Advances in Atomic and Molecular Physics, Volume 14

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.

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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

This Page Intentionally Left Blank

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, << AE,, the energy resolution of the experiment will be AE,, since electrons resulting from excitation of the two states cannot enter the analyzer at the same beam energy. In order to consider specific examples we must specify the energy width given by the cathode distribution. We take this width to be 0.3 eV. If we consider elastic scattering with no target states within 0.3eV of the target ground state (as will be the usual case for atomic targets) then we may set AE, = 0.3 and define the energy resolution of the experiment by AE,. We transform Eq. (20) into a number of counts at the detector by multiplying the counting time by the detector efficiency divided by the electron charge ( w e ) . The ratio of the signal achieved with the monochromator resolution AEl and energy analyzer resolution AE2 for the usual case with AE, = AE2 is given by

-

R = (AE1/AE2)3’2(t12/t2) (21) where t,, is the counting time used with 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 << 1 and E, << EL, EL E, . In order to get an idea of the contributions to the energy spread in the experiment, AEL, we differentiate Eq. (47). Thus

-

AEI,=AE,+

(i)

+ @)1'2

- AE,-

(?$)'j2-

cos 8 AEt (EeE,)"2 sin 8 Ad

For cos 8 = 0, AE; = AEe +

[ ( ~ ) 1 ' 2 c o s 9 AE,

(El -

AE,

+2j');(

-

1

(EeE,)"2sinl?A0

(49)

For m/M << 1 and reasonable values of A E e , the second term in Eq. (49) may be neglected with respect to AEe . The third term may be neglected in highenergy resolution crossed-beam experiments only if a high degree of angular collimation in both the electron and target beam is obtained. For a discussion of the broadening of resonance profiles due to the apparatus function, see, for example, Comer and Read (1973).

D. CALIBRATION TECHNIQUES Experimental studies of resonances have been used to obtain accurate calibration of electron energy scales and place cross-section measurements on

ELECTRON ATOM AND MOLECULE SCATTERING

29

an absolute basis. These calibration procedures will be discussed separately in this section. 1. Electron Energy Scale

Accurate electron energy scale calibration relies on the measurement of some feature, which occurs at a precisely defined energy, which is also precisely known. Cusps are features that have exactly the right properties to be used for such calibration. They are sharply defined in energy and they occur at precisely defined energies (inelastic thresholds). It has been shown by Wigner (1948) that all scattering cross sections exhibit cusp behavior at thresholds of new scattering channels. He also pointed out that the effects due to cusps were small and therefore probably not important. It is certainly true that cusp effects observed thus far are small and it is only because of the advances in technology that allow high-energy resolution electron-scattering measurements that cusp effects can in fact be observed (see Golden, 1976).It was postulated by Sanche and Schulz (1972a) that cusps existed on the 23S and 2's thresholds in their e-He transmission experiment. There is perhaps earlier evidence of the existence of cusps in the electron transmission experiments of Kuyatt et al. (1965) and Golden and Zecca (1970), but this evidence is inconclusive. The first unambiguous measurement of a cusp in differential electron-atom scattering is probably due to Andrick et al. (1972).They obtained data for the differential elastic electronsodium scattering cross section as a function of energy at various angles near the 32Pthreshold, which is shown in Fig. 11. The shape and magnitude are in general agreement with the close-coupling calculation of Moores and Norcross (1972). The total elastic cross section results of Moores and Norcross (1972)are shown in Fig. 12, together with the absolute measurement of both K. Rubin (unpublished data, 1972) and Perel et a!. (1962), and the results of Brode (1929)normalized to the calculation at 5 eV. It is readily seen from the figure that the error bars on the measurements are too large to show the cusp in the total cross section, which is about a 20% effect when given by the theory. In general, the differential elastic cross section may either increase or decrease both above and below the opening of a new scattering channel (see, for example, Baz et al., 1962). Thus the joining of these possibilities at threshold leads to four possible shapes, which will be referred to as positive or negative cusps and positive or negative steps. On the other hand, the total cross section must conserve flux and thus the total cross section may only decrease above the opening of a new channel, although it may either increase or decrease below. Therefore, only a positive cusp or a negative step is possible in this case. The equations developed by Cvejanovic et al. (1974) for

D . E. Golden

30

'-:;----2.5

2. I

1.6

Enerqy L e v )

1.6

2. I

2.5

Ervqy (eV)

FIG. 11. Differential cross sections of elastic e-Na scattering versus energy at various scattering angles (from Andrick et al., 1972) of (b) and (c) are enlarged against the scale of (a) by factors of 3.7 and 25, respectively.

the differential cross section near an inelastic threshold at energy Ethare (see also Herzenberg and Ton-That, 1975). bk- ' k z Re h,

E < E,h

--

do

(50)

+ bk- ' k z 1m h,

E > 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.

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Ross, M. H., and Shaw, G. L. (1961). Ann. Phys. [7] 13, 147. Rudd, M. E. (1964). Phys. Rev. Lett. 13, 503. Rudd, M. E. (1965). Phys. Rev. Lett. 15, 580. Rudge, M. R. H. (1973). Adv. A t . Mol. Phys. 9,47. Rudge, M . R. H., and Seaton, M. J. (1964). Proc. Phys. Soc. London 83,680. Rudge, M. R. H., and Seaton, M. J. (1965). Proc. R . SOC.London, Ser. A 283,262. Sanche, L., and Burrow, P. D. (1972). Phys. Rev. Lett. 29,1639. Sanche, L., and Schulz, G. J. (1972a). Phys. Rev. A 5 , 1672. Sanche, L., and Schulz, G. J. (1972b). Phys. Rev. A 6 , 69; see also Erratum in Phys. Rer. A 6 . 2500 (1972). Schowengerdt, F. D.. and Golden, D. E. (1975). Phys. Rev. A 11, 160. Schulz, G. J. (1959). Phys. Rev. 113, 816. Schulz, G . J. (1962). Phys. Rev. 125, 229. Schulz, G. J. (1963). Phys. Rev.Lett. 10, 104. Schulz, G. J. (1964a). Phys. Rev. Lett. 13, 583. Schulz, G . J. (1964b). Phys. Rev. 135, A988. Schulz, G. J. (1966). Proc. Int. Conf. Phys. Electron. A t . Collisions, 4th, 1965. Schulz, G. J. (1973a). Rec. Mod. Phys. 45, 378. Schulz, G. J. (1973b). Rev. Mod. Phys. 45,423. Schulz, G . J. (1976). In “Principles of Laser-Plasmas’’ (G. Beksfi, ed.), Chapter 2. Wiley (Interscience), New York. Schulz, G. J., and Asundi, R. K. (1965). Phys. Rev. Lett. 15, 946. Schulz, G. J., and Asundi, R. K. (1967). Phys. Rev. 158, 25. Schulz, G. J., and Fox, R. E. (1957). Phys. Rev. 106, 1179. Schwartz, C . (1961). Phys. Rev. 124, 1468. Seaton, M. J. (1953). Philos. Trans. R . Soc. London, Ser. A 245,469. Seaton, M. J. (1975). Adv. A t . Mol. Phys. 11,83. Shenstone, A. G., and Russell, H. N. (1932). Phys. Rev. 39,415. Siegert, A. J. F. (1939). Phys. Rev. 56, 750. Simpson, J. A. (1964). Rev. Sci. Instrum. 35, 1698. Simpson, J. A., and Fano, U. (1963). Phys. Rev. Lett. 11, 158. Sinfailam, A. L., and Nesbet, R. K. (1972). Phys. Rev. A 6 , 2118. Smith, A. J., Hicks, P. J., Read, F. H., Cvejanovic, S., King, G. C . M., Comer, J., and Sharp, J. M . (1974). J . Phys. B 7 , L496. Smith. K. (1966). Rep. Prog. Phys. 29, 373. Smith, K. (1971). “The Calculation of Atomic Collision Processes.” Wiley (Interscience), New York. Smith, K., McEachran, R. P., and Frasen, P. A. (1962). Phys. Rev. 125, 553. Smith, K., Golden, D. E., Ormonde, S., Torres, B., and Davies, A. R. (1973). Phys. Rev. A 8, 3001. Snow, R. P. (1975). Astrophys. J. 198, 361. Sobolev, N. N., and Sokovikov, V. V. (1966). JETP Lett. 4, 303. Spence, D. (1975a). J. Phys. B 8 , L42. Spence, D. (1975b). Phys. Rev. A 11, 1539. Spence, D., Mauer, J. L., and Schulz, G. J. (1972). J . Chem. Phys. 57, 5516. Stevenson, D. P. (1960). J. Am. Chem. SOC.82, 5961. Stolarski, R. S., Dulock, V. A. L., Watson, C. E., and Green, A. E. S . (1967). J. Geophys. Res. 1 2 , 3953. Sutcliffe, V. C., Haddad, G. N., Steph, N. C . , and Golden, D. E . (1978). Phys. Rev. A 17, 100. Swift, L. D. (1965). Br. J . Appl. Phys. 16, 837.

ELECTRON ATOM AND MOLECULE SCATTERING

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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 = <$\Iff0

-

E,J$\)

+ 2<$:Iff'

-

q$o>

(25)

choosing a trial function $\ that permits electrons i and j to correlate through xij while all other electrons occupy Hartree-Fock orbitals: $:(rl>

r 2 , r 3 , r4)

= (12)- l''s[%lsl

l ~ l ( ~ r12 ), $ 2 s ? ( r 3 ) ~ 2 s 1 ( r 4 )

+ X l s t 2 ~ t ( ~ y13, ) 4 1 s l ( r 2 ) 4 2 s 1 ( r 4 ) + X I S t 2 ~ 1 ( ~ r14>) 4 1 s 1 ( r 2 ) 4 2 s T ( r 3 )

+ X l s 1 2 s t ( 7 2 ,r3)d)lst(rl)42s i(r4) + X l s 1 2 s l ( ~ * 2r4)41st(r1)42st(r3) > + X2sT zsl(V3 > r 4 M l s t ( ~ l ) 4 1 s l ( ~ 2 ) 1

(26)

The pair functions Xij(rl, r2) are totally antisymmetric, with the operator S effecting a symmetric permutation of the one-electron coordinates. Accordingly, ) l i j does not contain any component of 4 k ( k # i,j). Hence <Xij(r, r')I+k(r)>

= 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<$O($1+))4?(ri) = 0 -

(67)

in which V t k are terms coupling the positive and negative frequency components 4!*(ri) of the response. In solving the TDHF equations, naturally one can choose either a variational or a numerical approach. For the first choice, the perturbed orbitals are expanded in a suitable set of basis functions Om, such as linear combinations of atomic orbitals, that are eigenfunctions of a hydrogenic Hamiltonian or are Hartree-Fock functions. [In the latter case the method is formally equivalent to the random phase approximation (Jamieson, 1971, 1973).] On substitution of the expansion a)

#!+(ri)

=

C

Krniiern

(68)

m=n+l

in the TDHF equations, a non-Hermitian matrix is obtained, whose eigenvalues wio (representing transition energies of the system) and eigenvectors $!* can be determined by standard techniques. The variational approach in a variety of forms has been applied to a range of atoms, and molecules (see Stewart et al., 1975; Yeager et al., 1975; Yeager and McKoy, 1975, and references therein). Unfortunately such techniques have a number of deficiencies, which can give rise to difficulties in certain cases: (a) As the number of electrons increases it becomes more awkward to diagonalize the non-Hermitian matrix unless approximations are made to eliminate some of the coupling terms, and thereby reduce the matrix size. In many cases this simplification can be performed in a straightforward way and excellent results obtained [for example, Lin (1974) on photoionization in the rare gases]. For systems where the coupling of several perturbed shells is important, it is more difficult to devise suitable approximations to the TDHF equation without introducing inconsistencies into the calculation. (b) Unless considerable care is taken in the selection of the basis set, variational convergence, especially for highly excited states, can be unsatis-

111

CALCULATION OF ATOMIC PROPERTIES

TABLE IX EXCITATION ENERGIES FOR BERYLLIUM, CALCULATED BY VARIATIONAL AND NUMERICAL SOLUTION OF THE TDHF EQUATION Be

Variational"

Variationalb

Numerical'

Observedd

2s2 ISo -+ 2s2p IP, 2s3p 'Po 2s4p 'Po 2s5p IP, 2s3d 'Do 2s4d ID,

0.17634 0.24706 0.27495 0.28767 0.25431 0.27828

0.1764 0.2490 0.3211 0.4881 0.2548 0.3003

0.17635 0.24707 0.27496 0.28775 0.25432 0.27829

0.1940 0.2742 0.3063

Stewart et al. (1975). Moore (1949).

Arrighini et a/.(1973).

-

0.2936 0.3134

Stewart (1975b).

factory. This is noticeable, for example, in the poor results obtained by Arrighini et al. (1973) for the beryllium isoelectronic sequence in Table IX. (c) If continunm functions are used in expansion Eq. (68), then in the determination of the matrix equations, integrals are encountered that are difficult to evaluate to high accuracy (Altick and Glassgold, 1963). Alternatively, if purely square-integrable 8, are employed, a series of poles rather than a continuous oscillator strength distribution is obtained for frequencies above the ionization potential. Clearly this is erroneous, though a promising development has appeared in the techniques by which continuum data can be extracted from calculations with Lzfunctions (Heller et al., 1973 ;Langhoff and Corcoran, 1974; Doyle et al., 1975). In this regard it may be observed that the observance of sum rules in the TDHF method may render it peculiarly suited to the application of these procedures (Stewart et al., 1974). For the above reasons it is attractive to attempt to solve the TDHF equations directly by numerical methods. B. NUMERICAL SOLUTION OF THE TDHF EQUATIONS

By expansion of the functions && in a central field form, the TDHF equations for atoms can be reduced to a set of coupled equations solely in the radial variable r, an example for neon being given by Stewart (197%). To obtain a numerical solution to these radial equations, the method selected must be capable of overcoming two problems. First, the method must be sufficiently economical that high accuracy can be obtained even for large sets of equations. Since the number of radial equations increases approximately as n, the number of electrons in the system, this is important unless attention is to be confined to small atoms or the atom has some special

Brian C . Webster et a/.

112

property such as a single pair of valence electrons, which renders a simplification of the equations. Second, the technique must be applicable near the poles wio in the vicinity of which the equations tend to become unstable owing to the dominance of the coupling terms V!+. Provided that the method can surmount these difficulties, the dynamic polarizability can be evaluated for frequencies below the ionization threshold from the relation n

'(0) =

1 <+!+ + 4!-lVil4?>

i= 1

where 0,is the excitation frequency, and f, the oscillator strength, the first term referring to bound-bound transitions, the second to bound-free transitions. Finally, in order for a numerical procedure to be applicable at frequencies above the ionization threshold, it is necessary for the numerical procedure to allow solution to be accomplished for the boundary conditions P i k ( r )+ 0, r + co,and kr

1

+ ( Z - n + 1)k-I In(2kr) - 2 + pi + 6, 171

-

,

r --+ x

(70)

where p, and 6, are the coulomb and noncoulomb phase shifts, respectively. Three numerical schemes that have been employed for solution of the T D H F equations are (a) the Numerov method by Jamieson (1970, 1971) in a study of bound and continuous spectral properties of two-electron atoms, (b) the piecewise continuous method by Alexander and Gordon (1972) with the use of the Aitken 6' procedure to obtain convergence close to the poles, (c) a simple three-point finite difference scheme with the use of the Aitken d2 process (or higher-order transformations) to secure convergence and Richardson extrapolation of the appropriate matrix elements to calculate properties (Stewart and Webster, 1974; Stewart, 1975b,d). It is difficult to assess the relative merits of these three numerical approaches in relation to a variational solution since only the helium atom has been treated by all of the methods. From the calculations listed in Table X for the ls2 'So -+ ls'np' 'Po transition energies and oscillator strengths for helium, satisfactory results are obtained by all of the approaches, any minor differences being attributable to the choice of zero-order functions. In Table IX for beryllium the numerical finite difference method furnished results in excellent accord with the variational solution of Stewart et al. (1975), although it required the calculation to be performed over a range

TABLE X EXCITATION ENERGIES A N D OSCILLATOR STRENGTHS (IN PARENTHESES) FOR (1 s2)'S,,--t(lsnp)n1P, TRANSITIONS OF THE HELIUM ATOMCALCULATED BY VARIATIONAL AND NUMERICAL SOLUTION OF THE TDHF EQUATION 2'P, Stewart and Watson (1975) Alexander and Gordon (1972) Jamieson (1971) Expt. Moore (1949), Wiese et a/. (1966)

3'P,

41P,

5'P,

6lP,

0.79697(0.2533)

0.86360(0.06955)

0.88721 (0.02879)

0.89822(0.01467)

0.90425 (0,0085)

0.79697 (0.2520) 0.79697 (0.2526)

0.86361 (0.07024) 0.86360(0.06973)

0.88722 (0.02898) 0.88721 (0.02898)

0.89822 (0.01478) 0.89821 (0.01478)

0.90422 (0,0085)

0.77988(0.2762)

0.84858 (0.0734)

0.87265 (0.0302)

0.8842 (0.0151)

0.8903 (0,0086)

-

Brian C. Webster et al.

114

of frequencies. For heavier atoms this need becomes less important, and the drawbacks of the variational method noted in Section IV,A become dominant. Among numerical procedures, the noniterative Numerov scheme could be difficult to apply to many-electron atoms due to the proliferation of equations, though if recast in an iterative form it could be competitive with the piecewise continuous method. Up to now, most numerical TDHF calculations have been performed using finite difference techniques, which though simple are most flexible. We now consider some of these applications. C. APPLICATIONS TO 3-, 4-, lo-, 12-, AND I~-ELECTRON ATOMS

Since the two-electron sequence has been discussed briefly, and is reviewed by Jamieson (1973) we turn directly to three- and four-electron atoms, for which the four coupled equations have been solved at frequencies both above and below the first ionization threshold by Stewart (1975b). [For the lithium sequence a restricted Hartree-Fock approach, due to Moitra and Mukherjee (1972), was adopted.] In Table XI we present the excitation energies, and in parentheses the corresponding oscillator strengths for the TABLE XI EXCITATION ENERGIES AND OSCILLATOR STRENGTHS (IN FOR THE 2s 'S, --t np 'P, TRANSITION I N THE PARENTHESES) THREE-ELECTRON SEQUENCE OBTAINED BY NUMERICAL SOLUTION OF THE TDHF EQUATION Numerical"

Experimental

0.0671 (0.7575) 0.1394(0.00406) 0.1645 (0.00384) 0.1760 (0.00237) 0.1823 (0.00143)

0.0679'(0.753)' 0.1409 (0.00552) 0.1662 (0.00480) - (0.00316)d - ( 0.00192)d

0.1452 (0.507) 0.4375 (0.0841) 0.5384 (0.0304)

0.1455 (0.505) 0.4397 (0.0804) 0.5411

0.2204 (0.369) 0.8704 (0.145) 1.1029 (0.0487)

0.2205 (0.366) 0.8792 (0.151) 1.1065

Li

2s 2so --t 2p 2Pn 3~ 'Pn 4~ 'Pn 5P 'Po 6~ 'Pn Be' 2s 2sn+ 2p 'P, 3P zPo 4~ 'Pn B'+ 2s zs, + 2p 'Po 3p2Pn 41, 'Pn a

Stewart (1975b). Moore (1949). Filippov (1931, 1932).

~

Wiese ef

trl.

(1966).

CALCULATION OF ATOMIC PROPERTIES

115

2s ' S o + np 'Po, transition of the lithium series. The accord with observation, and with other theoretical treatments, not noted in Table VI (Caves and Dalgarno, 1972; Sims et al., 1976) is very good since core polarization by a single optical electron is represented satisfactorily in TDHF theory (Jamieson, 1973, 1975a; Bersuker, 1960). In contrast, the excitation energies for the beryllium series given previously in Table IX are in poor agreement with experiment, as could be expected since no allowance is made for configuration interaction in the unperturbed function. To achieve greater accuracy for these and similar systems it is evidently necessary to use an extension of the TDHF method, such as proposed by Schneider et al. (1970). However, the theory does allow partly for the effects of configuration interaction upon ground- to excited-state oscillator strengths, the numerical results shown in Fig. 2 being in good agreement with the most refined computations of Victor and Laughlin (1973), Nicolaides et al. (1973), and Sims and Whitten (1973). In particular, for the 2s' 'So + 2s2p 'Po the numerical calculations yield an oscillator strength of 1.378 compared with the value of 1.372 obtained by Victor and Laughlin, and the experimental estimate of 1.38 _+ 0.12 due to Hontzeas et al. (1971). As can be observed in Fig. 2 the TDHF oscillator strengths display a regular variation with Z-' and show the asymptotic behavior discussed by Smith and Wiese (1971).

0.05

0.10 Z

-'

0.15

0.20

FIG.2 . The variation of the first three principal series oscillator strengths for the beryllium sequence, curves A, B, and C, respectively, as a function of Z - '.The solid line shows the required asymptotic behavior as Z + to for the 2s' ' S o + 2s2p 'Po oscillator strengths.

Brian C . Webster et al.

116

For these systems the TDHF equations have also been solved for continuum states below the second threshold. Phase shifts and photoionization cross sections for low ejected-electron energies are shown in Figs. 3 and 4. The results for lithium, as for the bound-state spectrum, indicate that core polarization is included correctly in the method, with excellent agreement being obtained with the model potential study of Caves and Dalgarno. For beryllium the photoionization cross section in Fig. 4 does not match experiment (Mehlmann-Balloffet and Esteva, 1969), due to the failure of the approach to represent autoionizing resonances of the type 2pns 'Po and 2pnd 'Po. At higher energies, as predicted by Jamieson (1971), resonances corresponding to inner-shell excitation appear since the V!, term permits coupling between 4is+ and the continuum orbital 4;s-. This is evident, for example, in the behavior of the phase shift in the vicinity of the first resonance (ls22s2'So -+ ls2s22p 'Po) to be seen in Fig. 5. The positions of such inner-shell excitation resonances have been confirmed by Stewart et al. (1975) in a variational study on four-electron atomic and molecular systems. Naturally, for the neon isoelectronic sequence computations are more difficult, with a set of ten coupled equations to be solved, and matrix methods become unsuitable unless simplifications are made (Amus'ya et al., 1971).

I

I

I

0.2

0.L

I

0.6

K2

FIG.3 . A comparison of the TDHF photoionization cross section for lithium (-,-) with the study by Caves and Dalgarno (---). The P-wave phase shift for elastic Li' . . e scattering is shown also by the solid line ( a . ~ . ) .

CALCULATION OF ATOMIC PROPERTIES

FIG.4. 1a.u.).

117

-) for beryllium

-01

I

O

r a

0.62

I I I

L.02

I ,

I

L.06

L.0L

I

L.08

KZ

FIG.5. The TDHF P-wave phase shift for Be' . . . e scattering in the region of the first resonance. corresponding to inner-shell excitation (a.u.).

Brian C. Webster et al.

118

TABLE XI1 COMPARISON WITH EXPERIMENTAL VALUESOF EXCITATION ENERGIES AND EFFECTIVE QUANTUM NUMBERS (IN PARENTHESES) FROM THE TDHF EQUATION, CALCULATED NUMERICALLY FOR THE NEONISOELECTRONIC SEQUENCE 2p6 'So

-+

2p53s 'Po

2p6 'So

-+

2p6 'So

2p53d 'Po

+ 2p54s

'Po

Calc. 0.6739 (1.683) Obs. 0.6190 (1.699)" 0.6744b 1.2782 (1.965) 1.2243 (1.973)

0.7944 (2.985) 0.7363 (2.988) 0.7947 1.5723(2.988) 1.5112(2.970)

0.7818 (2.698) 0.7268 (2.761) 0.7820 __ 1.5174(3.011)

Mg2+

2.0191 (2.136) 1.9659 (2.144)

2.4987 (2.981) 2.4336 (2.968)

2.5525 (3.153) 2.4997(3.181)

~

2.8990 (2.256) 2.8459 (2.263)

3.5705 (2.981) 3.51 15(2.986)

3.1242 (3.273) 3.6728 (3.297)

3.9180 (2.345) 3.8651 (2.351)

4.7833 (2.979) 4.7236 (2.984)

5.0818 (3.356) 5.0362(3.384)

7.8096 (2.515) 7.7639 (2.521)

9.2585 (2.976) 9.2050(2.983)

10.2958(3.525) 10.269 (3.555)

Ne

Na'

1

Si4

+

+

c17

+

3

Moore (1949).

Matsubara et a/. (1970).

Nevertheless, the finite difference approach still can be applied, and the bound-state properties of the neon sequence Ne-C1' have been investigated by Stewart (197%). Values for excitation energies and oscillator strengths are displayed in Tables XI1 and XI11 respectively. From the first of these tables it is apparent that as is usual with the T D H F method, accord of the excitation energies with experiment is only moderate. Yet interestingly, the effective quantum numbers, noted in parentheses, are in good agreement with their empirically determined counterparts. Turning to the oscillator strengths the approach again appears to provide accurate values, although the experimental data are rather sparse for comparison. The 2p6 'So + 2p5 3s 'Po excitation of neon has been examined mostly, and in this case the T D H F result of 0.159 is in harmony with the observed values of 0.16 f 0.014 (Korolev et a/., 1964) and 0.168 k 0.2 Lewis (1967). For heavier atoms it is useful to seek a simplification of the T D H F method, in particular it is not unreasonable to apply the method solely to the valence shell, the effect of the core being represented by a pseudopotential or some similar device (Stewart, 1974). The caveat should be attached that this type of approximation should not be applied at frequencies that are at all close to core excitation energies, as important synergic effects would be neglected +

119

CALCULATION OF ATOMIC PROPERTIES

TABLE XI11

COMPARISON BETWEEN CALCULATIONS OF THE OSCILLATOR STRENGTH FOR TRANSITIONS I N THE NEONISOELECTRONICSEQUENCE 2p6 'So Ne Na+ Mgz+ ~

1

+

3

Si4+ c 1 7+

+ 2p53s

0.159 0.213 0.231 0.238 0.240 0.236

'Po

0.161" 0.19' 0.21 0.20 0.19' 0.15

2p6 'So

-+

2p53d 'Po

0.024 0.20 0.468 0.787 1.103 1.866

0.016d 0.039' 0.16 0.48 0.84' 1.62

' Kastner et a/. (1967).

Aymar et al. (1970).

2p6 'So -+ 2p54s 'Po 0.026

0.013"

0.044

0.043 0.042 0.048

0.0201' 0.0167

' Crance (1973)

' Lawrence and Liszt (1969).

(Watson et al., 1976; Wendin, 1973).As an alternative to the use of a pseudopotential, the T D H F equations for the valence shells can be solved in the static-exchange field of the core with the perturbed orbitals being constrained to remain orthogonal to occupied core orbitals (Stewart, 1975d). In the case of magnesium, for example, only two rather than 12 coupled equations then have to be solved. Table XIV lists excitation energies and effective quantum numbers calculated in this manner for the magnesium isoelectronic sequence. The variation of the oscillator strengths for the transitions 3s' 'So -+3s3p 'Po and 3s' 'So + 3s4p 'Po with reciprocal atomic number Z - ' is shown in

TABLE XIV COMPARISON WITH EXPERIMENTAL VALUES OF EXCITATION ENERGIES AND EFFECTIVE QUANTUM NUMBERS (IN PARENTHESES) CALCULATED FROM A SIMPLIFIED TDHF EQUATION, FOR THE MAGNESIUM ISOELECTRONIC SEQUENCE 3s2 IS,

Mg Al' Si2 P3+

+

s4

+

c15

+

Ar6+ Fe14+ a

-

3s3p 'Po

3s2 'So

+ 3s4p

'Po

Calculated

0bserved"

Calculated

Observed"

0.1492 (2.194) 0.2653 (2.272) 0.3716 (2.356) 0.4734 (2.426) 0.5725 (2.482) 0.6697 (2.528) 0.7658 (2.567) 1.5111 (2.737)

0.1597 (2.031) 0.2726 (2.184) 0.3776 (2.298) 0.4792 (2.384) 0.5794 (2.448) 0.6786 (2.502) 0.7777 (2.546) 1.604 (2.72)'

0.2021 (3.133) 0.4599 (3.221) 0.7748 (3.324) 1.1427 (3.404) 1.5606 (3.465) 2.0260(3.511) 2.5457 (3.555) 8.4718 (3.737)

0.2248 (2.985) 0.4870 (3.125) 0.8040(3.251) 1.1731(3.347)

Moore (1949).

* Cowan and Widing (1973).

.-

~

8.616 (3.71)b

120

Brian C . Webster et al.

0

O O O

0.0

0

1

0.02

0.OL 2-1

0.06

_ I

0.08

FIG.6 . The variation of the TDHF oscillator strength with Z - ' for the magnesium sequence, for the transition 3s2 'So + 3s3p 'P, ( x ) and for the transition 3s' 'So --t 3s4p 'Po (0).

Fig. 6. A striking feature to be seen is the maxima and minima in f l and f 2 , respectively, at Al+, this behavior being corroborated independently by a refined CI study (Victor et al., 1976). For magnesium there is an indication of a deterioration in accuracy with f = 1.665 for the 3s2 'So + 3s3p 'Po transition, compared with experimental values of 1.76 & 0.09 (Lundin et al., 1973) and 1.86 & 0.3 (Smith and Liszt, 1971). This is due perhaps to the absence of any allowance for dd or pd configuration interaction in the ground or excited states. However, the TDHF value of 0.783 for the astrophysically important resonance transition in Fel4+ is in excellent agreement with values given by Froese (1964) and by Crossley and Dalgarno (1965) in studies that include extensive configuration interaction. The 18-electron sequence from C1- to Ti4+ has also been studied within the same approximation. In Fig. 7 the dynamic polarizability calculated for argon in the region of normal dispersion is compared with data derived from refractive index measurements (Chashchina et al., 1968). Accord with experiment is good, but as for neon, the TDHF method overestimates the position of the first pole at 0.4475 a.u., compared to 0.434 a.u. from observation. However, the oscillator strength for this transitibn is within the theoretical bounds of Aymar et al. (1970), 0.25 d f < 0.29, and compares with empirical estimates of 0.278 (Lewis, 1967) and 0.275 (Stacey and Vaughan, 1964). Further discussion of the properties of atoms in the magnesium and argon series is given by Stewart (1975d).

CALCULATION OF ATOMIC PROPERTIES

0.1

0.2

0.3

121

0.L

w(au)

FIG.7. The TDHF dynamic polarizability ct(o)of argon in the normal dispersion region (---) compared with the dynamic polarizability derived from refractive index measurements. (-.-) (a.u.).

V. Conclusion In the previous sections we have reviewed some applications of numerical procedures to obtain accurate values for atomic properties. We could have cited applications in more diverse areas, as one-center expansions for molecules (Stewart, 1973a, 1973c),the calculation of excitation cross sections by the impact parameter method (Jamieson, 1975b; for a review, see Briggs, 1976), or a simple solution of continuum and semicontinuum models for excess electrons in liquids (Carmichael and Webster, 1974; Carmichael, 1974; Webster and Carmichael, 1978). Yet even greater possibilities lie with time-dependent theories. Recently, a number of extensions of the TDHF method have been proposed to handle the general open-shell system (Armstrong, 1974; Dalgaard, 1975; Stewart, 1975e).This development probably offers the most promising line of advance in the subject, particularly for the study of photoionization. Even for closedshell atoms such as barium, complicated effects are apparent in dflde at high energies, and though there is broad agreement with experiment (Ederer et a/., 1975) substantial discrepancies exist between different RPA computations. Possibly only a direct numerical solution of the equations will be sufficiently accurate for successful applications of the T D H F approximation to complex systems (Wendin, 1973; Fliflet et al., 1974).

122

Brian C . Webster et al. ACKNOWLEDGMENTS

We should like to express our gratitude to Mrs. Jessie Rodgers for assistance with the figures, and to Mrs. Gina Christman and Mrs. Jeanne Boocock for their patience and skill in typing the manuscript. One of us (R.F.S.) would like to thank the Commonwealth Fund for the award of a Harkness Fellowship.

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Lahiri, J., and Mukherji, A. (1966a). J. Phys. SOC.Jpn. 21, 1178. Lahiri, J., and Mukherji, A. (1966b). Phys. Rev. 141, 428. Langhoff, P. W., and Corcoran, C. T. (1974). J . Chem. Phys. 61, 146. Langhoff, P. W., and Karplus, M. (1970). J. Chem. Phys. 53,233. Langhoff, P. W., Karplus, M., and Hurst, R. P. (1966). J. Chem. Phys. 44,505. Laughlin, C . , and Victor, G . A. (1973). At. Phys. 3,247. Lawrence, G . M., and Liszt, H. S. (1969). Phys. Rev. 178, 122. Lennard-Jones, J. E. (1931). Proc. Cambridge Philos. SOC.27, 469. Lewis, E. L. (1967). Proc. Phys. Soc., London 92, 817. Lin, C. D. (1974). Phys. Rev. A 9, 171. Linderberg, J., and Shull, H. (1960). J . Mol. Spectrosc. 5, 1. Luke, P. J., Meyerott, R. E., and Clendenin, W. W. (1952). Phys. Rev.85,401. Lundin, L., Engman, B., Hilke, J., and Martinson, I. (1973). Phys. Scr. 8, 274. McKoy, V., and Winter, N. W. (1968). J. Chem. Phys. 48, 5514. Mehlman-Balloffet, G., and Esteva, J. M. (1969). Astrophys. J . 157, 945. Moitra, R. K., and Mukherjee, D. K. (1972). Int. J. Quantum Chem. 6, 211. Moore, C. E. (1949). “Atomic Energy Levels.’’ Natl. Bur. Stand, Washington. D.C. Nesbet, R. K. (1958). Phys. Rec., 109, 1632. Nesbet, R. K. (1967). Phys. Rev. 155, 51. Nesbet, R. K. (1971). Int. J. Quantum Chem., Symp. 4, 117. Nicolaides, C. A., Beck, D. R., and Sinanoglu, 0. (1973). J . Phys. B 6, 62. Pekeris, C. L. (1962). Phys. Rev. 127, 509. Renken, J. H. (1971). Phys. Rev. A 3, 510. Richardson, L., and Gaunt, J. (1927). Philos. Trans. R. SOC.London, Ser. A 226, 299. Roothan, C. C. J. (1951). Rev. Mod. Phys. 23,69. Roothan, C. C. J., and Bagus, P. (1963). “Methods in Computational Physics,” Vol. 2. Academic Press, New York. Sabelli, N., and Hinze, J. (1969). J. Chem. Phys. SO, 684. Schaefer, H. F. (1972). “The Electronic Structure of Atoms and Molecules: A Survey of Rigourous Quantum Mechanical Results.” Addision-Wesley, Reading, Massachusetts. Schneider, B., Taylor, H. S., and Yaris, R. (1970). Phys. Rev.A 1, 855. Schulman, J. M., and Musher, J. I. (1968). J . Chem. Phys. 49,4845. Schulman, J. M. and Tobin, F. L. (1970). J. Chem. Phys. 53, 3662. Schwartz, C . (1962). Phys. Rev. 126, 1015. Shull, H., and Lowdin, P.-0. (1959). J . Chem. Phys. 30. 617. Silverman, J . N., and Bridgman, G . H. (1967). Rev. Mod. Phys. 39, 228. Sims, J. S., and Hagstrom, S. A. (1971). Phys. Rer. A 4, 908. Sims, J. S., and Whitten, R. C. (1973). Phys. Rev. A 8, 2220. Sims, J. S., Hagstrom, S. A., and Rumble, J. R. (1976). Phys. Rev.A 13, 242. Sinanoglu, 0. (1961). Proc. R . SOC.London, Ser. A 250, 379. Sinanoglu, 0. (1962). J . Chem. Phys. 86, 706 and 3198. Sinanoglu, 0. (1969). Adv. Chem. Phys. 14, 237. Singh, T. R. (1971). Chem. Phys. Lett. 11, 598. Slater, J. C. (1929). Phys. Rev. 34, 1293. Smith, K., Henry, R. J. W., and Burke, P. G . (1966). Phys. Rev. 147, 21. Smith, M. W., and Wiese, W. L. (1971). Astrophys. J . , Suppl. Ser. 23, 103. Smith, W. H., and Liszt, H. S. (1971). J . Opt. SOC.Am. 61, 938. Southwell, R. V. (1940). ”Relaxation Methods in Engineering Science.” Oxford Univ. Press, London and New York. Stacey, D. N., and Vaughan, J. M. (1964). Phys. Lett. 11, 105.

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Starkschall, G., and Gordon, R. G. (1971). J . Chem. Phys. 54, 663. Starkschall, G., and Gordon, R. G. (1972). J. Chem. Phys. 57, 3213. Sternheimer, R. M. (1957). Phys. Rev. 107, 1565. Sternheimer, R. M. (1970). Phys. Rev. A 1, 321. Stewart, R. F. (1973a). Ph.D. Thesis, University of Glasgow. Stewart, R. F. (197313). J . Phys. B6, 2213. Stewart, R. F. (1973~).Mol. Phys. 25, 1451. Stewart, R. F. (1974). J . Chem. Soc., Faradoy Trans. 2 70, 85. Stewart, R. F. (1975a). Mol. Phys. 29, 787. Stewart, R. F. (1975b). J . Phys. B 8 , 1. Stewart, R. F. (1975~).Mol. Phys. 29, 1577. Stewart, R. F. (1975d). Mol. Phys. 30, 745. Stewart, R. F. (1975e). Mol. Phys. 30, 1283. Stewart, R. F., and Watson. D. K . (1975). Unpublished work. Stewart, R. F., and Webster, B. C. (1973a). Chem. Phys. Lett. 19,462. Stewart, R. F., and Webster, B. C. (1973b). J . Chem. Soc., Foraday Trans. 2 69, 1685. Stewart, R. F., and Webster, B. C. (1974). J. Chem. Soc., Faraday Trans. 2 70, 524. Stewart, R. F., Laughlin, C., and Victor, G . A. (1974). Chem. Phys. Lett. 29, 353. Stewart, R. F., Watson, D. K., and Dalgarno, A. (1975). J . Chem. Phys. 63, 3222. Stwalley, W. C. (1970). J. Chem. Phys. 54,4517. . 1147. Szasz, L. (1962). PI1.v.s. Rri. 126, 169. and J . Moth. P h j ~ 3, Teachout, R. R., and Pack, R. T. (1971). At. Data 3, 195. Tuan, D. F.-T., Epstein, S. T., and Hirschfelder, J. 0. (1966). J. Chem. Phys. 44,431. Victor, G. A,, and Laughlin, C. (1973). Nucl. Instrum. Methods 110, 189. Victor, G. A., Stewart, R. F., and Laughlin, C. (1976). Astrophys. J., Suppl. Ser. 31, 237. Waller, I., and Hartree, D. R. (1929). Proc. R . Soc. London, Ser. A 124, 119. Watson, D. K., Stewart, R. F., and Dalgarno, A. (1976). J. Chem. Phys. 64,4995. Webster, B. C., and Carmichael, I. (1978). J . Chem. Phys. 68, 4086. Webster, B. C., and Stewart, R. F. (1972). Theor. Chim. Acta 27, 355. Weiss, A. W. (1961). Phys. Rev. 122, 1826. Wendin, G. (1971). J . Phys. B4, 1080. Wendin, G. (1973). J . Phys. B6,42. White, R. J. (1967). Phys. Rev. 154. 116. Wiese, W. L., Smith, M. W., and Glennon, B. M. (1966). “Atomic Transition Probabilities,” NSRDS-NBS4. Natl. Bur. Stand., Washington, D.C. Wilkinson, J. H. (1965). “The Algebraic Eigenvalue Problem.” Oxford Univ. Press, London and New York. Winter, N. W. (1970). Ph.D. Thesis, California Institute of Technology, Pasadena. Winter, N. W., and McKoy, V. (1970). Phys. Rev. A 2, 2219. Winter, N. W., Laferriere, A,, and McKoy, V. (1970a). Phys. Rev. A 2,49. Winter, N. W., McKoy, V., and Laferriere, A. (1970b). Chem. Phys. Lett. 6, 175. Yeager, D. L., and McKoy, V. (1975). J . Chem. Phys. 63,4861. Yeager, D. L., Nascimento, N. A. C., and McKoy, V. (1975). Phys. Rev. A 11, 1168.

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ADVANCES I N ATOMIC AND MOLECULAR PHYSICS, VOL.

14

(e, 2e) COLLISIONS

ERICH WEIGOLD and IAN E. McCARTHY Institute for Atomic Studies School of Physical Sciences The Flinders University of South Australia Bedford Park, Australia

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Experimental Methods ................................... 111. Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Derivation of the Scattering Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. The Distorted Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Discussion of Reaction Mechanism Approximations . . . . . . . . . . . . . . . . . D. The Atomic and Molecular Structure Problem . . . . .......... IV. Reaction Mechanism at Intermediate to High Energies ......... A. Noncoplanar Symmetric Kinematics . . . . . . . . . . . . B. Coplanar Symmetric Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. The (e, 2e) Cross Section for Atomic Hydrogen . . . . . . . . . . . D. Absolute Cross Sections for Helium ..................... V. Structure of Atoms and Molecules ............................... A. Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........ B. Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Ground-State s ....................................... VI. Conclusions . . . . . ........................................ References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127 130 139 139 142 147 147 151 158

164 164 167 173 176

177

I. Introduction The study of (e, 2e) collisions has expanded rapidly since the first coincidence measurements of the two emitted electrons in the electron impact ionization of helium by Ehrhardt et al. (1969).This and subsequent work by the group at Kaiserslautern (e.g., Ehrhardt et al., 1971, 1974; Jung et a/., 127 Copyright @ 1978 by Academic Press. Inc All rights of reproduction in any form reserved ISBN 0-12-003814-5

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1975) employed asymmetric kinematics at low energies, with the prime objective of investigating the reaction mechanism at low energies. Camilloni et al. (1972) were the first to use the (e,2e) reaction at high incident and outgoing energies as a tool for investigating the momentum distribution of the target (ejected) electrons. They used thin-film targets and coplanar symmetric geometry, in which the outgoing electrons have equal energies and make equal and opposite angles with respect to the incident direction. They observed the momentum distribution of the 1s electrons of carbon. The first detailed information on valence electrons was reported by Weigold et al. (1973). They used the noncoplanar symmetric geometry (in which the azimuthal angle 4 is the variable) to study the 3s and 3p orbitals of argon They found strong electron-electron correlation effects in the 3s orbital, and demonstrated the power of the technique in determining structure information. In the (e, 2e) reaction the momenta of both outgoing electrons are measured and particular electronic quantum states f of the residual ion can be selected by resolving their separation energies cf = E , - E , - E,, ignoring the ion recoil energy. The subscripts 0, A, B denote the ingoing and two outgoing electrons, respectively. For each state a profile of ion recoil momenta q may be determined by counting electron coincidences for chosen arrangements of outgoing energies and angles that define a set of values of q. When the incident and outgoing energies are sufficiently high, the electrons interact weakly with the residual system, and the recoil momentum can be closely connected to the momentum of the struck electron in the target system. The q profile can then give direct information on the square of the momentum space wavefunction of the struck electron. This is the decisive advantage of (e, 2e) spectroscopy over photoelectron spectroscopy, where q can only be varied by varying the incident photon energies, and q values normally associated with bound valence electrons (0-2 a.u.) can only be studied at energies too low for reliable reaction theory. In this review we shall concentrate on the symmetric (e,2e) reaction at high and intermediate energies, i.e., the regime of (e, 2e) collisions that can yield reliable information on the momentum distribution of the struck electrons for transitions to definite final ion states f . This means that we shall omit from the discussion several important experimental arrangements, in particular the work of Brion, Van der Wiel, and co-workers (Van der Wiel, 1973; Brion, 1975; Van der Wiel and Brion, 1973; Branton and Brion, 1974).Their technique uses asymmetric kinematics at high energies with the main aim of measuring partial oscillator strengths. At high incident energies and small scattering angle of the incident electron, the Born approximation is appropriate. The generalized oscillator strength, which describes the cross section, may be expanded in a power series of K Z

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129

about K = 0, where K = k, - k, is the momentum transfer, the subscripts 0 and A denoting, respectively, the incident and high-energy forward-emitted (scattered) electron (Inokuti, 1971; Kim, 1972). The zeroth-order term in the expansion is given by the optical dipole oscillator strength, which is directly proportional to the cross section for photoabsorption. The momentum transfer K obviously approaches zero for glancing collisions if the incident energy is sufficiently high and if the energy loss E , - E , is small, that is, if the ejected electron energies are low. In that case the generalized oscillator strength is well described by the optical oscillator strength and the (e, 2e) process is equivalent to the (y, e) process, the energy of the “photon” absorbed being E , - E , . A detailed treatment of the scattering geometry by Hamnett et al. (1976) showed the expected quantitative relationship with photoelectron spectroscopy. The most significant result of the analysis was that an angle of ejection of 54.7” corresponded to a magic angle at which the coincidence intensity is independent of the angular anisotropy factor p. A number of “photon simulation” experiments have been carried out in recent years by Brion, Van der Wiel, and co-workers, including “photoionization,” “photoabsorption,” “photofluorescence,” and “photoelectron” spectroscopy. Postcollision interaction effects have been observed in threshold photoelectron spectra (Wight and Van der Wiel, 1977). The distant electron-electron collisions observed in these (e, 2e) experiments at small momentum transfers contrast with the close electron-electron collisions observed in the symmetric arrangements, which seek to maximize the momentum transfers. In the forward-scattering experiments q cannot be unambiguously identified as the momentum of the “target” electron, and no attempts have been made to measure momentum profiles. Since the “photon” energy can be easily and cheaply varied, this approach of obtaining optical quantities is particularly advantageous in the X-ray and UV regions of the spectrum, where the only adequate continuum light source available is synchrotron radiation. Adequate synchrotron sources are, of course, very expensive. The (e, 2e) reaction has also been used to study autoionizing transitions by Weigold et al. (1975a). All previous measurements of autoionizing transitions involved the integration over at least the momenta of the undetected “emitted” or “scattered” electron. The above (e, 2e) experiment, on the other hand, gave information on the resonance and direct processes and their interference as a function of the momenta of the emitted electron for known values of the momentum transfer K = k, - k,. The advantages of using the (e, 2e) collision for investigating autoionizing transitions were first pointed out by Balashov et al. (1972a,b). Their discussion, based on the plane-wave Born approximation, showed that the angular correlation for the separated resonance should be symmetric about the momentum transfer

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Erich Weigold and Ian E . McCarthy

direction. Weigold et al. (1975a) found that the symmetry axis had rotated relative to the direction of the momentum transfer. Cvejanovic and Read (1974) studied the electron impact ionization of helium at total outgoing energies ( E A + EB)below 3 eV with a time-of-flight (tof) coincidence technique. Wilden et al. (1977) have used an electronelectron coincidence tof spectrometer to study the energy spectrum of electrons ejected from a chosen autoionizing state of a molecule. They were able to measure relative intensities of transitions between particular autoionizing levels and the various vibrational levels of the ion. Extension of these studies to the measurement of angular correlations should yield valuable information about the symmetries of the states concerned. It is outside the scope of this chapter to give a detailed review of all of these different types of (e, 2e) experiments. We shall concern ourselves with only those experiments designed to yield detailed information on the electronic structure of atoms and molecules. This is the regime of symmetric kinematics at intermediate and high electron energies. In Section I1 we outline some of the experimental techniques employed in these investigations. The basic theory necessary for describing the reaction mechanism and for extracting all the structure information is developed in Section 111. The theory for atoms is extended to cover the case of molecular targets. The reaction mechanism at intermediate to high energies is investigated in Section IV for atomic targets in both the coplanar and noncoplanar symmetric geometry. Finally, in Section V we outline some of the structure information that has been obtained on atoms and molecules. This includes electron-electron correlation effects in both the target and final ion states.

11. Experimental Methods Measurement of an (e, 2e) cross section involves the determination of the momenta of the incident particle (k,) as well as the momenta of the two outgoing electrons (kA and k,). The kinematic parameters defining the process are given in Fig. 1. A beam of electrons of energy Eo is incident on a target, assumed for simplicity to be at rest. Two electrons emerge from the ionizing collision with energies E A and E , in the directions given by the angular coordinates ( e A , (bA = n) and (O,, (b, = 4). Ignoring for convenience of notation the recoil energy of the ion, the energy and momentum conservation equations are, respectively, E

=E

q

=

+ E, = Eo - ES

A

k,

~

kA - k,

(1)

(e, 2e) COLLISIONS

131

FIG. 1. Schematic diagram of the kinematics of an (e, 2e) event

The sum E of the total outgoing energies is called the total energy, and q is the recoil momentum of the ion. The separation energy (or binding energy) E~ is the energy difference between the initial state of the target and the final ionic state f . A signal will only be observed at separation energies corresponding to ion states excited in the process. Since the two electrons produced in a single (e,2e) collision leave the interaction region simultaneously, fast timing techniques must be used to select the significant events from the many other electrons of momenta kA and k, produced in the interaction region and arriving at the detectors. A schematic diagram showing a typical (e, 2e) spectrometer arrangement is shown in Fig. 2. The electron beam is produced in an electron gun with suitable steering and collimating stages. The beam is usually dumped in a deep Faraday cup (FC) to suppress secondary electrons. The cup usually consists of an inner cup ( - 1 mm diam) and an outer cup ( 5-10 mm diam). The gun is then tuned to maximize the ratio of the current collected by the inner cup to that collected by the outer cup. The electron beam intersects an atomic or molecular beam, which enters the system through a small tube or multichannel array. The gas jet is often further collimated by skimmers located downstream from the multichannel array, although this is not necessary in the noncoplanar arrangement or any other arrangement in which the size of the interaction region does not depend on the angular setting of the moving detector(s). Two electron analyzers of solid angles QA and ClB select electrons of energies E , and En with energy resolution AEA and AEB at the angles OA and (On,4), respectively (see Fig. 1).The analyzers are usually electrostatic deflectors, either cylindrical mirrors (Weigold et ul., 1973) or hemispherical

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Erich Weigold and Ian E. McCarthy

132

XRRENT 4ONITOR SCALER

SCALER 1

LI

SCALER

I I I

SCALER

!

L L A n VAI

'1CONTROL

PRESSURE MONITOR DAC's

1

:

'

DAC 1

TO PUMP

STEPPI NG MOTOR DRIVES

-

-

PULSER PULSER

=+-

--

DISPLAY DRIVER

UNIBUS CRATE CONTROLLER

FIG.2. Schematic diagram of the (e, 2e) spectrometer and fast timing and data processing electronics. Not shown in the figure are the voltage supplies for the channel electron multipliers (CEM),analyzers, and retarding lenses.

deflectors (Ehrhardt et a/., 1969, 1971; Camilloni et a/., 1972). If the asymmetric kinematic arrangement is used and the ejected electron energies are low, they can be determined by tof analysis (Beaty et al., 1977; Wilden et al., 1977). The fast scattered electron gives the zero time signal for the tof analyzer, which usually consists of a retarding lens and a drift tube. The electrostatic deflector analyzers are also usually preceded by a retarding lens (e.g., Wannberg and Skollermo, 1977) in order to improve the energy resolution. The collimator and retarding lens is sometimes followed by an orthogonal set of steering electrodes placed near the entrance slit of the analyzer (McCarthy and Weigold, 1976; Hood et al., 1977). Recently tof spectroscopy has been reviewed by Raith (1976) and a discussion of various electrostatic deflector analyzers has been given by a number of authors (e.g., Hafner et a/., 1968; Risley, 1972; McCarthy and Weigold, 1976). The tof detector has an advantage in the study of autoionizing states since the

(e, 2e) COLLISIONS

133

whole of the resonance structure can be observed on the time-to-amplitude converter (TAC) output in the one measurement. The coincidence spectrometer must obviously be mounted in a vacuum system and isolated from the earth’s magnetic field and other stray magnetic fields. The residual magnetic fields are usually reduced to a satisfactory level (<5-10 mG) by surrounding the spectrometer with magnetic shielding constructed from a properly annealed high-permeability material and by using sets of Helmholtz coils to cancel out the earth’s field. Care must be taken to ensure that the materials used in the construction of the system are nonmagnetic. The electrons traversing the energy analyzers are detected by fast electron multipliers, such as continuous-channel electron multipliers (CEM), channel plates, Johnston electron multipliers, or dynode electron multipliers. (For discussion of the various detectors, see Baumgartner and Huber, 1976.) The pulses from the electron multipliers pass through fast preamplifiers and amplifiers, such as double-delay line amplifiers, before passing through fast timing discriminators. These can operate either in the leading edge, cross-over pick-off, or constant fraction of peak height mode. The constantfraction mode, when suitably adjusted, often leads to the best timing resolution. If CEMs are used as electron detectors it is important to operate them in the saturated mode (Baumgartner and Huber, 1976), to ensure a good pulse height spectrum at the output of the detector. The output from one timing discriminator provides the start pulse for the TAC. In the asymmetric arrangement, the start pulse is generally given by the fast electron. This must be the case if the tof of the slow electron is measured. The (delayed) pulse from the other discriminator provides the stop pulse. The pulse height distribution from the TAC can be monitored and measured by a multichannel analyzer. A typical pulse height distribution is shown in Fig. 3. The uniform background in this spectrum is caused by the arrival of uncorrelated electrons at the detectors. The peak superimposed on this background is the coincidence peak and corresponds to the arrival of a pair of electrons that originated in the same event. In the arrangement used at Flinders University the TAC output is fed to two single-channel analyzers (SCA), one set with its window covering the coincidence peak, typically < sec full width, whereas the other has its window set to record only background events. After subtraction of the background counts from the counts in the coincidence window, the number of counts is a measure of the triple differential cross section at the selected energies and angles. In the system shown in Fig. 2 the count rates are recorded by scalers on-line to a minicomputer. If the count rate in the coincidence window is N , and that in the background window is Nb and r = B/C is the ratio of the window widths, then

Erich Weigold and Ian E. McCurthy

134 500

t

400

VI

c

300

C 3 0

u

200

I00

+

0 20

I

I

I

40

60

80

1

160

I

I

180

200

Time Delay ( n s e c )

FIG.3. A typical pulse height distribution from the TAC output. The coincidence and background SCA windows are C and B, respectively.

the standard deviation in the true signal arrived at after background subtraction is (McCarthy and Weigold, 1976) 0,= ( N ,

+ Nb/r2)'I2

(3)

There is therefore an obvious advantage in using large background windows B while keeping the coincidence window to a minimum width (to reduce NJ. An accurate knowledge of r is essential since the signal count . . . rate is given by N,

=N,

-

N,/Y

(4)

Accurate measurements of r under normal experimental conditions are necessary before and after every run if it cannot be monitored during the run. Drifts in the ratio of window widths can seriously affect the inferred values of N , if the signal-to-background ratio is poor. For instance, if the ratio of signal to true background is 0.1, an error of 1% in r will produce a 1% error in the subtraction of the background signal, which in turn will produce a 10% error in the signal. This problem can be overcome by an arrangement introduced by Avida and Gorni (1967),which can ensure an accurate value of r. This arrangement has been used by Van der Wiel and co-workers (Ikelaar et al., 1971). A variation of it, which allows one to monitor the coincidence peak, is shown schematically in Fig. 4. Two different delays are used in one channel, one

(e, 2e) COLLISIONS

-

0

DELAY l k CHANNEL 2

-

0

135

CO'NC'DENC E UNIT RESOLVING TIME t

-

-

(Ns+Nb)

TA C

DELAY 2

ON-LINE SCALER

of which (delay 2) has the correct value for true coincidences, the other (delay 1) for accidental coincidences only. The output count rate from the coincidence unit is therefore the signal plus its underlying background N , + Nbas well as the purely background count rate N b , the time resolutions (i.e., window widths) being equal since they are processed by the same unit. The outputs from the first coincidence unit as well as those from delay 2 are then fed to the start and stop inputs of a TAC whose timing range must be greater than the time resolution 7 of the coincidence unit. The TAC output will then contain the signal as well as the underlying background coincidences. A minicomputer can determine from the outputs of the two timing units the true signal count rate N , . The standard deviation of N , is given by Eq. (3) with r = 1. Although this statistical error is larger than that obtained if a value of r >> 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

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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(<)]li) = 0

(23)

(24)

Substituting the multichannel expansion (with implied antisymmetry)

in the scattering Eq. (23) yields a set of coupled equations for the scattering functions f ; : (Ei - K

-

Yi)J; =

c yjjj

j#i

where Ei = E

-F,~,

vj

= (i(Vlj)

(27)

We formally eliminate the channels i # 0 from the set (26) by solving the equations for i # 0 and substituting the formal solutions fj in the right-hand side of the equation for i = 0. This yields the optical model equation (E - K

-

P)fO= 0

(28)

where

V

=

V,,

+ 2 j.k 1 0

Voj(E - K - V)slV,o

(29)

(e, 2e) COLLISIONS

143

In the case of atomic scattering we represent the atom by an independentparticle model. Apart from the electron-nucleus potential VN, the potential V is the sum of electron-electron potentials u. The wavefunctions [ i) are Hartree-Fock determinants of orbitals $ P . The optical model potential may be expanded in the multiple-scattering series vfO

= vNfO

+1

+


P(

v,

1


l+P

+ ..

->.

+ exchange term

(30)

where Go = ( E - K ) - ’

(31)

explicitly, this reduces to FfO

==

vN(r)fO(r)

+ c Sd3r’ Jd3rrr$:(r”)~(r

- r”, r’ - r“)tj,(r”)fo(r’)

P

+ 1 Jd’r’ c(

Jd3r”$3r”)T(r - r”,r’ - r”)$w(r’)fo(r’’)

(32)

P

where T is the electron-electron T-matrix calculated at the energy E , , which may include an average excitation energy, the sum p extends over orbitals occupied in the Hartree-Fock ground state, and the factor cx is determined by the Pauli exclusion principle for the particular target system. In view of the smallness of the coupling constant, T is usually evaluated only to second order. Apart from the first-order direct term all orders are nonlocal potentials, which may be approximated by equivalent local spherical potentials. The approximation of Furness and McCarthy (1973) for the first-order exchange term has been shown by Riley and Truhlar (1975) to give excellent phase shifts down to energies of a few tenths of an electron volt. This approximation is J 71E-- L 2 {

E-&-

[ ( E - VC)2+ a y > ,

cx2 = 47Lp

(33)

where p is the electron density, and V, the local central part of the potential,

The high-energy form of (33) is

%f

= 4np/lc2,

K’

= E - V,

(35)

Erich Weigold and Ian E. McCarthy

144

looK---100

z 101

0 + V

w

c

m m 10

m 0 a v

-a+

1

, 10

Z W

a

1.0

k D

0.1

.o 1 I

I

I

I

I

I

I

I

.Ol'

'

'

'

'

'

'

"

'

20 60 100 140 60 100 140 e (DEGREES) 0 (DEGREES) FIG.5. Elastic differential cross sections for electrons on (a) neon and (b) xenon compared with the optical model. From McCarthy et a/. (1977). 20

Similar considerations (B. H. Bransden, I. E. McCarthy, M. R. C. McDowell, and L. A. Morgan, unpublished, 1977) show that the imaginary part of the second-order term has the high-energy form ~m F2 = 2712p/K3

(36)

The energy dependence of this form is confirmed by a semiphenomenological fit to data for inert gases over a wide range of energies (McCarthy et al., 1977).

(e, 2e) COLLISIONS

145

The term describes absorption into unobserved channels. Its magnitude is fixed by requiring it to yield the correct total nonelastic cross section. The real second-order term, which describes polarization, decreases with increasing energy. It has no effect on the cross section at higher energies. At low energy the Temkin-Lamkin adiabatic approximation was used by McCarthy et al. (1977). Typical fits to differential cross-section data (with no free parameters) are shown for neon and xenon atoms over an energy range of a few hundred electron volts in Fig. 5. In this model the imaginary potential includes the density only of the valence particles, rather than the full density of Eq. (36). For elastic scattering it is necessary to have absorption in the model, but cross sections are not very sensitive to its localization in space. This is not necessarily true for the (e, 2e) reaction, which is localized by the bound-state orbital. However, the excellent predictions of differential elastic cross sections make it reasonable to use the wavefunctions as distorted waves in the (e, 2e) amplitude.

2. The Averaged Eikonal Approximation At energies of a few hundred electron volts, the distorted waves x'* ) calculated from the optical-model Eq. (28) may be approximated quite well by modified plane waves in a radially localized region (Furness and McCarthy, 1974).Since radial localization is provided in the (e, 2e) integrand by the bound-state wavefunction, it makes sense to try modified plane waves as a first approximation for computing the (e, 2e) amplitude. Figure 6 shows the surfaces of constant phase (wave fronts) of x(+)for 200-eV electrons on the xenon ion. Inside the potential the effective wave number k(1 p) is somewhat greater than k. An imaginary part iy is added to the wave number to simulate attenuation into unobserved channels. The averaged eikonal approximation is

+

X("(k,r)

= exp(-ykR)exp[i(l

+p

-

iy)k r

(37)

The function is normalized to unit amplitude at a distance R before the beam enters the scattering region. It will be seen in Section IV that (e,2e) angular correlations are quite well described by this approximation, provided that the parameters p and y are appropriately chosen. By solving the Schrodinger equation for a wave in a constant medium with potential P + i@, it is seen that

p = i7/2E,

y = W/2E

(38)

The effective potentials V and W are rather sensitive to the radial region concerned in a particular (e, 2e) reaction. They cannot be predicted a priori

Erich Weigold and Ian E. McCarthy

146

’r 1

-1

-2

FIG.6. Surfaces of constant phase calculated by the optical model for 200-eV electrons on Xe’. From McCarthy and Weigold (1976).

by fitting elastic scattering, where the relevant radial region is too broad for (37) to be a valid approximation. 3. The Factorization Approximation

The DWIA (16) involves a nine-dimensional integral. The integration may be reduced to three dimensions if the averaged eikonal approximation (37) is valid, since the particle coordinates r l , rz then occur in the distortedwave factor only as linear exponents. Making the transformation (13) to the r, R coordinate system the amplitude (16) factorizes as follows : M(kA

9

kB) =


T,(P2)lk’)(X~-)(kA)X~-)(kB)l(flg)Xb+)(kO))

(39)

The first factor in (39) is the half-off-shell Mott scattering amplitude. The second factor may be computed directly from optical model potentials by solving the distorted-wave equations in partial wave form and performing the radial integral and partial wave sums explicitly.

(e, 2e) COLLISIONS

147

C. DISCUSSION OF REACTION MECHANISM APPROXIMATIONS The factorized DWIA (39) can be computed. Its validity depends first on the validity of the factorization approximation. This approximation improves with increasing energy as the averaged eikonal (or plane-wave) approximation improves. The unfactorized DWIA (16) was derived from the quasi-three-body expression (lo), (11) by an argument that amounts to replacing the optical model potentials V, and V, by constants. While this argument is likely to be valid in localized regions of the reaction phase space, both it and the factorization approximation are likely to be more valid in regions where the energy conditions do not vary when the recoil momentum is changed. The noncoplanar symmetric mode satisfies this criterion. Since the solution of the Coulomb three-body problem is not understood, little can be said a priori about the validity of the basic expression (lo), particularly in view of the fact that the two-electron potential v(r) is a Coulomb potential. An interesting aspect of the (e, 2e) reaction is its use in testing approximations, since good optical model potentials are known and therefore it is reasonable to represent the reaction as a three-body problem. Note that at higher energies, where the factorization approximation is more valid, the phase of the Coulomb T-matrix is irrelevant in (10). It is the rapid phase variation that makes Coulomb amplitudes difficult to handle.

D. THEATOMIC AND MOLECULAR STRUCTURE PROBLEM 1. The Overlap Function

If the reaction mechanism is sufficiently understood in the limited phase space region over which the experiment is performed, the reaction may be considered as a measurement of the overlap function ( f l g ) for the target ground state Ig) and the observed residual ion state The overlap function may be computed directly from the one-particle Green’s function describing sudden ionization or electron attachment involving ( N - 1)-, N - , and ( N + 1)-particle systems:

If).

where gf(r) = ( f J 9 )

(41)

andhf,(r)is the overlap function for electron attachment. The poles of the Green’s function give the energies of bound states of the residual ion, or

148

Erich Weigold and Ian E. McCarthy

the positions and widths of the resonances that occur above the threshold for a second ionization. The poles and their residues are found numerically by application of perturbation theory to the solution of the equation of motion for the Green’s function (Pickup and Goscinski, 1973). The function is perhaps more easily understood in terms of configurationand Is), assuming that are bound interaction expansions of the states states or narrow resonances:

If)

If)

The basis configurations la) and Ip) are independent-particle-model determinants involving the creation of certain particles and holes in the groundstate determinant (0) for the N-particle system. The final state I f ) is expanded in basis configurations, which are regarded as holes i,bj in the target configurations ( p), with a Clebsch-Gordan coefficient C,,, ensuring that such basis configurations belong to the representation r of the point group of the system. For spherical systems (atoms) the sets of quantum numbers j , r, a represent the total angular momentum, parity, and magnetic projection for the hole I ) ~the , state If), and the configuration Ip), respectively. Sums over vectors in the representation (denoted by projection quantum numbers for atoms) are implied in the notation. The choice of basis configurations la) is not arbitrary for the (e,2e) reaction, since the reaction measures the overlap function (in momentum space). The optimum choice is the one that minimizes the number of configurations necessary to describe the q distribution. Another optimization procedure is the Hartree-Fock variational method. We show in Section IV,A that these two definitions of the optimum basis coincide within experimental error, giving an experimental significance to the single-particle orbitals i,bj, which comprise the determinants). . 1 The overlap function in coordinate space, is

or, in momentum space,

(fig> = C $ ? 4 g ) C j r a # j ( q )

(44)

jz

where the orbital t,bj(r) or its Fourier transform dj(q) describes the hole from which the electron was ejected. Most systems so far investigated have been closed-shell systems, in which it is reasonable to make the target Hartree-Fock approximation u y = 0,

a#O

(45)

(e, 2e) COLLISIONS

149

If this approximation is valid, the overlap function reduces to

If the plane-wave approximation for the mechanism is valid, the q distribution is exactly the square of the momentum-space orbital +j(q),provided there is no configuration interaction between hole states. If this is so there is only one term in the sum over j in (46). It is trivially true for atoms, since overlap of Rydberg states is removed by a redefinition of the basis in the process of optimization. For molecules it is sometimes possible for orbitals such as 3 0 and 40 to interact. For atoms the characteristic orbital c of the representation r is defined by

The Clebsch-Gordan coefficient Cjr0 is equal to nr-1'2, where n, is the degeneracy of the representation r. This definition also applies to electronic states of molecules where hole states do not significantly interact. Target ground state configuration interaction results in admixtures of different orbitals in the q distribution, so that its shape is not characteristic of a single orbital. This would be observed in the experiment if it were significant. It also results in the population of ion states that contain configurations forbidden in the transition from the ground-state configuration 10). An example is the population of the n = 2 states of helium. Observation of such states is a very sensitive test of ground-state correlations. 2. Cross Sections and the Spectroscopic Sum Rule

The magnitudes of cross sections are proportional to the spectroscopic factors Sif) in the target Hartree-Fock approximation, where

Normalization and closure properties of wavefunctions lead to the sum rule for spectroscopic factors,

c s y )= 1 s

(49)

and the definition of the single-hole energy Ec

=

c SLf

'Es

f

as the mean energy of eigenstates belonging to the representation r, weighted by the spectroscopic factors.

150

Erich Weigold and Ian E. McCarthy

Full considerations of normalization and antisymmetry (Austern, 1970) introduce a factor n,"' into the amplitude. This cancels the Clebsh-Gordan coefficient nr-'I2. The differential cross section is given by a sum over degenerate final states, which introduces a new factor of n, from the addition theorem. It is proportional to CJ = n

S ' I
'I$,xb' 9I'

(51)

States I f ) belonging to the representation Y can be identified by their characteristic momentum profile, which is approximately lq5c(q)(2 at high energies. The assignment to representations and the validity of the approximation used for the reaction mechanism are simultaneously verified by completion of the spectroscopic sum rule (49).

3. Molecules Since they have vibrational and rotational degrees of freedom, molecules exhibit a much greater variety of final states than atoms. In addition, gases at laboratory temperatures have a Maxwellian distribution over rotational states. Energy resolution available in the (e, 2e) reaction is such that it makes sense not to resolve vibrational and rotational states and to use closure in the analysis. Electronic states are studied. Making the Born-Oppenheimer and plane-wave approximations, vibrational and rotational closure results in the amplitude (McCarthy and Weigold, 1976)

The cross section involves a sum over degenerate electronic states labeled by p r . IF) and (G) are, respectively, the electronic ion and target states. The cross section is averaged over the vibrational ground state 10). Rotational closure leads to a spherical averaging over molecular orientations R. The right-hand side of (52) is independent of the initial rotational quantum number I , indicating that a Maxwellian average is unnecessary in describing relative cross sections in the same experiment. For molecules we are at present unable to make a distorted-wave approximation (apart from the averaged eikonal approximation), since the complications of the multicenter scattering problem make the computation of a scattering-wave function too difficult. We are therefore very interested in the (e,2e) reaction on atoms as an indication of the validity of simple approximations for f ' ) . The spectroscopic sum rule will give us an indication of the internal consistency of any reaction model that we make.

(e, 2e) COLLISIONS

151

For diatomic molecules, at least, the vibrational ground-state average will be most important for the lightest, hydrogen. The average has been performed explicitly in this case (Dey et al., 1975).The shape of the angular correlation was indistinguishable from the shape calculated for the equilibrium value of the vibrational coordinate. Calculations have therefore been performed for the equilibrium values only. The (e, 2e) reaction at high enough energy observes the spherically averaged square of the momentum-space electronic overlap function calculated at the equilibrium nuclear positions. This is directly related to the quantities calculated in the molecular-structure calculations of quantum chemistry. The discussion of Sections III,D,l and III,D,2 applies equally well to electronic states ot' molecules. In fact, the point group was not specified, in order to provide the required generality. Considerations of spectroscopic factors, the sum rule, and the definition of the single-hole energy are the same.

IV. Reaction Mechanism at Intermediate to High Energies In order to obtain information on the reaction mechanism the theory developed in Section 111 must be compared with measured cross sections. Since the approximations developed there are expected to be valid only at intermediate and high energies, the discussion will be limited to this region. The primary objective is to develop an understanding of the reaction mechanism for at least one kinematical arrangement, so that the reaction can be used to extract the structure information (Section 111,D). The absolute magnitude of the differential cross section is given in atomic units by (C. J. Noble, unpublished, 1977)

xaV

where denotes the usual procedure of summing over the final states and averaging over initial states, and symmetric normalization has been adopted, i.e.,

< k ( k ) = 6(k - k'), (rlr')

= 6(r - r'),

Jdklk)(k/

J

=

1

drlr)
152

Erich Weigold and Ian E. McCarthy

Making the factorization approximation (39) and the target Hartree-Fock approximation (46), this becomes

where

1 1 (k - k’I2 Ik + k’(’

r = 1/2k‘,

k’ = i ( k A - kB),

k

=gko

+ 9)

(57)

With symmetric kinematics the expression for I TI2 simplifies considerably, being nearly constant in the noncoplanar case. In the plane-wave limit for a closed-shell atomic target, (55) reduces to

where n is the number of equivalent target electrons, and (59)

A. NONCOPLANAR SYMMETRIC KINEMATICS The noncoplanar symmetric experiment involves the simplest region of phase space. The off-shell Coulomb T-matrix element corresponds to a collision of the incident electron with a bound electron moving perpendicular to the incident direction with momentum q, which varies as the azimuth 4 is varied, but is much less than the incident momentum k o . The energy p 2 therefore varies minimally with change of q, and all energies involved in the

(e, 2e) COLLISIONS I

c

1

I

153 I

V

H e ( e , 2 e ) He+ e =42.3"

:0.8

o

4OOeV

x

800eV

A

1200eV

.P 1.c c aJ

2V

I

200eV

d

:.

0.6

E % 0.4 u-

aJ

._ c

rg

5

0.2

Ls:

0 FIG.7. The q profile in noncoplanar symmetric geometry for the helium ground-state transition at the indicated energies, compared with the plane-wave theory using the HartreeFock orbital. From McCarthy and Weigold (1976).

DWIA [Eqs. (16) and (55)] are essentially constant. Any energy dependence of this approximation or of the factorization approximation (39)is minimized. In order to obtain a feeling for the validity of the reaction mechanism approximations, it is instructive first to try the simplest, the plane-wave approximation, on the simplest system studied in some detail, helium. If the plane-wave approximation is valid, (39) and (16) are identical and the shape of the momentum profile depends only on q, not on the total energy E [Eq. (56)]. Figure 7 shows that the momentum profile in this approximation has the same shape for values of E between 200 and 1200eV and that the shape is excellently described by the Hartree-Fock wavefunction (FroeseFischer, 1972). The configuration-interaction wavefunction of Joachain and Vanderpoorten (1970) gives an identical result. Encouraged by the situation for helium, we ask more detailed questions. Is the plane-wave theory adequate for a case where more than one singleparticle state can be observed? Figure 8 shows the momentum profiles for the 2p and 2s states of neon. In this case ion states at 21.6,48.5, and 59.88 eV are populated by the reaction, the last with a strength about 4% of the 48.5-eV state, and having the q profile corresponding to an s electron. The summed profile for the last two states is compared with that of the 2p- ground state.

-

Erich Weigold and Ian E. McCarthy

154

\, i-o

w

c

L

c,,,i,,,i,,,l,,,l,, 0

1

q(a.u.~3

FIG.8. The noncoplanar symmetric q profile for 2p and 2s orbitals of neon. The data have been normalized to the 2p distorted-wave (DW) peak at 600 and 1200eV and to the plane-wave (PW) peak at 2500eV. The eikonal approximation gives a cross section (dotted lines) that has lower magnitude but similar shape compared to PW. From Dixon et ul. (1978).

We can ask not only about the profile shapes, but also about the relative magnitudes. Since there is one unmeasured normalization, we are at liberty to choose one spectroscopic factor. In all observed cases the highest valence orbital is essentially unsplit. We therefore use it to define the unit spectroscopic factor. In Fig. 8 the data at each energy are therefore normalized to the 2p orbital by choosing S,, = 1. The profile shape is well described by the plane-wave approximation with Hartree-Fock orbitals for y < 2 a.u. For higher momenta the actual recoil momentum profile has more high-momentum components than the HartreeFock orbital. Possible sources of momentum transfer to the ion are elastic scattering before and after the elementary collision (described by distorted waves) and many-body correlations in the structure wavefunctions. = lOeV, = The eikonal approximation with realistic parameters 5 eV differs from the plane-wave approximation only in minor detail at very small y. The eikonal parameters p and y approach zero as E increases.

w

(e, 2e) COLLISIONS

155

The relative magnitude for the 2s and 2p states is badly predicted by the plane-wave theory. In terms of the eikonal approximation we could consider that the 2s orbital, whose rms radius is smaller than for 2p, should be assigned a larger value of the absorbing potential W . The approximation should improve as E increases, making y insignificant. This is seen to occur, but even at 2500eV the spectroscopic factor Szs is given by plane waves as only 0.8 instead of its correct value 1. The eikonal approximation can therefore account for the spectroscopic sum rule, but at the expense of a free parameter W2s/W2p.The approximation is not sufficiently detailed to describe the radial variation of the imaginary potential, and so this parameter cannot be fixed from the elastic cross section. The cross section calculated by the factorized DWIA [Eq. (39)], with optical-model potentials that describe elastic scattering very well, is illustrated in Fig. 8 for 600 and 1200eV. Distorted waves account for both the high-momentum profile shape and the relative magnitude, confirming the energy independence of the structure information involved in the calculation. We next ask whether the qualitative success of the plane-wave theory and the complete success of the factorized DWIA is repeated for the most difficult atom yet studied, xenon. Figure 9 shows that the success of the plane-wave and eikonal approximations in describing shapes extends now out to only about q < 1 a.u. We could generalize this to 4 < l/fT, where R is approximately the rms radius of the orbital. The shape of the q profile is considerably improved by distorted waves, but the relative magnitudes of 5p and 5s cross sections are in error by a factor of 2. The 5p strength is essentially all contained in the ion ground state, whereas the 5s cross section has been summed over all of the ion states containing the 5s- orbital (McCarthy and Weigold, 1976). In view of the success of the theory for smaller atoms, one must suspect the possibility that the basic ingredients of the calculation, in particular the imaginary part of the optical-model potential, are not sufficiently understood to make this case a valid test of the DWIA. Hood et al. (1977)and S. Dey and E. Weigold (unpublished, 1975)observed the relative magnitudes for exciting the spin-orbit split 5p,,, and 5p,,, ion states. Both measured ratios agree with the expected ratio of 2 within a few percent (see Fig. 18). Further questions can be asked about the description of the non-coplanar reaction. How sensitive are profile shapes to details of the orbital? Figure 10 shows a comparison of the 4 profile for the 3p orbital of argon calculated in the plane-wave approximation with three orbitals of varying degrees of sophistication. Essentially perfect agreement is obtained for the HartreeFock function of Froese-Fischer. The hydrogenic 3p orbital with variationally determined effective charge gives a much worse shape.

Xe 5s-' E-1200eV

m m

Xe 5p-'

L 0

4

E=LOOeV

-

t

v

1

0

2

q(a.u.)

FIG.9. The noncoplanar symmetric q profile for 5p and 5s orbitals of xenon. All Sp curves have been normalized to the distorted-wave result with the normalization factors shown. Other curves are plane-wave (PW) and eikonal (Eik). From Dixon et a/. (1978).

I

I

I

I

-

Ar

I

(3p-l)

e =45'

$ 1.0

.-

C

E = 8OOeV

3

x

h 0.8

c .-

\

D

-

* 0.6 I

\

\

C

.-0

;0.4

c

m VI

2

0

0.2

C

3

g(a;') FIG. 10. The noncoplanar symmetric q profile for the 3p orbital of argon at 800 eV computed for the different orbital models described in the text. From Hood et al. (1973).

(e, 2e) COLLISIONS

157

An important check on the smallness of the core-excitation contribution, B in Eq. (lo), is to verify that the shapes of the q profiles for all states belonging to the same representation r are similar. The case of the 3s representation of the argon ion is illustrated in Fig. 11. Here the 3s orbital is split into well-resolved eigenstates whose profiles are described quite well with the Hartree-Fock plane wave theory for q < 1. Shapes are similar except for the 48- and 51-eV transitions, which are well above the two-electron threshold and for which the counting rate is very low. If the plane wave mechanism were to be taken literally for Fig. 11, a structure explanation would have to be found for the enhancement of the

r)

q (a.u)

FIG. 11. Noncoplanar symmetric q profiles for different bound ( E < 43.6eV) and continuum ion states for argon at the indicated energies. The curves are the plane-wave theory with the 3s Hartree-Fock orbital. From McCarthy and Weigold (1976).

Erich Weigold and Ian E. McCarthy

158

cross section for q > 1. An admixture of d configurations in the ground state (g) would be an obvious candidate. However, the examples of neon and xenon have shown that profiles at high q are quite adequately accounted for by distorted waves. There is no evidence of any departure from the target Hartree-Fock or quasi-three-body approximations.

B. COPLANAR SYMMETRIC KINEMATICS The Coulomb T-matrix element for the coplanar symmetric experiment corresponds to a collision of the incident electron moving with momentum 4, either parallel or antiparallel to the incident direction. The energy p 2 therefore has its maximum sensitivity to changes in q, and this experiment is a strict test of the reaction mechanism assumed in a calculation. The first test is again helium, for which the 0 profile is illustrated in Figs. 12 and 13 at 200 and 1200eV, respectively. At 200eV the plane wave and eikonal

12

c

He

--

E=200eV

D.W.

---- P.W. ( x 0 . 4 6 ) Eik ( x 1.30)

1 -

50

40

e

60

70

(degrees )

FIG. 12. The 200-eV coplanar symmetric cross section for helium, normalized to the distorted-wave (DW) theory in the factorization approximation. Plane wave (PW) and eikonal (Eik) approximations are also illustrated. From Fuss et al. (1978).

(e, 2e) COLLISIONS

159

I

8 (degrees)

FIG. 13. As for Fig. 12 with E

=

1200 eV

theories give cross sections outside the limits of experimental error; the eikonal parameters V = 10 eV, W = 5 eV are not capable of bringing the theory into agreement. After the success of the DWIA for the noncoplanar experiment, it is quite surprising that it overcompensates for distortion effects, yielding a description of the 0 profile shape no better than plane waves. As expected, the theory improves at the higher energy, but there is still a discrepancy between the plane-wave theory, distorted-wave theory, and experiment. The case of argon at 400eV is shown in Fig. 14 in order to illustrate effects for larger systems. The experimental points have been arbitrarily normalized to the forward (0 < 0,) peak in the 3p distribution. As before, the summed valence s cross section is measured relative to the valence p cross section. Once again the DWIA overcompensates for distortion effects. In addition it produces a 5076 discrepancy in the relative magnitudes of the two 3p peaks, the peak at smaller 0 corresponding to a collision with an electron moving parallel to the incident electron, and the larger-angle peak corresponding to the antiparallel situation. The peak ratio is in fact much better described by plane waves.

Erich Weigold and Ian E. McCarthy

r Ar E=LOOeV

-D.W. - - - - RW. ( x 0.28) - - Eik (~0.57) .........

8 (degrees)

8 (degrees)

FIG.14. The 400-eV coplanar symmetric cross section for the valence orbitals of argon. Symbols and normalization are as for Fig. 12. The curve labeled MDW is calculated by switching o f the Coulomb potentials in the distorted waves. From Fuss et d.(1978).

These results are typical of the Flinders experiments, which have also been carried out for other inert gases at energies up to 1200eV (e.g., Fuss et a/., 1978). The results of coplanar, symmetric experiments at Frascati (A. Giardini-Guidoni, private communication, 1977) are similar. The trend that the distorted-wave theory yields an improved description of the 8 profile shape at higher energy is noted in all experiments. However, there is no detectable trend in the description of the ratio of profile magnitudes for valence p states and summed valence s states. The problem is, of course, how the valence p data should be normalized to the DWIA cross section: at the forward peak (as here) or the backward peak, or somewhere in between? The different normalizations will greatly affect the apparent discrepancy between the DWIA and the observed valence s cross sections. While the noncoplanar symmetric experiments on atoms give an excellent basis for the use of the factorized DWIA (or even the plane-wave approximation with suitable modifications) in atomic spectroscopy using this restricted region of phase space, the coplanar symmetric analysis shows that the factorized DWIA does not constitute a detailed reaction theory.

(e, 2e) COLLISIONS

161

There is no suggestion of a breakdown of the quasi-three-body approximation in the noncoplanar case, where it is just as likely to break down as in the coplanar case. We can therefore treat the reaction as an extremely sensitive test of theories of the three-body system with Coulomb forces. At the outset of the discussion of possible sources of error and improvements in the theory it must be emphasized that the apparent relative success of the plane-wave (or eikonal) model does not mean that it is a better theory than the DWIA. The electron-target and electron-ion potentials cannot be ignored. Also they cannot be adjusted, except by improvements to the optical model that result in equally good descriptions of elastic-scattering data. However the plane-wave success could be a factor in a future understanding of the relative failure of the factorized DWIA. The derivations of the DWIA (16) and the factorization approximation (39) both depend on the assumption that the space derivatives of the opticalmodel potentials can be ignored in comparison with the total energy E . The improvement of the approximations with increasing E confirms that our understanding is correct up to this point. Are Coulomb boundary condition effects important? Calculations with effective charges given by Eqs. (20) and (21) yield only minor shape adjustments, but nothing approaching the required changes (Fuss et al., 1978). It would be useful to remove the factorization approximation by calculating the nine-dimensional integral (16). Work is in progress on this (Koshel, 1976; I. R. Afnan, I. E. McCarthy, C. J. Noble, and G. J. Stephenson, unpublished, 1976). The effect would be to introduce the rapidly varying phase of the Coulomb T-matrix into the non-plane-wave part of the integration. There is one hint that adjusting the relative phases of partial matrix elements can improve the integral. This is the modified distorted-wave (MDW) calculation of Fig. 14, for which the Coulomb potential was switched off in the distorted waves. The main effect of this is to remove the Coulomb phase shift from each partial wave, thus modifying the phases of the partial matrix elements. The modification (for which there is no present justification) yields some improvement. FOR ATOMIC HYDROGEN C. THE(e, 2e) CROSSSECTION

The simplest ionization problem is obviously the case of atomic hydrogen, involving only three bodies, two electrons and a proton. The initial target wavefunction is known exactly, as is, of course, the overlap function in the matrix element [Eq. (16)l. It therefore provides a direct test of the approximation used to describe the reaction mechanism. Some results of an experimental study of the triple differential cross section for atomic hydrogen were recently presented by Weigold et al. (1977a).

Erich Weigold and Ian E. McCarthy

162

The shapes of the coplanar asymmetric cross sections were found to be in poor agreement with plane-wave Born and Coulomb-plane-wave Born (slow electron described by a Coulomb wave) calculations and the Coulombprojected Born results of Geltman and Hidalgo (1974). Coplanar cross sections with more symmetric kinematics are shown in Fig. 15. Here the outgoing energies were kept equal, but 0* was kept fixed and only BB varied. This was done in order to avoid any distortion due to the finite dimensions of the atomic hydrogen beam, the moving detector B viewing at all angles the whole of the interaction region, which was defined by the intersections of the atomic and electron beams and the viewing angle of the stationary detector. Although the two differential cross sections shown are not absolute, they are measured relative to each other. The solid and dashed curves are obtained using the factorized distorted wave and plane-wave impulse approximations, respectively, and the dotdashed curves using the plane-wave Born approximation (with exchange included). The two plane wave calculations are in very poor agreement with the data at “forward angles. This is hardly surprising since the energies of the electrons are all quite low. As was observed for helium (Fig. 12), the full distorted wave curves tend to lie on the other side of the experimental points compared with the plane

I

0

I

I

I

I

T

I

I

I

LO

60

20

I

‘

80

/ I

’

0

I

I

I

I

I

1

I I

20 degrees

I

I

I

I

LO

60

80

100

BS , FIG. 15. Angular correlations between emitted electrons in the ionization of atomic hydrogen by 113.6-eV electrons compared with the distorted wave (-) and plane wave (-) impulse approximations and the plane wave Born approximation (exchange included) From Weigold et ul. (1978).

(e, 2e) COLLISIONS

163

wave results. The effective charges Z , and 2, were chosen to be unity in the distorted wave calculations. This obviously violates the condition determining the effective charges [Eq. (20)], and more realistic (i.e., smaller) effective charges may lead to a significant improvement in the calculated cross sections, since the distorted waves in the outgoing channels are in this case pure Coulomb waves.

D. ABSOLUTE CROSSSECTIONS FOR HELIUM Absolute coplanar symmetric cross sections have recently been measured for helium by Stefani et al. (1978) at 0 = 44" (i.e., q z 0) in the energy range 200eV-4.5 keV. The data were put on an absolute scale by normalizing to the elastic cross sections of Jansen et al. (1975) [see Eqs. (6) and (7)]. Their measurements are shown in Fig. 16, together with some calculated cross

IIIIIIll

I

I

1 1 1 1 1 I

lo4

lo3

E,

I

I

I

102

(eV)

FIG.16. Absolute coplanar symmetric differential cross sections for the helium ground-state = 44 , The filled circles are the experimental points (Stefani et ul., 1978) with the estimated maximum errors indicated by the bars. The short dashed curve is the Born approximation cross section, the solid and long dashed curves are, respectively, the distortedand plane-wave impulse approximation cross sections, and the filled and open squares are. respectively, the Coulomb projected and Coulomb-plane wave Born results of Geltman (1974).

(e, 2e) reaction at 0

164

Erich Weigold and Ian E. McCarthy

sections. The short dashed line is the plane-wave Born approximation cross section (Glassgold and Ialongo, 1968;Vriens, 1970) with exchange included. The long dashed curve is the cross section given by the plane-wave impulse approximation [Eq. ( 5 8 ) ] , and the solid curve is the cross section obtained with the full factorized distorted wave off shell impulse approximation [Eq. (55)].In all cases we have taken S , = 1 for the ground state transition since S , > 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 <44

Kr(4s- I )

Hole state

Xe(5s

I)

Relative S:I’ e,2e ( q = 0)

XPS ( q >> 1a.u.)

0.96 <0.01 0.04(1) < 0.05

0.932)” 0.04 0.02

3s-’ 3p-’3d 3p-’4d

0.53 0.23(2) 0.13(2) 0.05(1) 0.08(1)

0.81 0.13(2) 0.06(2)

27.5 34

4s-’ 4p-’4d1

0.30 0.70(3)b

0.78

23.4 24.6 25.2 27.9 29.0 31.4 32.8 33.8

4s-’

0.32 0.03(1) 0.04(1) 0.132) 0.27(2) O.lO(2) 0.04(1) 0.05(2)

0.45 0.03(1) 0.03(1) 0.11(2) 0.24(2) 0.09(2) 0.03(1) 0.02( 1)

’

-

’ (ti) denotes an uncertainty of & O.On. I,

The peak at E

=

34eV is due to a number of unresolved ion states

0.22(2)b

(e, 2e)

COLLISIONS

167

eV. The photoelectron strengths are, on the other hand, obtained for regions very close to the nuclei (high q), since q is essentially given by the momentum of the outgoing electron. B. MOLECULES In view of the success of the noncoplanar symmetric reaction in elucidating atomic structure, it has been used to probe the electronic structure of molecules. In this case a complete distorted-wave theory would be computationally too difficult. The plane-wave theory (52) has therefore been used in the analysis of the experiments, with the expectation that it will describe the momentum profile shape for q < l/T, where T is roughly the rms radius of the atom-centered functions used as a basis for describing the molecular wavefunction. The validity of the plane-wave theory in the description of relative crosssection magnitudes for different ion states IF) can be assessed by experience. It is found, for example, that for CO and N2, relative magnitudes for several orbitals whose energies are about 20eV or less are well described at E = 400 eV, while the spectroscopic factor for more tightly bound states with E nearer to 40 eV is too small by about 20%. This discrepancy is removed by increasing the total energy E to 1200 eV (Dey et al., 1977; Weigold et al., 1977b).The standard energy for the analysis of molecules in the Flinders experiments is therefore 1200 eV. In all cases so far observed, the spectroscopic sum rule is satisfied at this energy. The case of ethane is illustrated in Fig. 17, where the noncoplanar symmetric cross sections at 400 and 1200eV are compared with plane-wave calculations. Although absolute cross sections were not measured in the experiment (Dey et a/., 1976), the relative magnitudes to the various ion eigenstates were obtained. The figure shows momentum profiles corresponding to the sums of cross sections for final states belonging to the same hole orbital. They are compared with the relative cross-section shapes and magnitudes calculated with the plane-wave theory using the orbital wavefunctions of Snyder and Basch (1 972). The q profile shape is well described by the plane-wave model up to q % 2a; ' for the less tightly bound orbitals and up to q = la; ' for the more tightly bound 2a1, orbital. Dey et al. (1976) observed considerable structure in the separation energy spectra at energies greater than that corresponding to the main 2a,, transition. Most of this structure was attributed to the 2al, hole orbital, and some to the 3al, orbital. Prior to this work there was some controversy over the PES identification of ground state of C2H6+,since the vibrational bands of the 3al, and leg orbitals overlapped. The photoelectron spectrum in the region of the overlapping leg and 3a1, bands is complicated by the Jahn-Teller effect, which

zg

a

Relative

1 N

Pi

differentlal cross section

RELATIVE DIFFERENTIAL CROSS SECTION

(e, 2e) COLLISIONS

169

splits the doubly degenerate leg-' state into two components. The (e,2e) angular correlations (Fig. 19) obviously remove any possible doubt about the identification of these orbitals. The case for acetylene, which is the simplest example of a triple rather than a single carbon-carbon bond, is shown in Fig. 20 and 21. The high-energy structure in the separation energy spectrum actually extends to 45 eV (Dixon et al., 1977a). The four main ion states observed are labeled in Fig. 20 by their hole orbital. The corresponding angular correlations are shown in Fig. 21. The relative magnitudes of the cross sections were measured at 400

150

Erich Weigold and Ian E . McCarthy

and 1200eV. For the 2gg transition, the 400-eV data were normalized to the 1200-eV calibration points, since it was found that absorption of the electron waves was relatively more important for this most tightly bound valence orbital at 400eV. Also shown for comparison in the figure (the crosses with error bars omitted) are the recent 400-eV results of Coplan et a/. (1977). Since the relative magnitudes were not measured in this experiment, they have been disposed on the graph to give agreement with the results of Dixon et al. Although both experiments used noncoplanar symmetric kinematics, quite different techniques were employed. The figure also shows the results of three plane-wave calculations using different approximations for the overlap function. The full curves are the cross sections obtained using the generalized overlap amplitudes derived from the one-particle Green’s function (Section 111,D). The one-particle propagator formalism introduces correlations in both the initial and final states, as well as relaxation effects, and therefore the spectroscopic strengths can be less than unity. The spectroscopic strengths for C2H, are found to be close to unity except for the 20; state at 23.7eV, for which S:f’ = 0.7 (Dixon et al., 1977a).In the neighborhood of poles with large pole strengths the Green’s functions are well approximated by quasi-particle propagators, and therefore the ionization process should be well described by the removal of a Hartree-Fock particle (Cederbaum et al., 1973). Figure 21 also shows the momentum distribution obtained from transforms of the molecular orbitals used in the generalized overlap amplitude calculation (dashed curves) and the orbitals of Snyder and Basch (1972), which are shown as dot-dashed curves where they differ from the dashed curves. The molecular-orbital transforms, which assume SLf) = 1, give a poorer fit to the data, particularly to those of the 20,’ transition. The shapes and relative magnitudes of the observed and calculated 20, and 30, momentum profiles are in good agreement. For the outer 171, orbital, however, the calculations all seriously underestimate the low-q contribution. This means that the probability of electrons being far from the nuclei is underestimated by the molecular orbital wavefunctions. For the innermost valence orbital, the 20,, the position is just the reverse, the molecular orbital wavefunctions underestimating the low momentum probability. A similar situation is observed for HzO. Figure 22 shows the observed momentum profiles compared with two plane-wave calculations, the solid curves being the cross sections obtained using the overlap function calculated from the one-particle propagator formalism and the dashed curves using the orbital wavefunctions of Snyder and Basch (1972). The 400-eV data for the innermost orbital (2a,) were again normalized upward to bring them into agreement with the 1200-eV data (Dixon et al., 1977b).

(e, 2e) COLLISIONS

171

FIG.22. Noncoplanar symmetric momentum profiles for water (Dixon el d.,1977b). The solid curves are the calculated cross sections using the overlap function obtained from the propagator formalism, and the dashed curves those obtained using the orbital wavefunctions of Snyder and Basch (1972).

The propagator formalism calculations lead to significantly better agreement with the data than d o the calculations using the SCF wavefunctions of Snyder and Basch. The shape disagreement between theory and experiment remains serious, however, for the momentum profile of the ground-state transition. This excess of low-momentum components in the momentum profile of the “lone pair” lb, orbital has also been observed by Hood et al. (1977), the two sets of experimental data being in excellent agreement. This is therefore another example of a case where the (e, 2e) reaction has uncovered a serious discrepancy in detail in quite sophisticated molecular-structure calculations. The (e, 2e) experiment is most sensitive to the spatially extended part of the wavefunction, whereas the most significant contribution to the energy arises from those parts of the orbital close to the nucleus, that is, the high-q

172

Erich Weigold and Ian E. McCarthy -

E=LOOeV

S e p a ra ti o n

energy

(eV)

FIG.23. The separation energy spectrum of PH3 (from Hamnett et a/., 1977). The peaks labeled 1-111 result from configuration mixing in the ion.

region. Although the spatially extended parts are most important chemically, they do not contribute significantly to the total energy, and their significance can therefore be underestimated in the variational calculations using minimum basis sets. Hood et al. (1976a) observed a similar discrepancy in the outermost orbital of ammonia, which is again of the lone pair type. The bonding of phosphine has been studied by Hamnett et al. (1977). Figure 23 shows the observed separation energy spectrum for PH,, and Fig. 24 the angular correlations for the two outermost bands. The peaks labeled 1-111 in Fig. 23 are the results of complex configuration interaction effects in PH:. In contrast to the results for ammonia, the momentum profile for the outermost lone pair 5a, orbital is well described by plane-wave calculations. Two molecular-orbital wavefunctions were used in the calculations, a one-center expansion calculation (solid lines) and a minimum basis set calculation to which a set of P 3d orbitals were added (dashed curves). The nature of the lone pair orbitals may explain the very different coordination chemistries of phosphine and ammonia (Hamnett et al., 1977). The application of (e, 2e) spectroscopy for investigating molecular structure is expanding rapidly. Recent work on molecules includes measurements on C 0 2 and SF, (Stefani et al., 1977), C,H4 (Dixon et al., 1977c; M. Coplan, private communication, 1977), and H,S (Brion et al., 1977), CO (Dey et a]., 1977), and formaldehyde (Hood et al., 1976b).

(e, 2e) COLLISIONS

q(au)

173

g(au)

FIG.24. Noncoplanar symmetric momentum profiles for the two outermost orbitals of P H 3 . The solid curve uses a one-center expansion wavefunction, the dashed curve a minimum basis set to which a set of P 3d orbitals is added. From Hamnett et a/. (1977).

In all cases studied in detail so far it is found that the q profile shape enables assignment of ion states to representations. The assignment is confirmed by satisfaction of the spectroscopic sum rule for E 2 1200eV.

C. GROUND-STATE CORRELATIONS The discussion of ion states so far has assumed that the Hartree-Fock independent-particle model is an adequate description of the target ground state. In fact, of course, there are always ground-state correlations. The overlap function is given by Eq. (44). If the coefficient ug) is nearly 1, then final states related to the hole states will be populated with high probability even if their coefficients t${) are quite small. Final states not related to the hole state are forbidden in the sense that their overlap function is of second order in the small coefficients and u$) for M # 0. Such states will, however, be populated for closed-shell targets although their cross sections will be so small that they are difficult to observe with the present ( 1 -2 eV) energy resolution in comparison with the allowed states that include the one-hole configuration. In the case of the helium ion, there is only one state for the one-hole configuration. It is the 1s state. Higher states are forbidden in the sense that they have no component consisting of one hole in the target HartreeFock ground state. Excitation of 11s ion states other than 1s can nevertheless occur in the Hartree-Fock approximation since atom and ion s states are

-

Erich Weigold and Ian E. McCarthy

174 0.10

-

I o

0.08 - A

I 0.1

I

8OOeV coplanar

He ( e . 2 e ) He'

-

1200eV noncoplanar

I

I

I

I

II4II 1.0

I

I

9 (a.u.!

t

I

I I I I I

I

]

10.0

FIG.25. The ratio of the n = 2 to n = 1 (e, 2e) cross sections for helium as a function of the momentum q. The curve is the calculated result using a high-quality correlated helium groundstate wavefunction. From Dixon et al. (1976). (Copyright by The Institute of Physics. Reprinted with permission.)

not completely orthogonal. The ratio of production for different states will, be independent of q, being approximately 2% for the n = 2 to n = 1 ratio. The population of the n = 2 states of the helium ion has been observed by McCarthy et a/. (1974) and Dixon et a / . (1976). It provides an extremely sensitive test of ground state correlations. Figure 25 shows the ratio of n = 2 to n = 1 states at different energies compared with the plane-wave theory with the overlap function calculated from the configuration-interaction wavefunction of Joachain and Vanderpoorten (1970). Agreement between theory and experiment is excellent. The ratio rises from approximately 0.008 at low q (i.e., in the outer regions of the atom) to a value of the order of 0.08 at q x 3 a i 1 (relatively close to the nucleus) before decreasing to an asymptotic value of z 0.05 at very high y. Ground state correlations have also been measured by the (e, 2e) technique in the simplest molecular case, namely, H, . Weigold et a / . (1977~) measured the relative intensities for exciting the lso,H: ground state and the 2p0,, 2p71,, and 2s0, ion states. Again, the ratio of the transition probability to the excited states relative to the ground state was found to depend strongly on q, rising from a value of 0.02 at small 4 . The noncoplanar symmetric momentum profiles obtained by them at different separation energies E are shown in Fig. 26. At E = 37 and 40.5eV the 2s0, and 2pn, states both contribute to the cross section due to their broad Franck-Condon overlaps. The data are compared with the shapes predicted by the simple H, groundstate configuration interaction wavefunction of McLean et a / . (1960), the calculated cross section magnitudes being in poor agreement with the data for the 2s0, and 2~71, transitions. The magnitude and shape of the 2p0,

(e, 2e) COLLISIONS

175

400

5c-

300

0

E

300 eV

0 W

v, v) v)

200

0

5

100

1

9 zW

c

o

a W LL

LL 0

w

1.0

2

c

a

a

0.15

0.5 0.25

FIG.26. Noncoplanar symmetric momentum profiles for hydrogen. The curves show the calculated 4 profiles using the simple configuration interaction H, ground-state wavefunction of McLean et a / .(1960). From Weigold et al. (1977~).

transition is however, well described by the calculation. Thus although the wavefunction of McLean et al. gives a very accurate description of the momentum profile of the ground-state transition, it does not give an adequate description of all of the excited-state cross sections. (e,2e) cross sections to ion states with a small overlap with the target state are very sensitive to the details of the wavefunctions. With improved energy resolution such states will be observed for larger atoms and molecules. Their q profiles will be characteristic of the holes qb! in the excited configurations a, quite often with a coherent sum over several terms so that the q profile will depend sensitively on the values of the small coefficients aig) and ti!) [Eq. (42)]. Such states will be important in (e, 2e) reactions on openshell systems such as alkali metal atoms.

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Erich Weigold and Ian E . McCarthy

VI. Conclusions We have discussed in some detail kinematically complete measurements of symmetric ionizing collisions initiated at medium to high energies by electrons incident on gaseous atomic or molecular targets. A distinction was drawn between symmetric kinematics and various asymmetric arrangements, which were discussed only briefly. The primary aim in low-energy asymmetric experiments lies in the investigation of the ionization process. Theoretical attempts to explain the data in this regime have on the whole not been particularly successful. On the other hand, glancing collisions at high incident energies are quite well understood, and the reaction has been used extensively by Brion, Van der Wiel, and co-workers in structure investigations, the reaction simulating various (y,e) processes. Symmetric ionization collisions enable ion-recoil momentum profiles to be recorded for resolved electronic states of the residual ion. According to the reaction theory (the distorted-wave off-shell impulse approximation) the momentum profiles are essentially proportional to the square of the momentum space orbital for each ejected electron averaged over experimentally degenerate substates multiplied by the half-off-shell Mott scattering cross section. In the noncoplanar symmetric arrangement, the Mott scattering term is essentially independent of angle, and the analysis of the data is particularly simple. In this geometry the reaction mechanism is sufficiently well understood to determine structure information uniquely. As well as measuring separation energies for individual ion eigenstates, the reaction measures the orbital shapes, determining directly the signature of the characteristic orbital of each ion eigenstate. It also obtains the spectroscopic factors, which give the probability of finding the single-hole configuration in the many-body wavefunction for the ion state. The analysis is tested by the sum rule: the spectroscopic factors must sum to unity for an orbital of given signature in the Hartree-Fock approximation for the (closed-shell) target state. One can also obtain the single-particle energy, defined as the centroid of the separation energies of ion eigenstates containing the characteristic orbital weighted by their spectroscopic factors. The cross sections to ion eigenstates with a small overlap with the target ground state are found to be extremely sensitive to electron correlation effects. Configuration interaction in both the target ground state and the final ion state is sensitively probed. Coplanar symmetric kinematics provides a more difficult test of the reaction theory. As the polar angle 8 is varied, the energy p 2 for the electronelectron T-matrix changes rapidly, so that every aspect of the theoretical amplitude [Eq. (39)] is tested. In this geometry one observes a breakdown

(e, 2e) COLLISIONS

177

of the factorized approximation, but the breakdown tends to heal as the energy is increased. The reaction theory therefore requires refinement, but it is nevertheless good enough to extract structure information. There is at present no evidence of the breakdown of the quasi-three-body picture, and the experiments provide a test for approximations to the Coulomb threebody problem. (e, 2e) spectroscopy should continue to grow rapidly in the next few years. Absolute cross sections of much improved accuracy are highly desirable. We should also see an extension of the technique to open-shell atoms and molecules, and significant improvements in energy resolution will make possible the study of much more complicated and chemically interesting molecules. ACKNOWLEDGMENTS We are grateful to M. Coplan and A. Giardini-Guidoni for providing us with some unpublished results, and to A. Dixon, S. T. Hood, and C. J. Noble for valuable suggestions and discussions.

REFERENCES Austern, N. (1970). “Direct Reaction Theory,” p. 289. Wiley, New York. Avida, R., and Gorni, S. (1967). Nucl. Instrum. & Methods. 52, 125. Balashov, V. V., Lipovetsky, S. S., and Senaskenko, V. S. (1972a). Phys. Lett. A 39, 103. Balashov, V. V., Lipovetsky, S. S., and Senaskenko, V. S. (1972b). Zh. Eksp. Teor. Fiz. 63,1622. Baumgartner, W. E., and Huber, W. K. (1976). J . Phys. E 9 , 321. Beaty, E. C., Hesselbachei, K . H., Hong, S. P., and Moore, J. H. (1977). J . Phys. B 10, 611. Branton, G. R., and Brion, C. E. (1974). J . Electron Spectrosc. Relat. Phenom. 3, 129. Brion, C. E. (1975). Radix. Res. 64,37. Brion, C. E., Hood, S. T., Hamnett, A., and Cook, J. (1977). Abstr. Pup., X ICPEAC 1977, p. 380. Camilloni, R., Giardini-Guidoni, A,, Tiribelii, R., and Stefan;, G. (1972). Phys. Rev. Lett. 29, 618. Cederbaum, L. S., Hohlneicher, G . , and von Niessen, W. (1973). Mol. Phys. 26, 1405. Coplan, M. A., Brooks, E. D., and Moore, J. H. (1977). Abstr. Pap.. X ICPEAC 1977, p. 378. Cvejanovic, S., and Read, F. H. (1974). J . Phys. B 7, 1841. Dey, S., McCarthy, I. E., Teubner, P. J. O., and Weigold, E. (1975). Phys. Rev. Lett. 34, 782. Dey, S., Dixon, A. J., McCarthy, I. E., and Weigold, E. (1976). J . Electron Spectrosc. Relut. Phenom. 9, 397. Dey, S., Dixon, A. J., Lassey, K. R., McCarthy, I. E., Teubner, P. J. O., Weigold, E., Bagus, P. S., and Viinikka, E. K. (1977). Phys. Reu. A 15, 102. Dixon, A. J., McCarthy, I. E., and Weigold, E. (1976). J . Phys. B9, L195. Dixon, A. J., McCarthy, I. E., Noble, C. J., and Weigold, E. (1978). Phys. Rev. A 17, 597. Ilixon, A. J., McCarthy, I. E., Weigold, E., and Williams, G. R. J. (1977a). J . Electron Spectrosc. Relut. Phenom. 12,239.

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Dixon. A. J.. Dey. S.. McCarthy. I. E., Weigold, E., and Williams, G. R. J. (1977b). Chem. Phys. 21,81. Dixon, A. J., Hood, S. T., and Weigold, E. (1977~).Abstr. Pap., X ICPEAC 1977, p. 384. Ehrhardt, H., Schulz, M., Tekaat, T., and Willmann, K. (1969). Phys. Rev. Lett. 22,89. Ehrhardt, H., Hesselbacher, K. H., Jung, K., and Willmann, K. (1971). Case Stud. At. Collision Phys. 2, 159. Ehrhardt, H., Hesselbacher, K. H., Jung, K., Schubert, E., and Willmann, K. (1974). J . Phys. B 7,69. Froese-Fischer, C. (1972). At. Data 4, 301. Furness, J. B., and McCarthy, I. E. (1973). J . Phys. B6,2280. Furness, J. B., and McCarthy, I. E. (1974). J. Phys. B 7 , 541. Fuss, I., McCarthy, I. E., Noble, C. J., and Weigold, E. (1978). Phys. Rev. A 17,604. Gelius, V. (1974). J. Electron Spectrosc. Relat. Phenom. 5,985. Geltman, S. (1974). J. Phys. B 7 , 1994. Geltman, S., and Hidalgo, M. B. (1974). J . Phys. B 7, 831. Giardini-Guidoni, A., Missoni, G., Camilloni, R., and Stefani, G. (1976). In “Electron and Photon Interactions with Atoms” (H. Kleinpoppen and M. R. C. McDowell, eds.), p. 149. Plenum, New York. Glassgold, A. E., and Ialongo, G. (1968). Phys. Rev. 175, 151. Hafner, H., Simpson, J. A,, and Kuyatt, C. E. (1968). Rev. Sci. Instrum. 39, 33. Hamnett, A,, Stoll, W., Branton, G., Brion, C. E., and Van der Wid, M. J. (1976). J . Phys. B 9,945. Hamnett, A,, Hood, S. T., and Brion, C. E. (1977). J. Electron Spectrosc. Relat. Phenom. 11, 263. Hood, S. T., McCarthy, I. E., Teubner, P. J. O., and Weigold, E. (1973). Phys. Rev. A 8,2494. Hood, S. T., Hamnett, A,, and Brion, C. E. (1976a). Chem. Phys. Lett. 39, 252. Hood, S. T., Hamnett, A,, and Brion, C. E. (1976b). Chem. Phys. Lett. 41, 428. Hood, S. T., Hamnett, A,, and Brion, C. E. (1977). J . Electron Speetrose. Relat. Phenom. 11, 205. Ikelaar, P., van der Wiel, M. J., and Tebra, W. (1971). J. Phys. E 4 , 102. Inokuti, M. (1971). Rev. Mod. Phys. 43,297. Jansen, R. H., deHeer, F. J., Luyken, K. J., and van Wingerden, B. (1975). J . Phys. B 9, 185. Joachain, C. J., and Vanderpoorten, R. (1970). Physica (Vtrecht)46, 333. Jung, K., Schubert, E., Paul, D. A. L., and Ebrhardt, H. (1975). J . Phys. B 8,1330. Kim, Y. K. (1972). Phys. Rev. A 6 , 666. Koshel, R. C. (1976). “Momentum Wave Functions-l976,” p. 227. AIP, New York. Lu, C. C., Carlson, T. A,, Malik, F. B., Tucker, T. C., and Neston, C. W., Jr. (1971). At. Data 3, 1. McCarthy, I. E., and Weigold, E. (1976). Phys. Rev. C 27, 275. McCarthy, I. E., Ugbabe, A., Weigold, E., and Teubner, P. J. 0. (1974). Phys. Rev. Lett. 33, 459. McCarthy, I. E., Noble, C. J., Phillips, B. A,, and Turnbull, A. D. (1977). Phys. Rev. A 15, 2173. McLean, A. D., Weiss, A,, and Yoshimine, S . (1960). Rev. Mod. Phys. 32, 211. Pickup, B. T., and Goscinski, 0. (1973). Mol. Phys. 26, 1013. Raith, W. (1976). Adv. At. Mol. Phys. 12, 281. Riley, M. E., and Truhlar, D. G. (1975). J . Chem. Phys. 63, 2182. Risley, J. S. (1972). Rev. Sci. Instrum. 43, 95. Rudge, M. R. H. (1968). Rev. Mod. Phys. 40,564. Snyder, L. C., and Basch, H. (1972). “Molecular Wave Functions and Properties.” Wiley, New York.

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Spears, D. P., Fischbeck, H. J., and Carlson, T. A . (1974). Phys. Rec. A 9, 1603. Stefani, G., Camilloni, R., Giardini-Guidoni, A,, Tiribelli, R., and Vinciguerra, D. (1977). Abstr. Pup., X ICPEAC 1977, p. 368. Stefani, G., Camilloni, R., and Giardini-Guidoni, A . (1978). PIzys. Leu. 64A, 364. Ugbabe, A., Weigold, E., and McCarthy, I. E. (1975). Phys. Rev. A 11, 576. Van der Wiel, M. J. (1973). At. Collision Processes Proc. Int. Con$ Phys. Electron. At. Collisions, 8th, 1973. Invited Lectures and Progress Reports, p. 417. Van der Wiel, M. J., and Brion, C. E. (1973). J . Electron Spectrosc. Relat. Phenom. 1, 309. Vriens, L. (1970). Physica (Vtrecht)47, 267. Wannberg, B., and Skollermo, A. (1977). J. Electron Speetrosc. 10, 45. Weigold, E. (1976a). In “Second International Conference on Inner Shell Ionization Phenomena” (W. Melhorn and R. Brenn, eds.), Invited Papers, p. 367. University of Freiburg. Weigold, E. (1976b). “Momentum Wave Functions-1976,” p. 84. AIP, New York. Weigold, E., Hood, S. T., and Teubner, P. J. 0. (1973). Phys. Rev. Lett. 30, 475. Weigold, E., Ugbabe, A,, and Teubner, P. J. 0. (1975a). Phys. Rev. Lett. 35,209. Weigold, E., Hood, S. T., and McCarthy, I. E. (1975b). Phys. Rev. A 11, 566. Weigold, E., Hood, S. T., Fuss, I., and Dixon, A. J. (1977a). J . PIzys. B 10, L623. Weigold, E., Dey, S., Dixon, A. J., McCarthy, 1. E., Lassey, K. R., and Teubner, P. J. 0. (1917b). J. Electron Speetrosc. Relat. Phenom. 10, 177. Weigold, E., McCarthy, 1. E., Dixon, A. J., and Dey, S. (1977~).Chem. Phys. Lett. 47, 209. Weigold, E., Noble, C. J., Hood, S. T., and Fuss, I. (1978). To be published. Wellenstein, H., Schmorantzen, H., Bonham, R. A,, Wong, T. C . , and Lee, J . S. (1975). Rel;. Sci. Instrum. 46, 92. Wight, G. R.,and Van der Wiel, M. J. (1977). J. Phys. B 10, 601. Wilden, D. G., Hicks, P. J., and Comer, J. (1977). Abstr. Pup., ICPEACConf:,IOth, 1977, p. 679. Wuilleumier, F., and Krause, M. 0. (1974). Phys. Rev. A 10, 242.

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I1

ADVANCES IN ATOMIC A N D MOLECULAR PHYSICS, VOL.

14

FORBIDDEN TRANSITIONS IN ONE- A N D TWO-ELECTRON ATOMS* RICHARD M A R R U S and PETER J . M O H R Materials and Molecular Research Diuision Lawrence Berkeley Laboratory Berkeley, Calqornia

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Preliminary Survey. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........ A. Forbidden Transitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Selection Rules. . . . . . . . . . . . . . . . .................. C. Multipole Transition Rates . . . . . .................. D. Forbidden Magnetic Dipole Transitions . . . . . . . . . . . E. Two-Photon Transitions ......................... F. Spin-Forbidden Transitions ...................... G . Nuclear-Spin-Induced Decays ...... ................... H. Electric-Field-Induced Transitions . . . . . . . . . ...................... I. Transitions in One- and Two-Electron Atoms. . . . . . . ........ 111. Magnetic Dipole Decay. . . . . . ............................. A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Initial Observations of the MI Decay 23S1 4 1' S o . . . . . . . . . . . . . . . . . . C. Astrophysical Significance . . . . . . . . ... ....... ... D. Laboratory Observations . . . . . . . . . ........................ E. Theoretical Studies of the MI Decay 23S, + I'S, . . . . . . . . . . . . . . . . . . IV. Magnetic Quadrupole Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Experiment.. ..................... V. Two-Photon Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. General Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . B. Two-Photon Decay of the 2S,,, State of One-Electron A C. Two-Photon Decay of the 2'S, State of Two-Elect D. Two-Photon Decay of the Z3S, State of Two-Electron Atoms . . . . . . . . E. Experiments on One-Electron Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. Experiments on Two-Electron Atoms. . . . . . . . . . . . VI. Intercombination Transitions ...................... A. General Considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

182 183

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186 187 188 188 188 189 189 192 194

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* This work was supported by the Division of Basic Energy Sciences, U.S. Department of Energy. 181 Copyright @ 1978 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-003x14-5

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Richard Marrus and Peter J . Mohr

B. Theory.. ............................. .................... C. Experiment.. . . . . ................................... VII. Nuclear-Spin-Induced ............................. A. Early Observations ....................................... B. Observation in Two-Electron Vanadium .................... VIII. Electric-Field-Induced Decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Metastability of the 2S,,, State in an Electric Field . . . . . . . . . . . . . . . . . B. Electric-Field-Perturbed Lifetime Measurement of the Lamb Shift . . . . C. Lifetime of the 21S0 State in an Electric Field . . . . . . . . . . . . . . . . . . . . . . D. Polarization of the Induced Radiation E. Angular Distribution of the Induced R References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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220

I. Introduction The notion of forbidden decays has its roots in the Ritz combination principle and the idea that the measured frequencies of all observed spectral lines can be expressed as the difference of two term frequencies. The fact that for any system not all the possible differences of the known terms are observed initially led to a separation of the possible radiative transitions into allowed and forbidden types. The well-known selection rules for electromagnetic decay, which provide a time-tested basis for this separation, predate quantum mechanics and were originally based on classical electromagnetic theory applied to the vector model of the atom. It is now possible to make a more refined distinction between the allowed and forbidden decays, and in the next section, we consider the appropriate selection rules in some detail. Because of the low transition rates associated with forbidden radiative decay, such radiation is generally observed only in environments of ultralow density, where collisional rates are comparably low. It is not surprising, therefore, that the first observations of forbidden transitions were not made in the laboratory, but rather in the study of the planetary nebula of Orion. These observations were made in the second half of the nineteenth century by the British astronomer Sir William Huggins. It is of some interest to note that for over 50 years the observed forbidden lines were thought to arise from transitions in elements (e.g., nebulium and coronium) not found on earth. As late as 1928 the concept of forbidden transitions in astronomical bodies was first introduced by Bowen (1928), who pointed out that these and other nebular lines could be ascribed to transitions in NII, 011, and 0111. Since Bowen’s work, numerous experimental observations of forbidden transitions in astrophysical sources have led to an extensive literature on the subject (Bowen, 1936; Borisoglebskii, 1958; Garstang, 1962a; Layzer and Garstang, 1968).

FORBIDDEN TRANSITIONS

183

Until about 15 years ago, there existed almost no laboratory observations of the forbidden transitions. More recently, activity in this area has been stimulated by two developments. The first of these is the realization that along an isoelectronic sequence, there is a very rapid increase with the atomic number 2 of the rate associated with a forbidden transition. For example, the rate associated with the A41 transition 23S1+ llSo in the two-electron system ranges from about 10-4secp' in helium to roughly 10'Osec-' in two-electron krypton. Hence, the need for a collision-free environment becomes much less critical at high Z. The second development is the proliferation of devices capable of producing highly ionized heavy atoms. Examples of such devices are the low-inductance vacuum spark (Feldman and Doschek, 1977),high-power lasers (Fawcett et al., 1967),plasma sources (Sawyer et al., 1963), and heavy-ion accelerators (Bashkin, 1976). The beamfoil technique has been of particular value in these studies, since transition rates can easily be measured by time-of-flight (tof) techniques. It is the purpose of this chapter to review recent theoretical and experimental work on forbidden transitions in the one- and two-electron systems. In Section I1 we consider the characterization of the forbidden transitions, and in subsequent sections we deal with each of the known transition types.

11. Preliminary Survey A. FORBIDDEN TRANSITIONS The most rapid mode by which an atom in an excited state can radiatively decay is by a fully allowed electric dipole transition. However, if a certain transition is prohibited by the selection rules for electric dipole decay, it is termed forbidden and may proceed by a higher multipole mode or by a multiphoton mode. Another type of transition that we shall also classify here as forbidden is an electric dipole transition that occurs only when the atom is influenced by a weak perturbation such as spin-orbit interaction, an applied electric field, or a nuclear magnetic moment.

B. SELECTION RULES The approximately forbidden transitions described above are, of course, to be distinguished from absolutely forbidden transitions, which are not possible at all because of general symmetry principles. The associated selection rules for electromagnetic radiation are well known, and we shall only state them here. For a spherically symmetric nuclear field, the rotational

184

Richard Marrus and Peter J . Mohr

invariance of the interaction between electrons and the electromagnetic field requires that the angular momentum of the initial atomic state J , the final atomic state J’, and the emitted photon L satisfy a triangle inequality, i.e.,

IJ

< L < J + J’

- J’I

(14

and that the z components M j , M J r ,and M , satisfy the relation

Mj

=Mj,

+ ML

(1b)

in order for the transition to be possible. The angular momentum of the photon may take the values L= 1,2,3,. . . , and the radiation field is either electric multipole E L or magnetic multipole M L in character, where L = 1 is dipole, L = 2 is quadrupole, L = 3 is octupole, etc. Another absolute selection rule on radiative transitions is imposed by the invariance of the nuclear field and the interaction between the electrons and the radiation field under parity inversion. There may be exceptions to this rule as a result of parity-violating weak neutral-current interactions, but none has been found at this time despite active searches that are still in progress. Parity invariance requires that a transition is forbidden unless the parity of the initial state is the same as the product of the parities of the final state and the multipole radiation field. The parities of the multipole fields are given by P(EL) = (- 1), and P ( M L ) = - P(EL).

C. MULTIPOLE TRANSITION UTES The problem of writing the electromagnetic field as an expansion in multipole moments is treated in many texts, and we refer the reader to Blatt and Weisskopf (1952) for a detailed discussion. In this section, we consider the order of magnitude and Z dependence of the corresponding transition rates. These can be obtained by simple semiclassical arguments such as those given by Jackson (1975). For electric multipole transitions of order L, the order of magnitude of the transition rate is given by

-

(24 where M. z 1/137 is the fine-structure constant, o and k are the (angular) frequency and (angular) wavenumber (w/c) of the emitted radiation, and a is a length that characterizes the size of the atom. For magnetic dipole transitions (rn is the electron mass) AEL

cro(ka)2L

FORBIDDEN TRANSITIONS

185

For transitions with An # 0 in few electron atoms, the parameters scale as follows: w ( Z c ~ ) ~ r n ca~ / h(Za)-'h/(rnc); , hence

-

-

-

AEL a(Zcr)2L+2mc2/h, A,,

- a(Za)2L+4mc2/h

(3)

From these estimates, it is clear that the higher multipole transitions are strongly inhibited because the wavelength of the radiation is long compared to the size of the atom. However, the forbiddenness becomes less severe for large Z, which makes high-2, few-electron atoms suitable systems for studying the higher multipole transitions.

D. FORBIDDEN MAGNETIC DIPOLE TRANSITIONS The forbidden magnetic dipole transition is a unique exception to the simple scaling estimates. As suggested by Eq. (2b), the lowest order rate is independent of the electron position operators. In particular, in the nonrelativistic limit, i.e., to lowest order in Za, the magnetic dipole rate is given by

where L and S are the total angular momentum and spin operators for the atom. Also, in this limit, L-S coupling may be assumed, in which case the wave function is written as a sum over products of coordinate space wave functions with fixed L and spin functions with fixed S. The operators L and S in (4) act only on the M , and M s indices, and so the matrix element in (4) will consist of a sum over products of the form: (scalar product of coordinatespace wave functions) x (scalar product of spin functions). Such products vanish unless An = 0, AL = 0, and AS = 0, where n is the principal quantum number, because of the orthogonality of the space and spin functions. When relativistic corrections are taken into account, Eq. (4) is no longer applicable and the selection rule may be violated by terms of order ( Z U )in~the ~ matrix element. We thus have A,,

E.

TWO-PHOTON

-

a(Zcr)"mc2/h

for An # 0

TRANSITIONS

Two-photon transitions, which are second order in the coupling of the radiation field to the electron, are inhibited for all Z because of the weakness of the coupling. The order of magnitude for two allowed electric dipole photon decay in few-electron ions (which is suggested by the simple picture

186

Richard Marrus and Peter J . Mohr

of two electric dipole rates divided by an energy denominator of the order of energy differences) is A,,,

-

cr2(Za)6rnc2/h

(6)

The selection rules for one-photon decays apply to two-photon decays if the angular momentum and parity of the single photon are replaced by the quantum mechanical vector sum of the angular momenta of the two photons and the product of parities.

F. SPIN-FORBIDDEN TRANSITIONS In the nonrelativistic limit, L-S coupling is exact and imposes an additional selection rule on electric dipole decays. In this limit, the coupling of electric dipole radiation to the electrons is independent of the electron spin and such decays occur only if AS = 0. In this case, the electric dipole selection rule for the total angular momentum J now applies to the orbital angular momentum L. This additional restriction may be violated to the extent that spin-orbit and other relativistic corrections mix states of the same J and different spin S. An important example is the 23P, state in heliumlike atoms. Electric dipole decay to the l'S, state is strongly inhibited at low 2, but at high Z the spin-orbit interaction matrix element between the 2'P, and 23P, states is comparable to the Coulomb energy difference between the states, and the decay 23P, + l'So proceeds with a rate comparable to an allowed electric dipole rate.

G. NUCLEAR-SPIN-INDUCED DECAYS If the nucleus has a nonzero spin I , the nuclear field is not spherically symmetric and the selection rules on J do not apply. However, the combined system of the nucleus and the atomic electrons is rotationally invariant and the selection rules on angular momentum apply to the total angular momentum F = I + J . On the other hand, J is approximately a good quantum number to the extent that the hyperfine splittings are small compared to the multiplet splittings, so that electric dipole transitions that are forbidden by selection rules on J , but allowed by selection rules on F are only weakly allowed. H. ELECTRIC-FIELD-INDUCED TRANSITIONS In an applied electric field, both the rotational and parity invariance of the atom are absent. However, the effects of the electric field are in general

187

FORBIDDEN TRANSITIONS

n=2

2 s1/2

ALz10

3

2

4

MHz

FIG. 1, Energy level diagram of the one-electron atom showing the forbidden decay modes.

weak, and electric dipole transitions that violate the field-free selection rules are only weakly allowed.

I. TRANSITIONS IN O N E -

AND

TWO-ELECTRON ATOMS

Figures 1 and 2 show energy level diagrams for the one- and two-electron atoms. Indicated also are the forbidden transitions that have been experimentally studied. In subsequent sections, we consider recent work on these transitions in more detail.

FIG.2. Energy level diagram of the two-electron atom showing the forbidden decay modes

188

Richard Marrus and Peter J . Mohr

111. Magnetic Dipole Decay A. INTRODUCTION There is one nonrelativistic magnetic dipole ,transition (see Section II,D) that has been observed in emission in one- and two-electron atoms. This is the transition between hyperfine levels of the ground state of hydrogen and deuterium observed in the interstellar gas. The most precise measurements of this line have been made with the hydrogen maser by studying the stimulated emission (Ramsey, 1969). As yet, there has been no observation of a forbidden magnetic dipole transition in a one-electron atom. However, for the one-electron case, the theoretical transition rate 2S1,, + lS,,, is accurately known. To lowest order in Zcc, the M1 rate is

AMl z (1/972)a(Za)'ornc2/h

(7)

W. Johnson (1972) has evaluated A,, for Z in the range 1-92 to all orders in Za.At Z = 92, the complete calculation gives a rate larger than the lowest order expression (7) by about a factor of 2. In the following sections, we discuss the forbidden magnetic dipole transition 23S1-,l'So in heliumlike atoms.

B. INITIALOBSERVATIONS OF THE M1 DECAY 23S1

---f

l'So

The first observations of the two-electron radiative decay Z3S1 l'So were made with X-ray spectrographs mounted in earth-orbiting satellites and rockets to study radiation from highly ionized atoms in the solar corona (Fritz et al., 1967; Rugge and Walker, 1968; Jones et a/., 1968, and references therein). A typical spectrum obtained this way is shown in Fig. 3. At the time, ---f

S K \ Y S FKO\I I t l E S L S

8 (deq)

FIG. 3. X-ray spectrum of the solar corona showing the spectra of heavy helium-like atoms.

189

FORBIDDEN TRANSITIONS

the observed M1 lines were not identified, since it was generally expected that the z3S, state decayed primarily by two-photon emission. Gabriel and Jordan (1969a) first pointed out that the unidentified lines in these spectra corresponded to the single-photon decay 23S1-+ l'So, thus prompting many detailed calculations of this process. They proposed that the M1 decay was being observed and that the rate was somewhat larger than earlier estimates by Breit and Teller (1940). Griem (1969, 1970) reestimated the magnitude of the A41 decay rate (see also Gabriel and Jordan, 1970) and showed that the M1 decay would be the dominant mode of decay for the 23S, heliumlike ions, confirming the suggestion of Gabriel and Jordan.

C. ASTROPHYSICAL SIGNIFICANCE Gabriel and Jordan (196913) noted that the forbidden line 23S1-+ l'So has substantial significance for the diagnostics of the low-density plasma associated with the solar corona. Using an appropriate model, they showed that the observed intensity ratio R = z(23S1-+ liS0)/1(23P1+ llSO)can be related to the electron density N , and a number of calculable atomic parameters. Specifically,

R=

(=9

A ( ~ ~+ s , I%,) [ N , c ( ~+~2 s3 ~ ) 4](1 F ) A ( ~ ~ S iiso) , B

where F ratio :

+ +

+

=

-+

+

C(l'SO+ 23S1)/C(11S0+ 23P) and B is an effective branching

A ( ~ ~-+ P 1'S0) , 3 A ( ~ ~+ P iiso) , + A ( ~ +~ 23s) P

B E -1

5 94

+-

2

4 2 3 ~ PS,) ~ 3+ ~ ~~ i s+~ ~ )( 2 +323s) ~

(8b)

4 is the photoexcitation rate from 23Sto 23PandC is the collisional excitation rate for the indicated process. Densities have been derived in this way for some solar active regions with values ranging up to 1 O I 3 cm- (Freeman et al., 1971). We note that in the higher density regime appropriate to most laboratory plasmas, this method is not useful, since collisions completely suppress the radiation from the 23S state. D. LABORATORY OBSERVATIONS The first laboratory observation of the decay 23Si + l'S, was made in a beam-foil experiment by Marrus and Schmieder (1970a). The 23S1 state of the two-electron atom was excited by passing the 10.2 MeV/nucleon argon,

190

Richard Marrus and Peter J . Mohr

sulfur, and silicon beams of the old Berkeley HILAC (heavy ion linear accelerator) through a carbon foil. The low-energy X-rays arising from the decay in flight of ions in this state were observed with high-resolution Si(Li)detectors. Identification of the transition was based mainly on the good agreement between the observed and predicted X-ray energies and also on the long lifetimes observed. Subsequent observations on other heliumlike atoms thru Z = 36 (krypton) have been made by the beam-foil method (See Table I). Observation of this decay in the afterglow of helium gas excited by an electrical discharge has also been reported (Moos and Woodward, 1973; Woodward and Moos, 1975). Here the transition was observed with the use of a UV monochromator in a situation where the radiative lifetime of the state is about lo4 sec. Substantial experimental effort has been devoted to attempts to confirm the predicted lifetime of this state. In the leading-order hydrogenic approximation, ~ ( 2 ~ scales s ~ as ) Z-" for heliumlike atoms. Hence, at sufficiently high Z , this lifetime is tractable to beam-foil tof measurements, even though it is about lo4 sec in ordinary helium. Early reported measurements in ArI6+ and CIl5+ (Schmieder and Marrus, 1970a; Cocke et al., 1973) seemed to indicate a discrepancy between the measured and theoretical values. HowTABLE I THEORETICAL A N D EXPERIMENTAL LIFETIMES FOR THE DECAY 23S, + 1's, IN TWO-ELECTRON ATOMS A,,(23S, + 1'S0)

2 16 17 18 22 23 26 36

7.86 x 1 0 ' " 699b 374 208 26.6 16.9 4.80 0.171

(9 & 3) x l o l l '

706 86d 354 & 24d 202 20" 25.8 1.3/ 16.9 i 0.7g 4.8 0.6g 0.20 & 0.06'

*

+

Drake (1971a). Johnson and Lin (1977). Woodworth and Moos (1975). Bednar et a / . (1975). '' Could and Marrus (19764. Could et a / . (1973). Could et a/. (1974).

"

'

191

FORBIDDEN TRANSITIONS

ever, measurements on the higher-2 ions Ti2'+, V2'+, Fe24+ and Kr34+ (Gould et al., 1973, 1974; Gould and Marrus, 1976a)yielded results that were in good agreement with theory. A remeasurement of the Cl"+ lifetime was undertaken by Bednar et al. (1975) in order to clarify the discrepancy at lower Z . By extending the earlier measurements to large foil-detector separation they found an apparent nonexponential character to the decay, and showed that at large foil-detector separations, the measured lifetime agrees with the theoretical lifetime (see Fig. 4). Measurements made on a beam of S14+ were also in accord with this result. Gould and Marrus (1976a) repeated their earlier measurement on Ar16+ and also noted agreement between the measured and theoretical lifetimes at the large foil-detector separations. With these values for z(2jSl) the beam-foil measurements are consistent with the calculations over the range from Z = 16 through Z = 36.

z3s, LIFETIME MEASUREMENTS , ,

1.4-,

1

,

0

1

1.2-

e 10.60.4-

0.2

*-

-

0 0

'

'

02

1

'

I

0.4

I

0.6

0

1

0.8

0

s 1 0

-

1.2

T/T,,, FIG.4. Apparent lifetime vs. foil-detector separation for beam-foil experiments on the 23S, lifetime.

The reason for the nonexponential character of the decay curve is not yet understood. However, Lin and Armstrong (1977) suggest that a small threeelectron component present in the beam can reproduce the results of Bednar et al. (1975). The M1 decay rate in ordinary helium was measured by Moos and Woodworth (1973); Woodworth and Moos (1975). Because of the long lifetime, tof measurement is not possible. However, they were able to obtain a value for the decay rate based on the observed intensity of the radiation and an independent determination of the density of metastable atoms. This result is in agreement with theory. The theoretical decay rate A(23S,) is thus verified over a range < A(23S1)< lo+" sec-', i.e., over 14 orders of magnitude. In this sense, the

192

Richard Marrus and Peter J . Mohr

theory of this decay is one of the best verified theories in all of atomic physics. A comparison of theory and experiment is shown in Table I.

E. THEORETICAL STUDIES OF

THE

M1 DECAY23S, + 1'So

In a study of metastable states of hydrogen and helium, Breit and Teller (1940) briefly considered the decays of the 23S, state. They pointed out that the two-photon decay to the l1S0 state would be substantially smaller than the corresponding two-photon decay 2S1,, -+ lS,,, of hydrogen because the helium decay requires a spin change that greatly reduces the rate. They also estimated the M1 decay rate of the 23S, state, but only considered the effect of additional configurations such as pp' and dd', which are present due to relativistic corrections to the wave functions. Such a model for the wave function together with the nonrelativistic transition operator predicts a rate that is many orders of magnitude smaller than the two-photon rate. The estimate of Griem (1969, 1970),mentioned above, was based on a relativistic calculation of the M 1 decay rate with the heliumlike rate approximated by the Dirac hydrogenic rate corrected by a statistical factor 2/3, with a reduced effective nuclear charge. The result is an M 1 decay rate that is much larger than the two-photon decay rate and consistent with the Gabriel and Jordan interpretation of the solar spectra. Stimulated by the solar observations and laboratory measurements, more refined calculations of the 23S1lifetime were soon carried out by C. Schwartz (unpublished work, 1970), Drake (1971a, 1972a), and Feinberg and Sucher (1971); see also Beigman and Safronova (1971). Since the MI decay rate for 23S, -+ l'So vanishes in the nonrelativistic limit, i.e., to lowest order in Za, these calculations were all aimed at obtaining the next nonvanishing term in the expansion of the rate in powers of Za.Schwartz's calculation derives the leading relativistic corrections to the Hamiltonian, for an atom interacting with a time-dependent external field to obtain an effective magnetic dipole operator. Drake applies the Foldy-Wouthuysen reduction to the relativistic Breit equation in a treatment similar to that of Schwartz. The most detailed treatment is that of Feinberg and Sucher, who begin with a two-particle relativistic Hamiltonian with the nuclear and interelectron Coulomb potential terms between positive energy projection operators. Additional contributions due to time-ordered terms involving the creation of virtual electron-positron pairs are then calculated as perturbations. The results of these calculations are in complete agreement and give, somewhat surprisingly, an expression with an effective operator between nonrelativistic wave functions for the matrix element. (One might expect a more complicated result involving sums over intermediate states.) The decay

FORBIDDEN TRANSITIONS

193

rate is given by

where ho = Ei - E , and the functions 4iand +f are the spatial portions of the nonrelativistic heliumlike wave functions. To evaluate A M l, it is still necessary to evaluate M numerically as a function of 2. This was done by Drake (197la), who applied a variational perturbation scheme that produces successive approximations to the wave function as a power series in Z - ' . Drake carried out the calculation including terms to relative order Z - ', which is sufficient to evaluate M at Z = 2, or any higher value. Feinberg and Sucher (1971) evaluated the matrix element for helium ( Z = 2) with a sixparameter Hylleraas-type wave function. Independent exact calculations of the leading Z - correction to the matrix element were carried out by Kelsey and Sucher (1975) and by P. J. Mohr (unpublished work, 1974). Both of these calculations were based on explicit expressions for the nonrelativistic Coulomb Green's function. The results are in exact agreement with each other and in agreement with Drake's variational result. An independent check of the accuracy of variational calculations of the leading term in Za has been made by Anderson and Weinhold (1975), who have obtained theoretical error bounds on the value of the relevant matrix element. Their results confirm Drake's evaluation of the leading order matrix element at Z = 18, giving an error bound of about 0.24% to the variational calculation. For large Z , higher order relativistic corrections to the decay rate should be considered, i.e., terms of order ( Z C Crelative )~ to the leading nonvanishing term. The relativistic correction amounts to about 7% at Z = 36. Calculations that include higher order relativistic effects have been carried out by Feneuille and Koenig (1972) and by Johnson and Lin (1974). The most accurate calculation at high Z has been done by Johnson and Lin (1976); see also Johnson et al. (1976); Lin et al. (1977), who applied a relativistic version of the random-phase approximation to calculate the rate for 23s1 -+ 1 1 s ~ . It is useful to summarize the theoretical results for this transition rate in terms of an explicit formula that is valid over a wide range of Z and that contains the leading terms in an expansion in Z-' and ( Z C (which ) ~ have a simple physical interpretation. Such a formula is given by

194

Richard Marrus and Peter J. Mohr

where the transition wave number k is approximated by

This formula reproduces the results of the detailed numerical calculations to within 1%over the range Z = 10to 50. The 2- term in the square brackets is the electron-electron interaction correction to the lowest order (in Za) matrix element and is discussed above. The (Za)' term is the relativistic correction in the Dirac hydrogenic approximation. This term has been evaluated by C . Schwartz (unpublished work, 1970),Lin (1975),and Johnson (quoted in J. Sucher, 1977). Additional corrections to the decay rate have been considered and shown to be small. Radiative corrections, involving one virtual photon, vanish for 2S,,, -+ lS,,, transitions in hydrogenlike ions to lowest relative order in Za (Lin and Feinberg, 1974; Drake, 1974). For the heliumlike atom case, this result applies to lowest order in Z-'. Kelsey (1976) calculated the order (Za)',~ ( Z Uand ) ~ a(Za)" , ln(Za)-' energy denominator and recoil corrections to the diagrams discussed by Feinberg and Sucher (1971) for the lowest order (Za)"transition amplitude. He found that only a(Za)" corrections arise in individual diagrams and that these cancel when summed.

'

IV. Magnetic Quadrupole Transitions A. THEORY

Mizushima (1964) first called attention to the possibility that magnetic quadrupole ( M 2 )radiation might be of astrophysical significance for atomic transitions that satisfy the selection rule AS = 1. The starting point for the calculations is the (nonrelativistic) result for the magnetic quadrupole transition probability:

where for N electrons

and

FORBIDDEN TRANSITIONS

195

Application of the formalism was made by Mizushima to the transition 'Xi + 'Xi of the hydrogen molecule. This work showed that M2 transitions compete with spin-orbit-induced electric dipole transitions for certain intercombination lines. Approximate probabilities for M2 transitions for many isoelectronic sequences (including helium) and molecules were calculated in subsequent work by Mizushima (1966). Garstang (1967) gave a tensor operator formalism for the calculation of magnetic quadrupole transition probabilities, and applied the method to the s2 'S0-sp 3P2 transition in general, and to eight specific cases (HeI, OVII, BeI, SiXI, MgI, ZnI, CdI, and HgI). He showed that in the first six of these cases, magnetic quadrupole radiation is more important than nuclear-spin-induced electric dipole radiation, while the converse is true for CdI and HgI. These calculations were extended by Garstang (1969) to spectral lines of interest in the deexcitation of atoms in the solar corona. In particular, he showed that three M2 lines, in Fe IX, Fe XVII, and Fe XXV, play an important role. He further noted that the magnetic quadrupole rate for decay of the 23P2 state to the l'So state exceeds the electric dipole rate to the 23S1 state for two-electron ions beyond heliumlike sulfur (Z = 16). This is a remarkable, and possibly unique, circumstance in atomic physics, where decay of a state by a magnetic quadrupole mode dominates over decay of the same state by a fully allowed electric dipole mode. This can be made plausible by considering the approximate Z dependence of the rates. Note that

+

A,,(~'P, + 2 ' ~ cc ~ )W : , ~ ( ~ ~ S , I Z~,12'P,)1~ ~

(134

scales approximately as Z, whereas

scales approximately as Z'. Both Garstang (1969) and Drake (1969) conclude that the rates become comparable near Z = 17. The calculations of Drake (1969, 1971b) of the two-electron M 2 rates are based on accurate nonrelativistic variational wavefunctions. He also shows that the Z-expansion results of Dalgarno and Parkinson (1967) for 'P, + 'So electric dipole transitions are easily modified to obtain the Z expansion of the corresponding 3P2+ 'So magnetic quadrupole transition and gives the first Z-expansion correction for the n3P, + 1'So transitions with n = 2 to 20. Similar nonrelativistic calculations were done by Jacobs (1972), who gives results for the n3P, + l'S, magnetic quadrupole decay rates, with n = 2 to 5, for members of the helium isoelectronic sequence up to Z = 10.

196

Richard Marrus and Peter J . Mohr

In addition, he compares the relative importance of the M2 and E l rates for the decays of the n3P, states. Corrections due to relativity were first pointed out by Could et al. (1974), who gave the leading relativistic correction to the matrix element. Based on this correction and the coefficient of the Z-' term in the Z expansion determined by Drake (l969,1971b), the M2 rate may be expressed as

[

lo(za)-2 1 + 0.245 - 0,64O(Z.)"]2 Z ~

(14a)

where the transition wavenumber k is approximated by 3 me k =-(Zr~)~--[l 8 h

-

1.06/2

+ 0.313(Zr~)~]

A different treatment based on the relativistic random-phase approximation (RRPA) has been given by Johnson and Lin (1976).Precise numerical solutions of the RRPA equations were obtained describing the 23P2-+ l'So transition, as well as other transitions. The calculated excitation energies and transition probabilities are in good agreement with accurate nonrelativistic calculations for low-Z elements. For intermediate- and high-Z elements, where relativistic effects are more important, the results should also be very accurate. Extensive comparison shows good agreement of the calculated forbidden transition rates with available beam-foil measurements, and the calculated transition energies show good agreement with several lines from the solar corona for high-Z (Z z 25) elements. We note that the M2 rates obtained by Johnson and Lin are in substantial agreement with the results of Eqs. (14a,b). Theory and experiment are compared in Table 11. Magnetic quadrupole decays play a role in the decays of three-electron ions in the metastable ( l s 2 ~ 2 p ) ~ P state , ~ , to the ground ( 1 ~ ~ 2 s ) ~state. S,~, The excited state is depopulated primarily by autoionization to the (ls2)'So + k2Fsi2state, where the ejected electron is in a continuum 2Fs,2 state. This is a rare situation, where a forbidden radiative decay competes with an autoionization process, since autoionization is generally more rapid than allowed E l decay. The radiative decay of the 4P5,2 state to the ( 1 ~ ~ 2state ~ )may ~ proceed s ~ ~ by ~ an M2 transition because AJ = 2 and the relative parity of the atomic states is odd. In the autoionization channel, the only energetically allowed final state is a two-electron atom in the (ls2)11Sostate plus an electron in a continuum state. Since the autoionization proceeds by internal electromagnetic processes, J and parity are conserved and the continuum electron state must have 'F5/2 character. Hence, in the nonrelativistic (L-S) coupling scheme, in which L and S are conserved, the transition is forbidden because AL = 2, AS = 1. Furthermore, spin-orbit

FORBIDDEN TRANSITIONS

I97

TABLE I1

THEORETICAL AND EXPERIMENTAL MAGNETIC QUADRUPOLE TRANSITION RATESFOR HELIUMLIKE IONSO

16 17 18 22 23 26

1.19(8)*.' I .96(8) 3.16(8) 1.66(9)b 2.39(9) 6.55(9)br'

1.7 k 0.3(8)d 2.7 k 0.3(8)' 2.3 f l(8)' 1.6 & 0.2(9)b 2.5 k 0.4(9)b 7.5 k 2(9)b

a Numbers in parentheses denote power of ten multiplying entry. Gould et al. (1974). Johnson and Lin (1976). Cocke et al. (1974b). Cocke et al. (1974a). Marrus and Schmieder (1972).

and spin-other-orbit perturbations are of vector character with respect to L and will not cause autoionization because AL = 2. As pointed out by Kroll(1961), the necessary changes in S and L may be induced by the tensor term of the spin-spin interaction, which thus causes the autoionization process. Hence, the low transition rate is associated with the weakness of the spin-spin interaction. For Z = 10 to 18, the branching ratio for M2 decays ranges from 2% to about 20%. The point at which the M2 mode begins to be larger than the autoionization mode occurs roughly in the range Z = 24 to 27. The rapid relative increase of M2 decays is due to the rapid 2 dependence ( Z 8 )of the M 2 decay rate. Cheng et al. (1974) have done Dirac-Hartree-Fock calculations of both the M2 radiative decay rates and the autoionization rates of the lithium sequence for Z = 3 to 26. In terms of G and F , the large and small components of the DHF radial wave functions, the M2 decay rate is

-

where

Richard Marrus and Peter J. Mohr

198

and k is the transition wave number. The M2 rates found by Cheng et a / . (1974) scale roughly as ( Z - 0.7)*. A nonrelativistic Z-expansion calculation of the three-electron M2 rate was done by Onello and Ford (1975). They evaluated the matrix element with L-S coupled configuration interaction wave functions with 41 configurations for the excited state. Their results differ from those of Cheng et al. (1974) by about 7% at Z = 18, but that difference is not significant for comparison to experiment since the M2 branching ratio at Z = 18 is only about 20%. See Cheng et al. (1974) and Haselton et al. (1975) for a comparison of theory and experiment.

B. EXPERIMENT The first observation in the laboratory of a magnetic quadrupole transition was made by Foote et al. (1925), who observed the 3040A line corresponding to the transition 'So -+ 3P2 in neutral zinc. The magnesium I transition at 4562.48 A was observed by Bowen (1960) in the planetary nebula NGC7027. This is probably the first and only observation of a magnetic quadrupole transition in an astronomical source. So far, all observations of M2 radiation from the 23P2 state of ions in the helium sequence have been made on ions in a beam excited by the beam-foil method. The first observation of this radiation was made by Marrus and Schmieder (1970b) on a beam of Ar16+, who also made a measurement of the lifetime of the state. Their result corresponds to a total transition rate of A(23P2) = (5.9 k 1.0) x 10' sec-'. The calculated transition rates for this state are AE1(23Pz)= 3.5 x lo8sec-' and A,,(23P,) = 3.2 x 10'sec-'. Hence the experimental result is consistent with a large M2 branching ratio and in rough agreement with the absolute rate. Subsequent measurements, particularly by Cocke et al. (1974a,b) and Gould et al. (1974), have extended measurements on the lI3P, state to S14+, C1151 Ti20+ V21+ , and FeZ4+. All of these measurements are based on observation of the X-ray transition z3P2+ l'SO. A sample spectrum is shown in Fig. 5 . Note that the decay 23P2 ]'So is not resolved by the Si(Li) detector and the presence of this transition must be inferred from the compound nature of the decay curve (Fig. 2). It is also possible to infer the presence of a large M2 branching from recent measurements by Davis and Marrus (1977) on the E l transition 23P2 23S,. This transition occurs in the ultraviolet and the lifetime has been obtained from a tof experiment using a high-resolution UV monochromator. Experimental results on the M2 rate are given in Table 11. Note that the M2 rate is obtained by subtracting the theoretical E l rate from the measured rate. 2

-+

--f

199

FORBIDDEN TRANSITIONS I

I

I M2,MI

v)

$1 I-

17.8 cm

29

NOISE

25.4 cm

un

-35.6 I

0

1

I

I I

3

4

5

I

I

I

1

2

ENERGY (keV)

FIG.5. Observed X-rays from a foil-excited beam of heliumlike argon. Three forbidden decay modes are simultaneously in evidence, with no evidence of allowed transitions. The M 2 and M1 peaks are not resolved.

V. Two-Photon Decay A. GENERAL PROPERTIES The possibility of an excited atomic state decaying by the simultaneous emission of two photons was first suggested by Maria Goeppert-Mayer (1929,1931). The process is second order in the interaction of the atom with the radiation field and proceeds through intermediate excited atomic states. The nonrelativistic expression for the transition rate from state i to state f with emission of one of the photons in the frequency range w to o dw is

+

Here o1and o2are the angular frequencies of photon 1 and photon 2, respectively; hwni= En - E i , and and E~ are polarization vectors associated with the photons. The following relevant physical properties of the two-photon process can be deduced from Eq. (16).

(1) Since the intermediate states are virtual rather than real states, the two photons are emitted in coincidence. (2) Energy conservation requires that h ( o , + w z )= Ei - E,. The photons have a continuous frequency distribution that is symmetric about the midpoint.

Richard Marrus and Peter J . Mohr

200

(3) For the decay S + S, the unpolarized transition probability predicts an angular correlation proportional to 1 + cos2 6, where 6 is the angle between the photon directions. (4) The total transition rate AZEl is obtained by integrating (16) over the frequency of the emitted photons:

where A ( o ) is averaged over initial-state spins and summed over final-state spins and photon polarizations. B. TWO-PHOTON DECAYOF

THE

2Sli2STATEIN ONE-ELECTRON ATOMS

Breit and Teller (1940) evaluated the two-photon transition rate for the decay 2S1,, lS,,, in hydrogen by directly summing over the discrete intermediate states and integrating over the continuum states, and obtained the estimate 6.5 < A < 8.7sec-'. A somewhat more accurate evaluation of Eq. (16) was made by Spitzer and Greenstein (1951), who also gave a point-by-point table of values for the frequent spectrum A ( o )of the emitted photons. The result for the transition rate was A = 8 . 2 3 ~ - ' .The transition rate was also calculated with similar methods by Shapiro and Breit (1959). More recent calculations, based on use of the nonrelativistic Coulomb Green's function, have been made by Zon and Rapoport (1968) and by Klarsfeld (1969a,b). With this technique, it is possible to obtain a semianalytic expression for the photon spectrum. Klarsfeld's result for the spectral and angular distribution of radiation is --f

Here 8 is the angle between the photon momenta, y #(y) = Q[2(1

= w / w i f ,and

+ 3y)-'"] + Q[2(4 - 3y)-'"]

(19)

with

where F is the Gauss hypergeometric function. Integration over angles and energy yields the decay rate

20 1

FORBIDDEN TRANSITIONS

Accurate numerical evaluation of (21) yields A

= (8.2294 f O.O0O1)Z6

sec-

(22)

[We have slightly modified Klarsfeld's result to take into account the more recent value cC1= 137.035987(29).] The above results are all based on the nonrelativistic (lowest order in Za) approximation. A relativistic calculation that gives the 2E1 rate for all Z up to Z = 92 and spectral distributions for representative values of Z has been done by W. Johnson (1972). Sample spectral distributions are illustrated in Fig. 6. For 2 = 18, the relativistic corrections reduce the 2E1 decay rate by about 1.5%. The higher multipole 2M1 decay mode was found to be much smaller (by a factor in hydrogen). Au (1976) has suggested that interference between 2E1 decay and higher multipole terms in the 2s 1 s decay can lead to an asymmetry in the twophoton angular distribution with an approximate relative magnitude of --f

&(ZCt)2.

c. TWO-PHOTONDECAYOF THE 2's0

STATE OF TWO-ELECTRON ATOMS

In two-electron atoms, two-photon decay is the primary decay mode of the 21S0 state. Breit and Teller (1940) suggested that the 2'S0 state lifetime in helium should be roughly the same as the 2 s state lifetime in hydrogen. Dalgarno (1966) first calculated the decay rate to be A = 46sec-'. The sum over intermediate states was evaluated with dipole matrix elements extracted 6-

I

I

0

0.2

0.4

I

I

0.6

0.8

1.0

Y

FIG.6 . Relativistic spectral distributions for the hydrogenic two-photon decay 2S,,* + lSl,2 at Z = 1,40, and 92. The spectral function I)(y, Z ) is defined by dA/djJ = (Za)6(9/1024)I)(jj, Z)Rj: y = O/W;,.

Richard Marrus and Peter J . Mohr

202

*

6.0

-

0

I

0.2 Of4 0.6 Q0 Fraction of endpoint frequency

1.0

FIG. 7. Frequency spectrum of the emitted photons in the decay of the 2 ' S , state of helium. Comparison is made with the corresponding spectrum for decay of the 2 S , , , state of hydrogen.

from the appropriate oscillator strengths. Dalgarno also gives the spectral distribution in tabular form. This is shown in Fig. 7 along with the hydrogenic two-photon spectrum. A value for the helium 2'S0 two-photon decay rate, within a factor of two of Dalgarno's (1966) value, was obtained by Dalgarno and Victor (1966), who used a Hartree-Fock approximation. A somewhat better value of 50seC' was obtained by Victor (1967), who employed the coupled HartreeFock method. Victor and Dalgarno (1967)reported a variational calculation of the lifetime z of the 2'S0 state in Li' with the result T = 5.15 x 10-4sec, with some uncertainty in the third figure. Extensive variational calculations were done by Drake et al. (1969) of the two-photon decay rates of the 21S0 and 23S, states for members of the helium isoelectronic sequence in the range 2 = 2 to 10. The infinite summations over intermediate states, including the continuum, were evaluated by replacing the true intermediate states by discrete sets of 30- to 50-term variational functions. A table of the spectral distribution of photons is given for each value of 2 (see also Jacobs, 1971). D. TWO-PHOTON DECAY OF

THE

2 3 S , STATEOF TWO-ELECTRON ATOMS

Various studies of the 2E1 decay of the 23S, state in heliumlike ions have been made. The two-photon transition rate for the 23S, state is much smaller than the two-photon rate of the 2'S, state, because a spin change is required in the former case. In an early study by Mathis (1957), an estimate of the z3S1 two-photon decay rate was made, but it was based on an incorrect formulation of the problem as pointed out by Drake and Dalgarno (1968) and by Bely (1968). Bely also estimated the two-photon rate to be A < lo-'' sec-'. More accurate calculations were done by Drake et al. (1969) and by Bely and Faucher (1969). Drake et al. did accurate variational calculations

203

FORBIDDEN TRANSITIONS

as described above for the 2'S0 decay. They calculated the rates for 2 = 2 to 10, finding A = 4.02 x sec- in helium, and point out that the rate, as a function of Z , increases as Z'O for low Z and increases as Z6 for high Z ( 25), where the singlet-triplet mixing becomes complete. Bely and Faucher considered only singlet-triplet mixing among states of the same principal quantum number. Their results which are consequently less accurate at low Z , are extended to larger Z ( - 30). Their result at Z = 31 is A = 6.28 x lo5 sec-', which is much smaller than the A41 decay rate (see Section 111).

-

E. EXPERIMENTS ON ONE-ELECTRON ATOMS The first successful experiments to detect and study the properties of radiative two-photon decay in any system were undertaken on the He' ion at the Columbia Radiation Laboratory (Novick, 1969). The experimental problems associated with these observations are many and formidable. For example, the lifetime of 1.9msec necessitates long flight paths, a problem exacerbated by the large cross section for collisional quenching and quenching that arises from motional electric fields. Moreover, the spectrum of emitted photons has an endpoint energy of 40.8 eV. Hence, the photons lie in the vacuum UV where detectors that combine high resolution and high efficiency do not exist. Despite these difficulties, an apparatus was constructed that permitted observation of the coincidences and verification of the 1 + cos2 f3 angular correlation between the two photons. These experimental results are exhibited in Fig. 8. In addition, a rough verification of the frequency spectrum was obtained by the use of broad-band photon filters. These measurements convincingly established the two-photon nature of the ,-NORMALIZATION POINT

j;, .4

I

I

I

I

225

247

270

5.2

67

90

112

135

157 180 ANGLE

202

292

FIG.8. The angular correlation of the coincident photons emitted in the decay of the 2SIi2 state of H e + .

Richard Marrus and Peter J . Mohr

204

decay and verified many of the essential details of the theory. Very recently, an accurate value for the lifetime of the 2S,,, state in He+ has been obtained by Hinds and Novick (1976)using a more sophisticated version of the original apparatus. Within the quoted accuracy of 0.3%, their result is in agreement with the prediction of Eq. (22). The beam-foil tof technique has been successfully used by Schmieder and Marrus (1970b) and by Cocke et al. (1974~)to observe features of the twophoton process in several hydrogenic ions thru argon (2 = 18).A schematic diagram of the apparatus used by Schmieder and Marrus is shown in Fig. 9. The coincidence results and a decay curve obtained by them for ArI7+ are shown in Figs. 10 and 11. Table 111gives a summary of the results obtained to date. Agreement with Eq. (22) is obtained in all cases. It is perhaps worth mentioning explicitly that experiments have not yet achieved the combination of precision and high Z necessary to observe the corrections to Eq. (22) arising from relativistic effects, radiative corrections, and contributions from the single-photon M1 mode. Much of the motivation for the early theoretical calculations on the 2S,,, state decay resulted from speculations concerning its astrophysical significance. For example, it was suggested long ago (Spitzer and Greenstein, 1951) that the two-photon decay of H(2Sl,,) is an important source of continuum emission in planetary nebulae. It has also been noted (Pagel, 1969) that the excess reddening of q-Carinae may arise from the metastable two-photon decay. However, the identification of this mode in astrophysical sources has not yet been firmly established.

A.0.C

ALTERUrr FOIL POSmONS

I

FIG.9. Experimental setup used in the study of the forbidden decays in Ar”+ and Ar16+.

10)

.. TI-12

, .

. -. ... .. ...

............. ,

,

,

I

3 0 0

I

,

,

..

... ... ..... ,

,

,

-I

,

,

,

,

,

lb)

. : .... . . .., .. ... '.

,

b a

*> -I,' ,.:- : .

-%. ,&: ,&; . .

,

-2

w m

. .

<.

-'TILE DIFRROENCE~ c&d f

E

...

.:\.

~.~':-~:.,,,.': ...

EI.E2

'

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

._..

4z w

>

2 a

E l t €2

W

*.

- -

.

.

I

I

1

2

,

,

ENEkGY tkrV)

,

I

5

6

FIG. 10. Coincidence spectra obtained for the photons emitted from the 2S,,, state of Ar"+.

Id

0

I

I

1

I

I

I

I

I

20

40

60

80

1

I

I

I

m

FOIL- DETECTOR SEPARATION

I

120

Ho

160

(~m)

FIG.1 1 . Decay curve obtained for the 2SIi2 state of Ar"+. A constant background is subtracted from all the points.

206

Richard Marrus and Peter J . Mohr TABLE 111 THEOKETICAL A N D EXPERIMENTAL LIFETIMES OF THE 2S,,, STATE I N HYDROGENIC ATOMS Z

2 8 9 16 18

q,,(nsec) 1.90 x lo6" 464 229 7.18 3.51

TeXp(nsec) (1.922 k 0.082) x 10" (1.903 0.005) x 10'' 453 43d 237 17d 7.3 0.7' 3.54 k 0.25'

*

Johnson (1972). Prior (1972). Hinds and Novick (1976). Cocke et al. (1974~). Marrus and Schmieder (1972).

F. EXPERIMENTS ON TWO-ELECTRON ATOMS A calculation of the lifetime of the 21S0 state in helium yields the value z(2lSO)= 19.5 x 10p3sec. Pearl (1970) and Van Dyck et al. (1971) have done tof experiments, using a beam of metastable helium, which were designed to measure this lifetime. In both experiments, the beams were excited to the metastable state by electron bombardment. In neither experiment were the decay photons observed. Rather, the lifetime was inferred by observing the decrease in the population of the metastable state with distance from the point of excitation. This population can be measured, since only atoms in the metastable states can give rise to Auger ejection of electrons when they impinge on a metal surface. To infer the 2'S0 population from the observed rate, it is necessary first to subtract counts arising from atoms in the 23S1 state. While similar in concept, the two experiments give results for the lifetime in apparent disagreement with each other. The result of Pearl is z(2lSO)= 38(8)msec while that of Van Dyck et al. is z(2lSO)= 19.7(1.0)msec. The latter result is in agreement with the calculated value. The two-photon mode of decay was established by a beam-foil experiment on the two-electron ion Ar16+(Marrus and Schmieder, 1972).While avoiding many of the difficulties of the helium experiment, beam-foil excitation introduces a new difficulty. The foil-excited beam necessarily contains substantial quantities of one-electron argon along with the heliumlike beam of interest. Suppression of the one-electron component is accomplished by

207

FORBIDDEN TRANSITIONS

I ’

1

I

I

I

I

I

I

I

3

4

1

TI- T2

...... *........ .. , 1’

ln

c c 3

0 0 ’c

-4

-2 0 2 -I I Time difference ( p sec

-3 1

I

I

I

I

I

I

0

..

El t E2 ......... ..... . ......*....*,..+ ........*.....L..*.

I

4.

1

2

- 1

3

4 Energy

5 6 (keV)

7

FIG. 12. Coincidence spectra observed for the photons emitted in the decay of ArI6+.

passing 412MeV Ar14+ emerging from the accelerator through a thin (10 pgm/cm2) carbon foil. The observed ratio Ar16+:Ar”+ z 6 : l is substantially improved over the equilibrium value of 1 :2. This enhancement is important, since photons from the decay of the 2S,,, state provide the largest source of background in the experiment on the two-electron beam. Results of the coincidence measurements are shown in Fig. 12. The timedelay spectrum for photons arriving in a pair of detectors viewing the beam shows a clear peak at zero time delay, thus establishing the existence of true coincidences. Moreover, the sum of the photon energies for those pairs arriving with zero time delay shows a clear peak at 3.12 keV. This is in good agreement with the theoretical energy for this decay and sufficiently well resolved from the one-electron two-photon energy of 3.34keV to convincingly establish that the coincident photons arise from the two-electron transition. The experimental technique of ion trapping has also been applied in the study of this decay. Prior and Shugart (1971) have created the metastable state of Li+ in a Penning trap using a pulsed electron beam. A schematic diagram of their apparatus is shown in Fig. 13. Measurement of the 2lSO lifetime was accomplished by measuring the photon intensity as a function of time after creation of the metastable state. Results of all measurements to date on the 2lSOlifetime are shown in Table IV.

208

Richard Marrus and Peter J, Mohr

FIG. 13. Schematic diagram of the apparatus used to trap Li+ and study the decay of the 2'S0 state.

TABLE IV THEORETICAL A N U EXPERIMENTAL VALUES FOR THE LIFETIME OF THE 2'S0 I N HELIUMLIKE ATOMS

2

3 18

19.5 x 10-30 513 x 2.35 x

+

(38 8) x 10-3c (19.7 1.0) x 10-3d (503 26) x (2.3 i 0.3) x lo-'/

Drake et al. (1969). G. W. F. Drake (private communication, cited by Marrus and Schmieder, 1972). Pearl (1970). Van Dyck et al. (1971). Prior and Shugart (1971). 1Marrus and Schmieder (1972).

FORBIDDEN TRANSITIONS

209

It has been noted by Gorshkov and Labzovskii (1974) that in the presence of hyperfine structure, the 2lSOstate can decay to the ground state by a single photon transition. This situation arises because of the hyperfine coupling of 2'S0 with 23S,. These authors further noted that if a parity-violating interaction should exist in the atomic Hamiltonian that is of the order of magnitude predicted by the Weinberg-Salam model for the weak interactions, then a large circular polarization should be associated with the accompanying decay photons. Unfortunately, the substantial experimental problems associated with this observation have so far prevented any test of this hypothesis.

VI. Intercombination Transitions A. GENERAL CONSIDERATIONS In ordinary helium, the 23P1state decays by a fully allowed electric dipole transition to the 23S1 state with a lifetime of sec. Insofar as total spin S is a good quantum number, direct decay to the l1S0 ground state by an electric dipole transition is forbidden by the AS = 0 selection rule. However, the spin-orbit and other relativistic interactions produce mixing of the 23P, state with 'P, states, so that the decay 23P1-+ 1'S0 can occur by electric dipole radiation. The first observation of the intercombination line in helium was made in an arc spectrum by Lyman (1924). The difficulty of making quantitative measurements on this transition in helium is perhaps best illustrated by the fact that it was not until 1972 that Tang and Happer established limits on the branching ratio B = A(23P1+ 11So)/A(23P,+ 23S,). They were able to set the limits 0.3 x lo-' d B < 1.8 x Recent interest in this transition has been stimulated mainly by two developments. During the past ten years, there have been many successful observations of the intercombination transition from heavy heliumlike ions present in the solar corona. Gabriel and Jordan (1969b) showed that the observed intensity ratios R = Z(23S1+ 11S0)/1(23P,+ l'So) can be used to obtain values of the electron density (see Section 111).During the same time period it has become possible to make laboratory studies of the properties of this decay in heavy heliumlike ions using the beam-foil method.

B. THEORY It was pointed out by Edlen (1951) that the ratio of the intercombination transition rate to the resonance rate A p 3 P l + 1'S0)/A(2lP1+ l'So) would

210

Richard Marrus and Peter J . Mohr

increase rapidly with increasing Z . Elton (1967) carried out a semiempirical calculation of the oscillator strengths of the l'S, -+ 23P, transitions for the heliumlike ions He1 to NeIX, which employs the observed energy splittings of the 3P levels and approximate off-diagonal matrix elements based on calculations of Araki (1937). Elton takes into account mixing of the 2'P and 23P states through diagonalization of the n = 2 energy submatrix, but omits the effectsof intermediate states of higher principle quantum number. These effects can be expected to be most important at low Z . Drake and Dalgarno (1969) have constructed a variational procedure that takes into account the contributions from the entire set of n'P states. This approach should give accurate results at low Z. They consider the heliumlike atoms with Z = 2 to 10 and show that the intercombination rate 23P, l1S0 becomes larger than the E l rate z3P, -+ 23S1for ions beyond CV. Johnson and Lin (1976) have calculated the intercombination transition rate using the relativistic random phase approximation for a wide range of Z values. In their results, an empirical correction factor is employed to correct for the poor 'P, -+ 3P, energy splitting determined by the RRPA method. Their corrected values are in general agreement with the Drake and Dalgarno (1969) results. It was suggested by Luc-Koenig (1974) and Laughlin (1975) that relativistic spin-dependent corrections to the p A form of the interaction operator were not included by Drake and Dalgarno (1969). Revised rates were given by Laughlin (1975);who included these effects. Subsequently, it was pointed out by Drake (1976) that provided the transition matrix element is expressed in E r (dipole length) form, these corrections are automatically included, and therefore the revisions are unnecessary. In a more recent work, Safronova and Rudzikas (1977) used S-matrix theory to do a relativistic calculation of the intercombination transition rate L I ( ~ ~-+PllS,) , for ions from Z = 2 to 100. Their comparison with the results of Elton (1967) for Z = 2 to 10 shows agreement over this range. -+

-

-

C. EXPERIMENT

There have been extensive observations of the X-rays associated with the intercombination transition from heavy heliumlike ions in the solar corona. The intensities of these observed lines have been important in determining electron densities (see Section 111). Laboratory measurements in heavy heliumlike ions were first reported by Sellin et al. (1968), who used 6 to 24 MeV nitrogen and oxygen ions from a tandem Van de Graff accelerator to measure the 23P, lifetime by the beam-foil tof technique. A later measurement by Sellin et al. (1970b) used a crystal spectrometer to resolve the line in oxygen and improve the measured lifetime. Subsequent tof measurements of this lifetime have been

FORBIDDEN TRANSITIONS

21 1

TABLE V THEORETICAL AND MEASURED TRANSITION RATESFOR 23P, + l1S0 (IN UNITSOF lo9 sec-')

0.140" 0.553"

7 8 8 8 9 9

1.85"

14

16 a

158' 587'

0.17 f 0.03' 0.58 f 0.05d 0.601 f 0.033' 0.601 f 0.42' 1.77 f 0.lq 1.77 f 0.07h 157 & 8' 637 f 73'

Drake and Dalgarno (1969).

'Johnson and Lin (1976). Sellin e t a / . (1968). Sellin et al. (1970b). Richard et al. (1973a). Moore et al. (1973). Mowat et al. (1973). Richard et al. (1973b). Varghese et a/. (1976).

made by Richard et al. (1973a) and Moore et al. (1973). Mowat et al. (1973) and Richard et al. (1973b) have made measurements in two-electron fluorine. Varghese et al. (1976) have succeeded in extending measurements on the intercombination transition to silicon and sulfur by constructing an apparatus capable of resolving lifetimes in the picosecond range. In order to compare the measured results with the theoretical intercombination rates, correction must be made for the electric dipole transition rate A,1(23P1 + 23S,). This is generally taken from the Tables of Wiese et al. (1966). Table V compares the theoretical and measured rates for the transitions that have so far been experimentally studied.

VII. Nuclear-Spin-Induced Decays A. EARLYOBSERVATIONS Early observations of forbidden transitions that were later identified as nuclear-spin-induced included absorption lines such as 1S0-3P, (Lord Rayleigh, 1927) and 'So-3P0 (Fukuda, 1926) in HgI. It was Bowen (quoted in Huff and Houston, 1930) who first suggested that the transitions were

212

Richard Marrus and Peter J. Mohr

due to perturbations of the atomic structure by the nuclear moments. This suggestion was confirmed by studies by Mrozowski (1938, 1945) and work by others. Extensive calculations of the nuclear-spin-induced transition rates were made by Garstang (1962b, 1967). Bigeon (1967) verified the calculations and made absorption measurements on the 'S0-3P0 line in 199HgIand "'HgI, which showed good agreement with Garstang's theoretical predictions. Interpretation of the 1S0-3P2lines requires the additional consideration of magnetic quadrupole radiation, which is discussed in Section IV. Because of the small relative magnitude of the hyperfine interaction compared with other atomic interactions, it usually plays no role in spontaneous radiative decay. However, certain special situations do exist that make possible the observation of hyperfine effects. In particular, in the simple atomic systems discussed in this chapter, measurable effects occur in the 23P multiplet of the two-electron system.

B. OBSERVATION IN TWO-ELECTRON VANADIUM Observation of the effect of nuclear perturbations on the spontaneous decay 23P2+ l1S0 in two-electron vanadium (2 = 23) has been reported by Gould et al. (1974). In this case, the theory is simplified because only two electrons are present, and an estimate may be made without empirical input for the atomic parameters. The calculation is outlined in Gould et al. (1974) and more details are given by Mohr (1976). In the presence of the nuclear spin I, the total angular momentum is F = J + I , and selection rules based on the J values for the states may be violated with an amplitude roughly proportional to the strength of the nuclear spin perturbation. In the case of vanadium ( I = g), the 23P2 state splits into a multiplet with F values $, 3, 4,2,y.Similarly, the l'So state has F = $and so electric dipole transitions may occur between the %,g,8 excited levels and the ground level. A sketch of the theory is as follows: The first-order perturbed wave function for the 23P, state is given by (23PlFM16H123P,FM) 1 2 3 ~F1M ) 1 2 3 ~ 2 ~=~1 ) 21 3 ~ 2 ~ ~ ) E ( ~ ~ P-, )E ( ~ ~ P , )

+

)23P2FM) 121P2FM) + (21P1FM16H E(23P2) E(2'P1) -

where mixing with only the 23P1and 21P1 states by the nuclear perturbation is included, and perturbations of the l1S0state are neglected. The zero-order states are sums of products of electron eigenstatesand nuclear spin functions, combined to give total angular momentum F with z component M . The

213

FORBIDDEN TRANSITIONS

23P1 and 21P1 states are intermediate coupling states that take into account the spin-orbit mixing of the L-S coupling states. These may be approximated by considering only spin-orbit mixing among the n = 2 states as was done by Elton (1967) in estimating the z3P1 intercombination decay rate. The rate for a dipole transition from the perturbed 23P2 state to the 1'So state is

where o is the transition frequency: ho = EQ3P2)- E(liSo). The predicted decay rates in vanadium for the states with F values 3, and are 1.0,1.8,and 1.7nsec- and the magnetic quadrupole decay rate is 2.4nsec- I . Because the matrix elements of 6H in Eq. (23), and hence the decay rates, depend on the F value of the initial state, the hyperfine levels are characterized by different lifetimes and the decay is no longer a single exponential. A sample decay curve taken in the vanadium experiment is shown in Fig. 14. After the effect of the M1 transition 23S, + 1'So is subtracted, the remaining component shows substantial deviation from a pure exponential. This can be fitted by a model that assumes that each of the hyperfine levels is

s,

',

Vt2'( I = 7/2

F I1/2

k io

io

50

-

I00

FOIL- DETECTOR SEPmATION nn

150

FIG.14. Decay curve showing the decay of the 23P, state of V 2 ' + . The effects of hyperfine mixing are in evidence.

214

Richard Marrus and Peter J. Mohr

populated according to the statistical weight 2 F + 1. With the calculated hyperfine induced decay rates, and the calculated value for the M2 decay rate, the theoretical decay curve is in good agreement with experiment. In the case of nuclear-spin-induced decays of the 23P0 state, estimates have been given by Mohr (1976) that show a rapid increase of the decay rate with increasing Z for Z in the range 9 to 29. This is to be expected because the main effect is associated with the intercombination decay rate of the 23P1 state, which has a rapid Z dependence (see Section VI). In vanadium, the nuclear-spin-induced decay rate is sufficiently fast (10" sec- ') that in an experiment employing beam-foil excitation, all decays would take place immediately at the foil. Unfortunately, the energy of this transition is sufficiently close to the energy of the intercombination transition that resolution of the two transitions is not experimentally feasible. However, observation at lower Z may be possible.

VIII. Electric-Field-Induced Decays A. METASTABILITY OF THE 2S1,, STATE IN

AN

ELECTRIC FIELD

A free hydrogen atom in the 2S1,, state decays primarily by two-photon emission with a rate that is much smaller than the E l transition rate from the 2P1,, state. However, in an applied electric field, the 2S,,, and 2P,,, states, which are separated in energy only by the Lamb shift, are relatively easily mixed, resulting in an increased decay rate for the 2SIl2 state. In a weak electric field, the 2S,,, state vector is approximately given by (neglecting the radiation width and mixing with states other than the 2P,,, state)

where the electric field is taken to be in the direction of the spin quantization axis. The electric dipole matrix element to the ground state is then nonvanishing,

and is proportional to the electric field strength E, and inversely proportional to the Lamb shift S = E(2S1,,) - E(2P,,,). As a result of this mixing, a new single-photon channel, associated with the possibility of an El transition from the 2Pt,, admixture to the ground state, is available. The radiation emitted in this decay mode is widely referred to as quench radiation and has

FORBIDDEN TRANSITIONS

215

been studied experimentally with the goals of understanding its properties and using it as a tool for measuring the Lamb shift in hydrogenic atoms. If the electric field is strong enough that the matrix elements of the external field potential are not small compared to the Lamb shift energy, then the "stationary" states are obtained by diagonalization of the 2 x 2 energy matrix. The mixing of the states then determines the decay rate according to (Bethe and Salpeter, 1957) A&

=

(1 + Q)l'2 - 1 2(1 + 9)1/'

where 8 = (2$eEa,/S),. This expression for the decay rate has been checked experimentally for hydrogen by Sellin (1964),who was able to observe deviations from the lowest order result A& z (8/4)A,, in strong (up to 500V/cm) electric fields. B.

ELECTRIC-FIELD-PERTURBED THE

LIFETIME MEASUREMENT OF

LAMBSHIFT

The discussion above neglects the radiation width of the 2P,,, state, which in hydrogen is about 10% of the Lamb shift. It also neglects the effect of the 2P3,, state, which is mixed with the 2S,,, state by the electric field; but this effect is small because the fine structure separation E(2P3,,) E(2S1,,) is an order of magnitude larger than S . Theoretical studies of the effects of an applied electric field on the 2S,,, state lifetime that consider the radiation width were made by Luders (1950)and Lamb and Retherford (1950). The latter authors considered a phenomenological approach in which the unperturbed states are given a decaying exponential time dependence, and solved the coupled equations for the time dependence of the states in an electric field. If the wavefunction for the atom is given by u ( t ) = a(t)u,e-'"J

+ b(t)ubepimhf+ c(t)ucep""='

(28)

where a = 2P,,,, b = 2Sli2,and c = 2P1',, then the equations for the time dependence of a, b, and c in an applied field are

Richard Marrus and Peter J. Mohr

216 where

, - oj, K j = (ileEz[i)/h

0.. = 0.

and yi is the fie 1-free decay rate of state i. For an atom initially in th 2SlI2 state, a(0) = c(0) = 0,b(0) = 1, the solution for b(t)is approximately b(t)= e - @ with the decay rate A&2 = 2 Re(,u) given by

Higher order terms in I&,12/0& are included in a discussion by Fan et al. (1967). Equation (30) has been used as the basis for determining the Lamb shift in several hydrogenic ions at high Z. In these quenching experiments A$S1Iz is measured for several electric fields and Eq. (30) is used to determine the Lamb shift oab. The first experiment of this type was done by Fan et al. (1967) on Liz+.A schematic diagram of the apparatus used in their experiment is shown in Fig. 15. The Li+ beam from a Van de Graaf generator is passed through a nitrogen-filled charge-exchange cell. About 29% of the beam emerging from the cell is in the one-electron charge state and some of these atoms are in the metastable 2S1,, state. These ions are passed into a quench region between a pair of electric field plates. Photons emitted in the region between the plates are viewed by a movable and a fixed detector. The fixed detector acts as a normalization and a decay curve is taken by varying the separation between the two detectors. From the measured decay length and the known beam velocity, A$S1I2 can be determined for a number of electric fields. (Lib)+ Beam / 3.39-MeV from Van de Graaff

il

Van de Graaff Magnet To Diffusion Fbmp

FIG. 15. Apparatus used to measure the Lamb shift in Liz+ by the electric field quenching technique.

FORBIDDEN TRANSITIONS

217

Lamb shift measurements based on electric field quenching have been carried out with higher-2 hydrogenic atoms by Kugel et al. (1972) on C 5 + , by Lawrence et al. (1972) and Leventhal et al. (1972) on 0 7 +and , by Gould and Marrus (1976b) on ArI7'. These subsequent experiments have used an important modification of the original technique. Instead of employing an electric field, a magnetic field B applied transverse to the beam was used. In this way, the atom experiences a motional electric field E in its rest frame given by

E = (1 - v ~ / c ~ ) - " ~ ( v x/ cB )

(31)

There are two advantages associated with this scheme. Unlike electric fields, magnetic fields can be measured conveniently and accurately, and since the velocity of ions from Van de Graaf accelerators is usually known to high accuracy, the motional electric field is precisely known. Moreover, with ion beams from an accelerator such that t/c > 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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Beigman, I. L., and Safronova, U. I. (1971). Zh. Eksp. Teor. Fiz. 60,2045; Sou. Phys.-JETP (Engl. Transl.) 33, 1102. (1971). Bely, 0. (1968). J . Phys. B 1, 718. Bely, O., and Faucher, P. (1969). Astron. Astrophys. 1, 37. Bethe, H. A., and Salpeter, E. E. (1957). “Quantum Mechanics of One- and Two-Electron Atoms.” Springer-Verlag, Berlin and New York. Bigeon, M. C. (1967). J . Phys. (Paris) 28, 51. Blatt, J. M., and Weisskopf. V. F. (1952). “Theoretical Nuclear Physics.” Wiley, New York. Borisoglebskii, L. A. (1958). Usp. Fiz. Nuuk 66, 603; Sou. Phys.-Usp. (Engl. Trans/.) 1, 211. (1959). Bowen, 1. S. (1928). Astrophys. J . 67, 1. Bowen, I. S. (1936). Rev. Mod. Phys. 8, 55. Bowen, I. S. (1960). Astrophys. J. 132, 1. Breit, G., and Teller, E. (1940). Astrophys. J. 91, 215. Casalese, J., and Gerjuoy, E. (1969). Phys. Rev. 180, 327. Cheng, K. T., Lin, C. P., and Johnson, W. R. (1974). Phys. Lett. A . 48,437. Cocke, C. L., Curnutte, B., and Randall, R. (1973). Phys. Rev. Lett. 31,507. Cocke, C. L., Curnutte, B., Macdonald, J. R., and Randall, R. (1974a). Phys. Reu. A 9, 57. Cocke, C. L., Curnutte, B., and Randall, R. (1974b). Phys. Rev. A 9, 1823. Cocke, C. L., Curnutte, B., Macdonald, J. R., Bednar, J. A,, and Marrus, R. (1974~).Phys. Rec. A 9, 2242. Dalgarno, A. (1966). Mon. Not. R . Astron. Soc. 131, 31 1. Dalgarno, A., and Parkinson, E. M. (1967). Proc. R . Soc. London, Ser. A 301,253. Dalgarno, A,, and Victor, G. A. (1966). Proc. Phys. Soc., London 87,371. Davis, W. A,, and Marrus, R. (1977). Phys. Rev. A 15, 1963. Drake, G. W. F. (1969). Astrophys. J. 158, 1199. Drake, G. W. F. (1971a). Phys. Reo. A 3, 908. Drake, G. W. F. (1971b). Astrophys. J . 163,439. Drake, G. W. F. (1972a). Phys. Rev. A 5, 1979. Drake, G. W. F. (1972b). Can. J. Phys. 50, 1896. Drake, G. W. F. (1974). Phys. Rev. A 9, 2799. Drake, G. W. F. (1976). J. Phys. B 9 , L169. Drake, G. W. F., and Dalgarno, A. (1968). Astrophys. J. 152, L121. Drake, G. W. F., and Dalgarno, A. (1969). Astrophys. J. 157,459. Drake, G. W. F., and Grimley, R. B. (1973). Phys. Rev. A 8, 157. Drake, G . W. F., and Lin, C. P. (1976). Phys. Rev. A 14, 1296. Drake, G. W. F., Victor, G. A., and Dalgarno, A. (1969). Phys. Rev. 180, 25. Drake, G. W. F., Farago, P. S., and van Wijngaarden, A. (1975). Phys. Rev. A 11, 1621. Edlen, B. (1951). Ark. Fys. 4,441. Elton, R. C . (1967). Astrophys. J . 148, 573. Fan, C. Y., Garcia-Munoz, M., and Sellin, I. A. (1967). Phys. Reu. 161, 6 . Fawcett, B. C . , Gabriel, A. H., and Saunders, P. A. H. (1967). Proc. Phys. Soc., London 90, 863. Feinberg, G., and Sucher, J. (1971). Phys. Rer. Lett. 26, 681. Feldman, U., and Doschek, G . A. (1977). A t . Phys. Proc. Int. Conf., Sth, 1976 p. 473. Feneuille, S., and Koenig, E. (1972). C . R . Hebd. Seances Acad. Sci., Ser. B214,46. Fite, W. L., Kauppila, W. E., and Ott, W. R. (1968). Phys. Rev. Lett. 20, 409. Foote, P. D., Takamine, T., and Chenault, R. L. (1925). Phys. Rev. 26, 165. Freeman, F. F., Gabriel, A. H., Jones, B. B., and Jordan, C. (1971). Philos. Trans. R. Soc. London, Ser A 270, 127.

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Fritz, G., Kreplin, R. W., Meekins, J. F., Unzicker, A. E., and Friedman, H. (1967). Astrophys. J. 148, L133. Fukuda, M. (1926). Sci. Pap. Inst. Phys. Chem. Res. Jpn. 4, 171. Gabriel, A. H., and Jordan, C. (1969a). Nature (London) 221,947. Gabriel, A. H., and Jordan, C. (1969b). Mon. Not. R. Astron. Soc. 145, 241. Gabriel, A. H., and Jordan, C. (1970). Phys. Lett. A 32, 166. Garstang, R. H. (1962a). In “Atomic and Molecular Processes” (D. R. Bates, ed.), p. 1. Academic Press, New York. Garstang, R. H. (1962b). J . Opt. Soc. A m . 52, 845. Garstang, R. H. (1967). Astrophys. J . 148, 579. Garstang, R. H. (1969). Publ. Astron. Soc. Puc. 81, 488. Goeppert-Mayer, M. (1929). Nuturwissenschufren 17,932. Goeppert-Mayer, M. (1931). Ann. Phys. (Leipzig) [5]9, 273. Gorshkov, V. G., and Labzovskii, L. I . (1974). Z / I .Eksp. Teor. Fis., Pis’nzu Red. 19. 768; JETP Lett. (Engl. Transl.) 19, 394 (1974). Gould, H., and Marrus, R. (1976a). Bull. A m . Phys. Soc. [2] 21, 84. Gould, H., and Marrus, R. (1976b). Ahstr., At. Phys., Int. Conf:, 5th, 1976 p. 207. Gould, H., Marrus, R., and Schmieder, R. W. (1973). Phys. Rev. Lett. 31, 504. Could, H., Marrus, R., and Mohr, P. J. (1974). Phys. Rev. Lett. 33, 676. Griem, H. R. (1969). Astrophys. J . 156, L103. Griem, H. R. (1970). Astrophys. J . 161, L155. Grisaru, M . T., Pendleton, N. H., and Petrasso, R. (1973). Ann. Phys. ( N . Y.) 79, 518. Haselton, H. H., Thoe, R. S., Mowat, J. R., Griffin. P. M., Pegg, D. J., and Sellin, I. A. (1975). Pliys. Rev. A 11, 468. Hinds, E. A , , and Novick, R. (1976). Ahstr. At. Phys. Int. Conr. 5th. 1976 on Atoinic Physics, Berkeley, Ju/y 26 - 30, p. 204. Holt, H. K., and Krotkov, R. (1966). Phys. Rev. 144, 82. Holt, H. K., and Sellin, I. A. (1972). Phys. Rev. A 6 , 508. Huff, L. D., and Houston, W. V. (1930). Phys. Rev. 36, 842. Jackson, J. D. (1975). “Classical Electrodynamics,” 2nd ed., p. 758. Wiley, New York. Jacobs, V. L. (1971). Phys. Rev. A 4,939. Jacobs, V. L. (1972). J . Phys. B5,213. Johnson, C. E. (1972). Bull. A m . Phys. Soc. [2] 17, 454. Johnson, C. E. (1973). Phys. Rev. A 7, 872. Johnson, W. R. (1972). Phys. Rev. Lett. 29, 1123. Johnson, W. R., and Lin, C. P. (1974). Phys. Rev. A 9, 1486 Johnson, W. R., and Lin, C. D. (1976). Phys. Rev. A 14, 565. Johnson, W. R., Lin, C. D., and Dalgarno, A,, (1976). J. Phys. B 9 , L303. Jones, B. B., Freeman, F. F., and Wilson, R. (1968). Nature (London) 219, 252. Kelsey, E. J., and Sucher, J. (1975). Phys. Ret.. A 11, 1829. Kelsey, E. J. (1976). Ann. Phys. ( N . Y . )98, 462. Kelsey, E. J., and Macek, J. (1977). Phys. Rev. A 16, 1322. Klarsfeld. S. (1969a). Lett. Nuoco Cirnento 1, 682. Klarsfeld, S. (1969b). Phys. Lett. A 30, 382. Kroll, N. M. (1961). Quoted in Pietenpol (1961). Kugel, H. W., Leventhal, M., and Murnick, D. E. (1972). Phys. Rec. A 6 , 1306. Lamb. W. E., Jr., and Retherford, R. C. (1950). Phys. Rev. 79, 549. Laughlin, C. (1975). J . Phys. B 8, L400. Lawrence, G. P., Fan, C. Y., and Bashkin, S. (1972). Phys. Rev. Letr. 28, 1612. Layzer, D., and Garstang, R. H. (1968). Annu. Rec. Astron. Astrophys. 6, 449.

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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.]

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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

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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

O
0<0<7L

I . Stationary Phase (Primitive, Semiclassical) Approximation

Under the normal molecular scattering conditions q(i.) is large and strongly dependent on I , and the A values of central significance for the theory are those at which the stationary phase condition is satisfied,

which may be seen to specify precisely those trajectories for which the classical deflection x(i) gives rise to the same scattering angle 8 = Ixl(mod 2n), given the semiclassical identity derived from Eq. (9,

This intimate connection between the phase shift and classical deflection is illustrated in Fig. 3 for a typical molecular system. Scattering experiments are normally performed on systems for which the interatomic well depth E is comparable with thermal energies. Otherwise, spectroscopic investigation is usually more convenient ;even the vibrationalrotational spectrum of Ar2 has recently been determined (Tanaka and Yoshino, 1970; Colbourn and Douglas, 1976).This means that in the normal experiment the rainbow angle xr in Fig. 3 satisfies xr < -71, so that the stationary phase condition (9) can be satisfied for real values of 2 only for M = 0, when there are typically three branches to the solution of Eq. (9), as illustrated in the lowest part of Fig. 3. A simple stationary phase evaluation of the integral for I&(@ at these points yields

f(4=

c

f,(@exp[irv(f3)1

(11 )

v=o,b,c

where fy(8) is the square root of the corresponding classical differential cross section f,(Q) = ( d ~ / d f i ) ;=~(bldb/dxl/sin ~ f3)'j2

(12)

and ~ " ( 0is)the phase of the integrand in Eq. (8)at L = A,. This triple-branched interference accounts for much of the structure observed in the differential cross section do/dQ = shown in Fig. 4, but corrections are required as 0 + 0 and 8 71 due to the breakdown of Eq. (6), and at angles close to the rainbow angle f3 N lxrl due to coalescence of the branches &(8) and --f

M . S. Child

236

I

I I I I I I I

I I

I

I I I I

I

I I

I I I I

I I I

I I

FIG.3. The semiclassical connection between the phase shift q, and classical deflection 0, for a Lennard-Jones potential. I, and I, denote the rainbow and glory angular momenta, respectively. Note the existence of the three I values, l,, I,, and lc, having a common scattering [Taken from Child (1974a) with permission.] angle 0 =

&(0). This invalidates the stationary phase approximation and leads to divergence of the a and b branch contributions to f ( 0 ) because Idb/dxI -+ co as x x r . +

2. Uniform Approximation

This behavior at the classical rainbow singularity is characteristic of the simplest (fold) catastrophe in Thorn’s (1969) classification. The proper

SEMICLASSICAL HEAVY-PARTICLE COLLISIONS

237

8 (dcg) FIG.4. Semiclassical interference structure in the elastic differential cross section. The rainbow angle, which is shown by the outermost point of inflection, lies at approximately 41’. [Taken from Berry and Mount (1972)with permission.] (Copyright by The Institute of Physics.)

uniform mapping is onto an Airy function (Berry, 1966). This is achieved by employing the equation 5.3

-

c(e)x+ y , ( q

= 2rl(4

+ A e + n/4

(13)

to define a variable transformation ,i(.~) over the a and b branch regions of E., with the parameters [ ( O ) and y,(d) determined by the requirement that the stationary phase points on the two sides of Eq. (13) should be in one-toone correspondence. This implies that c(6) and y,(6) may be expressed in terms of the same two stationary phase exponents ya(@ and y b ( 6 ) as those which appear in the primitive semiclassical approximation (11): Yr(O)

= +[Yb(O) =

{$[?b(s)

+

(14)

- ra(e)]}”3

(15)

Transformation to x as the integration variable on the right-hand side of Eq. (8) then yields (Berry, 1966)

f ( @ = L(Q)~

+ f,(d)~

X Ci~r(d)] P

XCiycCe)] P

where

+ f b ( e ) ] Ai[-l(0)]

= n”2(c”4(0)[fa(@)

+ i<-1’4(o)[h(o) - fb(@]

Ai’[-c(0)])

(16)

238

M . S . Child

and Ai( -x) and Ai’( - x) denote the Airy function and its first derivative (Abramowitz and Stegun, 1965). It is readily verified by use of standard asymptotic approximations for large [(O) that Eq. (16) reduces exactly to Eq. (1l), so that Eq. (16) may be taken to cover the entire intermediate angle region. The accuracy of this uniform Airy approximation is displayed in Fig. 4. The most important features are:

(1) The location of the rainbow angle, given by the outermost point of inflection, which depends on the ratio of the collision energy to the potential well depth. (2) The period of the slow oscillations (supernumerary rainbows), which arises from interference between the a and b branches of the deflection function. These therefore depend on the path difference between a and b type classical trajectories at the same scattering angle and on the de Broglie wavelength. The result is that the spacing between the supernumerary rainbows depends inversely on the product of the range of the potential and the square root of the reduced mass. (3) The period of the rapid oscillations, which arises from interference between the rainbow structure and the repulsive c branch component of the scattering amplitude. Extensions of the theory to cover the small-angle region where Eq. (6) becomes invalid are discussed by Bernstein (1966), Berry (1969),and Child (1974a). The above analysis applies to the scattering of heavy particles at energies above the orbiting limit, below which the occurrence of shape resonances can give rise to additional complications as qualitatively discussed by Buck (1976);the general theory in relation to the spectroscopic implications of these orbiting resonances has been reviewed by Child (1974b).

3. Diflraction Oscillations The scattering of light atoms raises special problems due to the small number of significant terms in Eq. (1).The dominant feature of the differential cross section is a series of roughly equally spaced “diffraction oscillations” bearing no relations to the classical rainbow angle as illustrated in Fig. 5 [see also Chen et al. (1973) for experimental differential cross sections for the scattering of helium on other inert gases]. These oscillations, which are observed even for a purely repulsive potential, have been attributed by Zahr and Miller (1975) to interference between the normal repulsive branch of the classical deflection function and other classically forbidden branches (arising from complex points of stationary phase), but no full semiclassical theory has been developed.

SEMICLASSICAL HEAVY-PARTICLE COLLISIONS

239

FIG. 5. Diffraction oscillations in the helium and neon differential scattering cross section. The persistence of regular slow oscillations at high angles cannot be accommodated within the semiclassical approximation. [Adapted from Chen et crl. (1973).]

4. Symmetry Oscillations

A final additional source of quantum oscillations arises from the exchange symmetry of identical particles, for which the scattering amplitude must be symmetrized or antisymmetrized for bosons and fermions, respectively, by the equation

f*(4= .f(4 f f ( .

-

4

(17)

The differential cross section is therefore necessarily symmetric in the forward and backward directions, and interference may arise between all branches off(0) with those off(71 - 0). The spacing of the resulting symmetry oscillations is again inversely dependent on the reduced mass, and so they are most readily observed in the scattering of light atoms. Several examples are discussed by Buck (1976). B. TOTALCROSSSECTION The theory of the total cross section (Bernstein, 1966: Berry, 1969) is based on the optical theorem, 471 o ( E ) = - Imf(0) k

=

471 k

1 (21 + I)sin2ql

1=0

The lower part of Fig. 3 shows that there are two contributions to the classical scattering at 6 = 0: one from large-impact parameters and the other from angular momenta close to Ig at which the b and c branches of the

240

M . S . Child

deflection function coalesce. Of these the former, o,(E) is dominant. Bernstein (1966) shows for an asymptotically inverse power potential V(r) - C(')/f, that the strong dependence of the high-angular-momentum semiclassical phase shifts,

-

Vl

-

+ 51 )1 - s

(19)

implies that the o,(E) cross section may be estimated as .nb*', where b* is the impact parameter [b* = (I* + 3)/k] at which qI falls below 0.5, in other words, at which the classical action falls below 0.5h. In this approximation, (20) , p ( s ) is a pure number where o is the collision velocity, o = ( ~ E / P ) " ~and [ p ( s ) = 8.083 for s = 6, for example]. Corrections are, however, required at very high energies because the limiting phase shift no longer belongs to the long-range scattering. o,(E)

= ~(S)[C"'/AV]~""')

1. Glory Oscillations

The second, glory contribution to o(E)may be evaluated by applying the stationary phase approximation to Eq. (18),because by the semiclassical correspondence relation (10) between the derivative of the phase shift and the scattering angle, the phase shift must pass through a maximum q,(E) at the glory angular momentum I,. This phase is therefore reflected in the relative phases of the two contributions to the total cross section (Bernstein, 1966): o ( E ) = ~(s)[C(')/AU]'/('- - (4n/k)f,(0)cos[2qg(E)- 3 ~ / 4 ]

(21)

where IfJ0)l has the same classical interpretation as lf,(O)I defined by Eq. (12). Experimental consequences of this general behavior are illustrated in Fig. 6. The velocity dependence of the first term in this expression provides the best experimental information on coefficient of the long-range attractive part of the potential. The total number of glory oscillations arising from the second term may be shown to count the number of rotationless bound states supported by the potential (Bernstein, 1966), and the spacing of these oscillations contains information on the product of the well depth and potential range parameter (Bernstein and La Budde, 1973, Greene and Mason, 1972,1973; Buck, 1976). 2. Symmetry Oscillations

Additional quantum oscillations in the total cross section may be caused by orbiting resonances, as observed by Schutte et al. (1972; see also Child,

SEMICLASSICAL HEAVY-PARTICLE COLLISIONS

24 1

10’

30 qv04

tt

3 2

I

0

I

0

005

-

I

I

I

I

0.10

015

(KIB)’”

0.20

0.25

FIG. 6 . Glory oscillations and orbiting resonance structure in the total cross section for the Lennard-Jones potential, V ( r )= - 4 ~ [ ( u / r ) ” - ( ~ / r ) ~at] , collision energy E , with K = E / E , A = ( 2 p E ) ’ ’o/h, B = 2puZ/hZ.[Taken from Buck (1976) with permission.] (Copyright by John Wiley & Sons, Ltd.)

1974b), and by exchange effects in the scattering of identical particles. The latter are relevant to the semiclassical inversion methods discussed below in providing experimental information on the energy dependence of the s-wave phase shift qo(E). The theory given by Helbing (1968) recognizes that the necessary symmetrization or antisymmetrization of the scattering amplitude [see Eq. (17)] introduces the possibility of interference with the backward glories at 8 = 71. Furthermore, any repulsive core potential always leads to backward scattering 8 = 71 at 1 = 0, and by the semiclassical correspondence relation (10) the phase shift curve must have a derivative of 71/2 at this point: q,(E) = qo(E)

+

$711 -

ic12 + . . .

(22)

Finally, 1 must change by 2 between successive partial waves because only even or odd 1 values contribute according to whether one applies Bose or Fermi-Dirac statistics. Hence, since the phase appears in (1) as exp(2iq,), successive terms combine exactly in phase apart from corrections due to the term d2in Eq. (22). It follows that the backward glory contribution to symmetrized or antisymmetrized total cross section also depends on this phase (Helbing, 1968),

M . S. Child

242

FIG. 7. Symmetry oscillations in the total helium-helium wattering cross sections. [Taken from Buck (1976) with permission.] (Copyright by John Wiley & Sons, Ltd.)

where tan4

=

[I

+2/(7c~)~’~]-~

(24)

and the upper and lower signs in Eq. (23) apply to Bose and Fermi statistics, respectively. This means that the symmetry oscillations for two isotopes obeying opposite statistics will be exactly out of phase, and that the positions of the maxima and minima may be used to determine the energy dependence of the 1 = 0, s-wave phase shift q,(E). This procedure has been applied to the scattering of two helium atoms, for which the experimental results (Feltgen et al., 1973) are shown in Fig. 7. There are no normal glory oscillations in this diagram because the potential well depth for He, is too small.

C. SEMICLASSICAL INVERSION PROCEDURES I . Firsov Inversion

The most complete semiclassical inversion procedure is that which makes use of the variation in the phase shift q ( A ) or classical deflection x([) derived from the differential cross section. In either case the radial integration variable in Eq. (5) is replaced by a new variable x, defined by the equation

x ( . )

= r2[l - V ( r ) / E ]

(25)

SEMICLASSICAL HEAVY-PARTICLE COLLISIONS

243

and the classical impact parameter b = A/k is introduced in place of A. Hence, for example, Eq. (10) may be written

X(b) =

6:

b(dy/dx)(x - b2)-l/’ dx

(26)

y(x) = h(x/r2) = In[l - V ( r) /E ]

(27)

where

Equation (26) is then inverted by an abelian transformation (Child, 1974a) to determine the function y(x) in the form

so that according to Eq. (27),

The inverse function x ( r ) determines the potential by means of Eq. (25). This classical procedure due to Firsov (1953) requires that the energy should be above the orbiting limit; otherwise dy/dx diverges over the integration range and the abelian transformation is invalid. A variant applied by Sabatier (1965), Vollmer and Kruger (1968), and Vollmer (1969) employs twice the derivative of the phase shift in place of ~ ( bin) Eqs. (28) and (29), as justified by Eq. (10). The extraction of the deflection function ~ ( l or ) the phase shift y(l) from the differential cross section presents some difficulty. The most satisfactory procedure due to Buck (1971) consists in optimizing the parameters in three separate piecewise approximations to the deflection function over the small-l, rainbow, and large-l regions, respectively. Application of this method to the high-quality data now available for mercury alkali systems (see Fig. 8) proves that the inverted potential is quite stable over a moderate range of collision energies, as shown in Fig. 9. Other methods related to parameterization of the phase shift have been suggested by Vollmer (1969), Remler (1971), and Klingbeil (1972). 2. Inversion of the s- Wave Phase Shift

Partial information about the potential may also be extracted from knowledge of the energy variation of the s-wave phase shift [l r-+ r

-

V(r)/E]’/’ dr - kr

+4 4

M . S . Child

244 25

20

i

15

I5

\

10

5

a

lo

tn,

IS

10

5

2:

2(

l!

/d

!

l

5

10

15

20

25

30

e

35

40

FIG.8. Measured c.m. differential cross sections for the scattering of sodium and mercury. [Taken from Buck and Pauly (1970) with permission.]

SEMICLASSICAL HEAVY-PARTICLE COLLISIONS

245

t 7 5

‘4

6

7

8

9

10

FIG.9. The Na-Hg potential obtained by inversion of the data in Fig. 8. Data are superimposed for the five different energies by use of the following symbols: 0,E = 2.87 x ergs; 0, E = 3.01 x ergs; A, E = 3.45 x E= ergs; 0, ergs; +, E = 3.13 x 4.02 x ergs. The solid line indicates the Lennard-Jones potential with the same potential minimum. [Taken from Buck and Pauly (1970) with permission.]

300

L

-c;

I

I

I

-

-

3I

2.0

I

2.5

I

3.0

rrk FIG. 10. The He4-He4 potential. Circles denote points obtained by inversion of the data in Fig. 7. The other curves are obtained by molecular beam scattering data: -, Farrar and Lee (1972); ---, Bennewitz et al. (1972); x x x , Gegenbach et a/. (1973); and by gaseous properties: -.-, Beck (1968). [Taken from Buck (1974) with permission.]

M . S. Child

246

which may be seen as a natural extension of the semiclassical quantum number derived from the Bohr quantization condition k n ( E )= -

jb[l - V(r)/E]1’2- 21 ~

7 1 0

An abelian transformation of Eq. (30) similar to that applied to Eq. (21) yields (Miller, 1969, 1971; Feltgen et al., 1973; Buck, 1974) the following expression for the energy dependence of the classical turning point a( U ) on the positive-energy part of the repulsive branch of the potential:

(32)

A comparison between the repulsive branch of the He, potential derived in this way from the symmetry oscillations shown in Fig. 5 (Feltgen et al., 1973) and that derived from the differential cross section (Farrar and Lee, 1972) is illustrated in Fig. 10.

111. Inelastic and Reactive Scattering The semiclassical treatment of inelastic and reactive scattering based on knowledge of exact classical trajectories is due to Pechukas (1969a,b), Miller (1970a,b), and Marcus (1970, 1971). The overall structure of the theory is similar to the Ford and Wheeler (1959a,b) analysis of elastic scattering. The first step is to represent the physical observable, in this case the S matrix, as an integral over a continuous family of classical trajectories. Marcus (1971) does this by exploiting the well-known connection between the Schrodinger and Hamilton-Jacobi equations (Born, 1960; Maslov, 1972) to obtain a generalized WKB form for the multidimensional wavefunction. This line has been reviewed in detail elsewhere (Child, 1974a, 1976a). The development that follows, due to Pechukas (1969a,b) and Miller (1970a,b),starts from the semiclassical limit of the Feynmann propagator (Feynmann and Hibbs, 1965). Other ramifications of this approach have been reviewed by Miller (1974, 1976b). The second step in the argument is to apply stationary phase and uniform approximation techniques to the reduction of the above integral to a form dependent only on the particular n, + n2 trajectories passing from the desired initial to the desired final state of the system. The techniques employed also show that the classical thresholds for a given process play the part of the caustics in the theory of elastic scattering, and more surprisingly that a

SEMICLASSICAL HEAVY-PARTICLE COLLISIONS

247

semiclassical description of classically forbidden events beyond these caustics may be obtained by analytical continuation of the classical equations of motion into complex time and coordinate domains. Examples given in Section II1,C show that this semiclassical description is remarkably accurate even for the most strongly forbidden classical events.

A.

~NTEGRALREPRESENTATIONS

The Feynmann propagator, denoted (qzlexp[-iff(t2

-

tl)/hllq,)

by Miller (1970a), which governs the quantum-mechanical evolution of a system from coordinatesq, at time t, to q, at time t,, is defined (Feynmann and Hibbs, 1965) by the equation (qzlexp[-iff(t,

-

~ l ) / h l ) q=J Jexp{iA[dt)l/hl d Path

(33)

where the integral is taken over all pathsq(t) leading from (qltl) to (q2t2),and A[q(t)] is the corresponding action A[q(t)l

=

1:'L[q(t), Mt),tl d t

(34)

with L[q(t), q(t), t] the classical Lagrangian (Goldstein, 1950), evaluated along the path q(t). This prescription is exact. The semiclassical reduction of Eq. (33) relies on the argument that the exponent A[q(t)]/h will typically be large under semiclassical conditions and sensitive to the path. Hence the dominant contribution to the integral will come from those paths around which the action is stationary. By Hamilton's principle (Goldstein, 1950) these are, of course, the classical paths from (qltl) to (q2t2).It is assumed for the sake of simplicity, that there is only one such path, in which case Eq. (33) reduces to
- tl

)lhllq 1 )

where S(q,t, ;q l t l ) is Hamilton's principal function obtained most frequently by integrating in Eq. (34) along the family of classical trajectories, although S(q2t2;q l t l ) may in principle also be derived by solving the HamiltonJacobi equation H [ V s,q]

+ (?S/c?t)= 0

(36)

where H(p, q) denotes the classical hamiltonian. The normalizing preexponent, containing the Van Vleck (1 928) determinant of second derivatives

248

M . S. Child

l?'S/dq, dq21, is chosen to be consistent with the unitarity of the propagator in the semiclassical limit (Fock, 1959; Miller, 1970a, 1974; Child, 1976a). The semiclassical Smatrix is derived by taking the limits t , + - 00, t2 + co, and contracting the above form for the propagator between the initial and final states +,hi)

= (qilni),

(37)

to obtain Sn,n, = J-mx

J% :

x ex~[iS(q2,ql)/hl+n,(q,) dq1 dq2

- t1)/h1lq1)(q11n2>dq1dq2

(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" =

<4p><+>

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 s