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Advances in

ATOMIC, MOLECULAR, AND OPTICAL PHYSICS

VOLUME 27

EDITORIAL BOARD

P. R. BERMAN New York University New York, New York

K. DOLDER The University of Newcastle-upon- Tyne Newcastle-upon-Tyne England M. GAVRILA F.O.M. Instituut voor Atoom- en Molecuulfysica Amsterdam The Netherlands M. INOKUTI Argonne National Laboratory Argonne, Illinois S. J. SMITH Joint Institute for Laboratory Astrophysics Boulder, Colorado

ADVANCES IN

ATOMIC, MOLECULAR, AND OPTICAL PHYSICS Edited by

Sir David Bates DEPARTMENT O F APPLIED MATHEMATICS AND THEORETICAL PHYSICS THE QUEEN’S UNIVERSITY O F BELFAST BELFAST, NORTHERN IRELAND

Benjamin Bederson DEPARTMENT O F PHYSICS NEW YORK UNIVERSITY NEWYORK,NEWYORK

VOLUME 27

@

ACADEMIC PRESS, INC.

Harcourt Brace Jovanovicb, Publishers Boston San Diego New York London Sydney Tokyo Toronto

This book is printed on acid-free paper. @ Copyright 0 1991 by Academic Press, Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.

ACADEMIC PRESS, INC. 1250 Sixth Avenue, San Diego, CA 92101

United Kingdom Edition published by ACADEMIC PRESS LIMITED 24-28 Oval Road, London NW1 7DX

ISBN 0-12-003827-7 ISSN 1049-25OX

PRINTED IN THE UNITED STATES OF AMERICA 90 91 92 93

9 8 7 6 5 4 3 2 1

Contents

CONTRIBUTORS

vii

Negative Ions: Structure and Spectra David R. Bates I. Atomic Anions 11. Diatomic Anions 111. Dipole-Supported States IV. Triatomic Anions V. Tetra-Atomic and More Complex Anions Acknowledgements References

1 2 23 39 44 59 69 69

Electron-Polarization Phenomena in Electron-Atom Collisions Joachim Kessler I. Introduction 11. Phenomena Governed by a Single Polarization Mechanism 111. Combined Effects of Several Polarization Mechanisms JV. Studies Still in an Initial Stage V. Conclusions Acknowledgements References

81 81 87 117 151 158 159 160

Electron-Atom Scattering I. E. McCarthy and E. Weigold I. Introduction 11. Formal Theory 111. Approximations for Hydrogenic Targets IV. Electron-Hydrogen Scattering V. Multielectron Atoms VI. Conclusions Acknowledgements References

165 165 166 175 182 189 198 198 199

Electron-Atom Ionization

I. E. McCarthy and E. Weigold I. Introduction 11. Theory of Ionization 111. Total Ionization Cross Sections: Asymmetries with Spin Polarized Atoms and Electrons IV. Double Differential Cross Sections V. Triple Differential Cross Sections VI. Conclusions Acknowledgements References

20 1 20 1 203 211 213 214 239 24 1 24 1

Role of Autoionizing States in Multiphoton Ionization of Complex Atoms V. I. Lmgyel and M . I. Haysak I. 11. 111. IV.

Introduction Quasienergy Method AIS Contribution Application of the Method to Calculation of the Two-Photon Ionization of Ca References

245 245 246 250 255 262

Multiphoton Ionization of Atomic Hydrogen Using Perturbation Theory E. Karule I. Introduction 11. Multiphoton Ionization of Atomic Hydrogen Within the Framework of Perturbation Theory 111. Sturmian Expansions IV. Analytical Continuation of the Transition Matrix Elements V. Theoretical Estimates and Experimental Data for Atomic Hydrogen References INDEX CONTENTS OF PREVIOUS VOLUMES

265 265 267 275 280 295 297 301 309

Contributors Numbers in parentheses refer to the pages on which the authors’ contributions begin.

David R. Bates (l), Department of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Belfast BT7 lNN, Northern Ireland M. I. Haysak (245), Uzhgorod Branch of the Institute for Nuclear Research, Academy of Sciences of the Ukraine, Uzhgorod, 294000, USSR E. Karule (265), Institute of Physics, Latvian SSR Academy of Sciences, Riga, Salaspils, USSR Joachim Kessler (8 l), Universitat Munster, Physikalisches Institut, WilhelmKlemm-Strasse 10, D-4400 Munster, West Germany

V. I. Lengyel(245), Uzhgorod University, Uzhgorod, 294000, USSR I. E. McCarthy (165,201), Electronic Structure of Materials Centre, School of Physical Sciences, The Flinders University of South Australia, Bedford Park, S.A. 5042, Australia E. Weigold (165, 201), Electronic Structure of Materials Centre, School of Physical Sciences, The Flinders University of South Australia, Bedford Park, S.A. 5042, Australia

This Page Intentionally Left Blank

ll

ADVANCES IN ATOMIC. MOLECULAR. AND OPTICAL PHYSICS. VOL. 27

NEGATIVE IONS: STRUCTURE AND SPECTRA DAVID R . BATES Department of Applied Mathematics and Theoretical Physics Queen’s University of Belfast Belfast. United Kingdom

I. Atomic Anions . . . . . . . . . . . . . A. Ground-State Electron Affinities . . . . . B. Excited States of Atomic Anions . . . . . C Effect of Electric and Magnetic Fields . . . D . Doubly Charged Anions . . . . . . . . I1. Diatomic Anions . . . . . . . . . . . . A. Few-Electron Systems . . . . . . . . B. Homonuclear Anions . . . . . . . . . C . Main Heteronuclear Family . . . . . . D . Hydrides . . . . . . . . . . . . . . I11. Dipole-Supported States . . . . . . . . . A . Theory . . . . . . . . . . . . . . . B. Experiment . . . . . . . . . . . . . IV. TriatomicAnions . . . . . . . . . . . . A . Systems of Isoelectronic Atoms . . . . . B. Dihydrides . . . . . . . . . . . . . C. Monohydrides . . . . . . . . . . . . D. Other Triatomic Anions . . . . . . . . V. Tetra-Atomic and More Complex Anions . . . A. AH;, Family . . . . . . . . . . . . B. Inorganic Anions . . . . . . . . . . . C. Organic Anions . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . References . . . . . . . . . . . . . . .

.

. . . . . . . . .

. . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . . . . .

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

. . . . . . . . . .

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

2 2 12 19 21 23 23 26 30 34 39 39 42

44

. . . . . . . . . . 44

. . . . . . . . . .

. . . . .

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

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

49 53 58 59 59 61 65 69 69

Massey once told me that of the monographs he had written. Negative Ions was his favourite. The third edition (Massey. 1976) covers research published up until April 1974. Massey (1979) has written an article updating it to August 1978. The present Chapter attempts further updating on the structure and spectra of negative ions (other than large clusters. see Mark and Castleman. 1985). Collisions and applications will be treated later. Since August 1978. great progress has made. mainly. but not entirely. due to laser photoelectron spectroscopy. to tunable laser photodetachment threshold studies. and to ab initio quanta1 calculations. For example. new stable atomic 1 Copyright Q 1991 by Academic Press. Inc. All rights of reproduction in any lorn reserved. ISBN 0-12-003827-7

David R. Bates

2

anions have been discovered (Section I.A. 1); some unexpected excited atomic anions have been investigated and found to have interesting properties (Section I.B.2); the predicted dipole-supported type of excited state (Section 1II.A) has been observed (Section 1II.B); and the spectroscopic constants of many diatomic (Section 11) and polyatomic (Sections IV and V) molecules have been determined with, in a number of instances, very high accuracy. Attention will be focused on the results (that is, the properties of the anions) rather than the methods used in obtaining them.

I. Atomic Anions A. GROUND-STATE ELECTRON AFFINITIES Experimenters are ahead of theorists in the determination of the groundstate adiabatic electron affinities (EAs) the sole exception being the special case of H for which the best value remains that provided by the renowned calculation of Pekeris (1962). The most widely used methods involve laser photodetachment electron spectrometry and photodetachment threshold studies with tunable lasers or conventional light sources. (See Drzaic et al., 1984; Mead et a!. 1984b.) The uncertainties are typically 2-20 meV but for 0 and S the electron affinities are known to within 1 x eV. Hotop and Lineberger (1985) have given an authoritative review of the field with recommended electron affinities of all atoms, other than the rare earths, up to Rn (atomic number 86). Table I cites their recommendations for the lighter atoms. As already noted, H is a special case. The best conventional EA measurement is that of Feldmann (1975) based on a photodetachment threshold TABLE I ELECTRON AFFINITIES (eV) OF LIGHTERATOMS He 0.007

0.099 Mn 0.869 f 0.01oY

Fe 0.934 f 0.011y

z N 0.374,362 f 0.000005' 0.18/, 0.01'

0 1.827,608 f 0.000021d 1.79', 1.76' P S 1.028 f O.Ol@ 2.314 f 0.003' 0.76k 2.22p, 2.35q Se 2.21 ;eo.03' measured 2.102 _+ 0.015"

{

co 0.671 f 0.010'"

Ni 0.481 f 0.007"

Source: 'Rackwitz et al. (1977); * Kasdan et al. (1975a); Al-Za'al et al. (1987); Schulz et al. (1982); Karo et al. (1978); Frenking and Koch (1986); 'Ortiz (1987a); 'Rosmus and Meyer (1978); Kalcher and Janoschek (1986); j Kasdan et al. (1975b); 'Zittel and Lineberger (1976); Janousek and Brauman (1981); " Olson and Liu (1980); " Lewerenz et al. (1983); Ortiz (1987b), q Ortiz (1988a); ' Stevens et al. (1981); ' Spence et al. (1982); 'Chapman et al. (1988); " Smyth and Brauman (1972); ' Freidho5 et al. (1986b); Stevens Miller et al. (1987); Stevens et al. (1983).

David R. Bates

36

TABLE XIX MEASURED (M) AND CALCULATED (C) SPECTROSCOPIC CONSTANTS OF HYDRIDE ANIONS ~~~

~

T,

rc

we

we%

Be

a,

Do

(eV)

(A)

(cm-')

(cm-')

(cm-')

(cm-')

(eV)

0 0 0 0.845 0.875

1.15 1.139 1.142 1.10 1.136

2620 2594 2577 3000 2650

83 80

13.96 13.91

0.670 0.660

-

-

-

73

14.05

0.620

-

0 0 0.57 0.60 1.05 1.06

1.57 1.568 1.50 1.563 1.50 1.560

1804 1850 2100 1876 2100 1908

-

-

-

46 40

7.05

0.255

2.99 2.94 3.29 3.23 3.30 3.26

0 0 0 3.61

1.040 1.039 1.043 1.040

OH-

0 0 0

SH-

0 0 0 0 0.10 0 0 0 0

Anion CH-

SiiH -

NH-

CrHMnHFeHCoHNiH-

State

Reference

-

-

7.09

0.215

3.44 3.32 ~

3.83

-

-

-

37

7.12

3191 3226 3173 3157

86 85 89 85

16.58 16.61 16.48 16.57

0.197 0.706 0.691 0.731 0.739

0.964 0.961 0.966

3738 3809 3732

91 94 99

19.12 19.23 19.05

0.772 0.766 0.795

(4.5) 4.62 4.56

1.343 1.348 1.346 1.75 1.75 1.82 1.74 1.67 1.61

2647 2642 2637

53 52 52

9.56 9.49 9.52

0.297 0.300 0.298

(3.92) 3.74

-

-

-

1125 1050 1250 1300 1430

-

~

-

-

-

-

-

-

-

~

~

-

-

3.60 ~

~

~

~

-

Source: (a) Kasdan et al. (1975a); (b) Rosmus and Meyer (1978); (c) Manz et al. (1986); (d) Kasdan et al. (1975b); (e) Lewerenz et al. (1983); (f) Miller and Farley (1987), Al-Za'al et al. (1987); (g) Rosenbaum et al. (1986); (h) Werner et al. (1983); (i) Gruebele et al. (1987); (j) Senekowitsch et al. (1985); (k) Stevens Miller et al. (1987); (I) Stevens et al. (1983).

X 3C-, u = 0 threshold. They observed that the signal decayed biexponentially with trap time, there being one excited state of lifetime 5.3-6.7 s and another of lifetime 1.75 & 0.15 ms. They identified the excited states as a 'A and X 'C-, v = 1, respectively, and the decay processes as the forbidden radiative transition a 'A + X 3X-,and the u = 1 + 0 infrared transition. In excellent agreement with the measurement of Okumura et al. Manz et al.

NEGATIVE IONS: STRUCTURE AND SPECTRA

37

(1986) have calculated that the lifetime toward the infrared transition is 1.96 ms. Neumark et al. (1985) obtained the infrared rotation-vibration spectrum of NH - in a coaxial ion-beam laser-beam spectrometer-the first anion infrared spectrum reported. They excited the u = 0 + 1 transitions of the anion. Autodetachment then ensued and the products were monitored. Neumark et al. resolved the fine structure and the A-doubling. They also measured the linewidths of the autodetachment resonances. For the upper A-doublet levels the linewidth increases by two orders of magnitude as the quantum number N goes from 2 to 11 with the corresponding lifetimes decreasing from 5 ns to 60 ps. In contrast, for the lower A-doublet levels, the linewidth scarcely varies with the rotation. Here the lines are so narrow that their widths may be attributed at least in part to unresolved fine structure. Consequently, the widths provide only lower bounds to the autodetachment lifetimes. These lower bounds are about 5 and 2.5 ns for the 21-13/2 and 21-11/2 spin-orbit states, respectively. Neumark et al, reason that the autodetachment occurs because of rotational-electronic coupling and that this coupling is much stronger for the upper A-doublet than it is for the lower A-doublet because of different orbital orientations. Similar high-resolution measurements by Miller and his associates (Al-Za'al et al. 1986, 1987, Miller and Farley, 1987), who used and isotopic substitution (14NH- and 15NH-) to extract the harmonic (0,) anharmonic (o,x,) contributions to the energy, gave the spectroscopic constants with high precision (Table XIX). It is striking how well the predictions of Rosmus and Mayer (1978) and Manz et al. (1986) agree with them. There is equally good agreement between the measured and calculated constants of the other hydrides to be considered. Illustrative of the scale and sophistication of the experimental effort Al'Za'al et al. (1987) measured a total of 114 transitions in the P, Q, and R branches of the fundamental band near 3000cm-' with an accuracy of 0.01 cm-'. Miller et al. (1987) determined the 14NH- and 15NH- hyperfine parameters. Manz et al. (1986) calculated that the lifetime toward the u = 1 + 0 transition is 3.3 ms. They also calculated that the lifetime of (A 'X+, u = 0) NH- toward a radiative transition to the X 21-1 state is 226 ns. They reckon that its lifetime toward autodetachment is probably an order of magnitude shorter. Owrutsky et al. (1985) observed OH- by direct absorption spectroscopy using the velocity-modulation-laser method (Gudeman and Saykally, 1984). In this and later work (Rosenbaum et al. 1986) they measured 23 transitions of the fundamental vibration of 160H- (band center near 3550 cm-') and of the "OH- isotopomer. Their deduced values of the spectroscopic constants

38

David R. Bates

(Table XIX) are very accurate. Werner et al. (1983) calculated the lifetime toward the u = 1 -+ 0 transition to be 7.3 ms and the dipole moment in the vibrational ground state to be 1.04 D. Helped by knowing the spectroscopic constants, Liu and Oka (1986) and Lui et al. (1987) searched for the pure rotational spectrum in the far infrared region. They found the J = 10 c 9 and J = 1 1 c 10 transitions within 0.008 and 0.016 cm- respectively, of the predicted positions. Bae et al. (1986) discovered that OH- ions produced from an H 2 0 + beam passing through a Cs vapor cell participate in slow autodetachment. They demonstrated that the autodetachment is due to vibrationally excited ions (u 2 5) in the ground electronic state. They calculated the lifetime z(u) toward the process, finding that for u = 5, 6, 7, and 8, ~ ( u is ) 24, 3.3, 0.6, and 0.2 ,us, respectively (consistent with earlier calculations by Acharya et al. 1984). Ab initio calculations on the effect of an electric field on OH- have been done by Pluta et al. (1988b) and by Adamowicz (1988). Using the velocity-modulation technique, Gruebele et al. (1987) observed transitions in the u = 0 -, 1 and o = 0 -, 2 bands of 32SH- and 34SH-. From their data they determined the spectroscopic constants (Table XIX). As in the cases of NH- and OH-, the accuracy achieved is impressive. Senekowitsch et al. (1985) have calculated that the lifetime toward the u = 1 + 0 transition is 13.3 ms and that the dipole moment in the u = 0 state is 0.273 D. As already indicated, HCl does not have a positive EA. However, there are states of HCl- whose energy is below that of the X 'C+ state of HCl at moderate and large internuclear distances and are therefore of interest in the context of dissociative attachment and associative detachment. ( c - Krauss and Stevens, 1981; Bettendorff et al. 1983.) ONeil et al. (1986) have carried out ab initio calculations on the 1 'X+ (Cl- H) and 1 'I7 (C1 H-) states using a basis set capable of describing both the diffuse nature of the hydride anion and the polarizability of the atomic constituents. Their results are expected to be accurate. They find that at all internuclear distances, the 1 'C' potential is below the asymptotic C1- + H energy and that it has a minimum at 2.1 A where the well depth is 0.08 eV; and they find that the 1 'TI potential is repulsive even at large internuclear distances. ONeil et al. did not do such extensive calculations on the 2 'Z'(C1 H-) state as on the others but judged that its potential is probably also purely repulsive. The predicted barrier to associative detachment on both the 1 'II and the 2 'Z' channel is 0.25 eV.

',

+

+

+


39

111. Dipole-Supported States A. THEORY The problem of the motion of an electron under the influence of a fixed finite dipole of moment dD, where D, as usual is the Debye unit lO-"esucm, has been treated by several groups (Wallis et al., 1960; Mittleman and Myerscough, 1966; Crawford, 1967; Turner et al. 1968; and others). The wave function is separable in elliptical coordinates and may be written

m9 = u4M(dexp( * W),

(25)

cp being the azimuthal angle, m an integer, and

A = (rl

+ r2)/R,

P = (rl

-rd/R

(26)

in which R is the separation of the poles and r l and r2 are the distances of the electron from them. It is characterized by quantum numbers (n,, n,, m) where n, and n, are the number of nodes in L(I) and M(p). A bound state is only possible if the dipole moment exceeds a critical value d(n,, n,, m)D. The entity d(n,, n,, m) is independent of n, but rises sharply as n, and m are increased from zero: thus d(n,, 0,O)= 1.625,

d(n,, 0, 1) = 9.64,

d(n,, 1,0) = 19.2.

(27)

This has an important consequence since the dipole moments of molecules are less than about 12D, usually indeed much less. Largely because of result (27), attention may be confined to (n,, 0,O)states-that is, to C states of the dipole-supported electron. Letting ~ ( n , )denote their binding energies, the general pattern as d is raised above the critical value 1.625 is that the largest, E(O), increases rapidly from zero while the ratios E(O)/E( 1) and E( 1)/~(2) decrease rapidly from infinity but remain large in the d range of interest. The adoption of a dipole that is fixed in orientation corresponds to making the Born-Oppenheimer approximation that the moment of inertia I of the molecule may be taken to be infinite. In reality, the rotation of the molecule is too rapid to be followed by the orbital motion of the electron when the binding energy is low. The effect of the invalidity of the Born-Oppenheimer approximation has been investigated by Garrett (1971, 1980, 1982). He naturally confined his attention to 2 states. Considering the case of a free rotor, Garrett (1980) set up and solved the coupled radial equations that arise

40

David R. Bates

when the Born-Oppenheimer approximation is not made. He found that the critical dipole moment is a decreasing function of I, the moment of inertia, and is an increasing function of R, the charge separation. The most important aspect of these variations is the effect of reducing I from infinity (the fixeddipole case) to values in the range (lo4 to lo6 mea ; ) encountered in practice. Results relating to this are given in Table XX. The critical dipole moments are enhanced (especially for the excited states). Garrett (1982) also carried out similar calculations on lithium hydride and the lithium halide anions but including in the potential the induced dipole term and a repulsive term tailored to make EA(LiC1) have the value measured by Carlsten et aZ. (1976). The binding energies in the excited states (Table XXI) are minute. Several revealing calculations on dipole-supported states have been done assuming that the Born-Oppenheimer (BO) approximation is valid. In comparing the results with those in Table XXI or assessing their reliability, it should be noted that the error due to this assumption is an increasing function of the ratio of the rotational constant to the binding energy. An anion in a dipole-supported excited state has virtually the same spectroscopic constants as the neutral molecule. The most wide-ranging investigation is that by Adamowicz and McCullough (1984a), who used the simple Koopman’s theorem (KT) approximation EA(XY)

E(XY) - E(XY-) = - E(IIU)

(28)

where ~ ( mis) the virtual (unoccupied) no orbital energy for the neutral molecule. This approximation neglects relaxation of the neutral-molecule TABLE XX DEPENDENCE OF CRITICAL DIPOLE MOMENT (IN DEBYE)OF FREEROTOR (CHARGE SEPARATION 4 a,) ON MOMENT OF INERTIAI (UNITSmeat) Critical dipole moment I 1 1 1 1 1

x 104

x 105 x lo6 x 107 x lo8

OD

n, = 0

nA = 1

n, = 2

2.14 1.91

3.85 3.29 2.91 2.65 2.48 1.625

5.1 4.3

-

1.82 1.625

Source:After Garrett (1980).

-

1.625

41

NEGATIVE IONS: STRUCTURE AND SPECTRA TABLE XXI CALCULATED ENERGIES (eV)

OF

DIPOLE-SUPPORTED STATESOF DIMERANIONS FREETO ROTATE LiH-

dipole moment (D) moment of inertia (me a;)

LiF-

5.9 2.6 x 104

nA

0 1

0 0

0.32 2.62 x 10-3 1.96 x 10-3 0.89 x 10-3

1

2 3 0 1

3

-

-

6.3 7.1 8.2 x 104 1.5 x 105 negative of energy 0.38 0.61 4.58 x 10-3 1.02 10-1 4.47 x 10-3 * 3.97 x 10-3 * 3.47 x 10-3 * 1.19 x 10-5 1.33 x 10-4 -

Source: After Garrett (1982). Note: --signifies that state is not bound;

LiI -

LiCI-

*

7.4 2.4 x 105 0.68 1.39 x 1.38 x 1.37 x 1.35 x lo-' 2.72 x 10-4

*

* signifies that calculations were not done.

orbitals upon anion formation and neglects electron correlation in anion and neutral. Relaxation is expected to be unimportant for the excited states, but there is evidence that electron correlation has a significant effect. Table XXII gives the results of the investigation and also Be for the relevant neutral molecules (from Huber and Herzberg, 1979). Predicted binding energies less than Be are bracketed. Adamowicz and McCullough (1984b) found that taking electron correlation into account increased the calculated binding eV. energy of the first excited state of LiH- from 1.9 x l o v 3eV to 2.6 x Comparison with the results in Table XXI suggests that the BO does not TABLE XXII FOR ANIONS COMPUTED KOOPMAN'S THEOREM BINDINGENERGIES

Anion

LiH -

dipole moment (D) Be (eV)

6.0 9.3 x 10-4

"1

0 1 2 3 4

0.18 1.9 x 10-3 (2.0 x 10-5) (2.0 x 10-7)

LiF-

LiCI-

6.4 7.3 1.7 x 8.8 x lo-' binding energy (eV) for J 0.27 0.41 3.6 x 10-3 8.0 x 10-3 (4.8 x 10-5) 1.6 x 10-4 (3 x (6 x (5 x 10-8)

5

Source: After Adamowicz and McCullough (1984a).

NaF-

MgO-

8.3

9.1 7.1 x 10-5

5.4 x lo-' =0 0.40

1.1 x 10-2 3.2 x 10-4 (9 x (2 x 10-7)

1.33 4.4 x 1 0 - 2 1.5 x 10-3 (4.8 x 1 0 - 5 ) (1.6 x (5 x 10-8)

42

David R. Bates

cause much error in the unbracketed values of Table XXII, but that the states having bracketed values are unbound with the possible exception of MgO (nl = 3). Adamowicz and Bartlett (1985, 1986a, 1988) have carried out more accurate BO approximation calculations on the first excited states of BeO-, LiF-, LiCl-, NaF-, and NaCl-. They got the binding energies to be 3.8 x lop3, 9 x 1.2 x and 2.1 x lO-’eV, respec2.3 x tively. Although the dipole moment of HF is 1.9D, which is rather above the critical value given in Eq. (27), Adamowicz and McCullough (1984a) found no bound state of HF- in their KT calculations. They deduced that the ground-state binding energy must be less than about lo-’ eV. Since Be is eV (Huber and Herzberg, 1979),there can be no bound state. The 2.6 x dipole moments of the hydrogen halides decrease in going down the Periodic Table, yet HI forms a stable anion (Section 1I.D). B. EXPERIMENT The first persuasive laboratory evidence for dipole-supported states came from high-resolution photodetachment cross section measurements on substituted acetophenone enolate anions (Zimmerman and Brauman, 1977; Jackson et al., 1979) and an acetaldehyde enolate anions (Jackson et al., 1981a). Wetmore et al. (1980) have done calculations on the latter that suggest an excited dipole-supported state is likely (indeed, would be expected, the dipole moment of the neutral molecule being 3.54 D according to Wetmore et al. or 3.19 D according to Huyser et al., 1982). The experimental research revealed a series of resonances (halfwidth about 1 nm) beginning at threshold. They were interpreted as being due to transitions to dipolesupported states that underwent autodetachment. Incidentally, the following EAs were reported: acetophenone g-fluoro, 2.215 eV, p-fluoro, 2.173 eV, 2.054 eV, P --C(CH~)~ 2.029 , eV, =-methyl, 2.027 ek, all f0.008eV p-H, (Jackson et al., 1979); CH,CHO, 1.824 eV, CD2CD0, 1.819 eV, both k0.005eV (Jackson et al., 1981a). Mead et al. (1984a) greatly extended the acetaldehyde enolate anion, CH, CHO -, work. Using ultra high-resolution photodetachment spectroscopy they detected about 50 narrow resonances near threshold and measured their positions and widths very accurately. From an analysis of their data they determined the molecular constants (A, B, C) for the excited anion,


43

finding them to be almost the same as those for the neutral radical and they determined the binding energy to be as low as between 5.1 x and 7.3 x 10-4eV. Both these results are characteristic of an excited dipolesupported state. The linewidths of the autodetaching levels increase rapidly with rotational quantum number. Marks et al. (1985) have given a neat demonstration of the essential role played by the dipole by an experiment on o- and p-benzoquine. The former species, OBQ, has an oxygen atom on each of two neighboring corners of the benzene hexagon, giving it a dipole moment of 5.1 D. The latter species, PBQ, is similar except that the oxygen atoms are on opposite corners of the hexagon so that the molecule is symmetric and without any dipole moment. On observing the photodetachment spectra, Marks et al. found threshold resonances for OBQ-, but not for PBQ- in accordance with expectations. They reported that EA(0BQ) = 1.62 eV and EA(PBQ) = 1.99 eV. The cyanomethyl anion CH,CN- and its deuterated analog have been investigated by Marks et a/. (1986), who measured the photodetachment spectra, and by Lykke et al. (1987), who applied the technique of autodetachment spectroscopy. Both groups found a dipole-supported state. Lykke et al., who had the higher resolution, observed as many as 5000 sharp features near the photodetachment threshold, all of which were assigned. They showed that the binding energy of the dipole-supported state of CH,CN- is less than 8.2 x eV. This corresponds to the energy of the lowest final state (the J = 9, K = 2 state) they could detect by its autodetachment. It is therefore an upper limit. Moreover, there may be one or more lower states that autodetach too slowly to be detected. A method for calculating the relative positions and widths of lines in the photodetachment spectra from the anions of polar molecules has been developed by Clary (1988). It is based on determining the rotationally adiabatic potentials on which the electron moves and finding the positions and widths of shape or Feshbach resonances on these potentials. Clary has applied his method to CH,CHO- and CH,CN- with signal success. Marks et al. (1988) have carried out high-resolution photodetachment spectroscopy measurements on the acetyl fluoride enolate anion CH,COFand its deuterated analog. The parent neutral, like the cyanomethyl radical, has a dipole moment of some 3-5 D. Marks et al. observed over 200 narrow resonances near the threshold and inferred that CH,COF- has a dipolesupported state having a binding energy of less than 4.3 x eV. Andersen et al. (1987) have studied FeO- by autodetachment spectroscopy. Their analysis of their data was facilitated by the theoretical work on

44

David R. Bates

the electronic structure and properties of FeO by Krauss and Stevens (1985). This showed that the dipole moment in the X 'A state is 3.4 D (about the same as that of CH,-CHO). There are numerous low-lying states. (cf: Cheung et al., 1984.) Andersen et al. observed five multi-transition bands and deduced that the ground state is an inverted 4A state and that near the threshold energy there are B 4A7/2 and A 4A5/2 states together with a C state of undetermined type. They noted that the increase with J of the autodetachment rates for the B 4A,/2 and C states is consistent with them being dipole-supported states but that the corresponding relatively slow increase for the A 4A5/2 state probably is not. The f0.0050-eV uncertainty in EA(Fe0) is reflected in the measured binding energies. In the case of B 4A7/2, the measured binding energy with respect to (5A4) FeO is 0.0037 0.0050 eV so that the actual binding energy may well be as low as would be expected for a dipole-supported state. However, in the case of A4A,/,, the measured binding energy with respect to ('A3) FeO is 0.0175 f 0.0050eV, which indicates an excited valence state. The assignment that Andersen et al. (1987) favor is A, valence; B and C, dipole-supported.

IV. Triatomic Anions The geometric configuration of the ground state of a polyatomic ion may not be the same as that of the neutral molecule. (See Massey, 1976.) According to the rules of Mulliken (1942,1958) and Walsh (1953), a triatomic molecule not containing hydrogen is linear in its ground state if it has 16 or fewer valence electrons; with 17 valence electrons the equilibrium configuration is bent with a bond angle of about 135" and with 18 valence electrons is about 120". The adiabatic electron affinity, which we always denote by EA, may differ markedly from the vertical electron affinity, which we will denote by VEA, if the configurations of the neutral molecule and its anion are not the same. (See Massey, 1976.)

A. SYSTEMS OF ISOELECTRONIC ATOMS As part of the study of H i clusters, SCF configuration interaction calculations on H; have been done by Rayez et al. (1981) and by Hirao and


45

Yamabe (1983). These show that the optimum geometry is linear with a bond length of 1.08 A but give the binding energy to be so small as to leave the existence of H; in doubt. Laser microprobe studies on carbon vapor (Furstenau et a1 1979) provide indirect evidence that Cn- clusters with n clusters with n up to 20 can exist as linear chains; and Mulliken's rules give that C; is linear. Sunil et al. (1984) have done MCSCF calculations. They deduced that EA(C,) is 2.0 eV. Confining their attention to the equilibrium geometry, which they took to have a C-C bond length of 1.283 A, they searched for a state, other than the X 211, state, that is stable toward autodetachment. They found none. However, they did not exclude the possibility that the 4Cu- state would be found to be stable if the C-C bond length were optimized. The azide anion N; is isoelectronic with N 2 0 and is linear in the X ' C l state. Jackson et al. (1981b) have generated it efficiently in an ion cyclotron resonance spectrometer by using azidotrimethylsilane, (CH,), SiN, as a precursor. It is a product of the fast ion-molecule reaction between (CH,)SiN- that results from the initial electron impact and azidotrimethylsilane. From photodetachment threshold measurements Jackson et al. determined that EA(N,) is 2.70 & 0.12 eV. Based in part on solid-state data, they gave r(NN) = 1.187 A, v 1 = 1350 cm-', v 2 = 640 cm-', and v , = 2020 crn-l., There have been a number of ab initio calculations. Botschwina (1986b) employed a larger basis set than did earlier investigators. He used the coupled electron pair approximation. Although he obtained results for six isotopomers, only those for the (14, 14, 14) form will be cited here. Botschwina got that r(NN) is 1.1911 A and stated that this is probably too large by about 0.005 A due mainly to the incompleteness of the basis set. He did not compute v 2 but got that v1 = 1295 cm-', v, = 1950 cm-', Be = 0.424 cm-', a,(sym) = 0.0016 cm-', and a,(asym) = 0.0039 cm-'. He also computed other entities, the most interesting being perhaps the integrated molar absorption intensities for the v, and v 1 v, bands. The value for the first of these is very high (6.6 x lo4 cm2 mol-'). The search region having been established by Botschwina's value of v,, Polak et al. (1987) used diode laser velocity modulation spectroscopy to detect and measure 34 transitions in the asymmetric stretch fundamental of N; in a NH, - N 2 0 discharge where it is produced by

+

NH; Note: v ,

3

v(sym), v2

+N20

--*

N3

= v(bend), v3 E v(asym).

+H20

(29)

46

David R. Bates

(Bierbaum et d., 1984). Their deduced values of the spectroscopic constants are in very good agreement with those of Botschwina. The only significant difference is that the measured v3 at 1986.47 cm-' is 36 cm-' greater than predicted. Novick et d. (1979a) have made a thorough study of the ozonide anion 0; by fixed-frequency-laser photoelectron spectrometry and tunable-frequencylaser photodetachment and photodestruction spectroscopy. Their photodetachment threshold set EA(0,) = 2.1028 & 0.0025 eV. They analyzed progressions in the spectra. Their photoelectron measurements gave v1 = 1040 & 60 cm-' while their photodetachment measurements gave more precisely v1 = 982 & 30 cm-'. Their photodissociation measurements gave v 2 = 550 f 50 cm-'. Earlier, Cosby et al. (1978) had carried out a tunablelaser photodissociation study of 0;that revealed considerable structure in the cross section. To explain this they invoked quasi-bound excited states 'A, or 'A, to which allowed dipole transitions from the X 'B, state can occur. Their observed structure is consistent with two alternative excitation energies, 2.163 and 2.146 eV. Values of v 1 and v 2 for the X 2B, state were obtained for each. Neither pair agrees with the corresponding values of Novick et al. (1979a). Further tunable-laser photodissociation measurements by Hiller and Vestal (1981) confirmed the structure. However, they found that all the experimental data are consistent with taking the 'A, and 'A, excitation energies to be 2.047 and 2.655eV. In particular v1 and vz of X 'B, then become 975 f 10 and 590 f 10 cm-' in excellent agreement with Novick et al. For the excited states Hiller and Vestal got that vl and v2 are 815 L 10 cm-' and 275 f 10 cm-'('A,) and 760 & 20 cm-' and 190 & 20 cm-'('A,). Dissociation of the excited states may occur along the reaction coordinate corresponding to the asymmetric stretching motion. Information on the geometry of X 'B, has been provided by Cederbaum et al. (1977). They used SCF calculations on O 3 as a starting point for the application of the Green's function method to determine directly the effect of attaching an electron. In encouraging agreement with the photodetachment measurement, they got that EA(0,) = 2.17 eV. They also calculated that the attachment would increase the bond length by O . x ( f r o m 1.27 to 1.37 A) and would decrease the angle between the bonds, 000, by 3.6"(from 117 to 113.4").Matrix work (Jacox and Milligan, 1972, Andrews and Spiker, 1973) gave that the harmonic frequency w3 = 800 cm-'. Noting that it also gave o1= 1010 cm-' and o3= 600 cm- we would expect o3* 770 cm- '. Nimlos and Ellison (1986) have studied the photoelectron spectroscopy of S ; and (keeping S the central atom) of S 2 0 - and SO; by a fixed-frequency laser beam. They determined that EA(S,) = 2.093 f 0.025 eV, EA(S,O) =

'


47

1.877 f 0.008 eV and EA(S0,) = 1.107 f 0.008 eV. von Niessen and Tomasello (1987) got EA(S,) = 2.34eV and Cederbaum et al. (1977) got EA(S0,) = 0.93 eV both using the Green's function method; and Hirao (1985) got EA(S0,) = 1.027 eV by a symmetry-adapted cluster CI calculation. In some cases Nimlos and Ellison (1986) were able to obtain harmonic frequencies from the few observed progressions and information on the geometry from a Franck-Condon analysis of the relative intensities. Table XXIII gives their results together with the theoretical results of Cederbaum et al. (1977) and Hirao (1985). For S; it also gives the frequencies measured by solution-phase Raman spectroscopy (Chivers and Drummond, 1972) and the bond length of the matrix-isolated ion measured by electron paramagnetic resonance spectroscopy (Lin and Lunsford, 1978); and for SO; it gives the frequencies and bond length measured by matrix infrared spectroscopy (Milligan and Jacox, 1971). The SO; comparison, like the similar 0; comparison, shows that matrix measurements provide useful information on the free anion. The bond energies of O,, S;, S 2 0 - , and SO; are collected together in Table XXIV. Hiller and Vestal (1981) deduced from their results that 0.747 f 0.013 eV is an upper limit to the bond energy of 0,. This is well below the value, 1.05 f 0.02 eV, commonly adopted. Whereas von Niessen and Tomasello (1987) had found that S; is bound only in the open (C,") form, von Niessen et al. (1989) found that Se; and Te; are also bound in the closed (D3h)form and in the case of the latter there is probably an excited state. If the C,, or D3hsymmetry be kept unchanged,

TABLE XXIII SPECTROSCOPIC CONSTANTS AND GEOMETRY OF S;, SO;,

AND $0-

Harmonic frequency (cm- ') Anion

s; so;

a1

535 985 944f48 ~

s,o-

W2

0 2

232 496 435f 100

585 1042

-

-

-

-

-

-

-

-

SS stretch

-

620 f 150

Bond length (A)

Bond angle (degrees)

2.10 1.49 1.523 f 0.020 1.49 1.50

-

s-s

2.010 f 0.020

Reference

-

a b

115.6 f 2 116.2 113.8

d e

-

-

-

C

C

Source: (a) Chivers and Drummond (1972), Lin and Lunsford (1978); (b) Milligan and Jacox (1971); (c) Nimlos and Ellison (1986); (d) Cederbaum et al. (1977); (e) Hirao (1985).

48

David R. Bates TABLE XXIV

BONDDISSOCIATION ENERGIES Anion 0;

s; $0-

so;

Products

+

0, 00;+ o

s, + ss; + s s, + 0s; + 0 so + sso- + s so + 0so- + 0

Do (eV) 1.74 2.78 2.87 3.26 4.88 4.67 3.31 4.26 5.37 5.71

Source: After Nimlos and Ellison (1986).

the calculated values of the VEA to the various negative ion states are as follows: ('B1) S ; , Czu,2.07 eV; ('A;) S ; , D3h, -0.035 eV; ('B1) Se;, Czu,2.03 eV; ('A;) Se;, D,,, 0.234 eV; ('B1) Te;, CZv,2.42 eV; ('A1) Te;, CZu,0.233 eV; ('A;) Te;, D,,,, 1.01 eV; and ('E') Te;, D,,,0.469 eV. Metallic triatomic anions have been studied because of the general interest in cluster ions. As decribed by Leopold et al. (1987), they may be prepared in a flowing afterglow ion-source incorporating a cold cathode dc discharge. Metal atoms and clusters are sputtered from the cathode (which is of appropriate material) by bombardment with Ar or other cations. Anions are formed by further interactions with the dense plasma. Basch (1981) has investigated Ag; using an ab initio relativistic effective core potential and SCF and CI methods. He established that the X 'Z; state is linear. Although the computed EA(Ag,) = 1.40 eV, Basch reckoned that the true value is close to the 2.0 eV Baetzold (1978) had obtained with the aid of semiempirical molecular orbital methods. He predicted that the anion has a 3B, state that has excitation energy of 0.85 eV and is equilateral with an angle of 122" between the bonds. +


49

A member of the family Cu, has been studied both experimentally and theoretically. Leopold et al. (1987) used fixed-frequency laser spectrometry. Bauschlicher et al. (1988) have carried out modified coupled pair functional calculations. The research of Leopold et al. gives that EA(Cu,) = 2.40 f 0.10 eV. Knowing that the dissociation energy D,(Cu, - Cu) = 1.08 0.19 eV(Hi1pert and Gingerich, 1980)and havingmeasured EA(Cu) = 1.235 f 0.005 eV and EA(Cu,) = 0.842 f 0.010 eV, Leopold et al. noted that D,(Cu, - Cu-) = Do(Cu3)

+ EA(Cu,) - EA(Cu) = 2.25 f 0.30 eV

(30a)

+ EA(Cu,) - EA(Cu,) = 2.64 f 0.30 eV.

(30b)

and D,(Cu - Cu;) = Do(Cu3)

The trimer bond is thus much stronger in the anion than in the neutral molecule even though the extra electron is expected to be in a nonbonding orbital. There is a similar difference between the azonide anion (Table XXIV) and ozone and in other cases. Following Gole et al. (1980) Leopold et al. attributed the effect to the greater stability of an electron when delocalized. By their calculations Bauschlicher et al. (1988) confirmed the suggestion of Leopold et al. that the X 'Xi state of Cu; is linear and that there is a state of about 0.9 eV excitation energy. They showed that this state is ,A; (equilateral triangle). From the photoelectron spectra of Ni;, Pd;, and Pt; Ervin et al. (1988) have determined that EA(Ni,) = 1.41 f 0.05 eV, EA(Pd,) 5 1.5 IfI 0.1 eV, and EA(Pt,) = 1.87 k 0.02 eV. It was found that each of the neutral trimers has multiple low-lying electronic states. The alkali trimer anions are linear by the Mulliken- Walsh rules. Calculations on the six that can be formed from Li and Na have been carried out by Ortiz (1988~).He used several approximations, the most refined being electron propagator theory carried to partial fourth order. For each anion he considered two doublet states of Z symmetry. The 'Ze state of the D,, symmetry anions lies below the 'Xu states. Table XXV gives Ortiz's main results. The VEA's are rather more than 1 eV greater than the EAs of the alkali atoms. Like the excitation energies, they differ little from system to system. B. DIHYDRIDES Senekowitsch and Rosmus (1987) have carried out ab initio calculations on HLiH- and its isotopomers. They found that the anions are linear, with

David R. Bates

50

TABLE XXV CALCULATED PROPERTIES OF ALKALI TRIMER ANIONS ~

~~

Trimer Li, Na, Li'Na Li,Na LiNa, LiNa,

Central atom

Anion bond lengths (A)

VEA (eV)

Vertical excitation energy of upper 'I: state of anion (eV)

Na Li Li Na

3.183 3.596 3.390 Li-Li = 3.146, Li-Na = 3.413 3.370 Li-Na = 3.352, Na-Na = 3.635

1.93 1.80 1.92 1.87 1.80 1.88

0.68 0.69 0.67 0.72 0.71 0.71

Source: After Ortiz (1988~).

r(LiH) = 1.734 A, and are stable, the vertical electron detachment energy being 3.10 eV and the dissociation energy toward LiH + H - being 2.34 eV. Their computed fundamental frequencies (in cm-') are v1 = 1014, v2 = 429, and v 3 = 1079. In early photoelectron spectroscopy of CH; (Zittel et al., 1976, Engelking et al., 1981) difficulties arose in connection with the accurate identification of vibrational hot bands. The difficulties were overcome in an investigation by Leopold et al. (1985) in which two improvements were incorporated. First, it used a flowing afterglow anion source. An advantage of this is that the anions, which may be formed initially with considerable internal energy, relax before photodetachment in collisions with the helium buffer gas and with molecules introduced into the helium. Second, it used an electron energy analyzer that could partly resolve the rotational structure. Both CH; and CD; were studied. Leopold et al. got that EA(CH,) = 0.652 & 0.006 eV and EA(CD,) = 0.645 k 0.006 eV. Bending modes of the anion in the X ,B, state being excited, analysis of the photoelectron spectrum enabled v 2 to be determined. Values of 1230 f 30 and 940 & 30 cm-' were obtained for CH; and CD;, respectively. From the absence of observed progressions in the 'A, CH, + e c ('B,) CH; systems Leopold et al., bracketed the geometry as HCH = 102 f 3" and r(CH) = 1.11 & 0.03 A. With the aid of a simulated photoelectron spectrum Bunker and Sears (1985) refined the angle to 103". Ab initio calculations (Shih et al., 1978, Kalcher and Janoschek, 1986; Nor0 and Yoshimine, 1989) gave the bond length and angle in the ranges 1.11-1.14 A and 100-104". Two such calculations predict the harmonic o,,and 03: 2517, 1550, and 2481 cm-' (Feller et al., 1982); frequencies ol, 2865, 1377, and 2904cm-' (Kalcher and Janoschek, 1986). A scaling

( A


51

procedure has been described (Colvin et al., 1983; Lee and Schaefer, 1984) whereby fundamental frequencies may be obtained from harmonic frequencies. However, it appears to be much less effective for anions than for cations (Lee and Schaefer, 1985). Fixed frequency-laser photodetachment measurements on the amide anion NH; by Celotta et al. (1972) have given that EA(NH;) = 0.779 f 0.037 eV. Fourth-order many-body perturbation theory calculations by Ortiz (1987b) gave 0.707 eV. Using velocity-modulation infrared-laser spectroscopy, Tack et al. (1986) measured 70 lines of the symmetric stretch fundamental (vl). Analysis of their results gave that for the lower (0, 0, 0) state the rotational constants (cm-') are A = 23.0507 (70), B = 13.0669 (30), C = 8.1211 (28); that for the upper (1,0,0) state they are A = 22.4090 (80), B = 12.8781 (40), C = 7.9484 (25); and that the band origin is 3121.935 (13). From the (0, 0,O) rotational constants they deduced that HNH = 102.1 (3.1)" and r(NH) = 1.041 (15) A. As they pointed out this geometry is virtually the same as that of the NH, radical. It is in good agreement with the geometry obtained from the self-consistent-field calculations of Lee and Schaefer (1985) who also obtained the harmonic frequencies (wl = 3509, o2= 1612, w3 = 3574 cm- l), the infrared intensities, and treated ND;. By means of the coupled electron-pair approximation Botschwina (1986a) has derived the rotational constants (A = 22.770, B = 13.054, C = 8.297), the harmonic frequencies (wl = 3288, w 2 = 1501, w 3 = 3367), and the fundamental frequencies (vl = 3108, v, = 1462, v3 = 3164). He also treated the isotopomers. Bishop and Pouchon (1987) have calculated the dipole polarizabilities and other electrical properties. Chipman (1978) has made a thorough investigation using a basis set that includes very diffuse orbitals to ascertain if an excess electron could be held to H,O in a dipole-supported state (Section 111). He found that on the Born-Oppenheimer approximation the binding energy would be only about eV and therefore discounted the possibility. It has been widely held that H 2 0 - is not a long-lived anion. However, de Koning and Nibbering (1984) have reported the observation of the anion, generated by

0-+ CH3NH2+ H 2 0 -

+ CH, =NH,

(31)

in their Fourier transform ion cyclotron resonance spectrometer. Their evidence is conclusive. They recorded an anion of mass 18.0104 f 0.0021 daltons (exact mass of H,O is 18.0105 daltons). On replacing l6O- by " 0 they recorded an anion of mass 20.0146 f 0.0021 daltons (exact mass of H2"0- is 20.0148 daltons) while on replacing CH3NH2 by CD3NH2 they

52

David R. Bates

recorded an anion of mass 19.0162 k 0.0020 daltons. (Exact mass of HDOis 19.0168 daltons.) The last result shows that in process (31) one of the H atoms of H 2 0 - comes from the methyl group and the other from the amino group. Werner et al. (1987) have examined the possibility of bound states of H 2 0 - with geometry other than the classical H 2 0 geometry. They did this by doing refined ab initio calculations on the potential energy surfaces of the three lowest bound states: the 12A'(2Z+)state and the 22A' and 12A"(211) states. The lowest bound asymptote of H 2 0 - is OH-('X+) + H(2S). The asymptote O-(2Po) + H2('Xl) and OH('II) + H-('S) lie above it by 0.25 and 1.08 eV. The calculations give that the 'X' and 'll states are nearly degenerate in the 0 - + H approach with local charge-quadrupole interaction minima about 0.2eV below the asymptote. The 'X' minimum is separated by a barrier from a saddle point on the OH- + H approach. From this saddle point H20('A1) e can be reached. The 211 minimum is separated by a barrier from a second lesser minimum on the OH + H approach. Werner et al. (1987) concluded that the system observed by de Koning and Nibbering (1984) is probably a charge-quadrupole bound O-..H2 cluster but is possibly a charge-dipole bound H-..OH cluster. Both clusters are linear with, in the former, r(0H) = 1.95 A, r(HH) = 0.78 A for 211 and r(0H) = 1.93 A, r(HH) = 0.77 A for 'X' while, in the latter, r(0H) = 1.04 A, r(HH) = 1.47 A. Ab initio calculations by Cremer and Kraka (1986) have led them to predict that FH; is a stable species with a peculiar structure. According to them it is best regarded as a F- and a H- anion held together by a rapidly oscillating proton and has a binding energy toward dissociation to F- H2 of 1.8 eV. Fixed-frequency-laser photoelectron spectrometry measurements by Kasdan et al. (1975b) have given that EA(SiH2) = 1.124 & 0.020 eV and that the bending frequency in the X 2Bl state of Si H; is 1200 160 cm-'. Using large basis sets and fourth-order many-body perturbation theory, Nguyen (1988) calculaEd that EA(SiH2) = 1.00 eV and that in the anion r(SiH) = 1.541 A and HSiH = 94.0". Measurements by Zittel and Lineberger (1976), similar to those on SiH; , have given that EA(PH2) = 1.271 0.010 eV. From the absence of significant off-diagonal transitions in the 2B,-'Al transition Zittel and Lineberger inferred that the geometry of ('A,)PH; is closely similar to that of (2B,)PH2 for which r(PH) = 1.429 A and the angle between the bonds is 91.7'. Ortiz (1987b) got that EA(PH2) = 1.160 eV by fourth-order many-body perturbation theory calculations. Using the SCF method and the coupled-electron-

+

+


53

pair approximation, Botschwina (1987) has computed the harmonic and anharmonic vibrational frequencies. The values (cm - I ) he obtained are w1 = 2296, w 2 = 1092, w 3 = 2293; v 1 = 2187, v2 = 1069, v 3 = 2182. He did not confine his attention to the fundamental modes and treated PD; in addition. Pople et al. (1988) have estimated EA(XH,) (X = Li through C1) by ab initio molecular orbital theory aided by a semiempirical correction procedure. The mean absolute difference between their estimation and measurement is only 0.05 eV for first-row compounds (12 comparisons) and only 0.08 eV for second-row compounds (1 3 comparisons). The estimates for EA(BH2) and EA(A1H2),which have not been determined in the laboratory, are 0.34 and 1.05 eV, respectively. Both anions have 'A, symmetry in the ground state; and both have states of 3B1symmetry and estimated excitation energies of 0.16 and 0.73 eV, respectively. From the photoelectron spectra of MnH;, FeH;, CoH;, and Ni H;, Stevens Miller et al. (1986) deduced that EA(MnH,) = 0.444 & 0.016 eV, EA(FeH,) = 1.049 & 0.014 eV,EA(CoH,) = 1.450 f 0.014 eV,EA(NiH,) = 1.934 f 0.008 eV. The observed spectra do not have structure. Stevens Miller et al. inferred that the anions and their parent molecules are linear and that the detachment causes no change in the bond length. They suggested that a dk+' a2 a2 + dk a2 u2 transition is involved the active electron being a nonbonding d-electron. This is consistent with the variation of the EA with atomic number. Two fixed-frequencylasers ofphoton energies 2.54and 2.71 eV were used. Neither produced detectable detachment from CrH; .Taking into account that the electron energy analyzer transmission falls rapidly below 0.2 eV kinetic energy Miller et al. concluded that EA(CrH,) > 2.5 eV. Since Cr is the element before Mn the high electron affinity might seem odd, but (as Miller et al. pointed out) CrH; has a d5 a2 a2 configuration and the half-filled d-shell has special stability. The photoelectron spectra of MnD;, FeD;, COD;, and NiD; were also studied. C. MONOHYDRIDES Janousek et al. (1979) made photodetachment cross section measurements on the acetylide anion HC; generated by the proton transfer reaction

+

C2H2 F-

+

HC;

+ HF

(32)

and did calculations on the photodetachment behavior near threshold. The combination yielded EA(HC2) = 2.94 f 0.10 eV. In good agreement with

54

David R. Bates

this they obtained 3.18 f 0.25 eV by the application of eighth-order perturbation theory. Another many-body perturbation theory calculation (Lime and Canuto, 1988) has given 3.15eV. The geometry and other properties (X 'C)HC; have been determined by the self-consistent field study of Lee and Schaefer (1985). The anion is linear with r(CC) = 1.223 A and r(CH) = 1.058 A; its harmonic frequencies (cm-') are o1 (CH stretch) = 3546, 0,= 620, o3(CC stretch) = 2029. Infrared intensities and results on DC; were also obtained by Lee and Schaefer. A convenient source of the anion of the hydroperoxyl radical is to allow HNO- (which may easily be produced from alkyl nitrites) to react with 0,: HNO- + 0, + HO; + NO (33) (De Puy et al. 1978). A threshold-photodetachment study by Bierbaum et al. (1981) lead to EA(H0,) = 1.19 f 0.01 eV. Photoelectron spectroscopy measurements on both HO; and DO; have been carried out by Oakes et al. (1985) using a fixed-frequency laser. These gave EA(H0,) = 1.078 k 0.017 eV (in conflict with the result of Bierbaum et al.) and EA(D0,) = 1.089 f 0.017 eV. The best value of EA(H0,) inferred from MRD-CI calculations is 1,069 f 0.05 eV (Vazquez et al. 1989a) which lends support to the result of Oakes et al. Only one active mode was detected by Oakes et al. in their electron spectra. From this mode they deduced that o3 (00 stretch) = 775 f 250cm-' for (X 'A') HO; and that wJ (00 stretch) = 900 f 250 cm-' for the deuterated anion. By modeling the Franck-Condon factors they obtained r(O0) = 1.50 A in HO;. They reasoned that the other bond length and the angle between the bonds must be almost the same as in HO, since otherwise the H - 0 stretch mode and the bending mode would have been a c w their electron spectra and they therefore took r(H0) = 0.97 A and HOO = 104". These bond lengths agree well with the values that Oakes et al. themselves got by ab initio work and, rather better, with the values that Cohen et al. (1984) and Vazquez et al. (1989b) got by more extensive ab initio work (Table XXVI). In this work the harmonic (Oakes et al.) and fundamental (Vazquez et al.) frequencies were also calculated (Table XXVI). Murray et al. (1986) have carried out fixed-frequency -laser photoelectron spectroscopy measurements on formyl anions produced from formaldehyde in a flowing afterglow by the proton-transfer reaction H - + H,CO --+ HCO- + H, (34) (Bohme et al. 1980). They deduced that EA(HC0) = 0.313 f 0.005 eV and EA(DC0) = 0.301 f 0.005 eV. Using the first of these values they showed


55

TABLE XXVI CALCULATED EQUILIBRIUM GEOMETRY AND VIBRATIONAL FREQUENCIES OF (X'A') HO;

Reference Oakes et al. (1985) Cohen et al. (1984) Vazquez (1989b) Oakes et al. (1985) Vazquez (1989b)

(A)

r(OH) (A)

HTO (degrees)

1.529 1.498 1.499 W', cm3662 vl, cm-' 3650

0.972 0.962 0.956 w2,cm1153 v 2 , cm-' 1170

98.5 99.8 98.5 w3,cm-' 821 vj, cm-' 883

'

'

that the bond strength toward HCO- -+ H- + CO is only 0.35 f 0.09 eV. They did not observe any hot bands. Assuming that the frequencies of the anion equal those of the parent molecule, they made a Franck-Condon analysis of the electron spectra. Knowing the HCO geometry, they estimated ,y)t in the anion r(CH) = 1.25 f 0.05 A, r(C0) = 1.21 f 0.02 A, and HCO = 109 k 2".Chandrasekhar et al. (1981) and Wasada and Hirao (1987) have performed ab initixalculations. The former got r(CH) = 1.166 A, r(C0) = 1.254 A and A C O = 110.0"; the latter got r(CH) = 1.203 A, r(C0) = 1.254 A, and HCO = 109.4'. Senekowitsch et al. (1987, 1988) have done ab initio calculations that gave EA(HCS) = 0.41 eV, that the equilibrium geometry of the anion is r(CH) = 1.111 A, r(CS) = 1.687 A, &S = 106', and that the fundamental frequencies (in cm-') are v1 (CH stretch) = 2648, v2 = 1140, v 3 (CS stretch) = 91 1. They also computed the components of the transition dipole moment, the integrated band intensities, and the radiative lifetimes of a number of the (rotationless) vibrationally excited states. For (1 0 0), (0 1 0), and (0 0 l), they got 3.5, 172, and 785 ms, respectively. Two hydrogen bihalide anions, FHF- and ClHCl-, have been studied experimentally and theoretically. Kawaguchi and Hirota (1986) produced FHF- by a hollow cathode discharge in a mixture of hydrogen and carbofluorides. They judged the source to be an abstraction reaction such as

F-

+ CHF2CF, + FHF- + C,F,.

(35)

Using a diode laser spectrometer they made high-resolution infrared measurements on (X 'Z:)FHFin an effective absorption path 10 m long. In the

56

David R. Bates

region between 1780 and 1853 cm- ' they observed 71 magnetic-field-sensitive lines of a 1848 cm-' band. Theorists (Barton and Thorson, 1979; Lozes and Sabin, 1979, Lohr and Sloboda, 1981) had established that in its ground state the anion has ' C : symmetry and has a linear symmetric configuration, and they had provided some information on the spectroscopic constants; and solid-state measurements (summarized by Kawaguchi and Hirota) give v 1 = 600-615, v 2 = 1199-1274, v 3 = 1284-1563 cm-'. Nevertheless, identification of the vibration-rotation lines whose frequencies had been measured was a difficult task. Originally Kawaguchi and Hirota thought they could determine v 3 directly and estimate v 1 and v 2 from the centrifugal distortion constant and the perturbation of Coriolis interaction between the v1 v2 and v 3 states. However, they noted that while their values of v1 and v 2 agree with the solid-state measurements, their value of v 3 does not (being 286cm-' higher than the upper limit to the solid-state range). Furthermore, their value of v 3 is 300-500cm-' higher than the ab initio vaues that later became available. These values are 1427cm-' (Janssen et al, 1986), 1334cm-' (Botschwina, 1987), and 1292 cm-' (Yamashita and Morokuma, 1987). Janssen et al. argued that the discrepancy is significant. In order to resolve the issue Kawaguchi and Hirota (1987) extended the range of the high-resolution infrared measurements on FHF- to the region between 1200 and 1350 cmIn it they detected three new bands. A careful analysis led them to conclude that these are the v 3 and v 2 bands and the hot v 1 + v 2 - v 1 band and that the 1848 cm-' band is not the v 3 band as had been supposed but the v1 + v 3 combination band. They deduced that v1 = 583, v 2 = 1286, v 3 = 1331, v1 v 3 = 1849 an-', Be = 0.342069 (21) cm-' and r(FF) = 2.27771 (7) A. The most successful of the ab initio calculations that cover all the frequencies is one by Janssen et al. (1986) giving v1 = 617, v 2 = 1363, v 3 = 1427, v1 + v 3 = 2055 cm-', r(FF) = 2.266 A while the most successful that omits only v 2 is that by Yamashita and Morokuma (1987) giving v 1 = 596, v 3 = 1292, v 1 + v 3 = 1813, and r(FF) = 2.280 A. The FHF- + H F + F- dissociation energy is 1.67 eV (Caldwell and Kebarle, 1985). An infrared-absorption study of ClHCl- has been made by Kawaguchi (1988). His measurements on lines of the v 3 band gave v 3 = 722.8965 (2) cm-'. He observed many lines in the 978 cm-' region that he attributed to the v 1 + v 3 combination band. Noticing that the v 3 band is perturbed by the v 2 band through Coriolis interaction, he deduced that v 2 = 792 (9) cm-'. His rotational constant gave that r(ClC1) = 3.14676 (5) A. Botschwina et al. (1988) have done ab initio calculations on the anion. They obtained that v1 = 308, v 3 = 768, v 1 + v 3 = 1031 cm-', the frequencies of 10

+

'.

+


57

other members of the (vl, v3) family, r(ClC1) = 3.132 A, and that the CHCl- -,HCl + C1- dissociation energy is 1.00 eV in close agreement with the experimental value of 1.02 eV (Caldwell and Kebarle, 1985). They calculated the (quite large) transition dipole moments of the nv, and v3 combination tones. In the case of the bending mode they went only as far as the harmoic approximation, getting 0,= 770 cm- Ab initio calculations by Ikuta et al. (1989) have given v1 = 310, v 2 = 877, v j = 709 cm-'. Sannigrahi and Peyerimhoff (1989) have done calculations on FHBr- and referenced work on other members of the family. Murray et al. (1988) have conducted a fixed-frequency-laser photoelectron spectroscopy study of six species of halocarbene anions generated in a flowing afterglow by H i abstraction reactions such as

'.

0-+ CH,F

+ HCF-

+ H,O

(36) (Tanaka et al. 1976; Dawson and Jennings, 1976). They found that AE(HCF) = 0.557 f 0.005 eV, AE(DCF) = 0.552 f 0.005 eV, AE(HCC1) = 1.213 & 0.005 eV, AE(HCBr) = 1.556 f 0.008 eV, AE(HC1) = 1.683 & 0.012 eV, AE(CF,) = 0.179 f:0.005 eV, AE(CC1,) = 1.603 f 0.008 eV; that the C-X stretch frequencies (cm-') are v3 = 745(30) HCF-, 730 (30) DCF-, 470 (30) HCCl-, 430 (40) HCBr-, 350(40) HCI-; and that the symmetric stretch frequency for CF; is v1 = 860(30)cm-'. From a Franck-Condon analysis they obtained information on the change in geometry that occurs on photodetachment. Knowing the neutral geometry, they hence deduced the anion geometry. However, they could not always avoid ambiguity though they accepted the expectation that in the anion bond lengths are longer and the angle between bonds is smaller than in the neutral. Table XXVII gives TABLE XXVII OF HALOCARBENE ANIONS GEOMETRY

Anion HCFHCClCF;

cc1;

r(HC) (A) 1.18 f 0.02 1.114 1.127 1.21 & 0.02

r(CX) (A) 1.48 f 0.02 1.537 1.492 1.99 f 0.02 1.45 f 0.02 or 1.44 f 0.02 1.92 f 0.02 or 1.91 f 0.02

HTX (degrees) 94 f 2 102.2 98.9 96 f 2 99 f 2" 92 f 2 103 f 2" 93 f 2 ~~~

(I

~

Two geometries are consistent with the available data.

Reference Murray et nl. (1988) Goldfield and Simons (1981) Tomonari et al. (1990) Murray et al. (1988)

58

David R. Bates

their results. In the case of HCF- it compares them with the results of the ab initio calculations of Goldfield and Simons (1981) and of Tomonari et al. (1990). The agreement is not close. Goldfield and Simons predicAat HCFhas a bound , A state with r(CH) = 1.080 A, r(CF) = 1.480 A, HCF = 115". They place the ground 'A" state too low in that according to them EA(HCF) > 0.94 eV. D. OTHERTRIATOMIC ANIONS Canuto (1981) has carried out ab initio calculations on (X 'A,) BeF;, finding that the equilibrium geometry, r(BeF) = 1.49 A, 130 f 3" differs markedly from that of (X 'X:)BeF,, r(BeF) = 1.41 A, FBeF = 180°, and that EA(BeF,) = - 0.44 eV while the vertical detachment energy is 0.53 eV. The anion is metastable. Fixed-frequency-laser photoelectron spectroscopy measurements by Oakes et al. (1983) have established that EA(CC0) = 1.848 f 0.027 eV. The Mulliken-Walsh rules give the anion to be linear. Oakes et al. observed a hot band from which they inferred o1fi 1625 cm-'. They reasoned that the ground state of CCO- is X 2 n . There may be a low ' X' state. By an ion-beam experiment Hopper et al. (1976) showed that N,O- is stable and that the dissociation energy D(N, - 0-)= 0.43 f 0.1 eV. Knowing EA(0) and D(N2 - 0),they deduced that EA(N,O) = 0.22 f 0.1 eV; and hence knowing D(N - NO) they futher deduced D(N - NO-) = 5.13 f 0.1 eV. Ab initio calculations by Yarkony (1983) have given that ( X 2 A ' ) N , K h a s the equilibrium geometry r(NN) = 1.211 A, r(N0) = 1.372 A, NNO = 126.0", and that the harmonic vibrational frequencies (cm-') are o1= 1610, w 2= 560, o3= 863. For comparison the equilibrium geometry of the parent neutral molecule in the X state is linear with r(NN) = 1.128 A, r(N0) = 1.184 A (Ahlrichs et al. 1975). Photoelectron spectroscopy measurements by Coe et al. (1986a) demonstrate that the vertical detachment energy is around 1.5 eV. When in the equilibrium configuration in the X 'Xl state, CO, is a symmetrical linear molecule with r(C0) = 1.16 A (Herzberg, 1945). Compton et al. (1975) have formed its anion (X 'A,)CO; by K(or Cs) + CO, -+ K'(or Cs')

+ CO;

(37)

and deduced that EA(C0,) = - 0.60 f 0.2 eV from the threshold energy for the process. Measurements by them gave that the lifetime of the anion toward


59

autodetachment is 90 f 20 ps. England (1981) has found by ab initio calculations that in its equilibrium configuration CO, has r(C0) = 1.22 A, n OCO = 136" and is in a potential well whose barrier has a height of around 0.2-0.3 eV. The fairly long lifetime arises from the slightness of the overlap between the CO; and COz vibrational wavefunctions. England got EA(C02) = - 1.15 eV, which is considerably more negative than the experimental value. By including polarization functions Yoshioka et al. (1981) raised EA(C0J to - 0.81 eV, which agrees with the value of Compton et al. (1975) to within their error bars. Using a dye-laser Woo et al. (1981) have studied photodetachment from (X 'A,) NO; and deduced that EA(N0,) = 2.275 & 0.025 eV. A FranckCondon analyKby them led to the equilibrium geometry that r ( x ) = 1.15 f 0.02 A,ONO = 119.5 f l.O"comparedwithr(N0) = 1.1934 A,ONO = 134.1' for the parent neutral molecule. Ab initio calculations on the anion by O = 117.1 & 1" and Harrison and Handy (1983) give r(N0) = 1.224 A, & are to be preferred to the values of Woo et al. because attachment of an electron is expected to increase the bond length.

V. Tetra-Atomic and More Complex Anions A. AH;, FAMILY By photoelectron spectroscopy Ellison et al. (1978) have found that EA(CH,) = 0.08 f 0.03 eV. Lee and Schaefer (1985) have carried out ab initio calculations on ( X 'A,)CH; that gave the equilibrium geometry to be n pyramidal with r(CH) = 1.09 A, HCH = 109.6O.They also predicted the harmonic frequencies and the infrared intensities. Consistent with a photodetachment measurement by Reed and Brauman (1974), ab initio calculations by Pople et al. (1988) give that EA(SiH,) = 1.35 eV. Using a Fourier-transform ion cyclotron resonance mass spectrometer, Kleingeld and Nibbering (1983) have demonstrated that H,O- first observed by Paulson and Henchman (1982) is a long-lived species in the gas phase. From reactions between the partly deuterated anion and neutral species they proved that the three H atoms are not equivalent and concluded that H,Ois best regarded as a (H- + H 2 0 ) cluster. It is generated by the reaction

David R. Bates

60

between the hydroxyl anion and formaldehide: OH-

+ C H , O e [OH--CH,O] +[H,O*HCO-] + H-*H,O + CO.

=$

[H--H,O*CO] (38)

Ion-beam measurements by Paulson and Henchman (1982) show that the binding energy toward dissociation to H- + H,O is 0.75 k 0.05 eV and that toward dissociation to OH- + H, it is 0.30 & 0.05 eV. A vertical detachment energy of 1.52 & 0.02 eV has been obtained by Miller et al. (1985). Ab initio calculations by Cremer and Kraka (1986) and by Chalasinski et al. (1987) have established that the anion has a bent equilibrium geometry. The latter, which are the more accurate, give that r(H0) = 0.972 A, r(0H) = 1.037 A, r(HH) = 1.431 A, % H = 99.4', O m = 173.3'. This is consistent with the conclusion of Kleingeld and Nibbering on the structure. Kleingeld et al. (1983) have shown that NH; can occur in the gas phase. It is formed in the sequence NH,

+ CH,O

+

NH,

+ HCO-

HCO-+NH,+CO+NH;.

(39)

Kleingeld et al. established the identity of the anion by an accurate measurement with their mass spectrometer. This gave 18.0350 & 0.0037 daltons (exact NH; mass is 18.0344 daltons); and when the "N isotope was used 19.0316 & 0.0041 daltons. (Exact "NH; mass is 19.0314 daltons.) According to ad initio calculations by Kalcher et al. (1984), the dissociation energy into NH, + H - is 0.36 eV. This is also the value that Coe et al. (1985) obtained by subtracting EA(H) of Table I from the vertical detachment energy 1.11 eV they measured by photoelectron spectroscopy. The ab initio calculations of Cremer and Kraka (1986) give a greater value 0.65 eV and the equilibrium geometry r(H,N) = 1.016 A, r(NH,) = 1.031 A, r(H2 H,) = 1.845 A, n n n H,NH, = 106", H,NH, = 104.6', NH,H, = 167.4' (Figure 1). Cremer and Kraka (1986) predict that CH; has only a short life.

N

H2

FIG.1. NH; equilibrium geometry.


61

B. INORGANIC ANIONS 1. Oxides (and Some Small Clusters)

The early research and the importance of CO; in the D-region has been described by Massey (1976, 1979). Ab-initio calculations by So (1976) have shown that the ground electronic state is a ,B, state, that it has C2"symmetry p r(C0) = 1.546 A, the unique w x t h e two equal r(C0) = 1.364 A, the u OCO = 131.4" (so that the two equal OCO = 114.3'). There have been substantial changes in the accepted values of relevant energies (Burt, 1972, Ferguson et al. 1973; Moseley et al., 1976; Beyer and Vanderhoff, 1976; Hong et al., 1977; Dotan et al., 1977; Wu and Tiernan, 1979). From a careful laser photodissociation study of CO; (which, incidentally, is the first anion discovered to undergo photodissociation in the gas phase), Hiller and Vestal (1980) determined that D(C0, - 0 - )= 2.258 f 0.008 eV. Hunton et al. (1985) investigated the dynamics of the photodissociation with the aid of a spectrometer allowing kinetic-energy resolved detection of photofragments. Their investigation included the dependence of the photofragment intensity on the laser power. Hunton et al. found that in the photon-energy range, 1.95-2.2 eV, of their laser, photodissociation of CO; is controlled by three electronic states correlating with CO, + 0 - :the ,B, ground state, a weakly bound 'A, state, and a repulsive ,B, state. According to their analysis, the photodissociation is initiated by a ,B, + 'A, transition. This may be followed by a 'A, -,*B, transition and dissociation; or alternatively a downward transition to a vibrationally excited level of the 'B, state followed by dissociation in collisions with the background gas after passage through the energy analyzer. Their measurements give that D(C0, - 0 - )= 2.27 eV in agreement with Hiller and Vestal (1980). Substitution in the relation EA(C0J = D(CO2 - 0 - )+ EA(0) - D(CO2 - 0)

(40)

with D(C0, - 0) taken to be 0.39 L 0.2eV (Benson, 1976) gives EA(C0,) = 3.34 k 0.2 eV which, as Hunton et al. (1985) noted, is much greater than the value 2.66 f 0.10 eV obtained by Hong et al. (1977) from a photodetachment-threshold measurement. The conflict seems unresolved. Using high-pressure mass spectrometry, Keesee et al. (1980) have studied ion-molecule association reactions of the form A-(B)S-i

+B

+

A-(B),

(41)

62

David R. Bates

over a range of temperatures determining some dissociation energies of interest. Among the values they obtained are the following (all in eV with error bars 0.01 eV or less): D(CO,*Cl-) = 0.35, D(CO,.NO;) = 0.40, D(CO,.CO;) = 0.31, D(C02*SO;) = 0.28; D(S02*C1-) = 0.95, D(S02.NO;) = 1.12, D(S02*SO;) = 1.04, D(SO,.SO;) = 0.58; D(H,O*Cl-) = 0.65, D(H20.NO;) = 0.66, D(H,O.CO,) = 0.61, D(H20.CO;) = 0.61, D(H,O.NO;) = 0.63. Time-of-flight mass spectrometer measurement by Klots and Compton (1977) have shown that CO,-CO; has a lifetime of at least 2 ms. According to a6 initio calculations of Fleischman and Jordan (1987), the most stable form has D,, symmetry with D(C0,-CO;) = 0.51 eV. The predicted EA (CO,.CO,) = - 0.31 eV and the predicted vertical detachment energy is 2.8 eV. Coe et al. (1987) have observed that the photoelectron spectra of the cluster anions NO-(N,O) and NO-(N20)2 strongly resemble that of NObut with the peaks broadened and shifted to successively lower electron kinetic energies. They interpreted the spectra in terms of an intact NOstabilized by nitrous oxide. They estimated the dissociation energies for NO-(N,O) + NO- + N,O and for NO-(N,O), + NO-(N,O) + N,O to be around 0.22 and 0.26 eV. A number of identifications of the metaphosphate anion PO; have been made. (cf: Meyerson et al., 1984.) Laboratory work by Henchman et al. (1985) has shown that like NO; (cf: Ferguson et al., 1979), it is highly stable and unreactive. Indeed, it is probably the more stable of the two species. Thus, from the heats of formation of NO, and NO;, Ferguson (1979) has found that EA(N0,) = 4.01 f 0.02 eV while from corresponding data Henchman et al. have found that EA(P0,) = 4.9 f 1.3 eV. Ab initio calculations (with D,,, symmetry assumed) give r(P0) = 1.475 A (O'Keefe et al. 1985; Rajca et al. 1987). The bonds are highly polar with negative charge on the 0 atoms. From measurements they made on the energy-dependence of the rate of

so- + so2 + so; + s,

(42)

Dotan and Klein (1979) inferred that EA(S0,) 2 2 . 7 x Ab initio calculations by Stanbury et al. (1986) give r(S0) = 1.483 A, OSO = 113.8", and the vibrational frequencies (in cm-' and stated to be probably about 10% too high) to be 1000, 604, 1175 (degenerate pair), 511 (degenerate pair). Posey and Johnston (1 988) have used pulsed photoelectron spectroscopy


63

to study the three species of anion with stoichiometry N,O; that may be generated by varying the neutral precursors in an electron beam-ionized free jet expansion. The spectra and photofragmentation properties indicate that, as might be expected, the species generated from 0, in N,, from NO in Ar, and from N,O are the cluster N 2 * 0 ; (dissociation energy 0.26 iz 0.03 eV), the resonance-stabilized NO-NO- (dissociation energy less than 1.19 eV), and the stable anion N,O; (dissociation energy less than 2.33 eV) similar to the isoelectronic COY.

2. Metal Clusters Although the general subject of cluster anions is not being covered in this chapters brief mention will be made of the anions of some small clusters of metallic atoms. Leopold et al. (1987) have obtained the photoelectron spectra of Cu,, n = 1-10. They found that EA(Cu,) tends to be an increasing function of n and to be higher if n is odd than if n is even. McHugh et al. (1989) have observed similar characteristics for clusters of other outer s shell atoms, specifically Nan n = 2-5, K, n = 2-7, Rb, n = 2-3, Cs,, n = 2-3: thus, EA(K,) = 0.83, EA(K,) = 0.95, EA(K,) = 0.95, EA(K,) = 1.05 eV (all with error bars of amount 0.10 eV). BonaEiE-Koutecky et al. (1989) have carried out ab initio calculations on the Nan- clusters investigated in the laboratory. They reproduced the observed photoelectron spectra remarkably well. They pointed out that the number of peaks in the spectra is greater if n is even than if n is odd and gave a simple explanation for this difference. The ground state of Nan- is a doublet if n is even and is a singlet if n is odd so that in the former case photodetachment transitions to singlets and to triplets occur whereas in the latter case only transitions to doublets occur. BonaEiE-Koutecky et al. also pointed out that linear geometries are energetically favorably for small anionic clusters because they minimize the Coulombic repulsion at the ends of the chain. In the case of Na;, there is little difference between the calculated energies of the linear and optimized rhombic structures. Comparison between the observed and computed spectra showed that rhombic Na; is only a minor constituent of the Na; generated by McHugh et al. (1989). BonaEiE-Koutecky et al., calculated that the trapezoidal planar form of Na; is more stable than the linear form by 0.08 eV. They judged that both isomers contribute to the observed spectrum of the anion.

64

David R. Bates

3. Silicon Compounds

Kalcher and Sax (1987) have done ab initio calculations relating to Si,H;. They found that EA(Si,H,) = 1.65 eV, that the ground state of the anion is ,At', and that there is a 4A'' state that is stable by 0.6eV. The predicted equilibrium configuration (Fig. 2) for ,A'' is r(SilSi2) = 2.252 A, r(Si,H) = 0 1.498 A, HSi,H = 103.9", 0 = 5.2" (where 0 is the angle the SilSi2line is out of the HSi H plane); and that for 4A'' is r(SilSi2) = 2.456 A, r(Si2H) = A 1.513 A, HSi,H = 99.3", 0 = 46.8'. The harmonic frequencies of both states were computed. Kalcher and Sax (1988) later investigated the anions XYH; with X, Y = C, Si. They found that EA(CSiH,) = 1.82 eV, EA(SiCH,) = 0.65 eV, and EA(SiSiH,) = 1.32 eV. The ground state of the anion is ,A, and CSiH; and SiSiH; have an excited 'E state predicted to be stable by 0.84 and 0.52 eV, respectively. Kalcher and Sax also give the calculated equilibrium configurations and the X- Y stretch harmonic frequencies.

.e

i--.

H FIG.2. Si,H; equilibrium geometry.

4. Fluorides

Considerable interest has been shown in the fluorides, partly because some are important as gaseous dielectrics and electron scavengers. Some have such high electron affinities that they are called superhalogens. Research on the sulphur hexafluoride anion SF, has a long history (see Massey, 1976) but it is only fairly recently that EA(SF,) has been determined reliably. Using the flowing afterglow method, Streit (1982) made an extensive study of ion-molecule reactions involving SF, and SF; and from the results was able to conclude that EA(SF,) = 1.0 f 0.2 eV. Grimsrud et al. (1985) have made A-

+ B s A + B-

(43)

equilibria measurements with a pulsed-electron high-pressure mass spectrometer. From their results they deduced that EA(SF,) = 1.05 f 0.1 eV (and also that EA(C,Fl4) = 1.06 f 0.15 eV). Combination of the value of


65

EA(SF,) gotten by Grimsrud et al. with data cited by Streit yields D(SF; - F) = 1.4 eV. Several ab initio calculations have been done. Those of Hay (1982) give that r(SF) = 1.710 A and that the symmetric stretch frequency is 652 cm-'. The corresponding values given by ab initio calculations of Klobukowski et al. (1987) are 1.704 A and 698 cm-'. The experimental EA is closely reproduced by Hay (1982) and by Miyoshi et al. (1988a), their values being 1.03 and 1.06 eV, respectively. Calculations on EA(MF,), where M is a d-shell metal atom, have been carried out by Hay et al. (1979) and Miyoshi et al. (1988b) using a model potential method and by Gutsev and Boldyrev (1983, 1984) using a discrete variational method. Values of EA(MF,) have been obtained in the laboratory by a number of methods; from thermochemical data and crystal lattice enthalpies (Burgess et al. 1974; Burgess and Peacock, 1977), from the threshold of reactions involving the anions (Beauchamp, 1976), from the thresholds for negative ion formation in collisions between MF, and alkali atoms (Dispert and Lacmann, 1977; Compton, 1977; Mathur et al. 1977; Compton et al., 1978), from charge-transfer studies (Webb and Bernstein, 1978; George and Beauchamp, 1979), and from equilibrium constant determinations (Pyatenko et al., 1980; Sidorov et al., 1982). Table XXVIII shows the results. It is evident that EA(MF;) may be much greater than EA(F) (Table I). The anion commonly has several excited states. For example, in the case of UF; there are states of excitation energy 0.57,0.86, 1.58, and 1.77 eV (Reisfeld and Crosby, 1965). According to the calculations of Sakai and Miyoshi (1987) and of Miyoshi et al. (1988b), some doubly charged anions are stable toward autodetachment in that they predict that EA (CrF,) = 2.44 eV and EA(Mo F;) = 0.58 eV. Doubtless each dissociates without a barrier into a pair of anions. Calculations by Alvarez et al. (1987) on Bri- are relevant. They show that this doubly charged anion is also stable toward autodetachment; and that although Bri- is linear (allowing the negative charges to be relatively far apart), it dissociates into Br- + Br; without a barrier. C. ORGANIC ANIONS Oakes et al. (1983) have made photoelectron spectroscopy measurements on HCCO- from which they determined that EA(HCC0) = 2.350 f 0.020 eV. They reasoned that the X 'C+ anion is linear and may be written HC = CO-.

David R. Bates

66

TABLE XXVIII ELECTRON AFFINITIES(eV) OF METALHEXAFLUORIDES MF, M Ti V Cr Mn Fe co Ni cu Zn

I

3d24s2 3d34s2 3d54s 3d54s2 3d64s2 3d74s2 3ds4s2 3d1'4s 3d1°4s2

7.5 6.7 5.0 5.9 7.0 6.8 6.9 6.1 5.8

l

l

8.2

a Mo

4d55s

...

5.4

5.4

3.8

4.6

C

e >4.5

f ~5.1

h

g 5.8

3.6 f 0.2

111

Hf Ta W Re Ir Pt Au Hg

5d26s2 5d36s2 5d46sZ 5d56s2 5d66s2 5d76s2 5d96s 5d1°6s 5d"6s2

U

5f36d7s2

0s

8.8 8.4 3.5 4.8 6.0 7.2 7.4 8.1 5.8 iv 7.1

b 4.9 f 0.5

3.7

d 25.1

>4.9

>4.3

3.5 >5.1

5.1 >5.1

7.8

*j 0.1

k ~ 5 . 8 6.3 f 0.5

Source: Calculation: i, Gutsev and Boldyrev (1984); ii, Miyoshi et al. (1988b); iii, Gutsev and Boldyrev (1983); iv, Hay et al. (1979). Experiment: a, Burgess et al. (1974), Burgess and Peacock (1977); b, Beauchamp (1976); c, Dispert and Lacmann (1977); d, Compton (1977); e, Mathur et al. (1977); f, Compton et al. (1978); g, Webb and Bernstein (1978); h, Sidorov et al. (1982); j, George and Beauchamp (1979); k, Pyatenko et al. (1980).

Using the coupled-electron-pair approximation, Botschwina (1987) found that the formate anionAC0; has CZv symmetry with r(CH) = 1.128 A, r(C0) = 1.126 A, and OCO = 130.2".From experience with carbon dioxide he suggested that his calculated value of r(C0) may be about 0.012 A too large. Botschwina also calculated the vibrational frequencies of the three totally symmetric modes. The values he obtained are v,(CH) = 2532, v2 (bend) = 730, v3 (CO) = 1318 cm-'. His corresponding values for the deuterated analog are v,(CD) = 1898, v,(bend) = 724, v3(CO) = 1292 cm-l. Self-consistent field calculations (Yarkony et al., 1974) on the methoxide anion CH30> given that in the X ' A state r(CH) = 1.12 A, r(C0) = 1.39 A, and OCH = 114".Engelking et al. (1978) have made fixed-frequency-


67

laser electron spectrometry measurements on this anion and its kin CD,Oand CH, S - . They hence got EA(CH,O) = 1.570 f 0.022 eV, EA(CD,O) = 1.552 f 0.022 eV, and EA(CH,S) = 1.882 & 0.024 eV. From measurements on hot bands they inferred that the symmetric H or D umbrella bend frequencies of CH,O- and CD,O- are 1075(100) cm-' and 915(100) cm-' and that the CS stretch frequency is 625(80) cm-'. Further research on the thiomethoxyl anion has been done by Janousek and Brauman (1980). Using a dye laser they carried out photodetachment cross section measurements thatledtoEA(CH,S) = 1.861 & 0.004 eVandEA(CD,S) = 1.858 & 0.006 eV. B ab initio calculations they found that r(CH) = 1.10 A, r(CS) = 1.80 A, HC - 5 3 = 103.6" and hence that the (prolate top) rotational constants are A = 5.47, B = 0.439 cm-' (CH,S-); A = 3.29, B = 0.350 cm-I (CD,S-) From a photodetachment threshold study of CH,CN- and CD2CNMarks et al. (1986) got EA(CH,CN) = 1.560k 0.006eV and EA(CD,CN) = 1.549 f 0.006 eV while from the photoelectron spectra Moran et al. (1987) got EA(CH,CN) = 1.543 & 0.014 eV and EA(CD,CN) = 1.538 & 0.012 eV. The cyanomethylradicalCH,CN is planar and symmetric. A Franck-Condon analysis together with multibody perturbation theory calculations they carried out led Moran et al. to conclude that the anion in the X ' A state may be described as CH, = C = N- but having the H atoms bent 30 & 5" out of the molecular plane with a barrier of only 0.012 k 0.006 eV to inversion; and that r(CN) = 1.162 A, r(CC) = 1.395 A, r(CH) = 1.078 A, = 117", & = 117", = 178". They computed the harmonic vibrational frequencies to be w1 = 2992, a, = 2077, 0,= 1396, w4 = 987, w5 = 550, w6 = 276, 0, = 2983, ug= 1032, w g = 425 cm-'. The most accurate information on the spectroscopic constants is provided by the autodetachment spectroscopy measurements of Lykke et al. (1987). Their values for the main spectroscopic constants are A = 9.29431(14), B = 0.338427(20). C = 0.327061(21)cm-'. In the course of their research on the excited dipole-supported state of the acetaldehyde enolate anion CH,CHO- ,Mead et al. (1984a) determined that the rotational constants of the ground state are A = 2.219(3), B = 0.3758(4), C = 0.3207(3) cm-'. From the smallness of the inertial defect

A

I, - I,

- Ib

(44)

they concluded that the anion is planar to within the experimental error of their data. Although they had made measurements on two isotropic variants, this did not provide enough information to fix the positions of all the atoms. In the case of the hydrogen atoms they took the positions in the radical from

68

David R. Bates

ab inicio calculations and the positions in the anion from measured values in neutral species of similar electronic structure. The equilibrium configuration (Fig. 3) they give is r(C,H,) = 1.100, r(C,H,) = 1.090, r(C,H,) = 1.100, r(C,C,) = 1.324, r(C,O) = 1.334; H E C , = 120.0, H C C , = 121.0, H E C , = 116.0, C E O = 129.4 (distances in A, angles in degrees). From their measurements in the acetyl fluoride enolate anion, CH,COF(Section 1II.B) Marks et al. (1988) found the rotational constants of the ground state (A' symmetry) are A = 0.38185(7), B = 0.35769(7), C = 0.1843(2)cm-l. They noted that the inertial defect, A of equation ("I) is, very small signifying that the anion is planar. Photodetachment measurements by Zimmerman et al. (1977)have given EA(CH,COF) = 2.22 & 0.09 eV. The photoelectron spectra of C,H, generated in an oxygen-methylacetylene (CH,C = CH) discharge and extracted as a beam has been investigated by Oakes and Ellison (1983). Proton transfer gives the isomeric anions

CH,C

= C-[39],

CH,CCH-[39],

(45)

the number in square brackets being the mass in daltons. Provided H - D scrambling does not occur, the two anions generated when the methylacetylene is replaced by methylacetylene d,, (CH,C = CD), or methylacetylene d,, (CD,C = CH), are CH,C

= C-[39],

CH,CCD-[40]

(46)

CD,C

= C-[42],

CD,CCH-[41].

(47)

or are

They could be mass-selected. With allene (CH, = C = CH,) instead of methylacetylene CH,CCH- is the only anion to be expected. Oakes and Ellison found no evidence of photodetachment from CH,C = C - by the 488-nm laser they used in most of the investigation nor by a less powerful 457.9-nm laser that were available. This knowledge allowed them to show that there is little H-D scrambling and that EA(CH,C = C) 2 2.60 eV.

FIG.3. CH,CHO- equilibrium geometry.


69

FIG.4. CH,CCH- equilibrium geometry.

They reasoned that EA(CH3 = C) is unlikely to differ much from EA(HC = C) (See Section 1V.C). From their spectra Oakes and Ellison also deduced that EA(CH,C E CH) = 0.893 f 0.025 eV, EA(CD,C = CH) = 0.907 f 0.023 eV, and EA(CH,C E CD) = 0.88 f 0.15 eV. Ab initio calculations by Wilmshurst and Dykstra (1980) give that the anion has a bent structure (Fig. 4) with r H C ) = 1.088, r(C,C,) = 1.288, r(C,C,) = 1.356, r(C,H,) = 1.079; 120.9, G C , = 175.9, = 116.6, 0 = 2.7 where 0 is the angle the C3C2 line is out of the H,C,H, plane (distances in A, angles in degrees). Chandrasekhar et al. (1981 ) have calculated the energies and equilibrium geometries of HCO-, FCO-, OCOH-, HCO;, NH,CO-, CH3CO-, NHCHO-, and CH,CHO-. Kalcher and Sax (1988) have calculated that EA(CCH,) = - 0.48 eV. Ellison et al. (1982) have determined the EAs of several alkoxide and enolate anions. The EAs of many large organic molecules have been tabulated by Drzaic et al. (1984).

&*

@,

ACKNOWLEDGMENTS

I thank the U.S. Air Force for support under grant AFOSR-88-0190.

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ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 21

ELECTRON-POLARIZATION PHENOMENA IN ELECTRON-A T O M COLLISIONP JOA CHIM KESSLER Universitat Miinster Physikalisches Institut Miinster, West Germany

. . . . . . . . . . . . . . . . . . . . . . . . Spin-Dependent Scattering Due to Spin-Orbit Interaction. . . Spin-Dependent Scattering Due to Exchange Interaction . . .

I. Introduction

11. Phenomena Governed by a Single Polarization Mechanism.

. . . .

. . . . . . .

A. B. C. Polarization Effects Caused by the Interplay of Fine-Structure Splitting with Exchange Scattering . . . . . . . . . . . . . . . . . 111. Combined Effects of Several Polarization Mechanisms . . . . . . . . A. Theoretical Description of Electron Scattering from U n p o l a r i d Atoms Having Angular Momentum . . . . . . . . . . . . . . . . B. Information Derived from Observation of the Scattered Electrons . . . C. Information Derived from Observation of the Atoms. . . . . . . . D. Information Derived from Simultaneous Observation of Electrons and Atoms.. . . . . . . . . . . . . . . . . . . . . . . IV. Studies Still in an Initial Stage. . . . . . . . . . . . . . . . . V. Conclusions . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

81 87 87 97 108 117 117 124 135 145 151 158 159 160

I. Introduction A new dimension has been added to the exploration of atomic forces: The methods of producing and detecting spin-polarized electrons have reached a stage where they can successfully be applied to investigations of the interactions of the spin and the magnetic moment of the electron. In conventional collision experiments, the spin-dependent interactions, such as the spin-orbit and exchange interactions, are masked by the much stronger Coulomb interaction. However, it is now possible to explore these weaker interactions by means of polarized-electron techniques. This is of considerable help in

* Dedicated to Dr. Klaus Jost, whose achievements and advice over a period of 30 years were invaluable for the success of my group. 81 Copyright 0 1991 by Academic Press, Inc. All nghts of reproduction in any form reserved. ISBNO-12-003827-7

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disentangling the role that the various interactions play in atomic collision processes. Before discussing the advances in this field we shall summarize the essential points that are necessary for understanding collisions in which polarized electrons play a role. This will be done in a rather intuitive way, a more rigorous treatment having been published previously (Kessler, 1985). A polarized electron beam is a beam with a preferential orientation of the electron spins as illustrated in Fig. 1, showing the cases of total and partial polarization. If the observation of the spin components along a particular direction yields N , electrons with spins parallel and N , electrons with spins antiparallel to that direction, then one defines P = N ? - Nl Nt + N l

as the component of the electron polarization vector P = ( P x ,P,, P,) in that direction. IPI is called the degree of the polarization. There are two basic types of scattering experiments suitable for analyzing spin-dependent interactions:

TOTAL POL. P=l

PARTIAL POL. P= N4-N$ N +N

+ *

FIG. 1. Total and partial electron polarization.

ELECTRON-POLARIZATION PHENOMENA

83

1. If polarized electrons are scattered from polarized light atoms and one observes a process of the type et

+A1 +el +A t ,

(2) then it is very likely that the spin flip of the collision partners has been caused by exchange between the incident electron with spin up and the atomic valence electron with spin down, because in light atoms other spin-dependent interactions are negligible. Accordingly, an analysis of the spin directions of the scattered electrons-such as a measurement of their polarization-yields direct quantitative information about the exchange interaction alone. Theoretically, the process given by Eq. (2) is described by the scattering amplitude g which is called the exchange amplitude. More precisely, the amplitude is defined to be - g (Kessler, 1985). Needless to say, the electrons may also be scattered by a direct process according to et +A1 +et +A1

(3)

where they do not change their spin directions; this is described by the direct scattering amplitude 1: 2. If one wants information on the spin-orbit interaction, one can scatter unpolarized electrons from spinless heavy atoms (Fig. 2) and observe the polarization phenomena occurring. The spin-orbit interaction of a scattered electron in the atomic field can be described by a term in the scattering potential that is proportional to the scalar product 1.s of the electron’s orbital and spin angular momentum. In the example of Fig. 2,I.s is positive for spinup and negative for spin-down electrons. As a consequence, the scattering potential (which results from the Coulomb plus the spin-orbit interaction) is different for electrons with spins parallel and antiparallel to 1. The different scattering potentials result in different scattering cross sections so that one obtains different numbers of spin-up and spin-down electrons in the scattered beam; in other words, the scattered beam is polarized. A quantitative example is given in Fig. 3 which also shows how the polarization of the scattered beam results from the difference between the cross sections for spinup and spin-down electrons according to Eq. (1). Since the polarization can be different from zero only if there is a nonvanishing spin-orbit interaction, it is clear that a polarization measurement yields direct quantitative information on this “weak force with conspicuous effects,” as Fano (1970) called the spin-orbit coupling in the title of one of his papers. Some of these conspicuous effects are the basis of the experimental techniques used in polarized-electron studies. The device commonly used for

Joachim Kessler

84

c UNPOLARIZED FIG.2. Polarization caused by different scattering potentials for spin-up ( 0 )and spin-down (@) electrons.

polarization measurement, the Mott detector, is based on the left-right scattering asymmetry of a polarized electron beam. Fig. 4 illustrates that, for spin directions normal to the scattering plane, the resulting scattering potential depends on whether the electrons are scattered to the left or to the right because the (small) contribution of the spin-orbit term 1.s to the potential differs in sign for the two directions. The resulting different scattering potentials give rise to different scattering intensities I, and I, to the left and to the right, from which the polarization P can be determined according to 1 I, - I, p=--

s I, + I,'

(4)

The so-called analyzing power (Sherman function) S is a complicated function of the electron energy, the scattering angle and the atomic number 2 of the target (the higher Z , the better). Its precise measurement is difficult.

85


"

XP

I

I

I 1I

I

I I

600

900

1500

lzoo

SCATTERING ANGLE

l w

e

FIG.3. Differential scattering cross sections ~ ( 0 )for spin-up and spin-down electrons, resulting in polarization P after scattering of an initially unpolarizedelectron beam. Example for elastic scattering of 300-eV electrons by mercury. (From Kessler, 1985).

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Joachim Kessler

POLARIZED FIG.4. Left-right intensity asymmetry caused by nonsymmetric contributions of the spin-orbit term to the scattering potential.

The most careful experimental analysis by Fletcher et al. (1986) resulted in an uncertainty of 5 % for S. The principal method of producing polarized electrons is also based on the spin-orbit interaction, which has the consequence that photoelectrons from atoms or solids may have significant spin polarization (Fano, 1969; Kessler, 1981, 1985; Heinzmann and Schonhense, 1985; Heinzmann, 1987). This is exploited in the GaAs source (Garwin et al., 1974) the principle of which is shown in Fig. 5. Irradiation of GaAs with circularly polarized light produces polarized photoelectrons. The direction of the photoelectron polarization coincides with the axis of the light beam, and the degree of polarization is usually between 30 and 40 %. The currents produced are generally less than 10 PA. Though there are other interesting techniques with specific advantages (for a summary see, e.g., Kessler, 1985), the GaAs source is the most widely used method of producing polarized electrons. Details of the preparation of a


87

GaAs (NEG.EL.AFFINITY) FIG.5. GaAs source of polarized electrons.

GaAs photocathode as well as the operation and performance of the source are described in a review by Pierce et al. (1980) and need not be repeated here.

11. Phenomena Governed by a Single Polarization Mechanism A. SPIN-DEPENDENT SCATTERING DUETO SPIN-ORBIT INTERACTION This chapter is concerned with processes that are conceptually simple. One of them is elastic electron scattering from spinless atoms. In this process, the spin-dependent effects are caused by the spin-orbit interaction alone. Though exchange may play a considerable role, it cannot produce polarization phenomena in elastic scattering from a spinless target. Elastic electron scattering by a spinless atom is described by two complex scattering amplitudes: the amplitude f = If1 eiY1,which is mainly determined by the Coulomb interaction, and the amplitude g = (g(eiY2describing spin

Joachim Kessler

88

flips of the electrons caused by the spin-orbit interaction of the scattered electrons in the atomic field.' In order to understand the scattering process completely one needs to know the scattering amplitudes completely, their moduli, and their phases. One piece of information, the sum of the absolute squares of the scattering amplitudes, is obtained by the conventional measurements of the cross section for scattering of unpolarized electrons ."(@ = lfI2

+ Id2

(5)

All the remaining information can be obtained from polarization measurements. This may be seen from the expression

for the final polarization P of an electron beam with initial polarization P = P, P, (see Fig. 6 ) that has been scattered by a spinless target (Kessler, 1985), where

+

If the incident beam is unpolarized, P = P,

P

= SA

+ P, = 0, one has from Eq. ( 6 ) (8)

FIG.6. Components P, parallel and P, normal to the scattering plane for arbitrary initial polarization P. (k and k' are the electron wave vectors before and after scattering; A =

w.)

'

Although it is common practice to use the same letter g, this amplitude has nothing to do with the g describing the completely different process of Eq. (2).


89

so that a measurement of the polarization P’ of the scattered beam, which has only a component normal to the scattering plane, yields the combination of scattering amplitudes denoted by S . With the preceding notation for the complex f and g one has

so that such a measurement yields informaion on the relative phase off and g. If the incident beam has a polarization P # 0, a measurement of the polarization components of the scattered beam yields the observables T and U . As seen from Eq. (6), T describes the change in length of the initial component P, whereas U describes the rotation of the polarization out of its intial plane spanned by P, and P,. Once the set of observables a,(e), S , 7; U is determined one has four equations to evaluate the moduli and phases of the scattering amplitudes f and g. The observables are, however, not independent of each other: from Eq. (7) one finds S2 + T 2 + U 2 = 1. Consequently, one can only determine the relatioe phase y1 - y2 as can be directly seen from Eq. (9) and the form

of U . This is in accordance with the concept of quantum mechanics that an absolute determination of the phases from an analysis of the scattered wave is impossible. Still, all of the four observables have to be measured since two measurements [ S a sin(y, - y 2 ) and U a cos(y, - y 2 ) ] are required for an unambiguous determination of y1 - y z . Since the measurements discussed yield I f l , 191, and y1 - y 2 thus giving the maximum possible information on the scattering process, we have a complete or “perfect” experiment in the sense defined by Bederson (1969) in a similar context. In the case of elastic scattering from an unpolarized target, there is another independent method of measuring the observable S. Instead of determining the polarization after scattering of an initially unpolarized beam according to Eq. (8), one can exploit the spin-dependence of the scattering cross section of a polarized electron beam a(e, 4 ) = o,(e)ci

+ S(~)P.AI

(1 1)

( 4 = azimuthal scattering angle). Reversal of either the polarization P or the vector A (e.g., change from left-hand scattering to right-hand scattering or

90

Joachim Kessler

vice versa) changes the sign in the last bracket. Denoting the cross sections for parallel and antiparallel orientation of P and A by at and al,respectively, one has the asymmetry

+

A = -O t - O1 = 1 PS - (1 - PS) = PS, at+a1 l+PS+l-PS

which shows that an asymmetry measurement also yields the Sherman function S if the incident beam polarization P is known. In fact, this procedure is usually more accurate than the determination of S from Eq. (8), because it requires intensity measurements in a single scattering process, whereas in the other case the polarization P‘ = S of the scattered beam must be analyzed by a second scattering process (double scattering). While numerous measurements of cross sections have been made for more than half a century and extensive measurements of S for more than two decades, complete electron-scattering experiments have only been performed in recent years. We shall not discuss here the “old” observables a,(@ and S. We mention only briefly that the knowledge of the Sherman function for spinless atoms has become quite satisfactory over the years. Discrepancies between theory and experiment below 100eV could in many cases be eliminated by recent theoretical approaches that take proper account of the atomic charge-cloud polarization and of exchange effects’ playing a significant role at lower energies (McEachran and Stauffer, 1986,1987; Haberland et al. 1986; Haberland and Fritsche, 1987). Some discrepancies are left at a few selected energies where the results are particularly sensitive to energy or angle, and in cases where comparisons with early experiments were made (Sienkiewicz and Baylis, 1988). Disagreement between some of the experimental results at low energies shows, however, that not all of the measurements are reliable. Determination of S by the two independent methods based on Eqs. (8) and (12) in the same laboratory would be crucial in these cases. Such experiments are underway and their first results give clear evidence of the reliability of recent theoretical methods (Garcia-Rosales et al. 1988). Measurements of the observables T and U,necessary for completion of the “perfect” experiment, have been made over the past few years with the apparatus of Fig. 7. The polarized electrons come from a GaAs source that is fixed in space. An electrostatic deflection system can be rotated about the Though, in elastic scattering from spinless targets, exchange processes cannot generate polarization phenomena, they can, of course, occur and affect the numerical values of the observables.


HeNe L A S E R A

MOTT

DETECTOR

91

fi

LIGHT MOOULATOR

900 GaAsP CAT1 WlEN FILTER FILTER LENS

Xe ATOMIC BEAM

l8Oo DEFLECTOR U

FIG.7. Measurement of the change of the electron-polarizationvector caused by scattering. (From Berger and Kessler, 1986).

target so that the scattering angle varies continuously. The rotation does not affect the direction of the initial polarization P because the electron spins are not affectedby the electrostatic deflection fields. The polarization vector P is always oriented along the axis of observation, which facilitates the data evaluation. The transverse components of the final polarization vector P are determined by the left-right asymmetry measured in the two pairs of counters of the Mott detector. Since such a detector is not sensitive to longitudinal polarization components, a Wien filter is introduced that can rotate the longitudinal component by 90" to become transverse, so that it can also be measured in the Mott detector. The outcome of such measurements made for xenon and mercury at energies between 25 and 350 eV will be discussed with two examples. Figure 8 displays the complete set of observables a,(@, S, 'I; and U for elastic scattering from mercury at 50 eV together with the most recent theoretical data based on the Dirac equation. From the data of T and U it can be seen that, thanks to the advent of efficient polarized-electron sources, it has now become possible to measure these observables with a statistical accuracy of a few percent. The evaluation of the complex scattering amplitudes from the complete set of observables is shown on the right-hand side of Fig. 8. While the samples presented here were selected from experiments where the angular

Joach im Kessler

92

";-1'--i

-0 L

-08

360

Y,-Y21DEGl

t

.

.

300

1

210

180 120

60

- 0O 8L

i__i 30

60 90 120 150 SCATTERING ANGLE IDEGI

FIG.8. Complete set of observables (absolute differential cross section and polarization parameters S, 'I; U )together with moduli and relative phases of scattering amplitudes for Hg at 50eV. Theory: - McEachran and Stauffer (1987); - - - Haberland and Fritsche (1987). Experimental points: Berger and Kessler (1986) and Holtkamp et al. (1987).

dependence of the observables was studied at fixed energies, other results are available where the energy-dependence was measured at fixed scattering angles (Mollenkamp et al., 1984). Figure 9 shows the complete set of observables and the evaluation of the scattering amplitudes for elastic scattering from xenon at the same energy of


T

I0

93

Y,-y,IDEGl 270

05

180 90 60 90 I20 150 SCATTERING ANGLE lOEGl

30

U

-0.5 30

60 90 120 150 SCATTERING ANGLE (DEG)

FIG.9. Complete set of observables (absolute differential cross section and polarization parameters S, 'I; U )together with moduli and relative phases of scattering amplitudes for Xe at 50 eV. Theory: McEachran and Stauffer (1986); - - - Haberland et al. (1986);. . . Awe et al. (1983). Experiment: 0 Berger and Kessler (1986); Mollenkamp et al. (1984) and Wiibker et al. (1982). Experimental cross sections used for the evaluation of the moduli If1 and (91: 0 Register et al. (1985); A Mehr (1967). ~

50 eV. Because the theoretical polarization parameters have narrow dips at some angles, the theoretical curves were convoluted with the experimental angular resolution ( f2" at 50 eV). This was not necessary for Hg because the peaks are broader there. Since the spin-orbit interaction of the scattered electrons is smaller in xenon than in mercury (atomic number 2 = 54 compared with Z = 80) one finds smaller values of the spin-flip amplitude g. Nevertheless, the polarization effects are at least as pronounced in xenon as

94

Joachim Kessler

in mercury, because the amplitude f is also smaller and in the interplay of f and g it is the relative size of the amplitudes that determines the significance of the polarization phenomena as an inspection of Eqs. (7), (9), and (10) shows. An obvious difference between f and g in the energy regime considered here can be seen from the examples given. The amplitude f has an interference structure typically occurring when several partial waves play a dominant role in the scattering process. In g such interference is only weakly present because there is only one dominant partial wave. This is because the spin-orbit interaction that is responsible for g is important mainly in the inner part of the atom; it decreases rapidly as the distance from the scattering center increases. Accordingly, it is only electrons with small orbital angular momenta 1 (corresponding to small impact parameters) that experience a spin-dependent force. On the other hand, for electrons with 1 = 0 the spin-orbit interaction vanishes completely. As a result, only the partial waves with I = 1 and, less importantly, 1 = 2 make a significant contribution to the spin-dependent effects. A quantitative substantiation of this explanation can be found in a table of phase shifts given by McEachran and Stauffer (1986). Another difference is that the absolute values of g are, on average, an order of magnitude smaller than those of f: This is because the spin-orbit interaction is weak compared with the Coulomb interaction. As a consequence g is much harder to measure than f, which explains why experimental data on the spin-flip amplitude have not been available previously and why the uncertainties of g are larger than those off: Comparison of the experimental and theoretical results shows that in particular the three polarization parameters S, 7: and U are well described by the present theories. After Walker’s pioneering application of relativistic scattering theory to low-energy electron scattering (Walker, 1971), the calculations have now reached a level where the treatment of critical problems such as electron exchange and distortion of the atomic charge cloud by the scattered electrons is sufficiently accurate and different approaches produce similar results. The earliest of the theoretical results used here for comparison (Awe et al., 1983) were obtained by solving the Dirac equation with an energy-dependent local exchange potential of the Slater type. Though the atomic charge-cloud polarization was not explicitly taken into account, the authors conclude that their potential “while too strong for describing exchange only, already contains an approximate description of charge polarization effects.” (Awe et al., 1983, p. 61 1.) In contrast, Haberland


95

et al. (1986, 1987) start from a generalized Kohn-Sham theory, treating the colliding particles as an excited quasiatomic system made up of N target electrons and one additional projectile electron. The (N + 1)-electron wave function describes all electrons as indistinguishable while the effective potential comprises all many-body effects governing the scattering, in particular the charge-cloud polarization of the atom. McEachran and Stauffer (1986, 1987) describe the scattering within the framework of relativistic Dirac-Fock theory and treat the exchange exactly, whereas they derive their adiabatic charge-cloud polarization potential from a nonrelativistic polarized-orbital calculation. An inspection of the examples presented here and of the much greater number of results published as of 1989 reveals that it is mainly in the cross sections that systematic discrepancies between theory and experiment still exist. In particular, at energies higher than about 100 eV, the experimental and theoretical polarization parameters agree very well with each other. One might suspect that the problem comes from experiment for the following reasons: The differences between absolute across sections measured by different groups at the large Z and 8 considered here amount to about 20 % and are often larger than the uncertainty limits claimed by the experimentalists (see Kessler, 1986). Polarization measurements, on the other hand, do not require knowledge of such quantities as absolute target densities and absolute scattering intensities, which are the main error sources of absolute cross section measurements, because they imply observation of a ratio of intensities (e.g., left to right). As a consequence, the accuracy of polarization measurements is usually higher than that of absolute cross section measurements. This experimental situation does not, however, explain why discrepancies are larger in the cross sections than in the polarization parameters, because the variance of 50% or more between the experimental and theoretical cross sections is clearly greater than the difference of 20 % between the experimental cross sections of different groups. Rather, it is very likely that the loss of electron flux into open inelastic channels that was neglected by the therories mentioned accounts for the fact that, while the theories give an excellent description of the shapes of the cross sections, their absolute values tend to be systematically too high. The discrepancies in the absolute cross sections are reflected in the discrepancies in I f 1 and 191. The experimental values of I f 1 are systematically lower than the theoretical ones, while for (91 a similar tendency is clearly evident. The scatter of the experimental amplitudes for xenon is caused by the scatter of the absolute cross sections of the various groups used for the

96

Joachim Kessler

evaluation. The evaluation of the relative phase y1 - y z does not, in contrast to the moduli, require knowledge about absolute cross sections: from Eqs. (9) and (10) one has S tan(y, - y2) = U -y

which, in conjunction with one of the relations (9) or (lo), permits an unambiguous evaluation of y1 - y 2 . Accordingly, the agreement between theory and experiment in the relative phases is much better than in the moduli, in fact in most cases it is excellent. When comparing the experimental and theoretical values one should keep in mind that we are considering here the complete set of observables yielding the maximum possible information on the scattering process. This is a much more stringent test of the theory than the usual case where one is concerned with only one of the observables and does not care about the others. Here one can see whether a theory properly describes the complete scattering process, rather than only particular aspect of it. From this point of view the theoretical results are very encouraging, in particular since it seems to be clear how the remaining discrepancies can be eliminated: inclusion of the loss of flux into inelastic channels by a more rigorous method than an additional absorption potential and-in the McEachran-Stauffer approach-use of a fully relativistic polarization potential are promising. Although it is obvious that for the processes considered here, it is necessary to treat the scattering problem relativistically (since spin phenomena are not described by the Schrodinger equation), it is worth noting that for observables not related to spin polarization there are also significant differences between relativistic and nonrelativistic calculations of low-energy electron scattering. For instance, even such gross quantities as the Ramsauer minimum in xenon below 1 eV agree much better with experiment when calculated with the Dirac equation instead of with Schrodinger-type equations (McEachran and Stauffer, 1988). This suggests that relativistic calculations should be regarded as the norm, especially for atoms with moderate to large nuclear charge where the electrons are accelerated to considerable energies while being scattered. We shall conclude this section with the remark that significant spinpolarization phenomena in atomic scattering are typical of electrons, but do not occur in positron scattering. The reason is that, due to the repulsive nucleus-positron interaction, positrons have a negligible chance of reaching the small nuclear distances that we have seen to be most important for the generation of spin effects by the spin-orbit interaction. Accordingly, the

97


polarization effects for positrons are orders of magnitude smaller (=f-s.

The last relation follows from the fact that the contributions of direct and exchange scattering to the process e t + A t +et

+Af

(36)

cannot be distinguished so that one has a coherent superposition of the amplitudes (Kessler, 1985, Chapter 4). Since according to Eq. (34b) there are three pairs of identical terms in Eq. (35), one obtains Eq. (14). In addition to the three observables Eqs. (30)-(32) one needs the following five observables in order to describe the change of the polarization vector caused by scattering of spin-1/2 particles from an unpolarized target:

121


where the upper sign refers to the x- and the lower to the y-component,

T,=

c

kl (f(M19; Mo$)f*(M19; MOB %ko(2Jo + 1) M,Mo - f(M 1 I. M 0 - ))f*(M 1I. M 0 - I2)) 2, 2,

(37b)

+ f(M13; MoW*(M13; Mo - $1). The polarization P after scattering of an electron beam with initial polarization P = P,Et + P y 9 + Pz2 is then

which elucidates the physical meaning of the observables given by Eqs. (37). T,, T,, and T, describe the change in length of the polarization components while U,, and U,, describe the rotation of the polarization components in the scattering plane. These are generalizations of the results for elastic scattering from a spinless target for which one finds from Eqs. (37) in conjunction with Eq. (34a) T,= T,=

IT:

T,= 1,

U,,= U,,= U

in accordance with Eq. (6). In that case one has to measure four observables for a complete experiment: absolute cross section, Sherman function (which we may assume to be S p obtained by a polarization measurement), and two polarization components. (For example if the initial polarization is P = P,S, measurement of the x and z components yields Tand U . ) In the present case one needs four additional observables: two in-plane components U,,P, and T,P, obtained with a different initial polarization P = Pz2, the component Py,and the asymmetry S,. Unlike the case of elastic scattering from a spinless target, in general these eight measurements do not suffice to completely determine the scattering amplitudes by which the process is described, because the number of amplitudes increases rapidly with increasing J , and J , . The measurements yield, however, the maximum possible information

Joachim Kessler

122

about the scattering process of the electrons. How the missing information on the behavior of the atoms can be provided will be the subject of Sections 1II.C and D. Here we shall discuss another special case. We consider an inelastic process leading from J o = 0 to J , = 0. Taking parity conservation, Eq. (34b), into account one has immediately from Eqs. (301, (321, and (37)

showing that the process is again determined by four observables instead of eight. To be specific, we consider the excitation of a 3P$state from the ground state 'S; in helium or mercury. Since the product of the even and odd parities is - 1, one has from Eq. (39a) S , = - S p , a result we have already found in the preceding chapter to hold for excitation of the He 3P, levels under the conditions of the fine-structure effect. Here we see that for 3P,excitation this relation holds quite generally, being solely a consequence of parity conservation. Another interesting result is obtained by inserting Eqs. (39a) and (39b) in Eq. (38) and inspecting the y component of the final polarization for a 0 + 0 transition p'

=

sp + rIo*rI,*P,

1 + rI,.rI,.spP,'

For a totally polarized beam with P , = 1, one obtains by using ni.ll: = 1 p =

n,*n,*sp + 1n,.n, = n,*n, 1 +rI,*n,~sp

or PI = -1 for the transition to 3P$.This complete reversal of the spin polarization by electron-impact excitation of 3P, has previously been discussed under special conditions (Hanne and Kessler, 1976; Hanne, 1976), while here it follows quite generally from symmetry arguments. It is also seen from Eqs. (39) that for 0 + 0 transitions between states of the same parity, the polarization P'of the scattered electrons reduces to the simpler expression of Eq. (6) which therefore holds not only for elastic scattering from spinless targets. The reason why for 0 + 0 transitions one needs only four observables for describing the scattering process is, of course, the small number of scattering amplitudes by which such a process is characterized.


123

Another case where less than the eight observables presented suffice for a full description of the scattered electrons is the general elastic-scattering process. This is because the time-reversal invariance of elastic scattering results in the following two relations between the observables:

s, = s,

(4 1a)

U,, - U,, = tan 8 ( T , - T,). In the special case of an elastic 0 -,0 transition, the first of these relations is according to Eq. (39a) a direct consequence of parity conservation. In the general elastic case, Eq. (41a) is a well-known consequence of time-reversal invariance. Less well known is relation (41b). For elastic scattering from oneelectron atoms its validity has been shown by Burke and Mitchell (1974) while its derivation for the general elastic case is rather intricate (Bartschat, 1989,appendix). The obvious consequence of the two relations (41) is that the number of observables by which the electrons scattered from an arbitrary target are characterized is reduced from eight to six in elastic electron scattering. The number of observables by which a collision process is characterized depends not only on the specific atomic transition, but also on the relevant interactions because these also determine the number of necessary amplitudes. We have seen in Section 1I.B that for light alkali atoms where spin-orbit interaction is neglected, one needs only four observables for describing elastic scattering. And if all spin-dependent interactions-and thus all polarization phenomena-are excluded, one is left with only one observable, the good old differential cross section ou = We have followed here Bartschat’s presentation in some length because it is an important generalization of all preceding work on electron scattering from unpolarized targets. Considering the long progress from the quite intricate polarization formulae for much simpler cases derived in the pioneer papers (Tolhoek, 1956) over the later simplifications (Motz et al., 1964; Kessler, 1969) to the preceding results, one feels that the theoretical formulation of scattering of spin-1/2 particles from unpolarized targets is now very satisfactory. A general description has been found that encompasses transitions between arbitrary atomic states without making special assumptions about the atomic interactions while still being lucid and practicable. It is an impressive example of how a complicated scattering problem can be cast in an elegant form by means of the density matrix formalism. The existence of such a general theoretical guideline is certainly an encouragement for experimental studies of polarization phenomena in inelastic scattering. Although a few fundamental experiments have been made

124

Joachim Kessler

(some will be presented in the next section), we are far from the goal of a series of systematic studies of the phenomena outlined here. Such measurements are, however, within reach of present techniques as can be inferred from the elastic experiments with closed-shell atoms outlined in Section 1I.A. Similar observations of the more complicated phenomena of elastic scattering from open-shell atoms can be made with the same technique and, despite the lower cross sections, studies of inelastic scattering should also be feasible in suitable angular ranges. In fact, the preparations for such experiments have stimulated the theoretical treatment of the general scattering problem. An idea of the size of the effects that can be expected is given by the preliminary theoretical results presented in Fig. 21 for electron-impact excitation of the (unresolved) 63P, levels of mercury at 20 eV incident energy. Although the data were obtained with the distorted-wave Born approximation (DWBA), which will be seen in the following to be unreliable at 20 eV, they certainly illustrate the qualitative behavior of the observables. One notes a significant difference between S , and S,, while T, and T, as well as U,, and U,, are almost identical, as in elastic scattering. On the other hand, T, deviates over the whole angular range strongly from its elastic value T, = 1. Dramatic changes of the polarization components described by the deviations of ' I , T,, , and T, from 1, which are an exception in elastic scattering from spinless targets (Kessler, 1985, Fig. 3.23), are seen to be the rule here. This is a result of the strong exchange contribution in the excitation of the triplet level from the singlet ground state. In experiments in which the fine structure is resolved there can be considerable differences between the polarization parameters T,, T,, T, and also between U,, and U z x .This is borne out by calculations made between 15 and 120 eV for electron-impact excitation of states of mercury from the ( 5 ~ ~ 6 s ) " ~states P " of xenon and the (6~6p)'*~P" their respective ground states. The results for 40 eV can be found in Bartschat and Madison (1988). B.

INFORMATION DERIVED FROM OBSERVATION OF THE SCATTERED ELECTRONS

We shall now discuss electron-scattering experiments with unpolarized target atoms possessing angular momentum in their ground state and/or in the excited state. Examples are scattering from open-shell atoms such as cesium, thallium, or bismuth as well as inelastic scattering from closed-shell atoms such as the noble gases or mercury. Since, in contrast to elastic

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

5

1

1

.

1

1

1

.

1

1

.

1

.

1

.

0.8

U

T 0.4

0.0

0.0

- 0.2

- 0,4 0

60 120 SCATTERING ANGLE (DEG)

180

0

60 120 SCATTERING ANGLE (DEG)

180

FIG.21. The set of observables containing the maximum possible information on inelastic scattering of 20-eV electrons with excitation of Hg 63P, (fine structure not resolved) (Bartschat, 1988).

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Joachim Kessler

scattering from spinless targets, systematic studies of this wide field have not yet been done we can only give a critical review of the sporadic results that will make evident the next steps to be taken. We start with elastic scattering, where measurements have been made with the aim of comparing the polarization of electrons elastically scattered from mercury (Z = 80), thallium (Z = 81), lead (Z = 82), and bismuth (2 = 83), which have the respective configurations ‘So, 2P1/2,3P0, and 4S3/2. From earlier calculations between 25 and 800 eV, not taking exchange into account (Fink and Yates, 1970), one does not anticipate significant differences in the polarization curves of the four elements above 25 eV. On the other hand, the “fine-structure effect” caused by the interplay of exchange scattering and atomic fine-structure splitting might be effective in thallium and lead because these atoms populate in their ground state only one of the fine-structure levels. This is a situation similar to that discussed in superelastic scattering (Section II.C), where the population of one fine-structure level is artificially produced by laser optical pumping. Since in the fine-structure effect exchange is essential, it can play a major role only at low energies. A small selection from the measurements made between 6 and 180 eV is presented in Fig. 22. It is a good example of how a reliable interpretation of experimental results for complex target atoms can only be given in conjunction with a quantitative theory. At first sight one might be tempted to interpret the results along the line suggested by the preceding discussion, because at low energies one finds significant differences between the polarization curves whereas at 180 eV they are almost identical. The theoretical analysis by Haberland and Fritsche (1987) showed, however, that these differences are caused by a different long-range behavior of the scattering potential: the distortion of the atomic charge cloud by the scattered electrons which plays a role at low energies (Section 1I.A) varies because of the different polarizabilities of atoms with different configurations. This explains the strong variation of the polarization curves of neighboring atoms without taking the fine-structure effect into account; it is sufficient to consider the spin-orbit interaction of the scattered electrons in the atomic field. An example for the opposite situation has been given in a model calculation by Bartschat (1987). In elastic scattering from boron (22P1/2)and carbon Q3PO)he finds significant polarization of the scattered beam (Fig. 23) caused solely by the fine-structure effect. (The calculation assumes that only the lowest fine-structure level is populated.) Polarization due to the spin-orbit interaction of the scattered electrons in the atomic field (Z = 5 and 6) is negligible as the comparative results for carbon show. Although this example

127


TI

Bi

Pb

1.0

,

I

,

,

,

P 0.5

0.0

- 0,s P

1.0

U

0,s

-0,s

U LO 80 120

LO 80 120

LO 80 120

LO 80 120

SCATTERING ANGLE (DEG) FIG.22. Angular dependence of the polarization P of electrons scattered elastically by Hg, TI, Pb, and Bi at 12.2 and 180 eV. Experimental points (Kaussen et al., 1987) and theoretical curves (Haberland and Fritsche, 1987).

is only a simplified estimate, it demonstrates that the fine-structure effect should be taken seriously as a polarization mechanism in elastic scattering from atoms where normally only the lowest fine-structure level is populated (in contrast to boron and carbon with their small intervals between the finestructure levels). Besides the experiment previously discussed, there have been two other elastic polarization experiments with open-shell atoms. Klewer et al. (1979)

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Joachim Kessler

0.4

S 0.0

-0.4

0

60

120

180

FIG.23. Sherman function for elastic electron scattering from boron (-) and carbon (- - -) atoms at an incident electron energy of 13.6 eV. The dotted curve for carbon shows the minute effect of including the spin-orbit interaction of the scattered electrons in the atomic field (Bartschat, 1987).

measured the polarization of initially unpolarized electrons that were elastically scattered from unpolarized cesium atoms between 13.5 and 20 eV from 35 to 110". Their polarization curves, which never reach values above 20%, are in complete disagreement with the theoretical data of Fink and Yates (1970) and Walker (1974). Second, McClelland et al. (1987a) measured the Sherman function for sodium at 54.4 eV between 20 and 135" as a by-product of their spin-asymmetry data presented in Section 1I.C. They found small values, the maximum being 3-4% near 110" scattering angle. This agrees approximately with the theoretical results of Fink and Yates (1970) and is in accordance with what has been found for other targets of low atomic number (Schackert, 1968; Hilgner and Kessler, 1969). It is not only experimental but also theoretical results on polarization phenomena accompanying low-energy elastic scattering from open-shell atoms that are sparse. Apart from the Sherman function S calculated under the simplifying assumptions (no exchange or charge-cloud polarization) of Fink and Yates (1970) between 25 and 800eV, there are exploratory calculations of S with the R-matrix method for cesium at energies up to 2.04 eV (Scott et al., 1984a, b) for thallium at 1.9 and 3.5 eV (Bartschat et al., 1984b), and for lead at 1.9 eV (Bartschat, 1985). Scott et al. also give some


129

data for the change of transverse polarization in elastic scattering of polarized electrons from unpolarized cesium. They show that below 2 eV a significant reduction of the polarization takes place, in particular at the resonance energies. Polarization phenomena in inelastic scattering from heavy targets have been studied for quite some time. Those reviewed in earlier summaries will only be listed here for completeness with a short comment. The electron polarization P(8) after inelastic scattering by mercury atoms has been measured by Eitel and Kessler (1971) and Hanne et al. (1972) for excitation of the 6lP1 level and the autoionizing 5d'6~'6p(~P,) state between 25 and 180 eV incident energy. The remarkable similarity of the polarization curves for elastic and inelastic channels at the higher energies studied is clear evidence for the two-step model of inelastic scattering (Massey and Burhop, 1969; Madison and Shelton, 1973; Bonham, 1974; Bartschat and Blum, 1982b). Electron polarization after excitation of Hg 63P, and 6'P1 has been measured by Franz et al. (1982) at several scattering angles and incident energies between 10 and 20 eV. Comparison of the results for the 63P,, and 63P, fine-structure levels indicates that two mechanisms are responsible for the polarization: spin-orbit interaction of the scattered electrons in the atomic field and exchange via the fine-structure effect. This was confirmed by measurements of the scattering asymmetry at 90" associated with impact excitation of the 63P, fine-structure levels by polarized electrons between the thresholds and 22 eV (Bartschat et al., 1981b; Wolcke et al., 1987). Another result of the measurements, which were also made for 6'P, excitation, is that the well-known resonance structure of the cross sections between 5 and 12 eV is strongly reflected in the asymmetry curves. Attempts to explain the experimental data by different theoretical approaches were reviewed by Bartschat and Burke (1988). As pointed out by Bartschat et al. (1985), measurement of the angular dependence of the asymmetry provides a better test of the theory than measurement of its energy-dependence at a fixed scattering angle. One of the reasons is that slight angular shifts between theoretical and experimental curves with strong angular-dependence may give rise to large differences when asymmetries versus energy are compared at fixed angles. The angulardependence of the asymmetry between 30 and 120" has therefore been measured for a great number of energies between threshold and 50 eV for excitation of the Hg 63P,, 63P,, and 6'Pi states by polarized electrons. The apparatus used for these measurements is shown in Fig. 24. A transversely polarized beam of electrons with polarization between 0.3 and

He-Ne LASER

MAGNETIC LENS AND SPIN ROTATOR

190' DEFLECTOR

MAGNETIC LENS

0 GaAsP CATHODE

*

SPECTROMETERS

ATOMIC BEAM FIG.24. Apparatus for measurement of asymmetry in inelastic scattering of polarized electrons. (Only one of the channeltronsis shown). (Borgmann et al., 1987).


131

0.4 and an energy spread of about 200 meV is focused onto a mercury vapor

beam by a lens system that decelerates the electrons from the transportation energy of 200 eV to the desired scattering energy. Two detection systems, each including an electrostatic spectrometer (Jost, 1979a, b) and a channeltron select the inelastic channels to be studied. They are rotatable around the scattering center. The inelastic scattering asymmetries A can be observed in two ways. One is to use a single detection channel to measure the scattered intensities with incident polarization first up and then down. An equivalent way is to observe the spin intensities in the left and right channels simultaneously. This made possible a consistency check of the measurements. Figure 25 gives a small selection of the experimental asymmetries A / P = S, normalized to the polarization P of the incident electrons at the lower, medium, and upper energies covered by the measurement. Similar results for excitation of the 63P, state of mercury (Dummler et al., 1990) are being published. At the three energies given, the various polarization mechanisms effective in this intricate collision process are not of equal importance. The lower the energy, the larger is the contribution of the exchange mechanisms. But even at the lowest energies, the polarization cannot be explained by the relatively simple exchange-based fine-structure effect discussed for helium in Section II.C, because for mercury the LS coupling scheme does not hold. The superposition of singlet and triplet wave functions in the intermediatecoupling scheme results in more complicated expressions for the asymmetries (Borgmann et al., 1987). Nevertheless, at the lower energies studied there is a and S , (63P2), as Eq. (24) clear tendency toward opposite values of S , (63P1) predicts in the case of asymmetry caused by the fine-structure mechanism alone with LS coupling. However, in the light of the experiments to be discussed in Section III.D, an interpretation along this line turns out to be inadequate. Inspection of the high-energy end shows a quite different tendency in the curves: namely, no matter which of the atomic states has been excited, the experimental asymmetries look similar. This is because at such energies the polarization phenomena are caused mainly by the spin-orbit interaction of the scattered electrons in the atomic field; polarization mechanisms based on exchange that depend on the atomic configuration are less important here. The angular-dependence of the asymmetries was measured at energies where no pronounced resonance features occur, though such curves would also be of great interest at the resonance energies, since they are of considerable help in classifying the resonances. However, in order to make such measurements meaningful, one must separate the great number of

Joachim Kessler

132

63P, SA

G3P,

6'P,

O"

6.5eV

20eV

50eV

FIG.25. Angular-dependence of the scattering asymmetry A / P = S, normalized to the incident electron polarization for excitationof the Hg 63P,, 6'P,, and 6lP, states. Experimental points (Borgmannet al., 1987) and theoretical curves - - - from R-matrix calculations(Bartschat from DWBA calculations (Bartschat and and Burke, 1986, and private communication), Madison, 1987). ~

resonance features between 8 and 12 eV from each other, which needs a better energy resolution than the 280 eV of the experiment described, An impression of the asymmetry resonances is given by Fig. 26 for scattering angles of 45 and 120". The measurements were made in steps of 100 meV and 15" between 30 and 120". The asymmetry measurements were accompanied by a theoretical effort to understand the data quantitatively. Different theoretical approaches were used within the framework of the R-matrix theory (Bartschat et al., 1984b;

133


45 O

0.0 - 0.2 -0,k 0.4 L " "

' A

0.2

120 O

0.0 -0.2 -0.4

7

9

11

13

7

9

11

13

7

9

11

13

ENERGY (eV) FIG.26. Energy-dependence of the scattering asymmetry A / P = S, normalized to the incident electron polarization for excitation of the Hg 6 3 P , , 63P2,and 6'P, states (Borgmann et al., 1987).

Bartschat and Burke, 1986) and the distorted-wave Born approximation (Bartschat et al., 1985; Bartschat and Madison, 1987). Examples are given in Fig. 25 showing how the R-matrix calculations approximate the experimental data at the lowest energies and that the DWBA calculations are adequate at the highest energies studied, except for excitation of 63P,. For a discussion of how the results depend on different theoretical approximations that also affect the relative contributions of the various polarization mechanisms, we refer to the original papers. Considering the complexity of the processes studied, the agreement between theory and experiment is encouraging. But there are still problems to be solved in this relatively new field! None of the experiments considered so far in this section have been concerned with measurements of observables describing the change of the electron polarization caused by scattering. In fact, there has been only one experiment of that type in which the target angular momentum played a role. This was the direct observation of the influence of exchange in mercury singlet to triplet excitation by measuring the change in polarization of the electrons causing the transition (Hanne and Kessler, 1976). In the notation of Eq. (38) this constitutes a measurement of the observable Ty, because P;/P, was measured for electrons scattered in the forward direction, where S p is 0. Since this experiment has been reviewed by several authors we mention it only for completeness, pointing out that for a direct measurement of

134

Joachim Kesder

exchange scattering one does not necessarily need polarized atoms! The method is presently applied for measuring the exchange contribution to electron-molecule scattering. Further measurements of the polarization parameters T and U in inelastic scattering are being prepared. This brings us to the results that exist only in theory or at best as laboratory notes. All inelastic polarization experiments (of the type discussed in this section) with unpolarized targets other than mercury belong to this category as well as experiments with heavy polarized targets where polarization phenomena are caused by both exchange and spin-orbit mechanisms. I list a few examples in order to show in which direction developments will go in the coming years. Tentative calculations of electron polarization S , and scattering asymmetry S A for excitation of the lowest 3ps4s 3P states in argon have been made by Hanne (1983) at 16 and 50eV. For excitation of the 62P fine-structure levels in cesium these parameters have been calculated with the R-matrix method at 1.632 and 2.04 eV incident electron energy (Scott et al., 1984a). According to the calculations, the relations Sp,A(’P1/2) = --2sp,A(’P3/2), which have been seen in Section 1I.C to be characteristic of the fine-structure effect, are approximately valid. This indicates that the fine-structure effect seems to be the dominant polarization mechanisms in this case. Such results can conveniently be checked by superelastic polarization experiments as discussed in Section 1I.C. The first numerical results on the polarization parameters T,, Ty, T,, U,, and U,, have been discussed at the end of the preceding section. Our treatment of combined effects of the various polarization mechanisms has only dealt with scattering from unpolarized atoms, because electronscattering experiments with polarized atoms have so far focused on spin effects due to exchange alone. This situation is now changing. Burke and Mitchell (1974) were the first to give a general theoretical treatment of elastic electron scattering from polarized one-electron atoms. Based on this paper, Farago (1974) pointed out the left-right asymmetry of unpolarized-electron scattering from polarized atoms due to interference between exchange and spin-orbit amplitudes. Numerical values of the asymmetry were calculated by Walker (1974) for cesium at electron energies between 1.427 and 100 eV. In the lower energy range he found significant asymmetries. A more recent calculation by Scott et al. (1984b), which focuses on the resonance effects in elastic scattering by cesium below 2 eV, contains results on the polarization of an initially unpolarized electron beam after scattering from polarized atoms.

135


Measurements of the asymmetries of polarized-electron scattering from polarized cesium atoms are also underway (Baum et al., 1989a). By suitable combination of the measured data it is possible to separately observe the influence of exchange and spin-orbit interaction as well,as their interference in the scattering process (McClelland et al., 1987a). Until now we have only discussed observables that are related to the scattered electrons. It is obvious from the considerations of Section A that only in some favorable cases are the measurements of these observables sufficient for a complete determination of all the amplitudes by which an electron-atom collision process is characterized: the number of amplitudes must be small enough. In order to obtain full information in the general case with several angular-momentum channels, one has to also observe the atoms participating in the collision process. This will be the subject of the following sections.

c. INFORMATION DERIVED FROM OBSERVATION OF THE ATOMS Observation of the state of the atoms after the collision yields essential information on their role in the processes considered. Since the lifetime of the excited atoms is generally short, one usually has to resort to indirect methods for their analysis. Only in the exceptional case of metastable hydrogen with its long lifetime could a direct spin analysis of the excited atoms be performed (Lichten and Schultz, 1959) yielding information on the role of exchange in the excitation process. Indirect information on the excited atomic state may be obtained by analyzing the polarization of the atoms after their return to the ground state, a technique used in the recoil-atom method of Bederson's group (Bederson, 1971; Rubin et al., 1969). A convenient and widely used source of information is the light emitted by the excited atoms, in particular its polarization. We now consider what can be learned from measurements of the polarization of the light emitted after impact excitation by polarized electrons. First, let us recall that the polarization of a radiation field is characterized by the three independent Stokes parameters Z(45") - I(135") ( = P2), " = Z(45") (135")

+

and q3 =

Z(Oo) - Z(90")

r(Oo) + Z(90")

( = PI),

q2

=

I@+)

- Z(u-) +

( = P3),

(42)

136

Joachim Kessler

where Z(ct) and I(o*)are the light intensities with linear polarization in the ct direction and with positive and negative helicities, respectively. A different notation also used in the literature is indicated by parentheses. In this section we deal with arrangements in which only the light is observed, but not the scattered electrons by which it was produced. It can easily be shown by symmetry arguments (Kessler, 1985, Section 4.6) that in such experiments, the impact radiation produced by unpolarized electrons can only have linear polarization q 3 , whereas the circular polarization qz and the linear polarization q1 can only be produced by spin-polarized electrons. Because of this close correlation between electron spin and the light polarizations q1 and qz these latter two observables enable one to obtain direct information about the spin-dependent interactions in electron-impact excitation. That is the main reason why they have been studied by several groups in the past few years. A typical experimental setup that has been used for measurements with alkali atoms is shown in Fig. 27. A beam of longitudinally polarized electrons (typical polarization 37 %, beam current a few PA) passes through a gas cell which extends 1 cm in the beam direction. This cell contains vapor of either sodium, potassium, rubidium, or cesium with densities between 10" and 10" atoms/cm3. It follows from axial symmetry that the radiation emitted in the forward direction z may only have circular polarization q z . After selection of the line under study by an interference filter, the light passes through a quarter-wave plate that transforms the 'o and o- components into two linearly polarized waves with polarizations perpendicular to one another. The waves are separated by a beam splitter (Foster prism) and simultaneously detected by the photomultipliers M 1 and M 2 . The polarization of the electron beam is measured from time to time with a Mott analyzer. An example of the results obtained with this apparatus for all the alkalis just mentioned is given in Fig. 28, showing the circular polarization of the fine-structure doublet in the principle series of potassium. Comparison with the theoretical results of Moores (1976) confirms the reliability of his closecoupling calculations. In the type of experiment discussed in this section, the circular polarization originates from the exchange interaction between the incident and the atomic electrons. This can be visualized as follows. Exchange means that, after the collision, some of the polarized incident electrons will be found in the excited atoms so that one has an atomic polarization, i.e., an angular-momentum orientation of the excited atomic state. Atomic polarization means that the magnetic sublevels M , are unequally populated. When the atoms decay, the transition rate from the various sublevels is determined

Source

Mot t- Analy ser

FIG.27. Apparatus for measurement of circular polarization of impact radiation produced by polarized electrons (Ludwig et al., 1986).

Joachim Kessler

138 0.11,

0.12~

. . .

I

I

,

. .

1

K

0-

1.5 2.0 2.5

3.0

IS

L.0

1.5

5.0 5.5

0

ENERGY (eV) ENERGY ( eV I FIG.28. Circular light polarization from potassium normalizedto incident electron polarization (longitudinal case) versus incident electron energy. Transitions 4’PII,, 3/2 + 4’S1,,. Experiment: + (Ludwig et al., 1986). Theory: 0 (Moores, 1976).

by their population so that the numbers of transitions that can obey the section rules A M j = - 1 and A M j = + 1 for emission of 0’ and 0 - radiation are different from one another; in other words; one has a circular polarization q2 # 0. From this argument, which has been theoretically established by Bartschat and Blum (1982a), it is evident that measurements of v2 yield information on the role of the exchange interaction in electron-impact excitation. This observable has therefore attracted much attention in both experimental and theoretical groups. Apart from the measurements mentioned previously, such measurements have also been done with transversely polarized electrons and transverse observation of the light along the electron-polarization axis in an arrangement similar to that of Fig. 33 (to be discussed in the following section), though without detection of the scattered electrons. Figure 29 shows one of the experimental results obtained with a cesium atomic beam target for the transitions 82S1/2-,62P1/2, 312. It is clearly seen that one finds q2( 1/2) = - 2v2(3/2) for the respective circular polarizations in the transitions to 62P1/2and 62P3/2. Since, for these transitions, the intensity ratio is given by the statistical factor 1 :2 (which is no longer fulfilled for the higher doublets (Fermi, 1930)), one has a vanishing circular polarization if the experiment does not resolve the fine structure:

ii2 = v2(1/2) + 2v2(3/2) = -2v2(3/2) + 2t12(3/2) = 0. In addition to alkali atoms, extensive measurements of the light polarization produced by polarized electrons have been made for mercury (Wolcke et al., 1983; Goeke et al., 1983). Much of this work has been treated in earlier reviews (Kessler, 1985; Bartschat and Burke, 1988) so that we shall not

139


0.20 0.16 0.12

0.08 0.04

0.00

0.00

- 0.04 - 0.08 - 0.12 3

5

7

9

11

ENERGY (eV) FIG.29. Circular light polarization from cesium normalized to incident electron polarization (transverse case) versus incident electron energy. Transitions 8’S,/, -P 6’P,/,, 3 1 2 . (Eschen et

al., 1989).

Joachim Kessler

140

discuss it here in detail. As an example, Fig. 30 presents the linear polarization q1 for the transition from 63P, to the ground state 6’S,. (Only in the transverse case is q 1 # 0, because in the longitudinal case one has axial symmetry.) The rich structure of the curve is typical of mercury and was also observed in the circular polarization y12. The features occur at the same energies where resonances due to short-lived Hg- states were observed in the electron-scattering cross sections. They are another example of the fact that resonances in the cross sections are reflected by resonances in all the polarization parameters such as scattering asymmetry, electron polarization, and polarization of the emitted light. Such resonances in the polarization effects may be utilized for the classification of the compound ion states as was first exploited by Reichert’s group (Albert et al., 1977). The analysis of the light polarizations measured in mercury (Wolcke et al., 1983) decided a controversy about the classification of the Hg compound state at threshold: it turned out that only the assignment J = 5/2 for the total angular-momentum quantum number of the 4.92-eV resonance is compatible with the light polarizations observed at this energy. Similarly, the circular polarization measured for the transition 62P1,2+ 6’S1/2 in cesium near 1.5 eV enabled Nass et al. (1989) to assign the configuration 3F, to the compound ion state Cs- formed at this energy. Apart from resonance configurations, there is a lot more information attainable from such measurements. If Eq. (14) is integrated over the electron-scattering angles, which are not observed in the types of experiments we are presently discussing, and if one denotes the three integrated terms on

5

6

7

8

9

1011

12

ENERGY (eV) FIG.30. Linear light polarization q l from mercury normalizedto incident electron polarization (transverse case) versus incident electron energy. Transition 6’P1 -+ 6’S,. Dotted curve: experimental (Wolcke et al. 1983); full line: theoretical (Bartschat et al. 1984a; Bartschat and Burke, 1986).


141

the right-hand side by D, E, and I , respectively, since they describe direct, exchange, and interference between direct and exchange scattering, then one has

Q, = i(D + E

+I)

(43)

where Q, is the integrated excitation cross section. As a consequence of the just mentioned essential role of exchange processes in the generation of circular polarization, the measurements of q z for the alkali atoms by Nass et al. (1989) enabled them to determine the exchange contributions E + I so that, in conjunction with measurements of Q, by other authors, the direct cross section D could also be determined. By combining the measured qz for the two fine-structure lines with experimental data by Enemark and Gallagher (1972) and Chen and Gallagher (1978) for the ’P excitation cross section Q, and the linear polarization q 3 (both with unpolarized electrons), it was even possible to evaluate the cross sections Qo and Q1 as well as Do and D, for excitation of the sublevels rn, = 0 and m, = 1 (or - 1). (Recall Q, = Qo + 2Q1, D = D o = 2D,.) The results thus obtained show that at low energies, excitation by exchange plays a considerable role for all the alkalis studied. This holds in particular for the cross sections Qo;at collision energies of one or two eV above threshold, only about 60 % of these cross sections are due to direct collisions. With increasing energy, the probability for exchange excitation decreases. This is also borne out by the results of Eschen et al. (1989) for transverse observation in cesium which show, moreover, that for excitation of the optically allowed 6’P3/2 state and the optically forbidden 82S1,2state there is no appreciable difference of the exchange contribution. The role of exchange in the excitation of alkali atoms from sodium to cesium is shown in Fig. 31, which gives the experimental data for the polarization transfer T = Pb/Pefrom the electrons to the atoms. Recalling the previous definitions of D and Q, as the integrals of If[’ and Q,, one has from Eq. (16’) (p. 99) T = Pa/Pe = 1 - D/Q, for excitation of unpolarized atoms (Pa = 0) by polarized electrons, which shows how. one also obtains the polarization transfer from measurements yielding D and Q,. It is seen from Fig. 3 1 that the exchange mechanism produces remarkable polarization transfers. T increases as the atomic number increases, i.e., as the binding energy of the outer electron decreases, reaching values up to 45 % for cesium. The maximum of the polarization transfer is reached for all the alkalis studied at low energies of roughly 1.5 times the threshold energy. Where theoretical results are available, they confirm the general trend outlined here. From the close-coupling calculations by Moores and Norcross (1972) for sodium and

142

Joachim Kessler

...

FIG.31. Polarization transfer T in collisions of polarized electrons with unpolarized atoms for excitation of the lowest fine-structure doublet 'P of alkali atoms. Cesium: 000 theory; experiment. Rubidium: +++ experiment. Potassium: 000 theory; mmm experiment. Sodium: uoo theory; In experiment.(From Nass et al., 1989; for theory see main text).

by Moores (1976) for potassium, numerical values for the polarization parameters and the sublevel-excitationcross sections could be extracted for comparison with the experimental data up to 5 eV. The polarization parameters for cesium were calculated up to 2eV with the R-matrix method in conjunction with the close-coupling approximation by Nagy et al. (1984) (improved results: Bartschat, 1989). The problem with close-coupling calculations is that the computational problems increase dramatically as the collision energy increases. Accordingly, one finds theoretical values only at very low energies. At these energies the results turn out to be in fair agreement with the experimental data. The situation for mercury is quite similar. The light polarizations q1 (Fig. 30) and q2 produced in the transition 63P1+ 61S0 after excitation by polarized electrons were reliably calculated with the R-matrix method for energies about 2 eV above threshold. This and other examples given in this section demonstrate that detailed aspects of the difficult problem of spindependent electron-atom collisions can, at least at low energies, be success-


143

fully tackled by the available theoretical methods, even in the case of heavy atoms-certainly a great step forward compared to the former situation. For further improvement of theoretical methods and models, a remarkable property of the light polarizations q 1 and q z can be exploited. These observables not only yield specific information on the spin-dependent interactions (as pointed out at the beginning of this section), they even yield different pieces of information. The close correlation between the circular polarization qz and exchange discussed before is paralleled by a close correlation between the linear polarization q and the spin-orbit interaction: in the type of experiment discussed in this section, q1 can be different from zero only if the excitation process is influenced by the spin-orbit interaction (Bartschat and Blum, 1982a). That is why significant values of q 1 were found in mercury (Fig. 30), but not in the alkalis. Even for cesium the predicted values of q l / P are usually smaller than 1 % (Nagy et al., 1984) so that its accurate measurement with an electron polarization P x 1/3 was beyond the reach of the experiment by Eschen et al. (1989). The fact that q 1 is sensitive to the spin-orbit interaction while q z is sensitive to exchange makes these observables very useful for specific tests of theoretical models: they enable one to distentangle the different spin-dependent interactions and provide stringent quantitative tests of the theoretical assumptions on these interactions. We shall finally discuss a more practical aspect of the light polarization produced by polarized electrons-its application for electron polarimetry. The overwhelming majority of polarization analyzers exploit the old principle of the Mott detector: from the scattering asymmetry of a polarized electron beam given in Eq. (4) one can determine the polarization P if the analyzing power S of the target is known (Ross and Fink, 1988). The exact absolute determination of S is a delicate problem that is frequently underestimated by experimental groups, who sometimes claim uncertainties as low as 1% for their polarization measurements. Few of the experimental papers are as painstaking as that of Fletcher et al. (1986), who showed that with conventional methods of calibrating S one can hardly reach uncertainties below 5 %. In order to reach the 1% limit one has to evaluate several independent scattering processes, taking advantage of relations such as Eq. (41a) that follow from basic principles and are therefore well established (Hopster and Abraham, 1988; Garcia-Rosales et al., 1988). Since this is not a convenient procedure, it is worth considering whether the light polarization produced by polarized electrons may be utilized for measuring their polarization (Farago and Wykes, 1969; Wykes, 1971;

144

Joachim Kessler

Eminyan and Lampel, 1980). Attention was first focused on mercury and atoms of similar electronic structure. Resonant transitions such as 63P,+ 6lSo were considered as well as nonresonant transitions such as 7'S, + 63P0which are preferable because the problem of self-absorption 1,2 does not then occur. Inspection of Fig. 32a shows, that for 73S1+ 63P0, the relationship between the electron polarization and the light polarization is quite complicated, depending strongly on the electron energy. The same situation was found for the circular polarization q2 of the transition 63P,+ 6'S, (Wolcke et al., 1983), showing a pronounced resonance feature near threshold related to that of q1 in Fig. 30. The strong energy-dependence of q2 means that there is a twofold drawback for the calibration of an electron polarization P : First, one needs to know the electron energy very precisely in order to determine P from the measured q 2 . Second, theoretical values sufficiently accurate for calibration are only available at threshold, and are no longer valid for the energies of a practical experiment above threshold. A way out of this dilemma follows a suggestion by Gay (1983) to use the transition 33PJ --+ P S 1 (388.9 nm) from the unresolved 33PJmultiplet of helium (Goeke et al., 1987). Fig. 32b shows the circular light polarization of this transition obtained with an electron beam of polarization P = 33%. There is a striking difference between the results obtained with mercury and helium, there being no significant energy-dependence of q z in helium. No influence of He- resonances on q2 was observed indicating that, in helium, even at resonance energies one has no spin flips from spin-orbit interaction. The relationship beween q2 and the electron polarization P is therefore simple and can easily be calculated near threshold. The gradual decrease of q2 at higher energies E is a consequence of cascading transitions that can set in

ENERGY (eV)

ENERGY (eV1

FIG.32. Circular light polarization produced by transversely polarized electrons versus incident electron energy. a) transitions 7%, --t 63P,, 1.2 in mercury; q2 normalized to electron polarization (Goeke et al., 1983). b) transitions 33P, -+ 23S, (unresolved) in helium (Uhrig et al., 1989).


145

after population of the 4%, level at 23.59 eV. Once qz(E) has been carefully measured, it can be used over a wide energy range for calibration of the electron polarization; because of the weak energy-dependence of qz the electron energy need not be very well defined. By using the same polarized electron beam for careful simultaneous measurement of q z at 23.3 eV, where cascading does not yet play a role, and the scattering asymmetry in a Mott detector, Uhrig et al. (1989) were able to calibrate the Mott detector with an accuracy better than 1 %. Since the theoretical relationship between q z and the electron polarization P can be calculated very reliably for the light helium atom, it is not a serious drawback that the calibration has to rely on a theoretical value. From our experience with different types of electron polarimeters we would certainly not use the helium polarimeter for routine measurements since its efficiency is much lower than that of a good Mott detector. For absolute calibration of a Mott analyser it is, however, very useful.

D. INFORMATION DERIVED FROM SIMULTANEOUS OBSERVATION OF ELECTRONS AND ATOMS In the two preceding sections we discussed electron-atom collision experiments in which either the scattered electrons or the atoms are observed. In the general case, none of these basic arrangements suffices to obtain full information on the collision process. One also needs experiments in which correlations between the colliding particles are observed. Investigations of that type, electron-photon coincidence measurements, have been made for quite some time with unpolarized electrons. (For reviews see, e.g., Blum and Kleinpoppen, 1979; Slevin, 1984; Andersen et al., 1988.) But only in recent years has it become possible to perform coincidence measurements with polarized electrons, too. The first results have now been obtained and more can be anticipated, since several laboratories are presently setting up such experiments in an effort to tap a powerful source of detailed insight into the functioning of spin-dependent mechanisms. Figure 33 gives an impression of such an experiment. The part of the apparatus used for the electron-scattering process is similar to that described in connection with Fig. 24. In the experiment we are now discussing, electrons that have excited the 63P, state of mercury (energy loss 4.9 eV) are selected by a spectrometer and detected by a channeltron in coincidence with the photons from the transition 63P,-+ 6'S, (253.7 nm). The photon detector consists of a linear polarization analyzer (two piles of ten quartz plates in

Joachim Kessler

146

He-Ne LASER

MAGNETIC LENS AND SPIN ROTATOR

/90°

DEFLECTOR

MAGNETIC LENS

ELECTRON-PHOTON COINCIDENCE

, ANALYSER ~~

FIG.33. Polarized-electron-photoncoincidence experiment (Goeke et al., 1989).

Brewster-angle position), a wavelength filter, and a photomultiplier. The photons can be detected either along the --x direction, as depicted in Fig. 33, or along the y direction. The acceptance angle of the photon analyzer is f22". The polarization of the electron beam was roughly 0.4, the exact value depending on the state of the GaAsP cathode. The beam current at the target was typically 5nA, its energy spread E 200 meV, and angular divergence f3". The mercury atomic beam was produced by effusion from a capillary of 1 mm diameter and had a density n I 5 x 10" cm-3. After (fast) amplification the electron and photon pulses were fed into constant-fraction discriminators. The fast negative electron pulses from the discriminator started a time-toamplitude converter (TAC) which was stopped by a delayed photon pulse. The TAC generated a pulse with a height proportional to the delay time. Each coincidence event was stored in a pulse-height analyzer. True coincidences form a decay curve corresponding to a lifetime of about 120 ns above a nearly constant background of chance coincidences. The total number of true coincidences was obtained from the area below the true coincidence peak. The idea behind the arrangment of Fig. 33 is the following. While in Section 1II.B inelastic scattering asymmetries of polarized electrons were


147

presented that had excited certain energy levels, the interest is now focused on the scattering asymmetries occurring when the energetically degenerate sublevels of different magnetic quantum number are excited. Such data are more helpful for guiding theory on its route of understanding the dynamics of the collision, than data that are averaged over the sublevels. A specific example is given by Bartschat et al. (1985) showing that different theoretical approaches may give completely different results for the individual sublevels, while the averages over the sublevels are about the same. In order to separate excitations of different sublevels from one another, one takes advantage of the fact that the radiation emitted by the decay of different sublevels has different polarization. Figure 34 illustrates that for observation normal to the z axis (=direction of incident beam in Fig. 33) transitions from the magnetic sublevel M , = 0 of 63P, to the 6'S, ground state produce linear light polarization parallel to the z axis whereas transitions from M , = & 1 produce linear light polarization perpendicular to z (circular for observation along z). By selecting photons of either parallel (intensity Ill) or perpendicular (II) polarization and observing each of them in coincidence with the electron by which it was produced, it was possible to distinguish between the electrons that had excited either M , = 0 or M, = k 1 substates. Let us clearly point out the difference from the experiments discussed in the preceding sections. While observation of the scattered electrons alone does not distinguish between excitation of different sublevels, measurement of the light polarization without regard to the scattered electrons does provide information on sublevel excitation, but only averaged over the electronscattering angle. The coincidence experiment averages neither over the

J

FIG.34. Polarization of light emitted from different magnetic sublevels for a transition 1 -+ J = 0.

=

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Joachirn Kessler

electron-scattering angle nor over the excited sublevels, thus yielding detailed data for exacting tests of theory. fP y ) and I,( P y ) produced by polarized electrons The four intensities I 11( with polarization fP, were used to evaluate the scattering asymmetries

of electrons that had excited the sublevels M , = 0 and M , = f 1, respectively, and the linear light polarization for unpolarized electrons

These quantities were measured at 8 and 15 eV incident electron energies with observation of the photons in the directions y and -x. The electronscattering angles could be varied from 5 to 120". In this chapter we can only highlight the basic idea of the rather involved theory describing the measurement (Bartschat et al., 1981a). The essential parameters are the state multipoles of the excited atoms. They are defined by certain characteristic combinations of the excitation amplitudes f of Section 1II.A and describe how the various atomic sublevels become populated by the excitation process, thus giving the polarization and alignment of the excited atomic ensemble, which determine the polarization of the emitted light. The three observables, Eqs. (44a-c), of the experiment can be written in terms of five normalized state multipoles, so that their measurement for two different directions (y and -x) yields six relations for a redundant determination of the state multipoles. From the state multipoles thus obtained for all the electron-scattering angles covered by the measurement, other quantities can be determined. Figure 35 selects one example from the results giving for 8 eV the scattering asymmetry S , of a totally polarized electron beam and the polarization S, after scattering of an initially unpolarized beam, if the sublevels M , = 0 or f 1 of 63P, in mercury are excited by the scattering process. S , and S , are now averaged over all emission angles of the radiation. It is worth noting that S, was determined without really measuring the polarization of the scattered electrons (which, due to the low efficiency of electron polarimeters, is not feasible in a coincidence experiment). This is possible because S , depends only on state multipoles that are determined by the experiment under discussion.

149


0.8

0.8

0.4

0.4

0.0

0.0 -0.4 sP(MJ>

0.8

0.8 L

0.4

0.4

0.0

0.0

-0.4

-0.4

0

30

60

90

0

30

60

90

SCATTERING ANGLE (DEG) FIG.35. Scattering asymmetry S , of totally polarized electron beam and polarization S , after inelastic scattering of an initially unpolarized beam for excitation of the sublevels M , = 0 and 1 of 63P,in mercury. Experimental data points and theoretical curves from R-matrix calculation (Bartschat, 1988, 1989). (The relation S, = S , for M = 0 is trivial since it follows from the definition of these quantities.) (Goeke et al., 1989).

The difference of the quantities in Fig. 35 for excitation of the different sublevels is considerable. The theoretical curves given demonstrate the successful application of the R-matrix (close-coupling) theory to a detailed aspect of a complex spin-dependent process. It must, however, be seen that the low energy of 8 eV is favorable for such close-coupling calculations. At 15 eV a reliable theoretical description of the measured data is much more difficult. At even higher energies the performance of the distorted-wave Born approximation improves, but at 8 eV the DWBA completely fails to describe the data of Fig. 35 (Bartschat et al., 1985; Bartschat, 1988).

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In this survey we can discuss neither further results of this investigation nor details of the physical conclusions drawn from them. Instead we shall summarize a few of the consequences concerning the interplay of the various spin-dependent interactions, without giving the full line of argument from which they follow. 1. Contrary to what was discussed in Section 1I.C for light atoms, for a heavy atom such as mercury it is not valid to conclude from certain simple relationships between the polarization effects associated with different finestructure levels, that exchange is the dominant polarization mechanism. From the approximation underlying such a conclusion it would follow that S,(M, = 0) x 0, a relation not in accord with the high measured value of S,(M, = 0) in Fig. 35. The asymmetries for different fine-structure levels at low energies having opposing signs, as mentioned in connection with Fig. 25, does therefore not necessarily imply dominance of the exchange interaction. 2. Since the internal spin-orbit interaction in mercury mixes the 3P, state with the 'PI state, the observables of the experiment contain interference terms between the exchange amplitude for excitation of the triplet state and the amplitude for singlet excitation. By comparison of S, and S p for M, = k 1 one finds regimes where the interference with the singlet amplitudes plays a dominant role [S,(M, = k 1) x SP(MJ= f l)] and others where (pure) exchange excitation of the triplet admixture is perceivable. 3. The q3 measurement shows that excitation of the singlet admixture, which takes place without spin flip, dominates at 15 eV and small angles. This can also be inferred from direct observation of spin flips using polarized electrons (Hanne and Kessler, 1976) and is in keeping with other results reviewed in the present paper, which show that at small scattering angles and energies not too low, spin-flip cross sections are no longer significant. It contradicts, however, an electron-photon coincidence measurement of Hg6'P, excitation (Murray et al., 1989) and findings of other groups for heavy noble gases (e.g., Nishimura et al., 1986) that should be similar. On the other hand, in the more accurate data of Plessis et al. (1988) for xenon and krypton the influence of spin-flip processes is already smaller, and from the results discussed at a 1989 symposium it seems that this tendency will continue: several authors (Zetner, 1989; McConkey, 1989; Hanne, 1989) warned that spurious spin-asymmetries may be simulated by an ill-defined scattering plane, as it frequently occurs at small scattering angles. 4. Determination of parameters that are sensitive to the spin-orbit interaction of the scattered electron in the atomic field such as cos E (Blum and


151

Kleinpoppen, 1983) indicates that at larger angles this force competes strongly with the other spin-dependent mechanisms. When these measurements are complemented by coincident observation of q 1 and q 2 , they will allow a comprehensive description of the dynamics of the Hg 63P1 excitation. So far the Stokes parameters q l , q2, and q3 have only been measured in coincidence with the polarized electrons that were scattered in the forward direction (Wolcke et al., 1984). From the measurements one obtains the alignment angle y (determined by q 1 / ~ 3 )and the orientation of the excited atomic charge cloud. Both of these quantities are zero if unpolarized electrons are used in this particular arrangement. While the experimental data cannot be described by a first-order theory such as the Bonham-Ochkur approximation (Bonham, 1982), which yields q I = 0, the R-matrix results are encouraging at the lowest energies studied (Bartschat and Burke, 1988). If one views the outcome of the coincidence experiments in conjunction with the results of the other types of experiment treated in the preceding sections, then a quantitative picture begins to form of the interplay between the various spin-dependent mechanisms in electron-impact excitation. The detailed and specific data provide insight into the excitation process at the most fundamental level, offer a stringent test of the regions of applicability of the different theoretical models, and are thus of considerable value for the further development of scattering theory. As yet, the number of investigations is small. This will, however, change in the near future since polarized-electron coincidence experiments are being prepared in laboratories in Europe, Australia, and the United States. Besides polarized-electron-photon coincidences, impact excitation with simultaneous observation of the scattered polarized primary electron and the secondary electron-i.e., polarized (e, 2e) processes-will also be studied in these investigations.

IV. Studies Still in an Initial Stage This survey has not only presented well-established results but also tried to indicate what can be expected in the near future. In such a new field there are, however, areas where only very preliminary steps have been made so that the time is not ripe for a review and future developments cannot be foreseen. We shall conclude this review with a brief outline of such areas.

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One example is the polarization of Auger electrons. In line with the objective of this series we shall restrict the discussion to Auger transitions of free atoms. Theoretically, it was found by Klar (1980) and Kabachnik (1981) that, under certain conditions, Auger electrons may be polarized. This may occur if either the incident electrons or target atoms are prepolarized. Another mechanism is provided by spin-orbit interaction which even in the case of inner-shell ionization of unpolarized atoms by unpolarized electrons, may result in a polarization of the electrons emitted in the subsequent Auger transition. As in photoionization of unpolarized atoms by unpolarized radiation, the polarization vanishes if averaged over the directions of emission. Consequently, one has to define a direction of observation in order to find Auger-electron polarization. In contrast to photoelectron polarization there are, however, further conditions that have to be fulfilled. T o obtain polarized Auger electrons the excited state must be aligned before the decay, and the final ion state must have nonzero angular momentum J # 0. This shows that polarization of Auger electrons is by no means the rule. Besides, the polarization is usually small since it is proportional to the alignment factor. Despite these restrictions, measurements (of Auger-electron polarization have appeal because they provide more complete and fundamental information about the Auger decay than the commonly studied energies and intensities of Auger lines. In particular, the amplitudes of the Auger decay via different channels and their relative phase shifts are obtained in such experiments. The potential of polarization and anisotropy measurements for yielding this information has so far hardly been exploited. A first step in this direction was made by Hahn et al. (1985) who studied the polarization of Auger electrons from selected lines of the MNN groups of krypton and xenon atoms. The essential parts of the apparatus, as shown in Fig. 36, are an electron gun used for inner-shell ionization of the atoms, a gas beam emerging from a capillary as the target, a cylindrical mirror analyzer of for separation of the Auger lines, and a convenresolution A E / E I 2 x tional Mott detector for polarization analysis at 120 keV. The energy of the primary electrons used for ionization was 1.5 keV. For the transitions studied the polarization of the Auger electrons is given by

where A,, is the alignment, P, and a, are the polarization and anisotropy parameters, respectively, P , is a Legendre polynomial, and ii is the normal to

M o t t analyser

FIGi. 36. Apparatus for measurement of Auger-electron polarization (Hahn

et al.,

198

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the reaction plane spanned by the momenta of incident and Auger electrons. The polarization measurements were carried out at the magic angle 8, = 125" so that, due to P2(8,) = 0, only the numerator in Eq. (45) is left. The quantity /?,,which depends on the transition matrix elements and contains information on the dynamics of the process, can then be directly determined from the measured polarization, if the alignment A,, is known. For determination of A , , the angular distribution of the Auger electrons has to be measured, which could be done by rotating the electron gun around the target. It is intimated by Eq. (45) that the Auger electron polarization will usually be small. In the present experiment with an alignment A,, of order 0.05 the polarization was below 1 % even for the favorable M , N , N , , , transition of krypton with 8, of order 0.1. Theoretical work has confirmed this magnitude of the polarization effect (Kabachnik and Sazhina, 1984, 1988; Blum et al., 1986). Though A,, may be enhanced by using heavy particles for the primary ionization and though Auger lines with higher /?, were theoretically predicted in the meantime (Kabachnik et al., 1988), it will be difficult to produce Auger electrons with a polarization much larger than a few percent in a collision between unpolarized electrons and atoms. It is therefore difficult to determine 8, with an accuracy that is sufficient to really provide useful information on the dynamics of Auger emission, because this means accurate measurement of both a small polarization and the alignment A z 0 . Thus, one has a source of information that is much harder to exploit, then, e.g., polarization measurements in photoelectron spectroscopy where polarization effects are much more prevalent and significant. It does not seem very likely that in Auger electron spectroscopy of free atoms polarization measurements will develop to the same import as in photoelectron spectroscopy. Another area with many open problems is the interaction of polarized electrons with optically active (or chiral) molecules. We shall not reconsider here the question of whether the origin of one-handedness in nature (e.g., only L-amino acids in most natural proteins, only D-sugars in carbohydrates, etc.) was caused by the longitudinally polarized electrons emitted in /? decay. The state of this problem has not changed since my previous review (Kessler, 1985). There are controversial results by different groups, and an unequivocal proof of stereoselective degradation of racemic mixtures could not be given then or now. A topic that has found much attention quite recently is polarization effects in elastic scattering of electrons from chiral molecules. Farago (1980, 1981) discussed polarization phenomena that have their origin in the fact that


155

chiral molecules do not possess space-reflection symmetry. One of the consequences is that the restrictions on the polarization components following from reflection symmetry in electron-atom scattering are no longer valid, so that scattering of unpolarized electrons from chiral molecules may result in longitudinal and transverse in-plane polarization components of the scattered beam. Since components lying in the scattering plane do not appear in scattering from (unpolarized) atoms their generation is a genuine molecular phenomenon and the spin-dependent mechanisms by which they are produced must be closely related to the molecular structure. Although a theoretical estimate of the size of the effects is difficult, it is clear that they are small. (cf: Blum, 1988, and further references given there.) It can, however, be hoped that the structure-related in-plane components resulting from the genuine electron-molecule interaction can be distinguished from in-plane components produced by the well-known spin-orbit interaction of the electrons with the isolated atoms in conjunction with intramolecular plural scattering (Hayashi, 1985, 1986, 1988). Some of the polarization effects in chiral substances are close analogues of well-known optical phenomena. The attenuation of a longitudinally polarized electron beam traversing an optically active substance depends on the beam polarization. This is a direct analogue of circular dichroism. As a result of differing attenuation of electrons with positive and negative helicity, an initially unpolarized beam emerges from such a substance with longitudinal polarization. If the incident beam is transversely polarized its polarization undergoes a rotation about the beam axis, the angle of rotation having opposite signs for left- and right-handed molecules. This is analogous to optical activity. While the aforementioned effects should be found in unoriented chiral molecules, chirality of the target is not required for the structure-related polarization effects if scattering from oriented molecules is considered. This is shown in a fundamental theoretical study by Blum and Thompson (1989), who derived a general framework of spin-dependent elastic electron scattering from oriented (chiral and nonchiral) molecules. They showed that typical chiral effects can also be obtained with optically inactive but oriented molecules in cases where a handedness is defined by the geometry of the experiment rather than by the structure of the target. For instance, a nonchiral oriented molecule may, by its structure, define a sense of rotation that in conjunction with suitable orientation of the incident-electron momentum defines a handedness. But even if chiral effects do not exist, structurerelated spin polarization can be produced by the coupling between the

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electron spin and the fields along the various molecular axes. If, for instance, unpolarized electrons are scattered from oriented linear molecules, the coupling between the moving magnetic moment of the electron and the electric field along the molecular axis is a possible mechanism for producing structure-related polarization of the scattered electrons. It is worth noting that polarization phenomena of the type discussed here have also been predicted for photoionization and collisional ionization of chiral or oriented molecules (Onishchuk, 1982; Cherepkov, 1983; Cherepkov and Kuznetsov, 1987). While the theoretical work showed that such polarization effects can, in principle, exist, present knowledge about their magnitude is sparse. Since it is too difficult to make computational predictions for real chiral models, Blum and Thompson (1989) made numerical estimates for model molecules containing one heavy and three light atoms. For forward scattering of 10-eV electrons from unoriented chiral molecules they found minute structurerelated polarization of order at best, while for oriented molecules, effects up to the order of lo-' were predicted. Experimental attempts to measure the effects must, of course, use arrangements in which the structure-related polarization is not masked by the polarization caused by electron-atom scattering alone. One can, for example, observe the polarization components in the scattering plane of initially unpolarized electrons, or their polarization after scattering in the forward direction, a quantity that also vanishes for symmetry reasons in electron scattering from (unpolarized) atoms. Two pioneering experiments were made by Farago and his collaborators. First, they searched for in-plane polarization components after scattering of unpolarized 25-eV electrons through 40-70 from right-handed (D) unoriented C,oH,,O and found that these components were below the detec(Beerlage et al., 1981). tion limit of 5 x In a more recent experiment Campbell and Farago (1987) studied the attenuation of a longitudinally polarized electron beam traversing optically active camphor vapor. The experimental arrangement is shown in Fig. 37. After emission from the GaAs cathode the electrons with longitudinal polarization P = 28% are deflected by 90" so that they pass through the Mott polarimeter with transverse polarization. The in-line polarimeter continuously monitors the polarization and the stability of the beam current. The polarimeter target H is a high-transparency gold-plated copper mesh which transmits 70 % of the beam without deviation. The transmitted beam is deflected by 9 0 so that its polarization becomes longitudinal again and passes through the gas cell K which is filled with (unoriented) camphor of one

157


Polarised electron source

Spectrometer

Mott polarimeter

FIG.37. Schematic diagram of the apparatus for studying electron optic dichroism. The polarized electron source and electron spectrometer are shown in vertical section, the Mott polarimeter in horizontal section. A: GaAs crystal, B: cleaving mechanism, C: HeNe laser, D: Pockels cell, E: Cs dispenser, F: outer cylinder of Mott polarimeter (grounded), G : inner cylinder of Mott polarimeter (30 keV), H: Au/Cu micromesh, I: channel multipliers measuring Mott asymmetry, J: 90" spherical deflector, K: gas cell, L: 180" hemispherical analyzer, M: channel multiplier measuring transmitted electron current (Campbell and Farago, 1987).

handedness (D- or L-enantiomer). The vapor pressure was adjusted to attenuate the beam by a factor between 4 and 5 at the electron energy of 5 eV used in the experiment. After leaving the gas cell, the beam passes through a hemispherical spectrometer that discriminates against electrons that have suffered either an angular deviation greater than 2" in elastic scattering or an energy loss greater than 0.2 eV. From the transmitted beam intensities Z(P) and I( -P) for electron polarization P parallel and antiparallel to the beam axis the transmission asymmetry A=

Z(P) - I( - P ) I ( P ) I( - P )

+

was determined. For the right-handed enantiomer the result was A(D) = (23 & 11) x l o w 4while for the L-enantiomer the authors found A(L) = (- 50 k 17) x The quoted uncertainties are single standard

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deviations. The measured values differ from one another by more than three standard deviations. Accordingly, the authors conclude that the existence of “electron optic dichroism” has been confirmed with a confidence limit exceeding 95 %. As a consequence, an initially unpolarized electron beam emerging from optically active camphor should be longitudinally polarized. The asymmetries are much larger than could be anticipated from the theoretical estimates made so far. One can speculate that resonant temporary-ion formation that would lengthen the time over which spin-orbit interaction takes place could enhance the polarization effect. In fact, resonance features have been found in the transmission spectrum of slow electrons in camphor, though at energies smaller than the 5 eV of the CampbellFarago experiment (Stephen et al., 1988). Further experimental work is certainly required before all the open questions will be answered. The predictions concerning structure-related polarization effects in scattering from oriented molecules are too recent to have been studied by experiment. Since oriented molecular beams can now be produced by means of electrostatic six-pole fields (Kaesdorf et al., 1985; Gandhi et al., 1987) or other methods (Westphal et al., 1989), such experiments are, in principle, feasible. It seems, however, that in this case also, the small magnitude of the effects will make reliable experiments difficult.

V. Conclusions We have seen how spin-dependent interactions in electron-atom collisions can be studied by experiments with polarized electrons. In such investigations the information is derived from measurements of spin-sensitive observables. This enables one to observe the (sometimes conspicuous) effects of the weak spin-dependent forces that are usually masked by the Coulomb-interaction. Such measurements separate different reaction channels over which an average is usually taken. In some cases, the influence of different spindependent interactions on the processes studied can be separately observed by measuring suitable observables (e.g., q 1 and q2 in Section 1II.C). For processes that are not too complicated one can obtain the maximum possible information. In elastic scattering from spinless atoms such “perfect experiments” in the Bederson sense (Bederson, 1969) have been performed; in other cases they are being planned. For processes that are too involved for perfect experiments to be performed, polarization experiments yield also a much


159

clearer idea of what is going on than the conventional experiments because they determine new observables that are essential for the collision dynamics. In many of the processes discussed, the analysis of the experimental data requires an elaborate theory of the experiment. Strong theoretical effort in the past years has resulted not only in advances in formal descriptions, but also in powerful numerical and computational methods. This has enabled theoretical groups to calculate sound numerical data in cases that, like spindependent inelastic processes in heavy atoms, seemed hopelessly difficult only a decade ago. The task is now to also obtain reliable numerical data in those regions that are not covered by the present theoretical methods. This chapter has had to be restricted to polarization phenomena in lowenergy collisions between free electrons and free atoms. Spin-polarization studies in photon-atom interactions and in solid-state and surface physics have had to be omitted, although much exciting work has been done in these fields. In fact, there are as of 1989 even more polarized-electron studies with surfaces and solids than with free atoms. As one of many examples I highlight Scanning Electron Microscopy with Polarization Analysis (SEMPA) which is developed in several countries as a powerful method for studying magnetic microstructure (Koike et al., 1986; Celotta and Pierce, 1986; Kirschner, 1988). In this method, the information contained in the polarization of secondary electrons from magnetic surfaces is exploited for imaging the magnetic domain structure with high resolution. Needless to say, there are close correlations between the topics discussed in this review and the polarization phenomena in the fields omitted. Sound knowledge of the role of spin-dependent interactions therefore not only is important for understanding collisions between free electrons and atoms, but also is the basis for a quantitative treatment of related problems in several other fields of physics.

ACKNOWLEDGMENTS

This review is based on a series of lectures I gave at Flinders University, South Australia. I gratefully acknowledge many discussions with my colleagues at Flinders and the warm hospitality I enjoyed there. In particular, I thank Prof. Erich Weigold and Dr. Anne-Marie Grisogono for carefully reading the manuscript and making many constructive suggestions. I would also like to thank my colleagues and coworkers, in particular Prof. G. F. Hanne, in the Sonderforschungsbereich 216 of the Deutsche Forschungsgemeinschaft,by which our research is supported, for their fruitful cooperation.

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I/II

ADVANCES IN ATOMIC. MOLECULAR. AND OPTICAL PHYSICS. VOL. 21

ELECTRON-A TOM SCATTERING I . E. McCARTHY and E . WEIGOLD Electronic Structure of Materials Centre School of Physical Sciences Flinders University of South Australia Bedford Park. Australia

I. Introduction . . . . . . . . . . . . . . I1. Formal Theory . . . . . . . . . . . . . A. Multichannel Two-Body Scattering Theory . B. Antisymmetrization . . . . . . . . . C. &Projected Integral Equations . . . . . D . The Distorted-Wave Representation . . . E. The Polarization Potential . . . . . . . I11. Approximations for Hydrogenic Targets . . . A . The R-Matrix Method . . . . . . . . B. Pseudostate Description of the Target . . . C . Perturbative Methods . . . . . . . . D. The Unitarized Eikonal-Born Series. . . . E. The Coupled-Channels-Optical Method . . IV. Electron-Hydrogen Scattering . . . . . . . A. Observables at 54.4 eV . . . . . . . . B. Integrated and Total Cross Sections at 100 eV C. Experimental Checks at Intermediate Energy V . Multielectron Atoms . . . . . . . . . . . A . Helium . . . . . . . . . . . . . . . B. Sodium. . . . . . . . . . . . . . . VI. Conclusions . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . References . . . . . . . . . . . . . . .

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165 166 166 169 170 172 173 175 175 177 177 179 180 182 183 185 187 189 190 194 198 198 199

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I Introduction The understanding of electron-atom scattering depends first on experiments that observe the essential features. which are electron momentum and spin projection and the energies and angular momentum characteristics of target states. The essential element of the theoretical treatment is that the target contains electrons that are identical to the electron projectile and can be knocked out. Hence we have at least the problem of the interaction of three charged bodies. Scattering of electrons by atomic hydrogen contains all 165

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Copyright 0 1991 by Academic Press. Inc All rights 01 reproduction In any lorm reserved . ISBN 0-12-W3827-7

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these features and will be used as the major example in the chapter. The extension of the principles to larger atoms will also be discussed. The chapter will concentrate on our present understanding of the problem using nonperturbative theoretical methods. No attempt will be made to be historically comprehensive. In the theory section, derivations that are in the well-known literature will not be repeated. Attention will be paid to the aspects that are specific to the problem of three charged bodies, of which two are identical. Here two milestones occurred in 1989. One is the treatment of electron exchange by Stelbovics and Bransden (1989) which, for discrete reaction channels, enables it to be incorporated into the formalism of twobody multichannel potential scattering. The other is the establishment of the boundary condition for three charged bodies by Brauner et al. (1989). The ionization continuum of the target is directly relevant to scattering between discrete target states because it absorbs much of the flux of electrons from scattering channels. It is not unusual for the total ionization cross section to be as much as 25 % of the total cross section. The understanding of ionization experiments and their use in observing target structure is discussed in the accompanying review.

11. Formal Theory We first consider the scattering of an electron beam by a hydrogen atom at a total energy E where relativistic effects can be neglected. This has all the essential difficulties associated with electron-atom scattering. The total electron spin S is conserved in the reaction and enters the formalism only in consideration of the Pauli exclusion principle for the two electrons. We shall later generalize our considerations to many-electron targets, where there are two states of spin S, except for singlet targets. We shall briefly consider the generalization to relativistic energies. A. MULTICHANNEL TWO-BODY SCATTERING THEORY For some formal purposes it will be useful to go back to the beginnings of formal two-body scattering theory as done for example by Gell-Mann and Goldberger (1953). In order to treat scattering by initially using the algebra of countable, normalized states, we enclose the whole electron-target system in a cubic box of side L and allow the intensity of the electron beam to build up

ELECTRON-ATOM SCATTERING

167

with a risetime T = l/c. We use atomic units (h = m = e = 1) throughout. The box states are counted by quantum numbers v, which label the sets of quantum numbers associated with the target and projectile electrons. To simplify the coordinates, we make the (nonessential) approximation that the nucleus is infinitely massive. If only the target system is in the box, the states are counted by the quantum number j for negative energies 6,. (The zero of energy is somewhat arbitrary in the box.) The initial state is the target ground state j = 0. For positive energies the states are counted by the box quantum numbers m,, my, m,. The Cartesian components of the momentum qj are q, = 2nmJL.. .etc.

(1)

The projectile always has positive energy. Its box quantum numbers are n,, n,,, n,. The Cartesian components of its momentum k, are

k,

.

= 2nnJL.. etc.

(2)

Gell-Mann and Goldberger discuss the limit L + 00 and E + 0 + ,in which the discrete momenta q, and k, tend respectively to q and k and the zero of energy is defined as the lowest energy for which the target is unbound. For some formal purposes we shall abbreviate the discrete projectile momentum by the notation k and include the positive-energy target quantum numbers m,, my, m, in the set j of target states giving a discrete notation for the target continuum. The quantum number v representsj, n,, ny, n,. It gives a discrete notation for the scattering continuum. Provided we can identify the target and projectile, we have the standard formalism of multichannel two-body scattering theory. We initially label the projectile electron as particle 1 with kinetic energy operator K, and electronnucleus potential u , .The target electron has a similar notation with subscript 2 and the electron-electron potential is v3. The total Hamiltonian H is partitioned as follows.

(3)

H=K+K where K =K ,

+ Kz +

02,

v = 0 , + 03.

(4) (5)

If the target is charged, the residual Coulomb potential is included with K , . A target state l j ) is defined by (cj

- K , - 0Z)lj)

= 0.

(6)

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In discussing the integral equations of scattering theory it will be convenient to use the discrete notation v for the scattering continuum. A channel state 10,)is defined by ( E , - K ) I @,)

(7)

= 0.

Using Eqs. (4) and (6), we may write (7)as

( E , - ~j

(8)

- KI)I@,) = 0

where

The Schrodinger equation for the whole scattering problem is ( E ( * ) - K)lYh*)) = VlYh”),

(1 1)

where the subscript zero indicates that the state vector lYh*)) developed from the entrance-exit-channel state I (Do) and the superscript L denotes outgoing-ingoing spherical-wave boundary conditions, generated respectively by positive-negative values of L, the energy width corresponding to the risetime of the beam. The integral equation that fully defines the state vector IYh*)) is

The T-matrix element for final channel I@,)

is defined by

(@,,ITIOO) = (@,I VlY‘b+’)= (YL-11 VI@,).

(13)

Expanding the operator K in its spectral representation I a,), we find by substituting (12) in (13) that the T-matrix element satisfies the complete set of coupled integral equations:

(@,I

TPO) = (@,,I Vl@O)

1

+ ~ v ( @ w vI@J l E ( + ) - E , (@,I

~ I @ J(14) .

Here we are using our discrete notation for the scattering continuum. Equation (14) is the Lippman-Schwinger equation of multichannel twobody scattering theory in the limit L + 00, L + 0 +. For electron-atom scattering there are two difficulties associated with (14). First, the target electron is identical with the projectile, so the states lYh*))


169

must be antisymmetrized in the electron position and spin coordinates. Second, when the box is removed we cannot continue the discrete notation j for the complete set of target states, since there is no continuum analog of a set of equations. The continuum analog of the positive-energy sum over j is an integral over the corresponding target momentum q. We may keep the discrete notation j , including the continuum, for the formal sum over target states, but we cannot close the set of integral equations over the continuum variable.

B. ANTISYMMETRIZATION Coupled integral equations for identical electrons are derived for a manyelectron target by McCarthy and Stelbovics (1983). For a one-electron target we replace V of (5) by V,, where

v, = v +

(-1)S(H - E)P,.

(15)

Here, P, is the space-exchange operator. Since singlet and triplet states are independent, we may drop the subscript S and keep the formalism leading to (14), with (15) replacing (5). There is, however, a serious difficulty with the formal set of coupled equations (14) using the antisymmetrizing potential (15). The exchange potential contains terms that are constant energies. These terms depend on overlaps of bound and scattering states, which are nonzero. The solutions are not unique off the three-body energy shell, although the difficulties do not occur on shell. Uniqueness is recovered by additional orthogonalization constraints. Such procedures have been reviewed by Burke and Seaton (1971). The intention of a Lippman-Schwinger equation formulation is to have an integral equation that incorporates all the boundary and other conditions. Bransden and Stelbovics (1989) showed that this is achieved for one-electron targets by replacing the target-state matrix elements of (15) by (il

Glj) = (ilo3(1

+ (-

1)’Pr)Ij)

+ 6ij[u, + Ek(1 - s 6ik)(Ci + ck - E ) ] . (16)

They showed that (15) is equivalent to (16) on the three-body energy shell, but off-shell stability is achieved only by using a class of potentials of which (16) is a good example. For many-electron targets we still use (15) to describe scattering (on-shell) experiments.

170


C. &PROJECTED INTEGRALEQUATIONS We are now ready to tackle the difficulty of closing the set of coupled integral equations (14) in the realistic case where the complete set of target states l j ) includes the continuum Iq). We use the optical potential formalism of Feshbach (1962). The channel projection operator P is defined by

p-

c l.MA

(17)

jsP

where P-space contains the target ground state and a finite set of excited states. The complementary projection operator Q projects the remaining discrete target states and the continuum.

P+Q=1.

(18)

We introduce the unit operator (18) into the Schrodinger equation (11) for physical (+) boundary conditions and operate on the left with P:

P(E‘+’ - K ) ( P

+ Q ) I Y r ’ )= PV(P + Q ) I Y r ’ ) .

(19)

We arrange (19) using P(E‘+’ - K)QIY6+’) = 0,

(20)

which follows from the fact that P commutes with E ( + )- K and PQ operating on (Yb+))gives zero since P and Q project different target states. The rearranged equation is

P(E‘+’ - K

V)PIY‘b+’>= PI/QlY‘b+’).

(21)

Q(E‘+’ - K - V ) Q I Y r ’ ) = QVPlYL+’).

(22)

-

Similarly, Using the identity Q2

=Q

and defining the appropriate inverse operator we formally solve (22) for Q I Y V ) ) , obtaining

We now define the complex polarization potential V(Q)by substituting (24) into the P-projected set of equations (21). P(E‘+’ - K - v - V‘Q))PIY6+))= 0,

(25)


where

171

VQ)is defined by

We are now in a position to derive a set of coupled integral equations for IYb+)) by solving the differential equation (25) with the boundary condition JYb+))= I@,)

for V = 0.

(27)

+ P ( E ( + )- K ) P P( V + V(Q))PIY p ) .

(28)

The corresponding integral equation is 1

I%+’)

=Po)

We now derive the Lippman-Schwinger equation for the P-space Tmatrix, which is defined by (mpI P TI 0,)= (apIP VI Y p ) .

(29)

The effect of defining the polarization potential VtQ)is seen from the equation

PV(P

+ Q)IYb+))= P ( V +

V‘Q’)PIYb+’),

(30)

obtained from (21) and (25). When operating on IYb+),P V Q is replaced by P V(Q)P. The P-space T-matrix (29) becomes

( @ J P T l @ , ) = ( @ J P ( V + V(Q))PIYb+)).

(31)

Substituting the right-hand side of the integral equation (28) for lYb+))in (3 l), we obtain the P-projected Lippman-Schwinger equation

(@PIPTI@,)= (@,IP(V+ V‘Q’)PI@,)

+~ v ( @ p I W +

1

V‘Q’)PI@v)E“- E , ( @ v I P T P o ) .

(32)

This set of equations is closed over the channels in P-space. Using the notation (9) for the channel states lav)and proceeding to the continuum limit, we rewrite it as

(kiJT,lOk,)

= (kiI

V, + V$)IOk,)

172


The potential for the P-space set of integral equations is the optical potential V, + VLQ).We have restored the spin subscript S indicating that antisymmetry is treated by using the definition (15, 16) of Vs. The partial-wave treatment of these equations and their numerical solution is given by McCarthy and Stelbovics (1983b). They are solved for each total angular momentum J in the form of a set of coupled one-dimensional Fredholm equations, with the integration approximated by Gaussian quadratures.

D. THEDISTORTED-WAVE REPRESENTATION A useful way of improving the convergence of the numerical methods used for evaluating (33) is to replace the plane-wave representation of the T-matrix (33) by the distorted-wave representation in which the asymptotic plane waves Ik) representing the projectile are replaced by eigenstates Ik(*)) of a local, central distorting potential U.The distorting potential is added to the channel Hamiltonian K (Eq. 4) and subtracted from the potential V(Eq. 5). In practice U is chosen to cancel as much of the projectile potential V as possible. A convenient choice is the ground-state average of V:

u = (01 VlO). The total T-matrix element is now 1953) by

(34)

Tsgiven (Gell-Mann and Goldberger,

where t is the t-matrix for elastic scattering by U and the distorted-wave 7’-matrix element is given by the coupled integral equations

(k(-)il T,IOkb+)) = (k(-)il V, + VkQ)- UlOkb+))

+ C jd”q(k(-)ilV, + VLQ) jaP

1

173


(36)

1

The complete set of eigenstates of U includes scattering states I k(*)) given by [)q2

and bound states

- K , - U]lk(*))= 0,

(37)

IA) given by [tl - K1

- U ] I A ) = 0.

If the target is charged, U must include the residual Coulomb potential.

E. THEPOLARIZATION POTENTIAL We replace the operators in the definition (26) of the polarization potential by numbers using the spectral representation IY t - ) ) of the spin-dependent Hamiltonian

Hs = K

+ Vs,

(39)

where K and V, are defined, respectively, by Eqs. (4) and (16). The subscript v is a formal discrete notation for the scattering continuum. Using the notation (9) for the channel states, the matrix element of the polarization potential (26) becomes

Note that the use of the time-reversed state in the representation ensures that the small quantity t in the denominator, which tends to zero in the limiting process, is positive definite.

174


It is useful to expand the notation Yt-), indicating bound and continuum target states explicitly.

I Yt-)) = I Y!-)(k)) for final discrete channel states I Ik),

(41)

I Yt-)) = lY(-)(k,k)) for final continuum channel state Ik’k).

(42)

We therefore have discrete and continuum contributions to the polarization potential (q’il VkQ)Yq>=

C (q’il W j q ) + (q’iI

VYW,

(43)

1eQ

where the superscript I indicates the ionization space.

Note that since P and Q are disjoint sets of target states, the factor 6,, in the definition (16) of V, makes it reduce in this case to V, = ul

+ u3[1 + (-

1)’ P,].

(46)

The polarization potentials (44,45) contain the exact 7’-matrix elements for real (on-shell) and virtual (off-shell) excitation of the states of Q-space. The potential matrix elements for real (imaginary part of the Green’s function) and virtual (real part of the Green’s function) excitations are Hermitian. The projected integral equation (33) with the definitions (44,45) of the polarization potential and (16) of the antisymmetric potential operator constitutes a formal rearrangement of the electron-hydrogen scattering problem in terms of a finite set of coupled integral equations with Fredholm kernels. It is not a solution since the three-body states Yt-) remain to be calculated. The Coulomb three-body problem cannot yet be solved in closed form. For a long time it could not even be properly defined since, although the Hamiltonian is very simple, the boundary condition for three charged particles had not been established. The large-separation wave function for three charged particles is not a product of plane waves since Coulomb forces act at infinite particle separations.


Brauner et al. (1989) showed that the function

(r’, rlYL)(k’,k)) = (27~)-~ exp[i$(k, k, r’, r)], where

$(k, k, r’, r) = k‘.r’ + k.r - q ln(k‘r’ + k.r’) - q’ ln(kr + k.r) + aln[Ik’- kllr’- rl + (k - k).(r’- r)], q’ = -l/k’,

q = -l/k,

a = l / l k - kl

obeys the Schrodinger equation for large distances t’ and t in the case of one-electron atom targets. The boundary condition is essentially the product of three Coulomb boundary conditions, one for each two-body subsystem. Brauner et al. give references to earlier literature on this boundary condition. In the next section we discuss approximations that have been made to various aspects of the formal theory. So far none of them is a true three-body approximation since the three-body Coulomb boundary condition has not been obeyed. Brauner et al. (1989) have obeyed the boundary condition in an ansatz used for calculating ionization cross sections (involving on-shell ionization amplitudes) with considerable success. This is described in the accompanying review of ionization.

111. Approximations for Hydrogenic Targets Since the Coulomb three-body problem cannot be solved in closed form, it is necessay to make approximations to various aspects of the formal theory. For scattering to low-lying discrete target states, approximations are made in a part of the relevant space that is removed as far as possible from the part that strongly affects the scattering. The problem thus approximated is solved exactly for the remainder of the space. A. THER-MATRIX METHOD In this method the space relevant to the approximations is the coordinate space of the target, where a box is literally put into the calculation. The box is spherical with a large radius. The method was first used by Wigner and Eisenbud (1947) for low-energy neutron scattering. For neutron scattering the box radius can be chosen so large that two-body boundary conditions

176


apply in the external region. This is not the case for the break-up channels of electron-atom scattering, but the box is made large so that the difficulty is driven to large distances, where two-body boundary conditions are used. The electron-atom application is explained (with references to earlier work) by Scott et al. (1989). The full electron-atom problem is solved, using the full Hamiltonian in the box, in terms of a large set of energy-independent two-electron basis functions, which have a specified logarithmic derivative at the box boundary. They are used to calculate the logarithmic derivatives of the external (twobody) wave functions on the boundary. The method as explained so far does not produce T-matrix elements that are anywhere near correct if the total energy is so high that the three-body continuum must be represented in the basis. The T-matrix element for each channel fluctuates widely about an average value with a pseudoresonance at each eigenvalue as the total energy is increased. The average T-matrix is used as the final result of the calculation. It is defined by (T(E))=

J

m

-m

d ~ ' p ( E- E')T(E') = T(E + ill,

(50)

where the averaging function p ( E - E') is a Lorentzian of width I, which must be large enough to include several pseudoresonances but small enough not to obscure fluctuations in the background T-matrix. p(E - E') =

1

I

n ( E - ,1)2

+ z2'

As the box radius and the number of basis states become larger, the average width and separation of the pseudoresonances become smaller, so I may tend to zero. In practice it is possible to perform such a large computation that the averaged T-matrix elements are insensitive to I. The averaging procedure represents the loss of flux into all the states not explicitly included in the external region. The corresponding S-matrix is not unitary and can be used to estimate the total cross section, including the total ionization cross section. The validity of the T-matrix averaging procedure has been confirmed by Slim and Stelbovics (1987), who showed in a separable potential model that the R-matrix method with T-matrix averaging reproduces the T-matrix elements obtained from the exact solution. The R-matrix method is particularly suited to incident electron energies below about 10 eV, where much of the interest in the reaction lies in the rapid fluctuations of cross sections with energy. The computational difficulty is


177

mainly in solving the internal problem, which is independent of the incident energy over a range of several eV. The energy-dependent boundary-condition problem is computationally fast. T-matrix averaging is not necessary at energies below the ionization threshold. B. PSEUDOSTATE DESCRIPTION OF THE TARGET If the target statesj in (14), with the definition (9) of lav),were discrete, there would be no difficulty in closing the set of coupled equations. The approximation method sets up a discrete pseudoproblem that can be solved to numerical convergence. Q-space is represented by a many-parameter ansatz in which the parameters describe orthogonal and normalized squareintegrable functions with low values of orbital angular momentum. These functions play the same part in the computation as discrete target states and are called pseudostates. Their parameters are chosen to reproduce a simplified sum and integral over Q-space that is known exactly. If the simplified problem is a reasonable scattering approximation, it is hoped that the parameters that describe it will also describe the related sum and integral over Q-space in the real problem. An example of such a simplified integral is the second Born amplitude in the closure approximation.

C. PERTURBATIVE METHODS The series obtained by iterating the integral equation (33) is the Born series. For discrete channels the Born series is known to be divergent in general (Stelbovics 1990). The Born limit at high energy nevertheless holds and can be recovered by appropriate rearrangement of the Born series. There is no reason to believe that the target continuum makes any difference to this. In an angular-momentum expansion of (33) the Born series converges for large values of the total angular momentum. The distorted-wave Born (DWB) series is obtained by iterating equation (36) for the distorted-wave T-matrix. The iteration has proved to be divergent in numerical examples for small total angular momenta (Bray and McCarthy, 1989). This includes all the half-shell T-matrix elements of (36). There is strong reason to believe that the distorted-wave Born series is rapidly convergent for T-matrix elements on the three-body energy shell, i.e. the ones that represent real scattering problems. The first order of (36) is the distorted-wave Born approximation (DWBA).

178


If (36) is to be iterated beyond the first order then VLQ),defined by (26), must itself be expanded in a perturbation series, which is at least of second order in V,. The second order for the whole problem is the distorted-wave second-Born approximation (DWSBA). The term of second order includes a sum over the complete set of target states except for i and 0. The dependence of the operator K of (4) on the energy eigenvalue of the target state is sometimes ignored in the second-order term, replacing each eigenvalue by a single average excitation energy. The closure theorem for target states then eliminates them, except for 0 and i. It is common to evaluate the DWSBA in this closure approximation. (See, for example, Kingston and Walters, 1980.) If the real part of the Green’s function in (36,26) is neglected, then we have only on-shell values of the integration variables. The resulting approximation is the unitarized distorted-wave Born approximation (UDWBA). A much-improved development of the DWBA for the excitation of channel i is to define P-space to include only 0 and i and to approximate the whole driving term of (36) including VkQ).The approximation to VLQ)that has proved possible up to now is to represent the total wave function Y t - )of (40) by the product of an exact target state (discrete or continuum) and a distorted wave calculated in a local, central potential W,. This is the explicit secondorder approximation (ESOA) of Madison (1989). Note that antisymmetry is ensured by the definition (15) or (16) of V,. The explicit use of exact targets states in the ESOA makes possible a test of the validity of the closure approximation. Madison et al., (1989) found errors of 50-100 % in the Q-space part of the 54.4-eV second-order amplitude for the 1s-2s excitation of hydrogen at larger scattering angles, for a P-space consisting of the Is, 2s and 2p states. However this amplitude is small where the errors are large and the overall error in the second-order amplitude introduced by closure has a maximum magnitude of 20% at 20°, but is generally less than 5 %. They conclude that closure is intrinsically not very accurate but is a reasonable approximation if lower discrete states are treated explicitly. A good idea of the validity of perturbative approximations may be obtained from a model problem with a finite set of channels. The problem of the 3s and 3p channels of sodium, considered as a hydrogenic (one-electron) target except for the inclusion of the exchange potential of the 10-electron core, has been calculated by Bray et al., (1989~).In this severely truncated channel space, the second-order approximations DWSBA and ESOA are identical. Figure 1 shows the comparison of the DWBA, DWSBA, and UDWBA with the exact solution for an incident energy of 54.42 eV. DWBA

179

ELECTRON-ATOM SCATTERING 103

5L.L2eV

e-SODIUM

3s

1

I

0 ongle ( d e g )

I 60

I

I 120

I

1 0

FIG. 1. Electron-sodium scattering at 54.42 eV in the 3s, 3p two-channel model. The exact coupled-channels differential cross sections (solid line) are compared with perturbative approximations to the distorted-wave representation. From Bray et al. (1989~).

is a fair approximation for the excited channel, but the absence in the elastic channel of provision for excitation makes it a bad approximation. This is corrected by UDWBA, which is an excellent approximation in both channels and somewhat better than DWSBA.

D. THEUNITARIZED EIKONAL-BORN SERIES The unitarized eikonal-Born series is a nonperturbative scattering method based on the many-body generalization of the Wallace amplitude (Wallace, 1973). The method is explained by Byron, et al. (1982). It will not be detailed here since it is not directly related to the formalism of Section 11, but rather to the Glauber (1959) approximation, which is valid essentially when the wavelengths of continuum particles are small in comparison with the distance over which potentials change appreciably. The generalized Wallace amplitude has the advantage that the corresponding S-matrix is unitary. Its disadvantage is that it does not account properly for long-range polarization potentials due to dipole excitations. This is remedied by replacing the second-order Wallace amplitude by the second

180


Born amplitude for the direct partitions of the three-body problem. Amplitudes for the exchange partitions are calculated from the direct amplitudes by comparing the corresponding Lippman-Schwinger equations. E. THECOUPLED-CHANNELS-OPTICAL METHOD

Like the pseudostate method, this method removes the approximations from the scattering channels (P-space) to Q-space. However, it calculates the Q-space sum and integral to numerical convergence using the exact discrete and continuum target states. This is done in the polarization potential (40). The approximation is that the three-body wave functions (41, 42) are products of the exact target state and an elastic scattering function (distorted wave) for the projectile, calculated in a relevant local central potential. This approximation to the three-body scattering function is the distorted-wave Born approximation. It has been known for a long time to describe real excitations quite well (Madison, 1979) and studies of ionization cross sections (McCarthy and Zhang, 1989), described in the accompanying review, show that it is a good model for the large values that dominate the Q-space sum and integral. Like the R-matrix and pseudostate methods, it involves twobody boundary conditions and does not include the electron-electron interaction at large final-state separations. Clearly an ansatz with the correct three-body boundary condition could be used for the ionization amplitudes in the polarization potential (49, for example that of Brauner et al. (1989), so that the optical potential method gives a straightforward way of treating the scattering problem with three-body boundary conditions. The computation is very difficult and has not yet been done. The on-shell driving terms of the integral equations (36) in the CCO method constitute the explicit second-order approximation, discussed in Section 1II.C. These terms have been evaluated using antisymmetric distorted-wave approximations to the optical potential (Madison, 1989). The integral equations have only been solved thus far using direct DWBA amplitudes in the polarization potential but full antisymmetry otherwise (Bray et al., 1989a, b). The CCO method that has been widely used for electron-atom scattering is that of McCarthy and Stelbovics (1980). Here the distorted wave is replaced by a plane wave for the continuum electron in (44)or the faster electron in (45). The bound target state is represented in the Hartree-Fock approximation (or exactly for hydrogen) in (44)and the slower electron is represented


181

by a Coulomb wave for hydrogen or a Coulomb wave orthogonalized to the relevant bound state for larger atoms. This approximation obeys the first criterion for the polarization potential. It produces total ionization cross sections that compare well with experiment at all energies, but particularly at energies greater than about eight times the ionization threshold (McCarthy and Stelbovics, 1983a). With some minor approximations for computational feasibility, this potential has been evaluated in the case of electron-hydrogen scattering at 54.4 eV (Ratnavelu, 1989). However, in general it has been necessary to make a further approximation reducing the computational labor to a range of about 10 points in the variable K, where

This is achieved by an angular-momentum projection, which in the case of the elastic channel is the equivalent local approximation. In general, if the orbital angular momentum quantum numbers of the target states j and i are 4 m and el, m',respectively, the approximation is (q'il ViQ)ljq) =

1 i"'C$' p' F Ut-J,t(K)I'&@),

8%"

(54)

where the coefficient C is a Clebsch-Gordan coefficient and the polarization potential calculation is done for the one-dimensional functions UL,tG,G(K) =

c CF"F'FSmi(gfilviQ)Ijq)i-d"Y~,m,,(B).

(55)

m"m'

This is called the half-on-shell polarization potential because of the restriction (53). The eight-dimensional integration (45,55) is done by the multidimensional diophantine method using Cartesian vectors. This requires an analytic integrand which is possible for direct ionization amplitudes but not exchange. A further approximation is the Bonham-Ochkur approximation for which the exchange amplitude is written as a product of the two factors that result from replacing the distorted waves by plane waves. The electron-electron potential factor is kept while the distorted waves are restored in the factor that contains the bound state. This approximation is not very good for some individual amplitudes, but the integration over kinematic variables reduces the sensitivity of the polarization potential to it, particularly on shell (McCarthy et al., 1981).

182


IV. Electron-Hydrogen Scattering Because it is the prototype for electron-atom scattering theories, hydrogen has received the bulk of the theoretical attention in the field. However, there are experimental difficulties for hydrogen in comparison with, say, inert gas targets. First, the hydrogen molecule must be dissociated. A difficulty unique to hydrogen is the degeneracy of states with the same principal quantum number, which means that techniques other than energy resolution must be used to distinguish, for example, between 2s and 2p excitations. The major problem in the measurement of elastic scattering of electrons by atomic hydrogen is the accurate determination of the dissociation fraction of molecules (d) in either a high-temperature oven (usually made of tungsten) or a gas discharge (usually RF or DC). The degree of dissociation can be obtained by measuring the HZ ion signal in a mass spectrometer as a function of temperature for an oven source, the mass spectrometer being placed in the path of the target beam. Without dissociation the H i signal should vary as due to the increase in velocity of the beam as the temperature is raised (Fite and Brackmann, 1958). With a discharge it is possible to thermalize the beam by making it pass through a teflon collimating tube before reaching the interaction region (Lower et al., 1987). It is then again possible to measure the dissocation with a mass spectrometer, or to measure the (e, 2e) cross-section ratios for H and H, (Lower et al., 1987). Having measured the dissociation fraction d, the elastic cross section for atomic hydrogen can then be measured relative to that for the molecular cross section

where S,(@ is the scattered signal at some temperature Twhere H, is highly dissociated and S R ( 6 ) is the scattered signal at a temperature TRwhere there is no dissociation. For a thermalized discharge source T = TR. Thus, it is important to have an accurate measurement of the molecular hydrogen cross section as well as the dissociation ratio. For inelastic scattering from hydrogen there is an additional difficulty due to the degeneracy of states with the same principle quantum number. For the n = 2 states, Frost and Weigold (1980) used the electron-photon technique to obtain the ratio of ls-2s and ls-2p excitation of atomic hydrogen by electron impact. They showed that it is possible to separate the long-lived 2s


183

metastable-state excitation from that of the short-lived 2p if the total n = 2 cross sections are known (Williams and Willis, 1975). AT 54.4 eV A. OBSERVABLES

Differential and integrated cross sections for Is, 2s, and 2p states and electron-photon correlation parameters for the 2p excitation have all been measured at 54.4eV. In addition we have total and total ionization cross sections, forming a complete set of data for testing theoretical descriptions for a Is, 2s, 2p P-space. Results will be given for several calculations. This energy has been considered an upper limit for low-energy methods and a lower limit for higher-energy methods. 1. The Intermediate-Energy R-Matrix Method (IERM) The intermediate-energy R-matrix method of Burke et al. (1987) has been implemented for integrated cross sections by Scott et al. (1989). Here the full calculation was done for total angular momentum J in the range 0 IJ I4, using 3500 basis states and a box radius of 25a,. For 5 IJ I16 a nine-state basis (three eigenstates and six pseudostates) was used. Corrections for all the truncations were applied using the plane-wave second Born approximation. 2. The Pseudostate Method ( P S ) Two large calculations have been reported using the pseudostate method. All observables for the Is, 2s, 2p P-space have been calculated by van Wyngaarden and Walters (1986) (vWW) using 8s-, 7p- and 6d-pseudostates to represent Q-space. The 63 parameters of the calculation were chosen to give a good representation of the second Born closure approximation for Qspace at 100 eV. Truncations were corrected using the distorted-wave second Born approximation. Callaway et al. (1987) (CUO) calculated integrated cross sections using states up to n = 3 in P-space and representing Q-space by an optical potential calculated from a 7s-, 5p-, 3d-, 2f-, and lg-pseudostate basis in which the Is, 2s, 3s, 2p, 3p, and 4f states were exact eigenstates (Callaway and Oza, 1985). 3. The Unitarized Eikonal-Born Series (UEBS)

Byron et al. (1985) consider 54.4 eV to be the lower limit for applicability of the UEBS method. Results for 54.4 eV have only been given in the form of curves for observables related to 2s and 2p excitations.

184


4. The Coupled Channels-Optical Method (CCO)

As of 1989, the fullest implementation of the coupled-channels-optical method has been by Ratnavelu (1989). Here P-space consists of nine states: 1,2,3,4s; 2,3,4p; 3,4d. Q-space contains only the continuum, for which the direct polarization potential is fully off-shell and nonlocal and the exchange polarization potential involves the Bonham-Ochkur approximation, which is calculated only in its near-on-shell range of approximate validity. This optical potential has been included for Is-ls, ls-2s, and 2s-2s couplings. The half-on-shell optical potential has been used for 1s-2p and 2p-2p. 5. Differential Cross Sections

Figure 2 shows the differential cross sections for Is, 2s and 2p states. For the 1s channel the experimental data (which are taken at 50 eV) agree, but the 54.4 eV CCO curve lies significantly below the data. For the 2s channel CCO and vWW are in close agreement, both curves lying significantly below the data. The UEBS curve is similar, but is not shown in order to avoid confusion. All theoretical curves give quite a good description of the 2p channel. UEBS (not shown) agrees with experiment at all points. For this channel lo2 10’

loo lo-’ 10-2

1

1

60

1

1

1

120

1

60

1

1

120

Scattering angle (deg)

FIG.2. Differential cross sections for 54.4 eV electron-hydrogenscattering in the Is, 2s and 2p channels. For Is (at 50 eV) the experimental data are due to Williams (1975) (full circles) and van Wingerden et al. (1977) (crosses). The 2s and 2p experimental data are those of Williams (1981). Theoretical curves are full: CCO (Ratnavelu, 1989), dashed: PS (van Wyngaarden and Walters, 1986). dotted: ESOA (Madison, 1989).

185


ESOA (Madison, 1989) has been included since it is the driving term of the full CCO approximation. 6. Integrated and Total Cross Sections

Table I shows the comparison of theory with experimental cross sections. The integrated cross sections used for the 1s channel were calculated by de Heer et al. (1977) from the experimental data of Lloyd et al. (1974) and Williams (1975) at 50 eV. For the 2s and 2p channels the quoted cross sections were obtained by van Wyngaarden and Walters, who applied cascade corrections to the experimental data of Kauppila et al. (1970) and Long et al. (1968), respectively. There is an absolute measurement by Williams (1981) for the 2p channel. Total cross sections are taken from the compilation of de Heer et al. (1977), where errors are about 10%. Most calculated cross sections are quite close to the experimental values. Only CCO gives an explicit total ionization cross section, which is about 20% too large. This is usual for the DWBA at energies several times the ionization threshold. 7. Electron-Photon Correlation Parameters

Figure 3 shows the electron-photon correlation parameters A and R. Here no calculation is adequate over the whole angular range. Characteristics of most calculations are that the backward minimum in 1 is too shallow and that the negative values of R are difficult to reproduce. B. INTEGRATED AND TOTAL CROSSSECTIONS AT 100 eV

The full CCO method has been implemented for integrated and total cross sections at 100 eV, with the exception of exchange terms in the polarization TABLE I INTEGRATED AND TOTAL CROSS SECTIONS FOR 54.4 eV ELECTRONS ON HYDROGEN (mi)

Channel 1s 2s 2P ion total

' 50 eV.

Experiment

IERM

vWW

CUO

CCO

1 .ma 0.056 f 0.005 0.12 f 0.03 0.77" 3.28"

1.1044" 0.0661 0.7210

0.990 0.0651 0.739

0.922 0.062 0.746

-

-

-

-

2.933

0.991 0.063 0.765 0.92 3.03

2.9


186

.2

0.8

'x 0.4

0

R

0.

-0.4

0

I

I

40

1

1

80

1

3

1

120

Scattering angle ( d e g 1 FIG.3. Electron-photon correlation parameters I , and R for the 2p excitation of hydrogen at 54.4 eV. Experimental data are. by Hood et al. (1979) (crosses up to 209, Weigold et al. (1980) (crosses beyond 2W), and Williams (1981) (full circles). Theoretical curves are as for Fig. 2.

187


potential (Bray et al., 1989b). It is expected that the omission of these exchange terms will have a minor effect on integrated cross sections. Exchange terms are of course implemented in the first-order potential. Table I1 compares various higher-energy theoretical methods with experiment. The quoted experimental 2s and 2p cross sections are those of van Wyngaarden and Walters (1986), obtained by making cascade corrections to experimental data. Other are taken from the compilation of de Heer et al. (1977). Experimental errors are roughly 10 %. A feature of the 100 eV data is the underestimation of the integrated elastic cross section by all calculations, which are in fairly close agreement among themselves. For other channels the calculations are in close agreement with each other and are within experimental error.

c. EXPERIMENTAL CHECKS AT INTERMEDIATE ENERGY An overview of the comparison between experiment and theory for n = 1 and n = 2 differential cross sections at 54.4,100, and 200 eV is given by Fig. 4. (Elastic cross sections for 54.4 eV are actually measured at 50 eV.) The CCO calculation (Lower et al., 1987) couples the six n = 1, 2, 3 channels (supplemented by 4s and 4p at 200eV). Half-on-shell optical potentials for the continuum are included for all channel couplings in the Is, 2s, 2p subspace. At 100 eV the optical potentials include all dipole excitations in this subspace via 4s and 4p intermediate states. The pseudostate calculation of van Wyngaarden and Walters (vWW) is also shown. Elastic cross sections are significantly underestimated by both calculations, which agree with each other. Inelastic cross sections are severely underestimated at larger angles. In view of the consistent disagreement of experiment

TABLE I1 INTEGRATED AND TOTAL CROSS SECTIONS FOR 100 eV ELECTRONS ON HYDROGEN (It& Channel

Experiment

UEBS

vWW

CCO

Is 2s 2P Reaction Total

0.588 0.039 0.62 1.76 2.18

0.465 0.038 0.60 1.77 2.24

0.480

0.457 0.0405 0.622 1.75 2.20

0.0404 0.638 1.65 2.13


188

1

0

LO

80

120

I

I

I

I

I

I

LO 80 120 0 cattering angle (degrees)

I

I

LO

I

I

80

I

,

120

,

1 3

FIG.4. Differential cross sections for electron-hydrogen scattering in the n = 1 and n = 2 channels. Solid curves: CCO (Lower et al., 1987), dashed curves: van Wyngaarden and Walters (1986). Full circles (Is): Williams (1975), crosses:van Wingerden et al(1977). Errors in the elastic experiments are similar. They are shown only for Williams. Full circles (2s + 2p): Williams and Willis (1975). From Lower et al. (1987).

with both calculations, independent measurements of the ratio of n = 1 to n = 2 cross sections were made at 100 and 200 eV (Lower et al., 1987). The elastic differential cross sections of van Wingerden et al. (1977) are about 10 % larger than those of Williams (1975). The measurements were made by entirely different methods. Williams measured the cross section for atomic hydrogen relative to helium. The absolute cross section for helium was obtained from a phase-shift analysis of experimental data near the 1s 2s’ 2S resonance. Van Wingerden et al. used the cross section relative to molecular hydrogen measured by Lloyd et al. (1974). The absolute cross

189


section for molecular hydrogen was determined by an absolute measurement using a gas cell with known pressure. The inelastic cross sections of Williams and Willis (1975) were put on an absolute scale in a similar way to the elastic cross sections of Williams. Table I11 gives the n = 1 to n = 2 differential cross section ratios for Lower et al. (LMW) and for van Wingerden et al. and Williams (elastic) relative to Williams and Willis (inelastic). The latter two ratios are denoted, respectively, by vW/WW and W/WW. The CCO, vWW, and UEBS (Byron et al., 1985) calculations are shown for comparison. All three measurements agree within experimental error, except that LMW support VWat 60". UEBS is considerably better than CCO and vWW at 100 eV, but at 200 eV the three calculations essentially agree with each other and show significant discrepancies with experiment at 45".

V. Multielectron Atoms Nonperturbative theoretical methods that have been applied to n-electron atoms are the R-matrix and CCO methods. Both cases involve CI expansions. The R-matrix method expands the (n + 1)-electron system in the box TABLE 111 THERATIOOF n = 1 TO n = 2 DIFFERENTIAL CROSS SECTIONS FOR ELECTRON-HYDROGEN SCAITERING Scattering angle (degrees)

E(eV) 100

Case

30

45

60

LMW W/WW

11.4 f 0.7 9.8 f 1.2 10.1 f 1.0 10.0 13.0 10.0 25.4 f 1.4 25.6 f 3.4 29.7 f 3.5 29.6 28.4 30.0

14.5 f 0.8 15.6 f 2.7 14.4 f 2.1 20.0 19.3 14.6 28.8 f 1.5 26.4 f 6.6 29.9 f 5.8 36.4 34.4 35.0

14.8 f 0.8 11.7 f 1.6 12.0 f 1.7 17.5 18.2 15.1 35.8 2.5 25.7 f 3.6 32.6 f 4.5 33.0 31.0 34.0

vwjww cco vww

200

UEBS LMW

w/ww vwjww cco vww UEBS

Note: Row headings are described in text.


190

in terms of an independent-particle basis and matches the internal wave functions to two-body (electron-atom) boundary conditions. The CCO method uses target eigenstates expressed as a linear combination of independent-particle configurations. Equations (36) can be taken over for the CCO method, with the atomic states 0, i, and j given by CI expansions. The method has been explained in detail by Bray et al. (1989~).The polarization potential has thus far been implemented only in the half-on-shell approximation (52-55), using the Hartree-Fock approximation to the atomic states. The formalism for the polarization potential in the case of two electrons outside a closed-shell has been given by McCarthy et al. (1988). Larger atoms require a relativistic calculation, at least to describe spinorbit coupling in the target and scattering states. Equations (36) may again be used, but now the channel-state vectors are eigenstates of the Dirac kinetic energy or distorted-wave Hamiltonian for the projectile and the relativistic target Hamiltonian. This alters the form of the angular-momentum expansion. The relativistic CCO calculation has not yet been implemented.

A. HELIUM The use of the R-matrix method to describe rapid energy variation of lowenergy total cross sections is illustrated by the calculation of the n = 2 metastable (2% 2%) excitation function of helium by Fon et al. (1990). Figure 5 compares 11-state and 19-state K-matrix calculations with the measurements of metastable yield by Buckman et al. (1983) and Bass (1988) from the 23S threshold at 19.8-24 eV. The 19-state calculation involves the following helium states: 1, 2, 3, 4's; 2, 3, 4%; 2, 3, 4'P; 2, 3, 43P; 3, 4'D; 3, 4jD; 4'F; 43F. The main experimental features are reproduced very well by both calculations. The 19-state R-matrix calculation has been used to find differential cross sections for several excitations at 29.6 eV by Fon et al. (1988). Results for n = 2 singlet states are given in Fig. 6, while n = 2 triplet excitations are shown in Fig. 7. Experimental data are due to Trajmar (1973), Truhlar et al. (1973), Cartwright et al. (1989), and Brunger et al. (1990). Reasonable

+

FIG.5. The 11-state (a) and 19-state (b) R-matrix calculations compared with the experimental data (c) of Buckman et al. (1983) and Bass (1988) for the integrated excitation cross sections of the metastable states 23S and 2% from the ground state of helium.

Y

m
I (I being the ionization potential of the atom) gives the expression for the amplitude, obtained in Kazakov et al. (1976) the resonance approximation for nonresonance case.

111. AIS Contribution Let us turn to our main subject of interest: many-photon processes. The advantage of our formalism is the fact that it allows one to obtain results simply and elegantly. The importance of AIS is based on the fact that they are “drowned” in the continuum spectrum and the interference of two channels-of direct ionization and through AIS-can lead to a rich structure in the cross section. Here we can single out two cases: through intermediate resonance and without it. (See Fig. 1.) We shall restrict ourselves to analysis of the first case. The analysis of such a process has led Andryushin et al. (1982) to the conclusion that with increasing intensity of the laser field, it is necessary to account for both the decay of AIS due to interelectron interaction r;; (autoionizing widths) and the possibility of AIS decay in the field by absorption of an additional photon. We shall take into consideration this possibility. Thus, our model is distinguished by two features: the accounting of many-photon absorption and the existence of two discrete and two continua spectra. Let us consider the model in which the ground state of an atom is in twophoton resonance c1 + 2 0 N E, with AIS. At the same time we take into account the AIS decay through both interelectron interaction and absorption of additional photons. The scheme of such a model is shown in Fig. 1. Then, the solution of the quasienergy equation (7) must be found by taking into consideration the fact that levels are degenerate. For this purpose, it is necessary to find the solution of Eq. (14), which at W = 0 has the form

4i = ~ , ( E ) P ~ ~ - z+’ wa z‘ ( ~ ) q+, J a e , ( ~ ~ ~ e , d e , r

MULTIPHOTON IONIZATION OF COMPLEX ATOMS

25 1

--------/-- - - - - - - - - %

11

0

/

9-

/

0

0

0

/

-4P

0

FIG.1. Level scheme of a Ca atom. Six AISs are considered to give contributions to the given energy region.

where ipl and cpz are wave functions of the ground and autoionizing states, and $e3 and i,he4 are wave functions of continua spectra in the region of energies, which correspond to the absorption of two and three photons by atom. Expansion (21) is similar to the Fano (1961) expansion for the case of two discrete spectra embedded in continua spectra. The formal solution of the equation can be written in the form I ~ E )=

(1 - G V - ' I 4 i > ,

(22)

where

G = c1 - I 4 X 4 1 l -

11

142x421

e3><e3 I de3 -

J Ie4><de41gE.

(23)

YE is the Green function, defined by (15). Substituting (23) into (22) we obtain a system of integral equations of the type (14) for defining coefficients a,(E) as described in the previous section. If we neglect the transitions

V. I. Lengyel and M. I. Haysak

252

between continua spectra, we obtain the following system of equations

- R,,a,(E)

+ ( E - EL? - R,,)a,(E)

- jRZ,,a&(E)dz, -

s

Rzz4az4(E)dZ4= 0,

- Re32a2(E)

+ ( E - e3)ae3(E)

= O,

- R e 4 1 a l ( E ) - Re42a2(E)

+ ( E - e4)ae4(E)

= O,

- Re31a1(E)

(24) where Rij = ((ilRIj))are the matrix elements of shift operator (29), E'P,' = c1 20, ELo; = E~ = E,,, e3 = E , + 20, e4 = E~ + 3 0 are final energies in the first and second continuum. Following Fano (1961), the two last equations can be solved and we obtain ae3(E)and ae4(E)in terms of a,@) and a,(E) P a e 3 ( ~= ) + z ( E ) ~ ( E- e3) [Re,,a,(E) + R ~ ~ ~ ~ Z ( E ) I Y

+

1 1

[G

ae4(E)=

P

(25) + z ( E ) ~ ( E- e4) [Re,,a,(E) + ~ e 4 2 a 2 ( ~ ) 1 ,

[G

where Z ( E )is an unknown real function, which we take to be the same in each equation. Substituting (25) into (24) we obtain the system of linear homogeneous equations for a,(E) and a,(E):

CE - E(P) - z(E)~,,I~,(Q - DL + z(E)~,,I~,(E) = 0, - [ A ~ ,+ z(E)~,,~u,(E) + [ E - &lo)- z ( E ) I - , , ~ ~ , ( E )= 0, (26) where rll =

rt2

IR1E=e31Z+ IR1E=e412, + IRZE=e4I 2

= IR2E=e312

9

MULTIPHOTON IONIZATION OF COMPLEX ATOMS

253

It is clear that the homogeneous system of equations (26) has nontrivial solutions only when the determinant of this system is equal to zero. This condition allows us to determine the unknown function Z(E). Coefficient al(E) can be defined on the basis of the normalization of & (28) Using the technique described in Fano (1961), we obtain the expression for a,(E): la1(E)l2= Cn2 + z2(E)1-'Cr11+ ~*r21 + ~ l - 1 2+ l ~ l ~ l - ~ (29a) ~l-~ 1 but IzlI = 1. Therefore, we can use the transformation of Gauss's functions which allows one to get functions with the inverse argument. Then for Tz an expression, convergent above the N-photon ionization threshold, may be written in the form

E. Karule

288

For TB we have an expression convergent in the whole energy range where ionization with one additional photon is possible,

In the frequency range where ionization by one extra photon is possible, 1/z2 is real and varies in the region - 1 I 1/z2 5 -0.17157. In the region where I l / z , I 0.38, the double series with arguments 1/z1z2and z1/z2have good convergence

-=

where a = q N + l + s + n - 3, c = qN - qN+1 + 2 - s - n - k, and

MULTIPHOTON IONIZATION OF ATOMIC HYDROGEN

289

These expressions for TA and D must be used in the region close to the (N - 1)-photon ionization threshold. To get T2 convergent in the vicinity of the N-photon ionization threshold, other expressions for D and TA have to be obtained.

c. ANALYTICALCONTINUATION NEAR THE THRESHOLD The more difficult task is to get convergent expressions for transition matrix elements near the N-photon ionization threshold. In the case when only two photons were involved, Fainshtein et al. (1984), using analytically continued Green's function (Manakov et al., 1984), did not carry out calculations near the photoionization threshold. Also Pad& approximants don't give satisfactory results near N-photon ionization threshold when the number of photons is greater than two (Klarsfeld and Maquet, 1979b). Near the N-photon ionization threshold, transition matrix elements may be continued analytically but other transformations for Gauss's functions must be used. Let us use a transformation that allows one to obtain hypergeometric functions with variable 1 - z2/z1 instead of z2/z1. Then Appell's function in expression (51) for T2 may be written in the form

p+n-L-2

(I-$)

+ z2(-p

-n

z,s

+ L + 2)

'

S is the double series in terms of Gauss's hypergeometric functions with variable 1 - z1/z2, which can be obtained using one extra transformation

E. Karule

290

To get double series that converge rapidly near the N-photon ionization threshold, first it is necessary to make a resummation:

f

= m=O

[(,+-

x zFl(p

(l)m(q - L)m n L - l),,,m!

(59)

+ n + L - 1 , l - q + L, 1 - q + L - m ;l/zlz2).

The next transformation must be applied to get hypergeometric series with argument 1 - l/z,z,:

+ n + L - 1 , l - q + L, 1 - q -t L-m; l/z,z,) = (- l)m(zlzz)L-q((p+ n + L - I),

,F1(p

x [(q - L)#J-1

(1

z11z2)-~-n-L+1-m ~

Then we have a double series in a form

where once more carrying out a resummation we have 9(q, L, P, n)

k=O

x zFl(1, b

k!

+ 2L - k, b; l/z2z3),

b =q +n

-L -

1.


29 1

In the vicinity of the N-photon ionization threshold, the analytic continuation of the double series yields the subsequent expressions for D and T A ;

D=1 U-LNZ

TA

=

1

z2(

X

&IN-

z)-l

(z1 z 2) L N - q N z l3- q N + I

1,

1-

(z2

- 1)

LN,q N ; n + s) + 2 - n - s)‘

(LN - q N + 1

From expressions (44),(54), (55) and (61), one can obtain expressions valid at the N-photon threshold, similar to the case of two-photon ionization (Karule, 1985). The analytic continuations of transition matrix elements have finite regions of convergence. But both analytic continuations also have some common region of convergence, which was used to test the accuracy of calculations. The convergence of all series is rather fast. For sets of intermediate states with negative energies, it is enough to retain 22 terms in any of the sums. For the set of states in continuum we have to retain 25-30 terms in a sum.

V. Theoretical Estimates and Experimental Data for Atomic Hydrogen A. IONIZATIONBY LINEARLY POLARIZED LIGHT The Sturmian transition matrix elements and their analytic continuations were used in MPI and AT1 calculations for N < 16, S = 0; N G 8, S = 1 (Karule 1975, 1978, 1988a, b). With respect to order of magnitude, AT1 processes differ insignificantly from “normal” MPI processes of the same order (Figs. 1 and 2). There are no extra resonances in AT1 dispersion curves due to intermediate states in the continuum. In AT1 experiments the ratio of the AT1 cross section to the ordinary MPI cross section at fixed value of the intensity of the laser light is measured. In

292

E. Karule

q4=65 4

3

2

0,

10-731

u ~

5- 10-74. L I

1

a

10-75

i

2800 3000 3200 3400 3600 A,i FIG.1. AT1 absorbing 4 + 1 photons.

1

6500 6700 6900

7100 A,

FIG.2. AT1 absorbing 8

+ 1 photons.

a

293


the first AT1 experiment for atomic hydrogen (Muller et al., 1986) Q~"")/Q1") = 3.12 lo-" is obtained for the ionization of atomic hydrogen in the ground state at 1 = 308 nm (3p resonance) and I = 10" W cm-'. The results of our calculation Q{"")/Q{") = 2.90 lo-" are in a good agreement with experimental data. Feldman et al. (1987) have measured the angular distribution of photoelectrons for MPI and AT1 of atomic hydrogen in the ground state at 1 = 355 nm (N = 4) and A = 532 nm ( N = 6).Calculated by me, the differential probability for ionisation by N + S photons is in a reasonable agreement with experimental data (Fig. 3).

8

I

\O

v)

+ z

6

1

I

'p

X = 355nm

. \I

N=4

\

h

I

I

1

I I Of

4 I

I

a

U

I

'0

\

n

v)

1

Y

4

3c:

0

k

* 2

0,

30

90

8 (deg)

150

FIG.3. Angular distribution of photoelectrons at 1 = 355 nm (N = 4). Solid line S = 0 (x experiment from Feldman et al., 1987), dashed line S = 1 (0experiment from Feldman et nl., 1987).

E. Karule

294

In Fig. 4 the variations of the ratio Q{""")/QI") with I is given at I = 10I3 W cm-' for N = 2, 3,. . . , 8 . In a semiclassical approximation an estimate of this ratio was carried out by Delone et al. (1983). They determined that the ratio of the lowest-order AT1 cross section to MPI cross section is proportional to Therefore, according to this semiclassical law we must have linear dependence for QiN+')/QIN) in Fig. 4, where a scale linear in 1210/3 is used. It is seen that dependence on is close to linear up to N = 6. The growing deviations from the linear dependence at higher N are due to the growing number of channels with different angular momenta of the photothat I calculated is growing with N but for the electron. The ratio QINfl)/QIN) considered frequencies at I = 1013 W cm-2 remains less than unity. This demonstrates that, along with comparison with experimental data, perturbation theory is valid for MPI and AT1 of atomic hydrogen in the ground state and N < 8 up to intensities of light of lOI3 W cm-'. Ionization of atomic hydrogen in excited states (no< 9) was studied by me only in the case of two-photon ionization by an excess photon (Karule, 1984, 1985). It may be calculated also by a semiclassical formula (Berson, 1981). Qsk/Z = 0.681 x

0

3

-

119/3.

l04t

(u

I€

n;'

0.8

rr)

9

0.6

II H t

0 A

2 *

0 \

-+

0.2

z

- 0 )

0

X (nm)

0.4

0

lLk

/Nl I

I

I

400 500

I

I

600

I

I

700

knm) FIG.4. Variation of the ratio QI"")/QI"'

with 1 at I = lOI3 W cm-' (scale linear for 110/3).

295


The ratio of total cross sections calculated by the semiclassicalformula and in the dipole approximation is given in Fig. 5. For linearly polarized light, that ratio is close to unity in the vicinity of the threshold and decreases with growing w.

B. IONIZATIONBY

CIRCULARLY POLARIZED

LIGHT

For circularly polarized light a semiclassicalapproximation (Berson, 1981) gives Q': = 1.28Qp. In the dipole approximation the ratio QJQI is dependent on frequency and tends to 1.28 only in the vicinity of the photoionization threshold. Therefore agreement between results of calculations in semiclassical approximation and dipole approximation in the case of two-photon ionization are not so good (Fig. 6) as in case of linearly polarized light. In the case of ionization from certain nl states, semi-classical estimates of cross sections do not agree well with perturbation theory even for linearly polarized light. When multiphon ionization from the ground state is considered, the ratio of the total cross sections for circularly versus linearly polarized light decreases rapidly with K , while the maximum theoretical value of this ratio increases with K as (2K-l)!!/K! (Klarsfeld and Maquet, 1972, Lambropoulos, 1972a, b). The maximum value may be reached if in the case of linearly as well as circularly polarized light the photoelectron leaves the atom exclusively with the greatest possible orbital angular momentum. This may take

I

-

1

1.0 \

Y

m-

0

0.8 I

10

I

I

30

I

I

I

I

50 70 X /n: (nm)

FIG.5. Variation of the ratio Qf/Q, with A/n&

I

I

90

I

E. Karule

296

1.4

0

1 .o

10

30

50 70 X/nE (nm)

90

FIG.6. Variation of the ratio QZk/Q, with A/ni.

N I

E

0

3 0.9

N=6

N c

0 c X

rc)

0.6

I1 H c

0 Y

0.3

0

\

n c

+

co 0

Y

460

490

X

5

550

(nm)

FIG.I., Variation of the ratio Q(6"'/Q(6) with A at I = 3 x circularly polarized light, dashed line for linearly polarized light.

Solid line for


297

place only for K ,< 3. In the case of linearly polarized light for K > 3, small orbital angular momenta always contribute significantly (Karule, 1988a). The ratio of the AT1 to MPI cross sections grows rapidly near N-photon ionization thresholds in the case of circularly polarized light. In Fig. 7 variation of Qk6+')/Qk6) with wavelength is presented at I = 3 x 1 O I 2 W cm-2. At the six-photon ionization threshold (A = 547 nm), QL6+')/Qk6) x 0.9 but Qi6+')/Qi6) z 0.1, so for circularly polarized light the first and second peaks of the photoelectron energy spectrum must be almost of the same size. Therefore, the photoelectron spectrum with equal height of peaks can be obtained by circular polarized light at lower intensities than by linearly polarized light. In the case of circularly polarized light for atomic hydrogen in the ground state and the frequencies considered (Karule, 1988) perturbation theory is also valid up to I = l O I 3 W cmP2 except in regions close to the N-photon (N > 6) ionization thresholds.

REFERENCES Agostini, P., Clement, M., Fabre, F., and Petite, G. (1981). J. Phys. B: Atom. Molec. Phys. 14, L49 1. Agostini, P., Fabre, F., Mainfray, G. Petite, G., and Rahman, N. K. (1979). Phys. Rev. Lett. 42, 1127. Arnous, E., Klarsfeld, S., and Wane, S. (1973). Phys. Rev. A7, 1559. Aymar, M., and Crance, M. (1979). J. Phys. B: Atom. Molec. Phys. 12, L667. Aymar, M., and Crance, M. (1980). J. Phys. B: Atom. Molec. Phys. 13, L287. Bebb, H. B. (1967). Phys. Rev. 153, 23. Bebb, H. B., and Gold, A. (1966). Phys. Rev. 143, 1 . Berson, I. (1981). Phys. Lett. MA, 364. Chang, T. N., and Poe, R. T. (1976). J. Phys. B: Atom. Molec. Phys. 9, L3 11. Chang, T. N., and Poe, R. T. (1977). Phys. Rev. A16,606. Chan, T. N., and Tang, C. L. (1969). Phys. Rev. 185,42. Costescu, A., and Florescu, V. (1978). "Abstr. 6th Intern. Conf. on Atomic Physics, Zinatne, Riga, p. 39. Crance, M., and Aymar, M. (1979). J. Phys. B: Atom. Molec. Phys. 12, 3665. Dalgarno, A., and Lewis, J. T. (1955). Proc. R. Soc. London, Ser. A 233, 70. Delone, N. B., Goreslavsky, S. P., and Krainov, V. P. (1983). J. Phys. B: Atom. Molec. Phys. 16, 2369. Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G. (1953). "Higher Transcendental Function" vols. 1, 2, McGraw-Hill, New York. Fainshtein, A. G., Manakov, N. L., and Marmo, S. I. (1984). Phys. Lett. lMA, 347. Feldman, D., Wolf, B., Wemhoner, M., and Welge, K. H. (1987). Z. Phys. D6, 293. Gao Bo and Starace, F. (1988). Phys. Rev. Lett. 61,403. Gao Bo and Starace, A. F. (1989). Phys. Rev. A39,4550.

298

E. Karule

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Index

A

spin operator, expectation values, 15 for laser tunable hf, 19 Li-, energy, 16 fine and hyperfine calculations, 17 1 349 nm, observation, 16 lifetime autodetachment, 16 line emission, 16 radiative detachment, 16 Mg-, energy, 18 lifetime, relativisitic autodetachment,

Alignment angle, sodium, 196-197 Analyzing power, 84, 143 Anions, see specific anions Antisymmetrization, 169 Asymmetry, see Scattering asymmetry Atomic excited bound state anions alkali pattern, 16 Ar evidence for, 18 metastable lifetime, 18 Be-, energy, 17 lifetime line emission, 17 metastable, 17 radiative detachment, 17 relativistic autodetachment, 17 1 365 nm, observation, 17 Ca-, energy, 18-19 metastable lifetime, 10, 18 photodetachment from, 18-19 C- and Si-, energy, 19 H-, lifetime, radiative detachment, 14-15 He-, energy, 15 lifetime autodetachment, 15 radiative detachment, 16

18

Ne-, evidence against, 18 Sc-, Cr- through Zn-, Y-, calculations, 19 Atomic ground state anions in electric field, 19-21 F-, CI-, 21 H-, 20 Li-, Na-, K-, 20-21 electron affinity (EA) ub initio calculations, 3-6, 8-11 alkali, 6, 11 alkaline earth, 3-6 Br, I, 11

cu, 10 H through Ar, 2-3 301

302

INDEX

rare earth, 7-8 Sc, 6 Xe, 6-7 in magnetic field, H-,21 size, 11 Atomic hydrogen, 265 wave functions, 267, 273-274 Atomic orientation, 111-113, 151 Atomic resonance state anions H-, 12-13 He-, 12, 14-15 Li-, 14 Wannier resonance, 12 Wannier TEIL, 14 Auger electron polarization versus photoelectron polarization, 154 Auger electrons, polarized, 152-154 Autoionizing state (AIS), 246

B Bethe ridge, 203, 207, 225 Binary encounter approximation, 204, 208 Born series distorted-wave, 177 divergence, 177 Boundary condition, Coulomb three-body, 174, 180 C Calibration of electron polarimeter, 144-145 Channel state, 168, 173-174 Chiral molecules, 154-155 orientation, 156 Circular polarization of emitted light, 136-139, 143-144, 147 as means of studying spin-dependent interactions, 136, 138, 158 for polarization analysis, 143-145 Coincidence experiments with polarized electrons, 145-147, 151 scattering asymmetry in, 148-149 Complete experiment, 89-90, 98-99, 108, 121 Configuration interaction (CI), 205, 208, 232, 237, 257 coupled-channels-optical method, 189-190, 194 helium, 193

R-matrix method, 190 sodium, 194 Continuum cross section helium, 193 hydrogen, 185 discrete notation, 167-169, 173 Convergence region, 288 Correlation between electron and light polarization, 136 Correlation parameters, electron-photon, hydrogen, 185-186 Coulomb boundary condition, three-body, 174, 180 Coulomb potential, 167 Coupled-channels-optical method configuration interaction, 189-190, 194 definition, 180 helium, 190, 193 hydrogen, 184-189 multi-electron atoms, 189 sodium, 194-197 total ionization cross section, 181, 193 Coupled pseudostate method, 225 Cross section, see also Triple differential cross sections (TDCS), 246, 270 S 293-294, 296 AT1 V ~ ~ S U MPI, differential, 210 helium, 191-192 hydrogen, 184, 188 ratio for hydrogen, 189 sodium, 195 direct, 106-107, 141 double differential, 211, 213 exchange, 106-107, 141 experimental, 291 integrated for one channel hydrogen, 185-187 sodium, 196 molecular, 182 polarization-dependence, 89, 97,118 semiclassical formula, 295-296 semiclassical versus quantum mechanical, 295-296 theoretical, 291 total, hydrogen, 185, 187 total ionization, 211 coupled-channels-optical method, 181 helium, 193

INDEX hydrogen, 185 R-matrix method, 176 total reaction, hydrogen, 183, 187, 189 two-photon, 271

D Definition of polarization, 82 Degree of polarization, 82 Depolarization, 98 Diatomic heteronuclear anions alkali halides, structure, 32 EAs, some spectroscopic constants of anions AgAu, CuAg, CuAu, 34 alkali halides, table, 33 A10, BO, BS, CN, CP, CSi, PO SIN, table, 31 CS, NO, NS, 32 FeO, SeO, TeO, 33-34 Diatomic homonuclear anions CZ- in astrophysics, 26 B X transition, 26-27 EA(CJ, 27 lifetime autodetachment, 27-28 radiative, 27-28 spectra, 26-27 spectroscopic constants, 27-28 Cl,-, I,-, Morse potentials, 29 resonance states, low, 30 Fz-, EA(Fz), 29 hyperfine coupling constants, 30 Morse potentials, 29 TU+ resonance, 29 He, metastable, 25 lifetime, autodetachment, 25 4Eu-, In. resonances, 25 H2-, state, stable, proposed, 24 resonances, 25 X2C.+, B2C,+ potentials, 23-24 LizA2C,+ state, 26 autodetachment from, 26 EA(Li,), 26 X2C.+ state, 26 metals, EA and w., 30

-

N2 -

evidence against long-lived, 28 TI,, zC. resonances, 28

303 Na2-, observation, 26 02-.

bound states, spectroscopic constants, 29 EA(Oz), 28 PZ EA(Pz), 30 ground state, spectroscopic constants, 30 Si,EA(Si2), 30

,E,+ state, excitation energy, 30 XZn. state, 30 XeZ-, observation, 30 Diatomic hydride anions CHexcited alA state, 34, 36 lifetime YE-, 36 forbidden IA 0, 36-37 infrared v = 1 EAs, table, 35 HCI-, potentials, 38 NHhyperfine parameters, 37 lifetime autodetachment from v = 1 level, 31 0, 37 infrared v = 1 OHelectric field, effect of, 38 lifetime autodetachment from v = 5 levels, 58 0, 38 infrared v = 1 spectrum pure rotation, 38 rotation-vibration, 37 SH-, rotation-vibration spectrum, 38 SiH-, excited a'A, blE+ states, 34, 36 spectroscopic constants, table, 36 Differential cross sections, see Cross section, differential Dipole-supported state anions experiment C6H&OCH3-, 42 CHZCHO-, 42-43 CH,CN-, 43 CH2COF-, 43 FeO-, 43-44 role of dipole, demonstration of, 43

--

-

304

INDEX

theory Born-Oppenheimer approximation (BO) results, 39, 41-42 invalidity of BO, effect of, 39-41 Koopman’s theorem results, 41 rotationally adiabatic potential, use of, 43 Dim-Fock wave functions, 236-237 Direct cross section, 106-107, 141 Discrete notation, continuum, 167-169, 173 Distorted wave Born approximation (DWBA), 124, 132-133, 149, 207, 225 definition, 177 polarization potential, 180 second-order C~OSUIX, 177-178 explicit, 178-180, 184, 186 unitarized, 178-179 Distorted wave impulse approximation (DWIA), 207, 225 Distorted wave representation, coupled integral equations, 172 Double differential cross section, 211, 213 Double series, resummation, 287, 290 Doubly charged anions atomic calculation, 22-23 experiment, 22 interest in, 21 molecular, 65

E Electron momentum spectroscopy, 228-239 argon, 229-235 helium, 229 hydrogen, 228 lead, 237 xenon, 236 Electron optic dichroism, 157-158 Exchange amplitude, 83, 120 interference with spin-orbit amplitude, 134-135 Exchange cross section, 106-107, 141 Exchange interaction, 83, 97, 108, 117, 150 connection with circular light polarization, 136, 138 Exchange potential, 169, 174, 193 Exchange scattering, 83, 97-108, 133, 141

Excited targets, 238 Explicit second-order approximation, 178-180, 184, 186 External field strong, 254 weak, 254

F Factorization approximation, 207 Feshbach projection operators, 170 Fine-structure effect, 110-117, 126-127, 129, 131, 134 Fine-structure splitting, 108-109, 113-114 Floquet theorem, 247 Future work, 198

G GaAs source of polarized electrons, 86-87 Generalized oscillator strength, 217 Glauber approximation, 179 Green function, 249, 267 analytic continuation, 273 eigenfunction expansion, 267 integral representation, 268 Sturmian expansion, 269, 274

H Hartree-Fock approximation, 190 sodium, 194 Hartree-Fock method, 246 Helium coupled-channels-optical calculation, 190, 193 differential cross sections, 191-192 R-matrix method, 190-192 total ionization cross section, 193 Hydrogen coupled-channels-optical calculation, 184-189 differential cross sections, 184, 188 ratios, 189 dissociation of molecular, 182 electron-photon correlation parameters, 183, 185-186 explicit second-order calculation, 184, 186 integrated cross sections, 185-187 molecular cross sections, 182 pseudostate calculation, 183-189 R-matrix method, intermediate energy, 183, 185-186

INDEX total cross section, 185 total ionization cross section, 185 total reaction cross section, 183, 187, 189 Hydrogen molecule cross section, 182 dissociation ratio, 182 Hypergeometric function Appell (generalized), 268, 277-279 confluent, 276-278, 282-283 Gauss, 277-279, 285, 289 integral representations, 278, 286 series expansion, 279, 283, 287 transformations, 289-290

I Information maximal possible, 89, 96, 121, 125 from Stokes parameters, 136, 143 In-plane polarization, 155-156 Integral equations, coupled distorted-wave representation, 172 multichannel formalism, 168 numerical solution, 172 partial wave treatment, 172 P-projected, 170-172, 174 Intermediate coupling, 131 Interplay of fine-structure splitting with exchange, 108-117, 150 Intramolecular plural scattering, 155 Ionization, two-photon, 246 Ca, 257 He, 257 Ion recoil momentum, 205 Ion-target overlap, 208

K Kohn-Sham theory, 95

L Laguerre polynomial, summation, 283 Left-right asymmetry, 86, 111, 131 Linear polarization of emitted light, 140, 143, 147 as means of studying spin-dependent interactions, 143, 158 Lippman-Schwinger equation, see Integral equations, coupled Longitudinally polarized electrons, 156

305

M Momentum transfer, 202, 204 Mott detector, 84, 91, 143, 145 calibration, 145 Multiphoton ionization, 265 above-threshold (ATI), 272, 281, 284 by circularly polarized light, 291-295 dispersion curves, 292 by linearly polarized light, 295-297 nonresonant, 265, 293 ordinary (MPI), 265 resonant, 293 threshold, 291, 297 N Negative ions, see specific anions

0 Observables complete set, 91-93, 96, 98, 125 number of, 121-123 Optical activity, 154-155 Optical limit, 217 Optical potential, see also Polarization potential, 170, 172, 181 Orbital angular momentum, orientation, 110, 118 Orbital energy, 209, 232 Oriented molecules, 155-156, 158 ionization, 156 Oriented targets, 238 P Pad&approximants, 272, 289 Partial polarization, 82 Perfect experiment, 89-90, 116 Perturbation techniques Green’s function method, 267-269 implicit summation, 267-268, 270, 272 variational, 267 Perturbation theory, 265, 275, 293 Perturbative approximations, validity, 178 Photoelectrons, 266 angular distribution, 293 energy spectrum, 266 polarized, 86 Plane wave impulse approximation (PWIA), 204

306

INDEX

Polarization angular-dependence, 85, 92-93, 109, 125, 127-128, 149 change, 89, 91, 98, 120-121, 124, 129, 133, 155 final, scattered electrons, 88, 97, 121 reversal by scattering, 122 rotation, 89, 98, 121, 155 similarity elastic and inelastic, 129 Polarization-dependenceof cross section, 89, 97, 118 Polarization measurement, accuracy, 95 Polarization potential, 171, 174 distorted-wave Born approximation, 180 half-on-shell, 181, 190, 194 Polarization transfer, 98, 141-142 Polarized atoms, 136 metastable, 106 scattering from, 83, 97, 108, 134-135 Polarized electrons, source, 86-87 Polarized light, see also Circular polarization of emitted light; Linear polarization of emitted light, 135 Positron scattering, 96-97 Pseudostate method, 177 hydrogen, 183-189

Q Quasienergy, 247

R Rabi frequency, 255 Radial integrals, 276-279 Recoil atoms, 106-107 Reflection invariance, 120 Reflection symmetry, 155 Relativistic effects, 236-237 Resonances, 131-133, 140, 158 AIS, 256 k-photon, 257 R-matrix method, 128, 132-134, 142, 149, 151, 175 helium, 190-192 hydrogen, 185 multielectron atoms, 189 Rydberg constant, 270 S Scattering asymmetry, see also Left-right asymmetry; Spin asymmetry, 86,

99, 109, 111-113, 119, 132-133, 149 spin-up-down asymmetry, 111, 131 Scattering theory formal, 166 multichannel two-body, 167 Secondary electrons, polarization, 159 Second Born approximation closure, 177 unitarized eikonal-Born series, 179 Semiclassical law, 294 SEMPA, 159 Separation energy, 202 Sherman function S, 84, 90, 119-120, 128 Singlet scattering, 103 Sodium, 196-197 configuration interaction, 194 coupled-channels-opticalcalculation, 194-197 differential cross sections, 195 integrated cross sections, 196 Stokes parameters, 196-197 Spectroscopic factor, 209 Spectroscopic sum rule, 209 Spin, total electron, 166 Spin asymmetry, 100-107, 115, 211 differential, 101, 103 integrated, 100 sensitivity, 103 Spin-dependent interactions, interplay, 117, 150-151 disentanglement, 143 Spin-dependent potential, 169, 174 Spin-dependent scattering, generalized theory, 117-124 Spin-flip amplitude, 87, 93-94, 120 Spinless atoms, scattering from, 87-97 Spin-orbit interaction in Auger effect, 152 connection with linear light polarization, 143 internal, 117, 150 scattered electron in atomic field, 83-84, 86-88, 94, 106, 108, 117, 150 Spin-up-down asymmetry, 111, 131 State multipodes, 148 Stokes parameters, 135, 151 sensitivity to spin-dependent interactions, 143 Sturmian functions, 269, 274 Sublevel excitation, 141-142, 147-149

INDEX Superelastic scattering, 114-116 Switching adiabatic, 247 instantaneous. 247

T Target Hartree-Fock approximation (THFA), 208 Tetra-atomic and more complex anions fluorides meta; hexafluorides, 65-66 SF,-, 64-65 hydrides CHI-, 59 CH5-, 60 H@, 59-60 NH4-, 60 organic compounds CzHg-, 69 C,H,-, 68-69 CH,CN-, 43, 67 CH,CO-, 66-67 CH,COF-, 43, 68 CH30-, 66-67 CH,S-, 67 HCCO-, 65 HCO,-, 66, 69 NHCHO-, 69 NHZCO-, 69 others, 43, 69 oxides CO,-, 61 NO,-, 62 NZOZ-, 63 PO,-, 62 SO,-, 62 silicon compounds CSiH,-, SiCH2-, 64 Si2H2-, 64 small clusters C1-, COB-, NO,-, SO,-, 61-62 C02.C02-, 62 CO,, H20, SO, with, 62 H20’NO3-, 62 metal (Cs, Cu, K, Na, Rb), 63 N,*OZ-. NOaNO-, 63 NO-(N,O). NO-(N,O),, 62 Three-body boundary conditions, 206 Time-reversal invariance, 123 Time-reversed states, 173

307

T-matrix averaging in R-matrix method, 176 distorted-wave representation, 172 elements and Stokes parameters, 196 P-projected, 171 Total energy, 202 Total ionization cross sections, see Cross section, total ionization Total polarization, 82 Transition probability, 270 rate, 270 Transition matrix elements, 275 analytic continuation, 272-273, 281-291 recurrence relations, 280 Sturmian expansion, 275-276 Transmission asymmetry, 157 Triatomic anions atoms isoelectronic Ag3-, 48 c,- , 45 cu,-, 49 Hg-, 44-45 NS-, 45-46 Na,-, Li3-, Li,Na-, LiNa,-, 49-50 Ni,-, Pd3-,Pt-,49 0 3 - , 46-48 Se,-, Te,-, 48 S,-, S 2 0 - , SOz-, 46-48 dihydrides AIH,-, BH2-, 53 CHZ-, 50-51 CoH,-, OH,-, FeH2-, NiHZ-, MnH,-, 53 FH2-, 52 HLiH-, 49-50 HZO-, 51-52 NHZ-, 51 PH2-, 52-53 SiH2-, 52 monohydrides CIHC1-, FHF-, 55-57 FHBr-, 57 halocarbenes, 57-58 H C - , 53-54 HCO-, 54-55, 69 HCS-, 55 HO,-, 54 Mulliken-Walsh rules, 44

308

INDEX

other systems BeF2-, 58 CCO-, 58 COZ-, 58-59 FCO-, 69 N 2 0 - , 58 NO2-, 59 Triple differential cross sections (TDCS), 214 absolute, 215 autoionization, 221 coplanar asymmetric kinematics, 203,225 electron momentum spectroscopy, 228 noncoplanar symmetric kinematics, 203, 227 threshold behavior, 218 Triplet scattering, 103 Two-step model, 129

U Unitarized distorted-wave Born approximation, 178-179 Unitarized eikonal-Born series, 179, 187, 189, 193, 225 Units, atomic, 167

w Wallace amplitude, 179 Wannier threshold laws, 218 Wave-function approximations, 206 Weak-coupling approximation, 218 Whittaker function, 268 Widths autoionizing, 247 field, 247 ionization, 254

Contents of Previous Volumes

Volume 1 Molecular Orbital Theory of the Spin Properties of Conjugated Molecules, G . G. Hall and A. T . Amos Electron Affinities of Atoms and Molecules, B. L. Moiseiwitsch Atomic Rearrangement Collisions, B. H. Bransden The Production of Rotational and Vibrational Transitions in Encounters between Molecules, K . Takayanagi The Study of Intermolecular Potentials with Molecular Beams at Thermal Energies, H. Pauly and J . P . Toennies High-Intensity and High-Energy Molecular Beams, J . B. Anderson, R. P. Andres, and J . B. Fenn Volume 2 The Calculation of van der Waals Interactions, A. Dalgarno and W . D. Davison Thermal Diffusion in Gases, E. A. Mason, R. J . Munn, and Francis J . Smith Spectroscopy in the Vacuum Ultraviolet, W. R. S. Carton The Measurement of the Photoionization Cross Sections of the Atomic Gases, James A. R. Samson The Theory of Electron-Atom Collisions, R. Peterkop and V. Veldre Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, F. J. de Heer

Mass Spectrometry of Free Radicals, S. N. Foner

Volume 3 The Quanta1 Calculation of Photoionization Cross Sections, A. L. Stewart Radiofrequency Spectroscopy of Stored Ions I: Storage, H. G. Dehmelt Optical Pumping Methods in Atomic Spectroscopy, B. Budick Energy Transfer in Organic Molecular Crystals: A Survey of Experiments, H . c. w o l j Atomic and Molecular Scattering from Solid Surfaces, Robert E. Stickney Quantum Mechanics in Gas CrystalSurface van der Waals Scattering, E. Chanoch Beder Reactive Collisions between Gas and Surface Atoms, Henry Wise and Bernard J . Wood Volume 4 H. S. W. Massey-A Sixtieth Birthday Tribute, E. H. S. Burhop Electronic Eigenenergies of the Hydrogen Molecular Ion, D. R. Bates and R. H . G. Reid Applications of Quantum Theory to the Viscosity of Dilute Gases, R. A. Buckingham and E. Gal Positrons and Positronium in Gases, P . A. Fraser

CONTENTS OF PREVIOUS VOLUMES

Classical Theory of Atomic Scattering, A. Burgess and I . C. Percival Born Expansions, A. R. Holt and B. L. Moiseiwitsch Resonances in Electron Scattering by Atoms and Molecules, P. G. Burke Relativistic Inner Shell Ionization, C. B. 0. Mohr Recent Measurements on Charge Transfer, J . B. Hasted Measurements of Electron Excitation Functions, D. W . 0. Heddle and R. G. W . Keesing Some New Experimental Methods in Collision Physics, R. F. Stebbings Atomic Collision Processes in Gaseous Nebulae, M. J . Seaton Collisions in the Ionosphere, A. Dalgarno The Direct Study of Ionization in Space, R. L. F. Boyd

Volume 6 Dissociative Recombination, J. N . Bardsley and M . A. Biondi Analysis of the Velocity Field in Plasmas from the Doppler Broadening of Spectral Emission Lines, A. s. Kaufman The Rotational Excitation of Molecules by Slow Electrons, Kazuo Takayanagi and Yukikazu Itikawa The Diffusion of Atoms and Molecules, E. A. Mason and T . R. Marrero Theory and Application of Sturmian Functions, Manuel Rotenberg Use of Classical Mechanics in the Treatment of Collisions between Massive Systems, D. R. Bates and A. E. Kingston

Volume 7 Volume 5 Flowing Afterglow Measurements of IonNeutral Reactions, E. E. Ferguson, F. C. Fehsenfeld, and A. L. Schmeltekopf Experiments with Merging Beams, Roy H . Neynaber Radiofrequency Spectroscopy of Stored Ions 11: Spectroscopy, H.G. Dehmelt The Spectra of Molecular Solids, 0. Schnepp The Meaning of Collision Broadening of Spectral Lines: The Classical Oscillator Analog, A. Ben-Reuuen The Calculation of Atomic Transition Probabilities, R. J . S. Crossley Tables of One- and Two-Particle Coefficients of Fractional Parentage for Configurations s’s‘”pq, C. D. H. Chisholm, A. Dalgarno, and F . R. lnnes Relativistic Z-Dependent Corrections to Atomic Energy Levels, Holly Thomis Doyle

Physics of the Hydrogen Master, C. Audoin, J . P . Schermann, and P. Grivet Molecular Wave Functions: Calculation and Use in Atomic and Molecular Processes, J . C. Browne Localized Molecular Orbitals, Hare1 Weinstein, Ruben Pauncz, and Maurice Cohen General Theory of Spin-Coupled Wave Functions for Atoms and Molecules, J . Gerratt Diabatic States of Molecules- QuasiStationary Electronic States, Thomas F. OMalley Selection Rules within Atomic Shells, B. R. Judd Green’s Function Technique in Atomic and Molecular Physics, Gy. Csanak, H . S. Taylor, and Robert Yaris A Review of Pseudo-Potentials with Emphasis on Their Application to Liquid Metals, Nathan Wiser and A. J . Greenfield


Volume 8 Interstellar Molecules: Their Formation and Destruction, D. McNally Monte Carlo Trajectory Calculations of Atomic and Molecular Excitation in Thermal Systems, James C. Keck Nonrelativistic Off-Shell Two-Body Coulomb Amplitudes, Joseph C. Y. Chen and Augustine C . Chen Photoionization with Molecular Beams, R. B. Cairns, Halstead Harrison, and R. I . Schoen The Auger Effect, E. H. S. Burhop and W . N. Asaad

Volume 9 Correlation in Excited States of Atoms, A. W . Weiss The Calculation of Electron-Atom Excitation Cross Sections, M . R. H. Rudge Collision-Induced Transitions between Rotational Levels, Takeshi Oka The Differential Cross Section of LowEnergy Electron- Atom Collisions, D. Andrick Molecular Beam Electric Resonance Spectroscopy, Jens C. Zorn and Thomas C . English Atomic and Molecular Processes in the Martian Atmosphere, Michael B. M cElroy

Volume 10 Relativistic Effects in the Many-Electron Atom, Lloyd Armstrong, Jr. and Serge Feneuille The First Born Approximation, K . L. Bell and A. E. Kingston Photoelectron Spectroscopy, W . C. Price Dye Lasers in Atomic Spectroscopy, W . Lange, J . Luther, and A. Steudel

Recent Progress in the Classification of the Spectra of Highly Ionized Atoms, B. C. Fawcett A Review of Jovian Ionospheric Chemistry, Wesley T . Huntress, Jr.

Volume 11 The Theory of Collisions between Charged Particles and Highly Excited Atoms, I. C. Perciual and D. Richards Electron Impact Excitation of Positive Ions, M . J . Seaton The R-Matrix Theory of Atomic Process, P. G. Burke and W . D. Robb Role of Energy in Reactive Molecular Scattering: An Information-Theoretic Approach, R. B. Bernstein and R. D. Leoine Inner Shell Ionization by Incident Nuclei, Johannes M . Hansteen Stark Broadening, Hans R. Griem Chemiluminescence in Gases, M. F. Golde and B. A. Thrush

Volume 12 Nonadiabatic Transitions between Ionic and Covalent States, R. K . Janeo Recent Progress in the Theory of Atomic Isotope Shift, J . Bauche and R.-J. Champeau Topics on Multiphoton Processes in Atoms, P. Lambropoulos Optical Pumping of Molecules, M . Broyer, G. Gouedard, J . C. Lehmann, and J . Vigue' Highly Ionized Ions, Ivan A. Sellin Time-of-Flight Scattering Spectroscopy, Wilhelm Raith Ion Chemistry in the D Region, George C. Reid


Volume 13

Volume 15

Atomic and Molecular Polarizabilities-A Review of Recent Advances, Thomas M . Miller and Benjamin Bederson Study of Collisions by Laser Spectroscopy, Paul R . Berman Collision Experiments with LaserExcited Atoms in Crossed Beams, I. V. Hertel and W . Stoll Scattering Studies of Rotational and Vjbrational Excitation of Molecules, Manfred Faubel and J . Peter Toennies Low-Energy Electron Scattering by Complex Atoms: Theory and Calculations, R. K . Nesbet Microwave Transitions of Interstellar Atoms and Molecules, W. B. Somerville

Negative Ions, H. S. W . Massey Atomic Physics from Atmospheric and Astrophysical Studies, A. Dalgarno Collisions of Highly Excited Atoms, R . F. Stebbings Theoretical Aspects of Positron Collisions in Gases, J . W . Humberston Experimental Aspects of Positron Collisions in Gases, T . C. G r i f J h Reactive Scattering: Recent Advances in Theory and Experiment, Richard B. Bernstein Ion-Atom Charge Transfer Collisions at Low Energies, J . B. Hasted Aspects of Recombination, D. R. Bates The Theory of Fast Heavy Particle Collisions, B. H. Bransden Atomic Collision Processes in Controlled Thermonuclear Fusion Research, H. B. Gilbody Inner-Shell Ionization, E. H. S. Burhop Excitation of Atoms by Electron Impact, 0. W . 0. Heddle Coherence and Correlation in Atomic Collisions, H . Kleinpoppen Theory of Low Energy Electron-Molecule Collisions. P. G. Burke

Volume 14 Resonances in Electron Atom and Molecule Scattering, D. E . Golden The Accurate Calculation of Atomic Properties by Numerical Methods, Brian C. Webster, Michael J . Jamieson, and Ronald F. Stewart (e, 2e) Collisions, Erich Weigold and Ian E. McCarthy Forbidden Transitions in One- and TwoElectron Atoms, Richard Marrus and Peter J . Mohr Semiclassical Effects in Heavy-Particle Collisions, M . S. Child Atomic Physics Tests of the Basic Concepts in Quantum Mechanics, Francis M. Pipkin Quasi-Molecular Interference Effects in Ion-Atom Collisions, S. V. Bobasheo Rydberg Atoms, S. A. Edelstein and T. F. Gallagher UV and X-Ray Spectroscopy in Astrophysics, A. K. Dupree

Volume 16

Atomic Hartree-Fock Theory, M . Cohen and R. P. McEachran Experiments and Model Calculations to Determine Interatomic Potentials, R. Diiren Sources of Polarized Electrons, R . J. Celofta and D. T . Pierce Theory of Atomic Processes in Strong Resonant Electromagnetic Fields, S. Swain


Spectroscopy of Laser-Produced Plasmas, M. H. Key and R. J . Hutcheon Relativistic Effects in Atomic Collisions Theory, B. L. Moiseiwitsch Parity Nonconservation in Atoms: Status of Theory and Experiment, E. N . Fortson and L. Wilets

Volume 17 Collective Effects in Photoionization of Atoms, M. Ya. Amusia Nonadiabatic Charge Transfer, D. S. F. Crothers Atomic Rydberg States, Serge Feneuille and Pierre Jacquinot Superfluorescence, M . F. H. Schuurmans, Q.H . F. Vrehen, D. Polder, and H. M . Gibbs Applications of Resonance Ionization Spectroscopy in Atomic and Molecular Physics, M. G. Payne, C. H . Chen, G. S. Hurst, and G. W . Foltz Inner-Shell Vacancy Production in IonAtom Collisions, C. D. Lin and Patrick Richard Atomic Processes in the Sun, P. L. Dufton and A. E. Kingston

Volume 18 Theory of Electron-Atom Scattering in a Radiation Field, Leonard Rosenberg Positron-Gas Scattering Experiments, Talbert S. Stein and Walter E. Kauppila Nonresonant Multiphoton Ionization of Atoms, J . Morellec, D. Normand, and G. Petite Classical and Semiclassical Methods in Inelastic Heavy-Particle Collisions, A. S. Dickinson and D. Richards Recent Computational Developments in the Use of Complex Scaling in Resonance Phenomena, B. R. Junker

Direct Excitation in Atomic Collisions: Studies of Quasi-One-Electron Systems, N . Andersen and S . E. Nielsen Model Potentials in Atomic Structure, A. Hibbert Recent Developments in the Theory of Electron Scattering by Highly Polar Molecules, D. W . Norcross and L. A. Collins Quantum Electrodynamic Effects in Few-Electron Atomic Systems, G. W . F. Drake

Volume 19 Electron Capture in Collisions of Hydrogen Atoms with Fully Stripped Ions, B. H. Bransden and R. K . Janev Interactions of Simple Ion-Atom Systems, J . T . Park High-Resolution Spectroscopy of Stored Ions, D. J . Wineland, Wayne M . Itano, and R. S. Van Dyck, Jr. Spin-Dependent Phenomena in Inelastic Electron-Atom Collisions, K. BIum and H . Kleinpoppen The Reduced Potential Curve Method for Diatomic Molecules and Its Applications, F . JenE The Vibrational Excitation of Molecules by Electron Impact, D. G. Thompson Vibrational and Rotational Excitation in Molecular Collisions, Manfred Faubel Spin Polarization of Atomic and Molecular Photoelectrons, N . A. Cherepkov

Volume 20 Ion-Ion Recombination in an Ambient Gas, D. R. Bates Atomic Charges within Molecules, G. G. Hall Experimental Studies on Cluster Ions, T. D. Mark and A. W . Castleman, Jr.


Nuclear Reaction Effects on Atomic Inner-Shell Ionization, W. E. Meyerhof and J.-F. Chemin Numerical Calculations on ElectronImpact Ionization, Christopher Bottcher Electron and Ion Mobilities, Gordon R. Freeman and David A. Armstrong On the Problem of Extreme UV and XRay Lasers, I. I. Sobel'man and A. V . Vinogradov Radiative Properties of Rydberg States in Resonant Cavities, S. Haroche and J . M . Raimond Rydberg Atoms: High-Resolution Spectroscopy and Radiation InteractionRydberg Molecules, J . A. C. Gallas, G. Leuchs, H. Walther, and H. Figger

Volume 21

Subnatural Linewidths in Atomic Spectroscopy, Dennis P. O'Brien, Pierre Meystre, and Herbert Walther Molecular Applications of Quantum Defect Theory, Chris H. Greene and Ch. Jungen Theory of Dielectronic Recombination, Yukap Hahn Recent Developments in Semiclassical Floquet Theories for Intense-Field Multiphoton Processes, Shih-I Chu Scattering in Strong Magnetic Fields, M . R. C. McDowell and M . Zarcone Pressure Ionization, Resonances, and the Continuity of Bound and Free States, R. M. More

Doubly Excited States, Including New Classification Schemes, C. D. Lin Measurements of Charge Transfer and Ionization in Collisions Involving Hydrogen Atoms, H. B. Gilbody Electron-Ion and Ion-Ion Collisions with Intersecting Beams, K . Dolder and B. Peart Electron Capture by Simple Ions, Edward Pollack and Yukap Hahn Relativistic Heavy-Ion- Atom Collisions, R. Anholt and Harvey Could Continued-Fraction Methods in Atomic Physics, S. Swain

Volume 23

Vacuum Ultraviolet Laser Spectroscopy of Small Molecules, C. R. Vidal Foundations of the Relativistic Theory of Atomic and Molecular Structure, Ian P. Grant and Harry M. Quiney Point-Charge Models for Molecules Derived from Least-Squares Fitting of the Electric Potential, D. E. Williams and Ji-Min Yan Transition Arrays in the Spectra of Ionized Atoms, J . Bauche, C. EaucheArnoult, and M . Klapisch Photoionization and Collisional Ionization of Excited Atoms Using Synchrotron and Laser Radiation, F. J . Wuilleumier; D. L. Ederer, and J . L. PicquP Volume 24

The Selected Ion Flow Tube (SIFT): Studies of Ion-Neutral Reactions, D. Volume 22 Smith and N. G. Adams Positronium-Its Formation and Inter- Near-Threshold Electron-Molecule action with Simple Systems, J . W . Scattering, Michael A. Morrison Humberston Angular Correlation in Multiphoton Experimental Aspects of Positron and Ionization of Atoms, S. J . Smith and G. Leuchs Positronium Physics, T. C. Grifith


Optical Pumping and Spin Exchange in Gas Cells, R . J. Knize, 2.W u , and W . Happer Correlations in Electron-Atom Scattering, A. Crowe

Volume 25 Alexander Dalgarno: Life and Personality, David R. Bates and George A. Victor Alexander Dalgarno: Contributions to Atomic and Molecular Physics, Neal Lane Alexander Dalgarno: Contributions to Aeronomy, Michael B. McElroy Alexander Dalgarno: Contributions to Astrophysics, David A. Williams Dipole Polarizability Measurements, Thomas M . Miller and Benjamin Bederson Flow Tube Studies of Ion-Molecule Reactions, Eldon Ferguson Differential Scattering in He-He and He' -He Collisions at KeV Energies, R. F. Stebbings Atomic Excitation in Dense Plasmas, Jon C.Weisheit Pressure Broadening and Laser-Induced Spectral Line Shapes, Kenneth M . Sando and Shih-I Chu Model-Potential Methods, G. Laughlin and G. A. Victor Z-Expansion Methods, M . Cohen Schwinger Variational Methods, Deborah Kay Watson Fine-Structure Transitions in ProtonIon Collisions, R. H . G. Reid Electron Impact Excitation, R. J . W. Henry and A. E. Kingston Recent Advances in the Numerical Calculation of Ionization Amplitudes, Christopher Bottcher

The Numerical Solution of the Equations of Molecular Scattering, A. C. Allison High Energy Charge Transfer, B. H. Bransden and D. P. Dewangan Relativistic Random-Phase Approximation, W. R. Johnson Relativistic Sturmian and Finite Basis Set Methods in Atomic Physics, G. W. F. Drake and S. P. Goldman Dissociation Dynamics of Polyatomic Molecules, T. Uzer Photodissociation Processes in Diatomic Molecules of Astrophysical Interest, Kate P. Kirby and Ewine F. van Dishoeck The Abundances and Excitation of Interstellar Molecules, John H . Black

Volume 26 Comparisons of Positrons and Electron Scattering by Gases, Walter E. Kauppila and Talbert S. Stein Electron Capture at Relativistic Energies, B. L. Moiseiwitsch The Low-Energy, Heavy Particle Collisions- A Close-Coupling Treatment, Mineo Kimura and Neal F. Lane Vibronic Phenomena in Collisions of Atomic and Molecular Species, V. Sidis Associative Ionization: Experiments, Potentials, and Dynamics, John Weiner, Francoise Masnou-Sweeuws, and Annick Giusti-Suzor On the p Decay of '*'Re: An Interface of Atomic and Nuclear Physics and Cosmochronology, Zonghau Chen, Leonard Rosenberg, and Larry Spruch Progress in Low Pressure Mercury-Rare Gas Discharge Research, J . Maya and R. Lagushenko


Volume 27 Negative Ions: Structure and Spectra, David R. Bates Electron Polarization Phenomena in Electron-Atom Collisions, Joachim Kessler Electron-Atom Scattering, I. E. McCarthy and E. Weigold

Electron-Atom Ionization, I. E. McCarthy and E. Weigold Role of Autoionizing States in Multiphoton Ionization of Complex Atoms, V. I. Lengyel and M . I. Haysak Multiphoton Ionization of Atomic Hydrogen Using Perturbation Theory, E. Karule

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