetic electron beam due to traversing a given path length through the gas studied. The apparatus of Golden and Bandel (1965a) is shown schematically in Fig. 5. The electrons are momentum selected by a combination of the three slits S1, S z , and S, and a uniform magnetic field perpendicular to the plane of the diagram.
PUMP OUT I
FIG.5. Schematic arrangement of the modified Ramsauer transmission apparatus of Golden and Bandel (1965a).
The Ramsauer experiment is an example of a transmission experiment. The schematic arrangement shown in Fig. 6 is for the generalized transmission experiment. The current to the collector I , (transmitted current) is given by where I , , is that part of the current entering the scattering chamber at electron beam energy E at zero gas density and reaching the collector, o,(E) is the total scattering cross section at energy E, and L is the path length of
ELECTRON ATOM AND MOLECULE SCATTERING
19
FIG.6 . Schematic arrangement of the generalized transmission experiment (from Golden, 1973).
the electron beam through the gas contained in the interaction region (scattering chamber) at density n. It should be noted that for onL > 1, multiple scattering begins to become significant. Therefore making onL >> 1 should be avoided when making quantitative measurements. With this in mind, a measurement of I , as a function n at constant E gives a measurement of o,(E). Similarly, the scattered current I s , assumed to be collected by the scattering chamber on its inside surfaces, is given by
',(El
= Zsn(E)
+ I c n ( E ) { 1 - ~ X P [- ot(E)nL]}
(25)
where Isn(E)is that part of the current reaching the inside surfaces of the interaction region at energy E which would reach them in the absence of scattering. Then by adding Eqs. (24) and (25),
Ic(E) + Is(E) = Icn(E)
+ Isn(E) = In(E)
(26)
Dividing by Eq. (24) yields
Equation (27) is the basis for quantitative transmission experiments regardless of the type of energy selection used. This equation accounts for fluctuations in I , and for variations of Inwith E for which the ratio Zn/Ic, remains a constant. However, it should be noted that Eq. (27) has implicitly assumed that scattering events at all angles are detected and contribute to the attenuation of the transmitted current. In many transmission experiments used to study resonances, the scattered current is not detected and Eq. (24) has been used as the basis of the measurement. When Eq. (24) is used, the transmitted current or the derivative of the transmitted current with respect to electron energy is measured as a function of electron energy at constant gas density. The sensitivity of low-energy electron transmission experiments has been discussed, for example, by Golden (1973). He has cautioned that when Eq. (24) is used, measured signals can fluctuate for reasons other than variations of o(E)or do(E)/dE with E. For example, fluctuations or variations of either I,, or n or both due to external effects are not accounted for. The most serious problem is possibly that variations in I,, due to space charge
D.E. Golden
20
and electrostatic lens effects, although possibly reproducible, are not accounted for by Eq. (24). See also Burrow and Schulz (1969a) and Spence et al. (1972). Let us now ask what fraction of the scattering events that do take place in the interaction region of a transmission experiment can be detected. A different range of scattering angles will be detected at different positions in the interaction region. The geometry of the interaction region, the presence or absence of an external magnetic field, and the potentials on the interaction region can effect this range. In addition, the angular form of the differential scattering cross section and therefore the electron energy considered will also have an effect. In order to obtain a quantitative estimate of these effects we introduce a function F(8, E ) , which will represent the fraction of scattering events in the interaction region that contribute to the attenuation of the transmitted current. This type of analysis has been given by Golden and Bandel (1965a) for a Ramsauer-type transmission experiment. Effects of finite energy resolution and spatial variation of electron current density have been discussed by Bederson and Kieffer (1971) and will not be discussed here. We consider the exponent in Eq. (24) to be g(E), which can be written
where do/dQ is the differential scattering cross section and dQ is the element of solid angle. The sensitivity to scattering y(E) can be defined as v ( E ) = d E ) / o , ( E )nl
(29)
The calculation of F(8, E ) is carried out for the cylindrical interaction region geometry shown in Fig. 7. The length of the interaction region is taken to be L bounded by entrance and exit apertures of diameter S. All scattering events at the angle 6 shown on the figure in the volume labeled 0 will exit from the interaction region and therefore will not be detected as scattering events. All scattering events at the angle 8 in the volume labeled 1 on the figure will be detected. Approximately half of the scattering events at the angle 8 in the volumes labeled 1/2 on the figure will be detected. If a scattering event is equally likely to occur any place within the volume nS2L/4, the fraction of scattering events detected is simply given from the various volumes multiplied by the appropriate fractional area divided by the total volume : 1--
S
2L tan 8’
S tan8 3 2L S tan8 < 2L
ELECTRON ATOM AND MOLECULE SCATTERING
21
L FIG. 7. Cylindrical interaction region geometry of thc generalized transmission experiment (from Golden, 1973).
In the absence of any other effects, the function F(8,E) is simply given by
f(8)and this function is plotted for various values of the parameter S/2L in Fig. 8 for values of 0 d 9 < 71/2. The curve for n/2 < 8 d .n is obtained by reflecting the curve shown about 8 = n/2. Since the integral g ( E ) contains a factor sin 8, the lack of detection for angles close to zero (or 71) is not serious for reasonable variations in do/dQ. The curves shown are similar to those given by Golden and Bandel (1965a). Now we consider what happens in an axial magnetic field. In such a case, the scattered electrons will perform helical motion about the direction of the magnetic field. The field helps to align the incident beam, but as it is better aligned by making the field stronger, the more scattered electrons are I
I
-
I
I
I
I
I
I
0.005, 0.01
1.0 0.8
F (e) 0.6 0.4
0.2 0.0
10
30
50 (DEGREES)
70
FIG.8. Fraction of scattering events leading to attenuation of the transmitted current as a function of scattering angle without a magnetic field ( B = 0) for S/2L = 0.005, 0.01, 0.02, 0.04 (from Golden, 1973).
22
D.E. Golden
trapped in the interaction region and thus the more q ( E )decreases. In order to estimate the effect of the magnetic field, it will be assumed for simplicity that the electrons enter the interaction region with their velocity vectors along the axis of the magnetic field (the symmetry axis in Fig. 6). If this is not the case, the electron path length in the interaction region will be a function of the ratio of transverse to longitudinal velocity components. This will introduce a distribution of path lengths and make more serious the effects discussed below. For the simplified case considered here, for elastic scattering to an angle 0 at velocity u, a transverse velocity u, is achieved such that u, = usin 0. This produces helical motion of radius r given by Y = (2mE’’2/e2B2)sin 0 = r,
sin 0
(31)
where m is the electron mass, e the electron charge, B the magnetic field strength, and E the electron energy, all in mks units, and 0 is the electron scattering angle. For a 20-eV electron in a magnetic field of IOOG, ro is 1.5mm. The process of elastic scattering in the magnetic field further reduces the fraction of scattering events by a factor f,(0), which has been crudely underestimated by Golden (1973) to be
P ,
J;(& E ) = 3 1,
r < S/2 r
> S/2
(32)
When r = S/2, some electrons scattered to angles near zero (or n) will still reach the collector (or go out the entrance to the interaction region) due to the helical motion. Thus Eq. (32) underestimates the small and large angle effect.Nevertheless this effect makes the fraction of scattering events detected a function of E as well as 8. Furthermore this effect is more important for inelastically scattered electrons. There is another effect that will be considered here that also leads to the loss of sensitivity to scattering. As the scattering angle increases toward n/2, a scattered electron spends a longer and longer time in the interaction region following a helical path of longer and longer length. As the path length becomes longer, the chances of being scattered many times becomes extremely large. To estimate this effect, we assume that a,nL = 1. If an electron travels along the magnetic field with velocity u and is scattered through an angle 8, on the average it will spend a time t in the interaction region after being scattered, where t = L/2v cos 8
(33)
In the time t it will travel a distance Do in a circle, where Do = L tan 0/2
(34)
ELECTRON ATOM AND MOLECULE SCATTERING
23
The average path length is independent of the magnetic field strength, provided that the diameter of the interaction region is sufficiently large compared to the electron beam diameter so that the electron does not hit a wall. The total distance traveled by an electron on the average D is given by
D
= +L(1
+ tang)
(35)
For 2L > D > L , approximately D / L of the electrons scattered once will be scattered twice, etc. We assume that half of the electrons scattered twice are thrown back along the axis and therefore not detected as being scattered. The electrons scattered farther off the axis in the second collision again have a probability of 1/2 of being scattered back into the beam in a third collision, etc. Generalizing this crude approximation to another fraction fi(8) and decreasing the scattering events detected,
The function fi(8) takes account of the fact that the path length of an electron approaches infinity as its scattering angle in a magnetic field approaches n/2. We then write for the magnetic field case
F(& E ) = f(8lft@,W 2 ( Q ,
(37)
The function F(9,E) is plotted as a function of 8 for 20eV electrons, for S / 2 L = 0.005, and for B = 100 and 250G in Fig. 9. This effect leads to a loss
1.0
-
0.6
-
F(B)
c E = l O O GAUSS
0.6 tE
= 2 5 0 GAUSS
0.4-
0.2-
0.0
FIG.9. Fraction of scattering events leading t o attenuation of transmitted current as a function of scattering angle for S!2L = 0.005, and for axial magnetic fields of 100 and 250 G (from Golden, 1973).
24
D.E. Golden
of sensitivity q of about a factor of 2 for angular distributions of Legendre form P,(cos 0) through P,(cos 0) for 20-eV elastically scattered electrons in a 100-G field. One might hope to recover some of the loss in sensitivity by applying a retarding potential to electrons leaving the interaction region. It is certainly true that the sensitivity to inelastic scattering is increased in this way. However, if an electron of energy Eiis elastically scattered to an angle 8 with respect to the incident direction and is decelerated to energy E , as it emerges from the interaction region, those for which sin28 > Ef/Ei do not reach the collector (or return to the cathode). Rather they are reflected back into the interaction region due to the retarding fields. If a magnetic field is also present, it serves to concentrate the reflected electrons back into the interaction region. In such a case, the interaction region becomes a multiple scattering trap for elastically scattered electrons and the effect approaches the case of scattering at 90" in the magnetic field discussed above. That is, as the trap becomes better, the sensitivity to elastic scattering approaches zero. While in certain circumstances axial magnetic fields and traps can be used, it is cautioned that one must be very careful in interpreting the results of such measurements. C. CROSSED-BEAM EXPERIMENTS
In crossed-electron-beam, neutral-target-beam experiments, important considerations are the ratio of target beam density to background gas density and the target beam profile. In experiments that study inelastically scattered electrons, these considerations can be more important than in experiments that study elastically scattered electrons. Since elastic-scattering cross sections are considerably larger than inelastic-scattering cross sections, unless precautions are taken, inelastic scattering in the target beam may be followed by elastic scattering from the background gas. Let us consider the effect of the simple distribution of target gas density shown in Fig. 10a. When the electron detector is at 742 with respect to the electron beam direction, (as shown in Fig. lob) the electron detector may be constructed so as to view only the region of enhanced target gas density. However, when the electron detector is placed at very small (or very large) angles with respect to the incident beam direction, (as shown in Fig. lob) the electron detector views a larger interaction length, which may be a complicated function of scattering angle studied. When one designs an apparatus to be able to study very small (and very large) scattering angles, it is tempting to move the detector further from the interaction region. However, this makes the situation worse. The effect of scattering from the background gas for electrons not scattered from the target beam results in a measurement of too
ELECTRON ATOM AND MOLECULE SCATTERING
25
much scattering at the smaller (or larger) scattering angles but from places other than the scattering center. This effect may be compensated for by a subtraction as was done, for example, by Andrick and Bitsch (1975).One may turn the target beam off and flood the chamber to the same background density as with the beam on, measure the angular distribution of scattered electrons, and subtract it from the measured angular distribution with the target beam on. This procedure contains several problems, which can in some cases be overcome with great care. First of all, if the background effect becomes comparable to the effect due to the target beam effect, obviously the result is statistically meaningless. Second, the effect due to multiple scattering cannot be compensated for by substraction. Finally, at high target gas flow rates, the effect of turning off the target gas beam may change the composition of the background gas. Thus the subtraction may introduce an error due to change in species. To illustrate some of these effects we consider the equations involved in a differential scattering experiment with a target gas beam from the work of Sutcliffe et al. (1978). When the target gas beam is on, we may write the scattered-electron count rate 2ile as
.
I,
N, = - p e
E
do -Je(O,) ‘dQ
+ Ti,
where I,/e is the number of incident electrons/sec in the electron beam, po the background gas density (which is assumed to be composed of the same species as the target gas), E, the efficiency of the electron detector, do/d!2 the particular cross section that is being detected, Ti, the count rate due to electronic noise, and
where p(z) is the spatially dependent density of the target particles, d o , the solid angle viewed by the electron detector, dz the element of path length of the electron beam through the interaction region viewed by the electron detector, etc. When the target gas beam is off we may write the electron count rate as
.
NL
where
I, e
=-PO€,
do
-
dQ
L,
26
D.E. Golden
If the target gas beam is sharply defined, that is, if it is narrower than the interaction length viewed by the detector, we may write
where pj(z) is the density in the interaction region due to the target beam jet, which sits on a constant background density po, and
J , = AQ,J LP Wd z Ij
Po
+ L,
(43)
where lj is the path length of the electron beam through the target beam jet. Since L , may be determined from the geometry (see, for example, Sutcliffe et al., 1978), integration of f i e and for the same period of time will allow a determination of the integral
if ri, may be neglected. Furthermore, the difference N , - N : may be used to eliminate the scattering from the background:
Thus relative values of do/dQ may be determined as a function of E and 8, from Eq. (45). Fluctuations in both the electron beam current and its spatial distribution as well as fluctuations in the target gas density and its spatial distribution make matters quite complicated. However, the effect of scattering from the background gas for electrons that have also been scattered from the target beam is not compensated for at all by the subtraction technique. If one is interested in using angular distributions to place absolute scales on cross sections (see Section 111, D) or in studying small structures (see Section IV) one should make the background density very small and try to improve the target beam source such that the ratio of n , to no in Fig. 10a is as large as possible. This means that target beam sources need a great deal of care in their design (see, for example, Lucas, 1972). The incident electron beam spatial distribution is another source of difficulty. In the measurement of elastic cross sections at small scattering angles one cannot separate the elastically scattered electrons from the incident beam. Even the measurement of inelastically scattered electrons may contain a component from the incident beam. In the work of Sutcliffe
ELECTRON ATOM AND MOLECULE SCATTERING
27
position
I e- beam gun at 9 0'
FIG.10. (a) Target beam density as measured by Andrick and Bitsch (1975). (b) Schematic representation of interaction region geometry (from Andrick and Bitsch, 1975).(Electron energy analyzer is on the left.) (Copyright by The Institute of Physics. Reprinted with permission.)
et al. (1978) at a scattering angle of 5", a background was observed beneath the energy loss peaks that was attributed to electrons from the wings in the angular profile of the incident electron beam. The measurements of the angular profile of the incident electron beam made by Sutcliffe et al. (1978) showed that the incident electron beam was 3.6" wide (full angular width). However, since the scattered intensity is very much lower than the incident beam intensity, a very small percentage of the incident beam current at 1.5 beam widths could be observed. For incident beams of wider angular spreads, this could be a very large effect leading to incorrect angular distributions. Furthermore, the work of Sutcliffe et al. (1978) showed that it was extremely important to measure the incident electron beam profile during the angular distribution measurements because small changes in electron gun voltages
D.E. Golden
28
could produce large changes in the electron beam profile. In fact, in the work of Sutcliffe et al. (1978) electron beams could be made with angular profiles as wide as 30". This kind of effect (if not accounted for) will alter measured elastic angular distributions such that they will have too much small angle scattering. The resulting integrated (total) cross sections will be too large. Another problem that warrants some discussion is the effect of the velocity distributions of a moving electron beam colliding with a moving target beam. Consider a beam of target particles of mass M traveling with velocity v, and interacting with a beam of electrons of mass m traveling with velocity 0, such that they intersect with angle 9. In a frame moving with the target, the velocity of the electron is given by v;2 = 0,"
f
0:
- 2v,v,
cos 9
(46)
The energy of the electron in this frame EQ is given by E i = E,
+ (m/M)E,
-
2(m/M)'~2(E,E,)''2 cos 0
(47)
In a crossed-beam experiment cos 9 = 0 and for m / M Eth
31
ELECTRON ATOM AND MOLECULE SCATTERING
FIG. 12. Total cross section for e-Na scattering from Moores and Norcross (1972). x , Experiment of Brode (1929), divided by 2; 0 ,experiment of Perel et al. (1962), normalized to the calculation of 5 eV and experiment of K. Rubin (unpublished data, 1972).
where
hk = (2mE)’”, h = fth(e)exp(-2i6O,h), and the scattering amplitude f(0) is given by
f(e) = f , h ( e )
hk2 = [ h ( E - J?&,)]’”
-k -$bk-’k2eXp(2id,th)
(51)
These equations have been used by Cvejanovic et al. (1974) to fit their measured differential elastic electron-helium scattering cross sections at 90” for energies near the 23S threshold. At 90” the P-wave phase shift does not enter [the D-wave was taken from Knowles and McDowell(1973)], and all higher phase shifts were neglected. Then b, a, and Eth are fitted to data to determine them. The measurements of Cvejanovic et al. (1974) are shown in Fig. 13 with the solid line being the best fit to the data using Eq. (50) with b, a, and Eth determined from the best fit. Once the fit has been made, one knows where 19.818eV is on the raw energy scale, and the energy scale is calibrated. This calibration procedure places the (1s, 2s’)’S resonance in e--He scattering at 19.37eV. Furthermore, the value of b leads directly to the threshold dependence of the Z3S excitation cross section
o(1’S
--f
23S) = 2nbkK’k2
The total elastic scattering cross section has been analyzed by Golden et al. (1974b) using a slightly different procedure. They have found the structure
D.E. Golden
32
L+ 19.70
lnddrm energy k V )
FIG. 13. Differential e--He elastic scattering cross section at 90" in the neighborhood of the 2's threshold at 19.818 eV, (from Cvejanovic et a[., 1974). (Copyright by The Institute of Physics. Reprinted with permission.)
ofthe S-matrix in the vicinity of the 23S threshold by continuing the analytic form of the S-matrix in the vicinity of the (1s 2s2)2 S resonance across the 23S threshold using effective range theory. This type of procedure was used, for example, by Burke (1965) to estimate the S-wave contribution to the 23S excitation cross section. The matrix element corresponding to elastic scattering is, from Eq (1l),
s0 -- e 2 i d o E - E, - i r / 2 E - E,
ir/2
where 6, is the nonresonant S-wave phase shift r = ky2K and the scattering matrix is continued across the threshold according to the prescription iK = k , . Since y 2 k is almost constant near the threshold, r N TK, where is a constant. Then the elastic scattering cross section is given by r2K2/4 ( E - E,)2 + T2K2/4
-K -r-( 2
( E - E,) sin 26, + r K sin26, ( E - E,)2 + f2K2/4
T 2 k : sin2 6 , r"k$/4 ( E - E , Tk,/2)2 ( E - E , fk,/2)'
+
+
The cusp is due to the third term in Eq. (53), which predicts a decrease in the elastic cross section above threshold and either an increase or a decrease below threshold depending on the sign of sin26,. The sine of 26, is negative in this case, so that the cusp corresponds to a negative step in the elastic cross section across the 23S threshold. The derivative of the transmitted current as a function of electron energy was fitted by Golden et al. (197413) to Eq. (53) to determine r, do, and E,. The calculated energy differentiated
33
ELECTRON ATOM AND MOLECULE SCATTERING
Electron energy (eV)
FIG.14. Calculated energy differentiated transmitted electron signal in He from 19.2 to 20 eV for S-wave scattering (from Golden el a/., 1974b).(Copyright by The Institute of Physics. Reprinted with permission.)
transmitted current using the best values of l-, do, and E, is shown in Fig. 14. Two peaks and a dip are associated with the (ls,2s2) 'S resonance, while the negative step in the elastic cross section appears in the spectrum as a peak. Table I shows a comparison between the features seen in the experimental data and the features calculated with Eq. (53) using the values of r, do, and E, determined from the best fit to the experimental data. As can be seen from the table the fit is excellent. It should be noted that the size of the peak corresponding to the negative step is only about 2% of the size of the major peak due to the (Is, 2s') 'S resonance. The calibration procedures of Cvejanovic et al. (1974) and Golden et al. (1974b) bring the position of this resonance into excellent agreement with TABLE I
POSITIONS AND
THE ~~~
PEAK HEIGHTS FOR THE z2S STATE OF HeTHRESHOLD OF THE 23S STATE OF He"
RELATIVE
~
P,
AI(Pi) AIU',)
19.825 19.825
0.014 0.015
~
Exp. Calc.
AND
Po
PI
19.34 19.34
19.43 19.42
AI(P,) AI(Po)
r
G,(deg)
E,
0.011 0.012
0.013
102
19.35
~.
a Values in electron volts. Table from Golden er a/. (197413). (Copyright by The Institute of Physics. Reprinted with permission.)
34
D . E. Golden
recent calculations for its position of 19.36eV by Temkin et al. (1972), of 19.4eV by Sinfailam and Nesbet (1972), and of 19.38eV by Ormonde and Golden (1973). The width determinations of 9meV by Cvejanovic et al. (1974) and 13 meV by Golden et al. (1974b)are also in good agreement with the widths determined by Temkin et al. (1972), Sinfailam and Nesbet (1972), and Ormonde and Golden (1973) of 14, 15, and 11.5meV, respectively. However, the spread in measured or calculated widths would indicate that the rough estimate of Simpson and Fano (1963)has not really been improved on by much more sophisticated and detailed work. 2. Absolute Cross-Section Determinations
Precise absolute total cross-section measurements have been made by Golden and Bandel (1965a) and precise absolute momentum transfer crosssection measurements have been made by Crompton et al. (1970). The principal limitation in such experiments has been the absolute measurement of pressure. The measurements of Crompton et al. are more precise principally because of the higher pressures used. It would be very useful if some cross section could be known very well so that it could be used as a standard against which all other cross-section measurements could be calibrated. However, attempts to relate total cross sections to momentum transfer cross sections or vice versa have shown that differences of about 10% exist for the simplest case of e--He scattering (see Bederson and Kieffer, 1971). From the discussion of the previous section, it would seem that the fitting of a resonance in a particular partial wave to an analytic form leads to a measurement of the nonresonant part of the phase shift in that partial wave at the resonance energy. One might therefore expect that the phase shifts of the other significant partial waves may be determined completely from the angular distribution at the resonance energy. This was first done in the case of e--He scattering by Andrick and Ehrhardt (1966) to obtain a,, 6 , , 6, at the (1s 2s') 'S resonance at 19.3eV. Gibson and Dolder (1969)improved the experimental signal to noise ratio and used a similar procedure to analyze their data to obtain the same three phase shifts. The method used by Gibson and Dolder is briefly outlined. The differential cross section is written do(Q,E)/dR = A' + B2 (54) where 1 " A =(21 + l)(COS26, - l ) P , C O S O 2k ,=o
-
c
.
m
ELECTRON ATOM AND MOLECULE SCATTERING
35
The resonance is represented by the addition of a phase shift 6,,
6,
=
-arccot2
E - E, ~
r
(55)
Gibson and Dolder calculated do/& for trial values of d o , 6,, and 6, with E, = 19.3eV at the six scattering angles studied and compared them to their measurements. The parameters were varied until the best fit was determined. Of course the procedure outlined above leads one to the conclusion that if one is at sufficiently high energy, one can simply fit observed experimental elastic angular distributions, at energies where there are no resonances, with a few partial waves and use the Born approximation to evaluate the higher order phase shifts if necessary. This procedure has been used by Andrick and Bitsch (1975) to place e--He elastic cross sections on an absolute basis. (see also Section 111,C). This procedure gives a total elastic cross section at about 19.3eV for e--He that is about 12% higher than that of Golden and Bandel (1965a) and about 9% higher than that of Gibson and Dolder (1969). At the lower energies studied, the procedure gives errors that are too large to afford comparisons. The error in the measurements of Golden and Bandel (1965a) was stated as a probable percentage error of *3% with a maximum error of +7%. The error in the total cross section given by Andrick and Bitsch at 19 eV is stated to be & 4%, which would say that the two total cross-section determinations are separated by about three times either probable error. However, Andrick and Bitsch (1975) did not measure the angular profile of their electron beam. They have stated that their measurements were unreliable at 10" and they did not publish their 10" data. If this was due to electrons from their incident electron beam, the work of Sutcliffe et al. (1978) would say that their measurements have an angularly dependent systematic error such that the small-angle measurements reported are too large. If the experiment measures too much smallangle scattering, the total cross sections calculated from their data would be too large. One might expect that this kind of systematic error would not affect their calculated momentum transfer cross sections as much, since the momentum transfer cross section is not sensitive to small angle scattering (see Section 111,C). Thus, while this method is promising regarding the measurement of absolute total and momentum transfer cross sections, it appears that much more careful measurements need to be done. Such future measurements of angular distributions should include both smaller and larger angles than those reported by Andrick and Bitsch (1975). A very recent experiment by Bullis (1977),using a modified Ramsauer apparatus with improved technique, has obtained very good agreement with the results of Golden and Bandel (1965a) for the e--He total elastic scattering cross section at small electron
36
D. E. Golden
energies. However, a further measurement by R. E. Kennerly (private communication, 1977) in a tof apparatus gives values that are about 15% higher than those of Golden and Bandel (1965a)at the lower electron energies (although the error analysis is not yet known). Thus it is fair to say that the situation is not very different from that in 1971, which led Bederson and Kieffer (1971) to conclude that the total elastic cross section for e--He was known to _+ 10%.
IV. Results In this section, the results of both theory and experiment are discussed and compared. The section is divided into four parts. In each part a separate target system is considered in detail. The targets considered in detail are H, He, H,, and N, , which are, respectively, the simplest atomic target to treat theoretically (the lightest one-valence-electron target and the system for which the most theoretical work has been done), the lightest two-valenceelectron target (the target for which the most experimental work has been done), the simplest molecular target, and the first molecular target for which resonances were shown to play a dominant role. A. e--H
The elastic closed-channel resonance in e-H scattering first observed by Schulz (1964a) at about 9.6 eV was confirmed with better energy resolution by Kleinpoppen and Raible (1965) and McGowan et al. (1965). The mechanism for the resonance was given by Gailitis and Damburg (1963) as due to a potential produced by the dipole coupling between the degenerate 2s and 2p states of hydrogen. They showed that, for large distances, this potential is sufficiently attractive to support an infinite number of resonance states below the n = 2 threshold. The lowest H - resonance state has been classified 'S at 9.56eV with a width of 47.5 meV by Burke and Taylor (1966). The experimental results of McGowan et al. (1965) indicated the existence of further resonances and those of McGowan et al. (1969) gave agreement with the position and width determined by Burke and Taylor (1966) for this lowest resonance (9.56 eV, 43.0 meV). More recently the transmission experiment of Sanche and Burrow (1972) has very clearly resolved the 3P and 'S closed-channel resonance states below the n = 2 threshold of hydrogen. The results of Sanche and Burrow (1972) are shown in Fig. 15. Recently a precision caleV and width (0.04717 f 2) x culation of the position (9.55735 k 5) x l o p 5eV has been given by Ho et al. (1977). A narrow shape resonance just
ELECTRON ATOM AND MOLECULE SCATTERING
9 .O
9.5 10.0 10.5 ELECTRON ENERGY (eV)
37
11.0
FIG.15. Derivative of the transmitted current in H for electron energies from 9 to 1 1 eV (from Sanche and Burrow, 1972). The points are the best fit using resonance positions and widths as parameters and folding in a Gaussian instrumental function of 70meV FWHM.
above the n = 2 threshold produces a very pronounced effect in the nearthreshold inelastic cross sections. The near-threshold results of Williams and Willis (1974) for the total electron impact excitation cross section for the 2p level of hydrogen are shown in Fig. 16. The previous measurements of McGowan et al. (1969) and the three-state plus 20-term correlation calculations of Taylor and Burke (1967) are also shown on the plot. No other resonances have been observed near but above the n = 2 threshold by Williams and Willis (1974), although McGowan et al. (1969) have argued that a peak in their results at 10.45eV was statistically real. However, the results of Williams and Willis (1974) confirm the pseudostate calculations of Geltman and Burke (1970), which showed a smooth behavior near 10.45eV. Very recently, resonances have been observed in the photodetachment of H - near 11eV by Bryant et al. (1977). The observations shown in Fig. 17 together with the calculations of Broad and Reinhardt (1976) include a broad feature at about 10.98eV and a very sharp feature at about 10.93eV. It should be noted that the energy scale is shifted relative to that of the electron impact excitation experiment discussed above due to the binding energy of the ground state of H-. Resonances have been observed in electron scattering for the reverse process (that is, in the decay channel H- + hv). The binding energy of H - has been calculated by Pekeris (1958) to be 0.754eV. Thus, if the sharp feature in Fig. 17 is identified with the 'S and/or
38
D.E . Golden
Electron energy (eV 1
FIG. 16. Cross section for 1 + (2p) electron impact excitation from H(1s) from about 10 to 11 eV measured by McGowan et al. (1969) (-) and Williams and Willis (1974)(. . .). The 3-state plus 20-term correlation calculations of Taylor and Burke (1967) have been folded with the experimental electron energy distribution of 0.07 eV FWHM (---) (from Williams and Willis, 1974).
ENERGY ( e V )
FIG. 17. Photodetachment cross section of H- for energies of 10.9 to 11.08eV from the measurements of Bryant et al. (1977). The solid line is the calculation of Broad and Reinhardt (1976).
ELECTRON ATOM AND MOLECULE SCATTERING
39
'P resonance placed at 10.178eV by O'Malley and Geltman (1965) in electron scattering, the photodetachment experiment gives 0.753 eV for the binding energy of the ground state of H-, in very good agreement with the calculation of Pekeris (1958). The open-channel resonance ( - 10.22eV in electron impact excitation of H(2p) and 10.978eV in the photodetachment of H - ) has been discussed by Macek (1967) (see Fig. 16). The openand closed-channel resonances have been discussed by Lin (1975) and Broad and Reinhardt (1976). The closed-channel resonance observed by Bryant et al. (1977) is possibly the highest energy closed-channel peak seen in Fig. 15. Attempts to observe the open-channel effect in the emission from an arc discharge plasma by Ott et al. (1975) and in stellar spectra by Snow (1975) have both been unsuccessful. The location of resonances near but below the a = 3 levels has been difficult experimentally because of the obscuring effects of molecular hydrogen resonances due to the failure to dissociate the target beam completely. The measurement of McGowan et a/. (1969) included a broad structure in the 1s -+ 2p excitation cross section at 11.89 & 0.02eV. The negative-ionneutral-collision measurements of Riseley et al. (1974) showed an unresolved peak in the ejected electron spectrum at 11.86 -+_ 0.04eV. Both of these experiments attribute the structure to the (3p2)'D state of H-. More recently the transmission experiment of Spence (1975a), which is relatively free of H, contamination of the H beam, has shown that only the one resonance is strongly coupled to the ground state of atomic hydrogen. This work gives the position of this state to be 11.86 0.03 eV. The excitation of the 2s and 2p states by election impact has been measured from threshold to about 12.6 eV by Williams (1976).The experimental results for the 2s excitation cross section are shown in Fig. 18 together with the results of a recent variational calculation using a basis set of six atomic eigenstates and eight pseudostates by L. A. Morgan, M. R. C . McDowell, and J. Callaway (unpublished data, 1977). The total excitation cross-section calculation has used the DWPO method of Callaway et al. (1975) to obtain the contribution from the partial waves L 2 4, and the calculated cross sections do not include any allowance for the finite energy resolution of the experiment. The general agreement is good, but the calculated structure becomes more pronounced relative to the experimental structure as the y1 = 3 threshold is approached. This is probably due to some extent to the finite energy resolution of the experiment ( 16 meV). The experimental results of Williams (1976) for the 2p excitation cross section are shown in Fig. 19 together with the results of the variational calculation of L. A. Morgan et al. (unpublished data, 1977). The positions and widths of the resonances as determined by the variational calculation are in good agreement with the positions and widths previously determined in the six-state close-coupling calculations of Burke et al. (1967).
-
-
D . E. Golden
40
0.3-
v)
= b
0.1to! ID
I
El 0.0 0.80
I
0.82
I
I
0.84 0.86 k i 2(RYDBERGS)
I
0.88
I
0.90
FIG. 18. Total cross section for the Is + 2s excitation in e- - H scattering from 10.9 to 12.1eV from Morgan et al. (1978). The experimental points are due to Williams (1976). The vertical bars indicate positions of resonances.
A sharp structure was observed by Walton et al. (1970) in the scattering of electrons from H - near 14.5eV, which was attributed to the formation of H - - with a lifetime of sec. The position and configuration of this structure has been made plausible by the stabilization calculation of Taylor and Thomas (1972). They give the position at 14.8eV with a width of 1eV and ascribe it to a (2s2,2p)’P configuration with a small admixture of 2p3. This problem has been further investigated by A. Herzenberg and D. TonThat (unpublished data, 1973), who have suggested that the geometry of
ki2 (RYDBERGS) FIG. 19. Total cross section for the 1s + 2p excitation in e--H scattering from 10.9 to 12.1eV (from Morgan et a!., 1978). The experimental points are those of Williams (1976).
ELECTRON ATOM AND MOLECULE SCATTERING
41
the H - state must be such that the proton is at the center of an equilateral triangle and the electron at the corners. However, at the present time a definitive calculation of the effect has yet to be done. ~
B. e--He The original observation of the (Is, 2s2)*Sresonance in e--He scattering is due to Schulz (1963),who observed about a 10%decrease in the elastically scattered current at 72" for an incident energy of about 19.3eV. This result is shown in Fig. 20. The existence of this resonance was confirmed by the experiments of Fleming and Higginson (1963), Simpson and Fano (1963), Golden and Bandel (1965a), Kuyatt et al. (1965), Ehrhardt and Meister (1965), and Andrick and Ehrhardt (1966). From consideration of the interference between a single-channel resonance and the background, Simpson and Fano (1963) classified the resonance as an S state and estimated its
18 5
is
19.5 20 Electron Energy, eV
20.5
FIG.20. The first observation of the ( 1 ~ , 2 s ~ resonance )~S in e--He scattering by elastic scattering at 72- (from Schulz, 1963).
D. E. Golden
42
width as about 10meV. The scattered electron angular distribution measurements of Ehrhardt and Meister (1965) and Andrick and Ehrhardt (1966) in the vicinity of the resonance have confirmed the conclusion of Simpson and Fano (1963), which led to its classification as a 'S state. The more recent calculations of Temkin et al. (1972), Sinfailam and Nesbet (1972), and Ormonde and Golden (1973) are in very good agreement with each other regarding the position and width of this resonance. These calculations give an average position of 19.38 k 20meV and an average width of 13.5 f 2 meV. These calculations are also in excellent agreement with the recent measurements of the position and width of this resonance by Cvejanovic et al. (1974) and Golden et al. (1974b). The measurements give an average position of 19.36 If: 20meV and an average width of 11 k 2meV. The more recent calculations by Berrington et al. (1975) using the R-matrix method are in close agreement with the calculations of Sinfailam and Nesbet (1972) and give the position and width of this resonance as 19.38 and 15.3meV, respectively. Once the resonance at about 19.3eV was classified He-(ls, 2s2)'S, it became necessary to ask about the existence of He-(ls2, 2s)'S. The question of a stable state of He-(ls2,2s) had been addressed early on by Wu (1936) when making electron affinity calculations, with a negative response. However, the early measurements of e--He scattering by Ramsauer and Kollath (1929) and Normand (1930) showed "fine structure" near 0.5 eV, which could have been due to a temporary negative-ion state. The effect observed in these early total cross section measurements is shown in Fig. 21 together with the total cross-section measurements of Golden and Bandel (1965a).
0
0.4
0.8
1.2 1.6 2.0 2.4 2.8 ELECTRON ENERGY (VOLTS)
3.2
3.6
4.0
FIG.21. The total e--He scattering cross section as measured by Golden and Bandel (1965a) between 0.3 and 4.0eV compared to the prior measurements of Ramsauer and Kollath (1929)and Normand (1930)(from Golden and Bandel, 1965a).
ELECTRON ATOM AND MOLECULE SCATTERING
43
While the effect is seen to be about 15-20% in the measurements of Ramsauer and Kollath (1929) or Nomand (1930), it must be less than 3% from the measurements of Golden and Bandel (1965a). The question was reopened by Schulz (1966), who reported making observations similar to those of Ramsauer and Kollath (1929) and Normand (1930). However, the much more sensitive measurements of Golden and Nakano (1966) showed that the cross section in helium at low energies was smooth, and that if a structure existed below He-(ls,2s2)’S, it must give rise to a change of less than two or three parts in lo4 of the total cross section. Another possibility for He-(ls2,2s)’S is the broad peak between 1 and 2eV in the total crosssection measurements of Golden and Bandel (1965a). If this peak is interpreted as a shape resonance, its width would be -50eV. Such a width corresponds to such a short time delay that the concept of a resonance becomes “fuzzy.” The e--He differential scattering experiment of Gibson and Dolder (1969) showed an additional resonance above He-(ls, 2~’)’s but below He(ls, 2 ~ ) ~ s . The structure observed at 19.45eV and designated ’P by Gibson and Dolder (1969) from a study of their scattered electron angular distributions can also be seen in the results of the electron transmission experiments of Kuyatt et al. (1965) and Golden and Zecca (1970). The results of Golden and Zecca (1970) showed five sharp and weak structures between He-(ls, 2 ~ ’ ) ~and s He(ls,2s)’S, which had not been found in the close-coupling results of Burke et al. (1969b). The Feshbach projection operator technique was applied by Temkin et al. (1972) and a multichannel variational procedure by Sinfailam and Nesbet (1972) to the calculation of e--He scattering. Both of these calculations have failed to show any of these additional structures as has the transmission experiment of Sanche and Schulz (1972a). The results of Golden and Zecca (1970) were obtained by using a signal averaged over about 80 sweeps of the energy domain studied and required about 20 hours running time in order to obtain an energy spectrum. Thus, the results of Golden and Zecca (1970) were possibly subject to long-term drift and consequent systematic error. Therefore, the same energy domain was studied by Golden et al. (1974b) using a double modulation technique in order to obtain both high-energy resolution and an energy-differentiated spectrum. These techniques allowed measurements with a substantially higher signal to noise ratio at higher energy resolution in a significantly shorter time than the technique used by Golden and Zecca (1970). The electron transmission energy-differentiated results of Golden et al. (1974b) obtained in a single energy sweep of about 0.5 hours running time are shown in Fig. 22 for electron energies between about 19.2 and 21.2eV. The energy scale on the figure was obtained using the procedure described in Section II1,D. A summary of the positions of the structures observed by
D . E. Golden
44
19.5
20.0
20.5
21.0
Electron energy (eV)
FIG.22. Measured energy-differentiated transmitted electron output signal in He- from 19.2 to 20.2 eV (from Golden et a/.. 1974b). (Copyright by The Institute of Physics. Reprinted with permission.)
Golden et al. (1974b) is shown in Table I1 together with the positions of the structures observed previously by Kuyatt et al. (1965), Ehrhardt et al. (1968), Pichanick and Simpson (1968), Gibson and Dolder (1969), and Golden and Zecca (1970). The features labeled 1'-1"' are associated with the 22S of He-. It is this resonance shape that controls the shape of the cusp on the 23S threshold (4-4). The separation between 1"' and 2 is 80meV, and 25 meV energy resolution was required to resolve the 1"' and 2 peaks in the work of Golden et al. (1974b). When the energy resolution was deliberately made poorer, the two features degenerated into one. The position of 2-2' is in good agreement with the position of this feature as observed by Kuyatt et al. (1965), Gibson and Dolder (1969), and Golden and Zecca (1970). When allowance is made for the differences in energy scale calibration (see Section III,D) the agreement is excellent. This resonance was designated 2P by Gibson and Dolder (1969). The widths of the extra structures (UFOs) have been estimated by Golden (1976) to be of the order of lOOpeV or less, so that one should consider spin-orbit effects in future calculations of resonances in this system. Similar effects have been observed by Heddle (1976, 1977) in the e--He excitation for higher n. This resonance was not observed by Sanche and Schulz (1972a). However, Golden et a2. (1974 ) have argued that the energy resolution of that experiment was only about 63 meV, which
TABLE I1
POSITIONS OF STRUCTURES I N e--He SCATTERING” Metastable production
Transmission experiments
Feature No
1-1‘-1”-1”‘ 2-2‘ 3-3’ 4-4 5-5‘
6-6 7-7’ 8-8‘ 9-9‘ 10-10 11-11’ 12-12’ 13-13’
Max
Zero
Min
19.32 19.34 19.36 19.43 19.51 19.55 19.69 19.73 19.825’ 19.93 20.10 20.14 20.23 20.27 20.40 20.44 20.45 20.61 20.69 20.86 20.82 20.98 21.02 21.24 21.19
Sanche and Schulz (1972a) Max Min
19.30 19.37
19.80 19.80
Golden and Zecca (1970) Max Min
Kuyatt et al. (1965)
19.30
19.40 19.31
19.43 19.58 19.818h 20.04 20.17 20.30
19.47 19.43 19.47 19.62 19.818h 20.10 20.21 20.35
Pichanick Gibson and and Simpson Ehrhardt Dolder (1968) e l a/. (1968) (1969)
19.37
19.30
20.34 20.58 20.62
20.59 20.93
Differential scattering
19.30
2’SHe-
19.45
2’PHe-
20.45
20.59 21.22 20.99
Designation
21.00
21.19
Values in electron volts. Table from Golden et a/. (1974b). (Copyright by The Institute of Physics. Reprinted with permission.) Calibration point.
State energy
z3S He cusp
19.818
2’P He2lS He cusp
20.614
23P He 2’D He2’P He
20.962 21.216
46
D.E. Golden
would be insufficient to separate the 1"' peak from 2. In addition, the experiment of Sanche and Schulz (1972a) was done in an axial magnetic field, which has been shown by Golden (1973) to decrease the sensitivity for the detection of elastic scattering (see Section 111,B). A more recent search for the additional resonance below the (Is, 2 ~ ) threshold ~s by Andrick and Langhans (1975) has also produced a negative result. They have studied the electrons scattered to 10" and have stated that if such extra resonances exist, their widths would have to be less than 10peV. However, the experimental procedure used by Andrick and Langhans (1975) places their result in doubt. The apparatus used by Andrick and Langhans has been described by Andrick and Bitsch (1975). In the work of Andrick and Bitsch (1975) a 50% correction was subtracted from the raw scattering data at 15" due to scattering from the background gas. The correction was measured with the gas beam off and with the chamber flooded to the same background pressure as when the gas beam was on. This subtraction does remove the effect of scattering from the background although it makes the statistics worse (see Section 111,C). However, from the work of Sutcliffe et al. (1978) one would expect scattering from the background gas at lo" to be significantly larger than at 15". It is not clear from the paper of Andrick and Langhans (1975) whether or not the subtraction technique was, in fact, used. If it was used, one would have to question the statistical significance of the results. On the other hand if it was not used, the results would certainly be subject to a large uncertainty. (We are talking about effects that are or less of the total scattering cross section.) Perhaps even more significant is the fact that the paper of Andrick and Bitsch (1975) states that measurements at 10"were unreliable and therefore were not included in their results. However, the measurements of Andrick and Langhans (1975) were made at 10" ! Therefore, one may conclude that the experimental evidence suggesting that there are additional resonances near the n = 2 levels of helium unaccounted for as yet by theory is on surer ground than the evidence suggesting that there are no additional resonances. However, the problem is clearly not as yet resolved. The excitation functions of many states of helium have been studied with high electron energy resolution by Heddle and co-workers (1973, 1974a,b, 1977; Heddle, 1977; Keesing, 1977). The work has shown that there are many resonances that affect the excitation functions. The features have been shown by Heddle (1977) to fall into three groups. One is a series of closedchannel resonances; another is a series of near-threshold shape resonances, which decay into states of high angular momentum; the third is composed of a number of sharp and weak features that do not belong to either of the other groups. The excitation functions of seven S states for 3 d n < 6 from Heddle (1977) are shown in Fig. 23.
47
ELECTRON ATOM AND MOLECULE SCATTERING
I
I
- --
. I
I
1 -
I .'
. ..
. .
:
'D
1;
'I
.....
..............
1
?s '
1 , :
i-
.'.. . .. . . .. .. ... ... . . ..... ................................................ . . . . . . . . . . . . . ..... .............. .... . . . . ..-. : , . . ........... :-.: . . . . . . . . . . . . . . . .................................... .......... . .
:
. .
I
-I- I
....... . .... ...:
,
'
. .
... . . . ... . . . . ................ ..
.. ..
..
'+i. 1 1
22.8
2:i.Z
23.6
I
. .. .. ............... 1 f f
21.0
.... I
tt'
I
I
0
2.4 4
energyleV
FIG.23. Excitation functions of the 33S, 3'S, 43S, 4'S, 5 j S , 5 ' S , and 6's states of He in order of threshold energy (from Heddle, 1977). The positions of the (ls,ns')'S, (ns,np)'P, (lD,npZ)'D, and 'S resonances are marked with solid lines. The broken lines are at the positions of resonances as a result of extrapotation.
The potential V of two electrons moving symmetrically about a central charge 2 may be expressed in terms of a single coordinates I as
where Z is the nuclear charge. The energy eigenvalues T for this potential are given by T = -R
( 4 2 - 1)2 1 8 n2
(57)
where R is the Rydberg constant and n is an integer. In the cases of H- and the doubly excited states of helium there is no core electron, and with the exception that the model predicts no fine structure, it gives reasonably accurate results. In the case of excited helium, the core electron cannot be ignored, and one may except Rydberg terms of the form T = -RZzf,,/(n - 1 3 ) ~
(58) Nevertheless, we may expect Zeffand 6 to be reasonably constant along a series of resonances. The values of n* = n - 6 plotted versus n are shown
D . E. Golden
48
in Fig. 24. Two series of resonances are immediately clear on the plot. They can be identified as ( l ~ , n s ’ ) ~ and S ( l ~ , n s , n p ) ~ It P . is interesting to note that an extrapolation of the second series to smaller values of n gives a (Is, 2s, 2p)’P resonance at 19.78eV. The uppermost diagonal line on the figure corresponds to the n’P terms of helium, which are the most loosely bound. One would expect closed-channel resonances to lie below this line. On the other hand, one would not expect any resonance to be more tightly bound than (Is, ns2)’S, and so the lowest line on the figure is a lower bound. When the bounds are established, there are clearly energy regions in which there are resonances outside the bounds. There are three features that are weak and sharp in Fig. 24 for 23.08 < E < 23.39eV, which is outside the bounds. These weak and sharp features are also seen in the 3D excitation functions. The weak features in the “non-Feshbach” energy regions are a problem. They are much too long-lived to be shape resonances and similar structures can be seen in the results of Sanche and Schulz (1972a) although they did not comment on them. As discussed previously, some additional weak features have been observed by Golden et al. (1974b) at lower electron
n*
n FIG.24. The position of resonances, expressed as an effective quantum number n*, given by ( - R/T)”’ plotted against successive integers n. 0 ,’S resonances; 0, ’P resonances; +, (Is, 3P2)’D resonance (from Heddle, 1977).
ELECTRON ATOM AND MOLECULE SCATTERING
49
energies. It has been shown by Read (1977) that by modifying the Rydberg formula for atomic states, accurate (within 1OOmeV) term energies are obtained for a wide variety of negative ions, neutral atomic states, positiveion states, negative-ion resonance states, and auto-ionizing atomic states for terms of the type [core] (nsZ)'S. The formula can also be used for terms of the type [core] ( n l , r ~ l ' ) ~ ~ +and ' L an extension of the formula used for configurations of the type [core] npN,for N = 1-6. The ionization of atoms in the near-threshold region has been treated classically by Wannier (1953) and extended by Vinkalns and Gailitis (1968). In these treatments, the electrons move classically when outside a reaction zone so that the trajectories from inside the reaction zone fill the available phase space with a smooth nonsingular probability. The results of this treatment are (1) that the ionization cross section near threshold has an energy dependence ( E - E,)1.127, where E is the incident energy and E, is the ionization threshold energy; (2) that the probability distribution P,(8, 2 ) of angles 012between the two outgoing electrons has a maximum at d12 = 71 and a width proportional to ( E - E,)1/4;and (3) that the distribution PEl2(E1) of the outgoing electron energies (El + E , = E - E,) is constant and independent of E , over the range 0 to E - E , . The early quantum-mechanical treatments by Rudge and Seaton (1964, 1965) gave the result that the ionization cross section has an energy dependence near threshold of ( E - E,) and that PE12(E1) is uniform. Further treatments by Temkin (1966), Temkin et al. (1968), and Kang and Kerch (1970) have given ionization cross-section threshold energy dependencies ( E - E,)" with 1 < n < 1.5, but have not given Po and PEI2. The most recent quantum-mechanical treatment by Rau (1971) gives predictions in accord with the classical predictions. The problem has also been formulated by treating the electron motion semiclassically, and using the wkb approximation by Peterkop and Liepinsh (1970) and Peterkop (1971). This gives the Wannier power law exponent to be 1.127, but does not give Po or PE12. Classical trajectories have been numerically integrated by Banks et al. (1969), Peterkop and Tsukerman (1970), and Griijic (1972) to yield values of n close to 1.127 and uniform distributions for P , but these calculations give no information about PBI2. The range of E - E, over which the theoretical predictions have validity was not given by any of the calculations. The experimental results of McGowan and Clarke (1968), Brion and Thomas (1968), Krige et al. (1968), and Marchand et al. (1969) all found values of n more or less close to 1.127 near threshold. However, none of these experiments gave P , or PE12 near threshold. More recently, this problem has been studied by Cvejanovic and Read (1974b). These results are consistent with the Wannier formulation with regard to the formation of a cusp at the ionization threshold, the form of Po and PE12, and the exponent of
D.E. Golden
50
the ionization cross section power law-energy dependence for a range of incident energies relative to the threshold energy from 0.2 to 1.7eV. These results confirm that correlation between the outgoing electrons is important as implied by the Wannier theory. In the work of Cvejanovic and Read (1974b) a coincidence tof technique was used to study the energy distribution and angular correlations for the two outgoing electrons in ionization events for which the energy above threshold was between 0.2 and 3.0eV. Two coincidence spectra are shown in Fig. 25 for excess energies of 0.37 and 0.60eV. The solid curves drawn through the experimental points are the best fits to the data assuming uniform energy distribution functions (which transform to the time distributions function shown). The right-hand side of the figure shows the deduced energy distribution functions. The deduced energy distribution function has been found to be uniform is the range of excess energies from 0.2 to 0.8eV. The function Po was measured at O I 2 = rc rc/9 and 57~16i-rc/9 for five values of E from 0.2 to 3.0 eV. Assuming that the form of Po is Gaussian, Po was shown to have the form Po K Ell4. The most precise measurement of Cvejanovic and Read (1974b) was the measurement of the exponent n in the ionization threshold energy dependence. For this work the interaction
200OC 1300k
1100/”
900
-
e
v
..
::p 08
ELECTRON ATOM AND MOLECULE SCATTERING
51
region was unshielded so that a small electrostatic field penetrated into the interaction region, and the analyzer was used in the "negative-energy'' mode described by Cvejanovic and Read (1974a). With these changes, the detection system has its highest sensitivity for electrons of 15 meV or less and the sensitivity for electrons of 50meV or higher is much reduced. If the energy distribution function is constant, the measured partial ionization cross section ok(E)is given by
where AE is the mean energy width of the partial ionization cross section detected. The yield of low-energy electrons obtained in this way is shown in Fig. 26. Below the ionization threshold, peaks are detected that appear at the thresholds of the infinite series of excited states converging to the ionization threshold. Above the ionization threshold, the yield is proportional to the partial ionization cross section 0; defined in Eq. (59). The curve drawn through the points is proportional to E0.'27. The mean of this and other measurements, found by the method of least squares, is 0.131 k 0.019 between 0.2 and 1.7 eV. Since the measurement is of n - 1 rather than
E,
(rV)
FIG.26. Yield of electrons of very low-energy electrons ( 550 meV) as a function of incident electron energy near the ionization threshold of He. The curve drawn through the points above the ionization threshold is proportional to EO.'*' where E is the energy excess above the ionization energy (from Cvejanovicand Read, 1974a).(Copyright by The Institute of Physics. Reprinted with permission.)
52
D . E . Golden
n, the precision of the measurement is improved over that of previous measurements. By using an improved version of the double-retarding potential difference technique of Knoop and Brongersma (1970), Spence (1975b) has observed the threshold cusp as a function of excess electron energy above the ionization threshold in e--He scattering. This work has shown that the cusp disappears for excess energies greater than about 2 eV. Thus it confirms some of the results of Cvejanovic and Read (1974a) using a different technique. The lowest doubly excited states of helium are the (2s2)'S at 57.82eV and the (2s, ~ P ) state ~ P at 58.34eV (Rudd, 1964, 1965). Closed-channel resonances associated with these states were first observed by Kuyatt et al. (1965) and confirmed by Golden and Zecca (1970) and Sanche and Schulz (1972a). The resonances were classified by Fano and Cooper (1965a) as (2?, 2p)'P at about 57.16 eV and the (2s, 2p')'D at about 58.25 eV. Eliezer and Pan (1970) used the stabilization method and Nicolaides (1972) used a variational projection operator method with correlation to obtain good agreement with the positions and assignments of these resonances. These resonances have also been observed in two-electron decay by Burrow and Schulz (1969b) using a trapped-electron method. The two-electron decay of these states causes structure in the energy dependence of He' as observed by Grissom et al. (1969) and Quemener et al. (1971). By using the multiconfiguration close-coupling model of Smith et a/. (1973), Ormonde et al. (1974) were able to calculate an additional triply excited resonance state (2s, 2p2)'Se at 59.4eV, which is about 300meV wide. This structure was predicted by Fano and Cooper (1965a) and may be seen in the trappedelectron results of Grissom et al. (1969), the optical excitation results of Heideman et al. (1966), and the helium ionization results of Quemener et al. (1971). A very broad resonance ( - 10 eV wide) has been observed by Crooks et al. (1972) in the differential excitation cross section for the 23S state of helium by electron impact at about 50eV. They observed a very deep broad minimum near 90" at 50meV impact energy. The integrated cross section produced a broad peak at that energy. Their analysis showed the feature to be due to interference between the various partial waves as well as a broad P-wave resonance. The existence of such a quasi-stationary state was calculated by A. Herzenberg and D. Ton-That (unpublished data, 1973) at about 45.9eV as a state of He- in which the three electrons are equidistant from the nucleus. The close-coupling calculations of Ormonde and Golden (1973)confirmed the results of Crooks et al. (1972)and predicted additional broad resonances at both lower and higher energies. The separation of resonances in the extra-electron system from resonances in the target system by experiment is made difficult by the fact that the ejected electrons from the autoionizing levels of the target are displaced in
ELECTRON ATOM AND MOLECULE SCATTERING
53
the near-threshold excitation region. This displacement of ejected electron energies was first observed by Hicks et al. (1974) while studying the electron impact excitation of the four lowest autoionizing levels in helium by operating their electron spectrometer in a constant energy loss mode while scanning the incident electron energy. This mode of operation excites the autoionizing levels with constant excess energy above their thresholds. Four constant energy loss spectra for helium at an ejection angle of 70" for various excess energies is shown in Fig. 27. As the incident electron energy increases, the ejected electron energies shift to higher energies. The apparent energy shift of the autoionizing states as a function of excess energy is shown in Fig. 28. Shifts in the energies of ejected electron peaks have been observed by Barker
1
y I I ' ji Energy above threshold :?"J
I 33 0
I
0-1e V
.;
I
I
34 0
I
I
35 0
E j e c t e d e l e c t r o n energy,
I 36 0
eV
_c
FIG.27. Constant-energy loss spectra of ejected electrons from the four lowest autoionizing levels of He at 70' for four values of excess energy, (from Hicks et a/., 1974).
D.E. Golden
54
I
I
1 0
I
I
1
I
0.2
I
I
I
I
0.4
I
I
0.6
(E lev)-’$
FIG.28. Energy shift of four autoionizing states of He as a function of excess energy (from Hicks et d., 1974).
and Berry (1966) and Gerber et al. (1973) in experiments in which the autoionizing states of helium were produced in collisions of He+ + He. In this case the ejected-electron energy was found to decrease as the incident-ion energy decreased. This is opposite to the effect observed in electron scattering. The ion-atom effect was explained qualitatively by Barker and Berry (1966) by either a molecular model or by a “post collision interaction” model. The post collision interaction model, which says that the incident particle interacts with the ejected electron, can explain qualitatively both types of experiment. When the autoionizing state decays, the ejected electron is in the field of the inelastically scattered incident projectile. For example, if the inelastically scattered particle is a 1-eV electron and the autoionizing state has a mean life of 5 x sec, the electron will travel a mean distance of about 3 nm before the state decays. The resulting coulomb potential between the scattered and ejected electrons is about 0.5 eV. If it is assumed
ELECTRON ATOM AND MOLECULE SCATTERING
55
that the speed of the scattered electron can be neglected compared to that of the ejected electron, the ejected electron must gain 0.5eV. This gain is balanced by a corresponding loss in energy of the scattered electron. The phenomenon was regarded by Barker and Berry (1966) as a dynamic screening effect in which the inelastically scattered particle screens the ejected electron from the nucleus. The energy change E depends on the lifetime of the decay of the autoionizing state and has a probability distribution that has a peak proportional to u - ’ . The energy resolution of the ion-atom experiments was insufficient to test the model. However, Hicks et al. (1974) took u to be the final velocity of the inelastically scattered electron in the absence of interaction between it and the ejected electron E l to be the final energy of the scattered electron (the excess energy), E , to be the energy of the autoionizing state of width w, and wrote the apparent energy E, as
where I
= me4/32nKoh2,with
K Othe permittivity of free space.
C. e--H, state. ~ Since this state has The ground state of H; is the (ls0 ,)~ (2 p o,) ~ C two bonding orbitals and one antibonding orbital, one should expect this state of H; to be weakly bound when compared to the ground state of H,(lso,)’ ‘C,(see Taylor and Harris, 1963). We should therefore expect the negative ion to be short-lived and unstable toward auto detachment (see Bardsley et al., 1966a; Eliezer et al., 1967). The experiment of Golden and Nakano (1966) showed that if the state of H; existed, it produced an effect of less than 2 or 3 parts in lo4 of the total scattering cross section. The total scattering cross-section measurements of Golden et al. (1966) are shown in Fig. 29 together with the elastic cross-section measurements of Linder and Schmidt (1971).The first observation of large vibrational excitation cross sections in the ground electronic state is due to Ramien (1931). The large vibrational excitation cross sections were later confirmed by Englehardt and Phelps (1963), Schulz (1946b), and Ehrhardt et al. (1968). The large vibrational excitation cross section was successfully interpreted in terms of a resonance model by Bardsley et al. (1966b). However, it should be pointed out that the indirect measurements of the ( u = 0 + 1) excitation by Crompton et al. (1969) have found an initial slope of 0.11A’ eV for this cross section, while the direct measurements of Schulz (1964b) found 0.6A2/eV and those of Ehrhardt et a!. (1968) found 0.21 A2/eV. These latter direct measurements give a slope in agreement with that determined by Englehardt and Phelps (1963). It was pointed out by Crompton et al. (1970)
D.E. Golden
56
0
1
1
~
1
*
"
'
'
1
~
~
-
FIG.29. Total cross section in H, of Golden et a/.(1966) and elastic portion from Linder and Schmidt (1971) from 0.3 to 12eV. The absolute magnitude of the elastic cross section is obtained by normalization to the value at 0.3 eV.
and also discussed by Golden et al. (1971) that if a reasonable rotational excitation cross-section energy dependence is assumed (consistent with the calculation of Chang and Temkin, 1969) a rather large discrepancy exists between beam and swarm experiments regarding the threshold behavior of the vibrational excitation cross sections. This discrepancy has still not been resolved. Furthermore, the calculations of Breig and Lin (1965) and Takayanagi and Geltman (1965) for the (v = 0 1) excitation give fairly good agreement with observation without invoking a compound state. The strongest evidence for the compound-state model is in the results of Schulz and Asundi (1965). They observed a dissociative attachment process in e--H, scattering with an appearance potential at 3.75 eV. The attachment process was found to have a strong isotope effect as seen in Fig. 30. These results may be interpreted in terms of a lifetime of 10-l4sec for the compound state, which corresponds to a width of almost 1 eV (see Massey, 1976, p. 312). However, it should be noted that this is a very small effect ( 2 x 10of the elastic cross section). The effect shown in Fig. 30 is closely reproduced in the results of Tronc et al. (1977). The work of Linder and Schmidt (1971) has shown that pure vibrational excitation has a PO wave dependence. These observations are shown in Fig. 31 together with the calculations of Henry and Chang (1972). When rotational excitation accompanies vibrational excitation, an isotropic angular distribution is observed. It was pointed out by Linder and Schmidt --f
-
57
ELECTRON ATOM AND MOLECULE SCATTERING
Electron energy, eV
FIG.30. Energy dependence of the total cross section for dissociative attachment in H,, HD, and D, near 3.7 eV (from Schulz and Asundi, 1967). The rising portion of the D-/D, curve shown as dashed was postulated to be due to the wings of the 'Zi resonance near 10eV.
0
20
40
60
80
I00
120
140
160
leo
SCATTERING ANGLE 8 "
FIG.3 1. Angular dependence of the cross-section ratio u (AJ = O)/o ( J = 1 + 3) for electron impact excitation of the 1' = 1 vibrational level of H, at 4.5 eV from the experiments of Ehrhardt and Linder (1968) (0); Linder and Schmidt (1971) ( O ) ,and from the calculations of Abram and Herzenberg (1969) (A), Henry (1970) (B), and Henry and Chang (1972) ( C )(from Henry and Chang, 1972).
D . E. Golden
58
(1971) that the results can be interpreted in terms of a direct and a resonant component to the phase shifts. The more recent theoretical work of Chang (1974) and Temkin and Sullivan (1974) shows that a combination of direct and resonant processes must be used to interpret the cross section for for c = 0 1 (possibly 1: = 0 -+ 2), while the cross section for I' = 0 + 3 is purely resonant. This can be seen in Fig. 32, where the experimental data of Linder and Schmidt (1971) and Wong and Schulz (1974) for the ratio o (AJ = O)/o ( J = 1 + 3 ) are shown as a function of scattering angle together with the calculations of Temkin and Sullivan (1974). More recently, the isotope effect of the c = 0 + 1 vibrational cross section and the isotope independence of rotational cross sections at 4 e V in H, and D, has been studied by Chang and Wong (1977). It has also been suggested by Frommhold (1968) that resonances are necessary to explain rotational excitation for energies from 10 to 100meV. This suggestion comes from the pressure dependence of the drift velocity, which was confirmed by Crompton and Robertson (1971). In a tof transmission experiment in H2, Land and Raith (1973) also observed structure at 24 meV. The observed structure lies below the first inelastic threshold in H,. However, in D, they observed their first structure above the first inelastic threshold. Rotational resonances have not been calculated by Henry and Lane (1969) at this energy, and so this effect is not explained at the present time. 8 .O
7
7.0 0
0
6.0
v = 1 : LINDER
8. SCHMIDT
v 1 1 : WONG 8, SCHULZ
z
5.0
- 4.0 L' 3.0 b -. 2.0
v
v=2:WONG&SCHULZ
0
v=3:WONG&SCHULZ
m f
I1
0 I1 .-
Q b
1.0
0 0
20
40 60 80 100 120 SCATTERING ANGLE (DEGREES)
140
-
FIG.32. Angular dependence of the cross section ratio G(AJ = O ) / o ( J = 1 3 ) for the electron impact excitation of the L' = 1, 2. and 3 vibrational levels of H,. The experimental data of Linder and Schmidt (1971) for L' = I and those of Wong and Schulz (1974) for L' = 1.~3 are compared to the calculation of Temkin and Sullivan (1974) (from Temkin and Sullivan, 1974).
ELECTRON ATOM AND MOLECULE SCATTERING
The first resonances observed in e--H,
59
scattering are due to Kuyatt
et a/. (1964) for energies between 11.6 and 13.3eV. These measurements
were confirmed and extended to include the isotope effect in D, by Golden and Bandel (1965b). The structures were observed by Menendez and Holt (1966) in the excitation of the c = 1 and c' = 2 vibrational states of the ground electronic state. The first observation of the decay of these resonances into electronically excited states of H2 was made by Heideman et al. (1966). The experiments of Kuyatt et al. (1966) and the calculations of Eliezer et al. (1967) showed that more than one electronic state of H - contributes to the observed structure. Five series of resonances have been identified by Comer and Read (1971) (a -+ e) and seven series could possibly be necessary to explain all of the data. A sample of some of the data of Comer and Read (1971) is shown in Fig. 33. ,
110
I15
I2Q
IPS
..
i)a
I
,-
I35
MQ
COLClSlON ENERGY (eV)
FIG.33. Differential vibrational excitation cross section in H, from 11 to 14eV at 85 (from Comer and Read. 1971). (Copyright by The Institute of Physics. Reprinted with permission.)
60
D . E. Golden
FIG.34. Isotope effect in dissociative electron attachment for 7 to 18eV in H,, HD, and D, (from Rapp et a/., 1965).
Structure has also been observed in the dissociative ionization cross section in this energy domain by Khvostenko and Dukelskii (1957), Schulz (1959), and Rapp et al. (1965). The results of Rapp et al. (1965) are shown in Fig. 34. There is a broad peak at about 10 eV (FWHM 3 eV) and a narrower peak at about 14eV (FWHM 1eV). The lower energy peak is mostly due to the formation of H; in the repulsive '2: state, which dissociates to Hand H in their respective ground states. The resulting H - has considerable kinetic energy, The potential energy curve for this state is given by the work of Bardsley et al. (1966a,b) and Chen and Peacher (1968). Fine structure was predicted in this broad peak by O'Malley (1966) and found by Dowel1 and Sharp (1967) at the same positions as the strong resonance series a observed by Kuyatt e l al. (1966). This structure is better resolved in the recent differential cross-section measurements of Tronc et al. (1977) shown in Fig. 35. The c series has been interpreted by Comer and Read (1971), and Joyez et al. (1973) as resulting from a Z ' : or 2rIuresonance. In light of the anisotropic behavior of the B 'Z: excitation cross section measured by Weingartshofer et al. (19701, the latter designation is preferred. The reexamination of these experimental results by Chang (1975) has concluded that the resonance in question is more probably 2Ag. More recently the Feshback projection operator technique has been used to calculate resonant states of H; by Buckley and Bottcher (1978). The calculations examined
-
-
ELECTRON ATOM AND MOLECULE SCATTERING
1
1 1 10.0
1
l
I I 11 0
I
I
I
I
I I 12 0
I
l
I
61
I 13 0
Incident electron energy (eVl
FIG.3 Energy dependence of H - formation by electron impact .A the 'El resonance state of H i between 9 and 13 eV at 50" (from Tronc et a[., 1977). (Copyright by The Institute of Physics. Reprinted with permission.)
seven resonance curves of H;, which are shown in Fig. 36. The 'CJ(1) state is found to be in good agreement with the previous calculations of Bardsley et al. (1966a) and Eliezer et al. (1967). The calculated ,C,'(l) state is attached mainly to the dissociating B 'CC,. state of H,, and is in agreement with the calculations of Bardsley et al. (1966a). This work also found a 'CC,. doublewelled resonance 'Ci(2) in the 10-12-eV region resulting from an avoided crossing with the higher 'CC,'(3) state. Since the widths ofthese two resonances have not yet been calculated, the exact behavior of the H; ion near this point is still uncertain. It is likely that the 'Zl(3) resonance is never realized and that the ion will follow the dotted path joining the attractive part of 2 C, + (3) to the first well of ,C,'(2). Then the 2Cc,'(2)/2C,'(3) resonance is probably responsible for series a. The 'C:(2') well is responsible for series b. The calculation located both a 'C:(2) resonance and a resonance, either of which could explain the experimental results for series c, but did not locate a ,A, resonance at the proper energy. The vibrational energy levels of the resonances are presented in Table 111. Since the variational principle used by Buckley and Bottcher (1978) gives only upper bounds for the eigenenergies, the levels lie above experimental values. In the future, the widths of the resonances will be calculated.
i
0
V
-0'
.
w
I
-0.
1
I
2.0
1
4.0
3
R (a,)
FIG.36. Potential energy curves of H, (full lines) and H, (dashed lines) versus internuclear separation (from Buckley and Bottcher, 1978).(Copyright by The Institute of Physics. Reprinted with permission.) TABLE 111 THEVIBKATIONAL LEVELS01-
0 1
2 3
11.57 11.89 12.17 12.43
THE
11.45 11.68 11.89 12.08
H; RESONANCE STATES'
11.82 12.15 12.46 12.75
11.98 12.30 12.60 12.88
~~
'I From Buckley and Bottcher (1978). (Copyright by The Institute of Physics. Reprinted with permission.)
ELECTRON ATOM AND MOLECULE SCATTERING
63
A higher energy series of resonances beginning at about 13.5eV was observed by Ehrhardt and Weingartshofer (1969)by studying the C'I'I, decay channel of H,. They attributed a .X ' ; symmetry to this series and classified it band f. Six members of this series were identified by Weingartshofer et al. (1970) between 13.6 and 14.9eV, which have been shown to decay preferentially to the C' II, state without change in vibrational quantum number. ' : state It is therefore very likely that the potential energy curve for this C of H; has the general shape and the same equilibrium internuclear separation as the C'n, or D'n, states. The designation of the f series resonances as being due to an H, state of 'XC,. symmetry is confirmed by the earlier calculations of Eliezer et al. (1967). The f series of resonances was studied further by Golden (1971). This work found 12 resonances between 13.6 and 16.3eV. Thus the f series of resonances proceeded across the threshold for the formation of H: and it was suggested that these structures could lead to the production of H:. It has been originally suggested by Stevenson (1960) that short-lived negative-ion states (10- 14-10-13 sec) would be necessary to explain his results for the electron impact ionization of H2 near threshold. The resonances observed by Golden (1971) with widths -80meV gave evidence for this point of view. Further work by Sanche and Schulz (1972a,b) confirmed the existence of the f series observed by Golden (1971) and suggested that the additional three structures observed by Golden (1971) were members of another series of resonances, which they labeled g. Further work by Schowengerdt and Golden (1975) found 16 members of the f series (nine above the H: threshold) and seven members of the g series (five above the H i threshold). In addition, the work of Schowengerdt and Golden (1975) showed that more than 20 vibrational levels of the 'Cp' state of H; exist. Thus one might expect H; formation to be significant within at least 6eV above the HZ threshold. The electron impact ionization threshold region of H2 has been the subject of considerable attention and controversy. The consequences of a direct-ionization model near threshold were considered by Krauss and Kropf (1957). In this model, excitation of each vibrational state of H i was assumed to be proportional to the incident energy (relative to the threshold energy). When the Franck-Condon principle was invoked, the calculated ionization cross section consisted of a series of straight-line segments beginning at each vibrational threshold of H:. Shortly thereafter, Stevenson (1960) remeasured the ionization cross section near threshold. He found a linear ionization cross section that was in agreement with the earlier measurements of Bleakney (1930, 1932). In addition, Stevenson (1960) found that the average appearance potential of H: and the initial kinetic energy of the protons in the dissociative ionization spectrum were in disagreement
64
D . E. Golden
with the expectations based on the conventional application of the FranckCondon principle. He found disagreement between the most probable internuclear separation for H l formed by electrons in the energy range 15-100eV and the potential energy diagrams given by Bates et al. (1953). The time required for the slowly moving nuclei to assume a new distribution of states in the field of three electrons was 10-'4-10-13 sec. The first highenergy resolution electron impact ionization studies (claimed energy resolution 30meV) of Marmet and Kerwin (1960) showed breaks in the cross section for the production of H:, which were interpreted as the thresholds for the individual vibrational states of H: in agreement with the model proposed by Krauss and Kropf (1957). In order to settle the discrepancy, Briglia and Rapp (1965) repeated the measurement (albeit with lower energy resolution) and found a linear ionization cross section near threshold, They also suggested that time delay mechanisms such as autoionization and intermediate negative-ion formation might be necessary to describe the ionization process. Further high-energy resolution work (claimed energy resolution 60 meV) by McGowan and Fineaan (1965) showed a nearly straight line for the ionization cross section near threshold. A break and a short linear region very close to threshold was interpreted as being due to competition between direct rotational and vibrational excitation of the molecular ion and autoionization. Further work by McGowan et al. (1968) showed a marked similarity between the first derivative of their electron impact data and the photoionization data of Dibeler et al. (1965). Their conclusion was that near threshold, the structure in the ionization cross section was due to autoionization of long-lived Rydberg states of H, (lifetimes 10-7-10-6 sec), and they rejected the temporary formation of H; as a contributing mechanism for formation of H i . However, the structures observed by Golden (1971) had lifetimes of about 10-14sec, and so a mechanism was available for ionization with a very short time delay in agreement with the suggestion of Stevenson (1960). The kinds of processes that can occur in electron molecule scattering leading to structure include
+ H, - t e + [H2]**+2e + H: e + H, + [Hi]* 2e + H: e + Hz +[H;]* + e + [Hz]** + 2 e + H: e
+
(61) (62)
(63) In the first process, the ionization proceeds via autoionization. The lifetimes of such autoionizing states have been shown to be of the order of lo-" sec by Faisal(l971) and Barnett et al. (1972). In the second process, the ionization proceeds via two electron decay of [H;]* and the important time here should be the lifetime of the [H;]*, which is about 10-l"sec. In the third
65
ELECTRON ATOM AND MOLECULE SCATTERING
process, the ionization proceeds via a two-step process and there are two lifetimes involved. The time delay is controlled by the second process because it has a much longer lifetime than that for the first process. However, a longer lifetime corresponds to a narrower structure. Therefore, the width to be observed for both (62) and (63) will be due to the formation of [H;]* and one cannot easily distinguish between the two processes. The positions of the structures observed by Boesten et al. (1974) were correlated with the members of the f series of resonances above the ionization threshold observed by Golden (1971). Thus, the ionization does proceed via (62) and/or (63). The structures observed in the electron transmission spectrum and the production of H: were also correlated with the positions of structure observed in the photoionization data of Cooke and Metzger (1964), Dibeler et al. (1965), and some of the structure in the higher resolution work of Chupka and Berkowitz (1969a,b).The structures observed by Boesten et al. (1974)and Schowengerdt and Golden (1975)together with the earlier electron ionization work of McGowan et al. (1968) and the photoionization work of Dibeler et al. (1965) are shown in Fig. 37. Weingartshofer et al. (1970) concluded that the ' C l state of H; vibrational members decay almost exclusively to the 'nustate of H2 with Av = 0, and the npn'II, Rydberg states were considered to be the significant states in the autoionization of of H, by Chupka and Berkowitz (1969a,b). A summary of the positions of
'I,
12
"ii
L
13
A
\/ I / \
v
l x 0.51
\ Dibelcr R c c s r Krouss
I
I
I
IL
5
15
.5
I
I
I
16
.S
17
~
E,llrVI
~~
4
FIG.37. Structures near the ionization threshold of H,; in electron transmission from Schowengerdt and Golden (1975):photoionization from Dibeler et u/.(1965);electron ionization from McGowan er a/. (1968) (from Schowengerdt and Golden, 1975).
TABLE IV POSITIONS OF STRUCTURES IN THE H;
AND
e- SPECTRUM"
H: Dibeler et al. (1965)
A1
v
Chupka and Berkowitz (1969a, b)
A1
H;
McGowan e t a ! . (1968) n = 1.127
n= 1
Present ionization f
g
0 1
Weingartshofer et al. (1970)
H; 13.63 13.93 14.20 14.47 14.70 14.92
H;
Golden (1971)
A1
Sanche and Schulz (1972b)
f
13.62 13.91 14.19 14.46 14.72 14.97
13.66 13.94 14.20 14.45 14.65 14.93
15.21
15.18
e
Present transmission
f
15.06
15.48 15.60
8 15.72 15.86 15.94 16.08
9 10
11
16.22 16.32 16.46
12 13 14 15 16 a
15.48 15.58 15.66 15.74 15.88 15.96 16.09 16.19 16.24 16.33 16.45
15.47 15.61 15.65 15.72 15.86 15.96 16.07 16.20
15.49 15.58 15.65 15.75 15.89 15.97 16.04 16.17 16.23
(15.60) (15.60) 15.66
15.43
15.44 15.59 15.66
(15.77) 15.87 16.03
15.57 15.74 15.96
16.43 16.61 16.76 16.92 17.10
From Boesten er al. (1974) and Schowengerdt and Golden (1975)
16.07
16.08
(16.12) 16.26
15.87
15.88
16.02 16.07
15.66
15.67 15.77
15.85
15.44
15.44
15.65
15.87 (15.93)
15.31
15.57
15.79
15.21
15.21 15.32
7
13.62 13.91 14.19 14.46 14.72 14.97
13.62 13.91 14.19 14.46 14.72 14.97 15.09
6
e
Empirical formula f
16.16 16.26
16.26
16.26
16.44 16.61 16.77 16.92
16.44 16.61 16.77 16.92 17.06
ELECTRON ATOM AND MOLECULE SCATTERING
67
these structures is presented in Table IV. The more recent work of Weingartshofer et al. (1975) indicates that the vibrational levels of the d3H, and k3n, states that lie above the H l threshold play an important role in the ionization of Hz by electron impact, due to autoionization. These states are populated by the decay of the close by resonances states (within 30meV) discussed above. These autoionizing states preferentially decay to the nearest H i vibrational level. Thus the two-step process seems to be preferred. In order to obtain further information that can unambiguously distinguish between one- and two-electron decay modes of H;, coincidence experiments will be necessary between H i ions and electrons with specific energy losses.
D. e--N, The earliest measurements of the total scattering cross section for e--N, by Ramsauer and Kollath (1931) showed a broad peak near 2eV. This peak was shown by Schulz (1962,1964b)to be composed of vibrational resonances. This work was confirmed and more members of this series of resonances were observed by Golden and Nakano (1966), Heideman et al. (1966), and Andrick and Ehrhardt (1966). Scattered electron angular distribution measurements were carried out by Ehrhardt and Willmann (1967), who showed that the scattering has a d-wave character. From Schulz's experimental work and by considering the isoelectric sequence O:, NO, N;, Gilmore (1965) calculated a potential energy curve for the 'lIgground state of N, at about 1.6 eV above the ground state of N2.The structures observed above 2eV are generally thought to be well understood in terms of the formation of the ground electronic state of N; in its various vibrational states. However, in the energy region above 1.8eV, there is disagreement between measurement and theory with regard to the frequency and number of the resonance peaks, which are shown in Fig. 38. The figure shows the total cross-section measurements of Golden (1966) and the calculations of Chandra and Temkin (1976).The calculation was made with a hybrid theory of electron-molecule scattering, which uses close-coupling calculations for the vibrational degrees of freedom, adiabatic-nuclei calculations for the rotational degrees of freedom, and experimental data to determine the position of the ground state of N;. While the agreement between the two curves is generally quite good, the calculated curve has only three peaks between 1.8 and 2.7eV, while the measured curve has four. A Temkin (private communication, 1977) has pointed out that this discrepancy might be due to the necessity of limiting the number of coupled states included in the calculation of the partial cross sections to three. Thus Chandra and Temkin (1976) did not get full convergence in all coupling indices. More
D.E. Golden
68
i . . . . . . . . . . . . . . d 0.0
0.5
LO
1.5
2.0
2.5
3.0
35 4.0
ELECTRON ENERGY (eV)
FIG.38. Total e--N, scattering cross section from 0.1 to 4eV; heavy line measurements of Golden (1966); light line calculations of Chandra and Temkin (1976) (from Potter et al., 1977).
recent calculations by Buckley and Bottcher (1978) and C. V. Sukumar (private communication, 1977) show more structures in the partial cross sections, but have considered a much smaller energy range than that of Chandra and Temkin (1976). Furthermore these latter calculations use the wavefunctions of Nesbet (1964), which predict an internuclear separation and a vibrational energy spacing both of which differ slightly from observed values. Thus, more theoretical work on this problem is indicated. In the region below 1.8eV there is disagreement between the various measurements with regard to the presence of structure in the total cross section. These structures were first observed by Golden (1966). However, they may also be seen in the results of Boness and Hasted (1966), and there is an indication of them in the work of Baldwin (1974); while Schulz (1964b) and Ehrhardt and Willmann find no structures below 1.8eV. The recent work of Potter et al. (1977) has shown that the presence of the low-energy structures depends on whether the population of vibrationally excited N, in the ground electronic state conforms to a Boltzmann distribution at room temperature, or is non-Boltzmann with an appreciable population in the various vibrational states. Indeed, some of the structures below 1.8eV have recently been observed by Michejda and Burrow (1976) after heating an N, beam in a microwave discharge before doing a crossed-beam transmission experiment. Since the electric dipole moments for homonuclear diatomic molecules are zero, spontaneous transitions from higher to lower vibrational states are forbidden by dipole radiation, and therefore the transition probability is determined by the electric quadrupole moments.
ELECTRON ATOM AND MOLECULE SCATTERING
69
The lifetime of the X'C; (u = 1) state is between 3 x lo6 and 1.7 x lo7 sec by electric quadrupole transition, and so the lifetime will be primarily determined by molecule-wall, molecule-molecule and/or electron-molecule collisions. Since the last type of collision can alter the internal energy of the molecule, it can cause the population distribution to differ substantially from a Boltzmann distribution. In fact, electron scattering from a Boltzmann distribution of N, molecules at room temperature for the energy range from 0.1 to 4eV has been shown by Potter et al. (1977) under proper conditions to lead to various inverted vibrational population distributions in the N2 electronic ground state. The population of vibrationally excited N, in the ground electronic state will cause resonances to appear at lower electron impact energies. This is probably the cause of the low-energy structure in the total cross-section measurements of Golden (1966). Recent electron tof total cross-section measurements in N, have been made by R. E. Kennerly (private communication, 1977). The calculations of Potter et al. (1977) predict that virtually no vibrationally excited N, was present in the work of R. E. Kennerly (private communication, 1977). One would therefore expect Kennerly's experiment to have an absence of structures below 1.8 eV, to have a slightly smaller total cross section, and the peak to valley separation of the structures above 1.8 eV to be larger. This is the qualitative difference between the two experimental results, but a more refined theory will be necessary in order to make more quantitative statements regarding the two measurements. Population inversion of the vibrational levels of the ground state of N2 is an important practical consideration with regard to lasers. Golden and Ormonde (1974) have suggested that inverted populations are likely due to electron impact excitation for impact energies close to negative-ion resonances such as the ground state of N;. Laser action on the 00'1 -10'0 vibrational-rotational transition of COz in an N2-C0, discharge was first reported by Patel (1964). This is one of the most efficient lasers built to date (- 8%). It was known from the work of Kaufman and Kelso (1958) and Dressler (1959) that a low-pressure nitrogen discharge is a very effective means of producing vibrationally excited ground-state nitrogen molecules. Patel (1966)estimated that 10-30% of the N, was in the X1C; (t. = 1) state, and postulated that the N,X'Zi ( 0 = 1, 2, . . .)molecules were primarily produced by electron-ion recombination, atomic recombination, and cascade processes. Since N2 is a homonuclear diatomic molecule, the lifetimes for the decay of these vibrationally excited states are very long. Since it has been shown by Morgan and Schiff (1963) that the deactivation rate of the vibrationally excited molecules by molecule-molecule and molecule-wall collisions is quite slow, this is an efficient energy storage mechanism. Thus, the mechanism of electron impact
D. E. Golden
70
excitation of homonuclear diatomic molecules coupled with vibrational energy transfer to molecules possessing permanent electric dipole moments was suggested by Patel (1964, 1966) as an attractive one for near infrared lasers. The criteria for selecting molecules with vibrational levels that are likely to produce laser systems of this type have been discussed in detail by Moore et al. (1967). However, the electron energy distribution in a nitrogen glow discharge measured by Swift (1965) was much narrower than Maxwellian, with a maximum in the range 1.5-2eV. Furthermore, as Swift raised the gas pressure, the maximum of the electron energy distribution shifted toward lower energy and the number of high-energy electrons in the tail of the distribution was greatly reduced. The consideration of the work of Swift (1965) and Schulz (1962, 1964b), led Sobolev and Sokovikov (1966) to interpret the results of Patel (1964) by postulating that the excess population of N, ' C l (ti = 1) is due to resonance electron impact excitation ' : (v = 0) state. They also suggested that for N, 'C: ( u = 4), from the C energy transfer to the upper laser level of the CO, molecule can take place. The work of Potter et al. (1977), which uses detailed calculations of electron impact excitation and deexcitation for the vibrational levels of the ground electronic state, shows that electron collisions in the range of 2.5eV do, in fact, provide a very efficient means of populating these vibrational levels. Table V shows the average percent populations calculated by Potter et al. (1977)for v = 0, 1,2, 3,4 levels in the energy ranges 2.0-2.5 eV and 1.9-3.0 eV after a time of 0.5msec with an electron current density of 1 A/cm2. The results have been obtained using a uniform distribution of electron energies in the energy range considered in keeping with the observations of Swift (1965). It should be noted that for electron energies near the resonance, 85-90% of the molecules will be in vibrationally excited states. The results TABLE V
AVERAGE POPULATION (2,) I N THE FIRST FIVE VIBRATIONAL STATES OF THE GROUND STATEOF N, FOLLOWING ELECTRON IMPACT"
Vibrational state
0 1 2 3 4
Average population &) E = 1.9-3.0eV
Average population (%) E = 2.0-2.5eV
8.6 15.6 15.1 13.8 14.1
10.4 20.3 17.1 16.0 13.2
* From Potter et a/. (1977). J , , = 1 A/cm2, t = 0.5 msec.
ELECTRON ATOM AND MOLECULE SCATTERING
71
also suggest that a monoenergetic electron beam or electrons in a discharge with strong peak between 2.0 and 2.5eV can very efficiently excite the N, molecules used to energy transfer to the C 0 2 . While resonant excitation involving discrete rotational levels has been resolved in H2(see Ehrhardt and Linder, 1968)where the rotational spacing is particularly large, in other molecules with smaller rotational spacings these effects are more difficult to observe. However, Read and Andrick (1971) and Read (1972) have shown that resonant excitation of discrete rotational levels should produce a broadening of the energy loss spectrum corresponding to a vibrational level that depends on the scattering angle and is characteristic of the contributing partial waves. The maximum value of AJ is restricted by selection rules. Whereas for the S-wave, AJ = 0 and little broadening will be observed; when A J # 0 is allowed, the broadening will be much greater. These effects due to rotational transitions have been observed by Comer and Harrison (1973) in resonant e--N2 scattering at an incident energy near the ,I-I, state of N; for scattering angles from 30 to 90". An energy loss spectrum corresponding to the X'C: (v = 1) exit channel is shown in Fig. 39. The measured FWHM for the unbroadened profile (from elastic scattering in helium) was about 24 meV, while the
I
0.26
0.30
0.34
Energy loss,(eV) FIG.39. Energy loss spectrum in N, corresponding to the '2; (1. = 1) exit channel for an incident electron energy of 3.05eV at 60 (from Comer and Harrison, 1973). The broadened profile is due to scattering. The experimental points are shown by circles, the apparatus function by squares, and the calculated profile by a solid line.
D.E. Golden
72
observed profile at this angle (60")was about 38 meV. The Doppler broadening in the experiment was found to be insignificant by showing that essentially the same unbroadened profile was obtained when the nitrogen target was replaced by either helium or argon. Structures have been observed in the range 7-11 eV by Brongersma and Oosterhoff (1967), Brongersma et al. (1969), Hall et al. (1970), and Sanche and Schulz (1972b). The derivative transmission spectrum of Sanche and Schulz (1972b) is shown in Fig. 40. All of the above authors concluded that the structures observed between 7 and 9eV were due to excitation of the vibrational levels of the B3n, state of N,. Between 9 and 11eV, 18 vibrational levels are observed that cannot be correlated with any states or combination of states of N, and must therefore be interpreted as coreexcited shape resonances (states of N;). The excitation of the A3C: state was studied by Mazeau et al. (1973a) from threshold (7.18 eV) to 12.5eV at scattering angles of 20, 70, and 90',
a' '1;
V-4
6
8 10 12 14 16
f
I'
J V=6
8 10 1214
band "a"
=O 1 2 3 4 5 6
I
l
a
l
l 9
l
l
l
10
l 11
l 2
E L E C T R O N E N E R G Y , (eV)
FIG.40. Derivative of the transmitted electron current vs. energy in N, from 7 to 12eV (from Sanche and Schulz, 1972b).
ELECTRON ATOM AND MOLECULE SCATTERING
I
E5
,
,
,
.
,
/
,
,
73
/
Irnxdd ekctm mtrqyw)
FIG.41. Differential e--N, excitation functions of the A3X: (G. = 3-6) states at 90' for incident energies from 8.2 to 9.7 eV (from Mazeau et a/., 1973a). (Copyright by The Institute of Physics. Reprinted with permission.)
and the excitation of the B3n,from threshold (7.77eV) to 13.2eV at scattering angles of 20,60, and 120". At 100-meV energy resolution and with a coarse energy step size, three broad overlapping peaks at 8.0, 8.8, and 9.6eV were observed in the excitation of the A3C: state at 90" for the levels 3 I v' I 6 is shown in Fig. 41 and for B3ng 1 I u' s 5 in Fig. 42. The oscillations were interpreted by Mazeau et al. (1973a) as core-excited shape resonances of rIg symmetry for the A state excitation a nu, a Cl,and/or a (3drIJN; state resonances for the B state excitation. Feshbach resonances are likely to occur below Rydberg excited states of molecules, and the lowest Rydberg state of N, has been given by Mulliken (1957)to be the E3C; state at 11.87 eV. Such a resonance was first observed by Heideman et al. (1966) at 11.48eV. This resonance may be seen in the work of Sanche and Schulz (1972b) shown in Fig. 40. The observations of Comer and Read (1971) gave the symmetry of this state as ' C l and produced two higher members of this resonance series at 11.75 and 12.02eV. This resonance series has been designated b and is a series of Feshbach resonances associated with the closed E3C; channel consisting of two 3sa, electrons with an N: 'C; core. The lowest of the b series has been observed by Kisker (1972) in the optical excitation of the C3n, state of N,, and structure has been observed near the thresholds of the emission functions of the second positive system of N, by Finn et al. (1972). A peak in the excitation of the E3C; state of N, just above threshold at 11.92 eV (50 meV above threshold) has been identified as a P-wave shape resonance associated with the E state. This resonance has been shown to be the first member of a
74
D . E. Golden
Incidmt electron enerqy (eV1 FIG.42. Differential e--N, excitation functions of the B'n, (1.' = 1-5) states at 90" for incident energies from 8.6 to 11.6 eV (from Mazeau et al., 1973a). (Copyright by The Institute of Physics. Reprinted with permission.)
series of shape resonances of ' 2 ; symmetry associated with the vibrational levels of the E state by Mazeau et al. (1973b). Structure in the excitation of the E state has also been observed by Lawton and Pichanick (1973) by studying the production of metastables. By studying the C3n,+ B3H,(0, 0) emission, Freund (1969) found a resonant excitation peak, which was attributed to cascade from the E to the C state. This process was further studied by Kurzweg ej al. (1973) in a delayed coincidence experiment, which showed that the lifetime of the C -+ B (0,O) decay due to direct excitation was 37.4 nsec, while the cascade contribution had a lifetime of 10 p e c at a pressure of 20 mTorr. The C -+ B (0,O) emission was studied further in a high electron energy resolution delayed-coincidence experiment by Golden et al. (1974a) in order to carefully study the resonance contribution. The gross features in the total emission function measured by Golden et al. (1974a) for 300 meV energy resolution are shown in Fig. 43. The previous results of Burns et al. (1969) and Kurzweg et al. (1973) all normalized at 14eV are also shown in Fig. 43. The result shown is also in good agreement with the previous measurements of Jobe et al. (1967) and Aarts and DeHeer (1969). At 16eV, the emission function has dropped by about a factor of 2 from its value at 14 eV. One might expect such a sharply peaked emission function since we are dealing with a triplet upper state. In the case of e--He triplet excitation, Ormonde and Golden (1973) have
ELECTRON ATOM AND MOLECULE SCATTERING 2 4 0 K ,
I
I
I
,
I
ELECTRON ENERGY
,
I
75
I
bV1
FIG.43. Total emission function for N,C3n, 4 B 3 n , (0,O) due to Golden et ul. (1974) at about 300 meV energy resolution for electron energies from about 11 to 16 eV. The total emission results of Burns et ul. (1969) (0)and the prompt emission results of Kurzweg et al. (1973) ( + ) normalized to the total emission results of Golden et a/. (1974a) at 14eV.
suggested that temporary negative-ion formation be considered as a mechanism for triplet excitation. In the resonant case, triplet excitation comes about from the spontaneous decay of the excited part of the three-electron configuration, and one does not need to flip a spin as in the case of direct triplet excitation. This mechanism allows larger triplet excitation cross sections. In the present case, many overlapping vibrational resonances are involved. Golden and Ormonde (1974) have suggested that at a negative-ion resonance one may expect the deexcitation cross section to be smaller than the excitation cross section by electron impact, so that an inverted population is possible. It should be noted that the Born calculation of Stolarski et a/. (1967) and the Ochkur-Rudge calculation of Cartwright (1970), neither of which can account for a resonance mechanism, give a much broader peck for the excitation of the C state. The total emission results of Golden et al. (1974a) at 100 meV energy resolution are shown in Fig. 44, and the delayed emission results in Fig. 45. One can see that the total emission results contains many features, while the delayed emission results is almost purely resonant. While many of the features observed in the metastable production experiment
D.E . Golden
76
ELECTRON ENERGY IeV)
FIG.44. Total emission function for N,C3H, + B'H, (0.0) due to Golden ri at 100meV energy resolution for electron energies from about 11 to 15.4eV.
UI. (
1974a)
of Lawton and Pichanick (1973),the shapes of the higher energy portion of the curves do not agree. This is probably due to the fact that the metastable detector used by Lawton and Pichanick (1973) did not discriminate against the strong UV emission coming from the C + B transition at the higher energies. The positions of the structures observed by Golden et a/. (1974a) are given in Table VI together with the positions of structures observed by
- 1200 z I
9 1000
e
N
E
600
v)
t-
z
3 0
400
0
200
FIG. 45. Delayed emission function for N,C3n, -+ B3H, ( 0 , O ) due to Golden et ol. (1974a) at 100rneV energy resolution for electron energies from about 11 to 15.4eV due to collisional energy transfer from the E3Z: state.
TABLE VI POSITIONS OF STRUCTURES IN
e--N,
SCATTERING FROM
11.48 TO 14.57eV"
Present
Desig.
21; h Threshold
Electron transmission (dips)
11.4S4 11.75 12.03 11.87
'z: shape Threshold
12.15 12.34 12.25
C 4B Total emission (peaks)
11.48g 11.75 12.02
C -+ B Delayed emission (peaks)
Electron transmission Elastic onlyb
Elastic and inelastic'
Elastic and inelastid
11.48 11.75
11.49 11.76
11.48 11.76 12.02 11.87
12.03 11.87
11.92 12.13 12.34 12.23
Differential scattering
11.92 12.12 12.33
Inelastic'
11.92
12.23
11.90(E.a) 12.15(E,a) 12.40(E,u) 12.21
'nu 'z: (.id
12.44 12.68 12.53 12.78 12.98 13.21 13.42 13.63 13.84 14.04 14.24 14.44 13.08 13.31 13.52 13.74 13.94 14.41 14.34 14.54
12.42 12.70 12.55 12.80 12.98 13.20 13.41 13.65 13.84 14.05 14.26 14.46 13.33 13.52 13.74 13.94 14.14 14.34 14.54
12.70 12.54 12.80 12.90 13.21
13.52
13.88
12.14(E) 12.40(E) 12.70(€) 12.54(E, a", C) 12.78(E,a". C) 12.99(E, a", C) 13.21(€, a") 13.45(E, a") 13.67(E, a") 13.83(E,a")
13.50 13.70
13.72(E, a")
12.64 12.87 13.00 13.23
From Golden ef ul. (1974a). Heideman ef al. (1966). Sanche and Schulz (1972b).
14.12 14.36 14.57 Comer and Read (1971). Mazeau ef al. (1973b).
12.59 12.80 13.03 13.24
13.52
~
a
12.15 12.23
2n" shape
Total metastable production'
~~
Lawton and Pichanick (1973). Calibration point.
78
D. E . Golden
Heideman et a / . (1966), Comer and Read (1971) Sanche and Schulz (1972b), Mazeau et al. (1973b), and Lawton and Pichanick (1973). The structures observed in the electron transmission results of Golden et al. (1974a) at 11.87 eV and 12.25 eV were attributed to cusps on the E state and a" state thresholds. The symmetry designations for the lowest energy series is due to Comer and Read (1971). The other symmetry designations are due to Mazeau et al. (1973b). An additional series (the last series listed in Table VI) was identified by Golden et al. (1974a). This transition was studied further by Burns et a/. (1976), who measured the collisional energy transfer rate from the E state to the C state to be 1.9 x lo3 sec-' mTorr-' and the collisional deactivation rate of the E state to be 3.8 x lo3sec-' mTorr-'. For energies above ionization, Pavlovic et al. (1972) interpreted their results in terms of resonances associated with doubly excited states of N, , which is in agreement with the works of Truhlar et a!. (1972). The ejected electron spectra produced by autoionizing transitions in N, was studied by Hicks et al. (1973) at constant energy loss. Due to the method used by Hicks et al. (1973) the structure observed was due to transitions between discrete states involving electron ejection. The main features observed in the electron emission range 0.6-1.4eV are three sets of levels of N, decaying to vibrational levels of the X2C' state of N:. These autoionizing states were observed in photon absorption by Ogawa (1964) and were referred to as NP1, NP2, and NP3. Other prominent features were assigned to the rn = 3,4 members of Hopfields Rydberg series converging to the B2X: ( c = 0) N: state. The autoionizing nature of all of these states has been discussed by Berkowitz and Chupka (1969). Other structures between 0.5 and 3.0eV cannot be assigned to any transition between known states of N, and N:. However, transitions to the X 2 Z l levels of N: seem likely. The situation seems to be similar to that in the formation of H t but more experimental work will be necessary in order to make more definitive statements.
REFERENCES Aarts. J . F. M.. and DeHeer, F. J . (1969). ~ ' h l W 7 .Ph.v.s. f x r r . 4. 116. Abram, R. A., and Herzenberg, A. (1969). Chem. Phys. L e f f .3, 187. Andrick, D. (1973). Adc. A t . Mol. Phys. 9. 207. Andrick, D., and Bitsch, A. (1975). J . Phys. B 8, 393. Andrick, D., and Ehrhardt. H. (1966). Z. Phys. 192. 99. Andrick, D., and Langhans, L. (1975). J . Phys. B 8 , 1245. Andrick, D., Eyb, L. D., and Hofmann, M. (1972). J . Phys. B5,L15. Auger, P. (1925). J . Phys. Radium [6] 6 , 205. Baldwin, G. C . (1974). Phys. Rev. A 9. 1225. Banks, D., Percival, I . V., and Valentine, N . A. (1969). VZth ICPEAC, p. 215. MIT Press, Cambridge.
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Takayanagi, K., and Geltman, S. (1965). Phys. Rev. 138, A1003. Taylor, A. J., and Burke, P. G. (1967). Proc. Phys. Soc. London 92, 336. Taylor, H. S. (1970). Adv. Chem. Phys. 18, 91. Taylor, H. S., and Harris, F. E. (1963). J. Chem. Phys. 39, 102. Taylor, H. S., and Thomas, L. D. (1972). Phys. Rev. Lett. 28, 1091. Temkin, A. (1966). Phys. Rev. Lett. 16, 835. Temkin, A., and Sullivan, E. (1974). Phys. Rev. Lett. 33, 1057. Temkin, A,, Bhatia, A. K., and Sullivan, E. (1968). Phys. Rev. 176. 80. Rev. A 5, 166. Temkin, A., Bhatia, A. K., and Bardsley, J. N. (1972). P/ZJLY. Tronc, M. Fiquet-Fayard, F., Schermann, C., and Hall, R. I. (1977). J . Phvs. B 10, 305. Truhlar, D. G . (1974). Adu. Chem. Phys. 25, 21 1. Truhlar, D. G., Trajmar, G., and Williams, W. (1972). J. Chem. Phys. 57,3250. Vinkalns, I., and Gailitis, M. (1968). Proc. Int. Con$ Phys. Electron. At. Collisions, 5th, 1967 p. 648. Walton, D. S., Peart, B., and Dolder, K. (1970). J. Phys. B 3 , L148. Wannier, G . H. (1953). Phys. Rea. 90, 817. Weingartshofer, A,, Ehrhardt, H., Hermann, V., and Linder, F. (1970). Phys. Rev. A 2, 294. Weingartshofer, A,, Clarke, E. M., Holmes, J. K., and McGowan, J. W. (1975). J . Phys. B 8, 1552. Wheeler, J. (1937). Phys. Rec. 52, 1107. Whiddington, R., and Priestley, H. (1934). Proc. R. Soc. London, Ser. A 145, 462. Whiddington, R., and Priestley, H . (1935). Proc. Leeds Philos. Lit. SOC.S r i . Sect. 3, 81. Wigner, E. P. (1948). Phys. Rev. 73, 1002. Wigner, E. P., and Eisenbud, L. (1947). Phys. Reu. 72,29. Williams, J. F. ( I 976). J . Phys. B 9, 15 19. Williams, J . F., and Willis, B. A. (1974). J . Phys. B7, L61. Wilson, W. S. (1935). Phys. Rev. 43, 536. Wollnik, H. (1967) In “Focusing of Charged Particles” (A. Septier, ed.), Vol. 2, pp. 163-202. Academic Press. New York. Wong. S. F., and Schulz, G. J. (1974). Phys. Rev. Lett. 32, 1089. Wu, T. Y. (1934). Phys. Rea. 46, 239. Wu, T. Y. (1936). Philos. Mag. [7] 22, 837. Yarlagadda, B. S., Csanak, G., Taylor, H. S., Schneider, B., and Yaris, R. (1973). Phys. Rec. A 7. 146.
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ADVANCES IN ATOMIC AND MOLECULAR PHYSICS, VOL.
14
THE ACCURATE CALC ULAT I 0N 0F A TOMIC PR 0PER TIES B Y NUMERICAL METHODS BRIAN C . WEBSTER Department of Chemistry Unicersity of Glasgow Glasgow, Scotland
M I C H A E L J . JAMIESON Department of Computing Science University of Glasgow Glasgow, Scotland
R O N A L D F. STEWART* Center for Astrophysics Harvard College Obsercatory Cambridge, Massachusetts
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Self-Consistent-Field Equations and Finite Difference Methods . . . . . . . . . . . 11. Time-Independent Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Pair Functions and Pair Correlation Energies . . . . . . . . . . . . . . . . . B. Dispersive Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Atomic Polarizabilities and the Geometric Approximation . . . . . . . . . . . . . . . 111. The Solution of Coupled Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Time-Dependent Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. TheTDHF Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Numerical Solution of the TDHF Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Applications to 3-, 4-, lo-, 12-, and 18-Electr V. Conclusion .................................. ............... References . . . . . . . . . . . . . . . . . . . . .......................
88 88
100
102 106 109
109 1I1 114
121 122
* Present address: ICI Corporate Laboratory, P.O. Box No. 11, The Heath, Runcorn, Cheshire WA7 4QE. England. x7 Copyright @ 1978 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN o - I ~ - ~ o ~ x I ~ - ~
Brian C . Webstev et al.
88
I. Introduction A. SELF-CONSISTENT-FIELD EQUATIONS AND FINITE DIFFERENCE METHODS 1 . A Brief Historical Overview
An exact solution of the Schrodinger equation for an atom possessing n electrons can be expressed in the form
41(rJ 42(rl) $0
= (n!)-1'243('1)
4n(v1)
41(r2)
4l(r3)
. . . 41(r,,)
42(r2)
42(r3)
42(yn)
43(r2)
43(r3)
43(rn)
4n(r2)
4n(r3)
*
.
4n(rn)
(2)
CALCULATION OF ATOMIC PROPERTIES
89
tions and a two-electron function x i j ( r i ,r j ) : a(i,j) = ( n ! ) -li2AJ41(r1)42(r2). . ‘4i-l(ri- 1 ) x
(3)
4j- ~ ( ~ j - ~ ) 4 i + l ( ~ i + ~ ) 4 j + l ( r j + l ) ~ i ~ ( r i , r j ) l
where A is an antisymmetry operator. The third term of Eq. (1) represents a three-electron approximation, and with a wave function in this form, Szasz (1962) formulated a general theory of correlated wave functions. Attempts to obtain correlated atomic functions are myriad, with several excellent reviews available (Silverman and Bridgman, 1967; Kelly, 1968; Sinanoglu, 1969; Nesbet, 1971; Handy, 1975). If in Eq. (1) t,bo refers to a Hartree-Fock basis, then the two-electron functions are provided by a set of equations originally derived by Sinanoglu (1961, 1962), though links to the work of the Russian school can be traced (Fock et al., 1940). Should the one-electron functions be determined simultaneously with the two-electron functions, then the one-electron functions satisfy equations developed by Nesbet (1958),and independently by Szasz (1962). A notable advance came with the report by McKoy and Winter (1968) of a numerical method for solving the first-order pair equations in the formulation of Sinanoglu for the helium atom. This stimulated a resurgence of interest in numerical techniques. The purpose of this chapter is to show that finite difference methods when linked with an effective extrapolation technique permit highly accurate solutions to be obtained for a variety of problems involving atomic interactions. We begin by exemplifying Richardson extrapolation in treating the interaction of a hydrogen atom with a point charge, for which the exact solution is known. Atomic units are used unless otherwise stated. 2. Richardson Extrapolation
Extrapolation methods have long been available (Fox, 1962). They represent a most efficient technique for obtaining accurate solutions to ordinary differential equations, partial second-order differential equations in two variables, or combined with sequence-to-sequence transformations, of which Aitken’s d2 transformation is the best known, they permit the solution of coupled equations in one variable. Hartree’s method for solving * an equation of the type (41, with specified boundary conditions
and where f ( x ) ,g ( x ) are known, can be expressed as
Brian C. Webster et al.
90
where 6 is a central difference operator and h the step length. Fox has shown that provided a polynomial expansion in h for the error is made, then the function p(x) is given by m
p ( ~ )= @(x, h)
+ C Ai(x)h2'
(6)
i= 1
Accordingly, the global truncation error in Hartree's method is O(h2).This error can be reduced significantly by applying Richardson extrapolation, bringing a deferred approach to the limit h = 0 (Richardson and Gaunt, 1927; Bolton and Scoins, 1956). As an indication of the accuracy afforded by this technique, the perturbation of a hydrogen atom by a point charge is apposite (Stewart, 1973a).The under the first-order perturbation equation (7) has an exact solution perturbation H'. The second-order energy can be represented by the multipole expansion of Eq. (10): ( H o - Eo)t+bl = (E' - H')$o $'
(7)
f R-('+')(: I + lc-) telx p ( - r ) P , ( c o s R )
= I=
1
For a numerical treatment of the problem, $' is written in the form a
$'
=
C
(TC-
1'2~-1fi(r)~I(cos Q)
(11)
1=1
After integration over the angular variables a family of radial equations is obtained for J;(r), the components of *1 :
Representing the differential by a second difference approximation, the problem is seen to be reduced to the solution of a set of linear algebraic equations, with a boundary condition upon f;(r) as J ( r ) + 0, r -+ co :
CALCULATION OF ATOMIC PROPERTIES
91
TABLE I THECOEFFICIENT B4 OF THE SECONDFOR A HYDROGEN ATOM ORDERENERGY PERTURBED BY A POINTCHARGE, a CALCULATED NUMERICALLY Grid size (strips)
'I
B4(h)
60 70 80 90 100 500
- 2.250473 -2.250019
Cutoffat r
= 20a.u.
B4(0)
- 2.251 314 - 2.250965
- 2.250000
- 2.250739
- 2.250000
- 2.250584
- 2.250000
An effective procedure for obtaining a solution of the linear equations is to apply a coordinate transformation r + x2 (Winter et al., 1970a), and use the method of successive overrelaxation. With a square root grid of 100 strips the coefficient of R - 4 is calculated to equal -2.250473 for a cutoff at r = 20 a.u., -2.249326 for a cutoff at r = 10 a.u., and -2.251314 if the strip size is reduced to 60 strips, the cutoff being 20 a.u. Such variability from the exact solution B4 = - 2.250 might suggest this method yields only a coarse result, to be improved by increasing the number of strips and maintaining a large cutoff distance. Table I lists values of B, for grid sizes of 60(10)100 strips, with a cutoff at r = 20 a.u. For comparison, B, is - 2.250019 over a 500-strip grid. If B,(h) is expanded as a series in even powers of h, a first extrapolant, using values of B,(h) at three different grid sizes, converges upon the exact result, as can be seen from the second column of Table I: B,(h)
=
B,(O)
+ C2h2+ C4h4 + C,h6 + . . .
(14)
A more sensitive test of accuracy is provided by the coefficients of the odd powers of R-", since these originate in the third order of the perturbation. Table I1 contains values of B , ,B , , and B , to show the influence of the cutoff distance. Exact values for B7 = - 53.25, B9 = - 886.5, and B1 = - 21217.5 (Kreek and Meath, 1969) are obtained when the cutoff equals 25 a.u. In our experience this numerical technique is of general applicability, resulting in highly accurate solutions to perturbation problems posed to third order. Some differing areas of application are described next.
,
Brian C. Webster et al.
92
TABLE 11 THECOEFFICIENTS B,. B,, B, FOR A HYDROGEN ATOMPERTURBED BY A POINTCHARGE, CALCULATED NUMERICALLY
10.0 15.0 20.0 25.0
-857.79528 -886.40380 - 886.49990 -886.50000
-52.835021 -53.249372 - 53.250000 -53.250000
- 19219.071
-21203.122 - 21217.475 -21217.500
11. Time-Independent Applications A. PAIRFUNCTIONS AND
PAIR CORRELATION
ENERGIES
I . The Pair Equation The calculation of pair functions for atoms usually has been confined to variational procedures. An admirable paradigm is presented by Byron and Joachain (1967a) in their study of four-electron systems. First we shall summarize the one-electron approximation as applied to the beryllium atom. The wave function $, for Be(ls)'(2s)' is $0
= (24)- " " ~ ~ l ~ t ( ~ l ) ~ l ~ l ( ~ ~ ) ~ Z s (15) ~ ~ ~ ~ ~ ~ 2
satisfying the Hartree-Fock equation Ho*o
=
(16)
Eo*o
with E,
= 241s)
+ 242s)
(17)
where ~ ( 1 s and ) 42s) are the one-electron orbital energies. Denoting the space part of the one-electron function by 4 ( d ;r), 41st(y) =
4(1s;r)x,
414(r) = 4 ( 1 s ; y ) P
These one-electron orbitals two coupled equations
4 2 s T ( r=) 4(2s;r)cc 92sl(r) =
4(2s;r)P
(18)
4(1s;r ) and 4(2s; r) are obtained by solving the
+ { T ( r )+ 2Vl5(r) + I/2s(r) - Ys(r)}4(2s;r ) = ~ ( 2 ~ ) 4 ( 2r )s ;
{ T(r)+ T/~\(Y) 2VZs(r)- Ks(r)i 4(ls; r ) = 4 l s ) 4 ( l s ; r )
(19)
CALCULATION OF ATOMIC PROPERTIES
93
with
and with similar definitions for I/zs(r) and V;s(r). Regarding the difference between the exact electron interaction and that represented by the Hartree-Fock potential as a perturbation H',
H'
=
ci > j rli j c4 --
V(Tk)
k=l
with
V ( r )=
VlSt(4
+
VlSl(4
+ V 2 s t ( r ) + V2slW
(23)
then the first-order energy E' = (Il/,lH'l$,> can be found simply, in terms of coulomb and exchange energies :
E'
= - V ( ~ SIS) ; - 4 V ( l ~ ; 2~ )V
+
( ~ S ; ~ S 2Ve(ls;2s) )
(24)
The sum E, + El is the Hartree-Fock energy, the difference between this quantity and the total nonrelativistic energy defining the atomic correlation energy. Byron and Joachain commence with the functional F[$'] :
Fl[Il/\l =
= 0,
k # i,j
(27)
Brian C. Wehster et al.
94
If F[V] is now developed and the functions xij(r,r') varied to minimize F[$'], a set of six pair equations is obtained, with each equation of the form
[HHF(r)+ HHF(r') - ei - ej]xij(r,r')
x (2)- ''2[4i(r)+j(rf)
-
4i(r'Mj(r)]
(28)
where eijis the contribution of the pair ij to the energy E'. Thus,
V(ls, 1s)
E'(1StlSJ)
=
e'(lsf2st) E'(lSf2SJ) E'(2St2Sl)
= E ' ( l S 1 2 S l ) = - V(lS,2S)
-
= E'(lSJ2Sf) =
-
=
-
+ V'(lS,2S)
V(ls,2s)
V(2s, 2s)
Comparison with Eq. (7) reveals that F[$'] has been minimized at $' = t,!I1, and Eq. (28) is indeed a first-order perturbation equation that might have been cast intuitively for each electron pair, interacting directly, under the field of the nucleus and the Hartree-Fock field of the other pair of electrons. With $' orthogonal to $o, the second-order energy E 2 is simply E 2 = ($'Iff1
-
E'II,~~)
or partitioned into contributions from each pair,
E 2 = C E$ i, .i
where
- I q r ) - Jqr') -
x [$i(rMj(r')
-
€6
>I
(2)- l i 2
4i(r')$j(r)l
Similarly, the third-order energy E 3 is E 3 = ($'Iff' or
- E'
I$')
\
(29)
CALCULATION OF ATOMIC PROPERTIES
95
if off-diagonal interactions manifest in E 3 are sufficiently small as can be neglected. It is to be expected that E 2 + E 3 will provide the major portion of the correlation energy. 2. Solution of the Pair Equation
Taking a partial wave expansion to simulate the pair function zij(rl, r 2 ,012) as suggested by Luke et al. (1952) and expanding r;; in a familiar way, the pair equation can be uncoupled into a set of equations for the partial wave coefficients U l ( r l ,r2). For a spherically symmetric pair as ( l ~ ) ~ ,
with r , , r > signifying the smaller and the greater of rl and r 2 , respectively. Assuming a hydrogenic form for the zeroth-order Hamiltonian, as compared with the Hartree-Fock Hamiltonian, the partial wave coefficients satisfy the equations
= ~ ( l sr1)P(1s, , r2)(~ ( l sIs) , - r;
l)
(35)
for the s-wave, and for higher waves
1
- 241s)
r',
Ul(r1,r2) = -P+lP(ls;rl)P(ls;rz)
(36)
>
where P(nl; r ) = rR(nl; r). These are the differential equations that McKoy and Winter (1968) solved , -+ 0; rl -+ 0, co; numerically, subject to the boundary conditions U l ( r l r2) r2 -+ 0, co.A second difference approximation is used for the partial differential operators :
with the boundary conditions expressed as u(R, r z ) = u(rl, R ) = 0, R being adjusted to facilitate solutions on a finite mesh. For example, with R = 5 a.u.,
96
Brian C. Webster et al.
and h = 0.25, there are 361 points on the mesh at which a solution need be constructed. O n this mesh a calculation of the s-wave for helium is accomplished in 3 sec (execution time on an IBM 7094) yielding the value E2(1= 0) = -0.12605 a.u. Extending the boundary to R = 6 a.u. has no significant effect on this energy: E2(l = 0) = 0.12607 a.u. By performing a Richardson extrapolation for E2(h)over a range of grid sizes,
+
E2(h)= E2(0) C2h2+ C4h4 + C6h6
+ ...
(38)
an accurate value for E2(1= 0) = -0.12531 a.u. is obtained. Winter and McKoy (1970) later refined this result on a square root grid, using a fourth difference approximation, to E2(l = 0) = - 0.125327 a.u. The limiting total energy of the helium atom for I = 0 is described by Shull and Lowdin (1959) as the radial limit. Schwartz (1962) placed this limit for helium at - 2.879028 f 1 x a.u. Extrapolating upon the s-limit energy, Winter et al. (1970a) predict an energy of - 2.879030, for the radial limit. Moreover, since Eq. (36) can be solved numerically with little effort, higher limits are located swiftly, a g-limit equaling - 2.90351 a.u. A classic result of Pekeris (1962) for the helium ground-state energy, is a benchmark, E = - 2.90372 a.u. It is Schwartz' contention that contrary to earlier opinion a configuration interaction type of trial function to represent U l ( r l ,r2) will converge only slowly and will require many terms to match the higher waves, Udr1,7.2) = 1.
c
m ,n
CI,m, nrlrZ(r;"Y.;
+ rlJ;)
the reason being that with increasing I, the function U l ( r l r2) , is concentrated ever more highly about rl = r 2 . Schwartz suggested that if the inclusion of the interelectronic coordinate r I 2 in a trial function proves intractable, at least a function that allows for some radial correlation should be selected, proposing for investigation a function of the type
C
Uz(rl,r2)=
C1,m,n~Ir2ym>r% exp[-2(rl
+ r2)I
(40)
1, m, n
Byron and .Joachain (1967b) conducted such an inquiry, and with a 30-term expansion of the radial correlated function, with - 1 d m + n d 4, calculated E 2 ( / = 0) = -0.125324a.u.andE' = C z E 2 ( / = ) -0.157656a.u.,verifyingan asymptotic relation for E 2 ( l )proposed by Schwartz (see Byron and Joachain, 1967b, footnote 8a; also see White, 1967) that
E2(I) = -(45/256)(1
+ &4[l
-
$(l
+ +)-' + O(IP4)],
( I >> 1) (41)
In contrast, the configuration-type function, Eq. (39), with 20-terms, + 17 < 7, yielded E2(I = 0) = -0.125031 a.u., E 2 = -0.155873 a.u., with
111
CALCULATION OF ATOMIC PROPERTIES
97
poor asymptotic behavior, E2(5)= - 1.18 x a.u., while from Eq. (41), E 2 ( 5 ) = - 1.84 x lOP4a.u. The numerical approach would seem to have resolved this difficulty for the higher waves. With finite difference methods, however, the situation rather is reversed with the computation of the s-wave constituting the major portion of a calculation. For open-pair situations, as the excited states of helium, He(ls,2s), 3S, and 'So, or L i ( l ~ ) ~ ( 2 where s), the s-wave function is diffuse, there are difficulties in securing an accurate result for the s-wave. Winter and McKoy (1970) report that for He(ls,2s), 3S, and 'So, a mesh of 14,161 points, at which the s-function is other than zero, is of insufficient size though requiring around 1 hour of computing time (JBM, 360/75). Accordingly, a procedure to adopt is to calculate the s-wave variationally and obtain the higher waves numerically, as suggested by Schwartz (1962). In Table I11 the partial wave contributions to the second-order energy for the ground state of the helium atom are compared in hydrogenic perturbation theory when a variational, numerical, and variational/numerical approach is adopted. For the variational/numerical study, the configurational form of TABLE I11 A COMPARISON OF PARTIAL WAVECONTRIBUTIONS TO THE SECOND-ORDER ENERGY,CALCULATED BY VARIATIONAL A N D NUMERICAL PROCEDURES IN HYDROGENIC hRTURBATION THEORY FOR THE GROUNDSTATE OF HELIUM
Variational"
Numericalb
Variational/ numerical'
-0.125334 - 0.026495 - 0.003906 -0.001077 - 0.000405 - 0.000183 - 0.000094 - 0.000053 - 0.000032 - 0.000021 -0.000014
- 0.125327
- 0.125327
- 0.026495
- 0.026497
- O.OO3905
- 0.003906
~
0 1 2 3 4 5 6 7 8 9 10 > 10 211
EZ
1
- 0.001076
-0.001077
- 0.000403
- 0.000406
-0.000181
- 0.0001 84
- 0.000092
- 0.000095
- 0.00005 1
- 0.000054
- 0.000030 - 0.000019
~
0.000032
- 0.000020
0.000052 0.000042 - 0.157656
~~
~
- 0.157579
Byron and Joachain (1967b). Winter and McKoy (1970). Stewart (1973a).
- 0.157651
Brian C. Webster et al.
98
function, Eq. (39),has been used: 54 terms, swiftly reproducing the numerical result for E2(0)calculated by Winter and McKoy (1970). The total secondorder energy E 2 = -0.157651 a.u. by this method, the difference from the variational energy of Byron and Joachain E 2 = -0.157656 a.u. mainly stemming from the s-wave representation. Partial wave contributions for 13 10 are estimated from the asymptotic relation, Eq. (41), and should this contribution be added to the result of Winter and McKoy then their energy E 2 = -0.157631 a.u. Total energies are listed in Table IV for the three methods, as applied to the helium atom. In hydrogenic perturbation theory, Eo = -4.0a.u., and E' = f1.25a.u. for the helium ground state, while the Hartree-Fock energy is - 2.86168 a.u. (Clementi and Roetti, 1974) compounded as E , = TABLE IV SECONDA N D THIRD-ORDER PERTURBATION ENERGIES FOR THE HELIUM ATOMGROUND STATE,CALCULATED BY VARIATIONAL AND NUMERICAL METHODS IN HYDROGENIC AND HARTREE-FOCK PERTURBATION THEORY, AND FOR THE HELIUM EXCITED STATES, He(ls, 2s), 'So, 3S, IN HYDROGENIC PERTURBATION THEORY
Variational
Numerical
Variationall numerical
Hydrogenic perturbation theory
He(1s)' IS,
EZ
- 0.157666"
- 0.157631h
- 0.1S7651'
E3 E
0.004349 -2.903317
0.008572 -2.899059
0.004374 -2.903278
Hartree-Fock perturbation theory He(1s)' 'So
E2
E3
E
-0.03725d - 0.00377 - 2.90270
- 0.037311'
-0.003665 - 2.902656
-0.03734' -0.00365 - 2.90267
Hydrogenic perturbation theory He(ls,2s)'S0
EZ E3
E He(ls, ~ s ) ~ S , E 2 E3 E a
- 0.1 14509"
- 0.1 144W
- 0.1 14499'
0.009415 -2.141445
0.009251 - 2.141586
0.0091SO -2.141700
- 0.047409" -0,004872 -2.176424
- 0.047406g -0.004876 -2.176425
- 0.047408' -0.004872 -2.176423
Winter and McKoy, (1970). Knight and Scherr (1963). Byron and Joachain (1967b). Winter et a / . ,(1970). Knight (1969). Winter (1970)
' Stewart, (1973a).
99
CALCULATION OF ATOMIC PROPERTIES
- 1.83592a.u. and E' = - 1.02576 a.u. Byron and Joachain (1967b) calculate values of -0.00085 and -0.00016 a.u. for the fourth- and fifth-order energies, respectively, which if subtracted from the result of Pekeris (1962), yields an energy of - 2.90271 a.u. for comparative purposes in Hartree-Fock perturbation theory. The variational/numerical approach, in which the s-wave is computed in a fraction of the time (10-2-10-3) of that in a total numerical computation, is seen to be highly competitive with accurate variational calculations. This is particularly in evidence for the excited states of He( Is, 2s), 'So and 3S,, where the comparison is with the variational study of Knight and Scherr (1963) and Knight (1969), entailing a large expansion of terms involving r 1 2 , the interelectronic coordinate to represent the pair function. Similarly, for four-electron systems the variational/numerical method permits a rapid calculation to be performed securing 80-90% of the correlation energy (Webster and Stewart, 1972). Table V presents a selection of results for the beryllium atom.
TABLE V THENONRELATIVISTIC ENERGY OF THE BERYLLIUM ATOM, BY A VARIETYOF METHODS CALCULATED
Method
Energy
Hartree-Fock" Hydrogenic perturbation theoryb Hylleraas function, 13-term' Multiconfiguration SCF, 10-termd Hylleraas function, 25-term' Hartree-Fock perturbation theory, to E 3 , variationalf Hartree-Fock perturbation theory, to E 3 , variational/numericaIg Transcorrelated functionh C.I., 55-term' C.I., 180-term' Bethe-Goldstonek Hartree-Fock perturbation theory, to E5,variational' C.I./Hylleraas function, 107-term"
- 14.5730
f a '
Ir
Correlation energy (7%
- 14.6579
0.0 76.2 81.4 86.5 90.0
- 14.6585
90.6
- 14.6593
91.4 91.7 93.2 96.7 97.7
- 14.6448 - 14.6497
- 14.6546
- 14.6596 - 14.6609 - 14.6642 - 14.6651
- 14.6655 - 14.6665
98.0 99.0
Clementi and Roetti (1974). Knight (1969). Karl (1966). Sabelli and Hinze (1969). Gentner and Burke (1968). Byron and Joachain (1967a). Webster and Stewart (1972). Handy (1969). ' Weiss (1961). j Bunge (1968). Nesbet (1967). Sims and Hagstrom (1971).
Brian C. Webster et a!.
100 - 0.10
- 0.08
/
Pair Correlation Energy, a.u
He,ls 1s
-0.06
;ls -0.01
i’
-0.02
-
~
0
6 7 8 9 10 Atomic Number FIG. 1. The variation of the (Is, Is), (2s,2s), and (Is, 2s) pair correlation energies (a.u.) with atomic number, for the beryllium isoelectronic sequence, and in full line the (Is, 1s) pair correlation energy for the helium isoelectronic sequence.
3
L
5
In Figure 1 is shown the variation with atomic number of the pair contributions to the correlation energy in the four-electron sequence Li--Ne6+, calculated by the variational/numerical method. Unlike the helium series, the ( 1 ~pair ) ~ contribution is seen to be nearly independent of atomic number, while the (2s, 2s) pair correlation energy follows a nearly linear dependence, as suggested by Linderberg and Shull (1960; see also Alper, 1969). B. DISPERSIVE INTERACTIONS Following from the example described in Section I,A,2, it is evident that the numerical approach could be well suited to the calculation of dispersion forces between atoms. Several methods have been proposed for determining dispersion coefficients (Dalgarno, 1967; Langhoff and Karplus, 1970; Starkschall and Gordon, 1971,1972) though none, to our knowledge, entails
101
CALCULATION OF ATOMIC PROPERTIES
a direct numerical solution of the problem. An exacting test is provided by the study of Deal (1972)on the long-range interaction between two hydrogen atoms in their ground state, Deal having obtained from a closed-form solution the coefficients B, of Eq. (10) to a very high accuracy. For two hydrogen nuclei a distance Ra.u. apart and with electron 1 associated with nucleus A, electron 2 with nucleus B, a perturbation H' can be expressed as
with H i , defining a dipole-dipole interaction, H i 2 + Hi, a dipolequadrupole interaction. Expressions for Mkl and 0 k l are given by Kolos (1967). Thus MI1 = ($)I/,, M,, = MI, = 1.0, M,, = (14/5)'12, and O I 1 = 6-112[Y;1Y;1 2YyY; + YiYi]. On expanding the first-order function $ I , maintaining the angular dependence of H I ,
+
substitution in the first-order perturbation equation (7) leads to a set of uncoupled elliptic partial differential equations for the coefficients uk,I ( r A 1 , rB2)of a familiar form:
=
-4.0Mk,ik,,rb, exp( - r,l)eXp(
(44)
-TB~)
From the coefficients Uk,l(rAl,rB2) the computation of the dispersion coefficients follows directly from B, = 4.0Mk1J U k l ( r A l ?
k + 2 1+2 r B 2 ) r A I rB2
exp(-rA,)exp(-rB2)dr,l
drB2
(45)
These equations have been solved numerically on a square root grid, employing a fourth difference approximation, reduced to second difference at the boundaries, a cutoff being selected for the interaction under consideration. Thus R = 20a.u. is a suitable cutoff for Ull(rAl,rB2).Accurate values for the B, coefficients are obtained through Richardson extrapolation. Dispersion coefficients B, for the interaction between two hydrogen atoms in their ground state and excited state (ls,2s) are given in Table VI. The agreement with the analytic results of Deal is quite remarkable. Other longrange atomic interactions, such as He-He, He-H, Li+-Li+, Li+-H, and
Brian C. Webster et al.
102
TABLE VI COEFFICIENTS
B,
GROUND
(Is, 1s) 'Z g , 'X " (ls,2s) ' Z8+' 'X+" +
+
( 1 S A
' Z J , 'XZ,.
INTERACTION ENERGY C,B,R -'BETWEENT W O HYDROGEN ATOMS IN THEIR STATE AND EXCITED STATE (IS, 2S), CALCULATED NUMERICALLY
FOR THE
B6
B8
BIO
BIO
dipole-dipole
dipole-quadrupole
dipole-octupole
quadrupole-quadrupole
- 6.4990267048"
- 124.399083573 - 124.399083583
-204,7355'; -204.736' - 148.769% - 148.769'
- 19,589,085
-2150.61437492 -2150.6143750, - 2,201,375.2
- 1135.214039898
- 6.4990267054b
- 19,588.6 - 16,607,735 - 16,607.2
- 2,077,542.6 -
' Stewart and Webster (1973a); Stewart (1973b)
- 1135.21403989, -308,301.36 -
-
Deal (1972)
- 227,987.44
Kolos (1967).
Lif-He, have been examined by Stewart (1973b)together with a consideration of three-body forces. The triple-dipole coefficient in the interaction H-H-H, for example, is calculated to equal 21.64246454,. C. ATOMIC POLARIZABILITIES AND THE GEOMETRIC APPROXIMATION The solution of the coupled Hartree-Fock equation for an atom in an external electric field has been achieved in only a few instances (Cohen and Roothan, 1965; Cohen, 1965; Billingsley and Krauss, 1972), recourse usually being taken to some procedure of simplification as in the coupled perturbed Hartree-Fock method (CPHF) and the Dalgarno uncoupled Hartree-Fock method (DUHF), reviews being given by Dalgarno (1962) and Langhoff et al. (1966). Other methods include the use of double perturbation theory, (Schulman and Musher, 1968) and many-body techniques (Kelly, 1964,1966; Caves and Karplus, 1969) while Sternheimer (1957, 1970) has followed a numerical approach. Static dipole polarizabilities of neutral atoms have been tabulated by Teachout and Pack (1971), who also give an extensive bibliography. Following Langhoff et al. (1966), for an n-electron atom subject to an external perturbation H1, representable by a sum of one-electron operators, the CPHF equations can be written
{ H V ) - €P)&(l)
+ (H'(1)
-
€:)40(1)
103
CALCULATION OF ATOMIC PROPERTIES
to be solved with the condition (@(4f)+ (+:I+?) = 0. Equation (46) remains unaffected if, in the summation, the terms i = j are suppressed, and HO(1) is reformulated accordingly as
(jti)
A simplification in the solution of Eq. (46) results now if all terms involving
4;
are discarded, other than that being calculated, the equation becoming
+
(HO(1) - €?)f#$(l) (H'(1)
-
€:)+;(1)
(48)
=0
with Ho(l) defined by Eq. (47). We refer to these equations, corresponding to method b of Langhoff et al. (1966), as the simplified coupled perturbed Hartree-Fock equations (SCPHF). In contrast, if the right-hand side of Eq. (46) is equated to zero, though the self-interaction terms are retained in the definition of ITo( l), then the DUHF approximation is defined. The polarizability cto calculated within this uncoupled approximation from the relation n
ctb
= 2.0
1 (+;lH'I+')
(49)
i= 1
can be corrected for lack of correlation in a manner described by Tuan et al. (1966). The first-order correction ctl to the polarizability cco for the helium isoelectronic sequence is m1
= 4(+1(l)+:s(2)lr~~l~1(l)+:s(2)) -
12( +1(1)+1(2)lrzl+:s(1)+:s(2D
(50)
Similarly, if the theory is formulated within the Hartree approximation, a first-order correction to the polarizability, (Schulman and Tobin, 1970) can be taken, in the helium sequence as a1 =
-
~(~1(~)+1(2)l~~~l+~~(~)+~s(2))
with
+:,(I)
= ( 4 ~ ) - ' / ~ 'P(1s; r;
rl)
(51)
For two-electron atoms, under a perturbation H'(1) = -r:P,(cos8) and with a first-order function +'(1) = (4n)- 'l2r; 'f;(rl)Pl(cos OJ, the coupled equation in radial form is 1 d 2 +--1(1 + 1) 2 dr2 2r:
Z rl
+
s
1
P2(ls;r')r;'dr' - ~'(1s) f;(rl)
104
Brian C. Webster et.al.
compared to the uncoupled Hartree-Fock method (UHF), for which
and the Dalgarno uncoupled method, for which
1
+ 2V(r,) - Ve(r,) - &'(Is) f;(rl) = riP(ls;rl)
(54)
where, as previously, V and V" are coulomb and exchange operators. These various approximations have been explored numerically, with accurate values for the multipole polarizabilities being calculated through Richardson extrapolation, in the isoelectronic sequences H - to Ne8+ (Stewart and Webster, 1973b), Li- to Ne6+ (Stewart and Webster, 1974), and the Ne, Mg, and Ar sequences (Stewart, 1975a), together with the use of model potentials (Stewart, 1974). The relative merits of each approximation can begin to be discerned in Table VII, listing dipole and quadrupole polarizabilities for helium and beryllium. While both of the uncoupled approximations, as can be expected, provide without correction very poor values for the dipole polarizability of helium, the selected value of Teachout and Pack being 1.3819 a.u., and a,(DUHF) = 0.9972 a.u., a,(UHF) = 1.4870,the correlation correction term a, brings a substantial improvement. Still greater accuracy can be achieved by recourse to the geometric approximation. With the polarizability expressed in a series a = a,
+ i a l + A2a2 + . . .
(55)
a geometrical summation yields a N a, = a,(l - a1/ao)-l. In this way, the uncoupled results are brought to within close range of the coupled values r,(DUHF) = 1.3167a.u., a,(UHF) = 1.3164, a(CPHF) = 1.3222a.u. For the beryllium sequence the SCPHF equations have been solved and are seen to be a competitive alternative for attaining to the coupled pertrubed result, a(SCPHF) = 45.566 a.u., a(CPHF) = 45.612 a.u., the value of Kelly (1964) r = 46.77 a.u. being a standard. Again, the geometric approximation with a,(DUHF) = 44.810 a.u. brings the uncoupled calculation to within a respectable margin of more arduous computations. It is our experience that this procedure, effectively for forcing convergence, is applicable generally. Static dipole polarizabilities for some neutral atoms and ions calculated in the uncoupled UHF approximation (Stewart, 1975a) and a simplified time-dependent Hartree-Fock study (Stewart, 1975d) are noted in Table VIII. The geometric approximation is seen to be quite applicable for larger systems; the static polarizability of magnesium, for example,
TABLE VII STATICPOLARIZABILITIES FOR HELIUM AND BERYLLIUM CALCULATED NUMERICALLY WITHIN A VARIETY OF APPROXIMATIONS DUHF
UHF
0.99722h 0.24195 1.2392 1.3167 1.8002 0.40685 2.2070 2.3258 30.556' 9.7199 40.725 44.810
1.4870, 1.4870" -0.1298 1.2942 1.3164 2.3591, 2.3606" -0.3410 2.3250 2.3255
45.566
CPHF
1.3222, 1.322,b
CHF'
1.3227
2.3260, 2.326b
45.612b
Other methods
1.2942"sd,1.3796'
2.3250d,2.3265", 2.4403'
45.5
45.28/, 46.77'
_____
Broussard and Kestner (1970),variational double pert. Lahiri and Mukherji (1966a,b), variational CPHF. Cohen (1965). Singh (1971 ), variational double pert. '' Davison (1966), correlated function. Gutschick and Mckoy (1973), variational H.F. ' Kelly (1964). many-body pert. th. Stewart and Webster (1973b). ' Stewart and Webster (1974). a
Brian C. Webster et al.
106
TABLE VIII
STATICDIPOLEPOLARIZABILITIES CALCULATED NUMERICALLY I N THE UNCOUPLEDHARTKEE-FOCK APPKOXIMAI lo\. AUD A SIMPLIFIEII HARTREE-FOCK APPROXIMATION TIME-DEPENUENT
F-
Ne Na+
7.4324 1.9978 0.8345
NaMg Al'
526.03 55.076 19.895
c1Ar K+
25.171 10.139 5.460
a
Stewart (1975a).
1.8049 0.2781 0.0808 260.21 16.035 4.488 1.8970 -0.1928 -0.3337 I, Stewart
9.2312 2.2559 0.9152 786.24 71.112 24.382 27.068 9.9463 5.1266
-
9.8161 2.3013 0.9239 1041 77.698 25.689
~
1200 81.06 26.76
27.222 9.9499 5.1458
30.46 10.68 5.428
(1975d).
at 77.698 a.u. being near to the simplified T D H F result of 81.06 a.u. and the fully coupled Hartree-Fock value of 81.25 a.u. (Kaneko and Arai, 1969). These values are slightly higher than obtained in other studies, as 74.9 a.u. by Stwalley (1970) or 72.8 a.u. by Laughlin and Victor (1973). The geometric approximation originated from the summing of higher order perturbation diagrams in a geometric series (Kelly, 1966, 1967; Schulman and Musher 1968; Caves and Karplus, 1969; Wendin, 1971), though the reason for its success has remained slightly obscure. Amos (1970)observed the relation of the geometric approximation to the FeenbergGoldhammer procedure for accelerating the convergence of a perturbation series, while recognizing the link to the screening approximation of Dalgarno and Stewart (1958). That the geometric approximation is equivalent to the use of a [1,0] Pade approximant has been noted by Broussard and Kestner (1970) (for the use of Pade approximants, see Brandas and Goscinski, 1970). We draw attention now to the geometric approximation as an example of the Aitken 6' transformation.
111. The Solution of Coupled Equations When several channels are considered, the calculation of polarizabilities, scattering matrices, and eigenvalues involves the solution of coupled differential equations. The coupling can be local, as in the collisional excitation of the rotational states of a diatomic molecule (Arthurs and Dalgarno, 1960),
CALCULATION OF ATOMIC PROPERTIES
107
or nonlocal, as in the collisional excitation of the electronic states of an atom or molecule (Burke, 1969), or indeed the Hartree-Fock and timedependent Hartree-Fock equations (Dalgarno and Victor, 1966). In certain cases, nonlocal coupling can be approximated by local coupling (Gordon, 1970). A set of N locally coupled equations can be solved by a generalized Numerov method. Thus the Numerov formula (Hartree, 1957)
d2@(X, h) = h2(1
+ i$d2)[f(X)@(X,
h)
+ g(x)]
(56)
can be applied to the set of equations
d 2 4 w l d x 2= [ W ) ] + ( x )
+ g(x)
(57)
where +(x) is an N-element vector of the unknown functions, g(.u) an N element vector, and [ V(.u)] an N x N matrix. Simple error analysis shows the leading error term to be 0(h4). At each step in the Numerov solution, a matrix must be inverted, which if performed directly is time consuming (Smith et al., 1966; Jamieson, 1971). Allison (1970) has taken advantage of the diagonally dominant nature of the matrix to develop a fast iterative inversion technique. In contrast, Gordon (1969) has introduced a method by which for a single differential equation the potential is represented by piecewiseacontinuous functions that over each step length are constant, linear, or quadratic, and for which independent analytic solutions of the equation are known. A detailed comparison of these methods is given by Allison. A serious difficulty can arise due to differing ranges of the solutions where more than one channel is closed, in that, while obtaining an independent trial inward solution, the outward solution becomes unstable. Such a situation occurs with the time dependent Hartree-Fock equations, in which positive and negative frequency components are involved. This problem can be avoided in limited circumstances by solving a matrix equation for a vector whose elements are the unknown functions at the pivotal points (see R. A. Buckingham, in Fox, 1962). The matrix elements are given by the terms in Hartree’s or Numerov’s formula and by the coupling terms. If the coupling is local, the method leads to inversion of a matrix with several zero elements, of which advantage may be taken. For nonlocal coupling the complete matrix has to be treated. The major disadvantage of the matrix method naturally is in the size of matrix to be inverted (or diagonalized in an eigenvalue problem) because of either the ranges of the solutions or the number of equations. Use of Richardson extrapolation enables larger step lengths to be taken, thereby reducing the matrix size, and since the coupling is usually small, iterative techniques (Wilkinson, 1965) or relaxation methods (Southwell, 1940) can
Brian C . Webster et al.
108
be adopted. The matrix method seems especially attractive for the coupled eigenvalue problem, since the eigenvalues satisfy a Bolton-Scoins relation, with the sign of the leading term dependent on the coupling. Yet the matrix problem is minor compared with the nonlinear problem of correcting a trial eigenvalue and starting conditions for a set of coupled equations. Gordon (1970) suggests a method by which only the trial eigenvalues need be adjusted. Possibly the most satisfactory procedure for circumventing the “ranges” difficulty is to solve the coupled equations as a diagonal set and introduce the off-diagonal coupling iteratively (Burke and Smith, 1962).This has the advantage of compactness and, provided the convergence is fairly rapid, is swifter than direct methods, Richardson extrapolation enabling accurate solutions of the diagonal set to be obtained. Nonlinear sequence-to-sequence transformations provide a method for accelerating convergence and indeed of obtaining convergence in difficult circumstances as arise near the poles of a response function. The iterative sequence is transformed into a sequence with faster convergence. A nonlinear transformation that appears promising for the calculation of response functions is suggested by analogy with the geometric approximation. If 40(x) and dl(x) are the zeroth and the first iterates to the solution for a particular channel, the transformation is although care is needed if the zeros of &(x) and 41(x)are close. This transformation is not expected to yield such rapid convergence for the function $(x) as the corresponding one for polarizabilities, u = 4 / ( 2 u , - uo)
(59) since the demonstration that Eq. (59) is a good approximation depends upon the weighting that is present in the integral for u (Jamieson and Ghafarian, 1975). It is interesting to compare the transformation implied by the geometric approximation with the Aitken 6’ process (for a discussion of the use of the Aitken procedure, see Renken 1971). For three successive iterates 40, 4 42,the Aitken transformation is
+ 40) and the geometric transformation applied to 42and 41is 4 = 42 - (42 - 4d2/(42- 241) d =4 2
-(42
-
dd2/(452- 24,
(60)
(61) The geometric approximation can be viewed therefore as the Aitken d2 transformation applied to successive iterates in a scheme where the zeroth iterate is identically zero, and the first iterate is the uncorrelated solution.
CALCULATION OF ATOMIC PROPERTIES
109
IV. Time-Dependent Applications A. THETDHF EQUATIONS The time-dependent Hartree-Fock (TDHF) method has been known for many years, an outline of the historical development being given by Jamieson (1973). However, it (or the equivalent random phase approximation, RPA) has been studied considerably in recent years, since the technique appears to offer the possibility of furnishing atomic and molecular properties to a high accuracy with much less effort than conventional procedures, as configuration interaction. For example, with the latter an excitation energy is evaluated as the small residual resulting from the addition of two quantities of opposite sign and almost equal magnitude, making it difficult to obtain very refined transition frequencies without considerable computation. Similar problems attend the determination of oscillator strengths because of the variation in accuracy of wavefunctions for different states. In contrast, the characteristic quantities in the TDHF method are excitation energies and transition densities giving directly the spectral properties that are of most interest. Additionally, the approach has the special property (which, naturally, must apply also for exact solutions of the Schrodinger equation) that f-value sum rules are obeyed, and that the individual oscillator strengths are independent of whether they are evaluated in a length, velocity, or acceleration formulation (Harris, 1969). Although a number of R P A studies were performed prior to 1966 (Altick and Glassgold, 1963; Herzenberg et al., 1964), most applications have appeared since the germinal paper of Dalgarno and Victor (1966) where the coupled TDHF equations were derived and applied to the helium atom. Following Dalgarno and Victor, for a closed-shell n-electron atom under an external time dependent perturbation H',
H'
=
[exp(iot) + exp( - iot)]
n
1 vi(ri)
(62)
i=l
the perturbed wave function can be expressed as
+ = A n 4i(r, t)exp(- iE,t) n
i= 1
with +i(rr
t ) = 4o(ri) + 2[4!+(ri)exp(iot) -
+/-(ri)exp(- iwt)] + O ( P )
(64)
&,(ri) being the unperturbed Hartree-Fock orbital, vi(ri)the space part of the one-electron perturbation, and ilthe magnitude of the perturbation. On
Brian C . Webster et al.
110
application of the Frenkel variation principle
subject to the orthogonality constraint
the TDHF equations are obtained.
( ~ -0EO w)4!dri)+ { J'!k(ri) + vi(ri) (401v!+ 140)T w> 1 is used, it is free of systematic errors due to drifts or uncertainties in r. The pulses from delays 1 and 2 must, however, be separated by more than the dead time of the coincidence unit. The (e,2e) experiment is usually carried out in two modes. In the first mode, the coincidence count rate is measured as a function of the separation energy at fixed angles, normally by keeping the outgoing energies [i.e., the total energy E in Eq. (l)] fixed and varying the gun energy. In the second mode, the angular correlation is measured for given separation energies. In the separation energy mode, the computer sets the analyzers at the required angles, and then sets the electron beam energy, records the counts in the coincidence and background scalers after being triggered by the preset scaler, subtracts the accidental coincidences from the coincidence counts to give the signal, calculates the statistical error, resets and restarts
136
Erich Weigold and Iun E. McCurthy
the scalers after setting the new beam energy. A cumulative result of counts versus energy is then obtained and displayed on an oscilloscope screen. The preset scaler can be triggered by either a clock, a given integrated electron or atomic beam current, one of the electron analyzers, or another monitoring device. In situations where only one of the analyzers is moved, such as in the noncoplanar symmetric arrangement, the fixed analyzer can be used to provide the preset signal. The computer can monitor the other parameters, such as electron current density, target gas pressures, and through ADCs all the analyzer and gun potentials and angular settings (not shown in Fig. 2), and if necessary reject data when these parameters do not conform with preset conditions. The angular correlations (recoil momentum distributions) are then measured for the ion eigenstates of interest by having the computer set the corresponding incident energies (keeping E fixed) and detector angles, maintaining a cumulative result of counts versus angles at each separation energy. The data are then printed out at regular intervals or on command. The advantage of the minicomputer is that it allows rapid variation of the angular settings of the analyzers, the energies of the incident electrons, and if necessary the analyzer pass band. The whole of the energy and angular range to be measured can therefore be covered quite rapidly with little loss in measuring time, and any long-term drifts are averaged out. A detailed review of various coincidence spectrometers has been given by McCarthy and Weigold (1976), and only a brief outline of the various techniques will be given here. The noncoplanar symmetric arrangement introduced by Weigold et al. (1973) sets dA and E A equal to OB and E,, respectively, and keeps them fixed while using 4 as the angular variable. This arrangement in which only one analyzer need be rotated has the advantage that the single count rates and angular correlations must be symmetric about the coplanar setting ( 4 = 0), thus providing a useful check on instrumental distortions. In addition, the size of the interaction region, which is determined by the intersection of the electron and target gas beams and the viewing angle of the fixed analyzer, is independent of the viewing angle of the moving analyzer providing that it is larger than that for the stationary analyzer. Another major advantage to this arrangement is that the theoretical description for the scattering process becomes relatively simple (see Sections I11 and IV), the shape of the angular correlation when plotted as a function of the momentum q should be independent of the energy to a very good approximation, and in fact it should be essentially given by the square of the momentum space wavefunction of the ejected electron. The noncoplanar symmetric geometry, being insensitive to the details of the reaction mechanism, is most suited for obtaining information on the structure of the target system.
(e, 2e) COLLISIONS
137
Several symmetric noncoplanar coincidence spectrometers have been used in the investigation of (e,2e) reactions. The first employed a moving slit in a large-angle cylindrical-mirror analyzer (Weigold et al., 1973, 1975b); this was followed by moving cylindrical mirror analyzers with retarding lenses (Dey et a!., 1976; Hood et al., 1977) and more recently by an arrangement using a cylindrical-mirror analyzer with multiple detectors (CEMs) at a number of different angles 4 (Coplan et al., 1977).The detectors (CEMs) were arranged so that every pair corresponded to a different set of momenta for the outgoing electrons intercepted by that pair. By multiplexing the signals from the detectors, only a single timing circuit was necessary. A microcomputer processed, stored, and displayed the information from the timing and pair identification circuits. The multiplexing system considerably improved the data rates over those obtained in conventional two-detector systems. For symmetric kinematics (kA = k,,B, = OB = 8) the magnitude of the ion recoil momentum is given by q = [(2k, cos 8 - k,)2
+
+ 4k3 sin2 8 ~ i n ~ ( + / 2 ) ] ” ~
(5)
and q + 0 as + 0 for a particular value 8, of 8, where H, 5 44, the inequality sign holding for all nonzero values of the separation energies. For energies high compared with the separation energy, 8, z n/4,and in the coincidence spectrometers referred to above the polar angles were all set in the range 42.3” < 8 < 45”. In contrast to the noncoplanar symmetric situation, the coplanar symmetric geometry (in which = 0 and 8 is varied) is more sensitive to details of the reaction mechanism (Sections 111 and IV) and to distortions of the incoming and outgoing electron waves. This experimental arrangement therefore provides a far more stringent test of the Coulomb three-body problem. In addition, with this geometry it is particularly simple to go over to asymmetric kinematics and to study the reaction over a larger region of the multidimensional space of the differential cross section. Detailed descriptions of coplanar spectrometers have been given by Ehrhardt et al. (1971), Giardini-Guidoni et al. (1976), Ugbabe et al. (1975), and McCarthy and Weigold (1976). In the coplanar symmetric arrangement both spectrometers must rotate about the interaction region, and it is therefore most important to ensure that the interaction region is well defined and contained in the field of view of both analyzers at all angular settings of the analyzers. The target gas beam density profile must therefore be well defined, and skimmers are usually placed between the collision region and the multichannel array providing the beam input. The absence of an angulardependent correction term can be ascertained by checking against a wellknown cross section (McCarthy and Weigold, 1976). If one of the analyzers
138
Erich Weigold and Ian E. McCarthy
is kept fixed, it is still necessary to ensure that the field of view of the moving analyzer is larger than the collision region, which may now be formed by the intersection of the electron and target beams and the viewing angle of the stationary analyzer. Recently Beaty et al. (1977) reported an apparatus that allowed @A,OB, and 4 to be varied. The scattering angle 8, was set by moving the gun about the collision centre. The ejected electron analyzer could be moved in three dimensions over a range of values of QB and 4. This work on helium was the first to report absolute differential cross sections, although with rather low precision. The relation between the differential cross section and the experimentally observable parameters when both analyzers view the full collision region is
where S is the true coincidence count rate, I , the incident electron rate, 1 the interaction length, n the target density, ARA and ARB the solid angles ofthe analyzers, AE the energy resolution, and E, and cB the overall efficiencies of the two analyzers (transmissions and detector efficiencies). The quantities ( n k iAQi) can be determined by normalization to a well-known absolute differential elastic cross section by
where I,] is the elastically scattered current, Zb the incident current, a,] the absolute elastic scattering cross section, and E ; the overall efficiency of the analyzer for elastically scattered electrons. This was assumed to be equal to ci by Beaty et al. (1977) and by Stefani et al. (1977b).This is not necessarily the case since the elastically scattered electrons all have the same energies, whereas the electrons in the (e, 2e) cross section have a continuum of energies, and this could lead to different edge effects in the transmission of the analyzer and associated electrostatic lenses. For atomic beams it is difficult to obtain an accurate measurement of the product of the target density and effective length. In their coplanar symmetric measurements on helium, Stefani et al. (1977b) determined nl by measuring the incoming gas flow and by using a gas beam scanning procedure similar to that described by Wellenstein et a/. (1975). Beaty et al. (1977) solved this problem by assuming the efficiencies of one of its analyzers was unity and by calculating its solid angle. Both groups claim a factor of 2 as the uncertainty in their absolute crosssectional values. Although both measurements were on helium, it is not possible to compare them because of the different kinematical regions studied.
(e, 2e) COLLISIONS
139
The target gas density and effective beam length can obviously be determined much more accurately if a “static” gas target is used. Such an experiment is being undertaken by van Wingerden and de Heer at the F.O.M. Instituut voor Atoom en Molecuulfysica, and it should lead to considerably improved estimates of the absolute cross sections,
111. Basic Theory A. DERIVATION OF THE SCATTERING AMPLITUDE 1. The Quasi-Three-Body Approximation
We discuss the (e, 2e) reaction in terms of two electrons with coordinates xi, i = 1, 2, and the ion resulting from the collision, which has internal coordinates (. The coordinate xi includes position ri and spin ci.Centerof-mass (c.m.) motion of the ion is neglected. The ion Schrodinger equation is IIEf
- HI(O1If)= 0
The (e, 2e) Schrodinger equation is
The electron-ion potentials are K, the interelectron Coulomb potential is U ( T ) , and the kinetic energy operators are Ki. It was shown by McCarthy and Weigold (1976), with approximations amounting to closure over target states and weak coupling between channels in both two- and three-body systems, that the (e, 2e) amplitude reduces, with implied antisymmetry, to where xl*) are optical model wavefunctions describing elastic scattering in the appropriate two-body subsystems with an implicit description (in an energy-dependent complex potential) of unobserved reaction channels. The ground state of the target system is 1s).The total Green’s function for the problem G ( - ) is G(-’(E) = lim [ E - i~ &+O+
-
(K, +K,
+ H I + V, + V2 + u)]-‘
(11)
The potential i? produces inelastic scattering from the ion. It gives rise to a term representing ionization by core excitation, which has been shown to
Erich Weigold and Ian E. McCarthy
140
be small (McCarthy and Weigold, 1976). Neglect of this term gives the quasi-three-body approximation. Although the optical model formulation takes into account excitation of unobserved channels, it is difficult to make it explicitly antisymmetric in all the electron coordinates. It is simpler to understand the effect of antisymmetry by restricting the ion to its ground state and describing the whole system by an independent-particle model. The (e, 2e) amplitude has been given in this approximation by Rudge (1968). Using a distorted-wave representation it is
x
C ~ * ( x 3 ., .’
>X”+l)(l- Pl,)xa-’*(k,,x,)X~-)*(kB,.~2)
+ ( n j - 1 ) $ * ( ~ 1 , ~ 4 , .. . ,xn+ 1 ) ~ ~ - ) * ( k A , ~ 2 ) ~ & - ) * ( k s , ~ 3 ) ] x
VY(”(k0,
XI,.
. , ,x,,+1)
(14
Here nj is the number of electrons of angular momentum j that are equivalent to the knocked-out electron, V is the difference between the total and distorted-wave hamiltonians, and P12is the exchange operator. The first term of (12) is the quasi-three-body term. The second term represents ionization by electron knockout from the core. It involves the overlap between an optical model wavefunction X&-)(kB,x3) and a bound orbital ~ , b ~ ( xwhich ~ ) , is small.
2. The Distorted- Wave Impulse Approximation The quasi-three-body approximation involves the three-body Green’s function G(-)(E).This is reduced to the two-electron Green’s function by making a Taylor expansion of the electron-ion potentials q(ri) about the electron-electron c.m. coordinate R.Quantities relevant to the relative, c.m. transformation are rl
=R
+ $r,
r2=R-&r ,
k
= &(kA- kB),
K=k,+kB,
k’ = *(k0 + q),
p 2 = (h2/m)k2 s2 = (fi2/4m)K2 q = ko - kA - k B
(13)
If we take only the zeroth-order term of the Taylor expansion the Schrodinger equation for the final-state distorted waves separates in the coordinates R, r, 5 (we neglect spin-orbit coupling so that the spin coordinates are relevant only to counting of antisymmetric states) : jE
-
CK, + K2 + VI(r1) + V20.2) + H,1 }X!P(rl)x&-)(r2)1f) { E - [ K , + K R + V,(R) + V2(R) + H ~ l ~ ~ ( - ) ( R ) ~ ( - )= ( r0) l ~(14) )
2
(e, 2e) COLLISIONS
141
where
+ Vl + V2)]f’(K,R) p +s [ p z - K,]&)(k,r) = 0, [s2 - (KR
2
=0 2
= E - E~
(15)
The distorted wave may now be commuted through the operator u in (10) to act directly on G(-), giving
M(~A k,),
=
(X~-)(kA)S11;)(ks))ThI(P2)I(flg)Xb+)(kO)) (16)
where T M is the antisymmetrized two-electron (Mott scattering) T-matrix,
Equation (16) is the distorted-wave impulse approximation (DWIA). We sometimes use the word “off-shell” to distinguish this approximation from a simple approximation where only the on-shell (at p 2 ) T-matrix was calculated. The present approximation involves the neglect of second-order (and higher) gradients of the optical-model potential in the symmetric (e, 2e) reaction. These gradients are quite small, except at the center of an atom, where the integrand is cut off by the bound-state orbital, except in the 1s case. Note that the amplitude (16) depends on the target and ion structure only through the overlap function (fig), since TMis independent of the internal ion coordinates 4.
3. Final-State Coulomb Forces The foregoing analysis is valid for short-range forces. In fact, there are Coulomb interactions between all three bodies in the final state. A method of circumventing this difficulty has been reviewed by Rudge (1968). Effective charges Z,, Z, for electrons detected at A and B are chosen so as to remove the logarithmic singularity in the phase of the (e, 2e) amplitude, which is, in atomic units, lim exp[i(l/k, P-
+ l/kB
-
l/\kA- k,( - Z,/kA - ZB/kn)h2Xp] (18)
30
where X 2 = k:
+ki
The condition determining the effective charges is For the symmetric (e, 2e) reaction we choose
Erich Weigold and Ian E. McCurthy
142
Since the actual charge of the ion is 1, there is a residual Coulomb potential that must be neglected beyond some cutoff radius Ro. For r < Ro it is included in the distorting potentials for calculating x('). The differential cross section is proportional to 0 = (f(kA,kR))2+ If(k,,kA)12 - Re[f(k,,k,)*f(k,,k,)l f(kA,kB)= - ( 2 4 exp[iA(k,, kB)]M'(kA,k,) A(k4, k,) = 2[(Z,/kA) W A l X ) + (Z,/k,) ln(k,/X)l ~
'"
(22)
The matrix element M' is equivalent to the direct term of (16). The inclusion of effective charges makes a small difference in practice and will be described later. B. THEDISTORTED WAVES I . The Optical Model j o r Electron Elustic Scattering We consider the elastic scattering of an electron from the ground state 10) of a system with a spectrum of eigenstates li), which includes the continuum. The Schrodinger equation for the scattering is [ E - K(r) - V(r, 5 ) - HI(()]Y(+)(r,5) = 0 I where the target is given by [Ei -
Hf( 0.99 for q < 0.5 a.u. (McCarthy et a/., 1974; Dixon et a/., 1976). Although the various calculated cross sections differ significantly from each other, the experimental errors are too large to allow definite conclusions to be drawn on which of the approximations best describes the cross sections. Improved absolute measurements, with errors of the order of 20-30%, are urgently needed. Nevertheless, within the accuracy of the measurements, the impulse approximation appears to give reasonable fits to the data. Also included in Fig. 16 are the Coulomb-projected Born (including exchange) and the plane-Coulomb-wave Born (without exchange) calculations of Geltman (1974)at 224.6 and 424.6 eV. They fall between the distortedwave impulse approximation and plane-wave Born approximation results.
V. Structure of Atoms and Molecules The atomic and molecular structure information obtained by the (e, 2e) reaction was extensively reviewed by McCarthy and Weigold (1976) and by Weigold (1976a,b). Much of the more recent work has been devoted to molecules. For atoms, we shall therefore only supplement briefly the discussion of Section IV. As we saw there, the noncoplanar symmetric geometry is particularly suited for structure determination. In addition, since the off-shell Coulomb T-matrix is nearly independent of the azimuthal angle Cp, the shape of the cross section is essentially given by the momentum distribution of the target electron and should therefore be nearly independent of energy when plotted as a function of q. This kinematical arrangement has therefore been extensively used in the investigation of target structure. A. ATOMS
The inert gases neon, argon, krypton, and xenon have all been studied in both the coplanar and noncoplanar geometries (see McCarthy and Weigold, 1976, for details). Separation energy spectra (e.g., Figs. 17 and 18) and angular correlations (e.g. Figs. 7-11) show that the ion ground states contain essentially all of the valence np-' strength (S!,:) z l), but that the ns-l strength is split among a number of ion eigenstates, especially for
(e, 2e) COLLISIONS
165
ELectron separation e n e r g y ( e V ) FIG. 17. Separation energy spectra for argon at 400 eV (from McCarthy and Weigold, 1976). The peaks are labeled by the dominant configuration of the ion eigenstate. The momentum profiles for the peaks are shown in Figs. 10 and 11.
Separation energy ( e V ) FIG. 18. The separation energy spectrum for xenon at 400eV (from Hood et ul., 1977). The arrows labeled 1-7 mark the positions of satellite peaks observed by Gelius (1974) in an X-ray PES spectrum.
Erich Weigold and Ian E. McCarthy
166
argon, krypton, and xenon. In fact the data, which are summarized in Table 1, show that for krypton and xenon the ion state usually identified with the hole state contains only one third of the spectroscopic strength. Although it is in one sense difficult to talk of independent particle orbitals when such strong electron-electron correlation effects are present, the shapes of the momentum distributions are very well described by means of Hartree-Fock wavefunctions [Eq. (46)]. This can be seen from Figs. 8-11. Also included in Table I are the normalized spectroscopic strengths obtained in various photoelectron (y,e) experiments (Spears et a/., 1974; Wuilleumier and Krause, 1974; Gelius, 1974). These differ markedly from the corresponding (e, 2e) values. The latter are all obtained at 4 = 0, that is, in the region where the electron is most probably to be found. Furthermore, they are independent of energy over the range of energies studied, 200-2500 TABLE I (e, 2e) A N D XPS SPECTROSCOPIC STRENGTHS NORMALIZED TO UNITYFOR THE VALENCE s HOLESTATESOF THE INERTGASES Separation energy for f + ion states
Dominant ion state configuration
Ne(2s-’)
48.5 55.9 59.9 > 60
2s2p-’3s1 2p-’3d1
Ar(3s- ’)
29.3 38.6 41.2 43.4 > 1a.u.)
0.96 0.1, magnetic fields are easily achieved corresponding to electric fields approaching lo6V/cm. Stable electric fields of this magnitude are difficult to obtain directly. The accuracy of the quenching experiments is limited by several factors. Among the more important are the following: (1) Background radiation arising mainly from radiative transitions from the n = 2 state of any heliumlike ions present in the beam. Background counts are also generated from collisions of the beam with any material present in the field of view of the detectors. (2) Deflection effects on the moving beam in the magnetic field. ( 3 ) Miscellaneous effects such as uncertainty in the beam velocity and edge effects in the magnetic field.
In spite of these and other difficulties, accuracies of about 1% in the Lamb shift have been achieved. Use of the quenching lifetime method beyond 2 h~ 25 is probably an unlikely prospect because of the following problems at higher 2 : (1) the electric field needed for a given fractional quenching increases rapidly with Z , because the Lamb shift increases rapidly with Z and the matrix element of the dipole operator X scales as 2-l; (2)the lifetime of the metastable state is rapidly decreasing with Z , making tof measurements more difficult.
c. LIFETIME OF THE 2lSo STATE IN AN ELECTRIC FIELD In helium, the electric field quenching of the metastable 21S0 state has been considered by a number of authors. Petrasso and Ramsey (1972) measured a transition rate of A = (0.926 +_ 0.020)E2 sec-', where E is the field strength in kV/cm. This result is in agreement with the value A =
218
Richard Marrus and Peter J . Mohr
(0.920 k 0.030)E2sec-', which they calculate, and with the earlier theoretical result of Holt and Krotkov (1966): A = (0.89 & 0.04)E2sec-l. The difference in the theoretical results is due to the electric field perturbation of the ground state l'So, which is included by Petrasso and Ramsey (1972) but not in the earlier work. The calculations are based on time-independent perturbation theory in which, to first order in the external field strength, the n'So state is written
where X = XI
+ X, . The decay rate is then
The values for the matrix elements were taken from various calculations of the relevant oscillator strengths. C. Johnson (1972) has pointed out that the time-dependent Bethe-Lamb theory, described above as applied to hydrogenlike ions in an electric field, gives a prediction 25% too large for the lifetime of the 2's state of helium in an electric field. The reason is that the Bethe-Lamb approach gives each state a phenomenological decay constant that does not take into account cross terms in the coherent sum over intermediate states. On the other hand, C. Johnson (1972) and Holt and Sellin (1972) note that the Bethe-Lamb theory works well for hydrogenlike 2S,,, decays, because the leading interference term between 2P1,, and 2P,,, intermediate states vanishes upon integration over photon directions. These points are discussed in more detail by Holt and Sellin (1972) and by Grisaru et al. (1973). Further calculations have been carried out by Jacobs (1971); quoted by Drake (1972b) and by Drake (1972b) who obtain A = 0.931E2sec-' and A = 0.932(1)E2sec- ',respectively. Drake employs a variationally determined discrete basis to evaluate the sum over intermediate states. A more accurate measurement was made by C. Johnson (1973), who found A = 0.933(5)E2sec-'. D. POLARIZATION OF THE INDUCED RADIATION It is of some interest to note that the angular distribution of the quench radiation plays a role in the determination of the absolute magnitude of the cross section 4 2 s ) for excitation of the 2Sl,, state of hydrogen by electron impact. Stebbings et al. (1960, 1961) measured 42s) by comparing the rate of quench radiation with the rate for excited 2P atoms. Lichten (1961) pointed out that the analysis of their data depends crucially on assumptions
FORBIDDEN TRANSITIONS
219
concerning the isotropy of the quench radiation. With neglect of the 2P3,, state this radiation is completely unpolarized and the angular distribution is isotropic. Measurements made by Fite et al. (1968) and Ott et al. (1970) showed that the radiation emitted perpendicular to the electric field direction has a 0.02, where I , and sizable polarization P = (I, - I , ) / ( I , + I,) = -0.30 I , refer to intensities for linear polarization parallel and perpendicular, respectively, to the electric field direction (see also Miliyanchuk, 1956, quoted in Borisoglebskii, 1958). Subsequent measurements by Sellin et al. (1970a) indicated a field strength dependence of the polarization fraction for strong fields. Spiess et al. (1972) have made a measurement that yields P = - 0.31 k 0.03 for weak fields (< 100Vjcm). The explanation for the polarization, is that although the 2P3,, state is about ten times farther in energy than the 2P1,, state from the 2S1,, state, its effect may not be neglected, and both states should be retained in the time-independent perturbation expansion for the 2S1,, state. The dipole matrix element for radiation to the ground state then consists of two terms, one involving the 2P,,, state and the other the 2P3,, state. The intensity, which is proportional to the dipole matrix element squared, is thus influenced by cross terms of the order of 20% of the 2P1,, term squared. An estimate by Fite et al. (1968) that neglected hyperfine structure yielded P = -32.9'4 in rough agreement with the observed value. Subsequent calculations, with basically the same approach, which included the effects of hyperfine structure, have yielded the following theoretical values: Casalese and Gerjuoy (1969) find P = -32.33%; Drake et al. (1975) have quoted P = -32.31%; and Kelsey and Macek (1977) obtained P = - 32.31%. All these results are consistent with the experiments. The polarization measurement may, of course, be regarded as a Lamb shift measurement in the same sense as the lifetime determinations, since the polarization P depends strongly on the Lamb shift interval. E. ANGULAR DISTRIBUTION OF THE INDUCEDRADIATION
Associated with the polarization of the decay radiation is an anisotropy in the distribution of the radiation relative to the electric field direction. Drake and Grimley (1973) have noted this and pointed out that a measurement of the anisotropy would be a way of measuring the Lamb shift. The anisotropy, as well as the polarization, arises mainly from the cross term between the 2P,,, and 2P,,, intermediate states. The relative magnitude of the cross term depends on the relative size of the energy denominators, which contain the Lamb shift. Measurement of the ratio R = ( I , , - II)/(IlI +IJ, where Illand I , are the intensity of radiation emitted parallel and perpendicular to the external electric field, has been made in hydrogen and in
220
Richard Marrus and Peter J. Mohr
FIG. 16. Apparatus used to measure the Lamb shift in hydrogen and deuterium using the anisotropy method.
deuterium by van Wijngaarden et al. (1974) and more accurately by Drake et al. (1975) to test the method as a scheme for measuring the Lamb shift. A schematic diagram of their apparatus is shown in Fig. 16. A 1-keV H + or D + ion beam enters a cell containing cesium vapor. The emerging beam contains neutral atoms, protons and H-, in addition to a usable component of hydrogen atoms in the 2S,,, state. Charged particles are removed from the beam by passing it through a region with a small electric field. The remaining beam is collimated and passed into an electric field created by a quadrupole electrode structure. A pair of ultraviolet photon detectors views the quench photons parallel and perpendicular to the applied electric field. Their results R , = 0.13901(12) and R , = 0.14121(14) are in good agreement with the calculated values R , = 0.139071 and R , = 0.141165 based on theoretical Lamb shift values, or conversely, these measurements determine the corresponding Lamb shifts to an accuracy of about 0.1% (Drake et al. 1975; Drake and Lin, 1976). Experiments on ions at higher Z are now underway. REFERENCES Anderson, M. T., and Weinhold, F. (1975). Phys. Rev. A 11, 442. Araki, G. (1937). Proc. Phys.-Math. SOC.Jpn. 19, 128. Au, C. K. (1976). Phys. Rev. A 14, 531. Bashkin, S. (1976). “Beam-Foil Spectroscopy.” Springer-Verlag, Berlin and New York. Bednar, J. A,, Cocke, C . L., Curnutte, B., and Randall, R. (1975). Phys. Rev. A 11,460.
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22 1
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Leventhal, M., Murnick, D. E., and Kugel, H. W. (1972). Phys. Rev. Lett. 28, 1609. Lichten, W. (1961). Phys. Rev. Lett. 6 , 12. Lin, C. D., Johnson, W. R., and Dalgarno, A. (1977). Phys. Rev. A 15, 154. Lin, D. L. (1975). Ph.D. Thesis, Columbia University, New York (unpublished). Lin, D. L., and Armstrong, L. Jr. (1977). Phys. Rev. A 16, 791. Lin, D. L., and Feinberg, G. (1974). Phys. Rev. A 10, 1425. Luc-Koenig, E. (1974). J . Phys. B7, 1052. Luders, G. (1950). Z. Naturforsch., Ted A 5, 608. Lyman, T. (1924). Astrophys. J . 60,1. Marrus, R., and Schmieder, R. W. (1970a). Phys. Lett. A 32,431. Marrus, R., and Schmieder, R. W. (1970b). Phys. Rev. Lett. 25, 1689. Marrus, R., and Schmieder, R. W. (1972). Phys. Rev. A 5, 1160. Mathis, J. S. (1957). Astrophys. J . 125, 318. Miliyanchuk, V. S. (1956). Doctoral Dissertation. L’vov State University, USSR. Mizushima, M. (1964). P/zj>s.Rev. 134, A883. Mizushima, M. (1966). .I. Pliys. SOC.Jpn. 21, 2335. Mohr, P. J. (1976). In “Beam-Foil Spectroscopy” (I. A. Sellin and D. J. Pegg, eds.), p. 97. Plenum, New York. Moore, C. F., Braithwaite, W. J., and Matthews, D. L. (1973). Phys. Lett. A 44,199. Moos, H. W., and Woodworth, J. R. (1973). Phys. Rev. Lett. 30, 775. Mowat, J. R., Sellin, I. A,, Peterson, R. S., Pegg, D. J., Brown, M. D., and Macdonald, J. R. (1973). Phys. Rev. A 8, 145. Mrozowski, S. (1938). Z. Phys. 108,204. Mrozowski, S . (1945). Phys. Ren. 67, 161. Novick. R. (1969). In “Physics of the One- and Two-Electron Atoms’‘ (F. Bopp and H . Kleinpoppen, eds.), p. 296, and other references cited therein. North-Holland Publ., Amsterdam. Onello, J. S., and Ford, L. (1975). Phys. Ren. A 11. 749. Ott, W. R . , Kauppila, W. E., and Fite, W. L. (1970). Phys. Rev. A 1, 1089. Pagel, B. E . J. (1969). Nature (London)221, 325. Pearl, A. S. (1970). Phys. Rev. Lett. 24, 703. Petrasso, R. and Ramsey, A. T . (1972). Phys. Rec. A 5 , 79. Pietenpol. J. L. (1961). Phys. Rev. Lett. 7, 64. Prior, M. H . (1972). Phys. Rev. Lett. 29,611. Prior, M. H., and Shugart, H. A. (1971). Phys. Rec. Lett. 27, 902. Ramsey, N. F. (1969). In “Physics of the One- and Two-Electron Atoms” (F. Bopp and H. Kleinpoppen, eds.), p. 218. North-Holland Publ., Amsterdam. Lord Rayleigh. (1927). Proc. R. SOC.London Ser. A 117, 294. Richard, P., Kauffman, R. L. Hopkins, F. F., Woods, C. W., and Jamison, K. A. (1973a). Phys. Rev. A 8,2187. Richard, P., Kauffman, R. L., Hopkins, F. F., Woods, C. W., and Jamison, K. A. (1973b). Phys. Rev. Lett. 30, 888. Rugge, H. R., and Walker, A. B. C., Jr. (1968). Space. Res. 8, 439. Safronova, U. I., and Rudzikas, Z. B. (1977). J . Phys. B 10, 7. Sawyer, G . A,, Bearden, A. J., Henins, I., Jahoda, F. C., and Ribe, F. L. (1963). Phys. Rev. 131, 1891. Schmieder, R. W., and Marrus, R . (1970a). Phys. Rev. Lett. 25, 1245. Schmieder, R. W., and Marrus, R. (1970b). Phys. Rev. Lett. 25, 1692. Sellin, I. A. (1964). Phys. Rev. 136, A1245. Sellin, I. A., Donnally, B. L., and Fan, C . Y . (1968). Phys. Rev. Lett. 21, 717.
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Sellin, I. A,, Biggerstaff, J. A,, and Griffin, P. M. (1970a). Phys. Rev. A 2, 423. Sellin, I. A,, Brown, M., Smith, W. W., and Donnally, B. (1970b). Phys. Rev. A 2, 1189. Shapiro, J., and Breit, G. (1959). Phys. Rev. 113, 179. Spiess, G., Valance, A,, and Pradel, P. (1972). Phys. Rev. A 6 , 746. Spitzer, L., Jr., and Greenstcin, J. L. (1951). Astrophys. J. 114, 407. Stebbings, R. F., Fite, W. L., Hummer, D. G., and Brackmann, R. T. (1960). Phys. Rev. 119, 1939. Stcbbings, R. F., Fite, W. L., Hummer, D. G., and Brackmann, R. T. (1961). Phys. Rev. 124, 2051 (E). Sucher, J. (1977). At. Phys. Proc. Int. Con/:, 5ih, 1976 p. 415. Tang, H. Y. S., and Happer, W. (1972). Bull. Am. Phys. Soc. [2] 17,476. Van Dyck, R. S., Jr., Johnson, C. E., and Shugart, H. A. (1971). Phys. Rev. A 4, 1327. van Wijngaarden, A,, Drake, G. W. F., and Farago, P. S. (1974). Phys. Rev. Lett. 33,4. Varghese, S. L., Cocke, C. L., and Curnuttc, B. (1976). Phys. Rev. A 14, 1729. Victor, G. A. (1967). Proc. Phys. Soc., London 91, 825. Victor, G. A., and Dalgarno, A. (1967). Phys. Rev. Lett. 18, 1105. Wiese, W. L., Smith, M. W., and Glennon, B. M. (1966). “Atomic Transition Probabilities,” Rep. No. NSRDS-NBS 4. U.S. Govt. Printing Office, Washington, D.C. Woodworth, J. R., and Moos, H. W. (1975). Phys. Rev. A 12, 2455. Zon, B. A., and Rapoport, L. P. (1968). Zh. Eksp. Teor. Fis., Pis’ma Red. 7 , 70; JETP Lett. (Engl. Trunsl.) 7 , 52. (1968).
ADVANCES IN ATOMIC AND MOLECULAR PHYSICS,
1
VOL. 14
SEMICLASSICAL EFFECTS IN HEAVY-PARTICLE COLLISIONS M . S . CHILD Department of Theoretical Chemistry University of Oxford Oxford, England
I. Introduction.. . . . . . .... ............... A. Experimental Bac ........................................ B. Theoretical Developments ........................................ C. Scattering in the Semiclassical Limit . . . . 11. Elastic Atom-Atom Scat A. Scattering Amplitude and Differential Cross Section. . . . . . . . . . . . . . . . . . B. Total Cross Section. . C. Semiclassical Inversio 111. Inelastic and Reactive Sc A. Integral Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Stationary Phase and Uniform Approximations ..................... ........... C. Classically Forbidden Events . . . . . . . . D. Numeridal Applications and Conclusio IV. Nonadiabatic Transitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. One-Dimensional Two-State Model. ......................... ...................... B. Inelastic Atom-Atom Scattering .......... C. Surface-Hopping Proc V. Summary . . . . . . . . . . . . . .......... ....... ............................. Refersnces
225 226 221
234
247 252 251 262 263 268 271 274 215
I. Introduction The past 15 years have seen major developments in the study of atomic and molecular scattering processes both from the experimental and theoretical points of view. The recent book by Levine and Bernstein (1974) offers a readable introduction. Among the most interesting of these has been the growing recognition of the semiclassical nature of the processes involved. This was first made apparent by Ford and Wheeler (1959a,b) in the case of elastic scattering, but its full significance has only recently been demonstrated by the work initiated by Pechukas (1969a,b), Miller (1970a,b), and Marcus 225 Copyright @ 1978 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-003814-5
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(1971).Subsequent developments have led to a coherent conceptual structure, the main elements of which appear sufficiently well established to justify the present review. There are, however, certain computational problems limiting general application of the theory, and it is also recognized that experimental conditions may lead to the averaging out of semiclassical effects in complex reactions. It may therefore be valuable first to give brief reviews of recent experimental developments, and of other important lines of theoretical research before turning to the main subject under review. A. EXPERIMENTAL BACKGROUND
One important class of experiments involves the scattering of molecular beams (Ross, 1966; Schlier, 1970; Fluendy and Lawley, 1973). Early limitations imposed by difficulties in detector design have been overcome by the development of high-intensity beam sources coupled with mass and spectroscopic analysis of the scattered particles. Atom-atom scattering crosssections derived in this way are illustrated in Figs. 5, 7, and 8, and analyzed in the text. Similar structure may also be evident in nonreactive atommolecule collisions, particularly if the reactant molecules are oriented by electric fields (Reuss, 1976).The chemically reactive scattering cross sections can seldom be resolved in such detail, but the measurement of product distributions as functions of velocity and scattering angle, from velocityselected reactants such as that illustrated in Fig. 1 is now possible for a large number of systems (Grice, 1976). This will be increasingly supplemented in the future by laser fluorescence analysis of the reaction products to provide information on the final internal energy distribution (Cruse el al., 1973; Pruett and Zare, 1976). Similar information on the product internal energy distribution may also be obtained by the infrared chemiluminescence techniques pioneered and largely developed by Polanyi for the study of very low-pressure gas reactions (see Polanyi and Schreiber, 1973, for a recent review). Figure 2 shows the type of information currently available by this technique (Ding et al., 1973). This shows product intensity contours as a function of final vibrational and rotational energy for two different reactant vibrational states (v = 0, 1). A number of systems studied in this way are now in use as commercial chemical lasers. The study of chemical laser intensities from one line to another also provides information on the relative populations of the product internal states (Berry, 1973). Another important application of laser technology has been to the study of energy transfer processes, particularly those involving the transfer of vibrational energy. Here one follows the quenching of either laser-induced fluorescenceor stimulated Raman scattering as a function of pressure (Moore,
SEMICLASSICAL HEAVY -PARTICLE COLLISIONS
90'
I
227
HCI from
0'
180 9 0'
FIG. 1. Contour maps of angle-velocity flux distributions in c.m. co-ordinates for the reaction product HX in the H + X, reactions. Direction of the incident hydrogen atom is designated 0 . [Taken from Herschbach (1973) with permission.]
1973; Bailey and Cruickshank, 1974; Lukasik and Ducuing, 1974). It is normally necessary to assume a thermal velocity distribution in the gas, but the accessible temperature range is considerably increased over that obtainable by shock tube and other more traditional techniques (Burnett and North, 1969). Studies of rotational relaxation leading to information on intermolecular anisotropies by spectral line broadening, molecular beam, and other methods (see Neilson and Gordon, 1973; Fitz and Marcus, 1973, 1975), have recently been supplemented by direct spectroscopic analysis of weakly bound Van der Waals complexes formed in the gas phase at artificially low translational temperatures (Klemperer, 1977). B. THEORETICAL DEVELOPMENTS The molecular scale of these events raises two types of problems for the theory. The first arises from the need to include quantum mechanical
IhI
’-4(k -0.097)
T;,-
300 K
FIG.2. Product internal state distributions for the H + C1, (ti = 0) and H + CI, ( L ‘ 2 I ) reactions. Contours give the measured rate constants as functions of the product rotational R’ and vibrational V ‘ energies. Note the bimodal character for u > 1. [Adapted from Ding rt al. (1973) with permission.]
SEMICLASSICAL HEAVY-PARTICLE COLLISIONS
229
effects in situations where the number of significant channels is large. For example, even the elastic scattering of two atoms may involve 100-loo0 significant partial waves, but the magnitude of the total cross section is determined by the uncertainty principle, and various readily observed interference effects, containing valuable information, can only be described quantum mechanically. The problem of the number of coupled channels becomes overwhelming for molecular collisions involving all but the lightest atoms. This is balanced to some extent by the averaging out of interference effects in the differential cross section, but interference might well remain significant in determining the vibrational distributions of chemical reaction products. The bimodal structure of Fig. 2b could be a case in point. There are also processes such as quantum-mechanical tunneling at the chemical reaction theshold and the transfer of vibrational and translational energy (Shin, 1976) under thermal conditions that are dynamically forbidden in classical theory and hence can only be accounted for by quantum mechanics. The second difficulty arises from the strength of chemical interactions compared with the relevant energy separations, and the resultant strong coupling between many channels. The cases of vibrational to translational energy transfer cited above and certain processes involving electronic energy or charge transfer are almost unique in being amenable to perturbation theory. The unifying concept in all approaches to these difficulties is the potential energy surface or, more accurately, the electronic energy surface in nuclear coordinate space visualized as being obtained by solution of the electronic Schrodinger equation within the Born-Oppenheim fixed nucleus approximation, although some processes of chemical interest involve nonadiabatic transitions from one surface to another (see Section IV). The present state of the theory is such that the principal qualitative features of at least the lowest energy surface can be reliably determined both for reactive (Baht-Kurti, 1976; Kuntz, 1976)and nonreactive (Gordon and Kim, 1972,1974)processes, but quantitative reliability can be expected only for systems involving the lightest atoms. An additional complication in the dynamical theory is therefore the need to employ a flexible functional form for the surface, consistent with the known qualitative form, which can be adjusted in order to bring the dynamical results into agreement with experiment. For the reasons given above, this scheme is feasible at present only within the realm of classical mechanics. Such Monte Carlo classical trajectory calculations have been the most important single aid to interpretation of chemical reactions studied by the molecular beam and infrared chemiluminescence techniques (Bunker, 1970; Polanyi and Schreiber, 1973; Porter and Raff, 1976). The other main lines of theoretical development bear on the question as to whether such a purely classical treatment can be justified. At one extreme
230
M . S. Child
there have been a number of accurate numerical solutions of the exact close-coupled quantum-mechanical equations for a variety of realistic model systems. The numerical techniques are discussed by Lester (1976). These benchmark studies relate in order of complexity to elastic-scattering phase shifts (Bernstein, 1960), the collisional excitation of harmonic and Morse oscillators (Secrest and Johnson, 1966; Clark and Dickinson, 1973), the scattering of rigid rotors (Shafer and Gordon, 1973; Lester and Schaefer, 1973), and more recently a spate of calculations on reactive systems, which have been reviewed by Micha (1976a). The latter are complicated by the necessity for a coordinate transformation from the reactants to products frame during the calculation, which raises acute problems in any full threedimensional study. At the time of writing, the only three-dimensional reactive calculations including both rotational and vibrational open channels have been for the H + H, reaction (Elkowitz and Wyatt 1975a,b; Schatz and Kuppermann, 1976). The most serious general complication in these exact calculations is the strong coupling between angular momenta associated with the internal and relative motions. Attention has therefore been concentrated on the development of decoupling schemes to reduce the number of coupled channels without serious loss of accuracy (McGuire and Kouri, 1974; Pack, 1974; De Pristo and Alexander, 1975, 1976).The spirit of this approach is similar to that which inspired Hunds’ cases in diatomic spectroscopy (Herzberg, 1950).Another general trend has been to reduce the labor of the calculation by the use of exponential approximations to the S matrix (Pechukas and Light, 1966; Levine, 1971; Balint-Kurti and Levine, 1970).The effort is little more than that required for a distorted wave perturbation calculation (Child, 1974a),but the unitarity of the S matrix is preserved. The sudden approximation (Bernstein and Kramer, 1966) is the simplest member of this family. A third device, borrowed from nuclear physics, is to introduce an imaginary “optical” term into the potential to suppress the need for calculation of quantities irrelevant to the process under investigation (Micha, 1976b). Finally there are several methods included under the general heading “semiclassical” that seek to retain the computational simplicity of classical mechanics without losing any essential quantum-mechanical characteristics. Conceptually the most interesting of these, to which the major part of this review is devoted, is the semiclassical S matrix method, stemming from the Feynmann path integral approach to quantum mechanics (Feynmann and Hibbs, 1965), as developed in the present context by Ford and Wheeler (1959a,b), Pechukas (1969a,b), Miller (1970a,b), and Marcus (1971). This differs from classical mechanics only by inclusion of a phase determined by the classical action (Goldstein, 1950) for the trajectory in question, and the development of special techniques to handle the resulting interference pat-
SEMICLASSICAL HEAVY-PARTICLE COLLISIONS
23 1
tern, which depend on the topological structure of the caustics of the classical motion (Connor, 1976a; Berry, 1976). The reader is referred to reviews by Berry and Mount (1972), Miller (1974, 1976b), Connor (1976b), and Child (1976a), which complement the account given here. A much older “classical p a t h procedure whereby the relative motion is assumed to follow a known mean classical trajectory while the internal motion is treated by quantum mechanics has recently been examined by Bates and Crothers (1970)and Delos et a/.(1972).This has the computational advantage of separating the internal from the orbital angular momentum and of reducing the equations of motion to a time-dependent form, with only one first-order equation for each channel. Use of this method is, however, restricted to situations in which the changes in the translational energy and angular momentum are small compared with their absolute values. Intermediate between these two philosophies there is a method due to Percival and Richards (1970) derived from the correspondence principle. Here the quantum-mechanical matrix elements appearing in the classical path equations are replaced by Fourier transforms taken over a fixed mean classical orbit for each internal motion in question. Thus the dynamical motion is again purely classical, but two mean trajectories, one for the internal and one for the relative motion, appear in place of the exact trajectories of the classical S matrix. The justification for this procedure lies in the use of classical perturbation theory, which may be particularly applicable to rotational energy transfer. Details of the method have been reviewed by Clark et a/. (1977). This completes the present brief review of developments in the dynamical theory. One other important but quite different, type of analysis, discussed at length by Levine and Bernstein (1976),concerns the information content of any given calculation or experiment, and the relation between statistical and dynamical behavior. The argument is that on purely statistical grounds the outcome of any event may be predicted from knowledge of the distribution of accessible phase space in the products region. Deviations from this distribution may therefore be attributed to dynamical considerations. Experience shows that these deviations may frequently be characterized by a small number of so-called surprisal parameters, which constitute the dynamical information content of the process. Similar arguments have also been used to extend the results of collinear collision calculations to threedimensional space. Detailed coverage of these developments may be found in the books by Levine (1969),Nikitin (1970), Eyring et a/. (1974, 1975),Fluendy and Lawley (1973),Child (1974a), and Miller (1976~). There are also a number of valuable review volumes covering both experimental and theoretical developments edited by Ross (1966), Hartmann (1968), Schlier (1970), Takayanagi (1973),
232
M . S. Child
and Lawley (1976). A number of recent reviews of the molecular collision theory literature by Levine (1972), Secrest (1973), George and Ross (1973), and Connor (1973e),may also be cited. The aim of the present report is to follow developments in the semiclassical S matrix version of the theory in relation to experimental measurements and exact quantum-mechanical calculations, where these are available. This will serve to emphasize the close interplay between classical and quantummechanical behavior required in the analysis of modern experiments. The semiclassical S matrix theory itself currently has some computational disadvantages, but its underlying philosophy has been of overriding importance in clarifying the nature of molecular collision processes. C. SCATTERING IN THE SEMICLASSICAL LIMIT Modern applications of semiclassical methods to heavy-particle scattering date from the work of Ford and Wheeler (1959a,b). The theory has been reviewed in detail by Berry and Mount (1972) in the context of elastic scattering and more generally by Miller (1974,1976b)and Child (1976a).The achievement has been to obtain quantum-mechanically accurate transition probabilities and collision cross sections by integrating the classical equations of motion. An obvious, but not necessary, starting point is the Feynmann path integral formulation (Feynmann and Hibbs, 1965), according to which the scattering amplitude may be represented as an integral over all possible phase-weighted classical trajectories relevant to the experiment in question, with the phase expressed in terms of the classical action (Goldstein, 1950). Recent progress lies in methods for the evaluation of this integral. The stationary phase (or saddle point) approximation yields a sum over the particular trajectories leading from the desired initial to the desired final state of the system. This “primitive semiclassical’’ approximation is adequate to account for most simple interference effects, but problems arise at the caustics or thresholds of the classical motion due to coalescence of two or more trajectories. This leads to divergence of the primitive semiclassical approximation, but the topology of the caustics obtained may be used to suggest a suitable mapping for a uniform evaluation of the integral by the methods of Chester et al. (1957), Friedman (1959), and Ursell (1965, 1972). The theory is particularly well developed for caustics with the structure of one or other of Thom’s (1969)elementary catastrophes (Berry, 1976; Connor, 1976a), but other situations can also be accommodated (Berry, 1969; Stine and Marcus, 1972; Child and Hunt, 1977). The difference between these uniform results and the primitive semiclassical approximations lies in the use of special rather than trigonometric functions to handle the interference,
SEMICLASSICAL HEAVY-PARTICLE COLLISIONS
233
but in every case the relevant classical information is derived from the particular initial- to final-state “stationary phase trajectories” for the problem in hand. The significance of these special trajectories has been further underlined by detailed investigation of classically forbidden events characterized by stationary phase points that are complex. The discovery of corresponding complex trajectories, obtained by integrating Hamilton’s equations from complex starting conditions along a complex time path (Kotova et al., 1968; Miller and George, 1972a,b, Stine and Marcus, 1972), has extended the concept of quantum-mechanical tunneling from coordinate to quantum number space. This is particularly important for the theory of molecular energy transfer because single vibrational excitations from the thermally occupied levels are dynamically forbidden by classical theory for almost all molecules at normal temperatures, in the sense that the classical maximum energy transfer is less than a vibrational quantum. Developments in the semiclassical theory of electronic energy transfer (nonadiabatic transitions) has also led to a complex classical trajectory treatment of systems with several degrees of freedom for the heavy particle motion. The intention of the following sections is to illustrate the main features of the theory in comparison, where possible, with available experimental results. The fundamental concepts are outlined in Section I1 by application to the theory of purely elastic scattering. The reader is referred for more detailed coverage to important recent reviews by Pauly and Toennies (1965, 1968), Bernstein (1966), Bernstein and Muckermann (1967), Schlier (1969), Beck (1970), Toennies (1974a), Pauly (1974), and Buck (1976). Section I11 describes extensions of the theory to cover molecular energy transfer and chemical reactivity, initiated by Pechukas (1969a,b), Miller (1970a,b), and Marcus (1970,1971). Here there is less scope for comparison with experiment although the measurement of inelastic differential cross sections is now becoming possible for favorable systems (Toennies, 1974a). Finally, developments in the theory of nonadiabatic transitions are discussed in Section IV. Related experimental measurements have recently been reviewed by Kempter (1976) and Baede (1976). Nikitin (1968), Crothers (1971),Delos and Thorson (1972), and Child (1974a) review in detail the most important theoretical lines of development.
11. Elastic Atom-Atom
Scattering
The semiclassical theory of purely elastic scattering is very fully developed. The general techniques are illustrated below by application to the scattering amplitude and the differential cross section. Short accounts are also given
M . S. Child
234
of the theory of the total elastic cross section, and of semiclassical techniques for direct inversion of experimental data to recover the scattering potential.
A. SCATTERING AMPLITUDE AND DIFFERENTIAL CROSSSECTION The theory relies on reduction of the standard expression for the scattering amplitude
by the methods of Ford and Wheeler (1959a,b), as extended by Berry (1966, 1969). It is assumed, unless otherwise stated, that the energy lies above the limit for classical orbiting (Child, 1974a).The first step is to use the Poisson sum formula to replace the sum by a combination of integrals,
jM(e) =
(ikl-1
Jox
-
1[exp(2iq,-
11 exp(2i~271)~,_,~,(cos0)di,(3)
where I is related in the semiclassical limit to the classical impact parameter b by the identity I = 1 + 3 = kb (4) The detailed semiclassical analysis relies on introduction of the WKB phase shift, the accuracy of which is well attested (Bernstein, 1960), and the following asymptotic approximation for PA- 1/2(cosO), valid for 2 sin 6 >> 1 :
P A - 1,2(cos0)
- (2/nI sin
Q ) l l 2 sin(i8
+ n/4)
(6)
Here k 2 = 2,uE/h2 and a denotes the classical turning point. Thus for angles at which Eq. (6) is valid, z
f(O) = (ik)-'(27~sind)-''~
1
M= -
[I,&(6) - ~ , ( e ) ] e x p ( - i ~ ~ ) (7) ~3
where I,.$(@
= JoX
A112exp{i[2q(A)+ 2 M h k 20 i-x/4]}dA
(8)
in which either the upper or the lower signs are to be taken together. Equation (8) displays the two main semiclassical characteristics. The integration over I corresponds, according to Eq. (4), to an integral over all
SEMICLASSICAL HEAVY-PARTICLE COLLISIONS
235
relevant classical trajectories, distinguished according to impact parameter, and the exponent in the integrand is determined by the quantity 2q(i) k id3, which may be identified with the classical action integral (Child, 1974a); the ambiguity of sign arises from the conventional restrictions
Odq1dq2
(38)
Here S(q2,4,) is used to denote the internal part of S(q2, t 2 ;q l , t , ) at times t , -+ - co, t2 + m, and it is assumed that the necessary integration over translational variables has been performed. For the sake of simplicity the argument will be developed in more detail only for a system with one internal degree of freedom, described by the Cartesian variables (p, 4). One further obvious step to complete the semiclassical description is to employ normalized WKB wave functions for the internal states (Landau and Lifshitz, 1965)
in which w is the local vibrational frequency. This reduces the S matrix element to a combination of four simpler terms:
SEMICLASSICAL HEAVY-PARTICLE COLLISIONS
249
A complete analytical description of the forced harmonic oscillator has recently been given (Pechukas and Child, 1976) in this Cartesian representation, but almost all numerical applications of the theory have employed angle action variables ( N , 8) (Goldstein, 1950)to describe the internal motion. These have the advantages that the action is closely related to the quantum number n; by the Bohr-Sommerfeld quantization condition N
= Qpdq
=
(. +
;)ll
(44)
for any oscillator example. Furthermore, since the action is a constant of the motion for the isolated system, the corresponding operator fi commutes with quantum hamiltonian E?. Hence the eigenfunctions of fl, namely,
(8ln) = $,(e)
= (271)- '/'ein'
(45)
are also eigenfunctions of fi. Transformation of the Cartesian wavefunction to this angle action form is traditionally achieved by use of the semiclassical unitary transform (Fock, 1959; Miller, 1970a; Marcus, 1973)
based on the generator F,(qd) of the corresponding classical transformation, which is designed to satisfy the equations (Goldstein, 1950). awaq
= p(4,
e),
=,/ad
=
-m, e)
(47)
where p ( q , d ) and N(q,8) are the momentum and action consistent with coordinate q and angle 8. Closer analysis (Child, 1976a) of the origin of Eq. (46) shows that an additional term -if3/2 should be included in the exponent. Equation (46) implies the following integral representation for the Cartesian wave function: =
SoZ" =
which is seen to bear an obvious relation to the previous WKB form when Eqs. (47) are integrated to yield Fl(4, 0) =
[P ( 4 , d ) 4
-
W q ,w 4
(49)
the first term being integrated along a line of constant N ( q , U). The precise connection is obtained by combining the result of stationary phase integration of Eq. (48) with its complex conjugate, because Eq. (45) erroneously
M . S . Child
250
implies the existence of two independent wave functions, to describe a bound one-dimensional motion. An angle rather than a Cartesian coordinate integral representation for the S matrix may now be obtained by substituting the above form for i,hn(q) in Eq. (38), performing the integrations over q1 and q2 by the stationary phase approximation. Considerable care is required in the analysis. The argument hinges on the fact that the principal function S ( q 2 ,q l ) itself plays the part of a classical generator of the dynamical transformation from initial ( p l ,ql) to final ( p 2 ,q 2 ) Cartesian variables, in the sense that W&l1= -Pl(q2,ql),
(50)
dS/&l2 = PZ(q2,ql)
where pl(q2,ql), for example, denotes the initial momentum consistent with a classical trajectory between coordinates q1 to q 2 , and similarly for the final momentum p 2 ( q 2 ,ql). The transformation will not be followed in detail, but it may be illuminating to describe the first step, which is to identify the stationary phase values of q1 and q2 as solutions of the equations
(as/%,)+ (aFl/dql) = 0,
(dSldq2)
-
(dFl/%,)
=0
(51)
or, according to Eqs. (47) and (50), -Pl(qZ,ql)
+ P(q1,81) = O,
PZ(q2,ql)
-
P(q2,82) =
(52)
These identities require that q1 and q2 should be chosen such that the initial and final momenta pl(q2,ql) and p 2 ( q 2 ,q l ) along the trajectory should be consistent with q1 and the set angle el, and q2 and the set angle 8,, respectively. The required final result is obtained by performing a quadratic expansion of the exponent about this point, but the immediate result will not be given because it has been found convenient in practical computations to replace the angle 8 by a modified variable -
0 = 8 - mR/P
(53) where ( P ,R ) denote the conjugate momentum and coordinate for the translation motion (Miller, 1970; Wong and Marcus, 1971).This new variable 0, which is a constant of the motion in the asymptotic regions, may be regarded as the conjugate variable to the action N in a system in which the time t rather than the translational variable R is employed as the second independent coordinate (Child, 1976a). The final expression obtained for the S matrix in this 8 representation may be written
25 1
SEMICLASSICAL HEAVY-PARTICLE COLLISIONS
where the function $d2, 0,) is closely related to the previous Cartesian action S(q, ,q l ) :
%L%) = S [ Y , ( ~ , ~ ~ l ) ~ 4 ,+ ~~ ~ 2, ~[ ~~1l~( 1~ , > ~ l ) > ~ l l - F1 [ q 2 ( 8 2 ,
el)>821
(55)
Defined in this way, g(8,,Gl) may also be shown by use of Eqs. (47) and (50) to act as a generator for the transformation from initial to final action angle variables, in the sense that
df,fT8,
= - Nl(8,,8,),
dg/ia8, = N2(8Z,Qi)
(56)
The above double-angle representation for S, 182 is clearly implied by the arguments of Fock (1959), Miller (1970a), and Marcus (1973), and has been explicitly displayed for the forced harmonic oscillator by Pechukas and Child (1976). The connection between the semiclassical phases in Cartesian and angle action systems has also been discussed by Fraser et al. (1975). A form similar to Eq. (54) but corrected by a factor K(n,)K(n,), where N K(n) = (2n)-1’4(n!/NNe-N)1’2,
=n
+7 1
(57)
has been derived from the standard generating function for Hermite polynomials (Abramowitz and Stegun, 1965) by Ovchinnikova (1974, 1975). Direct numerical evaluation of the S matrix by use of Eq. (54) would involve double quadrature over an integrand determined by knowledge of the classical trajectories between all combinations 0, --+ 0,. A simpler but less symmetrical form may be obtained (Miller, 1970b) by performing the integral with respect to by stationary phase. It is also permissible to replace the second integration variable 8, by Q1, because the end 8,of any classical trajectory may be taken to be functionally dependent on G, for any given initial action N , . The resulting “initial-value representation”
el
s,,,, = (2nl-l s,’” ( a ~ , / a ~exp[i4(8,)1 ),~~ d
~ ,
(58)
@(el)= S”[8,(N1,O,),0,] - N , B , + N,B,(N,
8,)
(59)
where
coincides with the integral form obtained by Marcus (1970, 1971) by generalized WKB solution of the Schrodinger equation, except that Marcus uses the symbol A for the phase, and an angle variable iij defined in the range (O,1) in place of 8. A number of numerical transition probabilities have been obtained by direct quadrature in this initial-value representation (Miller, 1970b; Wong and Marcus, 1971; Kreek and Marcus, 1974), but considerable effort has been devoted to developing uniform analytical approximations dependent
252
M . S . Child
only on knowledge of the special trajectories leading from the correct initial to the correct final action (or quantum number). The following section is devoted to these uniform approximations. A comparison between these uniform results and the above quadratures is given in Section II1,D. B.
STATIONARY PHASE AND UNIFORM
APPROXIMATIONS
The above integral representations for the S matrix bear a close resemblance to that employed for the scattering amplitude in the discussion of elastic scattering. First, the integrand depends on a quantity determined by the classical equations of motion, in this case the action a,) in Eq. (54) or the closely related phase (D(8,)in Eq. (58) and in the elastic scattering case and 8, at present, and the phase shift y~,. Second, the integration variables L in the previous case) span the full range allowed by the physical situation. Finally, within each integration range there are discrete values of the relevant variable ??"), say, or L"), which defines trajectories leading from the desired initial state to the desired final state of the system; these are shown below to correspond to the stationary phase points of the integrand. The purpose of the present section is show how a variety of uniform approximations for the S matrix dependent on knowledge of only these special trajectories may be constructed. Figure 11 may be used to clarify the discussion. This shows the final quantum number n2 as a continuous function of the initial phase angle at a constant value of n,. As can be seen there are always two stationary phase starting angles and elb)for values of n,, designated n!, say, lying in the range nmin< n! < n,,,, with c?n,/c'G, positive at @') and negative at It is useful to adopt this convention for the choice of label a or b. Values of n , outside this range are termed classically inaccessible, but may still be treated by extension of the classical theory as discussed in the following section.
s(Gl,
(el
sl
e'f)
s'p).
1. Stationary Phase Approximation
The simplest approximation dependent only on the stationary phase trajectories is obtained by simple stationary phase evaluation of Eq. (58) (Miller, 1970a), S,,,,
where
+ iz/4) + P,lt2exp(iQb/ti- iz/4)
= P,'I2 exp(i@.,/h
(60)
25 3
SEMICLASSICAL HEAVY-PARTICLE COLLISIONS
nma’
n2
,min
FIG.11. The final “quantum number” nz as a continuous function of the initial oscillator phase angle 8. nmaxand nmindenote the upper and lower classically accessible values. Note the existence of two trajectories, with initial angles Oc0) and 8(b)for classically accessible values of n 2 .
with the derivative evaluated at @’, and the sign of the term f n / 4 being that of (dn,/dO,),,. Pa and P b therefore have a simple classical interpretation as the density of classical trajectories at final quantum number n 2 . This primitive semiclassical result is the direct analog of Eq. (1l),derived for the simpler case of purely elastic scattering. It suffers from a similar defect in that it diverges at the caustics of the classical motion, which in this case are the classical thresholds nmin and nmax.Figure 11 shows that the two trajectories come together and that dn,/dO, = 0 at these points, so that Pa and P , go to infinity. The phase difference (Q, - @.,)/h governing the semiclassical oscillations in the transition probability
P,,
=
Pa + P b + 2(PaPb)”2sin[(@, - Qa)/h]
(62)
may be given a simple graphical interpretation, as shown in Fig. 12. This illustrates how the closed curve C, representing the initial asymptotic state of the system is translated and distorted to C; by the collision, subject by Liouville’s theorem (Goldstein, 1950) to conservation of the area enclosed. The dashed curve C, denotes a classically accessible final state n 2 . The points ( A , B ) and (A’,B’)the initial and final points, respectively, on the
M . S . Child
254
FIG. 12. The dynamical transformation in phase space induced by a typical collision. The curves C, and C2 describe asymptotic states with quantum numbers n , and n 2 . respectively. The collision induces the transition C , + C‘,, such that the phase difference between the two stationary phase trajectories AA’ and BB’ is given by the shaded area in the diagram.
n, + n2 trajectories. The relevant phase difference - @a may be shown (Pechukas and Child, 1976; Child 1978; Child and Hunt, 1977) to be equal to the area of the smaller segment into which the translate C; is divided by Cz. (Here it is assumed that n1 is the smaller of the two quantum numbers.) This means, since the area common to C, and C; is by the Bohr quantization condition equal to (2n, l)h, that the maximum phase difference in Eq. (62) is ( n , -$)Tc.In other words, there can be at most n, 1 maxima in the variation of P,, either as a function of final quantum number n2 at given energy or as a function of collision energy E at given n 2 .
+
+
+
2. Uniform Airy Approximation
The confluence of two stationary phase points leads to the simplest (fold) type of catastrophe in Thom’s (1969) classification, which is handled as in the case of the rainbow singularity in elastic scattering by means of a variable transformation O,(x), which maps the exponent in Eq (58) onto a function cubic in x : (63) @ ( ~ , ) /= h 9x3 - t(n,)x A(n,)
+
This lead to the following uniform Airy approximation (Connor and Marcus, 1971): S,,“, - ,1PeiA[(p;/2
+ PLi2)(li4Ai( - 5)
-
i(PAl2 - Pi/2)5-114 Ai’( - t)] (64)
SEMICLASSICAL HEAVY-PARTICLE COLLISIONS
255
with the parameters ( ( n 2 ) and A(n2)given in terms of the stationary phase exponents by
This reduces for ( >> 1 to the previous primitive semiclassical form given by Eq. (60).
3. Uniform Bessel Approximation Account has been taken in the above derivation of the presence of a maximum or a minimum in Fig. 11, but the periodicity of the function n2(8,,n,) has been ignored. This may become serious in cases of nearly elastic behavior, when the gap between nminand n,,, becomes so small that the stationary phase regions around the special trajectories overlap with both. This situation falls outside Thom’s (1969) catastrophe classification but it may be handled (Stine and Marcus, 1973), by a mapping 8,(y) such that @(e,)/h= A(n2)- 5(n2)cos 27ry
with m
=
In, -
-
2nmy
rill and A(n2)and C(nJ determined by the equations A = i ( @ b + @.,)/A (5’ - mz)1/2 + marccos(m/() = @.,)/h +(@b -
(67)
(68)
The final expression for the S matrix element is
S,,,, = ( ~ / 2 ) ” ~ e ’ ” [ ( P+ t ’ Pb”2)((2 ~ - m2)”“5,(() - i(P,”2
- Pb””((i“2
-
m”- 1/45, m(O1
(69)
where 5,(() and 5;(() denote the mth-order Bessel function and its first derivative. This also reduces to the primitive semiclassical form for 5 >> 1. Figure 13 gives a comparison between the exact transition probabilities and the above primitive semiclassical, uniform Airy Bessel approximations, for the special case of a forced harmonic oscillator for which the theory may be handled analytically throughout (Pechukas and Child, 1976). The parameter a measures the interaction strength. This illustrates the relatively crude nature of the primitive semiclassical approximation for transitions from n = 0 state. The uniform Airy approximation shows a marked improvement except, as expected, for the 0 - 0 transition at weak interaction strengths. The uniform Bessel approximation is seen to remedy this defect, but to give a progressively worse description as the interaction strength increases.
M . S . Child
256
\ \
\ \ ',,Primitive
\.
1
Airy
I I I I 1
I
\
\ \ \
\ \ \
I
I
I
2
FIG. 13. Comparison between the primitive semiclassical uniform Airy, uniform Bessel, and exact transition probabilities for the forced harmonic oscillator (a) 0 --t 1 transition; (b) 0 + 0 transition. The strength parameter c( is the fourier component of the forcing term at the oscillator frequency. [Taken from Pechukas and Child (1976) with permission.]
SEMICLASSICAL HEAVY-PARTICLE COLLISIONS
257
It is evident that the above approximations cover all eventualities, but that no single expression derived from Eq. (64) is universally applicable. More recently Child and Hunt (1977), following arguments similar to those of Ovchinnikova (1973, have derived a uniform Laguerre approximation from the double-integral representation (54), designed to be equally applicable to all oscillator excitation problems. This is more complicated to describe but as easy to apply as the uniform Bessel approximation. Comparison between all these forms is made in Section 111, D. Reference may also be made to a variety of other uniform approximations designed for use with systems having more than one degree of freedom, and for situations giving rise to more than two stationary phase trajectories (Connor, 1973a-d, 1974a,b; Marcus, 1972; Kreek et al., 1974, 1975). Two general discussions of the relevances of Thom’s (1969) catastrophe theory to the structure of uniform approximations have also been given (Connor, 1976a; Berry, 1976). C. CLASSICALLY FORBIDDEN EVENTS One of the most remarkable achievements of the theory (Miller and George, 1972a,b; George and Miller, 1972a,b; Stine and Marcus, 1972) has been to obtain a semiclassical description of events such as tunneling through a potential barrier or collisional excitation to vibrational states that are dynamically inaccessible by classical mechanics. The emphasis here is on dynamical inaccessibility, but not violation of any conservation law. There is no conservation law restricting motion to one side or other of a potential barrier; it is simply that the normal laws governing interconversion of kinetic and potential energy prevent the particle from passing through. Equally, there may be sufficient total energy to populate a given vibrational state, but the available interaction between oscillator and collision partner may be too weak to cause the relevant transition. The resolution of this paradox lies in analytic continuation of the classical equations of motion into the complex time plane and to complex values of any nonphysically measurable variables. This means that only real values of the internal action are acceptable because these correspond to the quantum numbers, but that the angle variables, which cannot be simultaneously measured in quantum mechanics, may be complex. This analytic continuation is already familiar in the WKB theory of one-dimensional tunneling based on the transmission factor exp( - J ( p ( d q ) determined by an imaginary momentum ilpl in the barrier region. It is also suggested by the presence of the maximum and minimum in Fig. 9 that analytic continuation of the solution of the equation %(9A
=
4
(70)
M . S . Child
258
which has two real roots in the classical region, will yield two complex solutions for the initial angle variable when ng is classically inaccessible. These ideas may be underlined by more detailed analysis of two soluble models. The first is the problem of passage through a quadratic barrier (Miller and George, 1972a), V ( q )=
(71)
-L 2 G 1 2
at a negative energy -AE, subject to the boundary condition p < 0 for t < 0. The solution of the classical equations is readily shown to be 4
-(2AE/lc)’12 coshw*t,
=
p
-(2AE/p)”’sinhw*t
=
(72)
where w* = (ti/p)1’2
(73)
The coordinate q therefore remains negative at all times, while the momentum changes sign at t = 0 as the particle bounces back from the barrier in accordance with classical experience. Suppose, however, that the time experiences an imaginary increment iz/w* during the motion, so that finally t = t‘
+ in/w*
(74)
with t’ real. Then according to Eq. (72) 4 p
= =
- ( ~ A E / K ) ”cosh(w*t’ ~ + in) = (2Af?/K)”2 cash w*t’ -(2AE/p)’” sinh(w*t’ + in)= (2AE/p)’12 sinhw*t’
(75)
The signs of both p and 4 have changed and the particle has passed through the barrier. It is readily verified on computing the action that the semiclassical phase associated with the motion simultaneously acquires an imaginary component “J
~ m [ ~ q ql)/til= ,, Im
J-
+ i n / g*
oo
p q dt
= 71 A E / ~ W *
(76)
giving rise to the correct first-order WKB transmission factor exp( - n AE/hw*) for the problem. There is, of course, a second complex conjugate trajectory that also passes through the barrier, but leads to an exponential increase in the amplitude of the wave function. This is rejected on physical grounds (Miller and George, 1972a). A more detailed analysis of this quadratic barrier passage problem has been given by Child (1976b). The second example is the forced harmonic oscillator, with hamiltonian H ( p , q ) = +p2
+ +q2
-
f(t)q
(77)
in a system of units for which the mass, force constant, and vibrational
259
SEMICLASSICAL, HEAVY -PARTICLE COLLISIONS
frequency are equal to unity. It is assumed that the forcing term vanishes at t = f co,that it is an even function of time, and that the system starts in the state ( N , , 0,) so that, at time t + - co, q = (2N1)l/,cos(t
+ O1),
p
=
-(2N1)112sin(t
+ 8,)
(78)
The classical equations may be shown to yield, at t + + K, q
= (2N,)'I2cos(t
+ 0,) + a sin t,
p
=
-(2N1)1/2 sin(t + 0,)
+ acos t
(79)
where c( is the Fourier component of the forcing function. The final action is therefore
N,
= +(p2
+ q 2 ) = N,
-
a ( 2 ~ , ) 'sindl /~
+ +a2
(80)
and the maximum and minimum classically accessible values, obtained at
O1
=
-n/2 and 8,
= 4 2 , respectively, are given
by
N 2 = (Nil2
(81)
Real values of N , outside this range may however be obtained by choosing the initial angle to lie along one or other of the lines 8, = f n/2 + i07 in the complex angle plane, so that
N,
=N
, 3- ~ 1 ( 2 N , ) "cash ~ 0;'
+ *a2
(82)
The semiclassical phase associated with these complex trajectories again acquires a progressively increasing imaginary part as N , moves away from the classical region (Pcchukas and Child, 1976). There are again two complex conjugate trajectories for each classically forbidden transition, one consistent with an exponentially small and the other with an exponentially large transition probability. The mathematical argument for rejection of the latter is somewhat clearer than in the tunneling case. It is that the integration path for stationary phase (or steepest descents) evaluation of the integral in Eq. (58) can pass through only one of the complex stationary phase points, and the chosen point is always that leading to an exponentially small value for the integral (see Child, 1976a).This has already been taken into account in obtaining the uniform approximations given in Section III,B, since these approximations are specifically designed t o bridge the classical threshold regions. This analytical discussion demonstrates the existence of physically meaningful complex solutions of the classical equations of motion. The numerical determination of such complex trajectories in real applications initially posed some stability problems (Stine and Marcus, 1972; Miller and George, 1972a,b), but the results obtained are in close agreement with exact quantum-mechanical values (see Tables I and 11). Calculations of this type are particularly relevant to studies of vibrational energy transfer and the
M . S . Child
260
chemically reactive exchange of light atoms, which are dominated in the thermal energy range by events that are forbidden by classical mechanics. D. NUMERICAL APPLICATIONS AND
CONCLUSIONS
Tables I and I1 list sample results for the excitation of harmonic and Morse oscillators subject to exponential interactions according to the models of Secrest and Johnson (1966) and Clark and Dickinson (1973). Results are given for the primitive semiclassical (PSC), uniform Airy, uniform Bessel, and uniform Laguerre approximations ; the heading Quadrature includes results for numerical quadrature in Eqs. (58), where these are available. Entries marked by an asterisk are classically inaccessible. A more extensive tabulation of this type, which also covers other less sophisticated approximations, has been given by Duff and Truhlar (1975). It is evident that the primitive semiclassical approximation is always relatively crude, at least for small n, values, but that the uniform Airy expression shows a marked improvement except for the diagonal n , + n , transitions at low collision energies. These are adequately covered by the TABLE I HARMONIC OSCILLATOR TRANSITION PROBABILITIES SECREST AND JOHNSON (1966) '
n,
n,
o*
0 1 2 3 4 1 2 3 4 5 2 3 4 5 6
0 0 0
o* 1 1 1 1 I* 2 2 2 2 2*
Primitiveb -
0.422 0.416 0.359 ~
0.290 0.009 0.168 0.285 -
0.208 0.020 0.165 0.262 ~
IN THE
MODELOF
Airyb
Bessel'
Laguerre'
Exactd
0.058 0.211 0.381 0.266 0.075 0.287 0.01 1 0.174 0.240 0.062 0.206 0.017 0.170 0.194 0.045
0.334 0.205 0.380 0.264 0.0851 0.284 0.012 0.175 0.239 0.0756 0.203 0.016 0.167 0.193 0.0367
0.0523 0.219 0.366 0.267 0.0887 0.281 0.010 0.170 0.240 0.0766 0.204 0.017 0.169 0.194 0.0370
(0.0599) 0.218 0.366 0.267 0.0891 (0.286) 0.009 0.170 0.240 0.0769 (0.207 0.018 0.169 0.194 0.0371
The energy unit is half the vibrational quantum; m = 2/3, OL = 3/10, E = 20. Entries marked with asterisk are classically inaccessible. Values in parentheses were obtained by difference. Child and Hunt (1977). Miller (1970b). Secrest and Johnson (1966).
26 1
SEMICLASSICAL HEAVY-PARTICLE COLLISIONS
TABLE I1
HARMONIC OSCILLATOR TRANSITION PROBABILITIES" n,
nz
Airy'
Bessel'
Laguerre'
O*
1 2 2 3 3
1.08(-1) 1.20(-3) 4.41(-2) 1.51(-5) 1.48(-3)
1.03(-1) 1.15(-3) 4.16(-2) 1.43(-5) 1.33(-3)
1.08(-1) 1.22(-3) 4.16(-2) 1.45(-5) 1.33(-2)
O*
1* 1*
2*
Quadratured ~
5.3(-2) 2.5(-4) 1.7(-3)
~
4.3(-2) 1.8(-6) 4.6(-4)
Exact' 1.07( - 1) 1.22(-3) 4.18(-2) 1.46(-5) 1.33(-3)
" The second column under Quadrature gives Pn>",;m = 2/3, d~ = 3/10, E = 8. Values in parentheses were obtained by difference. ' Stine and Marcus (1972). ' Child and Hunt (1977). Wong and Marcus (1971). Secrest and Johnson (1966).
uniform Bessel formula, but this decreases in accuracy as the transition probability falls below unity. The uniform Laguerre approximation is seen to be consistently more accurate than either the Airy or Bessel approximation. Finally, quadrature results give moderate accuracy for the classically accessible transitions but become progressively less accurate outside this region. Two results for the n , -+n2 and n2 -+ n , transitions are given in each case because the integral in Eq. (58) is not symmetric in n, and n 2 .The reason for the greater accuracy in the classically accessible case is probably that the integration contour is necessarily taken along the real GI axis and hence passes through the points of stationary phase if the transition is classically accessible but not if it is outside the classically accessible range. Overall, the Airy uniform approximation is generally recommended on grounds of simplicity, but the Laguerre uniform approximation is to be preferred for the highest accuracy. The above results refer to excitation of one internal degree freedom. The general theory is equally valid in more complicated situations, but application of the powerful uniform approximations is complicated by the necessity to find the special n, + n2 trajectories, the direct search for which becomes prohibitive for as few as three degrees of freedom. For this reason only a few fragmentary results have been reported for the vibrationally and rotationally inelastic scattering of a diatomic molecule (Doll and Miller, 1972). One way around this difficulty is to employ a partial averaging procedure whereby the rotational motion is treated by purely classical Monte Carlo techniques, and only the vibrational part of the problem is treated by the full semiclassical method (Doll and Miller, 1972; Miller and Raczkowski, 1973; Raczkowski and Miller, 1974). Another solution is to revert to numerical quadrature for the multiple-integral initial-value representation analogous to Eq. (58) (Kreek and Marcus, 1974). Despite these
262
M . S . Child
difficulties Fitz and Marcus (1973, 1975) have been able to develop a full semiclassical treatment of collisional line broading. The semiclassical theory has also been compared with exact quantummechanical results for the collinear (all atoms constrained to lie on a line) hydrogen atom exchange reaction (Duff and Truhlar, 1973; Bowman and Kuppermann, 1973). Two special problems have been identified. The first concerns quantum-mechanical tunneling in nonseparable systems, because the use of complex classical trajectories (George and Miller, 1972a,b) yields a reaction threshold above that obtained by quantum mechanics. This problem has been reinvestigated by Hornstein and Miller (1974) but the situation is still not satisfactory. Nevertheless, considerable progress has been made toward developing a reliable semiclassical version of transitionstate theory for the chemical reaction rate constant (Miller, 1975; Chapman et a/., 1975; Miller, 1976a, 1977).The second difficulty concerns the treatment of Feshbach resonances observed in this reaction but not adequately described by the semiclassical calculation of Bowman and Kuppermann (1973).Fuller analysis by Stine and Marcus (1974) shows that a quantitative description may be obtained by following a series of multiple collisions within a collision complex.
IV. Nonadiabatic Transitions The theory of nonadiabatic transitions applies to situations where the Born-Oppenheimer separation of nuclear and electronic degrees of freedom breaks down. The basic theory was formulated by Landau (1932), Zener (1932), and Stuckelberg ( I 932), but serious doubts on its general application were cast by the criticisms of Bates (1960) and Coulson and Zalewski (1962) concerning the inflexibility of the Landau-Zener model. Recent developments have been to obtain appropriate validity criteria and to increase the flexibility of the model by emphasizing its topological structure. This has led to emphasis on the significance of certain complex transition points at which the adiabatic potential curves intersect. The key papers on the two-state model are by Bykovskii et al. (1964),Demkov (1964),Dubrovskii (1964),Delos and Thorson (1972, 1974), and Crothers (1971),and the review by Nikitin (1968). Applications of the two-state theory to analysis of the inelastic differential cross section and generalizations to more complicated situations are outlined in Sections IV,B and IV,C. Particular attention is given to the description of two-state “surface hopping” processes in systems with several nuclear degrees of freedom. The theory due to Tully and Preston (1971)and extended by Miller and George (1972a,b) is based on the assumption that since any
SEMICLASSICAL HEAVY -PARTICLE COLLISIONS
263
classical trajectory must cut a one-dimensional section through the intersecting surfaces, any problem may be reduced to a combination of singlecurve crossings. The reader is referred to reviews by Tully (1976) for more detail of the theory and by Baede (1976)for a wider account of its application.
A. ONE-DIMENSIONAL TWO-STATE MODEL The time-independent equations for a typical two-state problem may be written
where
k,Z(R)= 2 p [ E - qi(R)]/h2 U i j ( R )= 2PLl/j(R)/h2
and V ( R )is the matrix of the electronic hamiltonian in the basis of asymptotic electronic states. It is assumed in what follows that Vll(R) < V22(R)at infinite separation. Equations (83) define the exact quantum-mechanical problem. An equivalent time-dependent semiclassical form may be based on the assumed knowledge of a classical trajectory R(T)with velocity variation v(z) for the relative motion, in terms of which Eqs. (83) may be reduced to
where the elements q j ( z )denote Kj(R)evaluated along R(z).The arguments used by Bates and Crothers (1970)and Delos et al. (1972)in justifying Eq. (86) make use of the approximation
which will be used below to relate a number of equivalent results. The above equations are in the diabatic picture [see Smith (1969) and Lichten (1963) for a precise definition]. The equivalent adiabatic representation is obtained by transforming to a parametrically time-dependent
M . S. Child
264
The necessary unitary transform may be written (Levine et al., 1969)
w=
(
cos 8(z), sin 6(z),
-sin Q(z) cos QT)
(89)
where the angle 8(z), which is also used below to define a new independent variable t, is given by t = COt28(T) = -[Vi,(T) - Vzz(T)]/21/12(2) (90) The equations of motion in this adiabatic representation become (Delos and Thorson, 1972)
The coupling therefore depends on the time derivative of the mixing angle O(T), and hence on the rate of change of the electronic wave function. The key quantity d%/dz may be written
thereby drawing attention to the times zC,z,*[which are necessarily complex according to Eq. (88)] at which the adiabatic terms V,(T) intersect, because do/& clearly diverges at these points. Their location continues to dominate the structure of the theory even in the mathematically more convenient diabatic representation (86), where their role is less immediately apparent. The most convenient development for present purposes is based on the variable t defined by Eq. (90) as the independent variable, in terms of which Delos and Thorson (1972) show by introduction of the functions T ( t )= Vl z[z(t)l h dt/dz yl( t ) = [T(t)]-
exp[
that Eq. ( 5 5 ) may be reduced to T2(t)(l+ t 2 )
-
iT(t) +
-
i J'to T(t')t'dt']c1[z(t)]
(93)
265
SEMICLASSICAL HEAVY-PARTICLE COLLISIONS
The transition points now lie at t = & i. Further simplification results from the Landau-Zener curve-crossing model defined by the equations
GZ(4
-
Vll(d V,,(z)
=(F, =
F,)[R(d
-
Rx1 = (Fl - F,)u(z
- 7,)
V,, = const
(95)
where u denotes the nuclear velocity and z, is the time at the crossing point. In this case T ( t )= 2V~,/hv(F, - F,) = T ,
= const
(96)
with the result that Eq. (94) reduces to an equation of Weber form (Abramocitz and Stegun, 1965):
dZY1 + [ T i ( l dt2
~
+ t2)
-
i T o ] y ,= 0
(97)
As emphasized by Delos and Thorson (1972), the same will be true of any model in which the function T(t) is constant over the effective transition region. The problem is therefore mathematically equivalent to transmission through a quadratic barrier - T i t 2 at the complex energy T i - iT,,and the solution may be expressed in terms of parabolic cylinder functions of complex order. Manipulation of the standard asymptotic forms of these functions yields the familiar Landau-Zener transition probability (Landau, 1932; Zener, 1932; Delos and Thorson, 1972), which is given below as a special case of a more general result. The generalization is due to Dubrovskii (1964) and was followed in a different form by Child (1971). It is based on the argument that deviations from the strict Landau-Zener model or from constancy of the function T ( t ) will not affect the fundamental complex barrier transmission structure of the problem, providing the product T2(t)(l+ t 2 )has only two zeros (transition points) close to the real axis. Hence it is permissible to map the general Eq. (94) onto the quadratic model (97) by use of a variable transformation due to Miller and Good (1953). Analysis of the model case is quite lengthy because account must be taken of transitions occurring during both inward and outward motion ( - 00 < z < 0 with u < 0, respectively). The final results for the S matrix elements take the following forms if the classical turning point, corresponding to z = 0, lies a region such that It1 >> IT: - iTol: S,, S,, S,,
+ (1 - e-2"6)exp(- 2 f
2ix)] exp(2iq1) = SZl= 2ieCff6(1- e-2n6)1'2sin(r+ x)exp(iG, + iq,) = [ e - 2 K 6+ (1 - e-'"')exp(2ir + 2ix] exp(2iq,) = [e-21Ld
-
(98)
M . S. Child
266
where the parameters 6, r,q l , q2,x,and qk may be expressed in the following equivalent forms, related by Eqs. (87), (90), and (93): 1
6 =Im 71
Ji T(t)(l + t
1 271
1 h
=-1m-p
1 2n
= - Im
r = - 2 Re =
-h
[V+(Z) -
V-(z)]dz
c
[k-(R) - k+(R)]dR
JR:
i
T(t)(1 + t 2 ) ' I 2 d t -
V+(t)]dz
J ~k +' ( R )dR -
~~c a-
a+
k - ( R )dR
x = arg T(i6) + 6 - 6 In 6 + n/4 q1 = y q2 = y +
(99)
I,o,
ReIJi"[V-(z) 1
Re[
y dt
p i
1
+ r = y - + R e [l rycc k+(R)dR a+ a+
-
= y+ -
- JR U -c
Re[JRC a + k + ( R ) d R-
s"' a-
1
k_(R)dR
1
(102)
k-(R)dR
where y * are the WKB phase shifts in the two adiabatic channels. It is readily verified that Eq. (99) reduces in the Landau-Zener approximation to (5Lz
=
To12 = V:,/hv(F, - F J
(103)
The derivation of these equations has followed Landau (1932), Dubrovskii (1964) and Delos and Thorson (1972), although the original result including the phase term r but not x was derived by Stuckelberg (1932) by a phase integral approach more recently discussed by Kotova (1969), Thorson et al. (1971),Crothers (1971),and Dubrovskii and Fischer Hjalmars (1974).Similar results have been obtained in the Landau-Zener model by transformations of Eqs. (83)to the momentum representation (Ovchinnikova, 1964; Bykovskii et al., 1964; Nikitin, 1968; Child, 1969; Bandrank and Child, 1970). The only differences are that the broken phase shifts ijl,q2 are replaced by the true diabatic WKB phase shifts y 1 and the Stuckelberg interference term r is given in the present notation in the mixed diabatic-adiabatic form (Bandrank and Child, 1970)
SEMICLASSICAL HEAVY-PARTICLE COLLISIONS
267
Reservations about this mixed prescription have been expressed by Crothers (1975), but numerical differences with the form given by Eq. (100) are likely to be small under conditions where Eq. (98) is valid. Recent efforts have been devoted to assessing the validity of the results in the light of important criticisms by Bates (1960) of the flexibility of the crude Landau-Zener model. It is clear from the above discussion that the number and positions of the complex transition points (z,,z,*) at which the nonadiabatic coupling (do/&) diverges is of paramount importance. These could be detected as points close to the real axis, where there is a rapid change in the composition of the electronic wave function. The validity of the Stuckelberg-Landau-Zener equations (99)-( 104) depends on (1) the existence of one complex conjugate pair (zc,z,*) on each (inward or outward) part of the trajectory, and (2) an adequate separation between these pairs, in the sense that the usual large argument asymptotic expansions for the parabolic cylinder functions may be applied in the intervening classical turningpoint region. The implication in terms of l- and 6 is that (Bykovskii et al., 1964)
r >> 1,
rp >> 5
(105)
These conditions break down at energies close to a curve crossing because -+ 0, but a perturbation formula valid for 6
$y) = (cll$l,) is given by t,b'f) = (cos Q)$(p) + (sin
(1 6 4
(16b)
= cos 0
(17)
while the hidden variables, which transform by the same unitary transformation, are related by = (cos Q) ltY)l Explicitly Eqs. (17)-(19) yield cos Q > /tib) cos 0
+
t,) the two excited quasi-molecular states connect directly to two inelastic scattering channels (for example, excitation, charge exchange, autoionization). It is important to note that the two quasi-molecular states 1 and 2 are populated coherently. When the internuclear separation passes through R = R, >> R , at t = t, the amplitudes of the two quasimolecular states involved are coherently mixed due to a long-range nonadiabatic interaction. It is obvious that the interaction at R = R , is important only on the way out since the inelastic amplitudes (1 and 2) are zero on the way in (t d
-t2).
The electronic wavefunction of the quasimolecular system on the way out between R,, and R, takes the form
+ bZ(tP2R
(2) Here Y i Rare the adiabatic wavefunctions of the quasi-molecule (the term 0 is omitted as it does not take part in interference). From r = to up to t = t,, the system develops adiabatically: y e ,
=h(Wll7
where bi(to)are the inelastic amplitudes at the instant of population t = t , , and E,(R) are adiabatic terms of the quasimolecule. At 1 = t , the system
R
-to
0
t0
Ro
p
Ro
RL
R
FIG.3 . Schematic representation of the quasi-molecular terms as a function of internuclear distance R. E , = E,(R) is the energy of the ground state; E , , = E l , J R ) is the energy of inelastic states: hatched areas represent the nonadiabatic interaction region; R, and R, ( R , = R, z R , ) are the internuclear distances of crossing points a, b, a', b'. The times - t 2 , -r,, t , , r 2 correspond to the crossings a, b, a', h'; t = 0 is a turning point of nuclear motion. t = ti and R = R, are the parameters of the long-range interaction region (1.
,
S . V . Bobashev
346
passes the long-range interaction area at R = R, (area d in Fig. 3) where the molecular wavefunctions transform to atomic wavefunctions. The wavefunction of the system immediately on the left of R, is
where b; = bi(t,) exp[
--; lz
E,(R)d t ]
Y,; are the molecular wavefunctions on the left of R , . On the right of R,
the wavefunction has taken the form
y e ,= b:Y:,
+ b2+yiR
(5)
where Y & are the molecular wavefunctions on the right of R, . As the particles recede, Y & transform to the atomic wavefunctions. The formation probabilities of the atomic states of interest (1’ and 2’ in Fig. 3) are given by squaring b: and b:: W, = (b:)’, W, = (b:)’. The relationship between b:, and b , is given by the 2 x 2 unitary matrix aiK,which is determined by the parameters of the nonadiabatic interaction at R = R, :
Thus, the probability of 1’ state formation is
Wl.= I~ll121bl(~o)(2 + I~l2l2lb~(~0)l2
The probability of 2’ state formation is given by a similar expression. The unitarity of transformation implies
+
+ I@2212 = 1,
a21 . at2 = - E l 1 . .Tz (8) This expression displays the interference term due to the nonadiabatic interaction at R = R , . Let us write ( ~ 1 1 ( 2 1%112
= 1%212
2a, . aT2bl(tO)b~(t0) = A exp(i